CSA S157.1-05 Alumninum Design Code (1)

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S157-05/S157.1-05

Strength design in aluminum/ Commentary on CSA S157-05, Strength design in aluminum

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Update No. 3 CAN/CSA-S157-05/S157.1-05 July 2009 Note: General Instructions for CSA Standards are now called Updates. Please contact CSA Information Products Sales or visit www.ShopCSA.ca for information about the CSA Standards Update Service. Title: Strength design in aluminum/Commentary on CSA S157-05, Strength design in aluminum — originally published February 2005 Revisions issued: Update No. 1 — February 2007 Update No. 2 — April 2007 If you are missing any updates, please contact CSA Information Products Sales or visit www.ShopCSA.ca. The following revisions have been formally approved and are marked by the symbol delta (Δ) in the margin on the attached replacement pages: Revised

S157-05: Contents and Clauses 2, 3.1, 3.2.1, 4.4.2, 4.4.6, 5.2, 5.3.1, 5.4, 5.4.1, 5.4.2, 5.6, 5.7, 11.2.4.3, 13.3.1.2, and A6 S157.1-05: Clause C11.2.4.3

New

None

Deleted

S157-05: Clauses 5.4.3 and 5.4.4 S157.1-05: Clause C5.4

CAN/CSA-S157-05/S157.1-05 originally consisted of 137 pages (xiii preliminary and 124 text), each dated February 2005. It now consists of the following pages: February 2005

v, vi, ix–xiii, 15–32, 35– 44, 47–54, 57–60, 63–68, 71–78, 81–108, and 111–124

February 2007

Cover, National Standards of Canada text, title pages (i, 1, and 2), and copyright page (ii)

April 2007

33, 34, 61, and 62

July 2009

iii, iv, vii, viii, 3–14, 45, 46, 55, 56, 69, 70, 79, 80, 109, and 110

• Update your copy by inserting these revised pages. • Keep the pages you remove for reference.

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© Canadian Standards Association

Δ

Strength design in aluminum/ Commentary on CSA S157-05, Strength design in aluminum

Contents

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Technical Committee on Strength Design in Aluminum x Preface xii

S157-05, Strength design in aluminum 1 Scope 3 2 Reference publications 4 3 Definitions and symbols 5 3.1 Definitions 5 3.2 Symbols and subscripts 6 3.2.1 Symbols 6 3.2.2 Subscripts 8 4 Materials 9 4.1 Alloys 9 4.2 Mechanical properties 9 4.3 Physical properties 10 4.4 Fasteners and welds 10 5 Limit states design 11 5.1 General 11 5.1.1 Serviceability limit states 11 5.1.2 Ultimate limit states 11 5.1.3 Fatigue life 11 5.2 Safety criterion 11 5.3 Loads for buildings 11 5.3.1 Specified loads and influences 11 5.3.2 Erection loads 11 5.3.3 Thermal effects 11 5.4 Load factors and combinations 12 5.4.1 Load factors 12 5.4.2 Load combination factors 12 5.4.3 — Deleted 5.4.4 — Deleted 5.4.5 Applications other than buildings 12 5.5 Resistance factors 12 5.6 Deflections and vibrations 13 5.6.1 Deflection 13 5.6.2 Vibration 13 5.7 Provisions to avoid progressive collapse 13 6 Methods of analysis and design 13 6.1 Analysis 13 6.2 Testing 13 7 Net area, effective section, and effective strength 13 7.1 General 13 7.2 Gross area 13

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S157-05/S157.1-05

7.3 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.5 7.5.1 7.5.2 7.5.3

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Net area 13 Effective section 14 General 14 Effective thickness at welds 14 Effective thickness after local buckling of flat elements 15 Deflection under service loads 15 Effective strength and overall buckling 15 General 15 Influence of welding 15 Influence of local buckling 15

8 Local buckling of flat elements 16 8.1 Buckling stress 16 8.2 Elements supported on both longitudinal edges 16 8.2.1 Edges simply supported 16 8.2.2 Influence of adjacent elements 17 8.3 Elements supported at one longitudinal edge only 17 8.3.1 Edge simply supported 17 8.3.2 Flanges of sections 18 8.4 Elements supported at one edge with a lip at the other edge 18 8.4.1 General shapes 18 8.4.2 Uniform thickness with simple lips 18 9 Resistance of members 19 9.1 Limiting slenderness for members 19 9.2 Members in tension 20 9.2.1 Tensile resistance 20 9.2.2 Oblique welds 20 9.3 Members in compression: Buckling 20 9.3.1 Normalized slenderness 20 9.3.2 Limiting stress 21 9.3.3 Buckling stress 21 9.4 Columns 22 9.4.1 General 22 9.4.2 Flexural buckling 22 9.4.3 Torsional buckling 23 9.5 Bending 24 9.5.1 Classification of members in bending 24 9.5.2 Moment resistance of members not subject to lateral-torsional buckling 24 9.5.3 Moment resistance of members subject to lateral-torsional buckling 25 9.6 Webs in shear 27 9.6.1 Flat shear panels 27 9.6.2 Stiffened webs 28 9.6.3 Web stiffeners 28 9.6.4 Combined shear and bending in webs 29 9.6.5 Web crushing 29 9.7 Members with combined axial force and bending moment 30 9.7.1 Axial tension and bending 30 9.7.2 Eccentric tension 31 9.7.3 Beam-columns 32 9.7.4 Eccentric compression 34 9.7.5 Shear force in beam-columns 35 9.8 Built-up columns 36 9.8.1 Spacing of connectors 36 9.8.2 Multiple-bar members with discrete shear connectors 36

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© Canadian Standards Association

Strength design in aluminum/ Commentary on CSA S157-05, Strength design in aluminum

C5.4 — Deleted C5.5 Resistance factors 79

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C6. Methods of analysis and design 79 C7. Net area, effective section, and effective strength 79 C7.3 Net area 79 C7.4 Effective section 80 C7.4.2 Effective thickness at welds 80 C7.4.3 Effective thickness after local buckling of flat elements 80 C7.4.4 Deflection under service loads 82 C7.5 Effective strength and overall buckling 82 C7.5.1 General 82 C7.5.2 Influence of welding 82 C7.5.3 Influence of local buckling 82 C8. Local buckling of flat elements 83 C8.1 Buckling stress 83 C8.2 Elements supported on both longitudinal edges 83 C8.2.1 Edges simply supported 83 C8.2.2 Influence of adjacent elements 84 C8.3 Elements supported at one longitudinal edge only 85 C8.3.1 Edge simply supported 85 C8.3.2 Flanges of sections 86 C8.4 Elements supported at one edge with a lip at the other edge 86 C9. Resistance of members 87 C9.1 Limiting slenderness for members 87 C9.2 Members in tension 87 C9.3 Members in compression: Buckling 87 C9.3.2 Limiting stress 92 C9.3.3 Buckling stress 92 C9.4 Columns 93 C9.4.2 Flexural buckling 93 C9.4.3 Torsional buckling 93 C9.5 Bending 94 C9.5.1 Classification of members in bending 94 C9.5.2 Moment resistance of members not subject to lateral-torsional buckling 94 C9.5.3 Moment resistance of members subject to lateral-torsional buckling 95 C9.6 Webs in shear 99 C9.6.1 Flat shear panels 99 C9.6.2 Stiffened webs 100 C9.6.3 Web stiffeners 101 C9.6.4 Combined shear and bending in webs 101 C9.6.5 Web crushing 102 C9.7 Members with combined axial force and bending moment 102 C9.7.1 Axial tension and bending 102 C9.7.2 Eccentric tension 102 C9.7.3 Beam-columns 103 C9.7.4 Eccentric compression 105 C9.7.5 Shear force in beam-columns 105 C9.8 Built-up columns 105 C9.8.1 Spacing of connectors 105 C9.8.2 Multiple-bar members with discrete shear connectors 105

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C9.8.3 C9.8.4 C9.9

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Double angle struts 106 Lattice columns and beam columns 106 Members in torsion 106

C10. Panels 106 C10.1 Flat panels with multiple stiffeners 106 C10.1.1 Axial compression 106 C10.1.2 In-plane shear 107 C10.2 Curved panels and tubes 107 C10.2.1 Axial compression 107 C10.2.2 Radial compression 107 C10.2.3 Shear 108 C10.3 Curved axially stiffened panels in axial compression 108 C10.4 Flat sandwich panels 108 C10.4.1 General 108 C10.4.2 Panel bending 108 C10.4.3 Panel buckling 108 C10.4.4 Skin buckling 109 C10.4.5 Core strength 109 C11. Resistance of connections 109 C11.1 General 109 C11.1.1 Connection types 109 C11.2 Mechanical fasteners 109 C11.2.1 General 109 C11.2.2 Fastener spacings 110 C11.2.3 Bolts and rivets in shear and/or tension 110 C11.2.4 Bolts and rivets in bearing 110 C11.2.5 Tear-out of bolt and rivet groups (block shear) 111 C11.2.6 Eccentrically loaded fastener groups 112 C11.3 Welded connections 114 C11.3.2 Butt welds 114 C11.3.3 Fillet welds 115 C11.3.4 Flare groove welds 118 C11.3.5 Slot and plug welds 118 C11.3.7 Stud welds 118 C12. Fatigue resistance 118 C12.1 Load cycles of constant amplitude 118 C12.2 Known load spectra 119 C12.3 Unknown load spectra 119 C13. Tests 119 C13.1 General 119 C13.2 Test methods 120 C13.3 Test procedures 120 C13.3.1 Confirmatory tests 120 C13.3.2 Performance tests 120 Tables C1 — C2 — C3 — C4 —

viii

Influence of adjacent elements: Comparison of theoretical and code values for m 85 Local buckling of flanges: Comparison of theoretical and code values for m 86 Various best-fit values for the coefficients λ o and α 89 Strength of fillet welds: Comparison of some code values for the factor k 116 July 2009 (Replaces p. viii, February 2005)

© Canadian Standards Association

Strength design in aluminum

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S157-05 Strength design in aluminum 1 Scope 1.1 This Standard applies to the design of aluminum alloy members and assemblies intended to carry a known load.

1.2 This Standard specifies requirements for the design of members to meet the requirements of the National Building Code of Canada using limit states design procedures.

1.3 This Standard contains rules to determine the ultimate resistance of aluminum members and connections, and may be used for the design of aluminum assemblies in general.

1.4 Where members designed in accordance with this Standard are intended for use in structures for which other standards apply, this Standard supplements such standards, as applicable. Note: Annex A lists some applications to which other standards apply.

1.5 Where this Standard does not provide design expressions or dimensional limitations that are applicable to a specific situation, a rational design may be used, based on appropriate theories, tests, analyses, or engineering experience. Note: See Annex B for common uses of alloys.

1.6 In CSA Standards, “shall” is used to express a requirement, i.e., a provision that the user is obliged to satisfy in order to comply with the standard; “should” is used to express a recommendation or that which is advised but not required; “may” is used to express an option or that which is permissible within the limits of the standard; and “can” is used to express possibility or capability. Notes accompanying clauses do not include requirements or alternative requirements; the purpose of a note accompanying a clause is to separate from the text explanatory or informative material. Notes to tables and figures are considered part of the table or figure and may be written as requirements. Annexes are designated normative (mandatory) or informative (non-mandatory) to define their application.

1.7 The expressions contained herein are dimensionally uniform, and any consistent system of units may be employed. Where dimensions are identified, SI units are used. Evaluation is normally conducted in newtons (N) for force, millimetres (mm) for length, and megapascals (MPa = N/mm2) for stress.

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© Canadian Standards Association

2 Reference publications

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This Standard refers to the following publications, and where such reference is made, it shall be to the edition listed below. Δ

CSA (Canadian Standards Association) CAN/CSA-S16-01 (R2007) Limit states design of steel structures S408-1981 (R2001) Guidelines for the development of limit states design

Δ

W47.2-M1987 (R2008) Certification of companies for fusion welding of aluminum

Δ

W59.2-M1991 (R2008) Welded aluminum construction AISI (American Iron and Steel Institute) Steel Products Manual, No. 13 Stainless and Heat Resisting Steels Aluminum Association Publication Aluminum standards and data, Metric SI, 2003 ASME International (American Society of Mechanical Engineers) International Boiler and Pressure Vessel Code — 2004 Edition

Δ Δ

ASTM International (American Society for Testing and Materials) A 307-07B Standard Specification for Carbon Steel Bolts and Studs, 60 000 PSI Tensile Strength A 325M-09 Standard Specification for Structural Bolts, Steel, Heat Treated 830 MPa Minimum Tensile Strength [Metric] B 695-04 Standard Specification for Coatings of Zinc Mechanically Deposited on Iron and Steel B 696-00 (2004) Standard Specification for Coatings of Cadmium Mechanically Deposited B 766-86 (2003) Standard Specification for Electrodeposited Coatings of Cadmium

Δ

F 467M-08 Standard Specification for Nonferrous Nuts for General Use [Metric] ISO (International Organization for Standardization) ISO 6361-2:1990 Wrought aluminium and aluminium alloy sheets, strips and plates — Part 2: Mechanical properties ISO 6362-2:1990 Wrought aluminium and aluminium alloy extruded rods/bars, tubes and profiles — Part 2: Mechanical properties

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© Canadian Standards Association

Δ

NRCC (National Research Council Canada) National Building Code of Canada, 2010

Δ

User’s Guide — NBC 2010: Structural Commentaries (Part 4 of Division B)

Strength design in aluminum

SAE International (Society of Automotive Engineers) SAE Handbook, Volume 1, Materials, 1991 United States Department of Defense MIL-HDBK-5G 1994 Chapter 3, “Aluminum,” Military Standards Handbook, Metallic Materials and Elements for Aerospace Vehicle Structures

3 Definitions and symbols 3.1 Definitions The following definitions apply in this Standard: Δ

Buckling stress, Fc — the compressive stress that causes buckling. Characteristic resistance, Rk — the maximum force, moment, or torque that a component can be assumed to be capable of sustaining. Effective section — a section in which elements, because of welding or local buckling, are reduced to their effective thicknesses. Effective strength, Fm — the reduced strength of an element, at the ultimate limit state, to account for the influence of local buckling or welding. Effective thickness, t’ — that portion of the thickness of an element, affected by welding or local buckling, deemed to be capable of carrying the yield strength. Elastic buckling stress, Fe — the theoretical stress that would initiate elastic buckling. Element — any flat or curved portion of a section, such as a flange or web, that can be treated as a plate. Factored compressive, tensile resistance, Cr , Tr — the product of the characteristic resistance and the resistance factor. Factored load — the product of the specified load and the load factor. Heat-affected zone (HAZ) — the zone of reduced strength in the metal adjacent to a weld (see Clause 11.3.6). Heat-treated alloys — those alloys for which improved mechanical properties are obtained by their response to heat treatment.

Δ

Importance factor, γ — Deleted Limit state — a condition of a structure in which the design function is no longer fulfilled. An ultimate limit state is represented by fracture, collapse, overturning, sliding, or uncontrolled deformation. A serviceability limit state is represented by unacceptable deformation or vibrations. Limiting stress, Fo — the compressive stress that limits the capacity of a column or beam (yield, local buckling, or postbuckling strength).

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Δ

© Canadian Standards Association

Load combination factor, ψ — Deleted Load factor, α — a factor by which a specified load is multiplied, as appropriate to a particular limit state, to take into account the variability of loads and load patterns.

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Normalized buckling stress, F — the value of Fc /Fo. Normalized slenderness, λ — the value of (Fo /Fe)1/2. Resistance factor, φ — a factor applied to the characteristic resistance to account for variations in material properties, product dimensions, fabrication tolerances, and assembly procedures, and to account for the imprecision of the predictor itself. Slenderness, λ — a factored geometric ratio used to determine the stress level required to cause buckling. Specified load (service load) — a load defined in the appropriate standard or as determined by the use of the structure. Tests — Confirmatory test — a test to confirm that an item has at least the required stiffness and/or resistance. Performance test — a test to determine the actual stiffness and/or resistance of an item. Ultimate resistance test — a test to determine the maximum load carried by an item prior to the attainment of an agreed-upon level of distress. Weld throat — the shortest distance through a fillet, groove, flare groove, or partial penetration butt weld. Work-hardened alloys — those alloys for which the mechanical properties are enhanced by work hardening.

3.2 Symbols and subscripts 3.2.1 Symbols The following symbols apply in this Standard, except as further defined in specific clauses: A = area of cross-section, mm2 a = breadth of element, mm = distance, mm = thickness of a bar, mm = panel dimension, mm = weld throat, mm B = bearing resistance, N b = width, mm C = compressive force, N = geometric property c = distance from the neutral axis of the gross cross-section to the extreme fibre, mm = distance from the centroid to the centre of rotation of a weld or bolt group, mm = lip width, mm D = dead load, N/m2

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© Canadian Standards Association

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d

do E e

F F Fm f G g

H h I J Io Ip K k L

Lm M m

N

n

P

Strength design in aluminum

= fastener diameter, mm = distance from the centre of rotation, mm = distance to extreme fastener, mm = mast width, mm = member depth, mm = sandwich panel thickness, mm = hole diameter, mm = elastic modulus, MPa = load due to earthquake, N = eccentricity, mm = end edge distance, mm = distance from the centre of bearing to the end of a beam, mm = normalized buckling stress, (Fc /Fo) = strength, MPa = effective strength, F 1/2 Fy , MPa = calculated applied stress, MPa = shear modulus, MPa = acceleration due to gravity, m/s2 = transverse fastener spacing, mm = row spacing measured perpendicular to the direction of the load, mm = total length of fillet weld, mm = torsional resistance, N•mm = web depth, mm = moment of inertia (second moment of area), mm4 = St. Venant torsion constant, mm4 = polar moment of inertia about the centroid, mm4 = polar moment of inertia about the centre of rotation, mm4 = effective length factor for columns = a factor = distance, mm = length, mm = live load, N/m2 = span, mm = effective length, mm = moment, N•mm = factor for the slenderness of plates = number of shear planes = number of lines of fasteners = force, N = number of chords = total number of cycles = total number of fasteners = bearing length for beams, mm = number of cycles = number of rows of fasteners = force per unit length, N/mm = force, N

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R

r r’ S s

T t tm V v W w X y Z α β Δ

Δ

γ η θ κ λ

λ ρ ψ φ υ

© Canadian Standards Association

= force on a fastener, N = radius, mm = resistance, N = radius of gyration, mm = radius of gyration of a built-up or effective section, mm = elastic section modulus (I/c), mm3 = longitudinal fastener spacing, mm = weld size, mm = row spacing measured in the direction of the load, mm = tensile force, N = influence resulting from temperature changes or differential settlement, N = thickness, mm = effective thickness, mm = shear force, N = shear flow (force per unit length), N/mm = load due to wind, N/m2 = flange width, mm = width of outstanding leg, mm = load, N = distance from the neutral axis of the gross cross-section to the centre of a weld, mm = plastic section modulus (first moment of area), mm3 = load factor = coefficient of thermal expansion = coefficient in buckling formula = ratio of the lip to flange width = Deleted = ratio of developed to net width = angle, degrees = rigidity per unit length of a weld = slenderness = normalized slenderness, (Fo /Fe)1/2 = mass density, kg/m3 = Deleted = resistance factor = Poisson’s ratio

3.2.2 Subscripts b = bending = bearing c = compression = chord D = dead e = elastic = effective f = factored = fastener = flange

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© Canadian Standards Association

Strength design in aluminum

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g k L m

= gross = characteristic resistance = live = effective = mean n = net o = limiting value = hole p = plastic = polar r = factored resistance s = shear = stiffener T = temperature t = tension = test = torsion u = ultimate v = v-axis W = wind w = web = welded = warping x = x-axis y = y-axis = yield

4 Materials 4.1 Alloys Aluminum alloys shall conform to the Aluminum Association publication Aluminum Standards and Data, ISO 6361-2 or ISO 6362-2, or US Department of Defense publication, MIL-HDBK-5G 1994, Chapter 3. Typical alloys and products to which this Standard applies are listed in Table 1. High-strength aircraft-type alloys are not included. Cast aluminum products are not included, as they require special study.

4.2 Mechanical properties 4.2.1 The ultimate strength, Fu , and yield strength, Fy , in tension, used for design purposes, shall be the minimum values specified for the alloy in the Aluminum Association publication Aluminum Standards and Data.

4.2.2 The characteristic mechanical properties for the base metal and for the heat-affected zone extending 25 mm in each direction from the centre of a weld (see Clause 11.3.6) to be used for design purposes for the alloys and products listed in Table 1 shall be as given in Table 2. Other mechanical properties shall be derived from these values according to the following ratios: (a) compression, yield strength — Fy; (b) bearing, ultimate strength — 2.0 Fu (see Clause 11.2.4); July 2009 (Replaces p. 9, February 2005)

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© Canadian Standards Association

(c) shear, ultimate strength — 0.6 Fu ; and (d) shear, yield strength — 0.6 Fy.

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Note: If the properties of the heat-affected zone due to welding are not known, they may be taken to be equal to those of the solution heat-treated condition for heat-treated alloys and to the annealed condition for work hardened alloys.

4.2.3 Mechanical properties for weld beads are given in Table 3. The strength of weld beads for combinations of filler and base metal alloys not included in Table 3 shall be subject to confirmation.

4.3 Physical properties Physical properties for the alloys listed in Table 1 shall be taken as follows: (a) coefficient of linear thermal expansion, α — 24 × 10–6/°C; (b) elastic modulus, E — 70 000 MPa; (c) Poisson’s ratio, υ — 0.33; (d) shear modulus, G — 26 000 MPa; and (e) mass density, ρ — 2700 kg/m3.

4.4 Fasteners and welds 4.4.1 Aluminum components shall be joined using aluminum, stainless steel, cadmium-plated steel, galvanized steel fasteners, or appropriate adhesives. Δ

4.4.2 Aluminum bolts shall be of an alloy conforming to the Aluminum Association publication Aluminum Standards and Data and shall be dimensioned according to ASTM F 467.

4.4.3 Aluminum welding wire shall be of an alloy conforming to the Aluminum Association publication Aluminum Standards and Data and shall be selected according to CSA W59.2.

4.4.4 Aluminum rivets shall be of an alloy conforming to the Aluminum Association publication Aluminum Standards and Data.

4.4.5 Stainless steel bolts shall be in the 300 series specified in AISI Steel Products Manual, No. 13. Δ

4.4.6 Carbon steel bolts shall conform to ASTM A 307 or A 325M or SAE Grade 5, and shall be protected by galvanizing or cadmium plating. Galvanizing shall conform to the requirements of ASTM B 695. Cadmium plating shall conform to ASTM B 696 or B 766.

4.4.7 Special fasteners, such as blind rivets, patented bolts, and adhesives, shall be subject to confirmation of their load-carrying capacity and of their chemical compatibility with aluminum in the operating environment.

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5 Limit states design

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5.1 General 5.1.1 Serviceability limit states The specified loads shall be used to compute deflections, requirements for camber, ranges of natural frequency, and other serviceability limit states.

5.1.2 Ultimate limit states When considering the ultimate limit states, the load effects such as moments, shear forces, and stresses shall be those due to the factored loads.

5.1.3 Fatigue life When considering fatigue life, the forces shall be those due to the specified loads. Δ

5.2 Safety criterion A building and its structure shall be designed to have strength and stability such that Factored resistance ≥ Effect of the factored applied loads Factored resistance shall be determined in accordance with Clauses 7 to 13 of this Standard, and the effect of factored applied loads shall be determined in accordance with Clause 5.4.

5.3 Loads for buildings Δ

5.3.1 Specified loads and influences The following loads and influences, as specified in the National Building Code of Canada, Part 4, shall be considered in the design of a building: (a) D — dead loads, including the mass of the member and all permanent materials of construction, partitions, and stationary equipment; (b) E — earthquake loads; (c) H — lateral earth, including groundwater; (d) L — live loads, including loads due to the intended use and occupancy of the building, movable equipment, and loads due to the pressure of liquid in containers; (e) P — permanent effects caused by pre-stress; (f) S — snow and rain; (g) T — influences resulting from temperature changes or differential settlement; and (h) W — wind loads.

5.3.2 Erection loads Provisions shall be made for loads imposed on the structure during its erection.

5.3.3 Thermal effects Provision shall be made for thermal movements or, where such movements are restrained, for the forces created, commensurate with the service conditions of the structure. Note: The temperature range for external structures is obtained from the National Building Code of Canada, Appendix C.

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5.4 Load factors and combinations

Δ

5.4.1 Load factors

© Canadian Standards Association

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Load factors for buildings shall be those specified in Part 4 of the National Building Code of Canada. Δ

5.4.2 Load combination The effect of factored loads for a building or structural components shall be determined in accordance with the load combination cases listed in Part 4 of the National Building Code of Canada and the applicable combination shall be that which results in the most critical effect.

Δ

5.4.3 — Deleted

Δ

5.4.4 — Deleted 5.4.5 Applications other than buildings For many applications, design rules are specified by a regulatory authority. In other cases, if limit states design is used, the load factors shall be established in such a way that, in conjunction with the resistance factors given in Clause 5.5, the required level of reliability is provided. Note: See Annex A for information on other applications and CSA S408 for guidance on limit states design.

5.5 Resistance factors

For general structures, the following resistance factors, φ, shall be used: (a) tension, compression, and shear in beams: on yield, φy = 0.9; (b) compression in columns: on collapse due to buckling, φc = 0.9; (c) tension and shear in beams: on ultimate, φu = 0.75; (d) tension on a net section, bearing stress, tear-out: on ultimate, φu = 0.75; (e) tension and compression on butt welds: on ultimate, φu = 0.75; (f) shear stress on fillet welds: on ultimate, φf = 0.67; and (g) tension and shear stress on fasteners: on ultimate, φf = 0.67.

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Δ

Strength design in aluminum

5.6 Deflections and vibrations Note: Commentary D of the User’s Guide — NBC, provides guidance on acceptable deflections and vibrations.

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5.6.1 Deflection Deflection due to the action of the specified loads shall not exceed the serviceability limits of the materials supported or the requirements set by the intended use.

5.6.2 Vibration The vibration of the structure and its components shall be acceptable for the intended use. Structural members subjected to wind forces shall be proportioned to ensure that sustained vibrations caused by vortex shedding, galloping, or wind gusts do not arise.

5.7 Provisions to avoid progressive collapse

Δ

To provide against progressive collapse caused by accidental loads, such as vehicle impact or explosion, the structure shall incorporate redundancy, alternative load paths, protection, or other means to limit the extent of the failure. Note: Commentary B of the User’s Guide — NBC, provides guidance on structural integrity.

6 Methods of analysis and design 6.1 Analysis The forces in components shall be determined by rational analysis. In general, elastic analysis shall be employed. If plastic or other forms of ultimate strength analysis are used, it shall be demonstrated that the structure satisfies the assumptions, such as the range of non-linear behaviour, made in the analysis.

6.2 Testing The adequacy of a structure or structural assembly may be determined by direct load tests in accordance with Clause 13.

7 Net area, effective section, and effective strength 7.1 General Both the drilling of holes and welding reduce the strength of a member. To accommodate this change, deductions shall be made from the gross cross-section to give a net or effective section. Strength reduction due to local buckling shall be represented by a reduction in the effective thickness or in the effective strength. (See Clauses 7.2 to 7.5.)

7.2 Gross area The gross area of a member shall be determined by summing the products of the thickness and the gross width of each element as measured perpendicular to the axis of the member.

7.3 Net area The net cross-sectional area of members in tension shall be the gross area less the sum of the hole diameters multiplied by the thickness (Σdot) in line across the section. For a chain of holes extending in any diagonal or zigzag line across a tension member (see Figure C1 in the Commentary), the net width of the part shall be obtained by deducting from the gross width the sum of the diameters of all the holes in the chain and adding, for each gauge space in the chain, a quantity given by

July 2009 (Replaces p. 13, April 2007)

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s2/4g where s = spacing of successive holes in the direction of the force g = transverse spacing of the two holes do = hole diameter t = thickness The least net width so obtained shall be used in calculating the net area. The staggered rupture line controls when s2 is less than 2gdo. The distance between holes shall not be less than 2.5d, where d is the fastener diameter. Clause 9.7.2 gives effective net areas for eccentrically loaded tension members.

7.4 Effective section 7.4.1 General Both welding and local buckling cause a reduction in the overall strength, which is accounted for by using an effective section to establish the bending resistance of a member. The geometric properties of the effective section shall be computed using the effective thicknesses of the elements according to Clauses 7.4.2 and 7.4.3. The section modulus of the effective section shall be multiplied by the yield strength of the base metal to give the limiting moment (characteristic resistance).

7.4.2 Effective thickness at welds 7.4.2.1 Plastic section modulus Where only parts of the cross-section are influenced by welding, as with longitudinal welds, the effective thickness, tm , of the metal in the heat-affected zone (see Clause 11.3.6), shall be determined as follows: tm = t (Fwy /Fy) ≤ t The section modulus of the effective section shall be multiplied by the yield strength of the base metal to give the limiting moment (characteristic resistance).

7.4.2.2 Elastic section modulus The effective thickness, tm , used in calculating the elastic section modulus to determine the moment at first yield in the base metal shall be given by tm = t (Fwy /Fy)(c/y) ≤ t where t = original thickness Fwy = yield strength in the heat-affected zone Fy = yield strength of the base metal c = distance from the neutral axis of the gross cross-section to the extreme fibre y = distance from the neutral axis of the gross section to the centre of the weld If local buckling occurs in a welded element, the influence of the weld may be neglected and the reduced thickness attributed to local buckling shall be used.

7.4.2.3 Deflections The gross cross-section of welded members shall be used for the calculation of deflections.

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m = number of shear planes A = nominal cross-sectional area of the fastener Fu = ultimate strength of the fastener material

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If the bolt threads are in a shear plane, the value of Vr shall be multiplied by 0.75.

11.2.3.2 Tensile resistance The factored tensile resistance of a bolt, Tr , shall be the lesser of the values given by (a) Tr = φf 0.75 A Fu ; and (b) Tr = φf A Fy where φf = fastener resistance factor A = cross-sectional area of the bolt based on the nominal diameter Fu = ultimate strength of the bolt material Fy = yield strength of the bolt material Rivets are not commonly used in tension.

11.2.3.3 Combined shear and tension For a bolt subject to both shear and tension, exclusive of tension due to tightening, the reduced factored tensile resistance, T’r , shall be given by the following:

T ′r = 1.25 Tr − kVf ≤ Tr where Tr = factored tensile resistance given in Clause 11.2.3.2 k = 1.8, or 1.4 when the bolt thread is excluded from the shear plane Vf = factored shear load on the bolt

11.2.4 Bolts and rivets in bearing 11.2.4.1 Bearing strength The factored bearing resistance, Br , of the connected material for each loaded fastener shall be the lesser of the values given by the following formulas: (a) Br = φu etFu; and (b) Br = φu 2dtFu where φu = ultimate resistance factor e = perpendicular distance from the hole centre to the end edge in the direction of the loading (not less than 1.5d ) t = plate thickness Fu = ultimate strength of the connected material d = fastener diameter

11.2.4.2 Lap joints For unrestrained lap joints in tension, the factored bearing resistance, Br , shall be the lesser of the values given by the following formulas: (a) Br = φu(t1 + t2) e Fu /4; and (b) Br = φu(t1 + t2) d Fu /2 ≤ φu 2dt1Fu where φu = ultimate resistance factor

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t1, t2 = e = Fu = d = Δ

© Canadian Standards Association

thicknesses of the plates, t1 < t2 distance from the hole centre to the end edge, but not less than 1.5d ultimate tensile strength of the connected material fastener diameter

11.2.4.3 Oblique end edges Where the end edge is oblique to the line of action of the tension force (see Figure C23 of the Commentary), the factored bearing resistance, Br , for a single bolt shall be the lesser of the values given by the following formulas: (a) Br = φu [e + (e – d) cos2 θ] tFu; and (b) Br = φu 2 dtFu where φu = ultimate resistance factor e = perpendicular distance from the hole centre to the end edge d = fastener diameter θ = angle made by the end edge with the direction of the force (when θ = 0°, d = do) t = plate thickness Fu = ultimate strength of the connected material do = hole diameter

11.2.5 Tear-out of bolt and rivet groups (block shear) 11.2.5.1 Tension: Rectangular patterns For a group of two or more fasteners in a rectangular pattern (see Figure C24(a) of the Commentary) resisting a force directed towards the edge, the factored bearing resistance, Rb , of the group of fasteners shall be the lesser of the values given by the following formulas: (a) Rb = φu [(m – 1)(g – do ) + (n – 1)(s – do ) + e]tFu; and (b) Rb = φu 2 NdtFu where φu = ultimate resistance factor m = number of fasteners in the first transverse row g = fastener spacing measured perpendicular to the direction of the force do = hole diameter n = number of transverse rows of fasteners s = fastener spacing measured in the direction of the force e = edge distance in the direction of force for the first row, but not less than 1.5d = 2d, when e > 2d t = plate thickness Fu = ultimate strength of the connected material N = total number of fasteners d = fastener diameter

11.2.5.2 Tension: Trapezoidal patterns For a triangular or trapezoidal group of fasteners in a staggered pattern (see Figure C24(b) of the Commentary) resisting a force directed towards the edge, the factored bearing resistance, Rb , of the group shall be the lesser of the values given by the following formulas: (a) Rb = φu [2 (m – 1)(g – do + s2/4g) + e] tFu; and (b) Rb = φu 2NdtFu where φu = ultimate resistance factor m = number of fasteners in the first transverse row

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13.3 Test procedures

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13.3.1 Confirmatory tests 13.3.1.1 Serviceability For confirmatory tests of serviceability under the action of the service loads, the specified limit state shall not be exceeded. Δ

13.3.1.2 Ultimate limit state For confirmatory tests to determine if the ultimate strength is adequate, the applied force shall be that due to the factored loads, as specified by Clause 5.4, divided by the appropriate resistance factor from Clause 5.5.

13.3.2 Performance tests 13.3.2.1 Characteristic resistance For performance tests, the highest load that the joint, member, or assembly can sustain without rupture, collapse, or excessive deformation shall be measured. The characteristic resistance shall be taken as one of the following, as appropriate: (a) If four or less items are tested, the characteristic resistance shall be the lesser of (i) the mean of the ultimate test loads multiplied by 0.9; and (ii) the lowest ultimate test load achieved. (b) If more than four items are tested, the characteristic resistance shall be the mean of the ultimate test loads minus one standard deviation. “Excessive deformation” shall be a measurable distortion that is agreed by the purchaser or the purchaser’s representative.

13.3.2.2 Adjustment for variation in yield and ultimate strength and dimensions If the yield or ultimate strength of the material of the test item exceeds the specified value, the measured test loads shall be multiplied by the following: (a) for the gross area in tension or bending, ⎛ A ⎞ Fy ⎜ ′⎟ ⎝ A ⎠ Fy′ and (b) for the net area in tension or bending, ⎛ A ⎞ Fu ⎜ ′⎟ ⎝ A ⎠ Fu′ For failure due to buckling, the test load shall be multiplied by

(

Fy / Fy′ + ( λ/1.5) 1 − Fy / Fy′

)

≤ 1

where A = nominal area A’ = actual area Fy = specified yield strength F’y = measured yield strength Fu = specified ultimate strength F’u = measured ultimate strength λ = normalized slenderness

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14 Fabrication 14.1 General Licensed for/Autorisé à Reid Costley, Cascade Engineering Group, Sold by/vendu par CSA on/le 10/27/2009. Single user license only. Storage, distribution or use on network prohibited./Permis d'utilisateur simple seulement. Le stockage, la distribution ou l'utilisation sur le réseau est interdit.

Fabrication practices shall follow those for steel, as in CAN/CSA-S16, except as otherwise modified by this Standard.

14.2 Tolerances 14.2.1 Fabrication tolerances shall be in accordance with CAN/CSA-S16, except as stated in Clause 14.2.2.

14.2.2 The tolerance on the end distance of bolt holes shall be – 0, +2 mm.

14.3 Layout and marking 14.3.1 Layout As the linear coefficient of expansion of aluminum is about twice that of steel, a temperature correction may be necessary in the layout of critical dimensions when using steel tapes in areas of unusually high or low temperatures. The correction shall be given by σ = 0.000012 L(20 – T) where L = length, mm T = ambient temperature, °C

14.3.2 Marking Hole centres may be centre-punched and cut-off lines may be punched or scribed. Centre-punching and scribing shall not be used where such marks would remain on fabricated material. Stamped erection marks may be used, but the impression shall be free from sharp corners and the mark shall not be located in highly stressed areas of the component. Paint or ink erection marks shall be used where fatigue is a consideration.

14.4 Forming 14.4.1 In general, forming of aluminum shall be carried out at room temperature. Should hot forming be unavoidable, the procedures shall conform to the requirements of CSA W59.2, and the post-forming mechanical properties shall be checked using hardness tests.

14.4.2 Bends shall be smooth, without sharp kinks. Cracks shall be cause for rejection if the crack lies in a zone that is stressed in service.

14.4.3 If straightening or flattening is carried out, it shall be done by a process and in a manner that does not injure the material.

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A.4.4 Cranes and hoists for materials Cranes and hoists for materials are generally regulated by the provincial labour authority. For mobile cranes, see CAN/CSA-Z150, and for tower cranes, see CAN/CSA-Z256.

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A.4.5 Lifts and equipment for people Lifts and equipment for people are regulated by the provincial labour authorities and sometimes by municipal regulations. For manlifts, see CAN/CSA-B311; for elevated work platforms, see CAN/CSA-B354.2; for portable ladders, see CSA CAN3-Z11; and for vehicle-mounted aerial devices, see CAN/CSA-C225.

A.4.6 Hoppers, bins, pipes, and conduits There is no general authority for hoppers, bins, pipes, and conduits; therefore, they may be considered as part of the larger unit (i.e., vehicle, building, or ship).

A.4.7 Pipeline systems For pipeline systems, see regulations on ecology in the relevant location and also CSA Z245.6.

A.5 Miscellaneous uses A.5.1 Medical equipment, appliances, and biomedical items Medical equipment, appliances, and biomedical items are usually controlled by specification; however, provincial medical associations should be contacted.

A.5.2 Sports equipment and safety items The controlling bodies for the sports may have regulations affecting the design of equipment. The following standards may be applicable: CAN/CSA-Z262.1 on ice hockey helmets, CAN/CSA-Z262.2 on face protectors and visors for ice hockey players, and CAN/CSA-Z94.1 on industrial protective headwear.

A.5.3 Plumbing products and materials For plumbing products and materials, the municipal authority should be referred to, together with CAN/CSA-B602.

A.5.4 Pressure vessels Pressure vessels are regulated by various authorities depending on the vessel’s use and whether they are static or mobile. The most generally accepted regulations are contained in CSA B51 and the ASME Pressure Vessel Code published by the American Society of Mechanical Engineers. Pressure vessel analysis is not covered by this Standard.

A.6 Reference publications This Annex refers to the following publications, and where such reference is made, it is to the edition listed below. CSA (Canadian Standards Association) B51-03 Boiler, Pressure Vessel, and Pressure Piping Code CAN/CSA-B311-02 Safety Code for Manlifts

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CAN/CSA-B354.1-04 Elevating Rolling Work Platforms

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CAN/CSA-B354.2-01 Self-Propelled Elevating Work Platforms CAN/CSA-B354.4-02 Self-Propelled Boom-Supported Elevating Work Platforms Δ

CAN/CSA-B602-05 Mechanical Couplings for Drain, Waste, and Vent Pipe and Sewer Pipe CAN/CSA-C225-00 Vehicle-Mounted Aerial Devices

Δ

CAN/CSA-S6-06 Canadian Highway Bridge Design Code S37-01 Antennas, Towers and Antenna-Supporting Structures S269.1-1975 (R2003) Falsework for Construction Purposes CAN/CSA-S269.2-M87 (R2003) Access Scaffolding for Construction Purposes CAN3-Z11-M81 (R2003) Portable Ladders

Δ

CAN/CSA-Z94.1-05 Industrial Protective Headwear CAN/CSA-Z150-98 (R2004) Safety Code on Mobile Cranes CAN/CSA-Z240 MH Series-92 (R2001) Mobile Homes

Δ

CAN/CSA-Z240 RV Series-08 Recreational Vehicles

Δ

Z245.6-06 Coiled Aluminum Line Pipe and Accessories CAN/CSA-Z256-M87 (R2001) Safety Code for Material Hoists CAN/CSA-Z262.1-M90 (R2002) Ice Hockey Helmets CAN/CSA-Z262.2-M90 (R2002) Face Protectors and Visors for Ice Hockey Players

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C5.3 Loads for buildings Appendix C to the National Building Code of Canada gives loads for the geographic location, and for the type and shape of the structure.

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Δ

C5.4 — Deleted C5.5 Resistance factors Characteristic resistances are predicted by the design expressions in the Standard, which have been developed on the basis of theory and computer simulation to represent the mean value of test results less two standard deviations. To increase confidence that the structure can resist the factored loads, additional factors are applied to the characteristic resistances. These resistance factors have been derived in such a way that in combination with the load factors for buildings, the reliability index is greater than 3 for members and 4 for connections. The reliability index, defined in CSA S408, is a measure of the extent by which the computed strength exceeds the probable load. The type of failure anticipated is also reflected in the choice of a resistance factor. Where there is an overall yielding of the cross-section, but no precipitous failure, a resistance factor of 0.9 is adopted. Where failure is of a more brittle type, with little evident distress prior to collapse, as in the case of rupture in tension at a net section, the factor is reduced to 0.75. Because of their importance, the value for fasteners is 0.67. These factors parallel those adopted for steel structures (see CAN/CSA-S16). In other applications, the resistance factors are held constant (as they are related to the predictors used to establish strength), while the load factors are chosen to provide the required level of reliability.

C6. Methods of analysis and design Elastic analysis is always permitted even though the force/deformation relationships for the components, up to failure, may not be linear. The justification is that so long as equilibrium is satisfied and there are no “brittle” components, the solution will be a lower bound. Caution may be needed in highly redundant lattice structures such as space trusses (Schmidt, 1976). In general, aluminum assemblies do not possess the range of ductility of structural steel. There is a lower spread between yield and ultimate strength, and a lower elongation at rupture. The reliance placed in steel design on the redistribution of stress after yielding cannot always be transferred to aluminum, and a clear grasp of the limits that can be exploited is essential for safe designs, if plastic or limit design methods are used. Because many components are not subject to direct analysis and design, proof by testing is acceptable in all cases.

C7. Net area, effective section, and effective strength C7.3 Net area For the design of the net section of plates in tension across a row of staggered holes (see Figure C1), the holes are deducted and s2/4g added for each space. This was shown by Brady and Drucker (1955) to correspond to an upper bound solution based on a theoretical plastic analysis.

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g

s

Figure C1 Net section at staggered holes (See Clauses 7.3 and C7.3.)

C7.4 Effective section C7.4.2 Effective thickness at welds To account for the reduction in strength at a longitudinal weld used for members in bending, it is assumed that the thickness of the heat-affected zone is reduced in the ratio Fwy /Fy. The plastic section modulus of the effective section gives the fully plastic moment capacity when multiplied by the yield strength of the base metal (Brungraber and Clark, 1962). Should the extreme fibre stress in the base metal be required to remain elastic, then the stress in elements nearer to the neutral axis will be lower and the effective thickness of the HAZ, given by t(Fwy /Fy)(c/y) ≤ t, reflects this fact, where y is the distance from the neutral axis and c is the distance to the extreme fibre. Any change in the location of the neutral axis is not taken into account in this calculation, with negligible error.

C7.4.3 Effective thickness after local buckling of flat elements In thin flat elements that have both longitudinal edges supported, the stress distribution does not remain uniform after initial buckling. As the force is increased, the stress towards the boundaries increases and, at the limit, reaches the yield strength. The magnitude of the total force can be represented by the product of the yield strength and an effective area, or as the product of the gross area and an effective strength. The device is artificial but useful. von Karman et al. (1932) suggested that the limiting force be given by R = bt (Fe Fy)1/2 = Fy bt (Fe /Fy)1/2 This is equivalent to taking an effective width given by b’ = b (Fc /Fy)1/2, which is assumed to sustain the yield strength. The effective width of each part in compression is used in calculating the properties of the effective section. This model proves to be unconservative, and better agreement with test results, over the full range of slenderness, is obtained by using the actual buckling stress, Fc , as adopted in this Standard (Marsh, 1998) rather than the elastic value, Fe , used by von Karman. A comparison with other methods and test results is shown in Figure C2 for steel and Figure C3 for aluminum. Further, an effective thickness, rather than an effective width, is adopted. Where the stress in an element varies from compression to tension, as in the web of a beam, the thickness of the entire element is reduced, thereby avoiding the trial and error needed to locate the centroid of an effective section when its position influences the effective areas, as occurs if an effective width is used. The result is conservative when compared with the value obtained for a web when only the portion in compression is reduced, but as the web makes a relatively small contribution to the overall bending strength, the resulting minor loss of economy is acceptable. This procedure is used for bending and for beam-columns that fail in the plane of bending.

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Commentary on CSA S157-05, Strength design in aluminum

C10.4.4 Skin buckling The skin of a sandwich panel behaves as a plate on an elastic foundation, for which the critical stress is given theoretically by

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Fc = 0.86 (EEcGc )1/3 where Ec and Gc = the elastic properties of the core Because this mode of buckling is very sensitive to imperfections, the factor is reduced to 0.5. Equating the modified expression to the Euler formula gives the slenderness λ = 4.5E1/3/(EcGc )1/6 This is used with the yield strength of the skin alloy to obtain the buckling stress.

C10.4.5 Core strength Bond between the skin and core must resist the shear force due to the lateral load and provide the tensile strength required to hold the skin to the core. This required tensile strength has been derived on the assumption of an initial bow in the skin equal to 0.001 times the wave length of the buckle.

C11. Resistance of connections C11.1 General C11.1.1 Connection types The Standard recognizes the use of a wide variety of fastening methods, proprietary or otherwise. It is not the intent of the Standard to limit the use of these methods, provided that the suitability of the material can be shown.

C11.2 Mechanical fasteners C11.2.1 General The Standard gives design procedures for bolts and solid rivets only. Mechanical fasteners are usually galvanized or cadmium-plated steel, stainless steel, or aluminum. The actual material of a fastener, given that it is of sufficient strength, is usually only of concern where corrosion might occur. In the case of rivets, the material must be capable of being upset without damaging the parent material.

C11.2.1.4 Slip-critical joints Pre-loaded bolts, permitting the joint to transfer the force by friction, are not commonly used in aluminum, but where rigidity under service loads is essential, there has been sufficient experimental work to provide recommendations (ECCS, 1978). Surface treatment, such as sanding, is required. It is preferred that all bolts act in bearing at the ultimate state. When the holes are slotted in the direction of the force and the level of security is to be maintained, a safety margin on the theoretical slipload in excess of 2 is provided.

C11.2.1.6 Maximum number of fasteners Longitudinally loaded rows of fasteners are not uniformly stressed, and although yielding will largely reduce this non-uniformity by the time the ultimate load is reached, a joint that has many bolts in line may possess a reduced overall load capacity. The limit of six fasteners (at a spacing of 3d, this gives a length of 15d) has been shown to give an acceptable performance (Francis, 1953).

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C11.2.2 Fastener spacings

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C11.2.2.1 Minimum spacing Minimum spaces between fasteners and edge distances are governed by (a) clearance for bolt heads and driving tools; (b) build-up of compressive stress generated by cold formed rivets; and (c) limits to the validity of the design formulas given.

C11.2.2.2 Maximum spacing Maximum spaces between fasteners that are treated as acting together may be governed by local plate buckling in compression. For rows of fasteners widely spaced across the direction of force, with s < 1.3 g, the plate buckles into waves with all edges fixed, represented by a slenderness λ = 1.3 g/t. For fasteners widely spaced along the direction of force, with s > 1.3 g, the plate buckles over the length s, fixed at the transverse lines of fasteners, for which λ = 1.7 s/t. For staggered patterns of fasteners, the zone with fixed edges is defined by two pairs of fasteners spaced at 2g and 2s. When s = g, this gives a value of λ = 1.9 s/t. Linear relationships are adopted as s/g→0 and g/s→0, to the limiting values of 1.3 g/t and 1.7 s/t, respectively.

C11.2.3 Bolts and rivets in shear and/or tension Although the von Mises’s criterion, which gives a shear yield strength equal to 1/ 3 times the tensile yield strength, does not apply to ultimate strengths of the “engineering” variety, the customary ratio of 0.6 is adopted, as it usually gives conservative values for the shear strength. A bolt in tension fails across the net section at the thread, which is approximately 0.7 times the gross area. The constrained region of the net section inhibits necking, leading to an ultimate tensile strength closer to the “true” value than to the “engineering” value. Some recognition of this is seen in the formula for the tensile resistance, Tk = 0.75AFu , where A is the gross area. Interaction between shear and tension follows steel practice in CAN/CSA-S16.

C11.2.4 Bolts and rivets in bearing C11.2.4.1 Bearing strength The stress exerted by the fastener on the wall of the hole (termed the bearing stress) is not, as such, a design consideration. Failure occurs due to the tearing of the material adjacent to the bolt hole and thus varies with the end distance for force directed towards an edge. The limiting force is related to the ultimate strength of the metal (Fisher and Struick, 1964; Marsh, 1979). No use is made of a “yield strength” in bearing. As the end distance, e, increases, the resistance increases, approaching a constant value after the edge distance exceeds approximately twice the bolt diameter. Calculations based on shearing along the two planes beside the hole lead to the resistance R = 2Fsu et. Using the Tresca criterion, Fsu = Fu /2, gives R = Fu et, a convenient expression that has been shown to be conservative. Bearing stress on the fastener itself is never a consideration (Hartman et al., 1944).

C11.2.4.2 Lap joints Unrestrained lap joints in tension are eccentrically loaded and distort so as to introduce tension in the fastener and thereby cause a reduction in strength. This is most pronounced when the plates are of equal thickness, in which case the bearing strength is halved. The expression used, based on tests for steel sheet (Baehre and Berggen, 1973), requires that the thicker sheet be three times the thickness of the thinner sheet if Clause 11.2.4.1 is to be valid for the thinner sheet. Δ

C11.2.4.3 Oblique end edges For force directed at an angle to an oblique end edge (see Figure C23), the formula for the resistance gives a transition between bearing failure, when the angle between the force direction and the end edge is 90°, and tension failure on the net section with d = do, when the angle is 0°.

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Update No. 2 CAN/CSA-S157-05/S157.1-05 April 2007 Note: General Instructions for CSA Standards are now called Updates. Please contact CSA Information Products Sales or visit www.ShopCSA.ca for information about the CSA Standards Update Service. Title: Strength design in aluminum/Commentary on CSA S157-05, Strength design in aluminum — originally published February 2005 Revisions issued: Update No. 1 — February 2007 If you are missing any updates, please contact CSA Information Products Sales or visit www.ShopCSA.ca. The following revisions have been formally approved and are marked by the symbol delta (Δ) in the margin on the attached replacement pages: Revised

Clauses 3.1, 7.4.2.1, 7.5.2, 8.2.2, 8.3.2, 9.7.4.2 and 11.2.4.3, and Table 4

New

None

Deleted

None

CAN/CSA-S157-05/S157.1-05 originally consisted of 137 pages (xiii preliminary and 124 text), each dated February 2005. It now consists of the following pages: February 2005

iii–xiii, 3, 4, 7–12, 19–32, 35–44, 47–60 and 63–124

February 2007

Cover, National Standards of Canada text, title pages (i, 1, and 2), and copyright page (ii)

April 2007

5, 6, 13–18, 33, 34, 45, 46, 61 and 62

• Update your copy by inserting these revised pages. • Keep the pages you remove for reference.

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© Canadian Standards Association

Strength design in aluminum

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NRCC (National Research Council Canada) National Building Code of Canada, User’s Guide — NBC 1995, Structural Commentaries (Part 4): Commentary A, “Serviceability Criteria for Deflections and Vibrations”; Commentary C, “Structural Integrity” SAE International (Society of Automotive Engineers) SAE Handbook, Volume 1, Materials, 1991 United States Department of Defense MIL-HDBK-5G 1994 Chapter 3, “Aluminum,” Military Standards Handbook, Metallic Materials and Elements for Aerospace Vehicle Structures

3 Definitions and symbols 3.1 Definitions The following definitions apply in this Standard: Actual buckling stress, Fc — the compressive stress that causes buckling. Characteristic resistance, Rk — the maximum force, moment, or torque that a component can be assumed to be capable of sustaining. Effective section — a section in which elements, because of welding or local buckling, are reduced to their effective thicknesses. Effective strength, Fm — the reduced strength of an element, at the ultimate limit state, to account for the influence of local buckling or welding. Effective thickness, t’ — that portion of the thickness of an element, affected by welding or local buckling, deemed to be capable of carrying the yield strength. Elastic buckling stress, Fe — the theoretical stress that would initiate elastic buckling. Element — any flat or curved portion of a section, such as a flange or web, that can be treated as a plate. Factored compressive, tensile resistance, Cr , Tr — the product of the characteristic resistance and the resistance factor. Factored load — the product of the specified load and the load factor. Heat-affected zone (HAZ) — the zone of reduced strength in the metal adjacent to a weld (see Clause 11.3.6). Heat-treated alloys — those alloys for which improved mechanical properties are obtained by their response to heat treatment. Importance factor, γ — an additional factor applied to the loads to provide an increased margin of safety against collapse, and which is related to the use and occupancy of the building. Limit state — a condition of a structure in which the design function is no longer fulfilled. An ultimate limit state is represented by fracture, collapse, overturning, sliding, or uncontrolled deformation. A serviceability limit state is represented by unacceptable deformation or vibrations. Limiting stress, Fo — the compressive stress that limits the capacity of a column or beam (yield, local buckling, or postbuckling strength). April 2007 (Replaces p. 5, February 2005)

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Load combination factor, ψ — a factor applied to combined loads, other than dead load, to take into account the reduced probability of loads from different sources acting simultaneously.

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Load factor, α — a factor by which a specified load is multiplied, as appropriate to a particular limit state, to take into account the variability of loads and load patterns. Normalized buckling stress, F — the value of Fc /Fo. Normalized slenderness, λ — the value of (Fo /Fe)1/2. Resistance factor, φ — a factor applied to the characteristic resistance to account for variations in material properties, product dimensions, fabrication tolerances, and assembly procedures, and to account for the imprecision of the predictor itself. Slenderness, λ — a factored geometric ratio used to determine the stress level required to cause buckling. Specified load (service load) — a load defined in the appropriate standard or as determined by the use of the structure. Tests — Confirmatory test — a test to confirm that an item has at least the required stiffness and/or resistance. Δ

Performance test — a test to determine the actual stiffness and/or resistance of an item. Ultimate resistance test — a test to determine the maximum load carried by an item prior to the attainment of an agreed-upon level of distress. Weld throat — the shortest distance through a fillet, groove, flare groove, or partial penetration butt weld. Work-hardened alloys — those alloys for which the mechanical properties are enhanced by work hardening.

3.2 Symbols and subscripts 3.2.1 Symbols The following symbols apply in this Standard, except as further defined in specific clauses: A = area of cross-section, mm2 a = breadth of element, mm = distance, mm = thickness of a bar, mm = panel dimension, mm = weld throat, mm B = bearing resistance, N b = width, mm C = compressive force, N = geometric property c = distance from the neutral axis of the gross cross-section to the extreme fibre, mm = distance from the centroid to the centre of rotation of a weld or bolt group, mm = lip width, mm D = dead load, N/m2

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5.6 Deflections and vibrations Note: Commentary A of the National Building Code of Canada, User’s Guide, provides guidance on acceptable deflections and vibrations.

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5.6.1 Deflection Deflection due to the action of the specified loads shall not exceed the serviceability limits of the materials supported or the requirements set by the intended use.

5.6.2 Vibration The vibration of the structure and its components shall be acceptable for the intended use. Structural members subjected to wind forces shall be proportioned to ensure that sustained vibrations caused by vortex shedding, galloping, or wind gusts do not arise.

5.7 Provisions to avoid progressive collapse To provide against progressive collapse caused by accidental loads, such as vehicle impact or explosion, the structure shall incorporate redundancy, alternative load paths, protection, or other means to limit the extent of the failure. Note: Commentary C of the National Building Code of Canada, User’s Guide, provides guidance on structural integrity.

6 Methods of analysis and design 6.1 Analysis The forces in components shall be determined by rational analysis. In general, elastic analysis shall be employed. If plastic or other forms of ultimate strength analysis are used, it shall be demonstrated that the structure satisfies the assumptions, such as the range of non-linear behaviour, made in the analysis.

6.2 Testing The adequacy of a structure or structural assembly may be determined by direct load tests in accordance with Clause 13.

7 Net area, effective section, and effective strength 7.1 General Both the drilling of holes and welding reduce the strength of a member. To accommodate this change, deductions shall be made from the gross cross-section to give a net or effective section. Strength reduction due to local buckling shall be represented by a reduction in the effective thickness or in the effective strength. (See Clauses 7.2 to 7.5.)

7.2 Gross area The gross area of a member shall be determined by summing the products of the thickness and the gross width of each element as measured perpendicular to the axis of the member.

7.3 Net area The net cross-sectional area of members in tension shall be the gross area less the sum of the hole diameters multiplied by the thickness (Σdot) in line across the section. For a chain of holes extending in any diagonal or zigzag line across a tension member (see Figure C1 in the Commentary), the net width of the part shall be obtained by deducting from the gross width the sum of the diameters of all the holes in the chain and adding, for each gauge space in the chain, a quantity given by

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s2/4g where s = spacing of successive holes in the direction of the force g = transverse spacing of the two holes do = hole diameter t = thickness The least net width so obtained shall be used in calculating the net area. The staggered rupture line controls when s2 is less than 2gdo. The distance between holes shall not be less than 2.5d, where d is the fastener diameter. Clause 9.7.2 gives effective net areas for eccentrically loaded tension members.

7.4 Effective section 7.4.1 General Both welding and local buckling cause a reduction in the overall strength, which is accounted for by using an effective section to establish the bending resistance of a member. The geometric properties of the effective section shall be computed using the effective thicknesses of the elements according to Clauses 7.4.2 and 7.4.3. The section modulus of the effective section shall be multiplied by the yield strength of the base metal to give the limiting moment (characteristic resistance).

7.4.2 Effective thickness at welds Δ

7.4.2.1 Plastic section modulus Where only parts of the cross-section are influenced by welding, as with longitudinal welds, the effective thickness, tm , of the metal in the heat-affected zone (see Clause 11.3.6), shall be determined as follows: tm = t (Fwy /Fy) ≤ t The section modulus of the effective section shall be multiplied by the yield strength of the base metal to give the limiting moment (characteristic resistance).

7.4.2.2 Elastic section modulus The effective thickness, tm , used in calculating the elastic section modulus to determine the moment at first yield in the base metal shall be given by tm = t (Fwy /Fy)(c/y) ≤ t where t = original thickness Fwy = yield strength in the heat-affected zone Fy = yield strength of the base metal c = distance from the neutral axis of the gross cross-section to the extreme fibre y = distance from the neutral axis of the gross section to the centre of the weld If local buckling occurs in a welded element, the influence of the weld may be neglected and the reduced thickness attributed to local buckling shall be used.

7.4.2.3 Deflections The gross cross-section of welded members shall be used for the calculation of deflections.

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7.4.3 Effective thickness after local buckling of flat elements

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7.4.3.1 Flat elements with both long edges supported For a flat element supported at both longitudinal edges, subjected in some part to compressive stress which causes local buckling, the effective thickness, tm , of the entire element at the ultimate limit state shall be taken as tm = t ( F )1/2 where t = element thickness F = normalized buckling stress for plates, from Clause 9.3.3, using a normalized slenderness, λ from Clause 9.3.1, with Fo = Fy , and the slenderness, λ, given by λ = mb/t where m = a factor from Clause 8.2 appropriate to the stress distribution and boundary conditions b = element width

7.4.3.2 Angles and flanges Angle sections, and outstanding elements supported along one long edge only, shall not be considered to possess postbuckling strength, and torsional or local buckling shall be deemed to lead to collapse.

7.4.4 Deflection under service loads Local buckling due to unfactored service loads is not extensive and the gross section properties shall be used when calculating deflections.

7.5 Effective strength and overall buckling 7.5.1 General Where welding, or local buckling with postbuckling strength, influences the flexural buckling of columns or lateral buckling of beams, the capacity shall be established by using the effective strength, Fm , given in Clauses 7.5.2 and 7.5.3. This effective strength represents the limiting stress, Fo , described in Clause 9.3.2, for use in Clause 9.4.1 to determine the mean stress to cause overall buckling. Δ

7.5.2 Influence of welding For members containing longitudinal welds, the effective strength, Fm , shall be given by Fm = Fy – (Fy – Fwy)Aw /A where Fy = yield strength of the base metal Fwy = yield strength of the heat-affected zone Aw = cross-sectional area of the heat-affected zone A = gross cross-sectional area This value of the effective strength, Fm , shall be used as the limiting stress, Fo , in Clauses 9.3.2(g) and 9.4.1, in conjunction with the gross cross-sectional area, when determining the resistance to overall buckling (see also Clause 9.4.2.2). Note that the location of the HAZ is not a factor (see Clause 11.3.6).

7.5.3 Influence of local buckling For flat elements supported on both long edges, subjected to a compressive stress that causes local buckling, the effective strength, Fm , of the element shall be given by Fm = F

1/2

Fy

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where F = normalized buckling stress for plates, obtained from Clause 9.3.3, corresponding to a normalized slenderness, λ , obtained from Clause 9.3.1, using Fo = Fy and the slenderness given by

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λ = mb/t

Fy

where m = a factor from Clause 8.2 appropriate to the stress distribution and boundary conditions b = element width t = element thickness = yield strength

This value of the effective strength, Fm , shall be used as the limiting stress, Fo , in Clause 9.4.1, in conjunction with the gross cross-sectional area, to determine the resistance to overall buckling.

8 Local buckling of flat elements 8.1 Buckling stress The buckling stress, Fc , of a flat element subjected to compressive stress shall be obtained using a slenderness, λ, given by λ = mb/t where m = local buckling factor given in Clause 8.2 or 8.3 b = element width (see Items (a) and (b) below) t = element thickness This value of λ shall be used to determine the normalized slenderness, λ (see Clause 9.3.1). Clause 9.3.3 shall then be used to determine the normalized buckling stress and hence the actual initial buckling stress, Fc , given by Fc = F Fy where F = normalized buckling stress Fy = yield strength The element width, b, shall be measured as follows: (a) For sections of uniform thickness, such as shapes formed from sheet, it shall be measured from the intersections of the median lines of adjacent elements, ignoring any corner radii. (b) For extruded sections with root fillets at the junctions, it shall be the distance from the tangent points of the root radii.

8.2 Elements supported on both longitudinal edges 8.2.1 Edges simply supported For a linear variation in stress across an element whose long edges are simply supported, the slenderness, λ, shall be given by λ = mb/t where m = 1.15 + f2 /2f1, when –1 < f2 /f1 < 1 = 1.3/(1 – f2 /f1), when f2 /f1 < –1

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where f1 = maximum compressive stress, negative f2 = stress at the other edge, positive when the stress is tensile b = element width (see Clause 8.1) t = element thickness A range of cases is illustrated in Figure C4 of the Commentary. Δ

8.2.2 Influence of adjacent elements Where an element of width b, subjected to nominally uniform compressive stress, is connected along both edges to other elements of width a that are also supported along their edges (see Figure C5 of the Commentary), the slenderness, λ, shall be given by λ = mb/t where for columns, when b/t ≥ a/t1 m = 1.25 + 0.4 (a/t1)/(b/t) ≤ 1.65 for components in bending, such as decking profiles, when a/t1 ≤ 2.5b/t m = 1.25 + 0.2 (a/t1)/(b/t) ≤ 1.65 where a = width of the supporting element t1 = thickness of the supporting element b = width of the supported element (see Clause 8.1) t = thickness of the supported element When a/t1 >2.5 b/t, the web shall be checked for buckling using Clause 8.2.1.

8.3 Elements supported at one longitudinal edge only 8.3.1 Edge simply supported For a linear variation of longitudinal stress across an element that has one edge simply supported and the other edge free, buckling is in the torsional mode and the slenderness, λ, shall be given by (a) when the maximum compressive stress is at the free edge (see Figure C6 of the Commentary),

λ=

2.5 (3 + f2 /f1 )1/2 w t

When f2/f1 < –3, buckling does not occur. Note: This Clause applies particularly to angle sections in bending about the weak axis.

(b) when the maximum compressive stress is at the supported edge (see Figure C6 of the Commentary), for f2/f1 > –0.28,

λ=

2.5 (1 + 3 f2 /f1 )1/2 w t

where f2 = stress at the other edge, positive when the stress is tensile f1 = maximum compressive stress, negative w = element width (see the definition of b in Clause 8.1) t = element thickness When f2/f1 < –0.28, Clause 8.2.1 shall be used.

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Δ

© Canadian Standards Association

8.3.2 Flanges of sections

The slenderness, λ, for flanges of channels and Z and I sections subjected to uniform compressive stress due to axial force or bending (see Figure C7 of the Commentary) shall be given by

λ=

[3 + 0.6 (at/wt1 )]w t

≤ 5 w/t

where a = breadth of the web element to which the flange is connected t = thickness of the flange w = width of the flange (see definition of b in Clause 8.1) t1 = thickness of the web element This slenderness, λ, with the yield strength, Fy , shall be used to determine the normalized slenderness, λλ , in Clause 9.3.1, to give the normalized buckling stress, F , from Clause 9.3.3. The local buckling stress is then Fc = F Fy , which shall be used as Fo when determining the overall buckling resistance of non-compact columns and beams.

8.4 Elements supported at one edge with a lip at the other edge 8.4.1 General shapes For the general case of a flange element attached at one longitudinal edge to a web element with a lip at the other longitudinal edge (see Figure C8 of the Commentary), subjected to uniform compressive stress, the slenderness, λ, used to obtain the buckling stress shall be given by

λ=

5(I p / J )1/ 2 ⎡1 + 5.3(Cw k )1/2 / J ⎤ ⎣ ⎦

1/ 2

where Ip = polar moment of inertia of flange and stiffener about the supported edge J = St. Venant torsion constant for flange and stiffener Cw = warping constant for rotation of the flange and stiffener about the supported edge = Is w 2 where Is = moment of inertia of the stiffener about the inside surface of the flange to which it is attached; this applies to all types of stiffener, including inclined lips and bulbs w = flange width measured from the intersection of the median lines of the flange and web k = the spring constant for the restraint provided by the connection between flange and web = 3 t13/16 (a + 0.5w) for channel and Z sections = 1.5 t13/16 (a + 0.5w) for I-sections where t1 = web thickness a = breadth of the web element to which the flange is connected

8.4.2 Uniform thickness with simple lips 8.4.2.1 For a shape of uniform thickness, with simple lips on the flanges (see Figure C8 of the Commentary), the slenderness, λ, shall be given by

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9.7.3.2 Members with biaxial moments, not subject to lateral-torsional buckling

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For members subjected to axial force combined with bending about both principal axes, where the same extreme fibre carries the maximum stress from both moments and the member fails in flexure, the limiting condition shall be given by

Myf Mxf Cf + + A Sx (1 − Cf / Cex ) Sy 1 − Cf / Cey

(

)

≤ φy Fo

where Cf = factored compressive force, not greater than the value of Cr given by Clause 9.4.1 Mxf = moment in the member due to the factored lateral load, about the strong axis Myf = moment in the member due to the factored lateral load, about the weak axis φy = resistance factor on the yield strength Fo = limiting stress (see Clause 9.3.2) A = gross area Sx = section modulus of the gross section about the strong axis Sy = section modulus of the gross section about the weak axis Cex = Aπ2 E/λx2 where E = elastic modulus λx = L/rx where L = unbraced length rx = radius of gyration of the gross cross-section about the strong axis Cey = Aπ 2 E/λy2 where λy = L/ry where ry = radius of gyration of the gross section about the weak axis

9.7.3.3 Members subject to lateral-torsional buckling For combined axial force and bending about the strong axis, when lateral buckling can occur, the combined factored axial load, Cf , and bending moment, Mf , shall satisfy

Cf Mf + ≤ 1 Cry Mr (1 − Cf / Cex ) where Cf = applied compressive force due to the factored loads Mf = moment due to the factored lateral load, or as calculated in Clause 9.5.3.3 Cry = factored resistance for failure about the weak axis, obtained from Clause 9.4.1 Mr = factored moment resistance obtained from Clause 9.5.3.1 or 9.5.3.2, as applicable Cex = elastic buckling force for bending about the strong axis = Aπ 2 E/λx2 where A = gross area

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E = elastic modulus λx = KL/rx where K = effective length factor (see Table 4 for typical values) L = unbraced length rx = radius of gyration of the gross cross-section about the strong axis

9.7.4 Eccentric compression 9.7.4.1 General case For general cases of eccentric compression, the following requirements shall apply: (a) For failure in the plane of bending, Clause 9.7.3.1(a) or (b) shall be used with a factored moment, Mf , given by Mf = 1.2eCf (b) For lateral-torsional buckling, Clause 9.7.3.3 shall be used with a moment, Mf , given by Mf = eCf where e = eccentricity Cf = factored axial force Δ

9.7.4.2 Single angle bracing members For single angle bracing members, (a) the factored compressive resistance, Cr , of discontinuous single angles connected through one leg shall be given by Clause 9.4.1 using

(

λ = λv2 + λt2

)

1/2

where λv = KL/rv where K = effective length factor (see Table 4 for typical values) L = unbraced length of member rv = minimum radius of gyration λt = 5 w/t where w = width of longer leg, see Clause 9.4.3.1.2 t = thickness of longer leg (b) the factored resistance, Cr , shall not exceed (i) φc 0.5AFy for single bolt connections; or (ii) φc 0.67AFy for double bolt or welded connections where A = gross cross-sectional area Fy = yield strength

34

April 2007 (Replaces p. 34, February 2005)

© Canadian Standards Association

Strength design in aluminum

m = number of shear planes A = nominal cross-sectional area of the fastener Fu = ultimate strength of the fastener material

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If the bolt threads are in a shear plane, the value of Vr shall be multiplied by 0.75.

11.2.3.2 Tensile resistance The factored tensile resistance of a bolt, Tr , shall be the lesser of the values given by (a) Tr = φf 0.75 A Fu ; and (b) Tr = φf A Fy where φf = fastener resistance factor A = cross-sectional area of the bolt based on the nominal diameter Fu = ultimate strength of the bolt material Fy = yield strength of the bolt material Rivets are not commonly used in tension.

11.2.3.3 Combined shear and tension For a bolt subject to both shear and tension, exclusive of tension due to tightening, the reduced factored tensile resistance, T’r , shall be given by the following:

T ′r = 1.25 Tr − kVf ≤ Tr where Tr = factored tensile resistance given in Clause 11.2.3.2 k = 1.8, or 1.4 when the bolt thread is excluded from the shear plane Vf = factored shear load on the bolt

11.2.4 Bolts and rivets in bearing 11.2.4.1 Bearing strength The factored bearing resistance, Br , of the connected material for each loaded fastener shall be the lesser of the values given by the following formulas: (a) Br = φu etFu; and (b) Br = φu 2dtFu where φu = ultimate resistance factor e = perpendicular distance from the hole centre to the end edge in the direction of the loading (not less than 1.5d ) t = plate thickness Fu = ultimate strength of the connected material d = fastener diameter

11.2.4.2 Lap joints For unrestrained lap joints in tension, the factored bearing resistance, Br , shall be the lesser of the values given by the following formulas: (a) Br = φu(t1 + t2) e Fu /4; and (b) Br = φu(t1 + t2) d Fu /2 ≤ φu 2dt1Fu where φu = ultimate resistance factor April 2007 (Replaces p. 45, February 2005)

45

S157-05

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t1, t2 = e = Fu = d = Δ

© Canadian Standards Association

thicknesses of the plates, t1 < t2 distance from the hole centre to the end edge, but not less than 1.5d ultimate tensile strength of the connected material fastener diameter

11.2.4.3 Oblique end edges Where the end edge is oblique to the line of action of the tension force (see Figure C23 of the Commentary), the factored bearing resistance, Br , for a single bolt shall be the lesser of the values given by the following formulas: (a) Br = φu [e + (e – do) cos2 θ] tFu ; and (b) Br = φu 2 dtFu where φu = ultimate resistance factor e = perpendicular distance from the hole centre to the end edge do = hole diameter d = fastener diameter θ = angle made by the end edge with the direction of the force t = plate thickness Fu = ultimate strength of the connected material

11.2.5 Tear-out of bolt and rivet groups (block shear) 11.2.5.1 Tension: Rectangular patterns For a group of two or more fasteners in a rectangular pattern (see Figure C24(a) of the Commentary) resisting a force directed towards the edge, the factored bearing resistance, Rb , of the group of fasteners shall be the lesser of the values given by the following formulas: (a) Rb = φu [(m – 1)(g – do ) + (n – 1)(s – do ) + e]tFu; and (b) Rb = φu 2 NdtFu where φu = ultimate resistance factor m = number of fasteners in the first transverse row g = fastener spacing measured perpendicular to the direction of the force do = hole diameter n = number of transverse rows of fasteners s = fastener spacing measured in the direction of the force e = edge distance in the direction of force for the first row, but not less than 1.5d = 2d, when e > 2d t = plate thickness Fu = ultimate strength of the connected material N = total number of fasteners d = fastener diameter

11.2.5.2 Tension: Trapezoidal patterns For a triangular or trapezoidal group of fasteners in a staggered pattern (see Figure C24(b) of the Commentary) resisting a force directed towards the edge, the factored bearing resistance, Rb , of the group shall be the lesser of the values given by the following formulas: (a) Rb = φu [2 (m – 1)(g – do + s2/4g) + e] tFu; and (b) Rb = φu 2NdtFu where φu = ultimate resistance factor m = number of fasteners in the first transverse row

46

April 2007 (Replaces p. 46, February 2005)

© Canadian Standards Association

Strength design in aluminum

Table 3 Mechanical properties for typical aluminum alloy weld beads

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(See Clauses 4.2.3, 9.2.1, 9.6.1.2, 11.3.2.2, 11.3.2.3, 11.3.3.1.1, 11.3.3.2.2, 11.3.3.2.3, 11.3.6, C11.3, and C11.3.2.1.) Ultimate tensile strength, Fwu , MPa Filler alloy 4043 5356

Base metal alloy 3003

3004

5052

5083

5086

5454

6061

6063

6351

100 100

150 150

— 170

— 260

— 235

— 220

170 190

120 120

170 190

April 2007 (Replaces p. 61, February 2005)

61

S157-05

© Canadian Standards Association

Table 4 Effective length factors, K

Δ

(See Clauses 9.4.2.1, 9.7.1.3, 9.7.3.3, 9.7.4.2, 9.7.5.2, 9.8.2, and C9.4.2.)

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Kv* (Single angles) Kx

Ky

1 bolt

2 bolt

AB

k

1

(1+2k) 3

(1+2k) 3

AB

1

1

0.8

0.7

AB

0.5

0.5

0.45

0.4

AB

0.33

0.43

0.33

0.33

AB

0.25

0.35

0.25

0.25

AB

0.5

1

0.5

0.45

AB

0.5

1

0.5

0.45

AB

0.45

0.5

0.4

0.35

Member Y

V

L

X

A

B

kL

A L

L

AC B

C

A L

C

T

T

C

B A C

T

T

C

B A C

T

T

C

B A L B A C

B T

A C

L T

B C = Compression, T = Tension, T = C

62

* = See Clause 9.7.4.2

April 2007 (Replaces p. 62, February 2005)

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Update No. 1 S157-05/S157.1-05 February 2007 Note: General Instructions for CSA Standards are now called Updates. Please contact CSA Information Products Sales or visit www.ShopCSA.ca for information about the CSA Standards Update Service. Title: Strength design in aluminum/Commentary on CSA S157-05, Strength design in aluminum — originally published February 2005 The following revisions have been formally approved: Revised

Outside front cover and title pages

New

National Standards of Canada text

Deleted

None

CSA S157-05/S157.1-05 originally consisted of 137 pages (xiii preliminary and 124 text), each dated February 2005. It now consists of the following pages:

• •

February 2005

iii–xiii and 3–124

February 2007

Cover, National Standards of Canada text, title pages (i, 1, and 2), and copyright page (ii)

Update your copy by inserting these revised pages. Keep the pages you remove for reference.

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CAN/CSA-S157-05

A National Standard of Canada (approved February 2007)

S157.1-05

Strength design in aluminum/ Commentary on CSA S157-05, Strength design in aluminum

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(approved February 2007)

S157.1-05 Strength design in aluminum/ Commentary on CSA S157-05, Strength design in aluminum

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ISBN 1-55397-817-X Technical Editor: Ted Koza

© Canadian Standards Association — 2004

All rights reserved. No part of this publication may be reporduced in any form whatsoever without the prior permission of the publisher.

(Copyright page replaced February 2007)

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(approved February 2007)

CAN/CSA-S157-05 Strength design in aluminum

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S157-05/S157.1-05 Strength design in aluminum/Commentary on CSA S157-05, Strength design in aluminum

Published in February 2005 by Canadian Standards Association A not-for-profit private sector organization 5060 Spectrum Way, Suite 100, Mississauga, Ontario, Canada L4W 5N6 1-800-463-6727 • 416-747-4044

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ISBN 1-55397-817-X Technical Editor: Ted Koza

© Canadian Standards Association — 2005

All rights reserved. No part of this publication may be reproduced in any form whatsoever without the prior permission of the publisher.

© Canadian Standards Association

Strength design in aluminum/ Commentary on CSA S157-05, Strength design in aluminum

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Technical Committee on Strength Design in Aluminum x Preface xii

S157-05, Strength design in aluminum 1 Scope 3 2 Reference publications 4 3 Definitions and symbols 5 3.1 Definitions 5 3.2 Symbols and subscripts 6 3.2.1 Symbols 6 3.2.2 Subscripts 8 4 Materials 9 4.1 Alloys 9 4.2 Mechanical properties 9 4.3 Physical properties 10 4.4 Fasteners and welds 10 5 Limit states design 11 5.1 General 11 5.1.1 Serviceability limit states 11 5.1.2 Ultimate limit states 11 5.1.3 Fatigue life 11 5.2 Safety criterion 11 5.3 Loads for buildings 11 5.3.1 Specified loads and influences 11 5.3.2 Erection loads 11 5.3.3 Thermal effects 11 5.4 Load factors for buildings 12 5.4.1 Load factors 12 5.4.2 Load combination factors 12 5.4.3 Importance factors 12 5.4.4 Effect of factored loads 12 5.4.5 Applications other than buildings 12 5.5 Resistance factors 12 5.6 Deflections and vibrations 13 5.6.1 Deflection 13 5.6.2 Vibration 13 5.7 Provisions to avoid progressive collapse 13 6 Methods of analysis and design 13 6.1 Analysis 13 6.2 Testing 13 7 Net area, effective section, and effective strength 13 7.1 General 13 7.2 Gross area 13

February 2005

iii

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S157-05/S157.1-05

7.3 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.5 7.5.1 7.5.2 7.5.3

© Canadian Standards Association

Net area 13 Effective section 14 General 14 Effective thickness at welds 14 Effective thickness after local buckling of flat elements 15 Deflection under service loads 15 Effective strength and overall buckling 15 General 15 Influence of welding 15 Influence of local buckling 15

8 Local buckling of flat elements 16 8.1 Buckling stress 16 8.2 Elements supported on both longitudinal edges 16 8.2.1 Edges simply supported 16 8.2.2 Influence of adjacent elements 17 8.3 Elements supported at one longitudinal edge only 17 8.3.1 Edge simply supported 17 8.3.2 Flanges of sections 18 8.4 Elements supported at one edge with a lip at the other edge 18 8.4.1 General shapes 18 8.4.2 Uniform thickness with simple lips 18 9 Resistance of members 19 9.1 Limiting slenderness for members 19 9.2 Members in tension 20 9.2.1 Tensile resistance 20 9.2.2 Oblique welds 20 9.3 Members in compression: Buckling 20 9.3.1 Normalized slenderness 20 9.3.2 Limiting stress 21 9.3.3 Buckling stress 21 9.4 Columns 22 9.4.1 General 22 9.4.2 Flexural buckling 22 9.4.3 Torsional buckling 23 9.5 Bending 24 9.5.1 Classification of members in bending 24 9.5.2 Moment resistance of members not subject to lateral-torsional buckling 24 9.5.3 Moment resistance of members subject to lateral-torsional buckling 25 9.6 Webs in shear 27 9.6.1 Flat shear panels 27 9.6.2 Stiffened webs 28 9.6.3 Web stiffeners 28 9.6.4 Combined shear and bending in webs 29 9.6.5 Web crushing 29 9.7 Members with combined axial force and bending moment 30 9.7.1 Axial tension and bending 30 9.7.2 Eccentric tension 31 9.7.3 Beam-columns 32 9.7.4 Eccentric compression 34 9.7.5 Shear force in beam-columns 35 9.8 Built-up columns 36 9.8.1 Spacing of connectors 36 9.8.2 Multiple-bar members with discrete shear connectors 36

iv

February 2005

© Canadian Standards Association

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9.8.3 9.8.4 9.9

Strength design in aluminum/ Commentary on CSA S157-05, Strength design in aluminum

Double angle struts 36 Lattice columns and beam columns 37 Members in torsion 37

10 Panels 38 10.1 Flat panels with multiple stiffeners 38 10.1.1 Axial compression 38 10.1.2 In-plane shear 39 10.2 Curved panels and tubes 39 10.2.1 Axial compression 39 10.2.2 Radial compression 40 10.2.3 Shear 40 10.3 Curved axially stiffened panels in axial compression 40 10.4 Flat sandwich panels 41 10.4.1 General 41 10.4.2 Panel bending 41 10.4.3 Panel buckling 41 10.4.4 Skin buckling 42 10.4.5 Core strength 42 11 Resistance of connections 43 11.1 General 43 11.1.1 Connection types 43 11.1.2 Sharing of loads 43 11.1.3 Materials 43 11.2 Mechanical fasteners 43 11.2.1 General 43 11.2.2 Fastener spacings 44 11.2.3 Bolts and rivets in shear and/or tension 44 11.2.4 Bolts and rivets in bearing 45 11.2.5 Tear-out of bolt and rivet groups (block shear) 46 11.2.6 Eccentrically loaded fastener groups 48 11.3 Welded connections 48 11.3.1 General 48 11.3.2 Butt welds 49 11.3.3 Fillet welds 49 11.3.4 Flare groove welds 52 11.3.5 Slot and plug welds 52 11.3.6 Influence of welds, heat-affected zone 52 11.3.7 Stud welds 52 12 Fatigue resistance 53 12.1 Load cycles of constant amplitude 53 12.2 Known load spectra 53 12.3 Unknown load spectra 53 12.4 Low cycle stress limit 53 12.5 Severe stress raisers 53 12.6 Thick material 53 12.7 Load factor 53 12.8 Resistance factors 53 13 Tests 54 13.1 General 54 13.2 Test methods 54 13.3 Test procedures 55

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13.3.1 13.3.2

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Confirmatory tests 55 Performance tests 55

14 Fabrication 56 14.1 General 56 14.2 Tolerances 56 14.3 Layout and marking 56 14.3.1 Layout 56 14.3.2 Marking 56 14.4 Forming 56 14.5 Cutting 57 14.6 Mechanical connections 57 14.6.1 Rivets 57 14.6.2 Bolts 57 14.7 Welding 58 15 Protection against corrosion 58 15.1 General 58 15.2 Contact with dissimilar materials 58 15.3 Cleaning and treatment of metal surfaces 59 15.4 Stress corrosion 59 Annexes A (informative) — Applications other than buildings 67 B (informative) — Common uses of aluminum alloys 72 Tables 1 — Typical aluminum alloys and products to which this Standard is applicable 59 2 — Tensile properties for typical aluminum alloys and products used in buildings 60 3 — Mechanical properties for typical aluminum alloy weld beads 61 4 — Effective length factors, K 62 Figures 1 — Categories of members and joints for fatigue design 63 2 — Fatigue stress ranges for design of aluminum components and connections 66

S157.1-05, Commentary on CSA S157-05, Strength design in aluminum Preface 75 Introduction 77 C1. Scope 77 C4. Materials 77 C4.2 Mechanical properties 77 C4.3 Physical properties 78 C4.4 Fasteners and welds 78 C5. Limit states design 78 C5.1 General 78 C5.2 Safety criterion 78 C5.3 Loads for buildings 79

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Load factors for buildings 79 Resistance factors 79

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C6. Methods of analysis and design 79 C7. Net area, effective section, and effective strength 79 C7.3 Net area 79 C7.4 Effective section 80 C7.4.2 Effective thickness at welds 80 C7.4.3 Effective thickness after local buckling of flat elements 80 C7.4.4 Deflection under service loads 82 C7.5 Effective strength and overall buckling 82 C7.5.1 General 82 C7.5.2 Influence of welding 82 C7.5.3 Influence of local buckling 82 C8. Local buckling of flat elements 83 C8.1 Buckling stress 83 C8.2 Elements supported on both longitudinal edges 83 C8.2.1 Edges simply supported 83 C8.2.2 Influence of adjacent elements 84 C8.3 Elements supported at one longitudinal edge only 85 C8.3.1 Edge simply supported 85 C8.3.2 Flanges of sections 86 C8.4 Elements supported at one edge with a lip at the other edge 86 C9. Resistance of members 87 C9.1 Limiting slenderness for members 87 C9.2 Members in tension 87 C9.3 Members in compression: Buckling 87 C9.3.2 Limiting stress 92 C9.3.3 Buckling stress 92 C9.4 Columns 93 C9.4.2 Flexural buckling 93 C9.4.3 Torsional buckling 93 C9.5 Bending 94 C9.5.1 Classification of members in bending 94 C9.5.2 Moment resistance of members not subject to lateral-torsional buckling 94 C9.5.3 Moment resistance of members subject to lateral-torsional buckling 95 C9.6 Webs in shear 99 C9.6.1 Flat shear panels 99 C9.6.2 Stiffened webs 100 C9.6.3 Web stiffeners 101 C9.6.4 Combined shear and bending in webs 101 C9.6.5 Web crushing 102 C9.7 Members with combined axial force and bending moment 102 C9.7.1 Axial tension and bending 102 C9.7.2 Eccentric tension 102 C9.7.3 Beam-columns 103 C9.7.4 Eccentric compression 105 C9.7.5 Shear force in beam-columns 105 C9.8 Built-up columns 105 C9.8.1 Spacing of connectors 105 C9.8.2 Multiple-bar members with discrete shear connectors 105

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C9.8.3 C9.8.4 C9.9

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Double angle struts 106 Lattice columns and beam columns 106 Members in torsion 106

C10. Panels 106 C10.1 Flat panels with multiple stiffeners 106 C10.1.1 Axial compression 106 C10.1.2 In-plane shear 107 C10.2 Curved panels and tubes 107 C10.2.1 Axial compression 107 C10.2.2 Radial compression 107 C10.2.3 Shear 108 C10.3 Curved axially stiffened panels in axial compression 108 C10.4 Flat sandwich panels 108 C10.4.1 General 108 C10.4.2 Panel bending 108 C10.4.3 Panel buckling 108 C10.4.4 Skin buckling 109 C10.4.5 Core strength 109 C11. Resistance of connections 109 C11.1 General 109 C11.1.1 Connection types 109 C11.2 Mechanical fasteners 109 C11.2.1 General 109 C11.2.2 Fastener spacings 110 C11.2.3 Bolts and rivets in shear and/or tension 110 C11.2.4 Bolts and rivets in bearing 110 C11.2.5 Tear-out of bolt and rivet groups (block shear) 111 C11.2.6 Eccentrically loaded fastener groups 112 C11.3 Welded connections 114 C11.3.2 Butt welds 114 C11.3.3 Fillet welds 115 C11.3.4 Flare groove welds 118 C11.3.5 Slot and plug welds 118 C11.3.7 Stud welds 118 C12. Fatigue resistance 118 C12.1 Load cycles of constant amplitude 118 C12.2 Known load spectra 119 C12.3 Unknown load spectra 119 C13. Tests 119 C13.1 General 119 C13.2 Test methods 120 C13.3 Test procedures 120 C13.3.1 Confirmatory tests 120 C13.3.2 Performance tests 120 Tables C1 — C2 — C3 — C4 —

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Influence of adjacent elements: Comparison of theoretical and code values for m 85 Local buckling of flanges: Comparison of theoretical and code values for m 86 Various best-fit values for the coefficients λ o and α 89 Strength of fillet welds: Comparison of some code values for the factor k 116

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Strength design in aluminum/ Commentary on CSA S157-05, Strength design in aluminum

Figures C1 — Net section at staggered holes 80 C2 — Effective width formulas: Comparison with test and code values for steel 81 C3 — Effective thickness formula: Comparison of test and code values for aluminum 82 C4 — Edges simply supported: Comparison of theoretical and code values for the factor m 84 C5 — Elements supported and partially restrained at both long edges 84 C6 — Values of m for elements simply supported on one longitudinal edge only 85 C7 — Flange elements partially restrained at the supported edge 86 C8 — Lipped flanges 87 C9 — Column buckling formula, fully heat-treated alloys: Comparison of test and code values 90 C10 — Column buckling formula, work-hardened alloys: Comparison of test and code values 90 C11 — Normalized buckling stress for columns and beams 91 C12 — Normalized buckling stress and postbuckling strength for plates 91 C13 — Types of angle section 93 C14 — Lateral-torsional buckling for 2014-T6 H-beams: Comparison of test and code 97 C15 — Lateral-torsional buckling of steel thin-wall I-beams: Comparison of test and code values 98 C16 — Stiffened shear webs 99 C17 — Stress orientation in thin shear webs remote from stiffeners 100 C18 — Web crushing in formed sheet sections 102 C19 — Eccentric tension members 103 C20 — Beam-columns: Comparison of test and code values 104 C21 — Built-up members with discrete interconnectors 106 C22 — Curved panel 107 C23 — Tear-out at oblique ends 111 C24 — Tear-out of fastener groups (block shear) 112 C25 — Eccentrically loaded fastener groups 113 C26 — Fillet welds 116 C27 — Eccentrically loaded fillet welds 117 Bibliography 122

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Technical Committee on Strength Design in Aluminum C. Marsh

Victoria, British Columbia

D. Beaulieu

Laval University, Québec, Québec

P. Bulson

Lymington, Hants, England

B. Chiang

City of Toronto, Toronto, Ontario

J.C. Clark

Burlington, Ontario

D. Cuoco

Thornton-Tomasetti Group, New York, New York, USA

A. de la Chevrotière

Technomarine International Management Inc., Repentigny, Québec

G.V. Francis

Vision Engineering and Design Inc., Oakville, Ontario

K. Gong

ALCAN International Limited, Kingston, Ontario

J.D. Hull

R.M.C.A., Mississauga, Ontario

D. Joseph

Atomic Energy Control Board, Ottawa, Ontario

J.R. Kissell

The TGB Partnership, Hillsborough, North Carolina, USA

E. Lerner

Sota Glazing Inc., Brampton, Ontario

D. Malcolm

Lavalin Engineers Inc., Willowdale, Ontario

B. Mandelzys

VICWEST Steel, Oakville, Ontario

S.S. McCavour

McCavour Engineering Limited, Mississauga, Ontario

R.M. Schuster

University of Waterloo, Waterloo, Ontario

S.P. Sunday

Alcoa, Richmond, Virginia, USA

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Chair

Associate

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Strength design in aluminum/ Commentary on CSA S157-05, Strength design in aluminum

C.R. Taraschuk NRC Canada, Ottawa, Ontario Associate

T. Koza CSA, Mississauga, Ontario Project Manager

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Preface This is the fourth edition of CSA S157, Strength design in aluminum. It supersedes the previous editions published in 1983, 1969, and 1962. Continuing research in North America and Europe, and the manner in which this Standard is used, required that important changes be made in the arrangement of the text and the design procedures adopted. The 1983 edition was divided into two parts: the first included formulas for the characteristic resistances of components; the second provided requirements specific to the use of aluminum in buildings. The 2005 edition is primarily for applications in buildings but is written in a form that may be used for all types of aluminum load-bearing components or assemblies. Expressions for the strength of structural components are based on the concept of limit states design. Requirements to satisfy the ultimate limit state form the core of the Standard. As a document referenced by the National Building Code of Canada, the load factors and resistance factors specified by that code are given. However, because the design expressions predict the characteristic resistance of components and connections, this Standard will be useful in any field of engineering in which known applied loads are to be supported. Serviceability limit states depend on the desired behaviour under service loads for each particular application and are not specified in this Standard. For components used in buildings, reference is made to Commentary A of the National Building Code of Canada. Procedures to determine the resistance to the various modes of buckling have been unified, making use of formulas relating the normalized slenderness to the normalized buckling stress. From research during the past decade, it has become evident that the straight-line buckling formula did not provide a reasonably uniform margin of safety over the full range of slenderness, and it has been replaced by a modified Perry formula, which is a more widely accepted model. The influence of welds and local buckling on the capacity of columns and beams is now fully accounted for. Means have been introduced to make direct use of the basic buckling curve for each alloy type in the design of flat elements with postbuckling strength, a subject which is now much more thoroughly treated. The design of welded joints has been expanded. A commentary, CSA S157.1, Commentary on CSA Standard S157-05, Strength design in aluminum (hereinafter referred to as the Commentary), is provided which, while not a mandatory part of the Standard, is essential to the implementation of the code requirements. In this Standard, cross-references to the Commentary are identified by the prefix “C” before the clause, table, or figure number. Annex A has been added to provide a review of the relevant authorities for the many applications of aluminum other than in buildings. This Standard was prepared by the Technical Committee on Strength Design in Aluminum, under the jurisdiction of the Strategic Steering Committee on Structures (Design), and has been formally approved by the Technical Committee. It will be submitted to the Standards Council of Canada for approval as a National Standard of Canada. February 2005 Notes: (1) Use of the singular does not exclude the plural (and vice versa) when the sense allows. (2) Although the intended primary application of this Standard is stated in its Scope, it is important to note that it remains the responsibility of the users of the Standard to judge its suitability for their particular purpose. (3) This publication was developed by consensus, which is defined by CSA Policy governing standardization — Code of good practice for standardization as “substantial agreement. Consensus implies much more than a simple majority, but not necessarily unanimity”. It is consistent with this definition that a member may be included in the Technical Committee list and yet not be in full agreement with all clauses of this publication. (4) CSA Standards are subject to periodic review, and suggestions for their improvement will be referred to the appropriate committee. (5) All enquiries regarding this Standard, including requests for interpretation, should be addressed to Canadian Standards Association, 5060 Spectrum Way, Suite 100, Mississauga, Ontario, Canada L4W 5N6.

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Strength design in aluminum/ Commentary on CSA S157-05, Strength design in aluminum

Requests for interpretation should (a) define the problem, making reference to the specific clause, and, where appropriate, include an illustrative sketch; (b) provide an explanation of circumstances surrounding the actual field condition; and (c) be phrased where possible to permit a specific “yes” or “no” answer. Committee interpretations are processed in accordance with the CSA Directives and guidelines governing standardization and are published in CSA’s periodical Info Update, which is available on the CSA Web site at www.csa.ca.

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S157-05 Strength design in aluminum

Published in February 2005 by Canadian Standards Association A not-for-profit private sector organization 5060 Spectrum Way, Suite 100, Mississauga, Ontario, Canada L4W 5N6 1-800-463-6727 • 416-747-4044

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Strength design in aluminum

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S157-05 Strength design in aluminum 1 Scope 1.1 This Standard applies to the design of aluminum alloy members and assemblies intended to carry a known load.

1.2 This Standard specifies requirements for the design of members to meet the requirements of the National Building Code of Canada using limit states design procedures.

1.3 This Standard contains rules to determine the ultimate resistance of aluminum members and connections, and may be used for the design of aluminum assemblies in general.

1.4 Where members designed in accordance with this Standard are intended for use in structures for which other standards apply, this Standard supplements such standards, as applicable. Note: Annex A lists some applications to which other standards apply.

1.5 Where this Standard does not provide design expressions or dimensional limitations that are applicable to a specific situation, a rational design may be used, based on appropriate theories, tests, analyses, or engineering experience. Note: See Annex B for common uses of alloys.

1.6 In CSA Standards, “shall” is used to express a requirement, i.e., a provision that the user is obliged to satisfy in order to comply with the standard; “should” is used to express a recommendation or that which is advised but not required; “may” is used to express an option or that which is permissible within the limits of the standard; and “can” is used to express possibility or capability. Notes accompanying clauses do not include requirements or alternative requirements; the purpose of a note accompanying a clause is to separate from the text explanatory or informative material. Notes to tables and figures are considered part of the table or figure and may be written as requirements. Annexes are designated normative (mandatory) or informative (non-mandatory) to define their application.

1.7 The expressions contained herein are dimensionally uniform, and any consistent system of units may be employed. Where dimensions are identified, SI units are used. Evaluation is normally conducted in newtons (N) for force, millimetres (mm) for length, and megapascals (MPa = N/mm2) for stress.

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2 Reference publications

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This Standard refers to the following publications, and where such reference is made, it shall be to the edition listed below. CSA (Canadian Standards Association) CAN/CSA-G164-M92 (R2003) Hot Dip Galvanizing of Irregularly Shaped Articles CAN/CSA-S16-01 Limit States Design of Steel Structures S408-1981 (R2001) Guidelines for the Development of Limit States Design W47.2-M1987 (R2003) Certification of Companies for Fusion Welding of Aluminum W59.2-M1991 (R2003) Welded Aluminum Construction AISI (American Iron and Steel Institute) Steel Products Manual, No. 13 Stainless and Heat Resisting Steels Aluminum Association Publication Aluminum standards and data, Metric SI, 2003 ASME International (American Society of Mechanical Engineers) International Boiler and Pressure Vessel Code — 2004 Edition ASTM International (American Society for Testing and Materials) A 307-04 Standard Specification for Carbon Steel Bolts and Studs, 60 000 PSI Tensile Strength A 325M-04b Standard Specification for Structural Bolts, Steel, Heat Treated 830 MPa Minimum Tensile Strength [Metric] B 695-04 Standard Specification for Coatings of Zinc Mechanically Deposited on Iron and Steel B 696-00 (2004) Standard Specification for Coatings of Cadmium Mechanically Deposited B 766-86 (2003) Standard Specification for Electrodeposited Coatings of Cadmium F 468-03a Standard Specification for Nonferrous Bolts, Hex Cap Screws, and Studs for General Use ISO (International Organization for Standardization) ISO 6361-2:1990 Wrought aluminium and aluminium alloy sheets, strips and plates — Part 2: Mechanical properties ISO 6362-2:1990 Wrought aluminium and aluminium alloy extruded rods/bars, tubes and profiles — Part 2: Mechanical properties

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Strength design in aluminum

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NRCC (National Research Council Canada) National Building Code of Canada, User’s Guide — NBC 1995, Structural Commentaries (Part 4): Commentary A, “Serviceability Criteria for Deflections and Vibrations”; Commentary C, “Structural Integrity” SAE International (Society of Automotive Engineers) SAE Handbook, Volume 1, Materials, 1991 United States Department of Defense MIL-HDBK-5G 1994 Chapter 3, “Aluminum,” Military Standards Handbook, Metallic Materials and Elements for Aerospace Vehicle Structures

3 Definitions and symbols 3.1 Definitions The following definitions apply in this Standard: Actual buckling stress, Fc — the compressive stress that causes buckling. Characteristic resistance, Rk — the maximum force, moment, or torque that a component can be assumed to be capable of sustaining. Effective section — a section in which elements, because of welding or local buckling, are reduced to their effective thicknesses. Effective strength, Fm — the reduced strength of an element, at the ultimate limit state, to account for the influence of local buckling or welding. Effective thickness, t’ — that portion of the thickness of an element, affected by welding or local buckling, deemed to be capable of carrying the yield strength. Elastic buckling stress, Fe — the theoretical stress that would initiate elastic buckling. Element — any flat or curved portion of a section, such as a flange or web, that can be treated as a plate. Factored compressive, tensile resistance, Cr , Tr — the product of the characteristic resistance and the resistance factor. Factored load — the product of the specified load and the load factor. Heat-affected zone (HAZ) — the zone of reduced strength in the metal adjacent to a weld (see Clause 11.3.6). Heat-treated alloys — those alloys for which improved mechanical properties are obtained by their response to heat treatment. Importance factor, γ — an additional factor applied to the loads to provide an increased margin of safety against collapse, and which is related to the use and occupancy of the building. Limit state — a condition of a structure in which the design function is no longer fulfilled. An ultimate limit state is represented by fracture, collapse, overturning, sliding, or uncontrolled deformation. A serviceability limit state is represented by unacceptable deformation or vibrations. Limiting stress, Fo — the compressive stress that limits the capacity of a column or beam (yield, local buckling, or postbuckling strength).

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Load combination factor, ψ — a factor applied to combined loads, other than dead load, to take into account the reduced probability of loads from different sources acting simultaneously.

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Load factor, α — a factor by which a specified load is multiplied, as appropriate to a particular limit state, to take into account the variability of loads and load patterns. Normalized buckling stress, F — the value of Fc /Fo. Normalized slenderness, λ — the value of (Fo /Fe)1/2. Resistance factor, φ — a factor applied to the characteristic resistance to account for variations in material properties, product dimensions, fabrication tolerances, and assembly procedures, and to account for the imprecision of the predictor itself. Slenderness, λ — a factored geometric ratio used to determine the stress level required to cause buckling. Specified load (service load) — a load defined in the appropriate standard or as determined by the use of the structure. Tests — Confirmatory test — a test to confirm that an item has at least the required stiffness and/or resistance. Performance test — a test to determine the maximum load that can be carried prior to the attainment of an agreed-upon level of distress. Ultimate resistance test — a test to determine the maximum load carried by an item prior to the attainment of an agreed-upon level of distress. Weld throat — the shortest distance through a fillet, groove, flare groove, or partial penetration butt weld. Work-hardened alloys — those alloys for which the mechanical properties are enhanced by work hardening.

3.2 Symbols and subscripts 3.2.1 Symbols The following symbols apply in this Standard, except as further defined in specific clauses: A = area of cross-section, mm2 a = breadth of element, mm = distance, mm = thickness of a bar, mm = panel dimension, mm = weld throat, mm B = bearing resistance, N b = width, mm C = compressive force, N = geometric property c = distance from the neutral axis of the gross cross-section to the extreme fibre, mm = distance from the centroid to the centre of rotation of a weld or bolt group, mm = lip width, mm D = dead load, N/m2

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d

do E e

F F Fm f G g

H h I J Io Ip K k L

Lm M m

N

n

P

Strength design in aluminum

= fastener diameter, mm = distance from the centre of rotation, mm = distance to extreme fastener, mm = mast width, mm = member depth, mm = sandwich panel thickness, mm = hole diameter, mm = elastic modulus, MPa = load due to earthquake, N = eccentricity, mm = end edge distance, mm = distance from the centre of bearing to the end of a beam, mm = normalized buckling stress, (Fc /Fo) = strength, MPa = effective strength, F 1/2 Fy , MPa = calculated applied stress, MPa = shear modulus, MPa = acceleration due to gravity, m/s2 = transverse fastener spacing, mm = row spacing measured perpendicular to the direction of the load, mm = total length of fillet weld, mm = torsional resistance, N•mm = web depth, mm = moment of inertia (second moment of area), mm4 = St. Venant torsion constant, mm4 = polar moment of inertia about the centroid, mm4 = polar moment of inertia about the centre of rotation, mm4 = effective length factor for columns = a factor = distance, mm = length, mm = live load, N/m2 = span, mm = effective length, mm = moment, N•mm = factor for the slenderness of plates = number of shear planes = number of lines of fasteners = force, N = number of chords = total number of cycles = total number of fasteners = bearing length for beams, mm = number of cycles = number of rows of fasteners = force per unit length, N/mm = force, N

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R

r r’ S s

T t tm V v W w X y Z α β γ η θ κ λ

λ ρ ψ φ υ

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= force on a fastener, N = radius, mm = resistance, N = radius of gyration, mm = radius of gyration of a built-up or effective section, mm = elastic section modulus (I/c), mm3 = longitudinal fastener spacing, mm = weld size, mm = row spacing measured in the direction of the load, mm = tensile force, N = influence resulting from temperature changes or differential settlement, N = thickness, mm = effective thickness, mm = shear force, N = shear flow (force per unit length), N/mm = load due to wind, N/m2 = flange width, mm = width of outstanding leg, mm = load, N = distance from the neutral axis of the gross cross-section to the centre of a weld, mm = plastic section modulus (first moment of area), mm3 = load factor = coefficient of thermal expansion = coefficient in buckling formula = ratio of the lip to flange width = importance factor = ratio of developed to net width = angle, degrees = rigidity per unit length of a weld = slenderness = normalized slenderness, (Fo /Fe)1/2 = mass density, kg/m3 = load combination factor = resistance factor = Poisson’s ratio

3.2.2 Subscripts b = bending = bearing c = compression = chord D = dead e = elastic = effective f = factored = fastener = flange

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Strength design in aluminum

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g k L m

= gross = characteristic resistance = live = effective = mean n = net o = limiting value = hole p = plastic = polar r = factored resistance s = shear = stiffener T = temperature t = tension = test = torsion u = ultimate v = v-axis W = wind w = web = welded = warping x = x-axis y = y-axis = yield

4 Materials 4.1 Alloys Aluminum alloys shall conform to the Aluminum Association publication Aluminum Standards and Data, ISO 6361-2 or ISO 6362-2, or US Department of Defense publication, MIL-HDBK-5G 1994, Chapter 3. Typical alloys and products to which this Standard applies are listed in Table 1. High-strength aircraft-type alloys are not included. Cast aluminum products are not included, as they require special study.

4.2 Mechanical properties 4.2.1 The ultimate strength, Fu , and yield strength, Fy , in tension, used for design purposes, shall be the minimum values specified for the alloy in the Aluminum Association publication Aluminum Standards and Data.

4.2.2 The characteristic mechanical properties for the base metal and for the heat-affected zone extending 25 mm in each direction from the centre of a weld (see Clause 11.3.6) to be used for design purposes for the alloys and products listed in Table 1 shall be as given in Table 2. Other mechanical properties shall be derived from these values according to the following ratios: (a) compression, yield strength — Fy; (b) bearing, ultimate strength — 2.0 Fu (see Clause 11.2.4);

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(c) shear, ultimate strength — 0.6 Fu ; and (d) shear, yield strength — 0.6 Fy.

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Note: If the properties of the heat-affected zone due to welding are not known, they may be taken to be equal to those of the solution heat-treated condition for heat-treated alloys and to the annealed condition for work hardened alloys.

4.2.3 Mechanical properties for weld beads are given in Table 3. The strength of weld beads for combinations of filler and base metal alloys not included in Table 3 shall be subject to confirmation.

4.3 Physical properties Physical properties for the alloys listed in Table 1 shall be taken as follows: (a) coefficient of linear thermal expansion, α — 24 × 10–6/°C; (b) elastic modulus, E — 70 000 MPa; (c) Poisson’s ratio, υ — 0.33; (d) shear modulus, G — 26 000 MPa; and (e) mass density, ρ — 2700 kg/m3.

4.4 Fasteners and welds 4.4.1 Aluminum components shall be joined using aluminum, stainless steel, cadmium-plated steel, galvanized steel fasteners, or appropriate adhesives.

4.4.2 Aluminum bolts shall be of an alloy conforming to the Aluminum Association publication Aluminum Standards and Data and shall be dimensioned according to ASTM F 468.

4.4.3 Aluminum welding wire shall be of an alloy conforming to the Aluminum Association publication Aluminum Standards and Data and shall be selected according to CSA W59.2.

4.4.4 Aluminum rivets shall be of an alloy conforming to the Aluminum Association publication Aluminum Standards and Data.

4.4.5 Stainless steel bolts shall be in the 300 series specified in AISI Steel Products Manual, No. 13.

4.4.6 Carbon steel bolts shall conform to ASTM A 307 or A 325M or SAE Grade 5, and shall be protected by galvanizing or cadmium plating. Galvanizing shall conform to the requirements of CAN/CSA-G164 or ASTM B 695. Cadmium plating shall conform to ASTM B 696 or B 766.

4.4.7 Special fasteners, such as blind rivets, patented bolts, and adhesives, shall be subject to confirmation of their load-carrying capacity and of their chemical compatibility with aluminum in the operating environment.

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5 Limit states design

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5.1 General 5.1.1 Serviceability limit states The specified loads shall be used to compute deflections, requirements for camber, ranges of natural frequency, and other serviceability limit states.

5.1.2 Ultimate limit states When considering the ultimate limit states, the load effects such as moments, shear forces, and stresses shall be those due to the factored loads.

5.1.3 Fatigue life When considering fatigue life, the forces shall be those due to the specified loads.

5.2 Safety criterion A building and its structure shall be designed to have strength and stability such that Factored resistance ≥ Effect of the factored applied loads Factored resistance shall be determined in accordance with Clauses 7 to 13 of this Standard, and the effect of factored applied loads shall be determined in accordance with Clause 5.4.4. In cases of overturning and uplift, no positive anchorage shall be required if the stabilizing effect of dead load, multiplied by a load factor of 0.85, is greater than the effect of the loads tending to cause overturning and uplift, multiplied by the appropriate load factors.

5.3 Loads for buildings 5.3.1 Specified loads and influences The following loads and influences, as specified in the National Building Code of Canada, Part 4, shall be considered in the design of a building: (a) D — dead loads, including the mass of the member and all permanent materials of construction, partitions, and stationary equipment; (b) E — earthquake loads; (c) L — live loads, including loads due to the intended use and occupancy of the building, movable equipment, snow, rain, soil or hydrostatic pressure, impact, and any other live loads stipulated by applicable building legislation; (d) T — influences resulting from temperature changes or differential settlement; and (e) W — wind loads.

5.3.2 Erection loads Provisions shall be made for loads imposed on the structure during its erection.

5.3.3 Thermal effects Provision shall be made for thermal movements or, where such movements are restrained, for the forces created, commensurate with the service conditions of the structure. Note: The temperature range for external structures is obtained from the National Building Code of Canada, Appendix C.

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5.4 Load factors for buildings

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5.4.1 Load factors

Load factors, α, for buildings shall be taken as follows: (a) dead loads, D: (i) in general — αD = 1.25; and (ii) in cases of overturning, uplift, reversal of load effect — αD = 0.85; (b) live loads, L — αL = 1.5; (c) wind loads, W — αW = 1.5; (d) thermal effects, T — αT = 1.25; and (e) earthquake loads, E — αE = 1.0.

5.4.2 Load combination factors The most unfavourable combined effect shall be determined when using the following load combination factors, ψ: (a) when only one of L, W, or T acts — ψ = 1.0; (b) when two of L , W, and T act — ψ = 0.7; and (c) when all of L , W, and T act — ψ = 0.6.

5.4.3 Importance factors

The importance factor, γ, shall be not less than 1.0, except that for buildings where it can be shown that collapse is not likely to cause injury or other serious consequences, it shall be not less than 0.8.

5.4.4 Effect of factored loads

The effect of factored loads is the structural effect due to the specified loads multiplied by load factors, α, defined in Clause 5.4.1, a load combination factor, ψ, defined in Clause 5.4.2, and an importance factor, γ, defined in Clause 5.4.3. The factored load combinations shall be taken as follows: (a) without an earthquake,

αDD + γψ (αLL + αWW + αTT) (b) for an earthquake, D + γ (E )

and (i) D + γ (L + E ) for storage and assembly occupancies; or (ii) D + γ (0.5L +E ) for all other occupancies.

5.4.5 Applications other than buildings For many applications, design rules are specified by a regulatory authority. In other cases, if limit states design is used, the load factors shall be established in such a way that, in conjunction with the resistance factors given in Clause 5.5, the required level of reliability is provided. Note: See Annex A for information on other applications and CSA S408 for guidance on limit states design.

5.5 Resistance factors

For general structures, the following resistance factors, φ, shall be used: (a) tension, compression, and shear in beams: on yield, φy = 0.9; (b) compression in columns: on collapse due to buckling, φc = 0.9; (c) tension and shear in beams: on ultimate, φu = 0.75; (d) tension on a net section, bearing stress, tear-out: on ultimate, φu = 0.75; (e) tension and compression on butt welds: on ultimate, φu = 0.75; (f) shear stress on fillet welds: on ultimate, φf = 0.67; and (g) tension and shear stress on fasteners: on ultimate, φf = 0.67.

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5.6 Deflections and vibrations Note: Commentary A of the National Building Code of Canada, User’s Guide, provides guidance on acceptable deflections and vibrations.

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5.6.1 Deflection Deflection due to the action of the specified loads shall not exceed the serviceability limits of the materials supported or the requirements set by the intended use.

5.6.2 Vibration The vibration of the structure and its components shall be acceptable for the intended use. Structural members subjected to wind forces shall be proportioned to ensure that sustained vibrations caused by vortex shedding, galloping, or wind gusts do not arise.

5.7 Provisions to avoid progressive collapse To provide against progressive collapse caused by accidental loads, such as vehicle impact or explosion, the structure shall incorporate redundancy, alternative load paths, protection, or other means to limit the extent of the failure. Note: Commentary C of the National Building Code of Canada, User’s Guide, provides guidance on structural integrity.

6 Methods of analysis and design 6.1 Analysis The forces in components shall be determined by rational analysis. In general, elastic analysis shall be employed. If plastic or other forms of ultimate strength analysis are used, it shall be demonstrated that the structure satisfies the assumptions, such as the range of non-linear behaviour, made in the analysis.

6.2 Testing The adequacy of a structure or structural assembly may be determined by direct load tests in accordance with Clause 13.

7 Net area, effective section, and effective strength 7.1 General Both the drilling of holes and welding reduce the strength of a member. To accommodate this change, deductions shall be made from the gross cross-section to give a net or effective section. Strength reduction due to local buckling shall be represented by a reduction in the effective thickness or in the effective strength. (See Clauses 7.2 to 7.5.)

7.2 Gross area The gross area of a member shall be determined by summing the products of the thickness and the gross width of each element as measured perpendicular to the axis of the member.

7.3 Net area The net cross-sectional area of members in tension shall be the gross area less the sum of the hole diameters multiplied by the thickness (Σdot) in line across the section. For a chain of holes extending in any diagonal or zigzag line across a tension member (see Figure C1 in the Commentary), the net width of the part shall be obtained by deducting from the gross width the sum of the diameters of all the holes in the chain and adding, for each gauge space in the chain, a quantity given by

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s2/4g where s = spacing of successive holes in the direction of the force g = transverse spacing of the two holes do = hole diameter t = thickness The least net width so obtained shall be used in calculating the net area. The staggered rupture line controls when s2 is less than 2gdo. The distance between holes shall not be less than 2.5d, where d is the fastener diameter. Clause 9.7.2 gives effective net areas for eccentrically loaded tension members.

7.4 Effective section 7.4.1 General Both welding and local buckling cause a reduction in the overall strength, which is accounted for by using an effective section to establish the bending resistance of a member. The geometric properties of the effective section shall be computed using the effective thicknesses of the elements according to Clauses 7.4.2 and 7.4.3. The section modulus of the effective section shall be multiplied by the yield strength of the base metal to give the limiting moment (characteristic resistance).

7.4.2 Effective thickness at welds 7.4.2.1 Plastic section modulus Where only parts of the cross-section are influenced by welding, as with longitudinal welds, the effective thickness, tm , of the metal in the heat-affected zone, shall be determined as follows: tm = t (Fwy /Fy) ≤ t The section modulus of the effective section shall be multiplied by the yield strength of the base metal to give the limiting moment (characteristic resistance).

7.4.2.2 Elastic section modulus The effective thickness, tm , used in calculating the elastic section modulus to determine the moment at first yield in the base metal shall be given by tm = t(Fwy /Fy)(c/y) ≤ t where t = original thickness Fwy = yield strength in the heat-affected zone Fy = yield strength of the base metal c = distance from the neutral axis of the gross cross-section to the extreme fibre y = distance from the neutral axis of the gross section to the centre of the weld If local buckling occurs in a welded element, the influence of the weld may be neglected and the reduced thickness attributed to local buckling shall be used.

7.4.2.3 Deflections The gross cross-section of welded members shall be used for the calculation of deflections.

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7.4.3 Effective thickness after local buckling of flat elements

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7.4.3.1 Flat elements with both long edges supported For a flat element supported at both longitudinal edges, subjected in some part to compressive stress which causes local buckling, the effective thickness, tm , of the entire element at the ultimate limit state shall be taken as tm = t ( F )1/2 where t = element thickness F = normalized buckling stress for plates, from Clause 9.3.3, using a normalized slenderness, λ from Clause 9.3.1, with Fo = Fy , and the slenderness, λ, given by λ = mb/t where m = a factor from Clause 8.2 appropriate to the stress distribution and boundary conditions b = element width

7.4.3.2 Angles and flanges Angle sections, and outstanding elements supported along one long edge only, shall not be considered to possess postbuckling strength, and torsional or local buckling shall be deemed to lead to collapse.

7.4.4 Deflection under service loads Local buckling due to unfactored service loads is not extensive and the gross section properties shall be used when calculating deflections.

7.5 Effective strength and overall buckling 7.5.1 General Where welding, or local buckling with postbuckling strength, influences the flexural buckling of columns or lateral buckling of beams, the capacity shall be established by using the effective strength, Fm , given in Clauses 7.5.2 and 7.5.3. This effective strength represents the limiting stress, Fo , described in Clause 9.3.2, for use in Clause 9.4.1 to determine the mean stress to cause overall buckling.

7.5.2 Influence of welding For members containing longitudinal welds, the effective strength, Fm , shall be given by Fm = Fy – (Fy – Fwy)Aw /A where Fy = yield strength of the base metal Fwy = yield strength of the heat-affected zone Aw = cross-sectional area of the heat-affected zone A = gross cross-sectional area This value of the effective strength, Fm , shall be used as the limiting stress, Fo , in Clauses 9.3.2(g) and 9.4.1, in conjunction with the gross cross-sectional area, when determining the resistance to overall buckling (see also Clause 9.4.2.2). Note that the location of the HAZ is not a factor.

7.5.3 Influence of local buckling For flat elements supported on both long edges, subjected to a compressive stress that causes local buckling, the effective strength, Fm , of the element shall be given by Fm = F

1/2

Fy

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where F = normalized buckling stress for plates, obtained from Clause 9.3.3, corresponding to a normalized slenderness, λ , obtained from Clause 9.3.1, using Fo = Fy and the slenderness given by

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λ = mb/t

Fy

where m = a factor from Clause 8.2 appropriate to the stress distribution and boundary conditions b = element width t = element thickness = yield strength

This value of the effective strength, Fm , shall be used as the limiting stress, Fo , in Clause 9.4.1, in conjunction with the gross cross-sectional area, to determine the resistance to overall buckling.

8 Local buckling of flat elements 8.1 Buckling stress The buckling stress, Fc , of a flat element subjected to compressive stress shall be obtained using a slenderness, λ, given by λ = mb/t where m = local buckling factor given in Clause 8.2 or 8.3 b = element width (see Items (a) and (b) below) t = element thickness This value of λ shall be used to determine the normalized slenderness, λ (see Clause 9.3.1). Clause 9.3.3 shall then be used to determine the normalized buckling stress and hence the actual initial buckling stress, Fc , given by Fc = F Fy where F = normalized buckling stress Fy = yield strength The element width, b, shall be measured as follows: (a) For sections of uniform thickness, such as shapes formed from sheet, it shall be measured from the intersections of the median lines of adjacent elements, ignoring any corner radii. (b) For extruded sections with root fillets at the junctions, it shall be the distance from the tangent points of the root radii.

8.2 Elements supported on both longitudinal edges 8.2.1 Edges simply supported For a linear variation in stress across an element whose long edges are simply supported, the slenderness, λ, shall be given by λ = mb/t where m = 1.15 + f2 /2f1, when –1 < f2 /f1 < 1 = 1.3/(1 – f2 /f1), when f2 /f1 < –1

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where f1 = maximum compressive stress, negative f2 = stress at the other edge, positive when the stress is tensile b = element width (see Clause 8.1) t = element thickness A range of cases is illustrated in Figure C4 of the Commentary.

8.2.2 Influence of adjacent elements Where an element of width b, subjected to nominally uniform compressive stress, is connected along both edges to other elements of width a that are also supported along their edges (see Figure C5 of the Commentary), the slenderness, λ, shall be given by λ = mb/t where for columns, when b/t ≤ a/t1 m = 1.25 + 0.4 (a/t1)/(b/t) ≤ 1.65 for components in bending, such as decking profiles, when a/t1 ≤ 2.5b/t m = 1.25 + 0.2 (a/t1)/(b/t) ≤ 1.65 where a = width of the supporting element t1 = thickness of the supporting element b = width of the supported element (see Clause 8.1) t = thickness of the supported element When a/t1 >2.5 b/t, the web shall be checked for buckling using Clause 8.2.1.

8.3 Elements supported at one longitudinal edge only 8.3.1 Edge simply supported For a linear variation of longitudinal stress across an element that has one edge simply supported and the other edge free, buckling is in the torsional mode and the slenderness, λ, shall be given by (a) when the maximum compressive stress is at the free edge (see Figure C6 of the Commentary),

λ=

2.5 (3 + f2 /f1 )1/2 w t

When f2/f1 < –3, buckling does not occur. Note: This Clause applies particularly to angle sections in bending about the weak axis.

(b) when the maximum compressive stress is at the supported edge (see Figure C6 of the Commentary), for f2/f1 > –0.28,

λ=

2.5 (1 + 3 f2 /f1 )1/2 w t

where f2 = stress at the other edge, positive when the stress is tensile f1 = maximum compressive stress, negative w = element width (see the definition of b in Clause 8.1) t = element thickness When f2/f1 < –0.28, Clause 8.2.1 shall be used.

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8.3.2 Flanges of sections

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The slenderness, λ, for flanges of channels and Z and I sections subjected to uniform compressive stress due to axial force or bending (see Figure C7 of the Commentary) shall be given by

λ=

3 + 0.6 (at/wt1 ) w ≤ 5 w/t t

where a = breadth of the web element to which the flange is connected t = thickness of the flange w = width of the flange (see definition of b in Clause 8.1) t1 = thickness of the web element This slenderness, λ, with the yield strength, Fy , shall be used to determine the normalized slenderness, λ , in Clause 9.3.1, to give the normalized buckling stress, F , from Clause 9.3.3. The local buckling stress is then Fc = F Fy , which shall be used as Fo when determining the overall buckling resistance of non-compact columns and beams.

8.4 Elements supported at one edge with a lip at the other edge 8.4.1 General shapes For the general case of a flange element attached at one longitudinal edge to a web element with a lip at the other longitudinal edge (see Figure C8 of the Commentary), subjected to uniform compressive stress, the slenderness, λ, used to obtain the buckling stress shall be given by

λ=

5(I p / J )1/ 2 ⎡1 + 5.3(Cw k )1/2 / J ⎤ ⎣ ⎦

1/ 2

where Ip = polar moment of inertia of flange and stiffener about the supported edge J = St. Venant torsion constant for flange and stiffener Cw = warping constant for rotation of the flange and stiffener about the supported edge = Is w 2 where Is = moment of inertia of the stiffener about the inside surface of the flange to which it is attached; this applies to all types of stiffener, including inclined lips and bulbs w = flange width measured from the intersection of the median lines of the flange and web k = the spring constant for the restraint provided by the connection between flange and web = 3 t13/16 (a + 0.5w) for channel and Z sections = 1.5 t13/16 (a + 0.5w) for I-sections where t1 = web thickness a = breadth of the web element to which the flange is connected

8.4.2 Uniform thickness with simple lips 8.4.2.1 For a shape of uniform thickness, with simple lips on the flanges (see Figure C8 of the Commentary), the slenderness, λ, shall be given by

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1/ 2

⎛ ⎞ ⎜ ⎟ ⎟ 5w ⎜ 1 + 3β λ= ⎜ ⎟ 1 / 2 t ⎜ ⎛ β3 (w / t )2 + 0.1⎞ ⎟ ⎜ 1 + β + 3.7 ⎜⎜ ⎟⎟ ⎟⎟ ⎜ ⎝ a/w + 0.5 ⎠ ⎠ ⎝ where t = flange thickness a = web depth w = flange width (see definition of b in Clause 8.1) β = c/w where c = lip width

8.4.2.2

For a shape of uniform thickness, with simple lips inclined at 45°, the slenderness, λ, shall be given by 1/ 2

⎛ ⎞ ⎜ ⎟ ⎟ 5w ⎜ 1 + 3β λ= ⎜ ⎟ 1 / 2 t ⎜ ⎛ 0.5β3 (w / t )2 + 0.1⎞ ⎟ ⎜ 1 + β + 3.7 ⎜⎜ ⎟⎟ ⎟⎟ ⎜ a/w + 0.5 ⎝ ⎠ ⎠ ⎝

These expressions may also be used for standing stiffeners and for compression in a lip caused by overall bending. In such a case the stiffener spacing is 2a. The value of λ cannot be less than 1.6 w/t or 5c/t.

9 Resistance of members 9.1 Limiting slenderness for members Where the proportions of a member are to be limited to avoid excessive deflection under incidental lateral loads and vibrations due to wind or machinery forces, the following limits shall be observed: (a) for compression members: (i) chords — KL/r < 120; and (ii) diagonals — KL/r < 150; (b) for tension members — KL/r < 250(1 + f /Fe)1/2; and (c) for members subjected to wind: (i) tubes — KL/r < 100; and (ii) double angles — w/t < 32 000/L. where K = effective length factor L = length r = radius of gyration f = minimum permanent tension stress Fe = π2E/λ2 where E = elastic modulus

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λ = slenderness w = the breadth of the wider leg of the component angles (see definition of b in Clause 8.1) t = the thickness of the wider leg of the component angles

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9.2 Members in tension 9.2.1 Tensile resistance The factored tensile resistance, Tr , of an axially loaded member shall be the least of the values given by the following formulas: (a) general — Tr = φy Ag Fy; (b) with mechanical fasteners — Tr = φu An Fu ; (c) with transverse butt welds — Tr = φu Ag Fwu ; and (d) with longitudinal welds — Tr = φy Am Fy . where φy Ag Fy φu An Fu Fwu Am

= resistance factor on the yield strength = gross cross-sectional area = yield strength of the base metal = ultimate resistance factor = the net area of the cross-section from Clause 7.3 = ultimate strength of the base metal = ultimate strength for full penetration transverse butt welds (see Table 3) = effective area of the welded section using Clause 7.4.2.1

9.2.2 Oblique welds

When θ is less than 45°, where θ is the angle between the weld and plane normal to the direction of stress, the ultimate strength shall be taken to be that of a transverse weld (see Clause 9.2.1(c)). When the angle between the weld line and a line perpendicular to the direction of force exceeds 45°, the weld shall be treated as longitudinal for yielding of the overall transverse cross-section in Clause 9.2.1 (see Item (d)). In this case the transverse width of the HAZ is 50/sinθ.

9.3 Members in compression: Buckling 9.3.1 Normalized slenderness The normalized slenderness, λ , shall be given by

λ = (Fo /Fe)1/2 = (λ /π) (Fo / E)1/2 where Fo = limiting stress given in Clause 9.3.2. Fe = elastic buckling stress = π 2E/λ2 λ = slenderness given in Clause 8, 9, or 10 E = elastic modulus The normalized slenderness, λ , shall be used in Clause 9.3.3 to determine the normalized buckling stress, F = Fc /Fo , from which the actual buckling stress, Fc , is obtained.

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9.3.2 Limiting stress The limiting stress, Fo , used in determining the buckling stress, Fc , shall be taken as one of the following: (a) Where there is no welding or local buckling, the yield strength of the base metal is:

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Fo = Fy (b) When there is local buckling in an outstanding flange, the buckling stress, Fc , given by Clauses 8.1, 8.3.2, and 9.3.3 is: Fo = Fc (c) When local buckling occurs in an element supported on two longitudinal edges, when the element is at the extreme fibre for the axis of flexure, the effective strength for the element, Fm , given by Clause 7.5.3 is: Fo = Fm = ( F )1/2 Fy (d) In lattice columns, for the evaluation of the overall buckling capacity, the buckling stress of a chord, Fcc , given by Clause 9.4 is: Fo = Fcc (e) When there is transverse welding at the ends of the member, the yield strength of the base metal (with a mean axial stress not greater than Fwu ) is: Fo = Fy (f)

When there is a transverse weld away from the ends, the yield strength of the heat-affected zone, Fwy , is: Fo = Fwy

(g) When there is longitudinal welding, the effective strength, Fm , from Clause 7.5.2 (see also Clause 9.4.2.2) is: Fo = Fm where Fo = limiting stress Fy = yield strength of the base metal Fc = overall buckling stress Fm = effective strength to account for local buckling or longitudinal welds F = normalized buckling stress Fcc = buckling stress for a chord in a lattice column Fwu = ultimate strength for full penetration butt welds Fwy = yield strength of the heat-affected zone

9.3.3 Buckling stress The buckling stress, Fc , shall be given by Fc = F Fo where F = normalized buckling stress 1/ 2

1⎞ ⎛ = β − ⎜ β2 − 2 ⎟ λ ⎠ ⎝

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where

⎡1 + α ( λ − λo ) + λ2 ⎤ ⎦ β= ⎣ 2 2λ where α = 0.2 for heat-treated columns, beams, and plates = 0.4 for non-heat-treated columns, beams, and plates λ = normalized slenderness, from Clause 9.3.1 λ o = 0.3 for columns and beams = 0.5 for plates Fo = limiting stress Note: These relationships are plotted in Figure C11 of the Commentary for columns and beams and in Figure C12 for plates.

9.4 Columns 9.4.1 General The factored compressive resistance, Cr , of an axially loaded member shall be given by: Cr = φ c A F F o where φc = resistance factor for column buckling (see Clause 5.5) A = gross cross-sectional area F = normalized buckling stress (see Clause 9.3.3) Fo = limiting stress (see Clauses 7.5 and 9.3.2)

9.4.2 Flexural buckling 9.4.2.1 General

For flexural buckling, the slenderness, λ, shall be given by λ = KL/r where K = effective length factor (see Table 4 for typical values) L = unrestrained length r = appropriate radius of gyration of the gross cross-section The slenderness, λ, shall be used in Clause 9.3.1 with the applicable limiting stress, Fo , from Clause 9.3.2, to obtain the normalized slenderness, λ . This normalized slenderness shall then be used in Clause 9.3.3 to obtain the normalized buckling stress, F , to give the compressive stress, Fc = F Fo , to cause buckling.

9.4.2.2 Influence of longitudinal welds The normalized buckling stress, F , obtained for longitudinally welded members (see Clause 9.3.2(g)) using Clause 9.3.3 shall be multiplied by the following additional factor: k = (0.9 + 0.1 |1 – λ |) ≤ 1 where |1 – λ | = the absolute value

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9.4.3 Torsional buckling 9.4.3.1 Pure torsion Licensed for/Autorisé à Reid Costley, Cascade Engineering Group, Sold by/vendu par CSA on/le 10/27/2009. Single user license only. Storage, distribution or use on network prohibited./Permis d'utilisateur simple seulement. Le stockage, la distribution ou l'utilisation sur le réseau est interdit.

9.4.3.1.1 General case To obtain the critical stress for pure torsional buckling, the slenderness shall be given by λt = π (EIP/GJ)1/2 = 5 (IP /J)1/2 where IP = polar moment of inertia about the shear centre J = St. Venant torsion constant The normalized slenderness from Clause 9.3.1 shall be used in Clause 9.3.3 to give the normalized buckling stress, F . The factored compressive resistance, Cr , shall be given by Cr = φc A F Fy where φc = resistance factor for column buckling A = cross-sectional area F = normalized buckling stress Fy = yield strength

9.4.3.1.2 Sections composed of radial outstands 9.4.3.1.2.1 For simple angle sections, T-sections, and cruciforms, the slenderness shall be calculated as follows: λt = 5w/t where w = the longest leg width: for extruded sections it is measured from the start of the root fillet; for formed angles, the leg width is measured from the intersection of the median lines of the adjacent walls (see Figure C13 of the Commentary) t = thickness Note: The bend radius has a small negative influence in formed angles.

9.4.3.1.2.2 For lipped equal angles of uniform thickness, the expression for the slenderness becomes

λt = 5

w t

⎡ 1 + 3β ⎤ ⎢1+β⎥ ⎣ ⎦

1/ 2

where w = the leg measured between the intersections of the median lines of the elements t = thickness β = c/w where c = the lip length measured from the median line of the leg Note: The addition of lips reduces the stress to cause torsional buckling in formed angles.

9.4.3.1.2.3 For bulb angles, Clause 9.4.3.1.1 shall be used.

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9.4.3.2 Torsion with restrained warping

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For members, such as channels and Z shapes, which possess warping rigidity, the slenderness, λt , shall be given by

λt =

5(I p / J )1/ 2 ⎡1+ 25Cw / JL2 ⎤ ⎣ ⎦

1/ 2

where Ip = polar moment of inertia about the shear centre J = St. Venant torsion constant Cw = warping constant L = effective length

9.4.3.3 Combined torsion and flexure For open sections symmetrical about one axis only, failure by flexure about the axis of symmetry combines with torsion. The slenderness, λ, shall be given by λ = λ1 [1 + (xo/ro)(λ2/λ1)2 ]1/2 where λ1 and λ2 = the slendernesses for the two modes of buckling, λ1 > λ2 (see Clauses 9.4.2.1 and 9.4.3.1) xo = distance from the centroid to the shear centre ro = polar radius of gyration about the shear centre

9.5 Bending 9.5.1 Classification of members in bending Cross-sections of members in bending are classified, according to the compactness of the elements of the cross-section, as follows: (a) Class 1 sections are those capable of undergoing plastic strain in compression without local buckling. The sections shall be symmetrical about the plane of bending, be fully constrained against lateral buckling, and have λ < 0.3 (i.e., b/t < 250/mFy1/2 ). (b) Class 2 sections are those capable of carrying moment up to the onset of yielding in compression without local buckling. The sections shall be such that λ < 0.5 (i.e., b/t < 420/mFy1/2). (c) Class 3 sections are those in which there is local buckling below the yield stress with or without postbuckling reserve. This occurs when λ > 0.5. For local buckling, λ is obtained using Clause 8. Note: Lattice beams and masts, in which chord buckling controls overall flexural buckling, are Class 3.

9.5.2 Moment resistance of members not subject to lateral-torsional buckling For members with no tendency to buckle laterally, the factored moment resistance, Mr , in the plane of bending shall be given by: (a) For Class 1 sections: (i) for compression fibres, Mr = φy ZFy = φy Mp (ii) for tension fibres, Mr = φu Zn Fu

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(b) For Class 2 sections: (i) for compression fibres,

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Mr = φy SFy = φy My (ii) for tension fibres, Mr = φu SnFu (c) For Class 3 sections: (i) when outstanding flanges, or chords in lattice masts, buckle in compression, Mr = φy S F Fy (ii) when flat compression elements have two long edges supported, Mr = φy SmFy where φy = resistance factor on the yield strength Z = plastic section modulus Fy = yield strength Mp = fully plastic moment φu = ultimate resistance factor Zn = net plastic section modulus Fu = ultimate strength S = elastic section modulus My = moment at first yield Sn = net elastic section modulus F = normalized buckling stress for flanges (see Clauses 8 and 9.3.3) or chords Sm = effective section modulus using the effective thicknesses (see Clause 7.4) For sections influenced by longitudinal welds, Mp and My shall be calculated using the appropriate effective section from Clause 7.4.2.

9.5.3 Moment resistance of members subject to lateral-torsional buckling 9.5.3.1 Members with lateral restraint of the tension flange only For members of all classes described in Clause 9.5.1, for bending about the strong axis (x-axis) with lateral restraint at the tension flange only, the factored moment resistance, Mr , shall be given by Mr = φy Sx F Fo where φy = resistance factor on the yield strength Sx = section modulus about the strong axis, F = normalized buckling stress, from Clause 9.3.3, using the limiting stress, Fo , and the slenderness, λ, given by (a) for the general case,

⎡ ⎤ Sx d λ= ⎢ 2⎥ ⎢⎣ 0.04 J + Cw / L ⎥⎦

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(b) for I-sections, channels, and plate girders,

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λ=

L/ry ⎡1 + 0.5(Lt/bd )2 ⎤ ⎣ ⎦

1/ 2

(c) for deep rectangular solid or hollow sections,

λ=

rx 2.8 ry ⎡0.64 + (d/L )2 ⎤1/ 2 ⎣ ⎦

(d) where the tension flange is firmly attached to roofing or cladding panels, with bending resistance, which can provide elastic restraint to the twisting of the member, the slenderness becomes ⎡ ⎤ 10Sx ⎥ λ= ⎢ 1/ 2 ⎢⎣ 0.4 J/d + (Iy Im / a ) ⎥⎦

Fo

1/ 2

where Sx = section modulus about the strong axis d = member depth J = St. Venant torsion constant Cw = warping constant for rotation about the point of restraint L = distance between points of full lateral restraint ry = radius of gyration about the weak axis, or, for unsymmetrical I-sections, of the compression flange plus 1/6 of the web area t = flange thickness b = overall flange width rx = radius of gyration about the strong axis Iy = moment of inertia about the weak axis of the member Im = moment of inertia per unit width of the supported medium a = distance between the parallel members supporting the medium = appropriate limiting stress from Clause 9.3.2

9.5.3.2 Unrestrained members For unrestrained members bending about the strong axis (x-axis), which are subject to lateral-torsional buckling, the factored moment resistance, Mr , for all classes described in Clause 9.5.1, shall be given by the formula in 9.5.3.1, in which the normalized buckling stress, F , is obtained from Clause 9.3.3 using the limiting stress, Fo , and the slenderness, λ, given by: (a) for the general case,

λ=

(

(Sx L )1/2

)

⎡Iy 0.04 J + Cw / L2 ⎤ ⎣ ⎦

1/ 4

(b) for I-sections, channels, and plate girders,

λ=

L/ry ⎡1 + (Lt/bd )2 ⎤ ⎣ ⎦

1/ 4

(c) for deep rectangular solid or hollow sections,

(

)

λ = 2.2 rx / ry (L / d )1/ 2

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© Canadian Standards Association

Strength design in aluminum

where Sx = section modulus about the x-axis L = distance between points of full lateral or torsional restraint Iy = moment of inertia about the weak axis J = St. Venant torsion constant Cw = warping constant about the shear centre ry = radius of gyration about the weak axis t = flange thickness b = overall flange width d = member depth rx = radius of gyration about the x-axis

9.5.3.3 Members with end moments For members subjected to a moment gradient, with factored end moments of Mmax and Mmin , the value of the moment, which shall not exceed the factored moment resistance, Mr , in Clause 9.5.3.1 or 9.5.3.2, as appropriate, shall be taken as Mm = 0.6 Mmax + 0.4 Mmin , but not less than 0.4 Mmax For members bent in double curvature, Mmin is negative. Clause 9.5.2 limits the maximum moment at a support.

9.6 Webs in shear 9.6.1 Flat shear panels 9.6.1.1 Buckling stress For a flat rectangular panel with boundary flanges or stiffeners, subjected to shear force, the initial buckling stress, Fsc , shall be given by Fsc = Fo F where Fo = shear yield strength = 0.6 Fy where Fy = tensile yield strength of the base metal

F = normalized buckling stress from Clause 9.3.3, obtained as follows: λ = (λs /π)(Fo /E )1/2 where λs = 1.4 (b/t)/[1 + 0.75(b/a)2]1/2 (see Figure C16 of the Commentary) where b = smaller panel dimension t = panel thickness a = larger panel dimension

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9.6.1.2 Limiting shear resistance at the boundaries The factored shear resistance per unit length, vr , of connections between the web and the flange, or of any seams in the web, or along attachments of stiffeners shall be the least of the appropriate values given by (a) vr = φy 0.6Fyt; (b) vr = φu 0.6Fwu t, but not greater than the factored resistance of the fillet welds; (c) vr = φf Rk /s, but not greater than φu 0.6(1 – do /s)Fu t, for seams with mechanical fasteners; and (d) vr = φu vk where φy = resistance factor on the yield strength Fy = tensile yield strength of the base metal t = panel thickness φu = resistance factor on ultimate Fwu = ultimate tensile strength in the heat-affected zone at a weld (see Table 3) φf = resistance factor on fastener Rk = characteristic resistance of a fastener s = spacing of fasteners along a line of connection do = hole diameter Fu = ultimate tensile strength of the base metal vk = characteristic ultimate shear resistance per unit length of a joint or seam (riveted or bonded)

9.6.2 Stiffened webs 9.6.2.1 Stiffener locations Stiffened webs shall have stiffeners at the points of support and at any local applied forces. Other transverse and longitudinal stiffeners may be added to increase the initial buckling stress.

9.6.2.2 Shear resistance For webs that buckle before yielding, the factored shear resistance, Vr , shall be the lesser of the values given by: (a) Vr = φy [2(Fsc vk t)1/2 – Fsc ] ht; and (b) Vr = φuvkh where φy = resistance factor on the yield strength Fsc = buckling stress in shear from Clause 9.6.1.1 vk = characteristic ultimate shear resistance per unit length of the boundaries or seams (see Clause 9.6.1.2) t = web thickness h = web depth (see Figure C16 of the Commentary) φu = resistance factor on ultimate strength For webs with combined longitudinal and transverse stiffeners, the value to be used for Fsc shall be that for the panel, in the section considered, with the lowest initial buckling stress.

9.6.3 Web stiffeners Stiffeners shall be designed as struts to carry a factored axial force, Nf , given by: (a) for transverse stiffeners, Nf is the greater of the shear force at the stiffener due to the factored loads and the local factored load applied to the top flange or the support reaction; and (b) for longitudinal stiffeners, Nf = v r a F

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where vr = factored ultimate shear resistance per unit length at the boundaries from Clause 9.6.1.2 a = length between transverse stiffeners F = normalized buckling stress from Clause 9.6.1.1 Stiffeners on one side only shall be treated as eccentrically loaded. A 25t width of the web plate may be assumed to act with the stiffener. If longitudinal stiffeners accept axial force due to overall bending, the force shall be added to that required for stability under shear forces.

9.6.4 Combined shear and bending in webs In continuous beams with web stiffeners over the supports, if no buckling is to occur, the limiting condition is

⎛ fsf ⎜ ⎜ φy Fsc ⎝

2

⎞ ⎛ f ⎟ + ⎜ bf ⎟ ⎜ φy Fbc ⎠ ⎝

⎞ ⎟ ⎟ ⎠

2

≤ 1

where fsf = factored applied shear stress φy = resistance factor on the yield strength Fsc = buckling stress in shear from Clause 9.6.1.1 fbf = factored compressive stress due to bending Fbc = local buckling stress due to compression caused by bending from Clause 9.3.3, using the slenderness, λ, from Clause 8.2.1 and the limiting stress, Fy If this condition is not satisfied, then the web shall be assumed to carry shear force only, satisfying Clause 9.6.1, and the flanges shall be assumed to carry the bending moment.

9.6.5 Web crushing The factored resistance, Cr , for a local compressive force acting in the plane of the web shall be determined using one of the following formulas: (a) for flat webs,

Cr = φy k (n + h )tFc′ ≤ φy ntFy where φy = resistance factor on the yield strength k = 0.5 [1 + e/(n/2 + h)] ≤ 1 where e = distance from the centre of bearing to the end of the beam n = bearing length h = web depth t = thickness of the web

Fc′ =

π2Et 2 ⎡ 1 − 4 h2 ⎣

(fbf / Fbc )2 ⎤⎦

where E = elastic modulus fbf = factored longitudinal compressive stress due to the overall bending moment Fbc = web buckling stress for bending from Clause 9.3.3, using the slenderness, λ, from Clause 8.2.1 and the limiting stress, Fy Fy = yield strength

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(b) for webs with bend radii at the corners (see Figure C18 of the Commentary),

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Cr = φy k (11 + 0.07 n/t) (1 – 0.0008 θ R/t) (Fy – fbf )t2 where φy = resistance factor on the yield strength k = 0.5 [1 + e/(n/2 + h)] ≤ 1 where e = distance from the centre of bearing to the end n = bearing length h = web depth t = thickness θ = acute angle between web and bearing surface, in degrees R = interior bend radius Fy = yield strength fbf = longitudinal compressive stress at the bearing point due to the factored moment

9.7 Members with combined axial force and bending moment 9.7.1 Axial tension and bending 9.7.1.1 Elastic behaviour The combination of factored axial tension force, Tf , and factored bending moment, Mf , which is limited by yield or rupture, shall be taken as Mf /Mr + Tf /Tr ≤ 1.0 where the values conform to one of the following: (a) Mr is from Clause 9.5.2(a)(i) and Tr is from Clause 9.2.1(a); or (b) Mr is from Clause 9.5.2(a)(ii) and Tr is from Clause 9.2.1(b). If welded, the effective thicknesses shall be used when computing the geometric properties. Holes in the tension zone shall be deducted from members in bending to give the net section modulus. No adjustment need be made in the position of the centroid of the cross-section.

9.7.1.2 Plastic behaviour Where a fully plastic condition is permitted, for a flat plate or solid bar, the limiting combination shall be given by

⎛T ⎞ Mf + ⎜ f ⎟ Mr ⎝ Tr ⎠

2

≤ 1

where the symbols are as defined in Clause 9.7.1.1.

9.7.1.3 Stability Where the tension force, Tf , helps to stabilize a member against lateral buckling caused by an applied moment, Mf , the combination for stability shall be given by 2

⎛ Mf ⎞ Tf ≤ 1 ⎜ ⎟ − M C ⎝ r⎠ e where Mr = factored resistance (moment) from Clause 9.5.3.2

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Ce = Aπ 2 E/λy2 where A = cross-sectional area E = elastic modulus λy = KL/ry where K = effective length factor (see Table 4 for typical values) L = unbraced length ry = radius of gyration of the gross cross-section about the weak axis

9.7.2 Eccentric tension The factored tensile resistance, Tr , of a member with eccentric end connections shall be the lesser of the values given by (a) Tr = φy Ag Fy /(1 + ec/r 2); and (b) Tr = φu Ae Fu where φy = resistance factor on the yield strength Ag = gross area Fy = yield strength e = eccentricity c = distance to extreme tension fibre r = radius of gyration for the axis of bending φu = ultimate resistance factor Ae = the effective net area, which shall be obtained using the appropriate formula from the following cases: (i) for general shapes, Ae = An /(1 + An e/Zn) (ii) for single angles (see Figure C19(a) of the Commentary), (1) for connections with single fasteners, Ae = (2g – do)t < (b – do)t (2) for connections with two or more fasteners, or welded connections, Ae = (2g + w/3 – do)t < (b + w/3 – do)t Note: The addition of lug angles to engage the outstanding leg has a negligible benefit.

(3) for channels held by the web (see Figure C19(b) of the Commentary), double angles attached to one side of the gusset, and T-sections bolted through the flanges, Ae = An – Af /2 (4) for double angles, with an angle each side of the gusset plate, Ae = An – Af /4

Fu

where An = net area of the cross-section (see Clause 7.3) e = eccentricity Zn = plastic section modulus for the net section g = distance from the hole centre to the longitudinal edge ≥ 1.25do do = hole diameter b = width of the connected leg t = thickness w = width of the outstanding leg Af = total area of the non-connected flanges = ultimate strength

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9.7.3 Beam-columns

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9.7.3.1 Members not subject to lateral-torsional buckling Where there is no tendency to buckle laterally, the limiting combination of factored axial load, Cf , and factored bending moment, Mf , shall be calculated using the gross section and shall be given by: (a) where compressive stress governs,

Mf C + f Sc (1 − Cf / Ce ) A

≤ φy Fo

(b) where tensile stress governs,

Mf C − f St (1 − Cf / Ce ) A

≤ φy Fy

(c) for members with applied end moments, the limiting combination at the supports shall be calculated using the following formulas: (i) when compressive stress governs, Cf /A + Mmax /Sc ≤ φyFy (ii) when tensile stress governs, Mmax /St – Cf /A ≤ φyFy where Mf = maximum moment due to the factored lateral loads = 0.6M1 + 0.4M2 > 0.4M1 for moment gradients

Sc Cf Ce

where M1, M2 = the applied end moments due to the factored loads; they are of opposite signs when they cause a reversal of curvature along the beam (i.e., the moments act in the same sense), (M1>M2) = section modulus for the extreme fibre in compression = applied compressive force due to the factored loads, which shall not exceed the value for Cr given by Clause 9.4.1 = A π2 E/λ2 where E = elastic modulus λ = L/r

A φy Fo St Fy Mmax

32

= = = = = =

where L = unbraced length r = radius of gyration in the plane of bending gross area resistance factor on the yield strength limiting stress (see Clause 9.3.2) section modulus for the extreme fibre in tension yield strength maximum factored end moment

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Strength design in aluminum

9.7.3.2 Members with biaxial moments, not subject to lateral-torsional buckling

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For members subjected to axial force combined with bending about both principal axes, where the same extreme fibre carries the maximum stress from both moments and the member fails in flexure, the limiting condition shall be given by

Myf Mxf Cf + + A Sx (1 − Cf / Cex ) Sy 1 − Cf / Cey

(

)

≤ φy Fo

where Cf = factored compressive force, not greater than the value of Cr given by Clause 9.4.1 Mxf = moment in the member due to the factored lateral load, about the strong axis Myf = moment in the member due to the factored lateral load, about the weak axis φy = resistance factor on the yield strength Fo = limiting stress (see Clause 9.3.2) A = gross area Sx = section modulus of the gross section about the strong axis Sy = section modulus of the gross section about the weak axis Cex = Aπ2 E/λx2 where E = elastic modulus λx = L/rx where L = unbraced length rx = radius of gyration of the gross cross-section about the strong axis Cey = Aπ 2 E/λy2 where λy = L/ry where ry = radius of gyration of the gross section about the weak axis

9.7.3.3 Members subject to lateral-torsional buckling For combined axial force and bending about the strong axis, when lateral buckling can occur, the combined factored axial load, Cf , and bending moment, Mf , shall satisfy

Cf Mf + ≤ 1 Cry Mr (1 − Cf / Cex ) where Cf = applied compressive force due to the factored loads Mf = moment due to the factored lateral load, or as calculated in Clause 9.5.3.3 Cry = factored resistance for failure about the weak axis, obtained from Clause 9.4.1 Mr = factored moment resistance obtained from Clause 9.5.3.1 or 9.5.3.2, as applicable Cex = elastic buckling force for bending about the strong axis = Aπ 2 E/λx2 where A = gross area

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E = elastic modulus λx = KL/rx where K = effective length factor (see Table 4 for typical values) L = unbraced length rx = radius of gyration of the gross cross-section about the strong axis

9.7.4 Eccentric compression 9.7.4.1 General case For general cases of eccentric compression, the following requirements shall apply: (a) For failure in the plane of bending, Clause 9.7.3.1(a) or (b) shall be used with a factored moment, Mf , given by Mf = 1.2eCf (b) For lateral-torsional buckling, Clause 9.7.3.3 shall be used with a moment, Mf , given by Mf = eCf where e = eccentricity Cf = factored axial force

9.7.4.2 Single angle bracing members For single angle bracing members, (a) the factored compressive resistance, Cr , of discontinuous single angles connected through one leg shall be given by Clause 9.4.1 using

(

λ = λ2y + λt2

)

1/2

where λy = KL/rv where K = effective length factor (see Table 4 for typical values) L = unbraced length of member ry = minimum radius of gyration about v-axis λt = 5 w/t where w = width of longer leg, see Clause 9.4.3.1.2 t = thickness of longer leg (b) the factored resistance, Cr , shall not exceed (i) φc 0.5AFy for single bolt connections; or (ii) φc 0.67AFy for double bolt or welded connections where A = gross cross-sectional area Fy = yield strength

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9.7.5 Shear force in beam-columns 9.7.5.1 Licensed for/Autorisé à Reid Costley, Cascade Engineering Group, Sold by/vendu par CSA on/le 10/27/2009. Single user license only. Storage, distribution or use on network prohibited./Permis d'utilisateur simple seulement. Le stockage, la distribution ou l'utilisation sur le réseau est interdit.

When lateral loads combine with concentric axial force, the maximum factored shear force, Vmax , shall be taken to be the larger of the values given by the following formulas:

(a) Vmax =

Vf ; and (1 − Cf / Ce )

(b) Vmax = Cf /40 where Vf = factored shear force due to lateral loads only Cf = factored compressive force Ce = Aπ 2E/λ2 where A = gross area of the section E = elastic modulus λ = KL/r where K = effective length factor (see Table 4 for typical values) L = unbraced length of the member r = radius of gyration

9.7.5.2 When the moment is due to an eccentric axial load, the maximum factored transverse shear force shall be taken to be the larger of the values given by the following formulas:

(a) Vmax =

5Cf e ; and (Ce / Cf − 1)L

(b) Vmax = Cf /40 where Cf = factored compressive force e = eccentricity Ce = Aπ 2E/λ2 where A = gross area of the section E = elastic modulus λ = KL/r where K = effective length factor (see Table 4 for typical values) L = unbraced length of the member r = radius of gyration

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9.8 Built-up columns

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9.8.1 Spacing of connectors In compression members composed of battened channels, double or quadruple angles stitch-bolted or tack-welded together, and latticed masts, the slenderness of the individual members between the interconnections shall not exceed 0.75 times that of the overall member.

9.8.2 Multiple-bar members with discrete shear connectors For compression members composed of two or more bars, connected together at discrete intervals by fasteners, battens, or welds, that buckle in the built-up plane so as to cause shear in the connectors (see Figure C21), there shall be at least four interconnectors: one at each end and two within the length of the member. The factored compressive resistance, Cr , shall be given by Clause 9.4.1 using Fo = Fy and the slenderness, λ, given by

(

λ = λo2 + λa2

)

1/2

where λo = KL/r ’ where K = effective length factor (see Table 4 for typical values) L = length of the member r ’ = radius of gyration of the total section for the built-up axis λa = a/r where a = distance from centre to centre of the interconnections r = radius of gyration of a single element bending in the plane of failure Interconnections shall be designed to resist a total shear force at each location of Cf /40, where Cf is the factored compressive force.

9.8.3 Double angle struts Where the torsional flexibility of double angles influences the stability, the factored compressive resistance, Cr , for combined buckling about the built-up axis and torsional buckling shall be obtained from Clause 9.4.1 using the slenderness, λ, given by

(

λ = λ12 + 0.5λ22

)

1/2

where λ1 = greater value of λf and λt λ2 = lesser value of λf and λt where λf = slenderness for flexural buckling given by Clause 9.8.2 λt = slenderness for torsional buckling = 5w/t where w = width of the longer leg (see Clause 9.4.3.1) t = thickness of the longer leg

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Strength design in aluminum

9.8.4 Lattice columns and beam columns The following requirements shall apply: (a) the slenderness, λ, of a lattice column shall be given by

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λ = L/r where L = length of column r = radius of gyration of the total cross-section The limiting stress, Fo , for use in Clause 9.3.3 shall be the buckling stress for the chord. (b) The limiting combination of factored axial load, Cf , and factored moment, Mf , for a lattice beam-column failing in the plane of bending, shall be given by

Cf Mf + ≤ Cr N kd (1 − Cf / Ce ) (c) The maximum shear force in a beam-column shall be given by

Vmax =

Vf (1 − C f / C e )

where Cf = factored axial force N = number of chords Mf = factored applied moment k = for square sections, 1.4 or 2, depending on the direction of bending (see Clause C9.8.4 in the Commentary) = for triangular sections, 0.85, 1.0, or 1.7, depending on the direction of bending (see Clause C9.8.4 in the Commentary) d = face width Ce = Aπ2E/λ2 where A = total area of all chords E = elastic modulus λ = L/r where L = unbraced length of column r = overall radius of gyration of the cross-section of the lattice member = d/2 for rectangular sections = d/ 6 for triangular sections Cr = factored compressive resistance of one chord, from Clause 9.4.1 Vf = factored applied shear force

9.9 Members in torsion The factored torsional resistance, Hr , shall be given by (a) for hollow sections, Hr = φy 1.2 A’ t Fy where φy = resistance factor on the yield strength

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A’ = area enclosed by the median line of the walls

t = minimum thickness Fy = yield strength

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(b) for solid compact bars, Hr = φy Aa Fy/5 where φy = resistance factor on the yield strength A = area of cross-section a = least dimension across the section Fy = yield strength (c) for open shapes (i) without warping restraint, Hr = φy (Σ wt3) Fy /5 tmax where φy = resistance factor on the yield strength w = width of individual wall t = thickness of individual wall Fy = yield strength tmax = maximum thickness (ii) with warping restraint, the total longitudinal stress due to the warping rigidity shall be added to any simultaneous bending stress. The total longitudinal stress due to the factored load shall not exceed φy Fy.

10 Panels 10.1 Flat panels with multiple stiffeners 10.1.1 Axial compression The factored compressive resistance, Cr , of a sheet with multiple stiffeners, loaded in the direction of stiffening, shall be obtained from Clause 9.4.1 using the appropriate value of the slenderness, λ, given by (a) for flat sheet with stiffeners, the lesser of (i) λ = L/r; and (ii) λ = 1.3 (b/r)(I/t3)1/4; and (b) for formed sheet, the lesser of (i) λ = L/r; and (ii) λ = 1.2 (b/r)(ηr/t)1/2 where L = panel length between transverse supports r = radius of gyration of the gross stiffened section b = panel width perpendicular to the direction of the stiffeners I = moment of inertia per unit width of the gross stiffened section t = sheet thickness η = ratio of original sheet width to the final width of a formed profile If the sheet buckles between the stiffeners (see Clause 8.2), the limiting stress, Fo , shall be the effective strength, Fm , for the buckled elements, obtained from Clause 7.5.3, which is used in Clause 9.3.3 to give the overall buckling force.

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10.1.2 In-plane shear The factored in-plane shear resistance, Vr , of a sheet with multiple transverse stiffeners shall be given by

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Vr = φy F (0.6Fy)ht where φy = resistance factor on the yield strength F = normalized buckling stress obtained from Clause 9.3.3, for plates of the appropriate alloy type, using the limiting stress Fo = 0.6Fy and the slenderness, λ, given by: (a) for flat sheet with stiffeners, λ = 0.8b (t/I3)1/8 (b) for formed sheet panels, λ = 0.8b/(η r 3t)1/4 where b = panel dimension in the direction of the stiffeners I = moment of inertia per unit width of the stiffened sheet η = ratio of original sheet width to the final width of a formed profile r = radius of gyration of the formed profile Fy = yield stress h = width of panel in the direction of the shear force t = thickness of sheet from which the panel is formed

10.2 Curved panels and tubes 10.2.1 Axial compression The buckling stress, Fc , of a tube or curved panel shall be obtained from Clause 9.3.3 using the limiting stress, Fo = Fy , and the appropriate value of the slenderness, λ, given by one of the following formulas: (a) for a tube, λ = 4 (R/t)1/2 [1 + 0.03 (R/t)1/2] (b) for a curved panel (see Figure C22 of the Commentary),

λ=

λ1 1/2

⎡1 + (λ1 / λ2 )4 ⎤ ⎣ ⎦

where R = radius of curvature t = sheet thickness λ1 = 3.3 (a/t)/[1 + (a/b)2] when a < b = 1.65 b/t when a ≥ b where a = panel length between circumferential stiffeners b = arc length of the panel between longitudinal stiffeners λ2 = slenderness given by Item (a) above

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10.2.2 Radial compression

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The buckling stress, Fc , for a radially loaded tube, with circumferential stiffeners spaced at a distance a apart, shall be obtained from Clause 9.3.3 using the limiting stress, Fo = Fy, and the appropriate value of the slenderness, λ, given by one of the following formulas: (a) when a/R > 3.3 (R/t)1/2, λ = 6 R/t (b) when a/R < 3.3 (R/t)1/2, λ = 3.3 (a/t)1/2 (R/t)1/4 (c) for long curved panels supported on straight longitudinal boundaries, (i) when b/R >π, λ = 6R/t (ii) when b/R < π, λ = (3.3R/t)/[(2R/b)2 – 0.1]1/2 where a = distance between circumferential stiffeners R = radius of curvature t = sheet thickness b = length of curved edge

10.2.3 Shear For a tube or curved panel, the normalized shear buckling stress, F = Fsc /Fsy , shall be obtained from Clause 9.3.3, using the limiting stress, Fo = Fsy = 0.6Fy , and the slenderness, λ, given by one of the following formulas: (a) for a tube, the lesser of the values given by the following formulas: (i) λ = 4 (R/t)5/8 (a/R)1/4; and (ii) λ = 6 (R/t)3/4 (b) for a curved panel,

λ=

λ1 1/2

⎡1 + (λ1 / λ2 )2 ⎤ ⎣ ⎦

where Fsc = buckling stress in shear Fsy = shear yield strength 0.6Fy Fy = yield strength in tension of base material R = radius of curvature t = sheet thickness a = distance between circumferential stiffeners λ1 = slenderness from Clause 10.2.3(a) for a tube of the same radius and length λ2 = slenderness from Clause 9.6.1.1 for a flat panel of the same proportions

10.3 Curved axially stiffened panels in axial compression For axial loading in curved stiffened panels with multiple longitudinal stiffeners, the normalized buckling stress, F , shall be obtained from Clause 9.3.3, using the limiting stress, Fo = Fy , and the slenderness, λ, given by:

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λ=

Strength design in aluminum

λ1 1/2

⎡1 + (λ1 / λ2 )2 ⎤ ⎣ ⎦

where λ1 = a/r where a = length of the wall between transverse stiffeners r = radius of gyration of the stiffened sheet λ2 = 5.7 (ηR/t)1/2 where η = ratio of original sheet width to the final width of the panel = 1, for a curved sheet with longitudinal stiffeners added R = radius of curvature of the element t = sheet thickness The wall between stiffeners shall satisfy the requirements of Clause 10.2.1.

10.4 Flat sandwich panels 10.4.1 General For sandwich panels composed of aluminum sheets bonded to a core material having a modulus of elasticity less than one hundredth of the value of the modulus of elasticity of the aluminum skin, the treatment outlined in Clauses 10.4.2 to 10.4.5 shall be used unless a full study of the composite action is made. Only the skin shall be assumed to resist axial loads, bending moments, and shear forces in the plane of the panel. The core resists shear forces across the thickness of the panel.

10.4.2 Panel bending The factored resisting moment per unit width of a panel shall be given by M r = φc F Fy td where φc F Fy = the factored buckling stress for the skin, obtained from Clause 10.4.4 t = skin thickness d = panel thickness

10.4.3 Panel buckling 10.4.3.1 Overall buckling controlled by the skins The factored compressive resistance, Cr , of a panel shall be given by Clause 9.4.1, where A is the cross-sectional area of both skins, and the slenderness, λ, is given by the following formulas: (a) for longitudinal edges unsupported, λ = 2 L/d (b) for longitudinal edges supported, (i) when L < b λ = 2 (L/d)/ [1 + (1.2L/b)3]1/2 (ii) when L > b λ = 1.2 b/d

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where L = panel length in the direction of the load d = panel thickness b = panel width When determining the normalized slenderness, λ , from Clause 9.3.1, the limiting stress, Fo , shall be the buckling stress for the skin obtained from Clause 10.4.4.

10.4.3.2 Influence of shear in the core To account for the shear flexibility of the core, the factored compressive resistance, Cr , obtained using Clause 10.4.3.1, shall be divided by (1 + Cr /Gc db) where Cr = factored compressive resistance Gc = shear modulus of the core material d = panel thickness b = panel width

10.4.4 Skin buckling The normalized buckling stress, F , for the skins shall be obtained from Clause 9.3.3, using Fo = Fy , and the slenderness, λ, given by λ = 4.5 E1/ 3/(EcGc)1/ 6 where E = elastic modulus of the skin Ec = elastic modulus of the core for stress perpendicular to the skin Gc = shear modulus of the core

10.4.5 Core strength 10.4.5.1 Shear The factored shear strength of the core, Fsr , and of the bond between the core and skin, shall be not less than Fsr = vf /d where vf = factored shear force per unit length, perpendicular to the skins (see Clause 9.7.5.1) d = panel thickness

10.4.5.2 Tension The factored bond strength between the skin and core material, and the factored tensile strength of the core material, for a direction of stress perpendicular to the surface, shall be not less than

(Ec Gc )1/ 2 300(1 − f/Fc ) where Ec = elastic modulus of the core for stress perpendicular to the skin Gc = shear modulus of the core f = applied compressive stress in the skin due to the factored load Fc = buckling stress for the skin obtained using Clause 10.4.4

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Strength design in aluminum

11 Resistance of connections

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11.1 General 11.1.1 Connection types Any suitable mechanical fastening, welding, special device, or other effective means may be used to join component parts together, provided that the fastening method is compatible with the service conditions. Fastening methods that are not covered by the requirements in Clause 11 shall be subject to evidence of suitability prior to use.

11.1.2 Sharing of loads Loads shall not be shared between mechanical fasteners and welds in the same connection.

11.1.3 Materials Fastening materials shall meet the requirements of Clause 4.

11.2 Mechanical fasteners 11.2.1 General 11.2.1.1 Fastener types Except where permitted by Clause 11.1.1, connections between aluminum members shall be made with one of the following types of fastener: (a) aluminum rivets and bolts; (b) zinc-coated or cadmium-plated steel bolts; or (c) stainless steel bolts.

11.2.1.2 Nominal dimensions The nominal diameter of the fastener shall be used in proportioning and spacing fasteners. “Spacing” shall refer to the centre-to-centre distance on any gauge line.

11.2.1.3 Bearing action All mechanical fasteners shall be designed to act in bearing when determining the ultimate limit state. The effective bearing area of fasteners shall be the fastener diameter multiplied by the thickness in bearing.

11.2.1.4 Slip-critical joints If preloaded bolts are to be used to develop resistance to service loads by friction, the surface shall be sanded or given an equivalent treatment to ensure that a coefficient of friction of at least 0.3 is available. Installation of the bolts shall be in accordance with CAN/CSA-S16. Under the action of the service loads, the force on a bolt to be resisted by friction shall not exceed V = 0.15 mA Fu where m = number of shear planes A = cross-sectional area of the bolt Fu = ultimate strength of the bolt material Under the action of the factored loads, the joint shall satisfy the requirements for bearing.

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11.2.1.5 Joint symmetry All connections and splices should be symmetrical about the axis of the connected members. Members of a framework meeting at a joint should be arranged so that their centroidal axes intersect at a point. Groups of fasteners at the ends of members that transmit forces into those members should have their centroids on the centroidal axis of the member. If these conditions are not practicable, provision shall be made for the effects of the resulting eccentricity, as given in Clause 11.2.6.

11.2.1.6 Maximum number of fasteners Lines of fasteners connecting axially loaded members shall not exceed 6 in number or 15d in extension, where d is the fastener diameter, without a demonstration that the anticipated strength will be realized.

11.2.2 Fastener spacings 11.2.2.1 Minimum spacing The centres of fasteners shall be not less than 1.25d from the edge parallel to the direction of loading, nor less than 1.5d from end edges towards which the load is directed. The distance between centres of fasteners shall be not less than 2.5d, where d is the fastener diameter. See Clause 11.2.5 for the influence of fastener spacing on resistance.

11.2.2.2 Maximum spacing For widely spaced fasteners, the factored compressive resistance, Cr , may be controlled by local buckling of the plate between fasteners. The local buckling stress, Fc , shall be obtained from Clause 9.3.3, using one of the following formulas to determine the value of λ: (a) where the fasteners are arranged in a rectangular pattern, the slenderness, λ, shall be the largest of the values given by the following formulas: (i) λ = 1.7 s/t; (ii) λ = 1.3 g/t; and (iii) λ = 3 e/t; and (b) where the fasteners are in staggered rows, the slenderness, λ, for the plate between fasteners shall be given by one of the following: (i) when g/s >1, λ = (1.3 + 0.6 s/g) g/t (ii) when g/s ≤ 1, λ = (1.7 + 0.2 g/s) s/t where s = row spacing measured in the direction of the load t = plate thickness g = row spacing measured perpendicular to the direction of the load e = edge distance

11.2.3 Bolts and rivets in shear and/or tension 11.2.3.1 Shear resistance For bolts and solid rivets, the factored shear resistance, Vr , of the fastener in bearing-type connections shall be given by Vr = φf 0.6 m A Fu where φf = fastener resistance factor

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Strength design in aluminum

m = number of shear planes A = nominal cross-sectional area of the fastener Fu = ultimate strength of the fastener material

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If the bolt threads are in a shear plane, the value of Vr shall be multiplied by 0.75.

11.2.3.2 Tensile resistance The factored tensile resistance of a bolt, Tr , shall be the lesser of the values given by (a) Tr = φf 0.75 A Fu ; and (b) Tr = φf A Fy where φf = fastener resistance factor A = cross-sectional area of the bolt based on the nominal diameter Fu = ultimate strength of the bolt material Fy = yield strength of the bolt material Rivets are not commonly used in tension.

11.2.3.3 Combined shear and tension For a bolt subject to both shear and tension, exclusive of tension due to tightening, the reduced factored tensile resistance, T’r , shall be given by the following:

T ′r = 1.25 Tr − kVf ≤ Tr where Tr = factored tensile resistance given in Clause 11.2.3.2 k = 1.8, or 1.4 when the bolt thread is excluded from the shear plane Vf = factored shear load on the bolt

11.2.4 Bolts and rivets in bearing 11.2.4.1 Bearing strength The factored bearing resistance, Br , of the connected material for each loaded fastener shall be the lesser of the values given by the following formulas: (a) Br = φu etFu; and (b) Br = φu 2dtFu where φu = ultimate resistance factor e = perpendicular distance from the hole centre to the end edge in the direction of the loading (not less than 1.5d ) t = plate thickness Fu = ultimate strength of the connected material d = fastener diameter

11.2.4.2 Lap joints For unrestrained lap joints in tension, the factored bearing resistance, Br , shall be the lesser of the values given by the following formulas: (a) Br = φu(t1 + t2) e Fu /4; and (b) Br = φu(t1 + t2) d Fu /2 ≤ φu 2dt1Fu where φu = ultimate resistance factor

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t1, t2 = e = Fu = d =

© Canadian Standards Association

thicknesses of the plates, t1 < t2 distance from the hole centre to the end edge, but not less than 1.5d ultimate tensile strength of the connected material fastener diameter

11.2.4.3 Oblique end edges Where the end edge is oblique to the line of action of the tension force (see Figure C23 of the Commentary), the factored bearing resistance, Br , for a single bolt shall be the lesser of the values given by the following formulas: (a) Br = φu [e + (e – d) cos2 θ] tFu; and (b) Br = φu 2 dtFu where φu = ultimate resistance factor e = perpendicular distance from the hole centre to the end edge d = fastener diameter θ = angle made by the end edge with the direction of the force t = plate thickness Fu = ultimate strength of the connected material

11.2.5 Tear-out of bolt and rivet groups (block shear) 11.2.5.1 Tension: Rectangular patterns For a group of two or more fasteners in a rectangular pattern (see Figure C24(a) of the Commentary) resisting a force directed towards the edge, the factored bearing resistance, Rb , of the group of fasteners shall be the lesser of the values given by the following formulas: (a) Rb = φu [(m – 1)(g – do ) + (n – 1)(s – do ) + e]tFu; and (b) Rb = φu 2 NdtFu where φu = ultimate resistance factor m = number of fasteners in the first transverse row g = fastener spacing measured perpendicular to the direction of the force do = hole diameter n = number of transverse rows of fasteners s = fastener spacing measured in the direction of the force e = edge distance in the direction of force for the first row, but not less than 1.5d = 2d, when e > 2d t = plate thickness Fu = ultimate strength of the connected material N = total number of fasteners d = fastener diameter

11.2.5.2 Tension: Trapezoidal patterns For a triangular or trapezoidal group of fasteners in a staggered pattern (see Figure C24(b) of the Commentary) resisting a force directed towards the edge, the factored bearing resistance, Rb , of the group shall be the lesser of the values given by the following formulas: (a) Rb = φu [2 (m – 1)(g – do + s2/4g) + e] tFu; and (b) Rb = φu 2NdtFu where φu = ultimate resistance factor m = number of fasteners in the first transverse row

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g do s e

= = = = = t = Fu = N = d =

Strength design in aluminum

fastener spacing measured perpendicular to the direction of the force hole diameter fastener spacing measured in the direction of the force edge distance in the direction of force for the first row, but not less than 1.5d 2d, when e > 2d plate thickness ultimate strength of the connected material total number of fasteners fastener diameter

Clause 7.3 gives the net area for tensile failure.

11.2.5.3 Block shear in webs At end reactions resisted by fasteners in the webs of beams (see Figure C24(c) of the Commentary), the factored resistance provided by block shear shall be given by Rb = φu {0.5 [(m – 1)(g – do) +e1] + (n – 1)(s – do) + e2 }tFu ≤ φu 0.5 (h – mdo)t Fu where φu = ultimate resistance factor m = number of fasteners in the first vertical row g = fastener spacing measured in the direction of the force do = hole diameter e1 = edge distance in the direction of force for the first row (not less than 1.5d) n = number of vertical rows of fasteners s = fastener spacing measured normal to the direction of the force e2 = end edge distance t = thickness Fu = ultimate strength of the connected material h = web depth

11.2.5.4 Tear-out of groups subjected to torque For a group of three or more equally spaced fasteners that lie on a circle (see Figure C24(d) of the Commentary) designed to resist a torque, Clause 11.2.6 shall apply but, additionally, the torque shall not exceed the value given by Mr = φu 0.5 NR(s – do)tFu where φu = ultimate resistance factor N = number of fasteners R = radius of the circle measured to the centre of the holes s = centre-to-centre distance between adjacent fasteners on the circle do = hole diameter t = plate thickness Fu = ultimate strength of the connected material

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11.2.6 Eccentrically loaded fastener groups

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11.2.6.1 Highest fastener force (elastic) For a group of equal strength fasteners subjected to a factored load, Pf , applied at an eccentricity, e, from the centroid of the fastener group (see Figure C25 of the Commentary), the procedure shall be as follows: (a) Determine the position of the centroid of the fastener group. (b) Determine the normal distance, e, from the centroid to the line of action of the applied force. (c) Determine the distance, c, from the centroid to the centre about which the bolt group rotates under the action of the eccentric load. This distance is measured along the line through the centroid of the fastener group perpendicular to the line of action of the applied force, on the side opposite to that of the applied force, and shall be given by

c=∑

xi2 + y i2 Ne

where xi, yi = the coordinates of the i th fastener relative to the centroid N = number of fasteners e = eccentricity of the applied load (d) The highest factored force, Rf , on a fastener shall be given by Rf = (Pf /N)(dm /c) where Pf N dm c

= = = =

factored load number of fasteners distance from the centre of rotation to the farthest fastener distance from the centroid to the centre of rotation

11.2.6.2 Factored resistance (fully plastic) The centre of rotation shall be determined as given in Clause 11.2.6.1. If the calculated centre of rotation falls near a fastener, the location of the fastener may be taken to be the centre of rotation. The limiting factored applied load, Pr , shall be given by

Pr = Rr

( ∑ di ) (e + c )

where Rr = factored resistance of each fastener di = distance from the ith fastener to the centre of rotation e = eccentricity of the applied force from the centroid of the fastener group c = the distance from the centroid to the centre of rotation

11.3 Welded connections 11.3.1 General 11.3.1.1 Because of the annealing of the metal adjacent to a welded joint, the strengths of both the weld metal and the heat-affected base metal shall be considered.

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Strength design in aluminum

11.3.1.2 Requirements for the maximum and minimum sizes of welds and for the effective throat dimensions of welds shall be as specified in CSA W59.2.

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11.3.2 Butt welds 11.3.2.1 Tension The factored tensile resistance, Tr , of a member containing a full penetration butt weld shall be the least of the values given by the formulas in Clause 9.2.1(a), (c), and (d). Note: Partial penetration butt welds are not recommended for joints carrying calculated forces.

11.3.2.2 Compression The factored compressive resistance, Cr , of a full penetration transverse butt weld that is fully constrained against buckling shall be the lesser of the values given by the following formulas: (a) Cr = φy A Fy; and (b) Cr = φu A Fwu where φy = resistance factor on the yield strength A = cross-sectional area of the connected material Fy = yield strength of the base metal φu = ultimate resistance factor Fwu = ultimate tensile strength of the welded joint (see Table 3)

11.3.2.3 Shear The factored shear resistance, Vr , of a full penetration butt weld shall be the least of the values given by the following formulas: (a) Vr = φy 0.6 A Fy; and (b) Vr = φu 0.6 A Fwu where φy = resistance factor on the yield strength A = nominal cross-sectional area of the butt joint Fy = yield strength of the base metal φu = ultimate resistance factor Fwu = ultimate tensile strength of the welded joint (see Table 3)

11.3.3 Fillet welds 11.3.3.1 Concentrically loaded fillet welds 11.3.3.1.1 The factored resistance per unit length, vr , of a concentrically loaded fillet weld shall be given by the formula vr = φf kaFwu For forces applied at an inclination to the weld direction, the components vx , vy , and vz of the factored force per unit length shall be such that (vx/0.6)2 + (vy/0.7)2 + (vz/0.8)2 ≤ (φf aFwu)2 where φf = fastener resistance factor k = a factor related to the direction of the applied force (see Figure C26 of the Commentary)

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a

© Canadian Standards Association

= 0.6 for direction x, along the axis of the fillet weld, i.e., in simple shear = 0.7 for direction y, perpendicular to the plate to which the connection is made = 0.8 for direction z, perpendicular to weld in the plane of the plate = weld throat (distance through a fillet weld, usually taken as s/ 2 ) where s = weld size

Fwu = ultimate strength of the weld bead (see Table 3)

11.3.3.1.2 The effective length, Lm , of a discontinuous fillet weld shall be given by Lm = L – 2a where L = the actual length of the fillet weld a = weld throat

11.3.3.2 Eccentrically loaded fillet welds 11.3.3.2.1 Moment in the X-Z plane For welds subjected to a factored eccentric load, Pf , in the X-Z plane (see Figure C27(a) of the Commentary), the procedure to determine the resistance shall be as follows: (a) In order to establish the maximum force per unit length, vf , in the weld, in the elastic range (to be used, for example, when predicting fatigue life), the procedure shall be as follows: (i) Determine the position of the centroid of the weld pattern. (ii) Determine the perpendicular distance, e, from the centroid to the line of action of the applied force. (iii) Determine the total length, H, of the median line of the weld. (iv) Calculate the polar moment of inertia, Ip = Ix+Iy, of the weld pattern about the centroid, using a constant weld throat, a. (v) Determine the distance from the centroid to the centre about which the weld rotates under the action of the eccentric load. This distance, c, is measured along the line through the centroid of the fastener group perpendicular to the line of action of the applied force, on the side opposite to that of the applied force, and is given by

c = rp2 / e where rp = (Ip /Ha)1/2 where Ha = total throat area of the weld e = eccentricity of the applied load (vi) The maximum force per unit length, vf , in the elastic range, shall then be given by vf = (Pf /H )(dm /c) where Pf = factored eccentric load H = total length of weld dm = the distance from the centre of rotation to the farthest point of the weld c = distance from the centroid to the centre of rotation of the welded joint (b) In order to establish the factored resistance, Pr , the procedure shall be as follows: (i) Use the procedure in Item (a) to determine the centre of rotation.

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(ii) Divide the weld into convenient straight elements on each side of the line passing through the centre of rotation and the centroid. (iii) Determine the distances, di , from the centre of rotation to the midpoints of the elements. (iv) The factored resistance, Pr , shall then be given by Pr = Σ(Li di vr)/(e + c) where Li = length of the ith element di = distance from the centre of rotation to the midpoint of the ith element vr = the factored resistance per unit length of the weld, obtained in Clause 11.3.3.1 using k = 0.6 e = eccentricity of the applied load c = distance from the centroid to the centre of rotation

11.3.3.2.2 Moment in the X-Y plane For double fillet welds subjected to eccentric loading in the X-Y plane (see Figure C27(b) of the Commentary), the weld shall be continued around the edges of the plate. The factored resistance, Pr , shall be given by

Pr =

ncr ntr L2 ≤ nsr L 2e (ncr + ntr )

where ncr = factored compressive resistance per unit length of the welded plate, taken as the lesser of the values given by (a) φu t Fwu; and (b) φy t Fy ntr = factored tensile resistance per unit length of the welded joint; taken as the least of the values given by: (a) φu t Fwu; (b) φr t Fy; and (c) φf a k’ Fwu where φu = ultimate resistance factor t = plate thickness Fwu = ultimate tensile strength in the heat-affected zone (using the tensile value from Table 3) φy = resistance factor on the yield strength Fy = yield strength of the base metal (see Table 2) φf = fastener resistance factor a = weld throat k’ = 1.4 [1 – (nx/nsr)2]1/2 where nx = shear force per unit length of the welded joint = Pr /L nsr = factored shear resistance per unit length of the welded joint, taken as the least of the values given by the following expressions: (a) φu t (0.6 Fwu); (b) φy t (0.6 Fy); and (c) φf 2a (0.6 Fwu) L = length of weld (plate length) e = eccentricity

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The value of k’ is determined by trial and error, as the value of nx is not known initially. In general design, when the influence of the shear force is small, and the fillet weld controls the strength, the factored moment resistance, Mr , may be given by

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Mr = φf (aL2/3)Fwu

11.3.3.2.3 Moment in the Y-Z plane For double fillet welds bending in the Y-Z plane (see Figure C27(c) in the Commentary), the factored resistance, Mr , shall be given by Mr = φf 0.7 a L (t + 0.7 a)Fwu where φf = fastener resistance factor a = weld throat L = length of fillet welded joint t = plate thickness Fwu = ultimate tensile strength of the weld bead (see Table 3) Single fillet welds shall not be subjected to calculated bending forces in the Y-Z plane.

11.3.4 Flare groove welds Where welds are to be made between rounded surfaces, as between round bars and at the corners of formed shapes, the procedures used shall have been demonstrated to give the required penetration and throat thickness. The requirements may be shown to be satisfied by measurement of the weld throat or by load tests. If measurement is made, the throat shall exceed that required for the design strength by 3 mm. If tests are made, there shall be at least three specimens made consecutively using the same procedure. The lowest value obtained shall be used as the characteristic strength.

11.3.5 Slot and plug welds A connection made by a fillet weld along the inside edge of a hole or slot is acceptable if the radii of the corners are not less than the thickness of the plate plus 5 mm. The weld shall extend around the full length of the inside edge of the hole. The length of the weld shall be taken as the length of the centroidal axis of the fillet. Holes and slots completely filled with weld metal shall not be used to carry calculated forces.

11.3.6 Influence of welds, heat-affected zone The heat of welding shall be assumed to affect a width of the base metal extending 25 mm in each direction from the centre of the weld. The properties of this heat-affected zone are given in Tables 2 and 3 for some common alloys. The influence of welding on overall strength is treated in Clauses 7.4.2, 7.5.2, 9.2, and 9.3.2.

11.3.7 Stud welds Aluminum stud welds shall be treated as fasteners, and the factored resistance shall be given by (a) tension — Rr = φf (π d 2/4)Fwu; and (b) shear — Rr = φf (π d 2/4)(0.6Fwu) where φf = fastener resistance factor d = stud diameter Fwu = ultimate tensile strength of the welded stud

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12 Fatigue resistance

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Note: Members and joints designed in accordance with the static strength provisions of this Standard and constructed so as to be free from sharp re-entrant corners and other severe stress raisers can be expected to withstand, without visible fatigue cracking, at least 100 000 repetitions of the maximum unfactored live load if not welded, and 20 000 if welded.

12.1 Load cycles of constant amplitude Where the cyclic stress ranges are of constant amplitude, the fatigue life for the categories of members and joints listed in Figure 1 shall be determined from Figure 2, with the stress range at 5 × 106 cycles used as an endurance limit. (Stress range is twice the amplitude.)

12.2 Known load spectra Where the cyclic stress ranges and the number of repetitions are variable but known, the fatigue life for the categories of members and joints listed in Figure 1 shall be determined on the basis that Σ(n/N) < 1 where n = the number of cycles at each stress range N = the number of cycles given in Figure 2 for that stress range If all the applied stress ranges are less than the stress range for 5 × 106 cycles, then the stress range for 5 × 106 cycles may be considered to be the endurance limit.

12.3 Unknown load spectra Where the cyclic stress ranges and their number of repetitions are variable and not sufficiently well known to formulate a spectrum, the stress range that is estimated to be exceeded 106 times shall not exceed the value given in Figure 2 for 5 × 106 cycles for the appropriate category of the member or joint.

12.4 Low cycle stress limit For any member or joint, the maximum applied stress, calculated using the net section where applicable, shall not exceed the factored tensile resistance defined in Clause 9.2.

12.5 Severe stress raisers For members and joints that, except for obvious stress raisers, would be in Category A of Figure 1, the nominal stress range shall be multiplied by the appropriate fatigue notch factor (effective stress concentration factor) and used in conjunction with Category A of Figure 2. If the appropriate notch factor is not available, the theoretical stress concentration factor may be used.

12.6 Thick material For material thicker than 16 mm, the stress range given in Figure 2 shall be multiplied by the factor (16/t)1/4 where t = material thickness, mm

12.7 Load factor

For fatigue, a load factor, α, equal to 1.0 shall be used.

12.8 Resistance factors 12.8.1

For members in which failure would not lead to collapse, a resistance factor, φ, equal to 1.0 shall be used.

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12.8.2

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For members in which failure would lead to collapse, the values of the range of stresses allowed in Clauses 12.2 and 12.3 shall be multiplied by a resistance factor, φ, equal to 0.75.

13 Tests 13.1 General 13.1.1 The adequacy of joints, members, and assemblies may be demonstrated by tests instead of design calculations, subject to approval by the engineer.

13.1.2 Only static load tests are considered in Clause 13. Dynamic loads tests and fatigue tests shall be carried out and reported according to accepted engineering procedures.

13.1.3 An application that is controlled by a regulatory authority may require proof of compliance by means of specified tests. If these tests are mandatory, then they shall take precedence over the requirements of Clause 13.

13.2 Test methods 13.2.1 Tests shall be carried out on representative joints, members, and assemblies that are identical in geometric and material features to those to be used in the intended application.

13.2.2 Where possible, tests shall be conducted according to accepted procedures, such as provided by an appropriate ASTM Standard to suit the requirements of the owner.

13.2.3 Tests shall be fully documented so that they may be independently reproduced. Documentation shall include values such as loading rate and load duration.

13.2.4 Confirmatory tests demonstrate that the strength and serviceability under the specified loads are acceptable.

13.2.5 Performance tests determine the maximum load that can be carried prior to the attainment of an agreed level of distress.

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13.3 Test procedures

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13.3.1 Confirmatory tests 13.3.1.1 Serviceability For confirmatory tests of serviceability under the action of the service loads, the specified limit state shall not be exceeded.

13.3.1.2 Ultimate limit state For confirmatory tests to determine if the ultimate strength is adequate, the applied force shall be that due to the factored loads, as specified by Clause 5.4.4, divided by the appropriate resistance factor from Clause 5.5.

13.3.2 Performance tests 13.3.2.1 Characteristic resistance For performance tests, the highest load that the joint, member, or assembly can sustain without rupture, collapse, or excessive deformation shall be measured. The characteristic resistance shall be taken as one of the following, as appropriate: (a) If four or less items are tested, the characteristic resistance shall be the lesser of (i) the mean of the ultimate test loads multiplied by 0.9; and (ii) the lowest ultimate test load achieved. (b) If more than four items are tested, the characteristic resistance shall be the mean of the ultimate test loads minus one standard deviation. “Excessive deformation” shall be a measurable distortion that is agreed by the purchaser or the purchaser’s representative.

13.3.2.2 Adjustment for variation in yield and ultimate strength and dimensions If the yield or ultimate strength of the material of the test item exceeds the specified value, the measured test loads shall be multiplied by the following: (a) for the gross area in tension or bending, ⎛ A ⎞ Fy ⎜ ′⎟ ⎝ A ⎠ Fy′ and (b) for the net area in tension or bending, ⎛ A ⎞ Fu ⎜ ′⎟ ⎝ A ⎠ Fu′ For failure due to buckling, the test load shall be multiplied by

(

Fy / Fy′ + ( λ/1.5) 1 − Fy / Fy′

)

≤ 1

where A = nominal area A’ = actual area Fy = specified yield strength F’y = measured yield strength Fu = specified ultimate strength F’u = measured ultimate strength λ = normalized slenderness

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14 Fabrication 14.1 General Licensed for/Autorisé à Reid Costley, Cascade Engineering Group, Sold by/vendu par CSA on/le 10/27/2009. Single user license only. Storage, distribution or use on network prohibited./Permis d'utilisateur simple seulement. Le stockage, la distribution ou l'utilisation sur le réseau est interdit.

Fabrication practices shall follow those for steel, as in CAN/CSA-S16, except as otherwise modified by this Standard.

14.2 Tolerances 14.2.1 Fabrication tolerances shall be in accordance with CAN/CSA-S16, except as stated in Clause 14.2.2.

14.2.2 The tolerance on the end distance of bolt holes shall be – 0, +2 mm.

14.3 Layout and marking 14.3.1 Layout As the linear coefficient of expansion of aluminum is about twice that of steel, a temperature correction may be necessary in the layout of critical dimensions when using steel tapes in areas of unusually high or low temperatures. The correction shall be given by σ = 0.000012 L(20 – T) where L = length, mm T = ambient temperature, °C

14.3.2 Marking Hole centres may be centre-punched and cut-off lines may be punched or scribed. Centre-punching and scribing shall not be used where such marks would remain on fabricated material. Stamped erection marks may be used, but the impression shall be free from sharp corners and the mark shall not be located in highly stressed areas of the component. Paint or ink erection marks shall be used where fatigue is a consideration.

14.4 Forming 14.4.1 In general, forming of aluminum shall be carried out at room temperature. Should hot forming be unavoidable, the procedures shall conform to the requirements of CSA W59.2, and the post-forming mechanical properties shall be checked using hardness tests.

14.4.2 Bends shall be smooth, without sharp kinks. Cracks shall be cause for rejection if the crack lies in a zone that is stressed in service.

14.4.3 If straightening or flattening is carried out, it shall be done by a process and in a manner that does not injure the material.

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Material thicker than 12 mm shall not be sheared.

14.5.2 When arc cutting is used in cases where fatigue resistance is important, at least 1 mm shall be mechanically removed from the edges.

14.5.3 Gas or flame cutting shall not be used.

14.5.4 If re-entrant cuts are used, they shall be filleted by drilling prior to cutting.

14.6 Mechanical connections 14.6.1 Rivets 14.6.1.1 Rivet diameters should normally be between t and 3t, where t is the thickness of the material.

14.6.1.2 The hole size should not exceed the rivet diameter by more than 0.8 mm for rivets with less than a 12 mm diameter, or 1.2 mm for larger rivets. Tighter fits may be required for special rivets. A larger clearance may be permitted if it can be shown that, after driving, the rivet fills the hole.

14.6.1.3 Rivets may be closed by a hammer or squeeze riveter. The head may be of any form that ensures that the upset rivet fills the hole and that no cracks develop in the head.

14.6.2 Bolts 14.6.2.1 Holes for bolted joints not subject to fatigue may be punched to the finished size in material 12 mm or less in thickness.

14.6.2.2 Material greater than 12 mm thick shall not be punched to finished size, but may be punched under size and reamed to size.

14.6.2.3 The hole size shall not exceed the bolt diameter by more than 1 mm for diameters less than 12 mm, and 1.5 mm for larger diameters.

14.6.2.4 Joints designed for rigidity under service loads or for fatigue conditions shall use pre-loaded bolts, turned-and-fitted bolts, or proprietary fasteners that prevent relative movement at the joint.

14.6.2.5 Holes for bolted joints in fatigue service shall be drilled or punched under size and reamed.

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14.7 Welding 14.7.1 Licensed for/Autorisé à Reid Costley, Cascade Engineering Group, Sold by/vendu par CSA on/le 10/27/2009. Single user license only. Storage, distribution or use on network prohibited./Permis d'utilisateur simple seulement. Le stockage, la distribution ou l'utilisation sur le réseau est interdit.

All welding shall be with an inert-gas-shielded arc process, such as tungsten inert gas (TIG), metal inert gas (MIG), or plasma arc, conforming to the requirements of CSA W59.2.

14.7.2 Welding operators and procedures shall be qualified according to CSA W47.2.

14.7.3 Welds between different alloys shall be carried out using an appropriate filler alloy as specified in CSA W59.2.

14.7.4 Stud welds shall satisfy the requirements of CSA W59.2. They may be installed using drawn arc, capacity discharge, or fillet welds. The fillet weld throat shall not be less than one-third of the stud diameter.

15 Protection against corrosion 15.1 General Surfaces shall be protected where (a) the aluminum alloy parts are in contact with, or are fastened to, bare steel, wood, concrete, or other dissimilar materials in moist conditions; or (b) the structures are to be exposed to an aggressive environment. Note: Structures of the alloys covered by this Standard are not ordinarily painted.

15.2 Contact with dissimilar materials Where aluminum alloys are in contact with, or are fastened to, dissimilar materials in the presence of moisture, the aluminum shall be kept from direct contact with the dissimilar material as follows: (a) Aluminum surfaces to be placed against bare steel shall be given one coat of primer and a layer of a suitable non-hardening joint compound capable of excluding moisture from the joint during prolonged service. Bare steel surfaces to be placed in contact with aluminum shall be painted with good quality priming paint, such as zinc chromate primer, followed by one coat of paint containing 0.2 kg of aluminum paste pigment per litre. Stainless, aluminized, hot-dip galvanized, or electro-galvanized steels placed in contact with aluminum need not be painted. (b) Aluminum surfaces to be in direct contact with wood, fibreboard, or other porous material that is expected to absorb water shall be given a heavy coat of alkali-resistant bituminous paint or other coating providing equivalent protection before installation. Aluminum surfaces in contact with concrete or masonry shall be similarly protected where moisture is present and corrosive materials can be trapped between the surfaces. Note: Where wood and aluminum are to be in permanent unprotected contact, the alloy and the species of wood should be selected taking into account the potential corrosion problems.

(c) Aluminum embedded in concrete shall be given a coating of alkali-resistant bituminous paint or shall be wrapped with a suitable plastic tape that is applied to provide protection at the overlaps. Aluminum shall not be embedded in concrete to which corrosive components, such as chlorides, have been added if the aluminum will be electrically connected to steel. (d) Water that comes in contact with aluminum after first running over a heavy metal, such as copper, may contain trace quantities of the dissimilar metal or its corrosion product that will cause corrosion of the aluminum. Protection shall be obtained by painting or plastic coating the dissimilar metal or by designing the structure so that the drainage from the dissimilar metal is diverted away from the aluminum.

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(e) Commercially available prepainted aluminum generally does not need additional painting, even when it is in contact with other materials such as wood, concrete, or steel.

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15.3 Cleaning and treatment of metal surfaces Prior to field painting of structures, all surfaces to be painted shall be cleaned immediately before painting by a method that will remove all dirt, oil, grease, chips, and other foreign substances. Exposed metal surfaces shall be cleaned with a suitable chemical cleaner such as a solution of phosphoric acid and organic solvents. If the metal is more than 3 mm thick, sandblasting may be used.

15.4 Stress corrosion Where a component is subjected to a permanent stress in an aggressive environment, adequate protection against stress corrosion shall be provided for zinc-bearing alloys and for the higher magnesium-bearing alloys, such as 5083, where a temperature greater than 60 °C may be encountered. Note: In these situations it is important to recognize that forming and machining operations and welding create residual stresses and that bolting creates permanent local stresses. The following design features should be avoided: rapid changes of sectional area; small bend radii for plate; small fillet radii; drive fits for hardware; and massive sections machined to thin areas.

Table 1 Typical aluminum alloys and products to which this Standard is applicable (See Clauses 4.1, 4.2.2, 4.3, and C4.2.)

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Sheet, plate

Bar, rod, wire, extruded shapes

Drawn tube

Bolts

Rivets

3003 3004 5052 5083 5086 5454 5754 6061

— 6061 6063 6082 6105 6351 7004 7020

6061 6063 6351 — — — — —

6061 — — — — — — —

1100 6053 — — — — — —

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Table 2 Tensile properties for typical aluminum alloys and products used in buildings Licensed for/Autorisé à Reid Costley, Cascade Engineering Group, Sold by/vendu par CSA on/le 10/27/2009. Single user license only. Storage, distribution or use on network prohibited./Permis d'utilisateur simple seulement. Le stockage, la distribution ou l'utilisation sur le réseau est interdit.

(See Clauses 4.2.2, 11.3.3.2.2, 11.3.6, C4.2, and C11.3.) Strength, MPa Base metal

Welded (HAZ)

Ultimate Fu

Yield Fy

Yield Fwy

115 140 165 160 220 240 190 235 275 305 250 275 220 250

70 115 145 60 170 190 110 180 125 215 125 195 125 180

35 35 35 60 60 60 70 70 125 125 100 100 90 90

Rivet wire 6053-T61

205

—

—

Extrusions, sheet, plate, and drawn tube 6061-T6

260

240

110

Extrusions and drawn tube 6063-T54 6063-T5 6063-T6

230 150 205

205 110 170

70 70 70

Extrusions 6351-T6

290

255

110

Product and alloy Work-hardened Sheet and plate 3003-H112 3003-H14 3003-H16 3004-H112 3004-H34 3004-H36 5052-H112 5052-H34 5083-H112 5083-H321 5086-H112 5086-H32 5454-H112 5454-H32 Heat-treated

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Table 3 Mechanical properties for typical aluminum alloy weld beads

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(See Clauses 4.2.3, 9.2.1, 9.6.1.2, 11.3.2.2, 11.3.2.3, 11.3.3.1.1, 11.3.3.2.2, 11.3.3.2.3, 11.3.6, C11.3, and C11.3.2.1.) Ultimate tensile strength, Fwu , MPa Filler alloy 4043 5356

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Base metal alloy 3003

3004

5052

5083

5086

5454

6061

6063

6351

100 100

150 150

— 170

— 260

— 235

— 220

170 190

120 120

170 190

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Table 4 Effective length factors, K

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(See Clauses 9.4.2.1, 9.7.1.3, 9.7.3.3, 9.7.4.2, 9.7.5.2, 9.8.2, and C9.4.2.) Kv* (Single angles) Kx

Ky

1 bolt

2 bolt

AB

k

1

(1+2k) 3

(1+2k) 3

AB

1

1

0.8

0.7

AB

0.5

0.5

0.45

0.4

AB

0.33

0.43

0.33

0.33

AB

0.25

0.35

0.25

0.25

AB

0.5

1

0.5

0.45

AB

0.5

1

0.5

0.45

AB

0.45

0.5

0.4

0.35

Member Y

V

L

X

A

B

kL

A L L

AC B

C

A L

C

T

T

C

B A C

T

T

C

B A C

T

T

C

B A L

B A C

B T

A C

L T

B C = Compression, T = Tension, T = C

62

* = See Clause 9.7.4.2

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© Canadian Standards Association

Strength design in aluminum

Category A Base metal with rolled or as-extruded surfaces or with equivalent machined surfaces, and with no obvious stress raisers.

Category B Full-penetration machine butt welds made from both sides, with light bead reinforcement and a bead toe tangent angle of not more than 30°; base metal net section stress in riveted or bolted double lap joints. Note: Higher properties as shown under B* (see Figure 2) may be used for high-quality butt welds made from both sides when reinforcement is dressed flush and weld soundness is established by full non-destructive inspection. (Continued)

Figure 1 Categories of members and joints for fatigue design (See Clauses 12.1, 12.2, 12.5, and C12.1.)

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Category C Longitudinally stressed, continuous, longitudinal fillet welds made without interruptions during welding; manual longitudinal or transverse butt welds made from both sides, with normal or heavy bead reinforcement. The transverse butt weld bead toe tangent angle shall not exceed 50°. Transverse butt welds with greater bead toe tangent angles shall be treated as Category D.

Category D Full-penetration butt welds made from one side, with or without permanent backing; base metal at load-carrying or non-load-carrying transverse fillet welds; longitudinally stressed, continuous, longitudinal fillet welds containing stops and starts; members with T-joints welded from two sides with full penetration. (Continued)

Figure 1 (Continued)

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Category E Members with T-joints welded from both sides without full penetration or from one side with full penetration; members with intermittent web-to-flange fillet welds; and members with butt or fillet welds fixing non-loaded longitudinal lugs; self-backed butt welds stressed transversely to the weld.

Category F Average stress on the throat of longitudinal or transverse fillet welds; stress in the base metal at the end of load-carrying longitudinal fillet welds.

Figure 1 (Concluded)

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Stress range, N/mm 2

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500

210

175

145

120 105

104 A

100 B*

85 75

50 C

D

10

105 106 100 80

B 75

50

5 x 10 6

60

36 37

20 E 24 27

F 16 18

10 12

7

5 107 108

Stress cycles

Figure 2 Fatigue stress ranges for design of aluminum components and connections

(See Clauses 12.1–12.3, 12.5, 12.6, and C12.1 and Figure 1.)

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Annex A (informative) Applications other than buildings Note: This Annex is not a mandatory part of this Standard.

A.1 General Aluminum may be used in many applications that are not associated with buildings and are not therefore regulated and controlled by the National Building Code of Canada. Some of these applications are regulated by federal or provincial authorities, while others are basically uncontrolled. Even in the latter case, design for strength is normally made, because of both practical considerations and the possibility of liability litigation. This Annex lists some applications, references some related CSA Standards, and refers to the regulatory authority. Where control is under provincial jurisdiction some investigation is needed, as the responsibilities and names of ministries in one province may differ from those in another. If a product for a non-building application requires licensing by an authority, it can be assumed that there are design criteria required for that product. A product used in the workplace that, if failure occurs, could cause injury is normally controlled by the ministry that regulates labour and working conditions. Criteria set by regulations other than the National Building Code of Canada may require the application of safety factors used with either yield or ultimate stresses. This may not be compatible with the limit state design methods in this Standard. The criteria may require dynamic or fatigue considerations to be the critical design conditions, particularly in applications involving movement of the product. Many authorities require testing as part of the qualification of the design; the passing of such tests is mandatory, and cannot be superseded by calculations. Many of these tests measure survivability during accident conditions, which is not part of normal design criteria. The following list of applications, authorities, and standards apply to the following four categories: (a) free-standing structures; (b) vehicles; (c) material moving and containment; and (d) miscellaneous uses.

A.2 Free-standing structures Free-standing structures usually require permits or licenses from the local authority. The licences will usually define the regulating criteria. The structures include (a) pylons, towers, and antennas (see CSA S37); (b) stairs and ramps; (c) amusement rides (see CAN/CSA-Z267); (d) bridges (see CAN/CSA-S6 and the provincial transport or highway ministry); (e) access scaffolding for construction purposes (see CAN/CSA-S269.2); and (f) falsework for construction purposes (see CSA S269.1).

A.3 Vehicles A.3.1 General Most vehicles, because of their interprovincial use, are required to meet standards and requirements set by federal agencies. The design of aircraft and ships is required to meet international requirements.

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A.3.2 Road vehicles The design of road vehicles is required to comply with the federal Motor Vehicle Safety Act and the Motor Vehicle Safety Regulations. The latter set standards and test requirements. Some of these regulations with application to aluminum structures are (a) 210 — seat belt attachments; (b) 214 — side door strength; (c) 215 — bumpers; and (d) 216 — roof intrusion protection. Many standards and methods are borrowed from ASTM, ANSI, ASME, and SAE.

A.3.3 Rail vehicles Rail vehicles are required to comply with the Railway Safety Act and its related regulations and orders. American standards, such as those of the American Association of Railroads (AAR), are in general use.

A.3.4 Watercraft These are regulated under the Canada Shipping Act, which is administered by the Canadian Coast Guard.

A.3.5 Recreational vehicles and mobile homes When these are to be used on public highways, they are required to meet the road vehicle criteria. Mobile homes may also be subject to the regulations of local authorities (see CSA Standards in the CAN/CSA-Z240 MH Series and CAN/CSA-Z240 RV Series).

A.4 Material moving and containment A.4.1 General The authority responsible for this category is difficult to determine. Movement and containment of dangerous materials is regulated federally. Movement of items vertically by crane, hoists, or lifts is under the control of provincial authorities. The regulations for moving people are different from those for moving materials.

A.4.2 Packaging and containment Only materials having dangerous properties have mandatory packaging requirements. These are specified in the Transportation of Dangerous Goods (TDG) Regulations. The nine classes of goods covered are specified by the Class numbers as follows: (a) Class 1 — explosives; (b) Class 2 — compressed gases; (c) Class 3 — flammable liquids; (d) Class 4 — flammable solids, spontaneously combustible; (e) Class 5 — oxidizers and organic peroxides; (f) Class 6 — poisonous and infective substances; (g) Class 7 — radioactive material; (h) Class 8 — corrosives; and (i) Class 9 — miscellaneous dangerous goods. With the exception of radioactive materials, packaging is governed at present by the TDG regulations. Radioactive materials are to be packaged and contained in accordance with the Packaging and Transport of Nuclear Substance Regulations (PTNSR), which fall under the Nuclear Safety and Control Act and which conform to internationally agreed-upon standards.

A.4.3 Freight containers Freight containers are required to meet international standards and modal requirements for rail and ships.

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A.4.4 Cranes and hoists for materials Cranes and hoists for materials are generally regulated by the provincial labour authority. For mobile cranes, see CAN/CSA-Z150, and for tower cranes, see CAN/CSA-Z256.

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A.4.5 Lifts and equipment for people Lifts and equipment for people are regulated by the provincial labour authorities and sometimes by municipal regulations. For manlifts, see CAN/CSA-B311; for elevated work platforms, see CAN/CSA-B354.2; for portable ladders, see CSA CAN3-Z11; and for vehicle-mounted aerial devices, see CAN/CSA-C225.

A.4.6 Hoppers, bins, pipes, and conduits There is no general authority for hoppers, bins, pipes, and conduits; therefore, they may be considered as part of the larger unit (i.e., vehicle, building, or ship).

A.4.7 Pipeline systems For pipeline systems, see regulations on ecology in the relevant location and also CSA Z245.6.

A.5 Miscellaneous uses A.5.1 Medical equipment, appliances, and biomedical items Medical equipment, appliances, and biomedical items are usually controlled by specification; however, provincial medical associations should be contacted.

A.5.2 Sports equipment and safety items The controlling bodies for the sports may have regulations affecting the design of equipment. The following standards may be applicable: CAN/CSA-Z262.1 on ice hockey helmets, CAN/CSA-Z262.2 on face protectors and visors for ice hockey players, and CAN/CSA-Z94.1 on industrial protective headwear.

A.5.3 Plumbing products and materials For plumbing products and materials, the municipal authority should be referred to, together with CAN/CSA-B602.

A.5.4 Pressure vessels Pressure vessels are regulated by various authorities depending on the vessel’s use and whether they are static or mobile. The most generally accepted regulations are contained in CSA B51 and the ASME Pressure Vessel Code published by the American Society of Mechanical Engineers. Pressure vessel analysis is not covered by this Standard.

A.6 Reference publications This Annex refers to the following publications, and where such reference is made, it is to the edition listed below. CSA (Canadian Standards Association) B51-03 Boiler, Pressure Vessel, and Pressure Piping Code CAN/CSA-B311-02 Safety Code for Manlifts

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CAN3-B354.1-M82 (R2003) Elevating Rolling Work Platforms

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CAN/CSA-B354.2-01 Self-Propelled Elevating Work Platforms CAN/CSA-B354.4-02 Self-Propelled Boom-Supported Elevating Work Platforms CAN/CSA-B602-99 Mechanical Couplings for Drain, Waste, and Vent Pipe and Sewer Pipe CAN/CSA-C225-00 Vehicle-Mounted Aerial Devices CAN/CSA-S6-00 Canadian Highway Bridge Design Code S37-01 Antennas, Towers and Antenna-Supporting Structures S269.1-1975 (R2003) Falsework for Construction Purposes CAN/CSA-S269.2-M87 (R2003) Access Scaffolding for Construction Purposes CAN3-Z11-M81 (R2003) Portable Ladders CAN/CSA-Z94.1-92 (R2003) Industrial Protective Headwear CAN/CSA-Z150-98 (R2004) Safety Code on Mobile Cranes CAN/CSA-Z240 MH Series-92 (R2001) Mobile Homes CAN/CSA-Z240 RV Series-99 Recreational Vehicles Z245.6-02 Coiled Aluminum Line Pipe and Accessories CAN/CSA-Z256-M87 (R2001) Safety Code for Material Hoists CAN/CSA-Z262.1-M90 (R2002) Ice Hockey Helmets CAN/CSA-Z262.2-M90 (R2002) Face Protectors and Visors for Ice Hockey Players

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CAN/CSA-Z267-00 Safety Code for Amusement Rides and Devices

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ASME International (American Society of Mechanical Engineers) International Boiler and Pressure Vessel Code — 2004 Edition Government of Canada Canada Shipping Act. SC 2001, c. 26. Motor Vehicle Safety Act. SC 1993, c. 16. Motor Vehicle Safety Regulations. CRC, c. 1038. Nuclear Safety and Control Act. SC 1997, c. 9. Railway Safety Act. RSC 1985 (4th Supp.), c. 32. Transportation of Dangerous Goods Regulations. SOR/2001-286.

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Annex B (informative) Common uses of aluminum alloys Note: This Annex is not a mandatory part of this Standard.

B.1 Extruded shapes The following alloys are used for extruded shapes: (a) 6061-T6 and 6351-T6 are medium-strength heat-treated alloys used in general structures, employed where load-bearing characteristics predominate. (b) 6063-T6 is a lower-strength heat-treated alloy with superior surface finish, employed where architectural requirements are of primary importance.

B.2 Sheet and plate The following alloys are used in sheets and plates: (a) 3003-H112, -H14, and -H16 are three tempers of a work-hardening sheet alloy used for roofing, siding, and ductwork. (b) 3004-H112, -H14, and -H16 provide greater strength than 3003 for general applications. (c) 5052 is the strongest of the general purpose work-hardening sheet alloys. (d) 5086 is used where high strength, weldability, and corrosion resistance are required, such as in tanks, trucks, and trailers. (e) 5454 is used where corrosion resistance is required at temperatures above 60 °C, such as in chemical-processing equipment. (f) 5083-H112 and -H321 are two tempers of a medium-strength sheet and plate alloy used for welded structures where higher welded mechanical properties are required. (g) 6061-T6 is a medium-strength heat-treated alloy used in general structures, employed where load-bearing characteristics predominate.

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CSA Special Publication

S157.1-05 Commentary on CSA S157-05, Strength design in aluminum

Published in February 2005 by Canadian Standards Association A not-for-profit private sector organization 5060 Spectrum Way, Suite 100, Mississauga, Ontario, Canada L4W 5N6 1-800-463-6727 • 416-747-4044

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Commentary on CSA S157-05, Strength design in aluminum

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Preface This is the first edition of CSA S157.1, Commentary on CSA S157-05, Strength design in aluminum. The objective of this Commentary is to assist users of the Standard by documenting the research background on which the various provisions of the Standard are based. It also provides descriptive information to give the reader an appreciation of the current approach to strength design in aluminum. It is to be used in conjunction with the Standard. The clause headings and numbers used in the Commentary refer to the identical clause headings and numbers in the Standard, but in the Commentary the clause numbers are given the prefix “C”. The Commentary contains cross-references to the Standard as well as cross-references within the Commentary. Several clauses in the Standard require no comment and hence no commentary is provided. In such instances, a gap in the numbering will appear at the appropriate locations in this Commentary. This Commentary on CSA S157 was prepared by the Commentary Subcommittee of the Technical Committee responsible for the preparation of CSA S157. February 2005 Notes: (1) Use of the singular does not exclude the plural (and vice versa) when the sense allows. (2) Although the intended primary application of this Special Publication is stated in its Scope, it is important to note that it remains the responsibility of the users of the Special Publication to judge its suitability for their particular purpose. (3) All enquiries regarding this Special Publication, including requests for interpretation, should be addressed to Canadian Standards Association, 5060 Spectrum Way, Suite 100, Mississauga, Ontario, Canada L4W 5N6.

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Commentary on CSA S157-05, Strength design in aluminum

S157.1-05 Commentary on CSA S157-05, Strength design in aluminum Introduction Two earlier standards for structural design in aluminum, CSA S157-1969, Structural Use of Aluminum in Buildings, and CSA S190-1968, Design of Light Gauge Aluminum Products, were based on working stress design in imperial units. These two standards were replaced by one standard, CAN3-S157-M83, Strength Design in Aluminum, using limit states design and SI units. Since the publication of that Standard in 1983, there have been a number of developments in the philosophy of structural design in aluminum and a sufficiently large body of new knowledge from experimental and research activities to warrant revisions to the Standard in the interest of uniformity of security, improved economy, and a consistency of design philosophy throughout the procedures presented. The current Standard provides instruction for the design of aluminum load-bearing components. In general, the procedures give the ultimate limit state, which is defined as the highest force the component or assembly will sustain prior to collapse or uncontrolled deformation. Design treatments have been selected to provide characteristic resistances that have a high probability of being exceeded. The level of precision of the methods and materials used in structural engineering, as well as the scatter of test results, is such that no more than two significant figures can be justified for the coefficients in the design formulas.

C1. Scope Because it is intended to be referenced by the National Building Code of Canada, CSA S157 includes requirements that are specific to buildings, and the load and resistance factors given are for the design of building components. However, the design procedures determine the ultimate resistances of members and connections; thus the Standard has general validity and is expected to be applied to all types of load-bearing aluminum assemblies for which there is no separate design code. Aircraft design, pressure vessel design, and other well-established fields have their own bodies of rules. CSA S157 is aimed at general engineering, which includes such applications as lattice towers, cranes, vehicles, rolling stock, and bridges.

C4. Materials C4.2 Mechanical properties The minimum mechanical properties of alloys specified in industry standards (Aluminum Association, Aluminum standards and data, 2003) are the ultimate tensile strength, tensile yield strength, elongation in tension, and, in some cases, the bend factor. These properties are established at values for which 99% of the material is expected to conform at a 0.95 confidence level. This is the A-basis defined in the US Department of Defense publication, MIL-HDBK-5G, 1994. In compression, the yield strength is taken to be equal to that in tension for fully heat treated plate alloys and extruded products. As a consequence of the Bauschinger effect, the yield strength of rolled products in work-hardened alloys may be lower in compression, but this is not expected to influence significantly the level of safety. Similarly, no account is taken of the variation of tensile and compressive properties for stress along and across the machine direction of rolled plate (MIL-HDBK-5G).

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In shear, the yield strength is related to the guaranteed tensile yield strength by the von Mises’s criterion, using 1/ 3 rounded to 0.6, which is reasonably consistent with the measured values. Theoretically, there is no justification to use von Mises, which is based on elastic behaviour, for the relationship at the ultimate strength, as the value of the tensile ultimate is an artificial stress. However, the factor of 0.6 applied to the ultimate tensile strength gives acceptable values that vary between –10% and +5% of the measured values for the ultimate shear strength (MIL-HDBK-5G). Measured values that can be guaranteed may be used. The ultimate bearing strength is based on the ultimate tensile strength (Fisher and Struick, 1964; Marsh, 1979). For force directed towards an edge, the capacity is governed by shear failure in the connected material up to an end edge distance in excess of two diameters. Beyond this, for practical purposes, the value remains constant. No use is made of a “yield strength” in bearing, as distortions are small and are usually of no concern at the ultimate limit state. Should a bolted splice be used at the centre of a member in compression, it is possible that the “softness” of a joint governed by bearing may influence the capacity of the strut, but the larger resistance factor used for joints makes this unlikely. The fastener itself is not subject to bearing failure (Hartman et al., 1944). “Elongation” and “bend factor” are not used in the Standard, but the values for the alloys in general use are such that ductile behaviour can be anticipated and plastic design methods may be appropriate. A number of the more popular alloys, ranging from those used for building sheet to the higher strength extruded alloys, are listed in Table 1. Although the actual specified properties vary with product and thickness, a single representative characteristic value for each temper of an alloy is given in Table 2 to facilitate design. There are many other alloy designations that closely match those included, and no restriction is placed on their use, subject to their being suitable for the intended application. Aircraft alloys, armour plate, and other special alloys have their own design rules. Welded properties are discussed in Clause 11.3.

C4.3 Physical properties The elastic modulus varies by less than 5% over the range of alloys listed, and a uniform value of 70 000 MPa is adopted. Poisson’s ratio for aluminum is approximately 1/3, making the shear modulus 26 000 MPa, to the same accuracy as the value used for the elastic modulus. The coefficient of thermal expansion varies between 23 × 10–6/°C for the 6000 series of alloys and 24 × 10–6/°C for the 7000 series. The higher value of 24 × 10–6/°C is used. Density varies between alloys by ±2% from the average value of 2700 kg/m3.

C4.4 Fasteners and welds There is no restriction on the type of fastener or material used so long as the characteristic strength is known and the material is compatible with the aluminum in the prevailing environment. Filler alloys for welds are restricted to those permitted in CSA W59.2.

C5. Limit states design C5.1 General Also known as “load and resistance factor design”, limit states design is the preferred method to match calculated strength to probable loading and is based on an analysis of the likely loads and the probability that the component or assembly will possess sufficient strength to sustain those loads. CSA S408 describes how load and resistance factors are selected to provide an acceptable reliability index (see Clause 5.4).

C5.2 Safety criterion The safety criterion states simply that the structure should be strong enough to make the probability of failure, under the action of the anticipated loads, very small.

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C5.3 Loads for buildings Appendix C to the National Building Code of Canada gives loads for the geographic location, and for the type and shape of the structure.

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C5.4 Load factors for buildings Factors to be applied to the loads specified for buildings are those given in the National Building Code of Canada.

C5.5 Resistance factors Characteristic resistances are predicted by the design expressions in the Standard, which have been developed on the basis of theory and computer simulation to represent the mean value of test results less two standard deviations. To increase confidence that the structure can resist the factored loads, additional factors are applied to the characteristic resistances. These resistance factors have been derived in such a way that in combination with the load factors for buildings, the reliability index is greater than 3 for members and 4 for connections. The reliability index, defined in CSA S408, is a measure of the extent by which the computed strength exceeds the probable load. The type of failure anticipated is also reflected in the choice of a resistance factor. Where there is an overall yielding of the cross-section, but no precipitous failure, a resistance factor of 0.9 is adopted. Where failure is of a more brittle type, with little evident distress prior to collapse, as in the case of rupture in tension at a net section, the factor is reduced to 0.75. Because of their importance, the value for fasteners is 0.67. These factors parallel those adopted for steel structures (see CAN/CSA-S16). In other applications, the resistance factors are held constant (as they are related to the predictors used to establish strength), while the load factors are chosen to provide the required level of reliability.

C6. Methods of analysis and design Elastic analysis is always permitted even though the force/deformation relationships for the components, up to failure, may not be linear. The justification is that so long as equilibrium is satisfied and there are no “brittle” components, the solution will be a lower bound. Caution may be needed in highly redundant lattice structures such as space trusses (Schmidt, 1976). In general, aluminum assemblies do not possess the range of ductility of structural steel. There is a lower spread between yield and ultimate strength, and a lower elongation at rupture. The reliance placed in steel design on the redistribution of stress after yielding cannot always be transferred to aluminum, and a clear grasp of the limits that can be exploited is essential for safe designs, if plastic or limit design methods are used. Because many components are not subject to direct analysis and design, proof by testing is acceptable in all cases.

C7. Net area, effective section, and effective strength C7.3 Net area For the design of the net section of plates in tension across a row of staggered holes (see Figure C1), the holes are deducted and s2/4g added for each space. This was shown by Brady and Drucker (1955) to correspond to an upper bound solution based on a theoretical plastic analysis.

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g

s

Figure C1 Net section at staggered holes (See Clauses 7.3 and C7.3.)

C7.4 Effective section C7.4.2 Effective thickness at welds To account for the reduction in strength at a longitudinal weld used for members in bending, it is assumed that the thickness of the heat-affected zone is reduced in the ratio Fwy /Fy. The plastic section modulus of the effective section gives the fully plastic moment capacity when multiplied by the yield strength of the base metal (Brungraber and Clark, 1962). Should the extreme fibre stress in the base metal be required to remain elastic, then the stress in elements nearer to the neutral axis will be lower and the effective thickness of the HAZ, given by t(Fwy /Fy)(c/y) ≤ t, reflects this fact, where y is the distance from the neutral axis and c is the distance to the extreme fibre. Any change in the location of the neutral axis is not taken into account in this calculation, with negligible error.

C7.4.3 Effective thickness after local buckling of flat elements In thin flat elements that have both longitudinal edges supported, the stress distribution does not remain uniform after initial buckling. As the force is increased, the stress towards the boundaries increases and, at the limit, reaches the yield strength. The magnitude of the total force can be represented by the product of the yield strength and an effective area, or as the product of the gross area and an effective strength. The device is artificial but useful. von Karman et al. (1932) suggested that the limiting force be given by R = bt (Fe Fy)1/2 = Fy bt (Fe /Fy)1/2 This is equivalent to taking an effective width given by b’ = b (Fc /Fy)1/2, which is assumed to sustain the yield strength. The effective width of each part in compression is used in calculating the properties of the effective section. This model proves to be unconservative, and better agreement with test results, over the full range of slenderness, is obtained by using the actual buckling stress, Fc , as adopted in this Standard (Marsh, 1998) rather than the elastic value, Fe , used by von Karman. A comparison with other methods and test results is shown in Figure C2 for steel and Figure C3 for aluminum. Further, an effective thickness, rather than an effective width, is adopted. Where the stress in an element varies from compression to tension, as in the web of a beam, the thickness of the entire element is reduced, thereby avoiding the trial and error needed to locate the centroid of an effective section when its position influences the effective areas, as occurs if an effective width is used. The result is conservative when compared with the value obtained for a web when only the portion in compression is reduced, but as the web makes a relatively small contribution to the overall bending strength, the resulting minor loss of economy is acceptable. This procedure is used for bending and for beam-columns that fail in the plane of bending.

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2.0

1.5 B CSA S136 B = 1.9 [ 1- 0.416 (E/Fy)1/2 (t/b) ]

1.0

CSA S157 B = (b/t) (Fc /E)1/2 von Karman B = 1.9

0.5

0.0

1

2

3

4

5

6

Note: From Lind et al., 1971.

Figure C2 Effective width formulas: Comparison with test and code values for steel (See Clause C7.4.3.)

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1.0 1/

NACA test results

(Fc /Fy)1/2

F 0.5 1/

0.0 0.0

0.5

2

1.0

1.5

2.0

Note: From Mofflin and Dwight, 1983.

Figure C3 Effective thickness formula: Comparison of test and code values for aluminum (See Clauses C7.4.3.)

C7.4.4 Deflection under service loads Only small portions of the length and cross-section are affected by local buckling under service loads, so it is reasonable to neglect this effect when checking deflections, as the specified limiting values are usually arbitrary and safety is not a concern.

C7.5 Effective strength and overall buckling C7.5.1 General Effective strength is that stress which, when multiplied by the appropriate geometric property of the gross section, gives the resistance.

C7.5.2 Influence of welding For members with longitudinal welds, the effective strength is the weighted average yield strength of the cross-section, obtained by summing the products of the areas of the different zones and their yield strengths, and then dividing the sum by the gross area (Brungraber and Clark, 1962).

C7.5.3 Influence of local buckling When local buckling reduces the strength of flat elements that are supported on both longitudinal edges, the expression for the resistance given in the discussion of Clause 7.4 can be interpreted as a normalized effective strength given by

F

m

= F

1/2

which, when multiplied by Fy , gives the limiting stress, Fo , to be used in the design of columns and beams against overall flexural or lateral buckling.

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C8. Local buckling of flat elements C8.1 Buckling stress Licensed for/Autorisé à Reid Costley, Cascade Engineering Group, Sold by/vendu par CSA on/le 10/27/2009. Single user license only. Storage, distribution or use on network prohibited./Permis d'utilisateur simple seulement. Le stockage, la distribution ou l'utilisation sur le réseau est interdit.

Theoretically, elastic buckling under the action of a compressive stress occurs when the stress reaches a value given by (Timoshenko and Gere, 1961):

Fe =

kπ2D b 2t

where k = plate buckling coefficient, related to the stress distribution and the boundary conditions D = Et 3 /12(1–ν 2) where E = elastic modulus ν = Poisson’s ratio b = element width t = element thickness By equating this to an expression of the Euler type,

k π2D b 2t

=

π2E λ2

a slenderness is obtained given by 1/ 2

⎛ 12(1− ν2 ) ⎞ λ= ⎜ ⎟ ⎜ ⎟ k ⎝ ⎠

b b =m t t

where m is a factor related to the coefficient k. This form of the slenderness, λ, is used for all local buckling in flat elements and leads to the normalized slenderness:

λ = (Fy / Fe )1/ 2 With this value, the normalized buckling stress, F , is obtained from the buckling formula for the alloy as described in Clause 9.3.3.

C8.2 Elements supported on both longitudinal edges C8.2.1 Edges simply supported For elements simply supported at both long edges, carrying a stress that varies linearly from f1 to f2 across the element, where f1 is the maximum compressive stress, the theoretical values for m, as derived from Timoshenko and Gere (1961), are compared in Figure C4 with those given by the simplified expressions used in the Standard.

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f1

f2 1

0

m

1.63

1.15

0.67

(1.15 +f2 /2f1)

1.65

1.15

0.65

f2 /f1 Theory Code

-1

f2 /f1

-2

-4

m

0.44

0.27

1.3 / (1-f2 /f1)

0.43

0.26

Figure C4 Edges simply supported: Comparison of theoretical and code values for the factor m (See Clauses 8.2.1 and C8.2.1.)

C8.2.2 Influence of adjacent elements Where there is a direct connection to adjacent elements, the boundary restraint increases the critical stress. For an element of width b, subjected to uniform stress caused by axial force or bending, in sections such as those shown in Figure C5, the theoretical values of m for sections of uniform thickness are compared with the clause values in Table C1.

b b

t

a t1

a t t1

Figure C5 Elements supported and partially restrained at both long edges (See Clauses 8.2.2 and C8.2.2.)

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Table C1 Influence of adjacent elements: Comparison of theoretical and code values for m Licensed for/Autorisé à Reid Costley, Cascade Engineering Group, Sold by/vendu par CSA on/le 10/27/2009. Single user license only. Storage, distribution or use on network prohibited./Permis d'utilisateur simple seulement. Le stockage, la distribution ou l'utilisation sur le réseau est interdit.

(See Clause C8.2.2.) a/b

0

1.0

2.0

Theory

Columns m

1.24

1.63

Code

1.25 + 0.4 a/b

1.25

1.65

Theory

Beams m

1.24

1.43

1.63

Code

1.25 + 0.2 a/b

1.25

1.45

1.65

C8.3 Elements supported at one longitudinal edge only C8.3.1 Edge simply supported For elements simply supported at one longitudinal edge and free at the other, carrying uniform compressive stress or a stress that varies linearly across the element, failure is by torsional buckling. The values of m given in the Standard were derived directly using the methods of Timoshenko and Gere (1961). The variation of m with the stress distribution is illustrated in Figure C6. These values pay no regard to any restraint at the supported edge and thus are applied strictly to equal angles and are conservative for most other practical cases. For the range f2 /f1 < –0.28, with compression at the supported edge, the torsional buckling mode is inhibited and Clause 8.2.1 is used.

t f1

f2

w

f2

f1

f2 /f1

-3

0

1

0

–0.28

m

0

4.3

5

2.5

1.0

Figure C6 Values of m for elements simply supported on one longitudinal edge only (See Clauses 8.3.1 and C8.3.1.)

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C8.3.2 Flanges of sections

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w

t

w

t

t1

t1

a

a

Figure C7 Flange elements partially restrained at the supported edge (See Clause 8.3.2.) When a flange is attached to a web, the constraint increases the critical stress. Whether the uniform stress in the flange is due to axial force or moment has little influence on the value of m for ratios a/w less than 2. For higher ratios the expression used is conservative for compressive stress due to moment. The expression m = 3 + 0.6 [(at)/(wt1)] has been chosen to match the values given by Stowell et al. (1952) with reasonable accuracy in the practical range, as is shown in Table C2 for sections of uniform thickness.

Table C2 Local buckling of flanges: Comparison of theoretical and code values for m (See Clause C8.3.2.) a/w

0

1

2

3

Theory

m

2.9

3.5

4.0

4.8

Code

3 + 0.6 a/w

3.0

3.6

4.2

4.8

Typical bend radii in formed sections have a negligible influence on the critical stress, and the full distance from the intersection of the median lines is used (Marsh, 2000). Local buckling in outstanding elements (i.e., supported along one side only) usually leads to collapse, and this is assumed to hold in the general case (Marsh, 2001). It follows that where flange buckling occurs in columns or beams that are subject to overall flexural buckling, this initial local buckling stress will be used as the limiting stress, Fo , in the column formula. Should a column or beam be held straight, as when supporting cladding or decking, the tendency to deflect laterally when a flange buckles can be inhibited and stress may be redistributed, thus allowing the flanges to develop postbuckling strength. There is no generally accepted formal analytical method to determine the postbuckling reserve in this case, and for uniformity of treatment, the expression used for elements supported on two longitudinal edges (as described in Clause 7.4.3) is adopted.

C8.4 Elements supported at one edge with a lip at the other edge The treatment for the influence of lips on the stability of flanges, derived from Sharp (1966) and Marsh (1990), models the flange as an element rotating about the intersection of the flange and web, with elastic restraint provided by the web. The moment of inertia of the lip is taken about the inner surface of the flange when computing the warping rigidity. This applies to all shapes of lip, including those with radiused corners, those at 45°, and bulbs (Figure C8). The flat width is not used.

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w

w

w

w

c

c

c

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t a

t1

w

a

Figure C8 Lipped flanges (See Clauses 8.4.1, 8.4.2.1, and C8.4.)

C9. Resistance of members C9.1 Limiting slenderness for members Limits to the slenderness of compression and tension members are historical. There is seldom any rational justification for the values, but, as no pressure has been exerted to have them changed, it is concluded that they are reasonable. Limits to the proportions of tubes and double angles subjected to wind are based on tests and experience. Two kinds of wind-induced oscillations are of concern. For tubular members, resonant vortex shedding causes oscillation across the wind direction. This is limited to a small range of wind speeds near the critical velocity and requires linear flow. Winds over 50 km/h are usually turbulent, and no response is expected (Simiu and Scanlan, 1978). For double angles, torsional oscillations can arise, which, once started, increase with rising wind speed. The limit is imposed to keep the natural frequency for torsional vibration above 50 Hz.

C9.2 Members in tension With the move to limit states design, it is increasingly recognized that the elastic limit does not necessarily limit the load-carrying capacity. If only a small portion is in the non-linear range, overall distortions are small and “uncontrolled deformations” do not occur. For this reason the strength of tension members is governed by one of two distinct conditions: (a) overall yielding of the gross section; or (b) rupture at the connection. The “yield” strength is usually that of the base metal but may be the weighted average yield strength for sections with longitudinal welds. Rupture may occur across the net section at the ultimate strength of the base metal, or across a weld at the ultimate strength of the weld metal. Where a transverse weld is oblique to the direction of stress, there is no change in resistance controlled by the ultimate strength of the weld up to an angle of 45o. For a greater angle, the limiting condition is taken to be the yielding across the section at the weighted average yield strength, recognizing the increased width of the HAZ across the oblique weld.

C9.3 Members in compression: Buckling There are two bounding conditions associated with failure in compression: (a) a limiting strength, Fo , usually the yield strength, Fy ; and (b) the elastic buckling stress, Fe = π2E/λ2. The elastic buckling stress is normalized with respect to the limiting strength, leading to the normalized slenderness, λ , given by

λ = (Fo /Fe)1/2 = (λ/π)(Fo /E)1/2

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where E = the elastic modulus λ = geometric ratio called the slenderness

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For simple compression members, λ = L/r where L = the length r = the radius of gyration Between these two boundaries of yielding and elastic buckling lies the actual buckling curve for a particular column profile and alloy. A widely accepted model for the relationship is the Perry-Robertson curve. According to the original version of this model, the limiting axial force, P, for a compression member with an initial bow of σ occurs when the compressive stress on the concave side of the member reaches the limiting stress, Fo . Thus

P Pσ + = Fo A S (1− P/Pe ) where Pe = the theoretical elastic buckling load This can be expressed as Fc + ηFc /(1 – Fc/Fe) = Fo where Fc = mean axial stress at failure Fe = elastic buckling stress η = σc/r 2 where c = distance from the centroid to the extreme fibre in compression Normalizing the relationship with respect to Fo , and extracting Fc /Fo , gives the normalized buckling stress as follows:

F = Fc /Fo = β – (β2 – 1 λ 2 )1/2 where β = (1 + η + λ 2 )/2 λ 2 Replacing η by α ( λ – λ o) gives β = [1 + α ( λ – λ o ) + λ 2 ]/2 λ 2 This modified form of η permits the creation of a range for λ < λ o in which buckling can be disregarded. Variations in the value of α permit the fitting of the curve to the actual behaviour of struts as determined by direct tests or computer simulation. Mazzolani in his papers and book (1985) provides a wealth of information, drawn from many sources, on the buckling of aluminum struts, which has been interpreted and used as the basis for the ECCS recommendations (1978) for aluminum structures. This interpretation is conservative and suited to European national standards, but the design stresses in some cases are well below those permitted in North American standards, prompting a review of the experimental evidence. Results from column tests are given in the form of normalized stresses, F = Fc /Fy , plotted against the normalized slenderness, λ . Values at two standard deviations below the mean from the tests provide the target relationship for the design expression adopted.

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Commentary on CSA S157-05, Strength design in aluminum

As the tests are mostly for I-shapes failing about the weak axis, which are known to give lower buckling stresses than those for strong axis buckling, the curve gives reasonably conservative values relative to the entire family of strut shapes. To find a best-fit curve, the Perry-Robertson formula was taken and an analysis made to determine the most suitable values of the parameters α and λ o . Using the test data, values for both variables may be established for a best fit (Column (1) in Table C3). However, to facilitate design in general, it is convenient to have as large a range of slenderness for which buckling need not be considered as can be accepted. It is proposed that this range be up to λ o = 0.3, and the test results provide evidence that this is reasonable. Holding this value constant, a best-fit value of α is then determined (Column (2) in Table C3). The values adopted in the Standard are given in Column (3).

Table C3 Various best-fit values for the coefficients λ o and α (See Clause C9.3.)

Fully heat-treated alloys α

λo

Work-hardened alloys α

λo

(1) α and λ o

(2)

(3) Adopted

0.134 0.127

0.178 0.3

0.2 0.3

0.357 0.339

0.334 0.3

0.4 0.3

λ o = 0.3

Figures C9 and C10 show the curves adopted with the test values. Included in the figures are the straight line formulas used in the previous Standard, which can be closely represented by the following formulas: (a) heat-treated:

F = 1.12 – 1.37 λ (b) not fully heat-treated:

F = 1.15 – 0.48 λ It is seen that, relative to the values in CAN3-S157-83, there has been a reduction in the design stresses, particularly in the zone for values λ of between 1 and 2. This reflects the fact that the earlier formula, while taking into account non-linear material properties, took no account of imperfections. This did not lead to “unsafe” designs but rather to a variation in the margin of safety over the range of slendernesses. In other standards for steel and aluminum, different buckling curves are sometimes used for variations such as those found in profile shapes (tube vs I-shape), directions of buckling (strong vs weak axis), symmetric and asymmetric shapes (I-shape vs channels), and modes of buckling (flexural vs torsional). Extruded aluminum shapes are available in such a wide variety that it is not practical to recognize them all as distinct problems. Furthermore, there is not sufficient evidence to explain the variations found and to justify individual recognition. For these reasons a single, reasonably conservative formula is adopted for each alloy type, based on the values for flexural buckling of an I-section about the weak axis, intended to cover all structural shapes and all modes of buckling of columns and beams. The curves are given in Figure C11. For the local buckling of plates, λ o = 0.5 with α = 0.2 for fully heat-treated alloys and 0.4 for not fully heat-treated alloys. Figure C12 gives the resulting curves, as well as the postbuckling strength discussed in Clause 7.5.3.

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Normalized buckling stress, F

Euler CAN3-S157-M83

0.8 = 0.2

0.6 o

= 0.3

0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Normalized slenderness, Note: From Mazzolani, 1985.

Figure C9 Column buckling formula, fully heat-treated alloys: Comparison of test and code values (See Clause C9.3.)

1.0 Normalized buckling stress, F

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1.0

Euler 0.8 CAN3-S157-M83 = 0.4

0.6 o

= 0.3

0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Normalized slenderness, Note: From Mazzolani, 1985.

Figure C10 Column buckling formula, work-hardened alloys: Comparison of test and code values (See Clause C9.3.)

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1.0

Normalized buckling stress, F

0.8 0.7 Fully heat-treated alloys 0.6 0.5 Work-hardened alloys 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Normalized slenderness,

Figure C11 Normalized buckling stress for columns and beams (See Clauses 9.3.3, C9.3, and C9.3.3.)

1.0 0.9

Postbuckling: Fully heat-treated alloys

0.8

Normalized buckling stress, F

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0.9

Work-hardened alloys

0.7 0.6 0.5

Initial buckling: Fully heat-treated alloys Work-hardened alloys

0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Normalized slenderness,

Figure C12 Normalized buckling stress and postbuckling strength for plates (See Clauses 9.3.3, C9.3, and C9.3.3.)

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C9.3.2 Limiting stress The Perry-Robertson formula makes use of a limiting extreme fibre stress equal to the yield strength. In many cases, however, a stress other than the yield strength controls the design, and the limiting stress may be viewed as the “squash load” stress, which, in these design procedures, is the mean compressive stress at failure in a short, concentrically loaded, pin-ended compression member. It is not the same as the “squash load” for a column between fixed platens. Influence of welding For transverse welds at the end of a member, where the heat-affected zone is fully constrained against local buckling or is too short to buckle, it has been demonstrated in tests (Marsh, 1988) that the weld can sustain a compressive stress as high as the ultimate tensile strength; thus, the stress in the weld is limited to the lesser of the yield strength of the base metal and the ultimate strength of the heat-affected zone. For transverse welds away from the ends, which affect the total cross-section, the yield strength, Fhy , in the heat-affected zone will limit the applied compressive stress. The question is whether this stress should be used as a simple cut-off to the unwelded buckling curve, which may not be conservative, or used as the limiting stress, Fo , in the buckling formula. The latter has been selected in this Standard. It is known to be conservative, but, as transverse welding at the midpoint of struts is rare, the economic cost of this simple method is small. With longitudinal welds, the weighted average yield stress, discussed in Clause 7.5.2, is used as Fo . This strength reduction represents the primary influence of longitudinal welds on the capacity of a column. However, there is an additional effect due to the creation of longitudinal residual stresses; this effect has been studied by Mazzolani (1985), who gives curves obtained using computer simulation of the influence of these inherent stresses on a fully annealed alloy (i.e., one in which there is no reduction in yield strength due to welds). This influence has been applied as an added factor for all alloy types. Treating the family of curves developed by Mazzolani (1985) as a statistical set, the values of interest are those occurring at λ = 1 and 2. The mean reduction in strength and coefficient of variation at λ = 1 are 0.076 and 0.03, respectively. At λ = 2 there is a negligible mean influence. The reduction factor adopted (see Clause 9.4.2.2) is the following: k = 0.9 + 0.1 |1 – λ | ≤ 1.0 This is approximately the mean less one standard deviation. In view of the high values used for residual stresses in the analyses and the superposition of this factor on the influence of the reduced yield strength, the resulting characteristic strength is considered to be sufficiently conservative. It is to be observed that although the weld reduces the yield strength, the residual stress at the weld is tensile, creating compressive stress elsewhere; thus, welds at the extreme fibre may not be the most detrimental. For this reason weld location is not taken into account in columns, although it may be considered in simple bending (see Clause 7.4.2). Influence of local buckling For outstanding flanges, local buckling is considered to precipitate collapse; thus, the local buckling stress, Fc , is the limiting stress, Fo . For flat elements supported at both long edges, there is a postbuckling reserve of strength and the effective strength, Fm = Fy F 1/2, derived in Clause 7.5.3, is used as the limiting stress, Fo .

C9.3.3 Buckling stress Figure C11 provides curves relating the normalized slenderness, λ , to the normalized buckling stress, F = Fc /Fo , for columns and beams. Figure C12 gives the curves for the normalized buckling stress and for the normalized effective postbuckling strength of plates (see Clause 7.5.3).

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C9.4 Columns

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C9.4.2 Flexural buckling Table 4 in the Standard gives effective length factors for a number of cases arising in lattice structures, derived primarily from the methods of Timoshenko and Gere (1961). The values for multiple cross-bracing are based on the work of Picard and Beaulieu (1989).

C9.4.3 Torsional buckling C9.4.3.1 Pure torsion The expression for elastic torsional buckling from Timoshenko and Gere (1961) is Ftc = GJ/Ip Equating this to the Euler stress, π2 E/λ2, using G = E/2(1+ ν) with ν = 0.33, gives a value of λ rounded to λ = 5(Ip /J )1/2 Non-linear material behaviour and imperfections are assumed to have the same influence in torsional buckling as they do in flexural buckling, permitting the use of the same normalized buckling formulas. Angle shapes may fail in torsion, but not by local buckling. For a simple angle, Ip = 2w3t/3 and J = 2wt3/3, giving λ = 5w/t. A root fillet (see Figure C13(a)) increases J and permits the use of a reduced value of w. Bulbs at the tips and root (see Figure C13(b)), if proportioned to increase the ratio J/Ip , improve the performance. A bend radius at the heel (see Figure C13(c)) reduces J, and hence the critical force, very slightly. In formed angle shapes, the “flat” width has no relevance, and simple lips (see Figure C13(d)) increase Ip more than J, and therefore reduce the critical stress. c

w

w

(a)

w

(b)

w

(c)

(d)

Figure C13 Types of angle section (See Clauses 9.4.3.1.2.1 and C9.4.3.1.)

C9.4.3.3 Combined torsion and flexure For sections symmetrical about one axis only, buckling is by flexure about the axis of asymmetry, or about the axis of symmetry combined with torsion. The interaction relationship used provides a reasonably close match to the correct relationship (Timoshenko and Gere, 1961) given by:

(Ix + Iy )f 2 Ip where Ix , Iy = f = F1, F2 = Ip =

− (F1 + F 2 )f + F1F 2 = 0

moments of inertia mean applied axial stress critical stresses for flexural and torsional buckling polar moment of inertia about the shear centre

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For open sections with no axis of symmetry, where flexure about both principal axes combines with torsional buckling, reference should be made to the literature to obtain the theoretical elastic value for mean applied stress that causes buckling. This stress can then be used in Clause 9.3.3 to obtain the actual buckling stress.

C9.5 Bending C9.5.1 Classification of members in bending Bending resistance may be limited by a fully plastic condition, as in compact sections, or by local buckling, as in thin-walled sections. Although the design procedures themselves will establish the limiting stress for a given set of proportions, it is convenient to classify the geometries that determine the types of behaviour under compressive stress. In practice, for beam sections, the most common limit is the yield strength at the extreme fibre. The behaviour is basically elastic with no significant permanent set. This is the design condition for Class 2 sections, in which local buckling will not occur prior to yielding. This requires that the normalized slenderness for the local elements be less than 0.5. Straining beyond the yield strength is not permitted because the drop in elastic modulus at yield may precipitate local buckling. Should the section be sufficiently compact, yielding need not lead to instability, and large strains in the plastic range are possible, permitting the development of the fully plastic moment. The requirement for this stability is that the normalized slenderness for local buckling be less than 0.3. For outstanding flanges, a typical slenderness for local buckling is given by 3.5(w/t), which means a normalized slenderness of 1.1(w/t)(Fy/E)1/2. When this is equal to 0.3, then w/t = 0.27(E/Fy)1/2, which is in keeping with the recommendation by Jombock (1974) that w/t be limited to 0.3(E/Fy)1/2. Class 3 sections are those in which the elements have a normalized slenderness greater than 0.5, and local buckling must be considered in the parts carrying compressive stress. This applies also to lattice masts in which chord buckling controls the strength.

C9.5.2 Moment resistance of members not subject to lateral-torsional buckling Bending resistance may be controlled by the compressive or the tensile stress; thus, for each class there are two limiting conditions to be considered. In Class 1 sections, the fully plastic moment of the cross-section can only be realized if the stress at the net section does not exceed the ultimate tensile strength. The two conditions — one at yield in the gross section and one at the ultimate strength in the net section in tension, or at a weld — are independent limits to the resisting moment. In Class 2 sections, the stress computed using the elastic section modulus of the gross section must not exceed the yield strength, while the tension stress computed using the elastic section modulus of the net section must not exceed the ultimate tensile strength. There is sufficient plastic strain available in aluminum alloys to develop the ultimate tensile strength across the net section. When computing the properties of the net section, the holes with fasteners in the compression zone are not deducted, and the small change in the position of the neutral axis may be disregarded. Again, the two conditions of yielding and rupture in tension are independent. In Class 3 sections, the tension stress is limited to yielding in the gross section or the ultimate tensile strength in the net section, while compression in outstanding flanges is controlled by local buckling, which usually precipitates overall buckling and thus limits the capacity. For box beams and other shapes with the compression element supported on two longitudinal edges, the effective thickness is used (see Clause 7.4.3.1) to give the effective section modulus, which is multiplied by the yield strength to give the limiting condition.

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C9.5.3 Moment resistance of members subject to lateral-torsional buckling

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C9.5.3.1 Members with lateral restraint of the tension flange only Members with the tension flange restrained may buckle laterally by rotating about the restraint. The method of Timoshenko and Gere (1961) is used to give the critical elastic moment, Me . In this type of buckling, the contributions to the resistance made by the lateral flexural stiffness and by the torsional stiffness are independent, and the critical stress is the sum of the individual values attributable to the two sources of rigidity; thus, the expression that gives the critical stress in the compression flange, due to the moment, can be broken into the components of flexural and torsional resistance, allowing the slenderness to be written as

1 λ

2

=

1 λ12

+

1 λ22

where

λ = effective slenderness λ1 = the slenderness for the flange in compression buckling over the length of the member = L/ry where L = length of beam ry = radius of gyration of the flange about the weak axis λ2 = the slenderness for torsional buckling of the member = 5d/t for I-beams The expression for λ2 is derived as follows: For buckling by twisting about the tension flange, resisted by St. Venant torsion only, the critical stress is closely approximated by

Fe =

GJ Ip

The torsion constant, J, is approximately three times that for one flange; thus, J = bt3. The polar moment of inertia about the restraint is approximately that contributed by the compression flange; thus, Ip = btd2. The shear modulus, G, is equal to E/2.66. Equating the critical stress to π2E/λ22 gives λ2 = 5d/t. When ry = 0.3b is used, it leads to the expression adopted in Clause 9.5.3.1(b). This procedure does not take into account any warping restraint at the ends. In the preceding formulas J = torsion constant b = flange width t = flange thickness Ip = polar moment of inertia about the point of restraint d = member depth G = shear modulus E = elastic modulus

C9.5.3.2 nrestrained members A member in bending that is restrained against torsion at the ends only may fail by lateral/torsional buckling. In this case, the critical stress is the geometric mean of the contributions made by the flexural and torsional rigidities when considered independently; thus, if either is zero, the critical stress is zero. The general formula for the slenderness to give the elastic critical stress under a uniform moment is

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λ=

© Canadian Standards Association

(Sx L )1/2 1/ 4

⎡Iy (0.04 J + Cw / L2 ) ⎤ ⎣ ⎦

where Iy = Ary2 = b3t/6 J = bt3 Cw = Iyd2/4 By using the following approximations for an I-beam Sx = Ad/2 where A = area of two flanges = 2bt and by extracting the flexural component, the slenderness can be expressed as

λ=

L/ry 1/ 4

⎡1+ (Lt/bd)2 ⎤ ⎣ ⎦

For deep rectangular solid or box members, the torsion constant, J, is approximately 4Iy , while the warping rigidity is taken to be zero. Using these values, the slenderness for lateral/torsional buckling reduces to λ = 2.2 (rx /ry )(L/d)1/2 where rx = radius of gyration about the strong axis ry = radius of gyration about the weak axis Lateral buckling of beams is treated using the same normalized buckling formula as that adopted for columns. A comparison with the test results from Clark and Jombock (1957) for heat-treated alloys is shown in Figure C14. Local buckling of the flanges requires the replacement of Fy by Fc . Figure C15 shows how predictions using this method compare with test results on thin steel I-beams (Kubo and Fukumoto, 1988).

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Commentary on CSA S157-05, Strength design in aluminum

1.0

Mk Mp

0.5

0.0 0.0

1.0

2.0

Mk = bending moment to cause lateral/torsional buckling Mp = fully plastic moment for the section = (Mp /Me)1/2 Me = theoretical elastic moment to cause lateral/torsional buckling Note: From Clark and Jombock (1957).

Figure C14 Lateral-torsional buckling for 2014-T6 H-beams: Comparison of test and code (See Clause C9.5.3.2.)

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1.0

Fk Fc

0.5

0.0 0.0

0.5

1.0

1.5

Fk = bending stress to cause lateral/torsional buckling Fc = stress to cause flange buckling = (Fc /Fe)1/2 Fe = theoretical elastic stress to cause lateral/torsional buckling Note: From Kubo and Fukumoto (1988).

Figure C15 Lateral-torsional buckling of steel thin-wall I-beams: Comparison of test and code values (See Clause C9.5.3.2.)

C9.5.3.3 Members with end moments Where the maximum moment occurs at the end, rather than near the midpoint of the member, the value of the equivalent uniform moment to be compared with the critical moment (see Galambos, 1988) is expressed as 0.6Mmax + 0.4Mmin If the curvature reverses along the length of the beam, Mmin is taken as negative. In no case is the equivalent moment taken as less than 0.4 Mmax . Other variations from the basic case, such as the location and distribution of the applied forces, have not been included, as their influence can be obtained from the literature (Timoshenko and Gere, 1961). Knowing the critical elastic moment, Me , the normalized slenderness is obtained directly from λ = (Mo /Me) 1/2 to give F from Clause 9.3.3 and hence the actual buckling moment as follows: Mc = F Mo where Mo = moment required to create the limiting stress, Fo

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Commentary on CSA S157-05, Strength design in aluminum

C9.6 Webs in shear

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C9.6.1 Flat shear panels Shear stress in a panel cannot exceed the yield strength in shear of the base metal of the web, Fsy = 0.6Fy , while the shear flow cannot exceed the ultimate strength in shear of the boundary connections of the web or of any seams in the web. The factored strength at the boundary, identified as vr , may be set by the ultimate strength of a weld, Fswut = 0.6 Fwut, by the strength of a line of rivets or bolts, R/s, where R is the ultimate resistance of a fastener and s is the spacing of the fasteners, by the shear strength along a line of holes, or by the strength of an adhesive bond. A

B b h

a

h

a b

A

Section A-A

B

Section B-B

Figure C16 Stiffened shear webs (See Clauses 9.6.1.1, 9.6.2.2, and C9.6.1.) Also limiting the strength is the instability of the web under the action of shear stress. For a rectangular panel, a × b, a > b, with stiffeners at the boundaries (see Figure C16), the initial elastic buckling stress (Timoshenko and Gere, 1961) is expressed as

Fse =

=

5.35[1 + 0.75(b/a )2 ]π2Et 2 12(1 − v 2 )b 2 π2E λ2

thus

λ=

1.4(b/t ) [1+ 0.75(b/a)2 ]1/ 2

The normalized slenderness is λ = (Fsy /Fse)1/2, which is used with the appropriate buckling curve for plates to obtain the normalized buckling stress, F . The actual buckling stress is then Fsc = Fsy F = (0.6Fy ) F It is assumed that the shear stress remains uniform across the web up to this value and the total factored shear resistance at this stage is given by: V = htFsc ≤ hvk where h = web depth t = thickness vk = strength of the boundaries If no buckling is to occur, the selection of a safety margin is at the discretion of the designer.

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C9.6.2 Stiffened webs Because of the influence of the boundary flanges, initial buckling due to shear in a thin web does not precipitate collapse. After buckling, the initially uniform shear stress changes its distribution along the boundaries; the stress at the compression corner remains close to the initial buckling stress, while the stress at the tension corner is capable of increasing until it reaches the limiting condition at the boundary (Marsh, 1982, 1988; Vilnay and Burt, 1988). At this stage the stress is pure shear, with no stress normal to the boundary. The total shear resistance is the area under the curve of the shear stress distribution times the thickness. For a hyperbolic distribution, this integral leads to a factored resistance of Vr = φ(FscFso )1/2ht = φFso F

1/2ht

where Fso = the maximum shear stress at the tension corner of the web Although this formula provides a convenient parallel with the formula for effective stress used for postbuckled plates in compression, it is too conservative. The maximum stress, Fso , cannot exceed the yield strength of the base metal or the ultimate strength of the welded or riveted connections, vr /t, but because of a “gusset effect” in the tension corner, there is a zone along which the limiting stress is maintained. The capacity of the web in shear (see Marsh, 1982) then becomes Vr = φ[2(FscFso )1/2 – Fsc ]ht To explain the postbuckling strength of panels with a high aspect ratio, Höglund (1971) showed that there is a change in the orientation of the stress field. The direction of the principal stress rotates, and while the principal compressive stress remains constant at the value that caused initial buckling, the principal tension stress increases in such a manner as to maintain zero stress normal to the boundaries. When the maximum shear stress reaches the shear yield strength, the shear force on a transverse plane section through the web is obtained from the Mohr’s circle in Figure C17, and is given by Vw = φ(2FscFso – Fsc 2 )1/2ht This leads to values sufficiently close to those obtained by the earlier formula (for the practical range of web dimensions) that it permits a single expression to be adopted for all panel aspect ratios. Fsy Fv

Fsc

Fsc

Ft

2

Ft

Fv Note: From Höglund, 1971.

Figure C17 Stress orientation in thin shear webs remote from stiffeners (See Clause C9.6.2.) Should the limiting shear strength be yielding in the base metal, as is true for steel and for annealed aluminum products, the shear resistance may be attributed to an effective shear strength: Fsm = Fsy (2 F

1/2

– F )

This value is multiplied by the gross web area to give the shear resistance. Alternatively, an effective thickness (Vilnay and Burt, 1988) may be given by t’ = (2 F

1/2

– F )t

multiplied by the shear yield strength.

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Note that up to this stage no use has been made of the notion of “diagonal tension”. With an increase in shear force beyond this stage, the yielded zones in the tension corners distort in shear, compelling the flanges to bend. The flanges have an independent bending strength as follows:

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Mo = bt 2Fy /4 where b = flange width t = flange thickness The moment resistance, Mo , of the flange to this imposed bending causes the development of some diagonal tension, which contributes to the shear resistance an amount given by: V = 2(MotFy )1/2 Because of the nature of aluminum construction, where the strength is controlled by welds or riveted seams, there may not be sufficient shear distortion to develop any contribution by diagonal tension before rupture at the boundary occurs (Evans and Hamoodi, 1987; Marsh, 1988). In order for this additional capacity to be available, the ultimate strength at the boundaries must exceed the yield strength of the basic web metal, as in some welded work-hardened alloys. When a web is divided by vertical and horizontal stiffeners, the panel with the highest slenderness at the section of interest determines the initial buckling stress, Fsc . This value limits the capacity of that section to transmit shear force, which dictates the mean shear stress across the entire panel.

C9.6.3 Web stiffeners Transverse and longitudinal stiffeners act in compression when the web is buckled, carrying a maximum force given by N = vrsk(1 – k)/(1 + k) where s = the length of the stiffener k = F 1/2 For transverse stiffeners, as k diminishes, this force approaches the shear load at the stiffener. To accommodate this force, the stiffener is designed as a compression member with a length equal to the web depth. For longitudinal stiffeners, the compressive force created in the stiffener after the web buckles is due to the difference between the shear stresses along the opposite sides of the stiffener. As k diminishes, this value approaches N = v rs F

1/2

The stiffener is designed as a compression member, spanning between transverse stiffeners to resist this force.

C9.6.4 Combined shear and bending in webs In the case of long unstiffened webs, the following relationship must be satisfied (Timoshenko and Gere, 1961) to avoid initial buckling caused by shear stress combined with bending stress: (fs /Fsc)2 + (fb/Fbc )2 < 1 where fs = applied shear stress Fsc = shear buckling stress fb = compressive stress due to bending Fbc = local buckling stress due to bending

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When a stiffened web buckles due to shear before the ultimate capacity is reached, the web is not taken into account when determining the resistance of the section to bending and only the flanges are assumed to provide the resisting moment.

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C9.6.5 Web crushing Failure of a web, under the direct local action of an applied force or reaction in the plane of the web, may be caused by (a) buckling of the web, with an upper limit of yielding along the contact line; or (b) local bending of the radiused web to flange junction in formed sheet sections (see Figure C18). Buckling of the web due to a local force is treated by Timoshenko and Gere (1961). For a load distributed along a short length, n, the force required to buckle that length of web as a column is added to the value for the local force. Dividing this total force by the nominal area (n + h)t gives the nominal buckling stress, Fe , for which the slenderness is given by λ = 2h/t. This value of M is used to obtain the buckling stress from the buckling curve for the alloy; this stress is reduced by the presence of compressive bending stress in the web. The reduced stress is then multiplied by (n + h)t to give the resistance. At the ends of a member, the nominal area is reduced linearly, over a length h, to (n+h/2)t. In formed sheet shapes, the radiused corners cause a serious reduction in local strength (see Figure C18). The formula adopted is from the 1983 edition of CSA S157 and parallels that used for steel, but with some simplifications based on tests in which the influence of the compressive bending stress on the crushing strength was also shown.

t

R

h

Figure C18 Web crushing in formed sheet sections (See Clauses 9.6.5(b) and C9.6.5.)

C9.7 Members with combined axial force and bending moment C9.7.1 Axial tension and bending In the general case, the limit is again provided by first yield in the gross section based on elastic analysis or rupture across the net section. In compact sections, the fully plastic condition provides the limit. The formula was developed for rectangular bars but gives acceptable values for most solid shapes (Valtinat and Dangelmaier, 1989). The influence of the tension force on the stability of beams is an extension of the formula used by Hill and Clark (1951) for beam-columns.

C9.7.2 Eccentric tension Figure C19 illustrates some of the arrangements considered. In general, the limiting conditions are first yielded in the gross section using elastic analysis, as expressed in Item (a) of Clause 9.7.2 of the Standard, and the attainment of the ultimate strength on an effective net area derived using a plastic analysis, as expressed in Item (ii)(1). In Clause 9.7.2(ii)(1), the strength of single angles loaded through one leg (Figure C19(a)) is from Marsh (1969), using the ultimate tensile strength at the net section.

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Commentary on CSA S157-05, Strength design in aluminum

Marsh also demonstrates that a lug (clip) angle attached to the outstanding leg, in an attempt to harness the strength of that leg, is ineffective. In Clause 9.7.2 (ii)(3), for a channel connected by the web (see Figure C19(b)), the fully plastic condition, if the ends were pinned, gives an effective area equal to the web area plus 0.4 times the flange areas. Allowing for some end fixity permits the effective net area to include half the flange areas. In Clause 9.7.2(ii)(4), for double angles bolted each side of a gusset plate, the gauge line is usually half the leg width from the heel. The resulting eccentricity is accounted for by a reduced net area.

w

g

b

(a)

(b)

Note: The addition of lug angles to engage the outstanding leg has a negligible benefit.

Figure C19 Eccentric tension members (See Clauses 9.7.2 and C9.7.2.)

C9.7.3 Beam-columns C9.7.3.1 Members not subject to lateral-torsional buckling Owing to the wide variety of member types that can act as beam-columns, a general method has been selected that recognizes the different limiting stresses, as in the cases of thin-wall sections and lattice masts. For failure in the plane of the applied bending moment, the maximum stress is limited to the value that leads to collapse, which might be the yield strength, the effective yield strength for welded members, the stress to cause local buckling that precipitates collapse, the effective strength where there is a postbuckling reserve, or the buckling stress for the chord of a lattice mast. The design formula adopted for the limiting condition is as follows: Cf /A + Mf /S(1 – Cf /Ce ) = φy Fo where Cf = the factored axial load A = cross-sectional area Mf = the factored moment S = section modulus Ce = the elastic axial resistance of the member as a column buckling in the plane of the applied bending moment Fo = the limiting stress Comparisons between predictions using this method and test results (Hill et al., 1956) are shown in Figure C20 in which Mo = SFo and Co = AFo. The specimens in these tests had some plastic reserve; hence, the evident conservativeness of the design formula.

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1.0

Cf Co

= Circular hollow sections = Square hollow sections 0.0 0.0

Mf Mo(1-Cf ) Ce

1.0

Note: From Hill et al., 1956.

Figure C20 Beam-columns: Comparison of test and code values (See Clause C9.7.3.1.)

C9.7.3.2 Members with biaxial moments, not subject to lateral-torsional buckling Members with moments applied about both axes are relatively rare in aluminum structures, and when they do occur, it is usually the maximum stress that controls, with no concern for torsional modes of instability. For those cases requiring a general solution, there is an established theory to give the critical maximum stress (Timoshenko and Gere, 1961) from which a value of the normalized slenderness can be obtained to give the normalized buckling stress.

C9.7.3.3 Members subject to lateral-torsional buckling Lateral-torsional buckling of beam-columns, which involves an instability in the weak direction (not in the plane of bending) requires a treatment different from that used for in-plane bending failure. The relationship used for steel was shown by Clark and Hill (1960) to apply equally well to aluminum: Cf /Cry + Mf /Mr (1 – Cf /Cex) = 1 where Cf = the factored axial force Cry = the factored axial resistance for buckling in the weak direction Mf = the factored applied moment Mr = the factored moment resistance for lateral buckling. Cex = the factored axial force to cause elastic buckling by bending in the strong direction

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This differs from the expression in Clause 9.7.3.2 in the first term, where Cf /Cry replaces Cf /AFo , and in the second term, where Mf /Mr replaces Mf /SFo.

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C9.7.4 Eccentric compression C9.7.4.1 General case In Clause 9.7.4.1(a), the beam-column formula of Clause 9.7.3.1 is used for eccentrically loaded columns. However, because the basic formula used for beam-columns failing in the plane of bending is derived for a sinusoidal moment distribution, while the moment due to an eccentric axial force is uniform, a factor is introduced representing the first term of the Fourier expansion for a square wave, i.e., 4/π ≈ 1.2. In Clause 9.7.4.1(b), since the analysis for lateral-torsional buckling is based on a uniform moment, there is no need to adjust the applied moment, and Clause 9.7.3.3 applies.

C9.7.4.2 Single angle bracing members Based on tests reported by Marsh (1969), and drawing on the theory of torsional-flexural failure under eccentric load (Timoshenko and Gere, 1961), the design formulas explicitly include the combined effects of the torsional and flexural components of the buckling mode of discontinuous single angle compression members loaded through one leg.

C9.7.5 Shear force in beam-columns In Clause 9.7.5.1(a), shear force in a beam-column is increased by the presence of an axial force in the same ratio that the moment is increased, and the same multiplier is used. In Clause 9.7.5.2(a), for eccentrically loaded columns, the end slope is given by: θ = Cf eL/2EI(1 – Cf /Ce ) Using Ce = π2 EI/L2 leads to a shear force of Vmax = Cf θ = 5 Cf e/(Ce /Cf – 1)L The minimum value of the shear force follows traditional practice.

C9.8 Built-up columns C9.8.1 Spacing of connectors The limiting of the local slenderness of individual bars is intended to ensure that shear flexibility, represented by the bending of the individual members between fasteners, will not dominate the action of the member.

C9.8.2 Multiple-bar members with discrete shear connectors Shear flexibility in battened or stitch-bolted double members, such as channels and angles (see Figure C21), is accounted for in the manner indicated by Timoshenko and Gere (1961) and Bleich (1952).

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Stitch fasteners

Battens

Figure C21 Built-up members with discrete interconnectors (See Clauses 9.8.2 and C9.8.2.)

C9.8.3 Double angle struts Double angles are asymmetrical sections that may fail by combining the modes of overall flexure, local flexure between stitch bolts, and torsion. In this combined mode, flexure, with the influence of the shear flexibility between fasteners, is treated as though it were a simple column. Clause 9.4.3.3 provides the required relationship with xo /ro ≈ 0.5.

C9.8.4 Lattice columns and beam columns Lattice columns are treated in the same manner as other columns and beam-columns, using the buckling stress of a chord as the limiting stress. The direction of bending influences the limiting moment, as the distance from the centroid to the extreme chord varies with the axis of bending. Shear flexibility is discounted as its influence is small. However, applied shear forces are augmented by the action of the axial force, and the lattice bracing must be designed for this increased shear force.

C9.9 Members in torsion It is expected that the full plastic resisting torque can be utilized in those sections that rely on St. Venant action in torsion.

C10. Panels C10.1 Flat panels with multiple stiffeners C10.1.1 Axial compression It is usual for panel stiffeners to run in the direction of loading, as transverse stiffeners are much less effective. Treating the panel as an orthotropic plate (Timoshenko and Gere, 1961) and neglecting the torsional stiffness (Lind, 1973; Sherbourne et al., 1971) leads to a buckling force per unit width given as follows:

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Ncr =

Commentary on CSA S157-05, Strength design in aluminum

2π2 (D1D2 )1/ 2 b2

where D1 = longitudinal rigidity D2 = transverse rigidity This gives the elastic buckling stress and is used to derive the formulas for the slenderness.

C10.1.2 In-plane shear Closely spaced multiple stiffeners in panels will usually be in the direction of the shorter dimension, and buckling under the action of shear force takes the form of single half-waves across the panel with short waves along the panel. For flat sheets with simple stiffeners attached to the surface, the behaviour is reasonably well defined, but in tests on panels formed from sheet, there has been no clear buckling stress, while the distortion of the profile of the panel at the boundaries also limits the load-carrying capacity. By adopting the classical theory of elastic shear buckling in orthotropic panels (Timoshenko and Gere, 1961), taking no account of the torsion constant, and using the plate buckling curves to introduce the effects of non-linear material behaviour and imperfections, the design procedure gives conservative predictions for the buckling strength, but the boundary strength will usually be determined by tests or by reference to the literature.

C10.2 Curved panels and tubes C10.2.1 Axial compression Based on the work of Clark and Rolf (1964), the critical stress for curved walls subjected to longitudinal compression (Figure C22) is calculated using a formula that reflects the increasing sensitivity of the buckling stress to imperfections as the ratio R/t increases. Clark and Rolf’s formula for the slenderness is used, in conjunction with the yield strength of the alloy, to give the normalized slenderness for entry into the plate-buckling formula.

b

a

Figure C22 Curved panel (See Clauses 10.2.1 and C10.2.1.)

C10.2.2 Radial compression Following Timoshenko and Gere (1961), the value for the slenderness is taken as that for a well-formed cylinder satisfying the usual commercial tolerances. While cylindrical tubes carrying radial compression are not as sensitive to imperfections as they are when axially loaded, if there is severe ovality, a full analysis should be made. On the other hand, the presence of circumferential stiffeners increases the stability and reduces this sensitivity.

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Internal pressure helps to increase the stability of a shell by inducing a tension stress and by reducing the imperfections. In these cases reference will be made to the literature.

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C10.2.3 Shear The formula is taken from Timoshenko and Gere (1961) to give the elastic buckling stress and is converted to a form that can be used with the normalized buckling curves for plates.

C10.3 Curved axially stiffened panels in axial compression The combined action of the curved wall and the stiffened wall acting as a column of length equal to the spacing of ring stiffeners is obtained by the simple summation of the stresses required to cause each type of buckling.

C10.4 Flat sandwich panels C10.4.1 General Sandwich panels considered in this clause are of the type composed of two thin aluminum skins bonded to a core material that may possess a relatively low elastic modulus. There are rigid core materials, such as wood, which are capable of fully stabilizing the skin and may contribute to the overall strength. Panels with cores of kraft paper honeycomb or formed sheet metal are designed using the special procedures for these products.

C10.4.2 Panel bending The bending strength of a panel is assumed to derive entirely from the skins, with no benefit from edge or internal framing members. The designer is free to consider such members as independent sources of strength.

C10.4.3 Panel buckling C10.4.3.1 Overall buckling controlled by the skins For the buckling of a complete panel in compression, the limiting extreme fibre stress, Fo , is that which causes buckling of the skin: (a) In the case of a panel with no support at the longitudinal edges, the panel is treated as a column. (b) When all the edges are supported, the panel behaves as a plate. The formula used is taken from Platema (1966).

C10.4.3.2 Influence of shear in the core In both the above cases the low shear modulus of the core will influence the buckling load. The final relationship between the ultimate load, Cu , the load that causes flexural buckling, Cr , and the load that would cause pure shear buckling, Gcdb, is

1 1 1 = + Cu Cr Gc db where Gc = shear modulus of the core d = panel thickness b = panel width

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C10.4.4 Skin buckling The skin of a sandwich panel behaves as a plate on an elastic foundation, for which the critical stress is given theoretically by

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Fc = 0.86 (EEcGc )1/3 where Ec and Gc = the elastic properties of the core Because this mode of buckling is very sensitive to imperfections, the factor is reduced to 0.5. Equating the modified expression to the Euler formula gives the slenderness λ = 4.5E1/3/(EcGc )1/6 This is used with the yield strength of the skin alloy to obtain the buckling stress.

C10.4.5 Core strength Bond between the skin and core must resist the shear force due to the lateral load and provide the tensile strength required to hold the skin to the core. This required tensile strength has been derived on the assumption of an initial bow in the skin equal to 0.001 times the wave length of the buckle.

C11. Resistance of connections C11.1 General C11.1.1 Connection types The Standard recognizes the use of a wide variety of fastening methods, proprietary or otherwise. It is not the intent of the Standard to limit the use of these methods, provided that the suitability of the material can be shown.

C11.2 Mechanical fasteners C11.2.1 General The Standard gives design procedures for bolts and solid rivets only. Mechanical fasteners are usually galvanized or cadmium-plated steel, stainless steel, or aluminum. The actual material of a fastener, given that it is of sufficient strength, is usually only of concern where corrosion might occur. In the case of rivets, the material must be capable of being upset without damaging the parent material.

C11.2.1.4 Slip-critical joints Pre-loaded bolts, permitting the joint to transfer the force by friction, are not commonly used in aluminum, but where rigidity under service loads is essential, there has been sufficient experimental work to provide recommendations (ECCS, 1978). Surface treatment, such as sanding, is required. It is preferred that all bolts act in bearing at the ultimate state. When the holes are slotted in the direction of the force and the level of security is to be maintained, a safety margin on the theoretical slipload in excess of 2 is provided.

C11.2.1.6 Maximum number of fasteners Longitudinally loaded rows of fasteners are not uniformly stressed, and although yielding will largely reduce this non-uniformity by the time the ultimate load is reached, a joint that has many bolts in line may possess a reduced overall load capacity. The limit of six fasteners (at a spacing of 3d, this gives a length of 15d) has been shown to give an acceptable performance (Francis, 1953).

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C11.2.2 Fastener spacings

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C11.2.2.1 Minimum spacing Minimum spaces between fasteners and edge distances are governed by (a) clearance for bolt heads and driving tools; (b) build-up of compressive stress generated by cold formed rivets; and (c) limits to the validity of the design formulas given.

C11.2.2.2 Maximum spacing Maximum spaces between fasteners that are treated as acting together may be governed by local plate buckling in compression. For rows of fasteners widely spaced across the direction of force, with s < 1.3 g, the plate buckles into waves with all edges fixed, represented by a slenderness λ = 1.3 g/t. For fasteners widely spaced along the direction of force, with s > 1.3 g, the plate buckles over the length s, fixed at the transverse lines of fasteners, for which λ = 1.7 s/t. For staggered patterns of fasteners, the zone with fixed edges is defined by two pairs of fasteners spaced at 2g and 2s. When s = g, this gives a value of λ = 1.9 s/t. Linear relationships are adopted as s/g→0 and g/s→0, to the limiting values of 1.3 g/t and 1.7 s/t, respectively.

C11.2.3 Bolts and rivets in shear and/or tension Although the von Mises’s criterion, which gives a shear yield strength equal to 1/ 3 times the tensile yield strength, does not apply to ultimate strengths of the “engineering” variety, the customary ratio of 0.6 is adopted, as it usually gives conservative values for the shear strength. A bolt in tension fails across the net section at the thread, which is approximately 0.7 times the gross area. The constrained region of the net section inhibits necking, leading to an ultimate tensile strength closer to the “true” value than to the “engineering” value. Some recognition of this is seen in the formula for the tensile resistance, Tk = 0.75AFu , where A is the gross area. Interaction between shear and tension follows steel practice in CAN/CSA-S16.

C11.2.4 Bolts and rivets in bearing C11.2.4.1 Bearing strength The stress exerted by the fastener on the wall of the hole (termed the bearing stress) is not, as such, a design consideration. Failure occurs due to the tearing of the material adjacent to the bolt hole and thus varies with the end distance for force directed towards an edge. The limiting force is related to the ultimate strength of the metal (Fisher and Struick, 1964; Marsh, 1979). No use is made of a “yield strength” in bearing. As the end distance, e, increases, the resistance increases, approaching a constant value after the edge distance exceeds approximately twice the bolt diameter. Calculations based on shearing along the two planes beside the hole lead to the resistance R = 2Fsu et. Using the Tresca criterion, Fsu = Fu /2, gives R = Fu et, a convenient expression that has been shown to be conservative. Bearing stress on the fastener itself is never a consideration (Hartman et al., 1944).

C11.2.4.2 Lap joints Unrestrained lap joints in tension are eccentrically loaded and distort so as to introduce tension in the fastener and thereby cause a reduction in strength. This is most pronounced when the plates are of equal thickness, in which case the bearing strength is halved. The expression used, based on tests for steel sheet (Baehre and Berggen, 1973), requires that the thicker sheet be three times the thickness of the thinner sheet if Clause 11.2.4.1 is to be valid for the thinner sheet.

C11.2.4.3 Oblique end edges For force directed at an angle to an oblique end edge (see Figure C23), the formula for the resistance gives a transition between bearing failure, when the angle between the force direction and the end edge is 90°, and tension failure, when the angle is 0°.

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e

Figure C23 Tear-out at oblique ends (See Clauses 11.2.4.3 and C11.2.4.3.)

C11.2.5 Tear-out of bolt and rivet groups (block shear) C11.2.5.1 Tension: Rectangular patterns; C11.2.5.2 Tension: Trapezoidal patterns; C11.2.5.3 Block shear in webs The formulas presented are simple extensions of the tear-out model for single fasteners (Marsh, 1979) and include the tension failure of material between the fasteners in line across the direction of stress (see Figure C24 (a) and (b)). “ Tear-out” applies particularly to tension failures. The term “block shear” is more commonly applied to shear failure in webs (see Figure C24(c)). Only the ultimate strengths in shear and tension along the planes of rupture are considered to be relevant.

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s

e

s

e

g

g

(a)

(b) s

e1 g h

s

e2

(c)

(d)

Figure C24 Tear-out of fastener groups (block shear) (See Clauses 11.2.5.1–11.2.5.4, C11.2.5.1–C11.2.5.4.)

C11.2.5.4 Tear-out of groups subjected to torque It is assumed that failure occurs along a single circular shear plane through the centres of the holes forming the perimeter of the group, using the Tresca criterion Fsu = Fu /2 (see Figure C24(d)). This type of failure has been demonstrated in tests and the treatment is known to be conservative, but there is no body of research data available.

C11.2.6 Eccentrically loaded fastener groups C11.2.6.1 Highest fastener force (elastic) For elastic behaviour, when a group of N fasteners is subjected to a factored force, Pf , applied at an eccentricity, e, the group is considered to carry a force, Pf , through the centroid of the group, and a moment, Pf e (see Figure C25). If each bolt has a spring constant, k, the group will translate bodily a distance

σ=

Pf Nk

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in the direction of the force Pf . The group will rotate through on angle given by

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θ=

Pfe KIo

where, assuming fasteners of unit area, the polar moment of inertia about the centroid may be expressed as Io = Σ r2 i where ri = distance from the centroid to the ith fastener Combining this translation and rotation, the centre of rotation of the group is at a distance, c, from the centroid such that σ = cθ, resulting in the formula

c =

Io Ne

The maximum force on a fastener is then

Rmax =

Pf (e + c )dmax

(Io + Nc ) 2

where dmax = the distance from the centre of rotation to the furthermost fastener Using Io = cNe, the formula for the maximum force on a fastener becomes Rmax = (Pf /N)(dmax /c)

dmax

O C

e P

c

Figure C25 Eccentrically loaded fastener groups (See Clauses 11.2.6.1 and C11.2.6.1.)

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C11.2.6.2 Factored resistance (fully plastic) At the ultimate resistance, the fasteners are all assumed to be fully plastic, and although it is not strictly valid, it is convenient to use the elastic centre of rotation (Marsh, 1982a) which leads to

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Pf (e + c) ≤ Rr Σdi where Rr = factored resistance of a fastener di = distance from the centre of rotation to the ith fastener The analysis used by Crawford and Kulak (1971) for the non-linear behaviour of steel bolt groups gives similar results.

C11.3 Welded connections For engineered assemblies, welding is carried out using manual or automatic TIG, MIG, or plasma arc methods. Because of the different levels of heat input among these methods there are some variations in the strength of the resulting joints. To accommodate these variations within a design standard is impractical because at the time the design is being prepared the method of welding may not be known, and it may differ between fabricators of the same product. It is thus reasonable to specify a single value for each alloy and combination of alloys, which can be expected to be realized by a qualified welder, regardless of the method used. In Tables 2 and 3, the proposed values are taken from CSA W47.2 or are compromises between the specified strengths in other documents (ANSI/AWS D12-83; ECCS 1978; Hill et al., 1962; Marsh, 1985, 1988; Moore et al., 1971; Soetens, 1987). If higher values can be justified for general practice, they may be used. A similar argument is used for the extent of the heat-affected zone (see Clause 11.3.6). This can vary widely between automatic MIG welds and manual TIG welds, but without an exact knowledge of the planned procedure, the designer must use a value that is reasonable for welds in general. A band 25 mm on each side of the weld (Hill et al., 1962) is now widely adopted because it is simple, typical, and the value adopted has little influence on the final design. Should welding procedures in a particular case give a consistently higher strength or narrower heat-affected zone, the design values may be revised, subject to the usual controls. For design purposes, the nominal dimensions of a weld bead are the plate thickness for butt welds, and the shortest distance through the throat for fillet welds with no regard paid to bead reinforcement or root penetration. Inspection after fabrication will establish whether welds of the required size and quality have been provided, using CSA W59.2.

C11.3.2 Butt welds Butt welds designed to carry calculated forces should be full-penetration groove welds. If partialpenetration welds are needed, they should meet the requirements of CSA W59.2, and the weld size should be sufficient to provide the design throat size, taking into account the reductions specified in CSA W59.2. The joint is composed of the weld bead and the associated heat-affected zone, which may have different strengths. Because the bead is narrow, yielding results in little overall distortion; thus, the ultimate tensile and shear strengths can be used.

C11.3.2.1 Tension For the resistance in tension, the ultimate strength in Table 3 is that of the combined weld metal and base metal, unless this exceeds the yield strength of the unaffected base metal (ECCS, 1978).

C11.3.2.2 Compression For the resistance in compression, in cases where the welded joint is constrained to remain straight, the rules for tension are used (ECCS, 1978). This condition occurs at the end of a compression member (Moore et al., 1971) or a T-joint carrying a moment (Marsh, 1985).

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Where instability can occur, the compressive stress is limited to the yield strength in the HAZ, as any local reduction in the elastic modulus can precipitate local buckling.

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C11.3.2.3 Shear At yield it is customary to use the von Mises’s criterion which limits the shear stress to Fy / 3 , and any combination of normal and shear stresses to

(Ft2 + 3Fs2 ) < Fy2 While von Mises’s criterion is recognized as a valid condition for yielding, it cannot be expected to apply to the “engineering” values used at the ultimate strength, for a number of reasons: (a) it is based on elastic theory; (b) the true ultimate strength on the reduced area is not, in general, available; (c) the “engineering” ultimate stress is artificial even at the point of highest load because the area is actually A/(1 + σ) where σ is the strain at maximum load; and (d) shear failure involves no “necking”. The result is that the ultimate shear strength of a fillet weld can be as high as 75% of the “engineering” tensile ultimate. In the Standard, 0.6Fu has been adopted as a reasonably conservative value, when compared with the measured values given in MIL-HDBK-5G.

C11.3.3 Fillet welds C11.3.3.1 Concentrically loaded fillet welds As there is no generally accepted method for computing the actual stress on the throat or interface of a fillet weld for different directions of loading, values obtained from direct tests and from the qualifying of weld procedures provide the basis for the design strengths adopted. For the pure shear parallel to the weld direction, the strength is taken as 0.6Fwu. Because different researchers have tested fillet welds loaded in different directions (Marsh, 1985; Moore et al., 1971; Soetens et al., 1987), the conclusions about the influence of the direction of force are not always in agreement. It has not been explained why a weld loaded as shown in Figure C26(c) is stronger than one loaded as shown in Figure C26(b), but tests have demonstrated this tendency. The resistance per unit length of a fillet weld is given by Vk = kaFwu where a = throat size Fwu = ultimate tensile strength of the weld bead

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Y

Y Z X

X

Z

(a)

(b)

(c)

Figure C26 Fillet welds (See Clauses 11.3.3.1.1 and C11.3.3.1 and Table C4.) The values for the factor k for the three directions of loading are given in Table C4, where they are compared with the values from other standards.

Table C4 Strength of fillet welds: Comparison of some code values for the factor k (See Clause C11.3.3.1.) Directions* Standards

X

Y

Z

S157-05 CAN3-S157-M83 CAN/CSA-S16 (steel) Eurocode 9 (aluminum) Aluminum Association

0.6 0.6 0.67 0.6 0.6

0.7 0.6 1.0 0.7 0.6

0.8 0.85 1.0 0.7 0.6

*For directions X, Y, and Z, see Figure C26.

For oblique load directions, the Standard uses a simple spherical relationship between the different strengths. Other standards take the calculated stresses on the weld throat due to the orthogonal components of the force and combine them using the von Mises’s criterion for yielding. As discussed in Clause 11.3.2.3, this method is somewhat artificial; in this Standard use is made of the actual measured strengths of fillet welds.

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C11.3.3.2 Eccentrically loaded fillet welds

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C11.3.3.2.1 Moment in the X-Z plane In Clause 11.3.3.2.1(a), an elastic analysis to determine the highest stress in a weld pattern subjected to eccentric load is required when fatigue is a consideration. The method used for this analysis is extended to give the limiting static load when the welds are fully plastic (Marsh, 1985). The procedure follows that for eccentrically loaded groups of fasteners. In a weld forming any pattern, let the length of the weld be H, the weld throat be a, the polar moment of inertia of the weld pattern about its centroid be Io , and the rigidity of a unit area of weld be κ. If the weld is subjected to a factored force Pf , applied at an eccentricity e (see Figure C27(a)), the body connected by the weld will be displaced in the direction of the force, a distance given by σ = Pf /κHa It will rotate through an angle given by θ = Pf e/κIo These two distortions can be represented by a rotation about a point at a distance c from the centroid given by c = σ/θ = Io /eHa The maximum shear stress in the weld throat is then given by Fs = Pf (e + c)dmax /(Io + Hac2) Using Io = ceHa, the formula for the maximum force per unit length of the weld becomes vf = (Pf /H)(dmax /c) X Y

Z e Pf

a

dm

Pf t

X e

O

s

t

Z

L

C c

(a)

(b)

Y

(c) Pf

Figure C27 Eccentrically loaded fillet welds (See Clauses 11.3.3.2 and C11.3.3.2.) In Clause 11.3.3.2.1(b), in order to obtain the ultimate capacity, for simplicity, the elastic centre of rotation is used even though the behaviour is non-linear. The applied factored moment, Mf , and the design moment resistance, Mr , can then be calculated using Mf = Pf (e + c) Mr = vr ΣLi di

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where Pf = vr = Li = di =

© Canadian Standards Association

the factored load the factored resistance per unit length of the weld the length of the ith element the distance from the centre of rotation to the centre of the ith element

Comparison of this method with the computer-generated values for the non-linear analysis of steel fillet welds (Butler, Pal, and Kulak, 1972) shows very close agreement between them.

C11.3.3.2.2 Moment in the X-Y plane For eccentrically loaded double fillet welded T-joints (see Figure C27(b)), it is necessary to consider different strengths in tension and compression. If Ft and Fc are the tensile and compressive strengths, respectively, in a rectangular section, L × t, assuming plastic behaviour, the ultimate moment is given by Mo = (tL2/2)Ft Fc /(Ft + Fc ) The tensile strength of the welds is reduced by the presence of the shear force, and use is made of the formula for combined forces (Marsh, 1988b).

C11.3.4 Flare groove welds Flare groove welds rely on the penetration achieved for their strength. As this varies with the radius of the surface, it is not easy to predict with confidence the final effective throat size, and it is required that the specified size be demonstrated in practice.

C11.3.5 Slot and plug welds Slot and plug welds, in which a slot or hole is filled with weld metal, are subject to shrinkage cracks and are not permitted.

C11.3.7 Stud welds Welded studs are usually of an alloy comparable to welding wire and similar properties will be obtained.

C12. Fatigue resistance C12.1 Load cycles of constant amplitude The fatigue design requirements of the Standard are based on a “stress-life” (S/N) approach in which fatigue performance is determined from S/N data for a selection of representative member and joint categories. These data were obtained from laboratory tests and relate the average stress to the fatigue life. The stress is that calculated using the gross section properties. The influence of local stress-raisers is accounted for in the curve for the particular case. The material in Clause 12.5 is new in this edition of the Standard. It allows the use of a fatigue notch factor, or stress concentration factor, with the S/N curve for the base material for the analysis of elements with stress-raisers. The S/N data provided in Figure 2 are essentially the same as in the previous edition of the Standard, but with some refinements and changes in presentation. To be consistent with other standards, the data are now presented as lines on a log-log plot, and the stresses are given in terms of range rather than semi-range. The lines are now extended to 104 cycles where they may be required for the analysis of welded joints. The stress ranges at 104 cycles were determined by linear extrapolation of the previous semi-log plots. Note that the stresses for such low cycles may be limited by static rather than fatigue requirements (see Clause 12.4). From 104 cycles the ranges are shown as straight lines to values at 5 × 106 cycles, which are unchanged from the previous edition of the Standard. Beyond 5 × 106 cycles the lines have been extended to 108 cycles with a slope of 0.1. This is for use only when estimating fatigue life in the case of spectrum loading. For constant amplitude loading, the range at 5 × 106 remains the endurance limit.

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Commentary on CSA S157-05, Strength design in aluminum

It should be noted that the S/N curves in Figure 2 do not recognize any difference in the fatigue performance of different alloys. This is a simplification that is supported by the experimental data for the higher cycles but ignores the differences that exist for the lower cycles. The influence of high residual and/or mean stress is included; thus, in cases where these stresses do not occur, the values in the Standard are conservative. Residual stresses, such as those created by peening, can be beneficial, but such improvements require special study. Fatigue stresses given in the Standard are for dry service conditions, and allowances should be made where the environment may be corrosive. Tests have shown, for example, that a continuous water spray can reduce performance by as much as 50% (steels and stainless steels show similar reductions). The categories of joints shown in Figure 1 cover many of the configurations that are likely to occur in practice, but as they cannot cover all the possible details, it often falls to the designer to make an appropriate selection or interpolation of categories, on the basis of the similarity of geometry, load path, and stress concentrations. Knowledge of the loads that act on a member in service is usually less certain than is the determination of the S/N curves. In the rare cases when the loads are of constant amplitude, such as those imposed by rotating conditions, the fatigue life or allowable stress range can be read directly from the S/N diagram, with the stress range at 5 × 106 cycles taken as an endurance limit. Other cases are discussed below.

C12.2 Known load spectra Where a stress spectrum can be determined by a cycle counting procedure such as the rainflow method, a cumulative damage analysis is used to determine the life, including all the stresses in the 5 × 106 to 108 cycle interval. Note that if all the stress ranges in the spectrum are below the fatigue strength for 5 × 106 cycles, no fatigue damage is expected.

C12.3 Unknown load spectra When the stress spectrum cannot be determined, the estimate of fatigue life is necessarily more conservative. This is ensured by limiting the stress range that is estimated to be exceeded 106 times to the value allowed for 5 × 106 cycles. An alternative procedure for fatigue design that is gaining favour, particularly in the automotive industry, is “strain-life” analysis. This method has the advantage of treating high-stress, low-cycle spectra more rationally than methods based on constant amplitude S/N diagrams, but it has the disadvantage of requiring empirically determined material constants and a detailed analysis of the local strain at the points of expected fatigue. There are not sufficient material data available for this method to be included in this Standard, but its use is not precluded. Additional information regarding “strain-life” analysis can be founded in the Fatigue Design Handbook, published by SAE. Another procedure for fatigue design, based on an analysis of crack propagation, is used extensively for aerospace structures. Both the stress-life and strain-life methods of analysis, which relate mainly to crack initiation, can be augmented by crack growth analysis to determine how much “reserve” life is available after crack initiation. For most applications addressed by this Standard, the appearance of a crack is considered to be failure.

C13. Tests C13.1 General Testing of aluminum joints, members, and assemblies, to check that the strength is adequate for the intended purpose or to confirm that the serviceability limits are satisfied, is allowed by the Standard in lieu of design calculations, subject to the agreement of the regulatory authority. Because most of the Standard is concerned with ultimate resistance, this clause deals only with static loads. Requirements for tests to establish impact resistance and fatigue life are outside the scope of the Standard.

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Tests may be used to qualify items before going into production or for the acceptance of a special assembly. Research test programs, to provide the basis for design procedures, are not covered by this clause. Where an industry requires testing to its own standards, such tests may replace those required by this clause. However, the safety requirements implied to satisfy this Standard cannot be waived.

C13.2 Test methods It is understood that the test procedures will accurately simulate the conditions of the intended service, and that the records of the test, including loading rate and duration, points of measurement for stress and deflection, and mode of failure, will be in sufficient detail to permit a third party to interpret the findings.

C13.2.4 Tests to destruction are not always necessary. Where it is only required to show that a component or assembly meets the specified strength and rigidity, confirmatory tests may be conducted.

C13.2.5 Tests to destruction will be carried out when the actual characteristic resistance of an item is to be established. These are performance tests, and the information they provide will determine under what conditions of loading the item may be used. It is sometimes difficult to establish the maximum load-carrying capacity. Distortion after yielding or a change in the mode of action, as when roofing panels develop membrane action, may be responsible for an increasing resistance with deflections that cannot be said to be uncontrolled. In this situation, there should be an agreement on the level of distress that will be taken to represent the ultimate limit state.

C13.3 Test procedures C13.3.1 Confirmatory tests C13.3.1.1 Serviceability The service load, with a factor of 1.0, is applied to the test item. As this test usually examines the rigidity of the item and safety is not a concern, the test is satisfied if the specified limit is not exceeded. There is no need for any added margins.

C13.3.1.2 Ultimate limit state Load and resistance factors in combination give the security that is expected of a structural item. The dead load and specified service loads are multiplied by the appropriate values of α and the characteristic resistance is multiplied by φ, according to the requirements of the Standard. It follows that the load capacity of the item should be at least equal to the action due to the factored loads divided by the resistance factor. The choice of resistance factor, φ, will be governed by the type of failure.

C13.3.2 Performance tests C13.3.2.1 Characteristic resistance The load sustained at the condition that is agreed to represent the ultimate limit state is measured. Because conducting a test removes some of the unknowns and reduces some of the variables that are used to determine the resistance factor, the characteristic resistance is taken as 0.9 times the mean value obtained. This is based on the assumption that tests of this nature show a coefficient of variation of around 0.1, making the resistance approximately one standard deviation below the mean test value. Should a sufficiently large number of tests be made, then a full statistical analysis will be used to find the mean and the standard deviation. A characteristic resistance given by mean less one standard deviation is felt to be justified because of the greater security offered by testing.

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C13.3.2.2 Adjustment for variation in yield and ultimate strength and dimensions When failure is clearly due to yielding, the load obtained will be reduced in the ratio of the specified yield strength to the measured yield strength. Should failure be due to rupture at a weld or bolted section, the ultimate strength must be used. In cases where instability occurs in the elastic range, deemed to be when λ > 1.5, no adjustment is needed, but should it occur in the elastic-plastic range, then the test value will be reduced by the factor

⎡F ⎛ F ⎞⎤ ⎢ y + λ ⎜ 1 − y ⎟⎥ ⎢F ′ 1.5 ⎜ Fy ′ ⎟⎠ ⎥ ⎝ ⎣ y ⎦ where Fy = specified yield strength Fy’ = measured yield strength No adjustment is made if the properties are lower than those specified, and the test is passed.

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Bibliography

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Note: The publications listed below are currently used by the industry.

Aluminum Association, Inc. Aluminum Design Manual, 2000 Aluminum Standards and Data, 2003 AWS (American Welding Society) ANSI/AWS D1.2-97 Structural Welding Code — Aluminum “Welding Aluminum”, Welding Handbook: Volume 4, Seventh edition, 1984 CSA (Canadian Standards Association) CAN/CSA-S16-01 Limit States Design of Steel Structures S408-1981 (R2001) Guidelines for the Development of Limit States Design W47.2-M1987 (R2003) Certification of Companies for Fusion Welding of Aluminum W59.2-M1991 (R2003) Welded Aluminum Construction European Committee for Standardization Eurocode 9-2001 Design of Aluminum Alloy Structures NRCC (National Research Council Canada) National Building Code of Canada, 1995 National Building Code of Canada, User’s Guide — NBC 1995, Structural Commentaries (Part 4), 1995 SAE (Society of Automotive Engineers) AE-10, Fatigue Design Handbook, Third edition, 1997 United States Department of Defense MIL-HDBK-5G 1994, Chapter 3, “Aluminum,” Military Standards Handbook, Metallic Materials and Elements for Aerospace Vehicle Structures Other Publications Baehre, R., and L. Berggen. (1973). Joints in Sheet Metal Panels. Document DB. Stockholm: National Swedish Building Research. Bleich, F. (1952). Buckling Strength of Metal Structures. New York: McGraw Hill. Brady, W.G., and D.C. Drucker. (1955). Investigation and limit analysis of net area in tension. Transactions of the ASCE, 120.

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Brungraber, R.J., and J.W. Clark. (1962). Strength of welded aluminum columns. Transactions of the ASCE, 127 (II).

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Butler, L.J., S. Pal, and G.L. Kulak. (1972). Eccentrically loaded welded connections. Journal of the Structural Division, ASCE 98 (ST5): 989-1005. Clark, J.S., and R.L. Rolf. (1964). Design of aluminum tubular members, Journal of the Structural Division, ASCE 90 (ST6): 259. Clark, J.W., and H.N. Hill. (1960). Lateral buckling of beams. Journal of the Structural Division, ASCE 86 (ST7): 175. Clark, J.W., and J.R. Jombock. (1957). Lateral buckling of I-beams subjected to unequal end moments. Journal of the Engineering Mechanics Division, ASCE 83 (EM3). Crawford, S.F., and G.L. Kulak. (1971). Eccentrically loaded bolt connections. Journal of the Structural Division, ASCE 97 (ST3). European Convention for Constructional Steelwork (ECCS) (1978). European Recommendation for Aluminum Alloy Structures, Committee T2. Evans, H.R., and M.J. Hamoodi. (1987). The collapse of welded aluminium plates girders — An experimental study. Thin Walled Structures 5(4). Fisher, J.W., and J.H.A. Struick. (1987). Guide to Design Criteria for Bolted and Riveted Joints. New York: John Wiley and Sons. Francis, A.J. (1953). The Behaviour of Aluminum Alloy Riveted Joints. Report No. 15. London: Aluminium Development Association. Galambos, T.V. (1998). Guide to Stability Design Criteria for Metal Structures. New York: John Wiley. Hartmann, E.C., G.O. Höglund, and H.A. Miller. (1944). Joining aluminum alloys. Steel, August 7. Hill, H.N., and J.W. Clark. (1951). Lateral buckling of eccentrically loaded I-section columns. Transactions of the ASCE, 116. Hill, H.N., J.W. Clark, and R.J. Brungraber. (1962). Design of welded aluminum structures. Transactions of the ASCE, 127 (II): 102. Hill, H.N., E.C. Hartmann, and J.W. Clark. (1956). Design of aluminum alloy members for combined end load and bending. Transactions of the ASCE, 121, 1. Höglund, T. (1971). Simply supported long thin I-girders without web stiffeners subjected to distributed transverse load. IABSE Proceedings, Colloquium, London. Jombock, J.R. (1974). Plastic design of aluminum beams. ASCE Annual Meeting, Kansas City. Kubo, M., and Y. Fukumoto. (1988). Lateral-torsional buckling of thin-walled I-beams. Journal of Structural Engineering, 114.4: 841–855. Lind, N.C. (1973). Buckling of longitudinally stiffened sheet. Journal of the Structural Division, ASCE 99 (ST7). Lind, N.C., N.K. Ravindran, and J. Power. (1971). A review of the effective width formula. Technical Note 6. Solid Mechanics Division, University of Waterloo. Marsh, C. (2001). Post-buckling capacity of outstands. Structural Stability Research Council Proceedings, Annual Technical Session. ________. (2000). Design procedure for the influence of bend radii on local buckling. Structural Stability Research Council Proceedings.

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________. (1998). Design method for buckling failure of plate elements. Journal of Structural Engineering, ASCE 124 (7).

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________. (1990). Influence of lips on local and overall stability of beams and columns. Structural Stability Research Council Proceedings, Annual Technical Session. ________. (1988a). The ultimate shear capacity of welded aluminum stiffened webs. Structural Stability Research Council Proceedings, Annual Technical Session. ________. (1988b). Strength of aluminum T-joint fillet welds. American Welding Society, Welding Research Supplement, 67(8). ________. (1985). Strength of aluminum fillet welds. American Welding Society, Welding Research Supplement, 64(12). ________. (1982a). Discussion on Brandt, G.D., “Rapid determination of ultimate strength of eccentrically loaded bolt groups”. Engineering Journal, AISC,19(4). ________. (1982b). Theoretical model for collapse of shear webs. Journal of Engineering Mechanics Division, ASCE, October. _______. (1979). Tear-out failure of bolt groups. Journal of the Structural Division, ASCE, October. ________. (1969). Single angles in tension and compression. Journal of the Structural Division, ASCE, May. ________. (1967). Background to CSA 190, “Design of Light Gauge Aluminum Products”. Engineering Institute of Canada, Engineering Journal, December. Mazzolani, F.M. (1995). Aluminum Alloy Structures. Second edition. London: Chapman and Hall. Moore, R.L., J.B. Jombock, and R.A. Kelsey. (1971). Strength of welded joints in aluminum alloy 6061-T6 tubular members. Welding Journal, 50(4). Picard, A., and D. Beaulieu. (1989). Theoretical study of the buckling strength of members connected to coplanar tension members. Journal of Civil Engineering, 16 (3). Platema, F.J. (1966). Sandwich Construction. New York: John Wiley. Schmidt, L.C. (1976). Space trusses with brittle-type strut buckling. Journal of the Structural Division, ASCE, July. Sharp, M.L. (1966). Longitudinal Stiffeners for Compression Members. Journal of the Structural Division, ASCE, October. Sherbourne, A.N., C. Marsh, and C.Y. Liaw. (1971). Stiffened plates in uniaxial compression. International Association for Bridge and Structural Engineering Publications, 31-I. Simiu, E., and R. Scanlan. (1996). Wind Effects on Structures. Third edition. John Wiley and Sons. Soetens, F. (1987). Welded connections in aluminum alloy structures. Heron, 32(1). Stowell, E.Z., G.J. Heimerl, C. Libove, and E.E. Lunquist. (1952). Buckling stresses for flat plates and sections. Transactions of the ASCE 117: 545-578. Timoshenko, S.P., and J.M. Gere. (1961). Theory of Elastic Stability. New York: McGraw Hill. Valtinat, G., and H. Dangelmaier. (1989). Zur plastischen tragfahigkeit kompacter alumiumquerschnitte. Teil II. Aluminium, 65. Vilnay, O., and C. Burt. (1988). The shear effective width of aluminium plates. Thin-Walled Structures, 6(2). von Karman, T., E.E. Sechler, and L.H. Donnell. (1932). Strength of thin plates in compression. Transactions of the ASCE 54 (APM-54-5): 53.

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