DIN 18800-02 - Structural Steelwork Design Construction - DIN (1990)

UDC 693.814.074.5

DEUTSCHE NORM

Structural steelwork Analysis of safety against buckling of linear members and frames

November 1990

DIN 18800 Part 2

Contents

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Page

Page

1 General ....................................... 2 1.1 Scope and field of application . . . . . . . . . . . . . . . . . . . 2 2 1.2 Concepts ..................................... 2 1.3 Common notation ............................. 3 1.4 Ultimate limit state analysis ..................... 3 1.4.1 General ..................................... 1.4.2 Ultimate limit state analysis by elastic theory .... 4 1.4.3 Ultimatelimit state analysis by plastic hinge theory 5 .2 imperfections.. ................................ 5 5 2.1 General ...................................... 5 2.2 Bow imperfections. ............................ 6 2.3 Sway imperfections ............................ 2.4 Assumption of initial bow and coexistent initial ........................ 7 sway imperfections . 3 Solid members ..... ........................ 7 7 3.1 General ...................................... 8 3.2 Design axial compression ...................... 8 3.2.1 Lateral buckling ............................. 3.2.2 Lateral torsional buckling*) ................... 8 3.3 Bendingabout oneaxiswithoutcoexistentaxial force 8 8 3.3.1 General ..................................... 3.3.2 Lateral and torsional restraint ................. 1O 3.3.3 Analysis of compression flange ................ 12 12 3.3.4 Lateral torsional buckling ..................... 3.4 Bending about one axis with coexistent axial force 13 3.4.1 Members subjected to minor axial forces ....... 13 13 3.4.2 Lateral buckling ............................. 14 3.4.3 Lateral torsional buckling ..................... 3.5 Biaxialbendingwith or coexistent axialforce 15 3.5.1 Lateral buckling .... ................... 15 16 3.5.2 Lateral torsional buckling ..................... 4 Single-span built-up members .................. 16 16 4.1 General ...................................... 17 4.2 Common notation ............................. 4.3 Buckling perpendicular to void axis .............. 17 17 4.3.1 Analysis of member .......................... 4.3.2 Analysis of member components .............. 17 4.3.3 Analysis of panels of battened members ........ 18 4.4 Closely spaced built-up battened members ....... 19 20 4.5 Structural detailing ............................ 5 Frames.. ...................................... 20 20 5.1 Triangulated frames ...........................

General.. ................................... 20 Effective lengths of frame members designed to resist compression. . . . . . . . . . . . . . . .20 5.2 Framesand laterallyrestrainedcontinuous beams . 22 5.2.1 Negligible deformations due to axial force ...... 22 23 5.2.2 Non-sway frames ............................ 23 5.2.3 Design of bracing systems .................... 5.2.4 Analysis of frames and continuous beams. ...... 23 5.3 Sway frames and continuous beams subject to 23 lateral displacement ........................... 5.3.1 Negligible deformations due to axial force . . . . . . 23 5.3.2 Plane sway frames ........................... 23 5.3.3 Non-rigidly connected continuous beams ....... 27 6 Arches ........................................ 27 27 6.1 Axial compression ............................. 27 6.1.1 In-planebuckling ............................ 6.1.2 Buckling in perpendicular plane. . . . . . . . . . . . . . . . 30 6.2 In-plane bending about one axis with coexistent axial force ............ 6.2.1 In-plane buckling .............. 6.2.2 Out-of-plane buckling ........................ 33 6.3 Design loading of arches ........ ....... 34 7 Straight linear members with plan thin-wailed parts of cross section . . . . . . . . . . . . . . 34 7.1 General ...................................... 34 7.2 General rules relating to calculations . . 7.3 Effective width in elastic-elastic method 7.4 Effective width in elastic-plastic method 38 7.5 Lateral buckling ............................... 38 7.5.1 Elastic-elasticanalysis ........................ 7.5.2 Analyses by approximate methods . . . . . . . . . . . . . 38 39 7.6 Lateral torsional buckling ....................... 39 7.6.1 Analysis .................................... 7.6.2 Axial compression ........................... 39 7.6.3 Bending about one axis without coexistent 39 axial force .................................. 7.6.4 Bending about one axis with coexistent axial force .......................... ... 39 7.6.5 Biaxial bending with or without coexistent 39 axial force .................................. Standards and other documents referred t o ........ 40 Literature.. ....................................... 40 5.1.1 5.1.2

*) Term as used in Eurocode 3. In design analysis literature also referred to as flexural-torsional buckling.

Continued on pages 2 to 41

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Page 2 DIN 18800 Pari 2

1 General 1.1 Scope and field of application (101) Ultimate limit state analysis This standard specifies rules relating to ultimate limit state analysis of the buckling resistance of steel linear members and frames susceptible to loss of stability. It is to be used in conjunction with DIN 18800 Part 1.

(102) Serviceability limit state analysis Aserviceability limit state analysis need only be carried out if specifically required in the relevant standards. Note. Cf. subclause 7.2.3of DIN 18 800 Part 1.

1.2 Concepts

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(103) Buckling Buckling is a phenomenon in which displacement,v orw,of a member occurs, or rotation, 9,occurs about its major axis, or both occur in combination. A distinction is conventionally made between lateral buckling and lateral torsional buckling.

Figure 1. Coordinates, displacement parameters and internal forces and moments (109) Section parameters cross-sectional area second order moment of area

A I

i=

radius of gyration

(104) Lateral buckling

IT

Lateral buckling is a phenomenon in which displacement,v or w, of a member occurs,or both occur in combination,any rotation, 9, about its major axis being neglected.

I, W

torsion constant warping constant elastic section modulus axial force in perfectly plastic state bending moment in perfectly plastic state bending moment at which stress u, reaches yield strength in the most critical part of cross section

(105) Lateral torsional buckling Lateral torsional buckling is a phenomenon in which displacements, u and w ,of a member occur in combination with rotation, 4, about its major axis, consideration of the latter being obligatory. Note. Torsional buckling, in which virtually no displacements occur, is a special form of lateral torsional buckling.

1.3 Common notation (106) Coordinates, displacement parameters, internal forces and moments, stresses and imperfections axis along the member (major axis) axis of cross section (In solid members, I, shall be not less than Iz.) displacement along axes x, y and z rotation about the x-axis initial bow imperfections in unloaded state initial sway imperfection of member or frame in unloaded state axial force (positive when compression) bending moments shear forces (107) Subscripts and prefixes characteristic value of a parameter k design value of a parameter d grenz prefix to a parameter identifying it as being a limiting (¡.e. maximum permissible) value vorh actual red reduced Note. The terms ‘characteristicvalue’and‘designvalue’are defined in subclause 3.1of DIN 18800 Part I.

Physical parameters E elastic modulus G shear modulus f y yield strength Note. See table 1 of DIN 18800 Pari 1 for values of E , G and f y , k.

NP1

Mp1 Mel

apl= MP1 plastic shape coefficient

Mel Poisson’s ratio moment ratio Note. The term ‘perfectly plastic state’ applies when the plastic capacity is fully utilized, although in certain cases (e.g. angles and channels), pockets of elasticity may still be present. Where cross sections are non-uniform or internal forces and moments variable, Npl,Mpl and Mel at the critical point shall be calculated.

M

v

(110) Structural parameters system length (of member)

1

axial force at the smallest bifurcation load, according to elastic theory

NKi

(E * I )

7 ~ *

s K = i T ; y , AK

=

SK

slenderness ratio

1

reference slenderness ratio

&=n/-&

aK - =AK = (3 non-dimensional slenderness in comNKi

x

pression reductionfactor according to the standard buckling curves as used in Europe member characteristic

(108)

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effective length *) of a linear member associated with N K ~

VKi =

NKi,d 7

distribution factor of system

*) Translator’s note. Common term as used in design analysis. In Eurocode 3 termed ‘buckling length’.

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DIN 18800 Part 2 Page 3 Table 1. Methods of analysis

I

non-dimensional slenderness in bending

Note 1. Where cross sections are non-uniform or axial forces variable, (E.I ) , NKiand SK shall be determined for the point in the member for which the ultimate limit analysis is to be carried out. In case of doubt, an analysis shall be performed for more than one point (cf. item 316). Note 2. The reference slenderness ratio, ila, for steel of thickness 40mm and less shall be as follows: 92,9 for ~t 37 where fy,k = 240 N/mm2, and 75,9for St 52 where fy,k = 360 N/mm2. Note 3. Calculations of in-plane slenderness ratios shall be made using as the values Of f y , ( E . 1).NKi and MKi asspecifiedinitems116and117eithertheircharacteristic values or their design values throughout. Note4. V K ~shall beof thesame magnitudefor all members making up a non-sway frame. Note 5. Where cross sections are non-uniform or internal forces and moments variable, M Kshall ~ be calculated for the point for which the ultimate limit state analysis is carried out. In cases of doubt, an analysis shall be performed for more than one point. (111) Partial safety factors YF partial safety factor for actions YM partial safety factor for resistance parameters Note. The values of YF and YM shall be taken from clause 7 of DIN 18800 Fart 1. Thus, the ultimate limit state analysis shall be carried out taking YM to be equal to 1,l both for the yield strength and for stiffnesses (e.g. E T , E - A , G - A Sand S).

.

Ultimate limit state analysis

1.4.1 General (112) Methods of analysis The analysis shall be take the form of one of the methods given in table 1, taking into account the following factors: - plastic capacity of materials (cf. item 113); - imperfections (cf. item 114 and clause 2); - internal forces and moments (cf. items 115 and 116); - the effects of deformations (cf. item 1 1 6); - slip (cf. item 118); - the structural contribution of cross sections (cf. item 1 1 9); - deductions in cross-sectional area for holes (ci. item

120). As a simplification, lateral buckling and lateral torsional buckling may be checked separately, first carrying out the analysis for lateral buckling and then that for lateral torsional buckling whereby, in the latter case, members shall be notionally singled out of the structural system and subjected t o the internal forces and moments acting at the member ends (when considering the system as a whole) and to those acting on the member considered in isolation. Details on whether first or second order theory is to be applied are given together with the relevant method of analysis. The analyses described in clauses 3 to 7 may be used as an alternative to those listed in table 1.

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internal forces and moments

Method

reduction factor for lateral torsional buckling

XM

1.4

Calculation of

resistances

according to Elastic-

I

Elastic theory

Elastic theory

Elastic

Plastic

theory

theory

plastic plastic

Note 1. Details relating to elasto-plastic analysis are not provided in this standard (cf. [i]), though this is permitted in principle. Note 2. In table 11 of DIN 18800 Part 1, the generic term ‘stresses’ is used instead of ‘internal forces and moments due to actions’. Note 3. The conditions of restraint assumed when individual members are notionally singled out of the structural system shall be taken into account when verifying lateral torsional buckling. Note 4. Simplified methods substituting those set out in clauses 3 and 4 are listed in table 2. (113) Material requirements The materials used shall be of sufficient plastic capacity. Calculations may be based on assumptions of linear elastic-perfectly plastic stress-strain behaviour instead of actual behaviour. Note. The steel grades stated in sections 1 and 2 of item 401 of DIN 18800 Part 1 are of sufficient plastic capacity. (114) Imperfections

Reasonable assumptions (e.g. as outlined in clause 2)shall be made in order to take into account the effects of geometrical and structural imperfections. Note. Typical geometrical imperfections are accidental load eccentricity and deviations from design geometry. Typical structural imperfections would be residual stresses. (115) Internal forces and moments The internal forces and moments occurring at significant points in the members shall be calculated on the basis of the design actions. As a simplification, the index d has been omitted in the notation of internal forces and moments. Note. Subclauses 7.2.1and 7.2.2of DIN 18800Part 1 specify rules for calculating design values of actions. (116) Effects of structural deformations Calculations of internal forces and moments usually make allowance for deformation effects on equilibrium (according to second order theory), using as the design stiffness values the characteristic stiffnesses obtained by dividing the nominal characteristics of cross section and the characteristic elastic and shear moduli by a partial safety factor YM equal to 1,l. The effect of deformations resulting from stresses due to shear forces may normally be ignored.

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design buckling resistance moment according to elastic theory from My without coexistent axial force

MKi,y

Page 4 DIN 18800 Part 2 Table 2. Simplified ultimate limit state analyses Internal forces and moments Solid members

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I I I

Simplified analyses as in

Failure mode

Lateral buckling

3.2.1

Lateral torsional buckling

3.2.2

3

Lateral torsional buckling

3.3.2, 3.3.3, 3.3.4

7, 8, 12, 14, 16, 21

I

Lateral buckling

I I

I 10

I

3.4.2

I

24

Lateral buckling

3.4.2

24

Lateral torsional buckling

3.4.3

27

Lateral buckling

3.5.1

28.29

Lateral torsional buckling

I

3.5.2

I

30

Built-uprmbers

N+M,

Lateral buckling

4.3

Lateral buckling

4.3

Note 1. In calculations of internal forces and moments according to second order theory, for example, the member characteristic,s,and the distribution factor, ~ j - ~shall i. be determined using the design stiffness,

(E* I)d. Note 2. Reference shall be made to the criteria set out in item 739 of DIN 18800 Part 1when deciding whether to base calculations on second order theory. Note 3. Deformations also occur as a result of joint ductility. Note 4. Deformations resulting from stresses due to shear forces shall be taken into account as specified in clause 4 for built-up compression members. (117) Analysis on the basis of design actions multiplied by YM As a departure from the specifications of items 115 and 116, internal forces and moments and deformations may also be calculated using the designvalues of actions multiplied bya partial safetyfactoryM of l,l,in which case the ultimate limit state analysis shall be carried out using the characteristic strengths and stiffnesses, substituting these (denoted by subscript k) for the design resistances (denoted by subscript d) in the equations in clauses 3 to 7. i be made, for Note 1. Calculations of e and v ~ shall example, using the characteristic stiffness, (E.I)k.

Note2. The alternative procedure set out in this item is especiallysuitable forthe global analyses described in clauses 5,6 and 7 but may also be used by analogy in clauses 3 and 4, giving the same results as would be obtained if yM were assigned to the resistance.To preclude the risk of confusion, it shall be stated explicitly in the analysis that this alternative procedure has been used. Note 3. See subclause 7.3.1 of DIN 18800 Part 1 for resistance parameters.

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31 to 38

(118) Slip Account shall be taken of slip in shear bolt or preloaded shear bolt connections in members and frames susceptible to loss of stability, using the values specified in item 813 of DIN 18800 Pari 1. Note. Due account shall be taken of slip if this greatly increases the risk of loss of stability. (119) Effective cross section If the full cross section of parts in compression is taken into consideration, their geometry shall be such that the grenz (blt)and grenz (dit)values specified in DIN 18 800 Part 1are complied with. If,for thin-walled members,these values are not complied with, the analyses shall be of lateral buckling with coexistent plate buckling of individual members, or of lateral torsional buckling with coexistent plate buckling, as specified in clause 7 of DIN 18800 Part 3 or Part 4. Note 1. The grenz(blt) values differ according to the method of analysis selected (see table 1).The grenz (blt) values for individual parts of plane cross sectionsare given in tables12,13,15and 18of DIN 18800 Part 1. Note 2. The grenz (dlt) values for circular hollow sections are given in tables 14,15and 18 of DIN 18800 Pari 1. Methods of analyses of circular hollow sections the geometry of cross section of which does not comply with these limits are not covered in this standard. (120) Deductions for holes Deductions for holes need not be made when determining internal forces and moments and deformations if it can be ruled out that premature local failure occurs as a result. 1.4.2 Ultimate limit state analysis by elastic theory (121) Analysis The loadbearing capacity may be deemed adequate if an analysis of the internal forces and moments according to elastic theory shows the structure to be in equilibrium and either one of the following applies.

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DIN 18800 Part 2 Page 5

(122) Internal forces and moments in bi-axial bending Where bi-axial bending occurs with or without co-existent axial force but without torsion, the internal transverse forces and moments occurring may be determined by superimposing those internal forces due to actions which result in moments M yand transverse forces V, and those resulting in moments M, and transverse forces V,. However, calculation of E for the total axial force due to all actions is necessary in both cases.

Limiting the plastic shape coefficient In cases where the plastic shape coefficient,apl,associated with an axis of bending is greater than 1,25 and the principles of first ordertheorycannot be applied,the resistance moment occurring as a result of Co-existent normal and transverse forces in a perfectly plastic member cross section shall be reduced bya factor equal to 1,25/aPl.Thesame principle shall be applied to each of the two moments in biaxial bending if apl,yis greater than 1,25or apl,zis greater than 1.25. Note. Instead of reducing the resistance moment, the actual moment may be increased by a factor equal to api/1,25. (123)

1.4.3 Ultimate limit state analysis by plastic hinge theory (124) The loadbearing capacitymay be deemed adequate if an analysis according to plastic hinge theory shows internal forces and moments (taking into account interaction) to be within the limits specified for the perfectly plastic state (plastic-plastic method). This only applies if the structure is in equilibrium. Item 123 gives information on limiting the plastic shape coefficient. Note. Interaction equations are given in tables 16 and 17 of DIN 18 800 Part 1.

Note 2. As well as geometrical imperfections, equivalent geometrical imperfections also cover the effect on the mean ultimate load of residual stresses as a result of rolling, welding and straightening procedures, material inhomogeneities and the spread of plastic zones. Other possible factors which may affect the ultimate load, such as ductility of fasteners, frame corners and foundations, or shear deformations are not covered. In the elastic-elastic method, only two-thirds the values specified forthe equivalent imperfections in subclauses2.2 and 2.3 need be assumed. Ultimate limit state analyses of built-up members as specified in subclause 4.3 shall, however, always be made using the full bow imperfection stated in line 5 of table 3. Note 1. A reduction by one-third takes account of the fact that the plastic capacity of the cross section is not fully utilized. The aim is to achieve on average the same mean ultimate loads when applying both the elastic-elastic and the elastic-plastic methods. Note 2. The analyses set out in subclause 4.3 are based on comparisons of ultimate loads obtained empirically or by calculation, which also justify the value of bow imperfection stated in line 5 of table 3 (cf. Note under item 402). The equivalent imperfections are already included in the simplified analyses described in clauses 3 and 7. (202) Equivalent imperfections The equivalent geometrical imperfections, assumed to occur in the least favourable direction, shall be such that they are optimally suited to the deformation mode associated with the lowest eigenvalue. The equivalent imperfections need not be compatible with the conditions of restraint of the structure. Where lateral buckling occurs as a result of bending about only one axis with coexistent axial force, bow imperfections need only be assumed with DO or W O in each direction in which buckling will occur. Where lateral buckling occurs as a result of biaxial bending with coexistent axial force, equivalent imperfections need only be assumed for the direction in which buckling will occur with the member in axial compression. In the case of lateral torsional buckling, a bow imperfection equal to 0,5 DO (cf. table 3) may be assumed. (203) Imperfections in special applications Where provisions for special applications are made in other relevant standards,with specifications deviating from those given in this standard, such specifications shall form the basis of the global analysis. Note. Imperfections relating to special applications are not covered in clauses 3 to 7.

2.2

2 2.1

Imperfections General

(201) Allowance for imperfections Allowance shall be made for the effects of geometrical and structural member frame imperfections if these result in higher stresses. For this purpose, equivalent geometrical imperfections shall be assumed, a distinction being made between initial bow (see subclause 2.2) and sway imperfections (see subclause 2.3). Note 1. Equivalent geometrical imperfections may, in turn, be accounted for by assuming the corresponding equivalent loads.

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Bow imperfections

(204) Individual members,members making up non-sway frames and members as specified in item 207, shall generally be assumed to have the initial bow imperfections given in figure 2 and table 3.

-t

LYJ2 "o

Figure 2.

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I

"0

Initial bow imperfections of member in the form of a quadratic parabola or sine half wave

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The failure criterion is not higher than the design yield strength, f y , d (elastic-elastic method), the specifications of item 117 being applied by analogy. At isolated points, the failure criterion may be 10% higher than design yield strength (cf. item 749 of DIN 18800 Part 1). The internal forces and moments (taking due consideration of interaction) are within the limits specified for the perfectly plastic state (elastic-plastic method). Note 1. See item 746 of DIN 18800 Part 1 for f y , d . Note 2. The elastic-plastic method allows for plastification in cross sections with the possibility of plastic hinges with full torsional restraint at one or more pointS.This permits the plastic capacityof the cross sections to be fully utilized, but not that of the structure. Note 3. The analysis shall be made using interaction equations (cf. tables 16 and 17 of DIN 18 800 Part l).

Page 6 DIN 18800 Part 2 Bow imperfections need not be assumed if members satisfy the criteria specified in item 739 of DIN 18800 Part 1. Table 3. Bow imperfections

Type of member

Solid member, of cross section with following buckling curve

1

a

2

b

-

imperfection, WO?u0

11300 11250

I I

3 1 4 1

5

t

Built-up members, with analysis as in subclause 4.3

If the criteria for first order theory set out in item 739 of DIN 18 800 Part 1 are met, reductions in the sway imperfections may be assumed.

11200 11150

In the above figure, L or L, is the length of the member or frame, and ppo or ~ 0 ,the ~ .sway imperfection of the member or frame. Figure 5. Ideal member or frame (chain thin line) and member or frame with initial sway imperfection (continuous thick line)

11500

Note. See table 23 for bow imperfections for arch beams.

Initial sway imperfections shall generally be calculated as follows (cf. item 730 of DIN 18800 Part 1): a) solid members: 1

po = -r1 200

r2

r1 =

is a reduction factor applied to members or frames, where 1, the length of the member, L, or frame, L,, having the most adverse effect on the stress under consideration, is greater than 5 m;

r2=1(í+t) 2

is a reduction factor allowing for IZ independent causes of sway imperfection of members or frames.

Figure 3. Equivalent stabilizing force for bow imperfections as shown in figure 2 (assuming equilibrium)

Figure 4. Assumptions for bow imperfections (examples)

2.3

Sway imperfections

Assumptions Sway imperfections as in figure 5 shall be assumed t o occur in members or frames which may be liable to torsion after deformation and which are in compression. (205)

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Calculations of 12 for frames may generally assume n to be the number of columns per storey in the plane under consideration. Not included are columns subjected to minor axial forces, ¡.e. with less than 25Oío of the axial force acting in the column submitted to maximum load in the same storey and plane. Note 1. Since, in calculations of shear in multictorey frames, initial sway imperfections are assumed to have the most adverse effect in the storey under consideration, the storey height, ¡.e. the total length of columns,L, shall be substituted for the length of the column in that storey for calculation of Il. In the other storeys, the height of the structure,L,, may be substituted for I (cf. figure 6). Note 2. Allowance for sway imperfections may also be made by assuming equivalent horizontal forces.

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b) built-up members as in figures 20 and 21 and subclause 4.3: 1 (2) po = -r l . r2 400 where

4 -I!!

DIN 18800 Part 2 Page 7

970

= r1Zö

1

970=r1Töö

Single

1 100.2 = r p -with 200

n =2

1

po,~= r 2 -with

member

n =4

200

fTfl

%.2

E rn

Po.1

970?2

970,l

'

970.2

VI 0,2 : x =

1

k

k = 0,5[I

+

i

q

+ a (XK - 0,2)+ nK]

as a simplification, in cases where AK x=

> 3,O:

1 -

í& + a) a being taken from table 4. AK

Buckling curve

a

b

a

0.21

0,34

C

0,49

d

0,76

0,5

(305) Further provisions for non-uniform cross sections and variable axial forces Where equation (3)is applied to members of non-uniform cross section andlor variable axial forces, the analysis shall be made using equation (3) for all relevant cross sections with the appropriate internal forces and moments, cross section properties and axial forces,NKi.and in addition the following conditions shall be met:

min M,12 0,05man M,l

(6)

3.2.2 Lateral torsional buckling (306) Members of uniform cross section with anytype of end support not permitting horizontal displacement, subject to constant -¡al force shall be analysed as specified in subclause 3 . 2 . 1 . 1 shall ~ be calculated substituting for N K i the axial force occurring under the smallest bifurcation load for lateral torsional buckling,with the reduction factor x being determined for buckling about the z-axis. I sections (including rolled sections) do not require ultimate limit state analysis with respect to lateral torsional buckling. Note. Torsional buckling is treated here as a special type of lateral torsional buckling. Bending about one axis without coexistent axial force 3.3.1 General (307) Ultimate limit state analysis shall be carried out as specified in subclause 3.3.4for bending about one axis, except in cases where bending is about the z-axis or the conditions outlined in subclause 3.3.2or 3.3.3are met.

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D,il

Note 2. Reference shall be made to the literature (e.g. [2]) for the use of equations (4a) to (4c).

3.3

Table 4. Parameters a for calculation of reduction factor x

2,O

Figure 9. Effective lengths of single members of uniform cross section (examples)

~

AK

1,0

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DIN 18800 Part 2 Page 9 Table 5.

Buckling curves 1

2

3

Type of cross section

Buckling about axis

Buckling curve

Hollow sections

z

Y-Y

Hot rolled

a

2-2

Y-Y

Cold formed

b

Welded box sections

eN@i Y-Y

b

2-2

Thick welds and

h,lty < 30

Y-Y

C

2-2

Rolled I sections

Y-Y

a

2-2

b

hlb > 1.2; 40 e t 5 80 rnm

Y-Y

b

hlb 5 1,2;

2-2

C

hlb > 1.2; t

s 40 mrn

t580mm

t>80mrn

Y-Y

d

2-2

Welded I sections

Y-Y

b

2-2

C

Y-Y

C

2-2

d

Channels, L, T and solid sections

z

z Y-Y

C

2-2

plus built-up members to subclause 4.4 Sections not included here shall be classified by analogy, taking into consideration the likely residual stresses and plate thicknesses. Note. Thick welds are deemed to have an actual throat thickness, a, which is not less than min t.

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2-2

Z

Pagel0 DIN 18800 Part 2 Lateral torsional buckling

\

-a

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0.8

I Figure 10. Reduction factors x for lateral buckling (buckling curves a, b, C and d) and obtained by equation (18) with n equal to 2,5

for lateral torsional buckling,

where

3.3.2 Lateral and torsional restraint (308) Lateral restraint

k,

Members with masonry bracing permanently connected to the compression flange may be considered to have sufficient lateral restraint if the thickness of the masonry is not less than 0.3 times the height of cross section of the member. Masonry,

XM

2

ka

is equal to unity for the elastic-plastic and plasticplastic methods or 0,35 for the elastic-elastic method; is to be taken from column 2 of table 6 if the beam is free to move laterally,orfrom column 3of table 6 if the beam is laterally restrained at its top flange.

Table 6. Coefficients ko

Compression flange Figure 11. Lateral restraint (masonry bracing) If trapezoidal sheeting to DIN 18 807is connected to beams and the condition expressed by equation (7) is met, the beam at the point of connection may be regarded as being laterally restrained in the plane of the sheeting. Tt2

+ GIT + EI, 0,25 12 S being the shear stiffness provided by the sheeting for beams connected to the sheeting at each rib. If sheeting is connected at every second rib only, 0,2.S shall be substituted for S. Note. Equation (7) may also be used to determine the lateral stability of beam flanges used in combination with types of cladding other than trapezoidal sheeting, provided that the connections are of suitable design.

(309) Torsional restraint I beams of doubly symmetrical cross section with dimensions as for rolled sections complying with the DIN 1025 standards series shall be considered as being torsionally restrained (¡.e. due to their axes of rotation being restrained) if the condition expressed by equation (8) is met. Note 1. Equation (8) is a simpler check which makes use of the characteristic values.

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DIN 18800 Part 2 Page 11 Note 2. When determining the actual effective torsional restraint,cb,k, any deformations at the point of connection between the supported beam and the supporting member shall be taken into consideration, e.g. by means of equation (9). 1

--

--

1

C8M,k

C@,k

1

1

COA,k

C@P,k

+-+-

CfiA,k is the torsional restraint due to deformationof the connection, that of trapezoidal sheeting being obtained by means of equation (11 a) or (11 b), substituting ?@&k from table 7;

(9)

with

where --`,,,`-`-`,,`,,`,`,,`---

cg,k

is the actual effective torsional restraint;

CbM,k

is the theoretical torsional restraint obtained by means of equation (10) from the bending stiffness of the supporting member (a), assuming a rigid connection:

vorh b

-I1,251 1O0

with 1,25

vorh b

-I 2,o 1O0

where

(1O)

vorh b is the actual flange width of the beam, in mm.

where

Cf. [3]for further details on the use of C@A,k. Cbp,k is the torsional restraint due to deformation of the supported beam section (cf. [4]).

k

is equal to 2 in the case of singlespan or two-span beams or 4 in the case of continuous beams with three or more spans: ( E . r a ) k is the bending stiffness of the supporting member; a is the span of the supporting member;

Note 3. Instead of applying equation (81, the actual effective torsional restraint, C@,k,may also be considered when determining the ideal design buckling resistance moment, M K ~ the , ~ , check then being carried out as specified in subclause 3.3.4.

Table Z Characteristic torsional restraint values for trapezoidal steel sheetins connections, assuming a flange width, Bolting to

Position of profile Line TOP

Bottom

top flange

bottom flange

Bolt spacing,

b,')

1

Washer diameter, inmm

in C'A,k7 kNmim

max bt3), in mm

2 b,')

40

40 40

40 120 120

I Sheeting subjected to suction

I 7

X

X

8

X

X

b,

l)

2,

X X

16

40

16

40

I

- rib spacing.

Ka - washer diameter irrelevant; bolt head to be concealed using a steel cap, not less than 0,75mm in wall thickness.

3) bt - flange width of sheeting. The values stated apply to bolts not less than 6,3mm in diameter, arranged as shown in figure 13, used with steel washers not less than 1,Omm thick, with a vulcanized neoprene backing.

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Page 12 DIN 18800 Part 2

i

Asimplified method using equation (14) may be used where equation (12) is not met:

Ip"

u

Figure 12. Torsional restraint (example)

0,843 M~ 5 1 '

Mpl,y,d

where

My

is the maximum moment;

x

isareductionfactorasafunctionofbucklingc_urvec or d, obtained by means of equation (4), for A. from equation (13),buckling curve d being selected for beams otherthan the rolled beams in line 1 oftableg, which are subject to in-plane lateral bending on their top flange. Equation (15) shall also be met by beams coming under this category:

I I

5 4 4 t

I

I

I

I

h being the maximum beam depth;

t being the thickness of the compression flange. Buckling curve c may be used in all other cases. Note. Calculations may be simplified bysubstituting fori,,g the radius of gyration of the whole section, i,.

Figure 13. Arrangement of screws in connections between beams and trapezoidal sheeting (example)

Lateral torsional buckling

3.3.4

(311) The ultimate limit state analysis of I beams, chan-

3.3.3 Analysis of compression flange (310) I beams symmetrical about the web axis, with a

nels and C sections not designed for torsion shall be by means of equation (16):

compression flange which is laterally restrained at a number of points spaced a distance c apart, do not require a detailed analysis for lateral torsional buckling if

My

is the maximum moment as specified in item 303;

XM

is a reduction factor applied to moments as a function of AM;

where II is the beam coefficient from table 9.

Axial force diagram

kC

Where there are moments My with a moment ratio, W, greaterthan 0,5,the beam coefficient,n,shall be multiplied by a factor k , from figure 14.

*Figure 14. Beam coefficient and associated factor k ,

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--`,,,`-`-`,,`,,`,`,,`---

where

(1 2)

DIN 18800 Part 2 Page 13 Table 9. Beam coefficient, n Line

n

Type of section Rolled

t

2.5

I

Welded

I1

2.0

Castellated

-

pmaxM maxM -1cp1

I

r

Moment diagram

I

I

I I 1.77

- 0,77

Calculations of beams not more than 60cm in height may be simplified by substituting equation (20)for equation (19).

=

MKi,y

1,32 b * t ( E * I , )

-a

1*h2

Notched

VI

16) Haunched*)

Figure 15. Beam dimensions qualifying for simplified analysis using equation (20) or (21)

-r

0,7

+ 1.8 min h

max h

min h 2 0,25 max h

Note 2. X M may also be taken from figure 10 if the beam coefficient, n,is equal to 2 5 Note 3. X M may be assumed to be equal to unityfor beams not more than 60cm in depth (see figure 15)and of uniform cross section provided that they satisfy equation (21):

bet

1 5k,

h

When flanges are connected to webs by welding, n shall be further multiplied by a factor of 0,8.

Note 1. Calculation of äM is only possible where the ideal design buckling resistance moment, M K ~ , ~is, known (cf. [5] and [6]). Equation (19)or (20) may be applied for beams of doubly symmetrical uniform cross section.

MK~,,,= C * NK~,,, (11,'

+ 0,25 Z; + 0.5 zP)

(19)

where

<

is the moment factor applicable to fork restraint at the ends, from table 10

zp

I, is the distance of the point of transmission of the in-plane lateral load from the centroid (positive in tension).

3.4 3.4.1

Bending about one axis with coexistent axial force

Members subjected t o minor axial forces

(312) Members subjected to only minor axial forces and meeting the condition expressed by equation (22) may be analysed for bending without coexistent axial force, as specified in subclause 3.3.

N X

*

< 0,l

(22)

Npl,d

3.4.2 Lateral buckling 3.4.2.1 Simplified method of analysis

(313) The analysis for lateral buckling of members pinjointed on both sidesand subject to in-plane lateral loading

--`,,,`-`-`,,`,,`,`,,`---

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fy,k

Note 4. Coefficient n allows for the effect of residual stresses and initial deformations on the service load but not the effect of the support conditions (these being allowed for by MKi,y).

Io + 0,039 1' * IT

=

240 -

f y , k being expressed in N/mm2.

NK~,,, is equal to n2.E . Izll'; c2

200

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Page 14 DIN 18800 Part 2 in the form of a concentrated or line load and with a maximum moment, M ,according to first order theory, may be analysed by means of equation (3), while substituting in equation (4 b) k from equation (23).

+ a (&

- 0,2) + 3;

+-

Item 305 shall be taken into consideration. 3.4.2.2 Equivalent member method (314) Analysis The ultimate limit state analysis shall be made applying equation (24) and using the buckling curves specified in subclause 3.2.1.

N

. Np1.d where x

+-ID'

e

Mpl,d

M

An

(318) Portions of members not subjected t o compression The analysis of portions of members which are not themselves subject to compression but which are required to resist moments due to being connected to members in compression shall be by means of equation (26). The yield strength of cross sections not in compression shall not be less than that of those in compression.

M d

+ An 1,15

E a reduction factor from equation (4), a function of and the appropriate buckling curve (see table 5), for displacement in the moment plane; is the uniform equivalent moment factor for lateral buckling taken from column 2 of table 11. Moment factors less than 1 are only to be used for members of uniform cross section whose end support conditions do not permit lateral displacement and which are subjected to constant compression without in-plane lateral loading; is the maximum moment according to first order elastic theory, imperfections being neglected;

AK

ßm

Note. If a more detailed analysis is required,the design of connections shall be based on the basis of the bending moment according to second order theory, taking into account equivalent imperfections.

isequal to-

N x'Npi,d

(1--

N

x 2 * 36,

x-Npl,d)

but not more than 0,l.

Note. A portion of a member not in compression could bea beam connected to columns in compression. (319) Movement of supports and temperature effects Any effects of deformations as a result of movement of the supports or variations in temperature shall be taken into consideration when calculating moment M . Note. Further information shall be taken from the literature k g . VI). 3.4.3 Lateral torsional buckling (320) Channels and C sections, and I sections of monosymmetric or doubly symmetrical cross section, exhibiting uniform axial force and not designed for torsion, with relative dimensionsas for those of rolled sections,shall be analysed for ultimate limit state by means of equation (27):

N

Item 123 shall be taken into account when calculating Mpl,d. For doublysymmetrical cross sections with a web comprising at least 18Yo of the'total area of cross section, M p l , d in equation(24) may be multiplied by a factor of 1,l if the following applies:

Note 1. Where the maximum moment is zero,equation (3) shall be applied instead of equation (24) for the ultimate limit state. Note 2. Calculations mayde simplified by substituting for A n either 0,25 x 2 .A$ or 0.1. (315) Effect of transverse forces Due account shall be taken of the effect of transverse forces on the design capacity of a cross section. Note. This may be achieved by reducing the internal forces and moments in the perfectly plastic state (e.g. as set out in tables 16 and 17 of DIN 18800 Part 1).

(316) Non-uniform cross section and variable axial forces Where cross sections are non-uniform or axial forces variable, the analysis shall be made applying equation (24) to all key cross sections, with all relevant internal forces and moments and cross section properties and the axial force, NK~, assumed as acting at these points. In addition, equations (5) and (6) in item 305 shall be met.

(317) Rigid connections In the absence of a more rigorous treatment, rigid connections shall be calculated substituting forthe actual moment, M , the moment in the perfectly plastic state, Mp1,d.

+

My

ky< 1

xM ' Mpl,y, d The following notation applies in addition to that given in subclause 3.3.4. xz is a reduction factor from equation (4), substituting AK,z for buckling perpendicular to the z-axis, where & z is equal to -the non-dimensional slenderness xz

'

Npl, d

E

associated with axial force;

N K ~ is the axial force underthe smallest bifurcation load

k,

associated with buckling perpendicularto the z-axis or with the torsional buckling load; is a coefficient taking into account moment diagram My and a K , z . It shall be calculated as follows:

ky=l

-

N

ay. but not more than unity,

xz * N p l , d

where ay = 0,15jK,z. B M , -O,%, ~ with a maximum of 0,9 where & M , is ~ the moment factor associated with lateral torsional buckling, from column 3 of table 11, taking intoaccount moment diagram My. Note 1. Due regard shall be taken, particularly in the case of channels and C sections, of the fact that this analysis does not take account of design torsion. Note 2. Tsections are not covered by the specifications of this subclause. Note 3. A k, value of unity gives a conservative approximation. Note 4. The torsional bending load plays a major role, for example, in members subject to torsional restraint.

--`,,,`-`-`,,`,,`,`,,`---

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DIN 18800 Part 2 Page 15

3.5

Biaxial bending with or without coexistent axial force

3.5.1

"Y

Lateral buckling

x *Npl,d

ay, with a maximum of 1,5

- 1). With a ( 2 ß ~-, 4) ~ + maximum of 0,8 where ßM,,and ßM,z are the moment factors ßM associated with lateral torsional buckling, from column 3 of table 11; taking into account moment diagrams My and M,; apl,yand ctPl,, are plastic shape coefficients associated with moment M y or M,. (Item 123 is not applicable here.)

ay = & y

The ultimate limit state analysis shall be made applying equation (28):

+ -. MY

NpLd

where

(321) Method of analysis 1

N

N

k,=1-

M, ky + -k, I 1

Mpl,y,d

(28)

MpL z, d

where x = min (xy, x),

is a reduction factor for the relevant buckling curve, from equation (4);

Myand M ,

are the maximum moments in first order theory (disregarding imperfections);

kY

is a coefficient taking -into account moment diagram My and AK,y It shall be calculated as follows:

Table 11. Moment factors 1

Moment diagram

2

3

Moment factors,

Moment factors, ßMs

ßm.

for lateral torsional buckling

for lateral buckling

&,

3 d moments --`,,,`-`-`,,`,,`,`,,`---

y, .;,

.:;. ... ,,. .. . .... . ,.. .:s ....:. 150

O

(32)

(33)

Note. The literature (e.g. [IO]) shall be consulted for internal compression and design bending. 4.3.2

Analysis of member components

4.3.2.1 Chords of laced and battened members (406) The global analysis of internal forces and moments acting throughout the member not resistant to shear gives an axial force,NG, in the chord undermaximum stressequal to the following:

NG shall be used for analysis of the part of a chord as specified in subclause 3.2,assuming pin-jointing on both sides. The slenderness ratio, aK,1. shall be obtained as follows:

where SK,1

Figure 18. Laced and battened members (examples)

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is the effective length of the part of a chord under maximum stress, usuallytaken to be the same as the length of the chord, a, between nodeS.The effective length of parts of laced members consisting of four angles shall be taken from table 13.

Note. The analysis may be made as specified in subclause 3.4for laced members as shown in columns 4 and 5 of table 13 where a is subject to transverse loading.

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Page 18 DIN 18800 Part 2 Lacing systems (38)

(407) The axial forces of web members making up lacing

systems shall be obtained from the total transverse forces, Vy,acting in the laced member.The effective length shall be taken from subclause 5.1.2. Note. The total transverse force required when considering a member in axial compression, shall be obtained from equation (33). 4.3.3

Analysis of panels of battened members

(408) Panels between two battens The panel between two battens resisting the maximum transverse force, rnax Vv, obtained from the global calculation shall be analysed by verifying the ultimate limit state of a chord subject to the following internal forces and moments:

end moment, transverse force,

m a r Vy a MG = -r

VG

=

2

rnax Vy ~

(37)

r

where

XB

is the position of the batten on the chord.

In the case of monosymmetric chord cross sections, the resistance moment, M , at the ends of the part of the chord shall be obtained from the mean of the moments f Mpl,NG derived from interaction equation (38). Note 1. The plastic design capacity of the chord cross section as obtained from the interaction equations may be utilized (cf. [9] and [lo]),the transverse force, VG, normally being neglected. Note 2. The moments of resistance, M,!,N~,occurring in the chords at their connections with battens are of different magnitude owing to their different directions. Failure of a panel does not occur until all M p ~ , values ~ G have been fully utilized (cf. [9]). Note 3. The moment axes shall also be taken to be parallel to the void axis in the case of angle chords.

Table 13. Effectwe lengths sK,1 and equivalent shear stiffnesses, s,*,d, of laced and battened members 1

2

3

4

5

6 Battened members

a

r:r z

y

y

z

SK; 1

+

1.28 a

1,52 a

.

a

a

a

.

Sz, d = m ( E A& cos a . sin2a (m = number of braces normal to void axis)

The effective lengths,sK,l,in columns 1and 2 onlyapply to angle-sectioned chords, the slenderness ratio,ili, being calculatI d on the basis of the smallest radius of gyration, i l . If, in special cases, fasteners are used which are likely to slip, this may be accounted for by increasing the equivalent geometrical imperfections accordingly. The information relating to Sg,d does not apply to scaffolding,which generally makes use of highly ductile fasteners which must be taken into account. Note. Further information on ductilityand slip of fasteners and on eccentricityat the connections between web members in laced members is given in the literature (e.9. [9]).

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--`,,,`-`-`,,`,,`,`,,`---

4.3.2.2

DIN 18800 Part 2 Page 19 (409) Battens Battens and their connections shall be designed for shear and the design moments (cf. table 14).

Table 14. Distribution of forces and moments in the battens of battened members 1

2

Cross section of built-up battened members

Continuity of packing may be taken into consideration when calculating the second order moment of area. When determining the area of cross section,A, this only applies when the packing is adequately connected to the gusset. The shear in the battens, connections or packing may be calculated fora transverse force equalling 2.5% of the compressive force in the battened member. (411) Star-battened angle members Built-up members. consisting of two star-battened angle members need only be checked for lateral displacement perpendicular to the.material axis (figure 20) by the following equation:

--`,,,`-`-`,,`,,`,`,,`---

(39)

Structural model

If the effective lengths of the two members are not the same, the mean of the two effective lengths shall be used. Angles with a cross section as shown in figure 20 b) may be verified by the following equation, the radius of gyration, io, of the gross cross section relating to the centroidal axis parallel to the longer leg:

.

lY

io =1.15

Moment diagram in the connection due to shear, T

Shear, T,in the connection This also applies for closely spaced built-up battened members as shown in figures 19,20and 21.The moments in the centroids of batten connections shall be taken into account. If packing plates are used to connect the main components in built-up battened members as shown in figures 19 and 21, it is sufficient to design the connection for resistance to the actual shear.

4.4 Closely spaced built-up battened members (410) Cross sections with one void axis Built-up members with cross sections as shown in figure 19 may also be treated as solid members as set out in clause 3 when calculating lateral displacement normal to the void axis, provided that either of the following conditions is satisfied: a) battens or packing plates positioned as specified in subclause 4.5 are not more than 15 i, apart; b) continuous packing plates are used,which are connected at intervals equal to 15 il or less apart.

a) r = 2

b) r = 2

Figure 20. Star-battened angle members Consecutive battens may be in corresponding or mutually opposed order. Shear may be determined as specified in item 410. Note. According to item 503, the effective lengths of diagonals or verticals in triangulated frames differ, depending on whether lateral displacement in or perpendicular to the plane of the frame is being considered. (412) Cross sections with two void axes Where built-up members as shown in figure 21 consist of main components with a clear spacing not or only slightly greater than the thickness of the gusset,the specifications applying to the built-up members in figure 19 shall be applied by analogy to the two void axes.

r=4 Figure 19. Built-up memebers with a void axis and a clear spacing of main components not oronlyslightly greater than the thickness of the gusset

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Figure 21. Closely-spaced built-up member with two void axes

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Page 20 DIN 18800 Part 2

Structural detailing (413) Retention of cross-sectional shape Where member cross sections have two void axes, the rectangular cross-sectional shape shall be retained by means of cross-stiffening. Note. Cross-stiffening may take the form of bracing,plates or frames.

tom chords are in the perpendicular plane, the effective length in that plane may be determined as for compressive forces which do not always act in the same direction. Note 1. Chords may be held in the perpendicular plane by a road deck, for example. Note 2. The effective length can be determined with the aid of figures 36 to 38.

(414) Arrangement of battens and packing plates Battened members shall be connected at the ends by battens.This also applies to laced members unless cross bracing is used instead. If built-up members are connected at the same gusset,due account shall be taken of the fact that the gusset will also function as an end batten or end packing plate. The other battens shall be spaced as equally apart as possible, the use of packing plates being permitted instead for the members shown in figures 19 and 21. The number of panels shall be not less than three, and equation (41) shall be satisfied: a - 5 70 (41)

/

N

/

/

Ib(

i1

5

A'

Vertical member held horizontally, non-rigidly connected at one side

Frames

5.1 Triangulated frames 5.1.1 General (501) Calculation of forces in triangulated frame members The forces acting in the members making up a triangulated frame may be calculated assuming nominally pinned member ends.Secondary stresses as a result of nodes may be disregarded. Where the cross sections of compression chords are nonuniform over their length,any load eccentricity in individual members may be disregarded if the mean centroidal axis of each cross section coincides with the centroidal axis of the compression chord. (502) Analysis of compression members Analysis of compression members shall be as specified in clause 3,4 or 7.

Effective lengths of frame members designed to resist cornpression 5.1.2.1 General (503) Rigidly connected members In the absence of a more rigorous treatment, the effective length, SK, of frame members which are rigidly connected using at least two bolts or by welding shall be 0.9 I for inplane buckling (42) and equal to unity for out-of-plane buckling (43).

Vertical member held horizontally, non-rigidly connected at both sides

5.1.2

Figure 22.

5.1.2.2

Non-rigidly connected triangulated frame members for out-of-plane buckling

Triangulated frame members supported by another triangulated frame member

(506) Connection at intersection At intersections, members shall be connected directly or

(504) Non-rigidly connected members In the absence of a more rigorous treatment, the analysis for the sway mode of vertical and diagonal members held horizontally by cross beams or transverse members providing non-rigid connection, is a function of the structural detailing involved.

via a gusset. if both members are continuous, the connection between them shall be designed to withstand a force acting in the perpondicu!ar plane equal to 10% of the greater compressive force.

Noie. The effective length, S K , ~ ,of triangulated frame members as shown in figure 22 for the sway mode in the perpendicular plane may be determined by means of the diagrams in figure 27.

(507) In-plane effective length The effective length for the sway mode in the plane of the triangulated member shall be assumed to be the system length to the node of the intersecting members.

(505) Members with one end allowing lateral dlsplacement and one or two non-rigidly connected ends Where verticals and diagonals in main triangulated frames also act as the columns of sway portal frames,and thsirbot-

(508) Out-of-plane effective length The effective length forthe sway mode in the perpendicular plane appropriate to the structural detailing involved may be taken from table 15.

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--`,,,`-`-`,,`,,`,`,,`---

4.5

DIN 18800 Part 2 Page 21 Table 15. Out-of plane effective lengths of triangulated frame members of uniform cross section in the perpendicular plane

1

I

2

3

1- _ SK =

4

1

3

z-1 ~

N-1,

I , 13 1 + I . 1: but not less than 0,5 Z

SK =

1

Y-

N

-

I,

1 +-

1

1,

+-

13

I . 1:

but not less than 0,5 Z

but not less than 0,5 1

Continuous compression member

Nominally pinned compression member

where

but not less than 0,5 I

4

N . Il vhere -

z-1

ir where the following applies:

N Dut not less than 0.5 1

--`,,,`-`-`,,`,,`,`,,`---

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Page 22 DIN 18800 Part 2 Solid truss members with elastic support at mid-length

5.1.2.3

5.2

Frames and laterally restrained continuous beams

(509) The out-of-plane effective length of solid truss

5.2.1 Negligible deformations due to axial force

members with elastic support at mid-length for the sway mode may be obtained by means of equation (44):

(511) The specifications of subclause 5.2 may be deemed applicable if the deformations due to axial force of the columns of frames and bracing systems are negligible,this being the case when equation (45) is met:

(44)

E * I > 2,5 S.L2 where

(45)

where is the system length;

E .I

N

is the maximum compressive force acting in the member ( N I or N2);

S

cd

is the frame stiffness with respect to lateral displacement of the points of connection of solid members and of columns forming part of the subframe in the perpendicularplane,this being equal to not less than 4 N I L

1

is the bending stiffness, is the storey stiffness, L is the overall height (see figure 25), of the bracing system or multistorey frame. If E -1or S varies over a number of storeys,their mean may be used.

I may be approximated using equation (46): B2

I=

1

-+-

(46)

1

Ali Are the width, B, and cross-sectional areas Ali and Are of the columns being as shown in figure 25.

Bracing system

Figure 23. Solid member and frame stiffness

Multistorey frame

Ali

Angles used as solid members in triangulated frames

5.1.2.4

(510) Where angle ends are nominally pinned (e.g. by means of a single bolt), the effects of eccentricity shall be taken into consideration.

L

B

Figure 25. Criteria for calculation of I by means of equation (46)

If one of the two angle legs is rigidly connected at the node, the effects of eccentricity may be disregarded and the analysis of lateral buckling as specified in subclause 3.2.1 carried o3t using the non-dimensional slenderness in bending, Ak, from table 16. Table 16. Non-dimensional slenderness in bending, 2

1

1

O-l

I

~~~

0,43 0,578 + 0,34

82 + 1,05 7,81

7,81- 6,29 + 9,78W 2

q = -1

1,70- 5 I+

23,s

0,57- 0,21 ?# f 0,07@

W2

0,85

b’=b for Apo g 0,673

(1

- 0,22/äp0)

ob

APO

for npo > 0,673

Effective width in elastic-elastic method

b where

Note. This is an assumption only,and is not based on actual fact since the actual stress distribution is non-linear.

ripa

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17)

+

23,8

(711) Stress distribution In the elastic-elastic method, calculations shall be on the basis of a linear stress distribution in the effective cross section.

(712) Determining the effective width The effective width shall be determined by means of equation (81)for cases in which plates (web or flange) are supported on both sides with constant compression and equation (82)for support on only one side. The assumption of support on both sides presupposes that the supporting construction is of adequate stiffness.

0,57

1,70

ends are restrained and moments with different signs are liable to occur here. If one of the ends is nominally pinned, ep or e, (see item 709) is equal to zero at this end. Note. An additional imperfection is to be assumed as a result of this increase in sway imperfection when the equivalent member method is applied.

7.3

0,57 - 0,21 q + 0,07W2

--`,,,`-`-`,,`,,`,`,,`---

q =1

0,7

=: b, but not exceeding b

(82)

APO

b

is the width of the thin-walled part of the cross section from table 26; is the non-dimensional slendernessrelating to plate buckling, obtained by means of equation (83): U

=

/G (ir

, in N/mm2;

ue

= 189800

t

is the thickness of the thin-walled part of the cross section;

Not for Resale

DIN 18800 Part 2 Page 37

k

is the buckling factor from table 26, the edge stress ratio, y.being a function of the stress distribution in the effective cross section. Where plates are supported on both sides, y may be calculated on the basis of the gross cross section of the part under consideration. The stress distribution shall be calculated on the basis of all internal forces and moments; u is the maximum compressive stress according t o second order theory acting at the long edge of the thin-walled part of the cross section, calculated on the basis of the effective cross section, and expressed in N/mm2.The long edge is taken to be an edge of the gross part of the cross section. if. in equation (83),u is assumed to be less than fy,d. u shall be substituted for fy,d in the analyses specified in subclauses 7.5.2.1 to 7.5.2.3. Note 1. Reference may be made to,forexample,subclause 3.10.2 of the DASt-Richtlinie(DASt Code of practice) 016 Bemessung und konstruktive Gestaltung von Tragwerken aus dünnwandigen kaltgeformten Bauteilen (Design and construction of structures with cold formed, thin-walled sections) for suitable stiffness of the supporting constructions for plate edges. Note 2. Where u is equal to fy,d, npo is equal to Xp from table 1 of DIN 18 800 Part 3. Note 3. u, shall be obtained thus:

u, =

5~' *

Table 27. Resolution of effective width

-

m U C

(u

f O

-1 5 * 5 1

b; = Q - b - k , b > = Q +b . k ,

a c m L c

O Q

where

3

Q =

n v)

1

=-

[(0.97+ 0,03W ) - (OJ6 + 0,06tp)/IpJ

&o

k,

=

k2

=

-0.04 q2+ OJ2 I#+ 0,42 +0,04 @ - 0,12 I# + 0,58

0-w

(Compression)

E . t2

12 b2 (1 - ,U)'

Fa>

inserting a Poisson's ration, ,u, equal to 0,3.

(u

C

u+W

(Compression)

(Tension)

O c

m

c L

O CL

0.

Effective flange width with u and VI=