Seismic Design of Steel Buildings

  ETABS  manual     Seismic  design  of  steel   building  accordance  to       Eurocode  3  and  8       -­‐Worked  examples  –  Hand  calculations    

Valentinos  Neophytou  BEng,  MSc  

        JULY  2013  

ABOUT THIS DOCUMENT

This publication provides a concise compilation of selected rules in the Eurocode 8, together with relevant Cyprus National Annex, that relate to the design of common forms of concrete building structure in the South Europe. It id offers a detail view of the design of steel framed buildings to the structural Eurocodes and includes a set of worked examples showing the design of structural elements with using software (CSI ETABS). It is intended to be of particular to the people who want to become acquainted with design to the Eurocodes. Rules from EN 1998-1-1 for global analysis, type of analysis and verification checks are presented. Detail design rules for steel composite beam, steel column, steel bracing and composite slab with steel sheeting from EN 1998-1-1, EN1993-1-1 and EN1994-1-1 are presented. This guide covers the design of orthodox members in steel frames. It does not cover design rules for regularities. Certain practical limitations are given to the scope. Due to time constraints and knowledge, I may not be able to address the whole issues. Please send me your suggestions for improvement. Anyone interested to share his/her knowledge or willing to contribute either totally a new section about Eurocode 8 or within this section is encouraged.

For further details: My LinkedIn Profile: http://www.linkedin.com/profile/view?id=125833097&trk=hb_tab_pro_top Email: [email protected] Slideshare Account: http://www.slideshare.net/ValentinosNeophytou

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List of contents

1.1 DESIGN AND ANALYSIS EXAMPLE OF STEEL FRAME WITH CONCENTRIC BRACING ................................................................................................................................. 7 1.1 LAYOUT OF STRUCTURE............................................................................................... 7 1.2 PRELIMINARY DESIGN................................................................................................... 9 1.2.1 PRELIMINARY DESIGN OF COLUMNS AND BEAMS ............................................ 9 1.3 MATERIAL PROPERTIES .............................................................................................. 11 1.3.1 MATERIAL PROPERTIES OF CONCRETE ............................................................... 11 1.3.2 MATERIAL PROPERTIES OF STEEL ........................................................................ 12 1.3.3 MATERIAL PROPERTIES OF STEEL AND CONCRETE AS DEFINE IN ETABS 13 1.3.4.1 MODELING REQUIREMENTS OF EC8 FOR CONCRETE MEMBERS............... 15 1.3.4.2 MODELING REQUIREMENTS OF EC8 FOR FLOOR DIAPHRAGMS ................ 15 1.3.4.3 MESHING OF SLABS ................................................................................................ 16 1.4 JOINT MODELING (EN1993-1-1,CL.5.1.2) ................................................................... 17 2.0 MODAL RESPONSE SPECTRUM ANALYSIS ............................................................. 20 2.1 STRUCTURAL TYPES AND BEHAVIOR FACTOR ACCORDING TO EN1998-11,CL.6.3 ................................................................................................................................... 20 2.2 DEFINE DESIGN HORIZONTAL RESPONSE SPECTRUM ........................................ 24 2.2.1 VERTICAL RESPONSE SPECTRUM (EN1998-1-1,CL.3.2.2.3) ................................ 24 2.2.2 HORIZONTAL RESPONSE SPECTRUM (EN1998-1-1,CL.3.2.2.5) .......................... 24 2.2.3 PARAMETERS OF ELASTIC RESPONSE SPECTRUM (EN1998-1-1,CL.3.2.2.5).. 25 2.2.3.1 GROUND INVESTIGATION CONDITIONS ........................................................... 29 2.2.3.2 IMPORTANCE FACTOR ........................................................................................... 29 2.2.3.3 DUCTILITY CLASS ................................................................................................... 30 2.3 ANALYSIS TYPES .......................................................................................................... 31 2.3.1 MODAL RESPONSE SPECTRUM ANALYSIS .......................................................... 31 2.3.1.1 ACCIDENTAL ECCENTRICITY .............................................................................. 32 2.3.2 LATERAL FORCE ANALYSIS REQUIREMENTS .................................................... 34 2.3.4 ESTIMATION OF FUNDAMENTAL PERIOD T1 ...................................................... 35 2.3.5 AUTOMATIC LATERAL FORCE ANALYSIS USING ETABS ................................ 36 2.3.6 USER LOADS - LATERAL FORCE ANALYSIS USING ETABS ............................. 38 Page 3

2.3.7 TORSIONAL EFFECTS ................................................................................................ 45 2.3.8 SUMMARY OF ANALYSIS PROCESS IN SEISMIC DESIGN SITUATION........... 46 3.0 DEFINE STATIC LOADS ................................................................................................ 47 4.0 SEISMIC MASS REQUIREMENTS ACCORDING TO EC8 ......................................... 48 4.1 MASS SOURCE OPTION ................................................................................................ 49 5.0 WIND LOADING ON STRUCTURE (EN1991-1-4:2004).............................................. 51 5.1 CALCULATION OF WIND LOAD ACCORDING TO EN1991-1-4:2004 .................... 51 5.2 APPLICATION OF WIND LOADING USING ETABS ................................................. 54 6.0 LOAD COMBINATION ................................................................................................... 59 7.0 DESIGN PREFERENCES ................................................................................................ 61 8.0 ANALYSIS AND DESIGN REQUIREMENTS FOR CONCENTRICALLY BRACED FRAMES ACCORDING TO EN1998-1-1,CL.6.7.2 .............................................................. 64 8.1 STEPS OF THE DESIGN DETAIL OF CONCENTRIC STEEL FRAMES ................... 65 8.2 CLASSIFICATION OF STEEL SECTIONS .................................................................... 66 8.3 DESIGN OF COMPOSITE SLAB UNDER GRAVITY LOADS .................................... 68 8.4 DESIGN OF COMPOSITE BEAM (WITH STEEL SHEETING) UNDER GRAVITY LOADS .................................................................................................................................... 72 8.5 DETAIL DESIGN OF STEEL COLUMNS UNDER GRAVITY LOADS...................... 79 8.6 DETAIL DESIGN RULES OF STEEL CONCENTRIC BRACED FRAMES (CBF) ACCORDING TO EUROCODE 8.......................................................................................... 87 8.6.1 DETAIL DESIGN RULES OF STEEL BRACING ACCORDING TO EUROCODE 8 .................................................................................................................................................. 87 8.7 DETAIL DESIGN RULES OF STEEL COLUMNS AND BEAMS ACCORDING TO EUROCODE 8 ......................................................................................................................... 88 8.8 DETAIL DESIGN RULES OF STEEL COMPOSITE MEMBERS ACCORDING TO EUROCODE 8 ......................................................................................................................... 89 8.9 DETAIL DESIGN RULES OF STEEL MOMENT RESISTANCE FRAMES (MRF) ACCORDING TO EUROCODE 8.......................................................................................... 90 8.9.1 DETAIL DESIGN RULES FOR MRF - DESIGN CRITERIA .................................... 90 8.9.2 DETAIL DESIGN RULES OF STEEL BEAM FOR MRF ........................................... 90 8.9.3 DETAIL DESIGN RULES OF STEEL COLUMN FOR MRF ..................................... 91 9.0 DESIGN OF STEEL FRAMES ......................................................................................... 92 9.1 DESIGN OF STEEL MEMBER OVERWRITES DATA................................................. 92

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9.2 DESIGN OF COLUMNS / BEAMS USING ETABS – GRAVITY LOAD ANALYSIS ONLY ...................................................................................................................................... 97 9.3 DESIGN OF STEEL COLUMN (GRAVITY DESIGN SITUATION) – HAND CALCULATIONS ................................................................................................................. 105 9.4 DESIGN OF STEEL COLUMN (SEISMIC DESIGN SITUATIONN) ......................... 118 9.4.1 DESIGN OF STEEL COLUMN (SEISMIC DESIGN SITUATION – HAND CALCULATION) .................................................................................................................. 124 9.5 DESIGN OF COMPOSITE BEAMS - HAND CALCULATIONS ................................ 128 9.5 DESIGN OF STEEL BRACING ..................................................................................... 145 9.5.1 MAIN CONFIGURATION OF DESIGN OF STEEL BRACING .............................. 145 9.5.2 SIMPLIFIED DESIGN OF FRAMES WITH X BRACING (EXTRACT FROM DESIGN GUIDANCE TO EC8) ........................................................................................... 147 9.5.3 MODEL IN ETABS ..................................................................................................... 148 9.5.4 DESIGN OF STEEL BRACING (GRAVITY/SEISMIC DESIGN SITUATION) – HAND CALCULATION....................................................................................................... 156 10.0 MODAL RESPONSE SPECTRUM ANALYSIS ......................................................... 170 10.1 SET THE ANALYSIS OPTIONS ................................................................................. 170 10.2 EVALUATE THE ANALYSIS RESULTS OF THE STRUCTURE ACCORDING TO THE MODAL ANALYSIS REQUIREMENTS ................................................................... 171 10.2.1 ASSESS THE MODAL ANALYSIS RESULTS BASED ON THE EN1998 ........... 172 11.0 SECOND ORDER EFFECTS (P – Δ EFFECTS) ACCORDING TO EN1998-11,CL.4.4.2.2 ........................................................................................................................... 173 11.1 DISPLACEMENT CALCULATION ACCORDING TO EN1998-1-1,CL.4.4.2.2 ..... 174 11.2 INTERSTOREY DRIFT................................................................................................ 174 11.3 CALCULATION OF SECOND ORDER EFFECT USING ETABS ........................... 175 11.3.1 INTERSTOREY DRIFT DISPLACEMENT ............................................................. 176 11.3.2 TOTAL GRAVITY LOAD PTOT ................................................................................ 178 11.3.2 TOTAL SEISMIC STOREY SHEAR VTOT ............................................................... 180 12.0 DAMAGE LIMITATION ACCORDING TO EN1998-1-1,CL.4.4.3 .......................... 184 12.1 CALCULATION OF DAMAGE LIMITATION .......................................................... 185 ANNEX - A .......................................................................................................................... 186 ANNEX A.1 - ASSUMPTIONS MADE IN THE DESIGN ALGORITHM (MANUAL OF ETABS – EC3 & EC8) .......................................................................................................... 186

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A1.1:LIMITATION MADE IN THE DESIGN ALGORITHM (MANUAL OF ETABS – EC3&EC8) ............................................................................................................................. 187 ANNEX –B: STEEL DESIGN FLOWCHARTS .................................................................. 188

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1.1 Design and analysis example of steel frame with concentric bracing

1.1 Layout of structure

Figure 1.1: Plan view

Figure 1.2: Side Elevation (4) & (1)

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Figure 1.3: Side Elevation (A) & (D)

Table 1.1: Dimensions of the building Dimensions

Symbol

Value

Units

Storey height

h

3.0

m

Total height of the building

H

9.0

m

Beam length in X-direction

lx

5.0

m

Beam length in Y-direction

ly

5.0

m

Building width in X-direction

Lx

15.0

m

Building width in Y-direction

Ly

15.0

m

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1.2 Preliminary design Table 1.2: Seismic design data Data

Symbol

Value

Units

-

3

-

Reference peak ground acceleration on type A ground, agR. Importance class

agR

0.25

m/s2

γI

1.0

-

Design ground acceleration on type A ground

ag

0.25

m/s2

Design spectrum

-

Type 1

-

Ground type

-

B

-

Seismic zone

Structural system

Steel frame with concentric bracing

Behavior factor

q

4.0

-

1.2.1 Preliminary design of columns and beams Preliminary design of steel beam Design data: Span of beam

Lx := 5000mm

Bay width

wbay := 5000mm

Overall depth of slab

h := 130mm

Loading data: −3

Density of concrete

γ c := 25kN⋅ m

Loads of floor per meter

g floor := γ c⋅ h ⋅ Lx = 16.25⋅ kN⋅ m

Live load

q office := 2kN⋅ m

Live load per meter

q service := q office⋅ Lx = 10⋅ kN⋅ m

−1

−2 −1

Partial factor for actions: Safety factor are obtain from Table A.1(2)B EN1990 Permanent actions, γ G Variable actions, γ Q Total load

γ G := 1.35 γ Q := 1.5 −1

Ed := γ G⋅ g floor + γ Q⋅ q service = 36.94⋅ kN⋅ m

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Material properties: −2

Young Modulus of Elasticity

Es := 210kN⋅ mm

Structural steel (clause 6.1(1) EN 1993 1-1)

γ M0 := 1.0

Structural steel properties: −2

Yield strength, fy

fy := 355N ⋅ mm

Ultimate strength, fu

fu := 450N ⋅ mm

Yield strength of reinforcement, fyk

fyk := 500N ⋅ mm

−2 −2

Deflection limitation: Deflection limit - General purpose

Lx

F :=

300 3

300⋅ Ed ⋅ Lx

Second moment area required

Ireq :=

Second moment area provided (IPE240)

Iprov := 3892cm

(

Check_1 := if Iprov > Ireq, "OK" , "NOT OK"

3

4

= 1.718 × 10 ⋅ cm

384⋅ Es 4

)

Check_1 = "OK"

Moment resistance check: 2

Design moment (Fixed end)

MEd :=

= 76.953⋅ kN⋅ m 12 MEd 3 W pl.y.req := = 216.769⋅ cm fy

Plastic modulus required

3

W pl.y := 324.4cm

Plastic modulus provided (IPE240)

(

Check_2 := if Wpl.y > Wpl.y.req, "OK" , "NOT OK"

Ed ⋅ Lx

)

Check_2 = "OK" Weak Beam - Strong column -Capacity design: 3

Plastic modulus of column required

W pl.y.c.req := 1.3⋅ W pl.y = 421.72cm

Plastic modulus of column provided (HE220A)

W pl.y.c := 515cm

(

Check_3 := if Wpl.y.c > Wpl.y.c.req , "OK" , "NOT OK"

3

)

Check_3 = "OK"

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1.3 Material properties ETABS: Define > Material properties 1.3.1 Material properties of concrete

Design requirement Poisson ratio is equal to v = 0 (cracked concrete) and v = 0.2 (un-cracked concrete) as (EN1992-1-1,cl.3.1.3). Table 1.3: Concrete properties (EN 1992, Table 3.1) C16/20

C20/25

C25/30

C30/37

Property Data for concrete

(N/mm2)

(N/mm2)

(N/mm2)

(N/mm2)

Mass per unit Volume

2,5E-09

2,5E-09

2,5E-09

2,5E-09

Weight per unit volume

2,5E-05

2,5E-05

2,5E-05

2,5E-05

29000

30000

31000

33000

0

0

0

0

10E-06

10E-06

10E-06

10E-06

Charact. ConcCyl. Strength, fck

16

20

25

30

Bending Reinf. Yield stress, fyk

500

500

500

500

Shear Reinf. Yield stress, fyk

500

500

500

500

Modulus of Elasticity Poisson’s Ratio (cracked concrete) Coeff. of thermal expansion

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1.3.2 Material properties of steel

Table 1.4: Material properties of steel Material Properties

Symbol

Value

Units

References

Mass per unit Volume

γs

7.85E-09

kg/mm3

EN1991-1-1,table A.4

Weight per unit Volume Modulus of Elasticity

γs

7.70E-05

N/mm3

EN1991-1-1,table A.4

Es

210,000

N/mm2

EN1993-1-1,cl.3.2.6(1)

Poisson’s ratio

ν

0.3

-

EN1993-1-1,cl.3.2.6(1)

α

1.2x10-5 per K (for T ≤ 100oC)

K

EN1993-1-1,cl.3.2.6(1)

α

1.2x10-5 per K (for T ≤ 100oC)

K

EN1993-1-1,cl.3.2.6(1)

G

≈81,000

N/mm2

EN1993-1-1,cl.3.2.6(1)

fy

275

N/mm2

EN1993-1-1,table 3.1

fu

430

N/mm2

EN1993-1-1,table 3.1

Coeff of Thermal Expansion (Steel structures) Coeff of Thermal Expansion (Composite ConcreteSteel structures) Shear Modulus Characteristic yield strength of steel profile Ultimate strength

Table 1.5: Strength vales of steel sections (EN1993-1-1,table 3.1) Nominal thickness of the element t (mm) Steel grade

t ≤ 40mm

40mm < t ≤ 80mm

Grade

fy (N/mm2)

fu (N/mm2)

fy (N/mm2)

fu (N/mm2)

reference

S235

235

360

215

360

EN 10025-2

S275

275

430

255

410

EN 10025-2

S355

355

510

335

470

EN 10025-2

S450

440

550

410

550

EN 10025-2

Note: You may use the product standard instead of those given in EN1993-1-1

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1.3.3 Material properties of steel and concrete as define in ETABS

Figure 1.4: Material properties of concrete (C25/30)

Figure 1.5: Material properties of steel (S275)

1.3.4 Slab modeling

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Table 1.6: Slab properties Data

Symbol

Value

Units

Slab depth

hs

170

mm

Diameter of stud

d

19

mm

haw

152

mm

fu

430

N/mm2

Height of stud Tensile strength of stud

ETABS: Define > Wall/Slab/Deck Sections/Add new deck Figure 1.6: Deck section properties

Press “Set Modifier” in order to modify the slab properties

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1.3.4.1 Modeling requirements of EC8 for concrete members 1. Unless a more accurate analysis of the cracked elements is performed, the elastic flexural and shear stiffness properties of concrete and masonry elements may be taken to be equal to one-half of the corresponding stiffness of the un-cracked elements (EN1998-1-1,cl.4.3.1(7)). Figure 1.7: Modified “Stiffness Modifiers”

1.3.4.2 Modeling requirements of EC8 for floor diaphragms ETABS: Select > Wall/Slab/Deck section > Select Deck ETABS: Define > Diaphragms ETABS: Select “D1” (Rigid diaphragms) 2. When the floor diaphragms of the building may be taken as being rigid in their planes, the masses and the moments of inertia of each floor may be lumped at the centre of gravity (EN1998-1-1,cl.4.3.1(4)).

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1.3.4.3 Meshing of slabs ETABS: Select > Wall/Slab/Deck section > Select Deck ETABS: Assign > Shell area > Auto Object Auto mesh option When you have a composite beam floor system, ETABS, by default, automatically meshes (divides) the deck at every beam and girder. This allows ETABS to automatically distribute the loading on the deck to each beam or girder in an appropriate manner. Figure 1.8: Meshing of composite slab

Figure 1.9: Meshing of normal slab

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1.4 Joint modeling (EN1993-1-1,cl.5.1.2)

(1) The effects of the behavior of the joints on the distribution of internal forces and moments within a structure, and on the overall deformations of the structure, may generally be neglected, but where such effects are significant (such as in the case of semi-continuous joints) they should be taken into account, see EN 1993-1-8. (2) (2) To identify whether the effects of joint behavior on the analysis need be taken into account, a distinction may be made between three joint models as follows, see EN 1993-1-8, 5.1.1: – simple, in which the joint may be assumed not to transmit bending moments. – continuous, in which the behavior of the joint may be assumed to have no effect on the analysis. – semi-continuous, in which the behavior of the joint needs to be taken into account in the analysis.

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Table 1.7: Example of joint types Simple joint

Continuous Fixed joint

Semi- continuous joint

ETABS: Pin joint in ETABS The pin-joint in ETABS can be achieved by selecting the members that you assumed to be pinned in the analysis process. This can be done as follow: Select member > Assign > Frame/Line > Frame Releases Partial Fixity Figure 1.10: Pinned joint (both ends)

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ETABS: Fixed joint in ETABS The fixed-joint in ETABS can be achieved by selecting the members that you assumed to be fixed in the analysis process. This can be done as follow: Select member > Assign > Frame/Line > Frame Releases Partial Fixity Figure 1.11: Fixed joint

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2.0 Modal Response Spectrum Analysis

2.1 Structural types and behavior factor according to EN1998-1-1,cl.6.3 Table 2.1: Structural types and behavior factor q-factor DCM DCH

Structural Type Moment resisting frames (MRF)

4

αu/ α1 =1.1

5αu/ α1

αu/ α1 =1.2 (1 bay) αu/ α1 =1.3 (multi-bay)

dissipative zones in beams and column bases Concentrically braced frames (CBF)

4

4

2

2.5

Dissipative zones in tension diagonals V-braced frames (CBF)

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Dissipative zones in tension and compression diagonals Frames with K-bracing (CBF)

Not allowed in dissipative design

Eccentrically braced frame (EBF)

4

5αu/ α1

2

2αu/ α1

4

4αu/ α1

αu/ α1 =1.2 dissipative zones in bending or shear links Inverted pendulum system

αu/ α1 =1.0

αu/ α1 =1.1

dissipative zones in column base, or column ends (NEd/Npl,Rd < 0.3) Moment-resisting frames with concentric bracing (MRF) + (CBF)

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αu/ α1 =1.2 dissipative zones in moment frame and tension diagonals Moment frames with infills Unconnected concrete or masonry infills, in contact with the frame

Connected reinforced concrete Infills

Infills isolated from moment frame

2

2

See EN1998-1-1,table 5.1

4

5αu/ α1

Structures with concrete cores or walls

See EN1998-1-1,table 5.1

Note: If the building is non-regular in elevation (see EN1998-1-1,cl.4.2.3.3) the upper limit values of q listed above should be reduced by 20 %

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Table 2.2: Values of behavior factor for regular and irregular structure Structural type

Regular in plan

Irregular in

Regular in plan

Irregular in

Irregular in

Regular in plan

Irregular in

and elevation

plan / Regular

/ Irregular in

plan &

plan / Regular

/ Irregular in

plan &

in elevation

elevation

elevation

in elevation

elevation

elevation

DCM

DCH

DCM

DCM

DCM

DCH

DCH

DCH

Single storey portal

4.0

5.5

3.2

3.2

3.2

5.25

4.4

4.2

One bay multi-storey

4.0

6.0

3.2

3.2

3.2

5.5

4.8

4.4

Multi-bay, multi-storey

4.0

6.5

3.2

3.2

3.2

5.75

5.2

4.6

Diagonal bracing

4.0

4.0

3.2

4.0

4.0

4.0

3.2

3.2

V-bracing

2.0

2.5

1.6

2.5

2.5

2.5

2.0

2.0

2.0

2.0

1.6

2.0

2.0

2.0

1.6

1.6

Moment resisting frame

Concentrically braced frame

Frame with masonry infill panels

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2.2 Define design horizontal response spectrum

2.2.1 Vertical response spectrum (EN1998-1-1,cl.3.2.2.3) The vertical component of the seismic action should be taken into account if the avg>0.25g (2.5m/s2) in the cases listed below: •

for horizontal structural member spanning 20m or more,

•

for horizontal cantilever components longer than 5m,

•

for horizontal pre-stressed components,

•

for beams supporting columns,

•

in based-isolated structures.

2.2.2 Horizontal response spectrum (EN1998-1-1,cl.3.2.2.5) For the horizontal components of the seismic action the design spectrum, Sd(T), shall be defined by the following expressions:

0 ≤ 𝑇 ≤ 𝑇! : 𝑆! 𝑇 = 𝑎! ∙ 𝑆 ∙

!

𝑇! ≤ 𝑇 ≤ 𝑇! : 𝑆! 𝑇 = 𝑎! ∙ 𝑆 ∙

!.!

𝑇! ≤ 𝑇 ≤ 𝑇! : 𝑆! 𝑇 = 𝑎! ∙ 𝑆 ∙

!

+! ∙ ! !

!

!.! !

(ΕΝ1998-1-1,Eq. 3.14)

2.5 𝑇! 𝑞 𝑇

                                                                                     ≥ 𝛽 ∙ 𝑎! 𝑇! ≤ 𝑇 ≤ 4𝑠: 𝑆! 𝑇 = 𝑎! ∙ 𝑆 ∙ ≥ 𝛽 ∙ 𝑎!

!

− ! (ΕΝ1998-1-1,Eq. 3.13)

(ΕΝ1998-1-1,Eq. 3.15)

!.! !! !! !

!!

(ΕΝ1998-1-1,Eq. 3.5)

Design ground acceleration on type A ground:

ag=γIagR

Lower bound factor for the horizontal spectrum: β=0.2 Note: the value of q are already incorporate with an appropriation value of damping viscous, however the symbol η is not present in the above expressions. Page 24

2.2.3 Parameters of elastic response spectrum (EN1998-1-1,cl.3.2.2.5) Table 2.3: Parameters of Type 1 elastic response spectrum (CYS NA EN1998-1-1,table 3.2) Ground

S

TB (s)

TC (s)

TD (s)

A

1.0

0.15

0.4

2.0

B

1.2

0.15

0.5

2.0

C

1.15

0.20

0.6

2.0

D

1.35

0.20

0.8

2.0

E

1.4

0.15

0.5

2.0

Type

Note: For important structures (γI>1.0), topographic amplification effects should be taken into account (see Annex A EN1998-5:2004 provides information for topographic amplification effects).

ETABS: Define > Response spectrum function Select EUROCODE8 Spectrum

Add New Function

1.

Peak ground acceleration agR=0,25g,

2.

Type C or D for building within category of importance I and II,

3.

Define two response spectrum cases if the factor q is different in each direction,

Page 25

4.

Modify the existing values of elastic response spectrum case in order to change it into the design response spectrum.

Convert the existing elastic response spectrum case to design response spectrum case

Figure 2.1: Response Spectrum to EC8

PERIOD   ACCELERATION   T   Sd(T)   0.0000   0.2000   0.1000   0.1917   0.1500   0.1875  

g  =   β  =   SoilType  =   q  =   αgR   =  

9.81   0.2   B   4.00   0.25  

m/sec2   -­‐   -­‐   -­‐   -­‐  

0.2000   0.4000  

0.1875   0.1875  

S  =   TB   =  

1.20   0.15  

-­‐   sec  

0.6000  

0.1563  

TC   =  

0.50  

sec  

0.8000  

0.1172  

TD   =  

2.00  

sec  

1.0000   1.5000   2.0000   2.5000   3.0000   4.0000   5.0000   6.0000   8.0000   10.0000  

0.0938   0.0625   0.0469   0.0300   0.0500   0.0500   0.0500   0.0500   0.0500   0.0500  

       

 

sec   T  =   0.50               Data  for  soil  type  -­‐  Type  Spectrum  1     index   Soil  Type   S   TB     1   A   1   0.15     2   B   1.2   0.15   3   C   1.15   0.2   4   D   1.35   0.2   5   E   1.4   0.15          

 

    TD   2   2   2   2   2  

TC   0.4   0.5   0.6   0.8   0.5    

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Figure 2.2: Amendment Response spectrum (q = 4)

Page 28

2.2.3.1 Ground investigation conditions Table 2.4: Geological studies depend on the importance class (CYS NA EN1998-1-1, NA 2.3 / cl.3.1.1 (4)) Importance class of buildings Ground

I

II

III

IV

A

NRGS

NRGS

RGS

RGS

B

NRGS

NRGS

RGS

RGS

C

NRGS

NRGS

RGS

RGS

D

NRGS

NRGS

RGS

RGS

E

NRGS

NRGS

RGS

RGS

Type

NRGS: Not required geological studies RGS: required geological studies if there is not adequate information 2.2.3.2 Importance factor Table 2.5: Importance classes for buildings (ΕΝ1998-1-1,table.4.3 and CYS NA EN19981-1,cl NA2.12) Importance

Buildings

class I II

Buildings of minor importance for public safety, e.g. argricultural buildings, etc. Ordinary buildings, not belonging in the other categories.

Important

Consequences

factor γI

Class

0.8

CC1

1.0

CC2

1.2

CC3

1.4

CC3

Buildings whose seismic resistance is of III

importance in view of the consequences associated with a collapse, e.g. schools, assembly halls, cultural institutions etc. Buildings whose integrity during earthquakes

IV

is of vital importance for civil protection, e.g. hospitals, fire stations, power plants, etc.

Page 29

CC1: Low consequence for loss of human life, and economic, social or environmental consequences small or negligible. CC2: Medium consequence for loss of human life, economic, social or environmental consequences considerable. CC3: High consequence for loss of human life, or economic, social or environmental consequences very great

2.2.3.3 Ductility class

Table 2.6: Requirement for importance class relate to ductility class (CYS NA EN19981-1,cl NA2.16 & cl.5.2.1(5)) Importance

Zone 1

Zone 2

Zone 3

I

DCL

DCL

DCL

II

DCM/DCH

DCM/DCH

DCM/DCH

III

DCM/DCH

DCM/DCH

DCM/DCH

IV

DCH

DCH

DCH

class

DCL: Ductility class low. DCM: Ductility class medium. DCH: Ductility class high.

Page 30

2.3 Analysis types

2.3.1 Modal Response spectrum analysis

Table 2.7: Requirements of modal response spectrum analysis according to Eurocode 8 Requirements

Values

References

YES / NO

ΕΝ1998-1-1,table 4.1

Regular in elevation

NO

ΕΝ1998-1-1,table 4.1

Sum of the effective

≥ 90%

Regular in plan

modal masses

≥ 5% of total mass

EN1998-1-1,cl.4.3.3.1(3)

k ≥3.√n

Minimum number of

k: is the number of modes

modes

EN1998-1-1,cl.4.3.3.1(5)

n: is the number of storey Tk ≤ 0.20sec

Behaviour factor q

Tk: is the period of vibration of

EN1998-1-1,cl.4.3.3.1(5)

mode k. Fundamental period

Tj ≤ 0.9 Ti

SRSS

Tj ≥ 0.9 Ti

CQC

Accidental eccentricity

See section 2.1.1.1

EN1998-1-1,cl.4.3.3.2.1(2) EN1998-1-1,cl.4.3.2

1. Independently in X and Y direction, 2. Define design spectrum, 3. Use CQC rule for the combination of different modes (EN1998-1-1,cl.4.3.3.3.2(3)) 4. Use SRS rule for combined the results of modal analysis for both horizontal directions (EN1998-1-1,cl.4.3.3.5.1(21)). 5. Modal Combination: “Complete Quadratic Combination” (CQC) can be used if the Tj ≤ 0,9 Ti (EN1998-1-1,cl.4.3.3.3.2(3)P).

Page 31

2.3.1.1 Accidental eccentricity

Accidental eccentricity of each storey cause of uncertainties location of masses have been taken into account 5% (EN1998-1-1,cl.4.3.2). Moreover, if there are masonry infills with a moderately irregular and asymmetric distribution in plan, is doubled further in Eurocode 8 (i.e., to 10% of the storey orthogonal dimension in the baseline case, or 20% if accidental torsional effects are evaluated in a simplified way when using two separate 2D models).

Table 2.8: Summary of accidental eccentricity Percentage of

Geometry

Asymmetric

Masonry infills

accidental

of model

distribution of mass

(Regular/Irregular)

eccentricity

(3D/2D)

(Regular/Irregular)

5%

3D

Regular

Regular

10%

3D

Irregular

Irregular

20%

2D

-

-

Note: Accidental eccentricity is automatically included during response-spectrum analysis in ETABS, though equivalent static-load procedures are also available for manual evaluation. Note that floor diaphragms must be rigid, otherwise torsional effects are not substantial. ETABS implements an efficient and practical approach while formulating dynamic response from accidental eccentricity. After the response-spectrum load case is run, the X and Y acceleration at each joint location is determined, then multiplied by the tributary mass and the diaphragm eccentricity along either Y or X. The larger absolute value of these resultant moments (m*Xacc*dY or m*Yacc*dX) is then applied as torsion about the joint location. Static response is then added to response-spectrum output to account for the additional design forces caused by accidental eccentricity.

Page 32

Define > Response spectrum cases Note: Add two response spectrum cases: EQX and EQY as showing below (figure 9).

Figure 2.3: Response Spectrum case Data for EQY& EQX

Page 33

2.3.2 Lateral force analysis requirements Table 2.9: Requirements of lateral force analysis according to Eurocode 8 Requirements

Values

References

YES / NO

ΕΝ1998-1-1,table 4.1

Regular in elevation

YES

ΕΝ1998-1-1,table 4.1

Ground acceleration

0.10-0.25g

Regular in plan

Spectrum type

1

CYS NA EN1998-11:Seismic zonation map EN1998-1-1,cl.3.2.2.2(2)P

A,B,C,D,E Ground type

Normally type B or C can be used

EN1998-1-1,cl.3.1.2(1)

normal condition Lower bound factor for the horizontal design spectrum

λ = 0.85 if T1 ≤ 2TC and more than 2 storey

EN1998-1-1,cl.4.3.3.2.2(1Ρ)

λ=1.0 in all other case

Behaviour factor q

Fundamental period Accidental eccentricity

Concrete DCM

q= 1.5 – 3.90

EN1998-1-1,cl.5.2.2.2(2)

Concrete DCH

q= 1.6 – 5.85

EN1998-1-1,cl.5.2.2.2(2)

Steel DCM

q= 2.0 – 4.00

EN1998-1-1,cl.6.3.2(1)

Steel DCH

q= 2.0 – 5.85

EN1998-1-1,cl.6.3.2(1)

T1≤4Tc T1≤2,0s

EN1998-1-1,cl.4.3.3.2.1(2)

See section 2.1.1.1

EN1998-1-1,cl.4.3.2

Table 2.10: Equivalent Static Force Case Load case name

Direction and Eccentricity

% Eccentricity

EQXA

X Dir + Eccen. Y

0.05

EQYA

X Dir – Eccen. Y

0.05

EQXB

Y Dir + Eccen. X

0.05

EQYB

Y Dir – Eccen. X

0.05

Page 34

2.3.4 Estimation of fundamental period T1

Table 2.11: Estimation of fundamental period T1 Reference structure

Period T1

Exact formula for Single Degree of Freedom Oscillator. Mass M lumped at top of a vertical cantilever of height H. Cantilever mass MB = 0.

𝑀𝐻! 𝑇! = 2𝜋 3𝐸𝐼

Exact formula for Single Degree of Freedom Oscillator. Vertical cantilever of height H and of

𝑇! = 2𝜋

total mass MB.

0.24𝑀! 𝐻! 3𝐸𝐼

Exact formula for Single Degree of Freedom Oscillator. Mass M lumped at top of a vertical cantilever of height H and of total mass MB.

𝑇! = 2𝜋

𝑀 + 0.24𝑀! 𝐻! 3𝐸𝐼

Approximate Relationship (Eurocode 8). Ct = 0,085 for moment resisting steel space frames Ct = 0,075 for eccentrically braced steel frames Ct = 0,050 for all other structures

𝑇! = 𝐶! 𝐻!/! H building height in m measured from foundation or top of rigid basement.

Approximate Relationship (Eurocode 8). d : elastic horizontal displacement of top of building in m under gravity loads applied

𝑇! = 2 𝑑

horizontally.

Page 35

2.3.5 Automatic Lateral force analysis using ETABS ETABS: Define > Static load cases Figure 2.4: Apply the Equivalent Static Force Case

Figure 2.5: Modify the Equivalent Static Force Case

Note: The seismic forces should be applied only above the top of the basement

Page 36

Fundamental period (EN1998-1-1,Eq.4.6) T1=CtH3/4 (For heights up to 40m)

Value of Ct(EN1998-1-1,cl.4.3.3.2.2(3)) Ct = 0.085 (for moment resisting steel frames) Ct= 0.075 (for moment resisting concrete frames) Ct= 0.05 (for all other structures) (EN 1998-1-1:2004, cl. 4.3.3.2.2(3)) Ct= 0.075/√ΣAc(for concrete/masonry shear wall structures) (EN 1998-1-1:2004, Eq. 4.7)

Fundamental period requirements (EN1998-1-1,Eq.4.6) T1≤4TCT1≤2sec IF this

YES

LATERAL FORCE ANALYSIS

Correction factor λ(EN1998-11,cl.4.3.3.2.2(1Ρ)) λ=0.85 if T1≤2TC and more than 2 storey λ=1.0 in all other case

Horizontal seismic forces (according to displacement of the masses) s! ∙ m ! F! = F! ∙ s! ∙ m ! (EN 1998-1-1:2004, Eq. 4.10)

Ac= Σ[Ai·(0,2+(lwi/H2))] (EN 1998-1-1:2004, Eq. 4.8) NO

RESPONSE SPECTRUM ANALYSIS

Design spectrum Sd(T)(EN1998-11,cl.3.2.2.5) 0≤T≤TB TB≤T≤TcTC≤T≤TD TD≤T

Base shear(EN1998-11,cl.4.3.3.2.2) Fb=Sd(T1).m.λ (EN 1998-1-1:2004, Eq. 4.5)

Seismic mass(EN1998-11,cl.3.2.4) ΣGk,j/g”+”ΣψE,i.Qk,i/g (EN 1998-1-1:2004, Eq.3.17)

Horizontal seismic forces (according to height of the masses) z! ∙ m ! F! = F! ∙ z! ∙ m ! (EN 1998-1-1:2004, Eq. 4.11)

Page 37

2.3.6 User loads - Lateral force analysis using ETABS Geometrical data

Span of the longitutinal direction

Lx := 15m

Span of the transverse direction

Ly := 15m

Span of each beam

Lb := 5m

Span of each bracing

Lt := 5.831m

Height of each column

hc := 3m

Total heigh of building

H := 9m

Area of floor for each storey

A f := Ly ⋅ Lx = 225m

Number of floors

Nf := 3

Number of beams IPE240 at each floor

Nb := 24

Number of beams IPE180 at each floor

Ns := 9

Number of columns HE280A at each floor

Nc := 16

Number of TUBE sections D127-4 at each floor

Nt := 8

2

Page 38

Dead load −1

Weight of steel column HE280A

g c := 76.4kg⋅ m

Weight of primary beams IPE240

g p := 30.7kg⋅ m

Weight of secondary beams IPE180

g s := 18.8kg⋅ m

Weight of steel beams TUBE-D127-4

g t := 12.38kg⋅ m

Slab thickness

hs := 170mm

Weigth of concrete

γ c := 25kN⋅ m

Weight of slab

g slab := γ c⋅ h s = 4.25⋅ kN⋅ m

Weigth of finishes

g fin := 1kN⋅ m

−1

−1 −1

−3 −2

−2

Total dead load

(

3

)

Gk.storey := ⎡ gc⋅ Nc ⋅ hc + g p⋅ Nb ⋅ Lb + g s ⋅ Ns ⋅ Lb + gt⋅ Nt ⋅ Lt g + g slab ⋅ A f + g fin⋅ A f⎤ = 1.267 × 10 ⋅ kN ⎣ ⎦

Total dead load

(

3

)

Gk := ⎡ g c⋅ Nc ⋅ h c + g p ⋅ Nb ⋅ Lb + g s ⋅ Ns ⋅ Lb + g t⋅ Nt ⋅ Lt g + g slab ⋅ A f + g fin⋅ A f⎤ ⋅ Nf = 3.802 × 10 ⋅ kN ⎣ ⎦

Live load Combination coefficient for variable action

ψEi := 0.3

Live load

q k := 2kN⋅ m

Total live load

Qk := qk⋅ Af = 450⋅ kN

Total gravity load per storey (EN1998-1-1,cl.3.2.4(2)P)

FEd.storey := Gk.storey + ψEi⋅ Qk = 1.402 × 10 ⋅ kN

Total gravity load per storey (EN1998-1-1,cl.3.2.4(2)P)

FEd := Gk + ψEi⋅ Qk ⋅ Nf = 4.207 × 10 ⋅ kN

Seismic mass

S_mass :=

−2

(

(

FEd g

3

)

3

)

5

= 4.29 × 10 kg

Page 39

Horizontal design response Spectrum (EN1998-1-1,cl.3.2.2.5) Behaviour factor q (EN1998-1-1,cl.6.3) Lower bound factor (EN1998-1-1,cl.3.2.2.5(4)P)

q := 1.5

Seismic zone (CYS NA EN1998-1-1, zonation map)

Seismic_zone := "3"

β := 0.2

agR :=

0.15g if Seismic_zone 0.2g if Seismic_zone 0.25g if Seismic_zone

Importance factor (CYS NA EN1998-1-1,cl. NA2.12)

Value of Ct (EN1998-1-1,cl.4.3.3.2.2(3))

Fundamental period of vibration (EN1998-1-1,cl.4.3.3.2.2(3)) Type of soil (EN1998-1-1,cl.3.1.2(1))

= 2.452

m s

"2"

2

"3"

Importance_factor := "II" γ I :=

Design ground acceleration on type A (EN1998-1-1,cl.3.2.1(3))

"1"

0.8 if Importance_factor

"I"

=1

1.0 if Importance_factor

"II"

1.2 if Importance_factor

"III"

1.4 if Importance_factor

"IV"

m ag := γ I⋅ agR = 2.452 2 s

Value_Ct := "OTHER" Ct :=

0.085 if Value_Ct

"MRSF"

0.075 if Value_Ct

"MRCF"

0.05 if Value_Ct

"OTHER"

= 0.05

3⎤ ⎡ ⎢ ⎥ 4 ⎢ ⎛ H ⎞ ⎥ T1 := ⎢Ct⋅ ⎜ ⎟ ⎥ s = 0.26s ⎣ ⎝ m ⎠ ⎦

Soil_type := "B"

Value of parameters describing the Type 1 elastic response spectrum (EN1998-1-1,table 3.2) Soil factor, S

S :=

1.0 if Soil_type

"A"

1.2 if Soil_type

"B"

1.15 if Soil_type

"C"

1.35 if Soil_type

"D"

= 1.2

Page 40

Lower limit of the period, TB

TB :=

Upper limit of the period, TC

TC :=

Constant displacement value, TD

Corection factor λ (EN1998-1-1,cl.4.3.3.2.2(1)P)

TD :=

λ :=

0.15s if Soil_type

"A"

0.15s if Soil_type

"B"

0.20s if Soil_type

"C"

0.20s if Soil_type

"D"

0.40s if Soil_type

"A"

0.50s if Soil_type

"B"

0.60s if Soil_type

"C"

0.80s if Soil_type

"D"

2.0s if Soil_type

"A"

2.0s if Soil_type

"B"

2.0s if Soil_type

"C"

2.0s if Soil_type

"D"

0.85 if T1 ≤ 2TC ∧ Nf > 2

= 0.15s

= 0.5s

= 2s

= 0.85

1 otherwise

Check the fundamental period of vibration requirements (EN1998-1-1,cl.4.3.3.2.1(2))

(

Check_1 := if T1 ≤ 4TC ∧ T1 ≤ 2s , "Lateral force analysis" , "Response spectrum analysis"

)

Check_1 = "Lateral force analysis"

Design spectrum for elastic analysis (EN1998-1-1,cl.3.2.2.5(4)P)

⎡ 2 T1 ⎛ 2.5 2 ⎞⎤ S1e T1 := ag ⋅ S⋅ ⎢ + ⋅ ⎜ − ⎟⎥ 3 TB ⎝ q 3 ⎠ ⎣ ⎦

( )

2.5 S2e T1 := ag⋅ S⋅ q

( )

( )

S3e T1 :=

ag ⋅ S⋅

( )

S4e T1 :=

( )

−2

( )

−2

S3e TC = 4.903⋅ m⋅ s

2.5 TC if β ⋅ ag ≥ ag ⋅ S⋅ ⋅ q T1

⎛ 2.5 TC⋅ TD ⎞ 2.5 TC⋅ TD ⎜ ag⋅ S⋅ ⋅ ⎟ if ag⋅ S⋅ ⋅ ≥ β ⋅ ag q 2 ⎟ q 2 ⎜ T1 T1 ⎝ ⎠

( β ⋅ ag )

−2

S2e TB = 4.903⋅ m⋅ s

2.5 TC 2.5 TC ⋅ if ag ⋅ S⋅ ⋅ ≥ β ⋅ ag q T1 q T1

( β ⋅ ag )

S1e( 0) = 1.961⋅ m⋅ s

2.5 TC⋅ TD if ag ⋅ S⋅ ⋅ ≤ β ⋅ ag q 2 T1

( )

Page 41

m S4e T1 = 72.642 2 s

( )

(

(

(

Se( T) := if T < TB, S1e( T) , if T < TC, S2e( T) , if T < TD, S3e( T) , S4e( T)

)))

T := 0.01sec , 0.02sec .. 4sec

8

6

Se( T )4

2

0

0

1

2

3

4

T

Design spectrum acceleration

Se :=

S1e( 0) if 0 ≤ T1 ≤ TB

( ) S3e( TC) S4e( T1) S2e TB

Seismic base shear (EN1998-1-1,cl.4.3.3.2.2(1)) Seismic base shear on each bracing Note: 2 bracing on each direction

if TB ≤ T1 ≤ TC

= 4.903

m s

2

if TC ≤ T1 ≤ TD if TD ≤ T1 ≤ 4s

T1 Fb := S_mass ⋅ Se⋅ ⋅ λ = 464.519kN ⋅ s Fb Fb.bracing := = 232.259kN ⋅ 2

Page 42

3

Mass per storey

mi := FEd.storey = 1.402 × 10 kN

Heigth at roof level

z3 := 9m

Heigth at level 2

z2 := 6m

Heigth at level 1

z1 := 3m 4

Total mass:

Σmi_zi := FEd.storey ⋅ z3 + FEd.storey ⋅ z2 + FEd.storey ⋅ z1 = 2.524 × 10 kN⋅ m

Lateral force at roof level (EN1998-1-1,Eq.4.11)

mi⋅ z3 F3 := ⋅ F = 232.259kN ⋅ Σmi_zi b

Lateral force at level 2 (EN1998-1-1,Eq.4.11)

mi⋅ z2 F2 := ⋅ F = 154.84kN ⋅ Σmi_zi b

Lateral force at level 1 (EN1998-1-1,Eq.4.11)

mi⋅ z1 F1 := ⋅ F = 77.42kN ⋅ Σmi_zi b

F := F3 + F2 + F1 = 464.519kN

Check lateral force per storey

(

Check_2 := if F ≠ Fb , "OK" , "NOT OK"

)

Check_2 = "OK"

Table 2.12: Summary table of the lateral force results

Story STORY1 STORY2 STORY3

Heigth                              Mass                                     zi                                         mi                                     zi*mi (m) (kN) 9 6 3 TOTAL

1402 1402 1402 4206

12618 8412 4206 25236

Fb                        F=Fb(zi*mi)/           (kN) Σzi*mi 464.52 464.52 464.52

232.26 154.84 77.42 464.52

Moment   M=F*zi   (kNm) 2090.34 929.04 232.26 3251.64

Accidental   Length  of   eccentricity   floor  Lx=Ly ei=0.05L 15 15 15

0.75 0.75 0.75

Torsional   Moment  due  to   moment   SRSS   M=F*ei     MSRS=√Mx^2+My^2   (kNm) (kNm) 174.195 246.3489315 116.13 164.232621 58.065 82.1163105

Page 43

ETABS: Define > Static load case > Figure 2.6: Define manually the lateral forces

Figure 2.7: Define manually the lateral forces/moments per storey

Page 44

2.3.7 Torsional effects FLOW CHART OF TORSIONAL EFFECTS

Carry out Lateral force analysis/ Response spectrum analysis

𝑒! = +0.05 ∗ 𝐿!

𝑒! = −0.05 ∗ 𝐿!

𝑒! = +0.05 ∗ 𝐿!

𝑒! = −0.05 ∗ 𝐿!

𝑀! = 𝑒! 𝐹!

𝑀! = 𝑒! 𝐹!

SRSS rule

𝑀!"!! =

𝑀! ! + 𝑀! !

Page 45

2.3.8 Summary of analysis process in seismic design situation Importance class/Ductility class

I

II

III

IV

DCL

DCM DCH

DCM DCH

DCH

Ignore “topographic amplification effects”

Consider “topographic amplification effects” Ignore

Consider

Slopes Preferences > Steel frame design Figure 7.1: Steel frame design preferences

1 2 3 4 5

6

Page 61

Table 7.1: Steel frame design parameters Note 1: Reliability class Class section classification according to EN1998-1-1,cl.6.5.3(2) 1. Depending on the ductility class and the behavior factor q used in the design, the requirements regarding the cross-sectional classes of the steel elements which dissipate energy are indicated in table below (EN1998-1-1,cl.6.5.3(2).

Ductility class

Reference q factor Lower

q factor

limit DCM DCH

Cross-Section Class

Upper limit

1.5<

q

≤2

Class 1, 2 or 3

2.0<

q

≤4

Class 1 or 2

4.0<

q

Class 1

Note 2: Frame type See section 2.0 of this manual Note 3: Gamma factors Partial factors Resistance of cross-sections whatever the

Values

Reference

γΜ0=1.00

EN1993-1-1,cl.6.1(1)

γΜ1=1.00

EN1993-1-1,cl.6.1(1)

γΜ1=1.25

EN1993-1-1,cl.6.1(1)

class Resistance of members to instability assessed by member checks Resistance of cross-sections in tension to fracture Note 4: Behavior factor See section 2.0 of this manual Note 5: System Omega Omega Factor (System Overstrength Factor) axial load member: (𝛀 = 𝑵𝒑𝒍,𝑹𝒅 /𝑵𝑬𝒅 ) Omega factor may different for each diagonal member. Page 62

1. Run the design analysis with the Ω=1 2. Find the Npl,Rd and NEd of the bracing member and then overwrite the omega factor for each diagonal member separately and then re-run the analysis.(Ω=1). Note: Omega factor should be limited to the following for all diagonal members

(

Check_16 := if Ωmax ≤ 1.25Ωmin, "OK" , "NOT OK"

)

Note 6: Vertical deflection limits STEEL MEMBERS (CYS NA EN1993-1-1,table NA.1) Vertical deflection

Limits

Cantilevers

wmax L/180

Beams carrying plaster or other brittle finish

L/360

Other beams (except purlin and sheeting rails) Purlins and sheeting rails

L/250

General use

To suit cladding L/300

ETABS deflection limits DL limit, L/

360

Super DL+LL Limit, L/

360

Live load Limit, L/

360

Total Limit, L/

360

Total Camper Limit, L/

360

Page 63

8.0 Analysis and design requirements for Concentrically braced frames according to EN1998-1-1,cl.6.7.2 Analysis requirements according to EN1998-1-1,cl.6.7.2

Beams & Columns 1. Under gravity load conditions, only beams and columns shall be considered to resist such loads, without taking into account the bracing members (EN1998-11,cl6.7.2(1)P). Diagonal members 2. The diagonals shall be taken into account as follows in an elastic analysis of the structure for the seismic action: a) in frames with diagonal bracings, only the tension diagonals shall be taken into account, b) in frames with V bracings, both the tension and compression diagonals shall be taken into account (EN1998-1-1,cl6.7.2(2). 3. Taking into account of both tension and compression diagonals in the analysis of any type of concentric bracing is allowed provided that all of the following conditions are satisfied: a) a non-linear static (pushover) global analysis or non-linear time history analysis is used, b) both pre-buckling and post-buckling situations are taken into account in the modeling of the behavior of diagonals and, c)

background information justifying the model used to represent the behavior of diagonals is provided (EN1998-1-1,cl6.7.2(3).

Page 64

8.1 Steps of the design detail of Concentric steel frames Table 8.1: Detail steel frame design Design step

Description

number Step 1

Design of slab under gravity loads (without CBF bracings) considering columns as fixed supports

Step 2

Design columns under gravity loads (without CBF bracings)

Step 3

Design beams under gravity loads (without CBF bracings)

Step 4

Check concentric bracings under gravity loads combination

Step 5

Accidental torsional effects

Step 6

Second order effects (P-Δ) (P loads are those taken in the definition of the seismic mass “m”)

Step 7

Check of beams and of concentric bracings under gravity loads combination

Step 8

Design of concentric bracing under seismic combination of loads with the accidental torsional effects and P-Δ effects taken into account

Step 9

Check of beams and columns under seismic combination of loads with bracings overstrength factors Ω and with second order effects taken into account

Step 10

Re-run the analysis with the modified overstrength factors Ω

Page 65

8.2 Classification of steel sections

Table 8.2: Section classification (EN1993-1-1,cl.5.5) Classes Class 1

Analysis type Plastic analysis

Description Section can form a plastic hinge with the rotation capacity required from plastic analysis, without reduction of the resistance

Class 2

Plastic/ Elastic analysis Section can develop its plastic moment capacity, but has limited rotation capacity.

Class 3

Elastic analysis

Section in which the stress in the extreme compression fiber of the section, assuming an elastic distribution of stresses, can reach the yield strength, but local buckling is likely to prevent the development of the plastic moment capacity.

Description of detail

Equations

References

requirements fy. :=

fy if t < 16mm −2

fy − 10N ⋅ mm

−2

fy − 20N ⋅ mm

Reduction of yield and ultimate strength of sections

if 16mm < t < 40mm if 40mm < t < 80mm

EN10025-2 fu. :=

fu if t ≤ 16mm −2

fu − 10N ⋅ mm

−2

fu − 20N ⋅ mm

ε - Factor

ε :=

if 16mm < t ≤ 40mm if 40mm < t ≤ 80mm

EN1993-1-1,Table 5.2

235 fy

Depth of a part of section for internal compression

EN1993-1-1,Table 5.2

cw := h − 2⋅ tf − 2⋅ r

(I-sections) Class_type web :=

"CLASS 1" if

cw tw

≤ 72⋅ ε

Section classification for web element

"CLASS 2" if 84⋅ ε <

cw

"CLASS 3" if 105⋅ ε <

tw cw tw

≤ 83⋅ ε

EN1993-1-1,Table 5.2

≤ 124⋅ ε

Page 66

Depth of a part of section for oustand flange

cf :=

(I-sections) Class_type flange :=

(b − tw − 2.r )

EN1993-1-1,Table 5.2

2

"CLASS 1" if

cf tf

≤ 9⋅ ε

Section classification for flange element

"CLASS 2" if 9⋅ ε <

cf

"CLASS 3" if 10⋅ ε <

tf cf tf

≤ 10⋅ ε

EN1993-1-1,Table 5.2

≤ 14⋅ ε

Page 67

8.3 Design of composite slab under gravity loads Table 8.3: Detail design of composite slab (with steel sheeting) Partial factor Partial factor of longitudinal shear in composite slabs

Value

References

γvs = 1.25

CYS EN1994-11cl.2.4.1.2(6)P

Partial factor for shear connector

γv = 1.25

CYS EN1994-11cl.2.4.1.2(5)P

Partial factor for steel reinforcement

γs = 1.15

CYS EN1992-1-1,table 2.1

Partial factor of concrete

γc = 1.5

CYS EN1992-1-1,table 2.1

γM0 = 1.0

CYS EN1993-1-1,cl 6.1(1)

Equations

References

Minimum nominal thickness of profile steel sheets

t ≥ 0.70mm

CYS EN1994-1-1,cl.3.5(2)

Minimum depth of slab

h ≥ 90mm

EN1994-1-1,cl.9.2.1(2)

Depth of concrete slab above steel sheeting

hc ≥ 50mm

EN1994-1-1,cl.9.2.1(2)

As.prov ≥80mm2/m

EN1994-1-1,cl.9.2.1(4)

Spacing of the reinforcement bars

s = min{2h,350mm}

EN1994-1-1,cl.9.2.1(5)

Maximum height of steel decking

hp ≤ 85mm

EN1994-1-1,cl.6.6.4.2(3)

b0 ≥ hp

EN1994-1-1,cl.6.6.4.2(3)

d ≤ 20mm

EN1994-1-1,cl.6.6.4.2(3)

Partial factor of structural steel Description of detail requirements

Minimum steel reinforcement in both direction

Minimum width per ribs Diameter of stud that welded in the sheeting

Page 68

For holes provided in the sheeting, the diameter of the stud Maximum overall height of stud

Design stage

Description of checks

d ≤ 22mm

EN1994-1-1,cl.6.6.4.2(3)

hsc ≤ hp +75mm

EN1994-1-1,cl.6.6.4.1(2)

Equations

References

Resistance verifications of metal decking at the construction stage Moment resistance of steel sheeting Concrete compressive strength Design yield strength Bending resistance of metal decking

From manufacture data

-

fcd = fck / γc

EN1994-1-1,cl.2.4.1.2(2)P

fyo,d = fyp / γM0

-

MEd / MRd 1.2

"a" if tf < 40mm "b" if 40mm < tf < 100mm

Buckling curve if

h b

EN1993-1-1,table 6.2

≤ 1.2

"b" if tf ≤ 100mm "d" if tf > 100mm

Imperfection factor a

EN1993-1-1,table 6.1

Page 81

αy :=

0.1 if Buckling_class_Y

"ao"

0.21 if Buckling_class_Y

"a"

0.34 if Buckling_class_Y

"b"

0.49 if Buckling_class_Y

"c"

0.76 if Buckling_class_Y

"d"

Φ = 0.5 [1 + α (λ – 0.2) + λ2

Value to determine the reduction factor χ Reduction factor χ

χ=

Design buckling resistance of a compression member Buckling length

1 Φ + Φ ! − λ! 𝜒𝐴𝑓! 𝑁!,!" = 𝛾!! )

≤ 1,0

See: Figure 1: Effective length columns

Elastic critical force for the relevant buckling mode based on the

𝑁!".! =

gross cross sectional properties Non dimensional slenderness

λ! = Buckling_class_Y :=

if

h b

𝐸! 𝐼! 𝜋 ! 𝐿!".! ! 𝐴𝑓! 𝑁!".!

EN1993-1-1,cl.6.3.1.2(1) EN1993-1-1,cl.6.3.1.2(1) EN1993-1-1,cl.6.3.1.1(3) Design Guidance of EC3) EN1993-1-1,cl.6.3.1.2(1)

> 1.2

"a" if tf < 40mm "b" if 40mm < tf < 100mm

Buckling curve if

h b

EN1993-1-1,table 6.2

≤ 1.2

"b" if tf ≤ 100mm "d" if tf > 100mm

Imperfection factor a

EN1993-1-1,table 6.1

Page 82

αz :=

0.1 if Buckling_class_Z

"ao"

0.21 if Buckling_class_Z

"a"

0.34 if Buckling_class_Z

"b"

0.49 if Buckling_class_Z

"c"

0.76 if Buckling_class_Z

"d"

Value to determine the reduction factor χ

Φ = 0.5 [1 + α (λ – 0.2) + λ2

Reduction factor χ

χ=

Design buckling resistance of a compression member

≤ 𝜒 ≤ 1,0 Φ + Φ ! − λ! 𝜒𝐴𝑓! 𝑁!,!",! = 𝛾!! )

(

Non dimensional slenderness

)

λ := max λ y , λ z

Check the bukling effects if can be ignored and only cross section check is adequate

1

EN1993-1-1,cl.6.3.1.2(1) EN1993-1-1,cl.6.3.1.2(1) EN1993-1-1,cl.6.3.1.1(3) EN1993-1-1,cl.6.3.1.2(1)

Check := if (λ < 0.2, "Ignored buckling effects" , "Consider buckling effects" ) EN1993-1-1,cl.6.3.1.2(4)

Lateral torsional buckling interaction check 2

Elastic critical moment for lateral torsional buckling

Mcr := C1⋅

π ⋅ Es ⋅ Izz

(k⋅ Lcr)2

2 I

2

( cr) t k w 2 ⋅ ⎛⎜ ⎞⎟ ⋅ + + (C2⋅ zg) − C2⋅ zg 2 ⎝ kw ⎠ Izz π Es ⋅ Izz k⋅ L

G⋅ I

NCCI: SN003a-EN-EU

Effective length factor (Pinned End)

k = 1.0

NCCI: SN003a

Factor for end warping

kw = 1.0

NCCI: SN003a

Coefficient factor C1 (Load condition: UDL)

C1 := 1.88 − 1.40ψ + 0.52ψ

(

Check_5 := if C1 ≤ 2.7, "OK" , "NOT OK"

Coefficient factor C2

)

C2 = 1.554

NCCI: SN003a

zg = 0m

NCCI: SN003a

Distance between the point of load application and the shear centre (load applied on centre)

NCCI: SN003a

2

Page 83

Buckling_curve_Z :=

Lateral torsional buckling curves

"a" if

h b h

"b" if

αLT :=

b 0.21 if Buckling_curve_Z

≤2

EN1993-1-1,table 6.4

>2

"a"

Imperfection factors for lateral torsional buckling curves

0.34 if Buckling_curve_Z

"b"

0.49 if Buckling_curve_Z

"c" "d"

Non dimensional slenderness for lateral torsional buckling

0.76 if Buckling_curve_Z W pl.y ⋅ fy λ LT := Mcr

Value to determine the reduction factor χLT

can be ignored

(

(

)

EN1993-1-1,cl.6.3.2.2(1)

2

Check_6 := if λ LT < λ LTO, "Ignored torsional buckling effects" , "Consider torsional buckling effects"

EN1993-1-1,cl.6.3.2.2(1)

)

⎛ MEd.y ⎞ 2 Check_7 := if ⎜ < λ LTO , "Ignored torsional buckling effects" , "Consider torsional buckling effects" ⎟ ⎝ Mcr ⎠

Moments due to the shift of the centroidal axis for class sections 1,2 & 3 Characteristic resistance to normal force of the critical cross-section Characteristic moment resistance of the critical cross-section

EN1993-1-1,cl.6.3.2.2(1)

φ LT := 0.5⋅ ⎡1 + αLT⋅ λ LT − 0.2 + λ LT ⎤ ⎣ ⎦ 1 χ LT := 2 2 φ LT + φ LT − λ LT

Reduction factor for lateral-torsional buckling Check if the lateral torsional buckling

EN1993-1-1,table 6.3

ΔM Ed.z := 0 ΔM Ed.y := 0 NRk := fy ⋅ A My.Rk := fy ⋅ Wpl.y Mz.Rk := fy ⋅ Wpl.z

EN1993-1-1,cl.6.3.2.2(4)

EN1993-11,cl.6.3.3(4)/table 6.7 EN1993-11,cl.6.3.3(4)/table 6.7 E1993-1-1,cl.6.3.3(4)/table 6.7)

Page 84

MEd.y1

ψy :=

MEd.y2 MEd.y2 MEd.y1

Ratio of end moments

M Ed.z1

ψz :=

M Ed.z2 M Ed.z2 M Ed.z1

if −1 ≤

if −1 ≤ if −1 ≤

if −1 ≤

MEd.y2 MEd.y2 MEd.y1 M Ed.z1 M Ed.z2 M Ed.z2 M Ed.z1

≤1

≤1

EN193-1-1,Table B2) ≤1

≤1

Cmy := 0.6 + 0.4⋅ ψy Cmz := 0.6 + 0.4⋅ ψz

Equivalent uniform moment factor

EN1993-1-1,table B.1&B.2

NEd ⎞⎤ ⎤⎤ ⎛ ⎥⎥ , Cmy⋅ ⎜ 1 + 0.8⋅ ⎟⎥ NRk ⎥⎥ NRk ⎟⎥ ⎢⎢ ⎢ ⎜ χ y⋅ χ y⋅ ⎢⎢ ⎢ ⎜ γ M1 ⎥⎥ γ M1 ⎟⎥ ⎣⎣ ⎣ ⎦⎦ ⎝ ⎠⎦ N N ⎡⎡ ⎡ ⎛ ⎞⎤ Ed ⎤⎤ ⎥⎥ , Cmz⋅ ⎜ 1 + 1.4⋅ Ed ⎟⎥ kzz := min⎢⎢Cmz⋅ ⎢1 + ( 2λ z − 0.6) ⋅ NRk ⎥⎥ NRk ⎟⎥ ⎢⎢ ⎢ ⎜ χ ⋅ χ ⋅ z z ⎢⎢ ⎢ ⎜ γ M1 ⎥⎥ γ M1 ⎟⎥ ⎣⎣ ⎣ ⎦⎦ ⎝ ⎠⎦

⎡⎡

⎡

(

)

kyy := min⎢⎢Cmy⋅ ⎢1 + λ y − 0.2 ⋅

Interaction factors

MEd.y1

NEd

EN1993-1-1,table B.1&B.2

kyz := 0.6kzz kzy := 0.6kyy NEd

Combined bending and axial compression

xy ⋅ NRk γ M1

+ kyy ⋅

MEd.y + ΔM Ed.y χ LT⋅

My.Rk γ M1

+ kyz⋅

Mz.Ed + ΔM Ed.z M z.Rk

EN1993-1-1,Eq.6.61

γ M1

Page 85

NEd

Combined bending and axial compression

χ z ⋅ NRk γ M1

+ kzy⋅

MEd.y + ΔM Ed.y χ LT⋅

My.Rk γ M1

+ kzz⋅

MEd.z + ΔM Ed.z M z.Rk

EN1993-1-1,Eq.6.62

γ M1

Note: This equations is applicable only for I and H sections with section class 1 and 2 Note 1: The shear area is for rolled I and H sections, load parallel to web

Page 86

8.6 Detail design rules of steel Concentric Braced Frames (CBF) according to Eurocode 8

8.6.1 Detail design rules of steel bracing according to Eurocode 8

Description Overstrength factor used in design

Value

References

γov = 1.25

CYS EN1998-1-1cl.6.2(3)P

(

Non-dimensional slenderness (X bracing)

Check_6 := if 1.3 < λ y < 2, "OK" , "NOT OK"

)

EN1998-1-1,cl.6.7.3(1)

Non-dimensional slenderness (one diagonal)

λ ≤ 2.0

EN1998-1-1,cl.6.7.3(2)

Non-dimensional slenderness (V bracing)

λ ≤ 2.0

EN1998-1-1,cl.6.7.3(3)

Non-dimensional slenderness (V,X & one bracing) Yield resistance check Check Ω factor Check Ω factor

(

Check_5 := if Ns ≥ 3, "Consider limitation (As EC8)" , "Ignore limitation (As EC3)"

(

Check_15 := if NEd ≤ Npl.Rd , "OK" , "NOT OK"

Class_type_req :=

Ductility class require for seismic design

)

Npl.Rd Ω. := NEd Check_16 := if Ωmax ≤ 1.25Ωmin, "OK" , "NOT OK"

(

EN1998-1-1,cl.6.7.3(4) EN1998-1-1,cl.6.7.3(5) EN1998-1-1,cl.6.7.3(8)

)

EN1998-1-1,cl.6.7.3(8)

"CLASS 1 , 2 or 3" if 1.5 < q ≤ 2 ∧ Ductility_class "CLASS 1 or 2" if 2 < q ≤ 4 ∧ Ductility_class "CLASS 1" if q > 4 ∧ Ductility_class

)

"DCM"

"DCM"

EN1998-1-1,cl.6.5.3(2)

"DCH"

Page 87

8.7 Detail design rules of steel columns and beams according to Eurocode 8 Description Overstrength factor used in design Yield resistance check

References

γov = 1.25

CYS EN1998-1-1cl.6.2(3)P

(

Check_15 := if NEd ≤ Npl.Rd , "OK" , "NOT OK"

)

EN1998-1-1,cl.6.7.3(5)

Npl.Rd Ω. := NEd NEd. := NEd.G + 1.1⋅ γ ov⋅ Ω⋅ NEd.E

Check Ω factor Minimum resistance requirement, NEd Class_type_req :=

Ductility class require for seismic design

Value

EN1998-1-1,cl.6.7.3(8) EN1998-1-1,cl.6.7.4(1)

"CLASS 1 , 2 or 3" if 1.5 < q ≤ 2 ∧ Ductility_class "CLASS 1 or 2" if 2 < q ≤ 4 ∧ Ductility_class "CLASS 1" if q > 4 ∧ Ductility_class

"DCM"

"DCM"

EN1998-1-1,cl.6.5.3(2)

"DCH"

Page 88

8.8 Detail design rules of steel composite members according to Eurocode 8

Description Minimum concrete strength Steel reinforcement class Minimum degree of connection Reduction factor

Value

References

C20/25 – C40/50

CYS EN1998-1-1cl.7.2.1(1)

B or C

EN1998-1-1,cl.7.2.2(2)

η ≤ 0.8 kt = 0.75

EN1998-1-1,cl.7.6.2(3) EN1998-1-1,cl.7.6.2(4)

kt = kt * kr Profiled steel sheeting with ribs transverse to the

EN1998-1-1,cl.7.6.2(6)

supporting beams is used, the reduction factor

fy :=

Yield strength of steel

"DCM " ∧

"fy=235" if 1.5 < q ≤ 4 ∧ Ductility_class

"DCM " ∧ 0.27 < x

"fy=355" if q > 4 ∧ Ductility_class

"DCH" ∧

"fy=235" if q > 4 ∧ Ductility_class

"DCH" ∧ 0.20 <

Class_type_req :=

Ductility class require for seismic design

x

"fy=355" if 1.5 < q ≤ 4 ∧ Ductility_class

d

d

≤ 0.27 x d

EN1998-1-1,cl.7.6.2(8)

≤ 0.20 x d

≤ 0.27

"CLASS 1 , 2 or 3" if 1.5 < q ≤ 2 ∧ Ductility_class "CLASS 1 or 2" if 2 < q ≤ 4 ∧ Ductility_class "CLASS 1" if q > 4 ∧ Ductility_class

≤ 0.36

"DCM"

"DCM"

EN1998-1-1,cl.6.5.3(2)

"DCH"

Page 89

8.9 Detail design rules of steel moment resistance frames (MRF) according to Eurocode 8

8.9.1 Detail design rules for MRF - Design criteria

Description Below design criteria apply to (Bottom – Top)

Value

References

Single/Multi-story buildings

EN1998-1-1cl.6.6.1(1)

∑MRc ≥ 1.3MRb

EN1998-1-1,cl.4.4.2.3(4)

Value

References

𝑀!" ≤ 1.0 𝑀!".!"  

EN1998-1-1,cl.6.6.2.(2)

Moment capacity (where fixed support is provided)

8.9.2 Detail design rules of steel beam for MRF

Description Moment capacity verification VEd = VEd.G + VEd.M Design shear force

Shear capacity verification Axial capacity verification

Where VEd.M = (Mpl.Rd.A + Mpl.Rd.B)/L 𝑉!" ≤ 0.5 𝑉!".!"   𝑁!" ≤ 0.15 𝑁!".!"  

EN1998-1-1,cl.6.6.2.(2)

EN1998-1-1,cl.6.6.2.(2) EN1998-1-1,cl.6.6.2.(2) Page 90

8.9.3 Detail design rules of steel column for MRF

Description Overstrength factor used in design Check Ω factor (derivate from all beam with moment connection)

Ω!"# =

Value

References

γov = 1.25

CYS EN1998-1-1cl.6.2(3)P

!!".!"

MEd.E : Lateral force

!!".!  

EN1998-1-1cl.6.6.3(1P)

Design axial compression force

NEd = NEd.G +1.1γvoΩ NEd.E

NEd.E : Lateral force

EN1998-1-1cl.6.6.3(1P)

Design bending moment

MEd = MEd.G +1.1γvoΩ MEd.E

MEd.E : Lateral force

EN1998-1-1cl.6.6.3(1P)

VEd = VEd.G +1.1γvoΩ VEd.

VEd.E : Lateral force

EN1998-1-1cl.6.6.3(1P)

Design shear force Design shear force verification

𝑉!" ≤ 0.5 𝑉!".!"  

EN1998-1-1cl.6.6.3(4)

Page 91

9.0 Design of steel frames

9.1 Design of steel member overwrites data

Figure 9.1: Steel design result of the member

Overwrites Page 92

Figure 9.2: Steel frame design overwrites for Eurocode 3

1 2 3 4 5 7

6

8 9 10 11 12

Page 93

Table 9.1: Steel frame design overwrites for Eurocode 3 Explanation of Steel frame design overwrites for Eurocode 3 Note No.

Parameter

1

Effective length factor

Values

kyy

2

Moment coefficient

kzz

Page 94

3

Bending Coefficient (C1)

4

Moment coefficient Overstrength factor

5

used in design1

Npl.Rd Ω. := NEd

Omega gamma 6

factor

γov = 1.25

Compressive/Tensile 7

capacity Major bending

8

capacity, Mc3Rd Minor bending

9

capacity, Mc2Rd Buckling resistance

10

moment

Page 95

Major shear capacity, Vc3Rd

11

Minor shear

12

Notes:

capacity, Vc2Rd

1

Ω is not calculated automatically by the program. Rather, its value can be overwritten by the user through design Preference and Overwrites.

Page 96

9.2 Design of columns / beams using ETABS – Gravity load analysis only STEP 1: Analyze > Run Analysis STEP 2: Design > Steel frame design > Select design combo… Note: Under gravity load conditions, only beams and columns shall be considered to resist such loads, without taking into account the bracing members (EN1998-1-1,cl6.7.2(1)P).

Design combination at ULS STATIC 1. STATIC 10.

1.35DL + 1.5LL 1.00DL + 0.3LL Figure 9.3: Gravity load combination at ULS

Design combination at SLS DSTLD 1. DSTLD 2.

DL + LL DL

Page 97

Figure 9.4: Gravity load combination at SLS

Figure 9.5: Steel design under gravity load ONLY

Write click on each member in order to check it individually Column name: C2 Storey level: Storey 1

Page 98

Figure 9.6: Steel design result of the member

Worst case combination Figure 9.7: Ultimate moment results under worst case combination ETABS: Display > Show tables

Page 99

Take the ultimate moment and shear force from the above table and place them into the Excel spreadsheet or Mathcad file in order to verify the steel design results of ETABS.

Press the button summary

Table 9.2: Summarize of design values required to carry out the design of steel member Results

Design value

Symbol

Design axial force for gravity load combination (G+0.3Q)

NEd.GV

344.75

Design moment at y-y at end 1 (seismic load combination)

MEd.GV.y1

-1.293

Design moment at y-y at end 2 (seismic load combination)

MEd.GV.y2

3.195

Design moment at z-z at end 1 (seismic load combination)

MEd.GV.z1

-0.173

Design moment at z-z at end 2 (seismic load combination)

MEd.GV.z2

-0.142

Shear forces at y-y at end (seismic load combination)

VEd.GV.y

-0.01

Shear force at z-z at end 1 (seismic load combination)

VEd.GV.z

-1.63

(kN)

Page 100

Design results of ETABS

ETABS/HAND ETABS HAND (see section 9.3)

Description of comparison Equation 6.62 in EC3

Results 0.160 0.135

Page 101

ETABS/HAND

N.c.Rd

N.t.Rd

N.pl.Rd

ETABS

2675.75

2675.75

2675.75

HAND (see section 9.3)

2675.75

2675.75

2675.75

ETABS/HAND

Curve

Alpha

LambarBar

Phi

Chi

Nb.Rd

y-y

z-z

y-y

z-z

y-y

z-z

y-y

z-z

y-y

z-z

y-y

z-z

ETABS

“b”

“c”

0.340

0.490

0.268

0.454

0.548

0.66

0.976

0.868

2610

2322

HAND (see section 9.3)

“b”

“b”

0.340

0.340

0.248

0.42

0.539

0.625

0.983

0.918

2630

2534

Page 102

M.c.Rd

ETABS/HAND

M.v.Rd

M.b.rd

y-y

z-z

y-y

z-z

ETABS

305.8

142.45

305.8

142.45

302.05

HAND (see section 9.3)

305.8

142.45

305.8

142.45

305.80

ETABS/HAND

Curve

AlphaLT

LambdaBarLT

PhiLT

ChiLT

C1

Mcr

ETABS

a

0.21

0.255

0.538

0.988

2.532

4694

HAND (see section 9.3)

b

0.34

0.24

0.535

0.986

2.532

4679

ETABS/HAND

kyy

kyz

kzy

kzz

ETABS

0.442

0.582

0.964

0.970

HAND (see section 9.3)

0.441

0.576

0.265

0.96

Page 103

ETABS/HAND

V.c.Rd

V.pl.Rd

η

1234

504

1.2

1156

504

1.0

y-y

z-z

ETABS

504

HAND (see section 9.3)

504

Page 104

9.3 Design of steel column (Gravity design situation) – Hand calculations

1. Rolled I - section 2. Limit to class 1 and 2 section 3. Column not susceptible to torsional deformations Length of column

hc := 3m

Total axial load on column, NEd

NEd := 344.798kN

Shear force y-y axis

VEd.y := 0.011kN

Shear force z-z axis

VEd.z := 1.626kN

Design moment y-y axis

MEd.y1 := 3.195kN⋅ m

Design moment y-y axis

MEd.y2 := −1.293kN⋅ m

Maximum moment

MEd.y := max MEd.y1, MEd.y2 = 3.195kN ⋅ ⋅m

Design moment z-z axis

MEd.z1 := −0.142kN⋅ m

Design moment z-z axis

MEd.z2 := −0.173kN⋅ m

Maximum moment

MEd.z := max MEd.z1, MEd.z2 = −0.142⋅ kN⋅ m

(

)

(

)

Section properties: Depth of section,h:

h := 270mm

Width of section,b:

b := 280mm

Thickness of web, tw: Thickness of flange, tf :

tw := 8mm tf := 13mm

(

)

Thickness of element

t := max tw, tf = 13⋅ mm

Second moment of area z-z:

Izz := 47630000mm

Second moment of area y-y:

Iyy := 1.367⋅ 10 mm

4

8

4

2

Cross section area, A:

A := 9730mm

Radius of section:

r := 24mm

Heigth of web, hw

hw := h − 2tf − 2r = 196⋅ mm

Page 105

3

2

Area of the web

A w := h w⋅ tw = 1.568 × 10 ⋅ mm

Warping Constant, Iw:

Iw := 753.7⋅ 10 ⋅ mm

Torsional Constant, IT:

It := 635000mm

Plastic Modulus, Wply

W pl.y := 1112000mm

Plastic Modulus, Wplz

W pl.z := 518000mm

Elastic modulus, E:

Es := 210kN⋅ mm

Yield strength of steel , fy:

fy := 275N ⋅ mm

Ultimate strength, fu:

fu := 430N ⋅ mm

Shear modulus

G := 81kN⋅ mm

9

6

4 3

3

−2

−2 −2

−2

fy :=

fy if t ≤ 16mm −2

fy − 10N ⋅ mm

Reduction of yield and ultimate strength of sections EN10025-2 fu :=

−2

fy − 20N ⋅ mm

−2

if 16mm < t ≤ 40mm

fy = 275⋅ N ⋅ mm

if 40mm < t ≤ 80mm

fu if t ≤ 16mm −2

fu − 10N ⋅ mm

−2

fu − 20N ⋅ mm

−2

if 16mm < t ≤ 40mm

fu = 430⋅ N ⋅ mm

if 40mm < t ≤ 80mm

Partial safety factor Resistance of cross-sections whatever the class (CYS EN1993-1-1,cl 6.1(1)) Resistance of members to instability (CYS EN1993-1-1,cl 6.1(1)) Resistance of cross-section in tension (CYS EN1993-1-1,cl.6.1(1))

γ M0 := 1 γ M1 := 1 γ M2 := 1.25

Section classification ε :=

For section classification the coefficient ε is: For a flange element:

235 fy

= 0.924

−2

N⋅ mm

Page 106

Class_type flange :=

"CLASS 1" if

cf tf

cf :=

≤ 9⋅ ε cf

"CLASS 2" if 9⋅ ε <

tf

"CLASS 3" if 10⋅ ε <

cf tf

"CLASS 1" if

cw tw

Class_type flange = "CLASS 2" ≤ 14⋅ ε

≤ 72⋅ ε cw

"CLASS 3" if 105⋅ ε <

(

= 112⋅ mm

cw := h − 2⋅ tf − 2⋅ r = 196⋅ mm

"CLASS 2" if 84⋅ ε <

Class_type := if Class_type flange

2

≤ 10⋅ ε

For a web element:

Class_type web :=

(b − tw − 2.r )

tw cw tw

Class_type web = "CLASS 1" ≤ 83⋅ ε

≤ 124⋅ ε

Class_type web , Class_type flange , "ADD MANUALY"

)

Class_type = "ADD MANUALY"

Tension resistance (cl.6.2.3) Design plastic resistance of the cross section (EN1993-1-1,cl.6.2.3(2)

A ⋅ fy 3 Npl.Rd := = 2.676 × 10 ⋅ kN γ M0

Design ultimate resistance (EN1993-1-1,cl.6.2.3(2b))

0.9A ⋅ fy 3 Nu.Rd := = 1.927 × 10 ⋅ kN γ M2

Design tension resistance (EN1993-1-1,cl.6.2.3(2))

Nt.Rd := min Nu.Rd , Npl.Rd = 1.927 × 10 ⋅ kN

Check tension capacity

Check1 := if ⎜

(

⎛ NEd ⎝ Nt.Rd

)

3

⎞

≤ 1.0, "OK" , "NOT OK" ⎟

⎠

Check1 = "OK"

Page 107

Compression resistance (cl.6.2.4) Compression resistance of steel section (EN1993-1-1,cl.6.2.4(1))

Nc.Rd := Npl.Rd = 2.676 × 10 ⋅ kN

Check compression capacity (EN1993-1-1,cl.6.2.4(1)P)

Check2 := if ⎜

3

⎛ NEd

⎞

⎝ Nc.Rd

≤ 1.0, "OK" , "NOT OK" ⎟

⎠

Check2 = "OK"

Bending resistance (cl.6.2.5) Moment resistance of steel section at Y-Y (EN1993-1-1,cl.6.2.5(2)

Mc.Rd.y :=

Moment resistance of steel section at Z-Z (EN1993-1-1,cl.6.2.5(2)

Mc.Rd.z :=

W pl.y⋅ fy

= 305.8⋅ kN⋅ m

γ M0 W pl.z⋅ fy γ M0

= 142.45⋅ kN⋅ m

Shear resistance (cl.6.2.6) Factor for shear area (EN1993-1-1,cl.6.2.6(g)) Shear area of steel section (EN1993-1-1,cl.6.2.6(3)) A v :=

A vy if A vy > η ⋅ tw⋅ hw η ⋅ tw⋅ h w if A vy < η ⋅ tw⋅ h w

Shear resistance of steel section Y-Y (EN1993-1-1,cl.6.2.6(2)) Shear area of steel section (EN1993-1-1,cl.6.2.6(3)) Shear resistance of steel section Z-Z (EN1993-1-1,cl.6.2.6(2))

η := 1

(

)

Avy := A − 2⋅ b ⋅ tf + tw + 2r ⋅ tf 3

2

A v = 3.178 × 10 ⋅ mm

Vpl.Rd.y := A v ⋅

fy ⋅ ( 3)

−1

= 504.575kN ⋅

γ M0

3

2

A vz := 2⋅ b ⋅ tf = 7.28 × 10 ⋅ mm

Vpl.Rd.z := 2⋅ b ⋅ t f ⋅

fy ⋅ ( 3)

−1

γ M0

3

= 1.156 × 10 ⋅ kN

Page 108

Check if the verification of shear buckling resistance required or not (EN1993-1-1,cl.6.2.6(6)) ⎛ hw ⎞ ε Check := if ⎜ < 72⋅ , "Not required shear buckling resistance" , "Required shear buckling resistance" ⎟ η ⎝ tw ⎠ Check = "Not required shear buckling resistance"

Bending and shear interaction check (cl.6.2.8) Strong axis Y-Y VEd.y

−5

Interaction check 1

vy :=

Reduced yield strength

⎛ 2VEd.y ⎞ ρ := ⎜ − 1⎟ = 1 ⎝ Vpl.Rd.y ⎠

Vpl.Rd.y

= 2.18 × 10 2

Reduced design plastic resistance moment (EN1993-1-1,cl.6.2.8(5))

2 ⎛ ρ⋅ A w ⎞⎟ ⎜ ⎜ W pl.y − 4t ⎟ ⋅ fy w ⎠ ⎝ M c.Rd.y := if v y > 0.5 γ

M0

M c.Rd.y if v y < 0.5

Mc.Rd.y = 305.8kN ⋅ ⋅m

Weak axis Z-Z VEd.z

−3

Interaction check 1

vz :=

Reduced yield strength

⎛ 2VEd.z ⎞ ρ := ⎜ − 1⎟ = 0.994 ⎝ Vpl.Rd.z ⎠

Vpl.Rd.z

= 1.407 × 10 2

2 ⎛ ρ⋅ A w ⎞⎟ ⎜ ⎜ W pl.z − 4t ⎟ ⋅ fy w ⎠ ⎝ Reduced design plastic resistance moment M c.Rd.z := if v z > 0.5 γ (EN1993-1-1,cl.6.2.8(5)) M0

M c.Rd.z if v z < 0.5

Mc.Rd.z = 142.45kN ⋅ ⋅m

Page 109

Check combination of axial and bending (EN1993-1-1,cl.6.2.1(7))

⎛ NEd

Check_1 := if ⎜

⎝ Npl.Rd NEd

Unity factor Npl.Rd

+

+

MEd.y Mc.Rd.y MEd.y Mc.Rd.y

+

+

MEd.z Mc.Rd.z MEd.z Mc.Rd.z

⎞

≤ 1.0, "OK" , "NOT OK" ⎟

⎠

= 0.138

Check_1 = "OK"

Bending and axial force interaction check (cl.6.2.9)

⎡ (A − 2b ⋅ tf )

⎤

Factor a

a := min⎢

Factor n

n :=

Factor β

β :=

Coefficient 1

c1 :=

Coefficient 2

c2 :=

Coefficient check

c := max c1, c2 = 1.599

A

⎣

NEd Npl.Rd

, 0.5⎥ = 0.252

⎦

= 0.129

5n if 5n ≥ 1

=1

1 otherwise NEd 0.25Npl.Rd NEd 0.5hw⋅ tw⋅ fy

(

= 0.515

= 1.599

)

Strong axis Y-Y Mc.Rd.y⋅ ( 1 − n ) M := if c > 1 N.y.Rd Reduced design value of the resistance to 1 − 0.5a bending moments making allowance for the Mc.Rd.y if 0 ≤ c ≤ 1 presence of axial forces (EN1993-11,cl.6.2.9.1(5))

MN.y.Rd = 304.764kN ⋅ ⋅m

Page 110

MN.z.Rd := Mc.Rd.z if n ≤ a Weak axis Z-Z Reduced design value of the resistance to 2 ⎡ n − a ⎞ ⎤ bending moments making allowance for the Mc.Rd.z⋅ ⎢1 − ⎛⎜ ⎟ ⎥ if n ≥ a ⎣ ⎝ 1 − a ⎠ ⎦ presence of axial forces (EN1993-1-

1,cl.6.2.9.1(5))

MN.z.Rd = 142.45kN ⋅ ⋅m

Check combination of bi-axial bending (EN1993-1-1,cl.6.2.9.1(6)) a β ⎡⎡⎛ M ⎤ ⎛ MEd.z ⎞ ⎤⎥ ⎢⎢ Ed.y ⎞ ⎥ Check_1 := if ⎜ ⎟ + ⎜ ⎟ ⎥ ≤ 1.0, "OK" , "NOT OK"⎥ ⎢⎢ M ⎣⎣⎝ N.y.Rd ⎠ ⎝ MN.z.Rd ⎠ ⎦ ⎦ a

β

⎛ MEd.y ⎞ ⎛ MEd.z ⎞ + = 0.316 Unity factor ⎜ MN.y.Rd ⎟ ⎜ MN.z.Rd ⎟ ⎝ ⎠ ⎝ ⎠ Check_1 = "OK"

Bucking interaction check (cl.6.3) Strong axis Y-Y Status of effective length

Effective_Length := " Pinned Fixed"

Effective length factor (Guidance of EC3)

k :=

0.7 if Effective_Length

"Fixed Fixed"

0.85 if Effective_Length

"Partial restraint"

0.85 if Effective_Length

" Pinned Fixed"

1 if Effective_Length

Buckling length of column (fixed end) Euler Buckling at y-y axis (EN1993-1-1,cl.6.3.1.2(1)

= 0.85

"Pinned Pinned"

Lcr := k hc = 2.55m

Ncry :=

Es ⋅ Iyy⋅ π Lcr

2

2 4

= 4.357× 10 ⋅ kN

Page 111

Slenderness parameter at y-y axis (for class 1,2 and 3 cross-section) (EN1993-1-1,cl.6.3.1.2(1) Buckling curve (EN1993-1-1,table 6.2)

Buckling_class_Y :=

λ y :=

if

h b

A ⋅ fy Ncry

= 0.248

> 1.2

"a" if tf < 40mm "b" if 40mm < tf < 100mm if

h b

≤ 1.2

"b" if tf ≤ 100mm "d" if tf > 100mm

Buckling_class_Y = "b" αy :=

Imperfection factor (EN1993-1-1,table 6.1)

0.1 if Buckling_class_Y

"ao"

0.21 if Buckling_class_Y

"a"

0.34 if Buckling_class_Y

"b"

0.49 if Buckling_class_Y

"c"

0.76 if Buckling_class_Y

"d"

αy := 0.34

2⎤

Value to determine the reduction factor χ (EN1993-1-1,cl.6.3.1.2(1))

φ y := 0.5⋅ ⎡1 + αy ⋅ λ y − 0.2 + λ y ⎣

Reduction factor χ (EN1993-1-1,cl.6.3.1.2(1))

χ y :=

Reduction factor χ check

Check1 := if χ y ≤ 1.0, "OK" , "NOT OK"

(

)

1 2

φy +

φ y − λy

2

⎦ = 0.539

= 0.983

(

)

Check1 = "OK"

Design buklcing resistance (EN1993-1-1,cl.6.3.1.1(3)) Buckling resistance of compression member check (EN1993-1-1,cl.6.3.1.1(1))

Nb.Rd.y :=

χ y⋅ A⋅ fy γ M1

⎛ NEd

Check2 := if ⎜

3

= 2.63 × 10 ⋅ kN

⎝ Nb.Rd.y

⎞

, "OK" , "NOT OK" ⎟

⎠

Check2 = "OK"

Page 112

Weak axis Z-Z Buckling length of column (fixed end)

Lcr := k⋅ hc = 2.55m

Euler Buckling at y-y axis (EN1993-1-1,cl.6.3.1.2(1)

Ncrz :=

Slenderness parameter at y-y axis (for class 1,2 and 3 cross-section) (EN1993-1-1,cl.6.3.1.2(1) Check if the buckling may be ignored (EN1993-1-1,cl.6.3.1.2(4))

λ z :=

Es ⋅ Izz⋅ π Lcr

A ⋅ fy Ncrz

(

2

2

4

= 1.518 × 10 ⋅ kN

= 0.42

)

Slenderness parameter

λ := max λ y , λ z

MinimumEuler Buckling

Ncr := min Ncry , Ncrz

⎛

NEd

⎝

Ncr

Check_2 := if ⎜ λ < 0.2 ∧

(

)

< 0.04, "Ignored buckling effects" , "Consider buckling effects"

⎞ ⎟ ⎠

Check_2 = "Consider buckling effects"

Buckling curve (EN1993-1-1,table 6.2)

Buckling_class_Z :=

if

h b

> 1.2

"a" if tf < 40mm "b" if 40mm < tf < 100mm if

h b

≤ 1.2

"b" if tf ≤ 100mm "d" if tf > 100mm

Buckling_class_Z = "b"

Imperfection factor (EN1993-1-1,table 6.1)

αz :=

αz := 0.34

0.1 if Buckling_class_Z

"ao"

0.21 if Buckling_class_Z

"a"

0.34 if Buckling_class_Z

"b"

0.49 if Buckling_class_Z

"c"

0.76 if Buckling_class_Z

"d"

Page 113

2

Value to determine the reduction factor χ (EN1993-1-1,cl.6.3.1.2(1))

φ z := 0.5⋅ ⎡1 + αz⋅ λ z − 0.2 + λ z ⎤ = 0.625 ⎣ ⎦

Reduction factor χ (EN1993-1-1,cl.6.3.1.2(1))

χ z :=

Reduction factor χ check

Check_3 := if χ z ≤ 1.0, "OK" , "NOT OK"

(

)

1 2

φz +

2

= 0.918

φ z − λz

(

)

Check_3 = "OK" χ z⋅ A ⋅ fy

Design buklcing resistance (EN1993-1-1,cl.6.3.1.1(3))

Nb.Rd.z :=

Buckling resistance of compression member check (EN1993-1-1,cl.6.3.1.1(1))

Check_4 := if ⎜

γ M1

3

= 2.457 × 10 ⋅ kN

⎛ NEd ⎝ Nb.Rd.z

⎞

, "OK" , "NOT OK" ⎟

⎠

Check_4 = "OK"

Lateral torsional buckling check (cl.6.3.2) Effective length factor, k (SN003a-EN-EU) Factor for end warping, kw (SN003a-EN-EU)

k = 0.85 kw := 1.0 MEd.y2

= −0.405

Ratio of the smaller and larger moment

ψ :=

Coefficient factor C1

C1 := 1.88 − 1.40ψ + 0.52ψ = 2.532

(SN003a-EN-EU) Coefficient factor C1 check (SN003a-EN-EU)

MEd.y1

2

(

Check_5 := if C1 ≤ 2.7, "OK" , "NOT OK"

)

Check_5 = "OK"

Coefficient factor C2

C2 := 1.554

(SN003a-EN-EU) Distance between the point of load application and the shear centre

zg := 0m

Page 114

Elastic critical moment for lateral torsional buckling (SN003a-EN-EU) 2

Mcr := C1⋅

π ⋅ Es ⋅ Izz

(Lcr)2

2 I

2

( cr) t k w 2 3 ⋅ ⎛⎜ ⎞⎟ ⋅ + + ( C2⋅ zg) − C2⋅ zg = 4.679× 10 ⋅ kN⋅ m ⎝ kw ⎠ Izz π 2Es ⋅ Izz L

G⋅ I

Lateral torsional buckling curve (EN1993-1-1,table 6.4)

Buckling_curve_Z :=

"b" if "c" if

h b h b

≤2 >2

Buckling_curve_Z = "b"

Imperfection factor for lateral torsional (EN1993-1-1,table 6.3)

αLT :=

0.21 if Buckling_curve_Z

"a"

0.34 if Buckling_curve_Z

"b"

0.49 if Buckling_curve_Z

"c"

0.76 if Buckling_curve_Z

"d"

αLT = 0.34

Non dimensional slenderness (EN1993-1-1,cl.6.3.2.2(1))

λ LT :=

W pl.y ⋅ fy Mcr

= 0.256

Value to determine the reduction factor (EN1993-1-1,cl.6.3.2.2(1))

φ LT := 0.5⋅ ⎡1 + αLT⋅ λ LT − 0.2 + λ LT ⎤ = 0.542 ⎣ ⎦

Reduction factor for lateral-torsional buckling (EN1993-1-1,cl.6.3.2.2(1))

χ LT :=

(

1 φ LT +

2

)

2

2

= 0.98

φ LT − λ LT

Check_6 := if ⎛⎜ χ LT ≤ 1 ∧ χ LT ≤

⎜ ⎝

1 2

, "OK" , "NOT OK" ⎞⎟

⎟ ⎠

λ LT

Check_6 = "OK"

Parameter λ LTO (EN1993-1-1,cl.6.3.2.3(1)) Design buckling resistance moment (EN1993-1-1,cl.6.3.2.1(3))

λ LTO := 0.4 Mb.Rd := χ LT⋅ W pl.y ⋅ γ

⎛ MEd.y

Check_7 := if ⎜

⎝ Mb.Rd

fy

= 299.741kN ⋅ ⋅m

M1

⎞

≤ 1, "OK" , "NOT OK" ⎟

⎠

Page 115

Check if the lateral torsional buckling Check_7 = "OK" be ignored (EN1993-1-1,cl.6.3.2.2(4)) MEd.y ⎛ ⎞ 2 Check_8 := if ⎜ λ LT < λ LTO ∧ < λ LTO , "Ignored torsional buckling effects" , "Consider torsional buckling effects" ⎟ Mcr ⎝ ⎠ Check_8 = "Ignored torsional buckling effects"

Combine bending and axial compression cl.6.3.3 Moments due to the shift of the centroidal axis for class sections 1,2 & 3 (EN1993-1-1,cl.6.3.3(4)/table 6.7)

ΔM Ed.z := 0 ΔM Ed.y := 0

Characteristic resistance to normal force of the critical cross-section (EN1993-1-1,cl.6.3.3(4)/table 6.7)

3

NRk := fy ⋅ A = 2.676 × 10 ⋅ kN

Characteristic moment resistance of the critical cross-section (EN1993-1-1,cl.6.3.3(4)/table 6.7)

My.Rk := Mc.Rd.y = 305.8kN ⋅ ⋅m Mz.Rk := Mc.Rd.z = 142.45kN ⋅ ⋅m

ψy :=

MEd.y1 MEd.y2 MEd.y2 MEd.y1

Ratio of end moments (EN1993-1-1,Table B2) ψz :=

M Ed.z1 M Ed.z2 M Ed.z2 M Ed.z1

if −1 ≤

if −1 ≤

if −1 ≤

if −1 ≤

MEd.y1 MEd.y2 MEd.y2 MEd.y1 M Ed.z1 M Ed.z2 M Ed.z2 M Ed.z1

≤1

≤1

≤1

≤1

Equivalent uniform moment factor

Cmy := 0.6 + 0.4⋅ ψy = 0.438

Equivalent uniform moment factor

Cmz := 0.6 + 0.4⋅ ψz = 0.928

Page 116

⎡⎡

⎡

⎢⎢ ⎢⎢ ⎣⎣

⎢ ⎢ ⎣

(

)

kyy := min⎢⎢Cmy⋅ ⎢1 + λ y − 0.2 ⋅

NEd

N ⎤⎤ ⎛ ⎞⎤ ⎥⎥ , Cmy⋅ ⎜ 1 + 0.8⋅ Ed ⎟⎥ = 0.441 NRk ⎥⎥ NRk ⎟⎥ ⎜ χ y⋅ χ y⋅ ⎥ ⎥ ⎜ γ M1 γ M1 ⎟⎥ ⎦⎦ ⎝ ⎠⎦

Interaction factors NEd ⎤⎤ N ⎡⎡ ⎡ ⎛ ⎞⎤ (EN1993-1-1,table ⎥⎥ , Cmz⋅ ⎜ 1 + 1.4⋅ Ed ⎟⎥ = 0.96 kzz := min⎢⎢Cmz⋅ ⎢1 + ( 2λ z − 0.6) ⋅ NRk ⎥⎥ NRk ⎟⎥ B.1&B.2) ⎢⎢ ⎢ ⎜ χ ⋅ χ ⋅ z z ⎢⎢ ⎢ ⎜ γ M1 ⎥⎥ γ M1 ⎟⎥ ⎣⎣ ⎣ ⎦⎦ ⎝ ⎠⎦ kyz := 0.6kzz = 0.576 kzy := 0.6kyy = 0.265

EN1993-1-1,Equation 6.61

⎛ NEd

Check_9 := if ⎜

⎜ ⎜ ⎝

χ y ⋅ NRk

+ kyy ⋅

MEd.y + ΔM Ed.y χ LT⋅

γ M1

NEd

Unity factor

+ kyy ⋅

χ y ⋅ NRk

My.Rk

+ kyz⋅

M z.Rk

γ M1

χ LT⋅

⎞

≤ 1.0, "OK" , "NOT OK" ⎟

⎟ ⎟ ⎠

γ M1

MEd.y + ΔM Ed.y

γ M1

MEd.z + ΔM Ed.z

My.Rk

+ kyz⋅

γ M1

MEd.z + ΔM Ed.z M z.Rk

= 0.135

γ M1

Check_9 = "OK"

EN1993-1-1,Equation 6.62

⎛ NEd

Check_10 := if ⎜

⎜ ⎜ ⎝

χ z ⋅ NRk

+ kzy⋅

χ LT⋅

γ M1

NEd

Unity factor

MEd.y + ΔM Ed.y

χ z ⋅ NRk γ M1

+ kzy⋅

My.Rk

+ kzz⋅

MEd.z + ΔM Ed.z

γ M1

MEd.y + ΔM Ed.y χ LT⋅

My.Rk γ M1

M z.Rk

⎞

≤ 1.0, "OK" , "NOT OK" ⎟

⎟ ⎟ ⎠

γ M1

+ kzz⋅

MEd.z + ΔM Ed.z M z.Rk

= 0.142

γ M1

Check_10 = "OK"

Page 117

9.4 Design of steel column (Seismic design situationn)

Column name: C2 Storey level: Storey 1

Page 118

Step 1: Option > Preferences > Steel frame design

Modify the existing “System Omega”. The omega factor is equal to the minimum section overstrength factor of concentric bracing. See below:

Note: the minimum value of Ω is calculate over all the diagonals of the braced frame system

Step 2: Design > Steel frame design > Select design combo…

Figure 9.7: Lateral/gravity load combination at ULS

Page 119

Figure 9.8: Gravity load combination at SLS

Ultimate limit state (ULS)

Static load combination STATIC 1. STATIC 2. STATIC 3. STATIC 4. STATIC 5. STATIC 6. STATIC 7. STATIC 8.

1.35DL + 1.5LL 1.35DL + 1.5LL + 0.75WINDX 1.35DL + 1.5LL - 0.75WINDX 1.35DL + 1.5LL + 0.75WINDY 1.35DL + 1.5LL - 0.75WINDY 1.35DL + 1.5WINDX + 1.05LL 1.35DL - 1.5WINDX – 1.05LL 1.35DL + 1.5WINDY + 1.05LL Page 120

STATIC 9. 1.35DL - 1.5WINDY – 1.05LL STATIC 10. DL + 0.3LL Seismic load combination for “Modal Analysis” SEISMIC 1. SEISMIC 2. SEISMIC 3. SEISMIC 4. SEISMIC 5. SEISMIC 6. SEISMIC 7. SEISMIC 8.

DL + 0.3LL + EQX + 0.3EQY DL + 0.3LL + EQX – 0.3EQY DL + 0.3LL - EQX + 0.3EQY DL + 0.3LL - EQX – 0.3EQY DL + 0.3LL + EQY + 0.3EQX DL + 0.3LL + EQY – 0.3EQX DL + 0.3LL - EQY + 0.3EQX DL + 0.3LL - EQY – 0.3EQX

Serviceability limit state (SLS)

DSTLD 1. DL + LL DSTLD 2. LL ETABS: Display > Show Tables

Select all combinations

Table 9.3a: Analysis results of gravity load combination (STATIC 10: G + 0.3Q) Story  

Column  

Load  

Loc  

P  

V2  

V3  

T  

M2  

M3  

STORY1   STORY1  

C2   C2  

STATIC10   STATIC10  

0   1.38  

-­‐245.17   -­‐244.13  

-­‐0.28   -­‐0.28  

-­‐0.27   -­‐0.27  

0   0  

-­‐0.43   -­‐0.055  

0.001   0.389  

Page 121

STORY1   C2   Note:  P  =  NEd.G    

STATIC10  

2.76  

-­‐243.1  

-­‐0.28  

-­‐0.27  

0  

0.321  

0.776  

Table 9.3b: Analysis results of seismic action (MODAL EQX / EQY) Story  

Column  

STORY1   C2   STORY1   C2   STORY1   C2   STORY1   C2   STORY1   C2   STORY1   C2   STORY1   C2   STORY1   C2   STORY1   C2   STORY1   C2   STORY1   C2   STORY1   C2   Note:  P  =  NEd.E  

Load  

Loc  

P  

V2  

V3  

T  

M2  

M3  

EQX   EQX   EQX   EQX   EQX   EQX   EQY   EQY   EQY   EQY   EQY   EQY  

0   1.38   2.76   0   1.38   2.76   0   1.38   2.76   0   1.38   2.76  

38.99   38.99   38.99   33.61   33.61   33.61   3.55   3.55   3.55   2.6   2.6   2.6  

29.66   29.66   29.66   26.3   26.3   26.3   2.72   2.72   2.72   1.89   1.89   1.89  

0.49   0.49   0.49   1.15   1.15   1.15   8.97   8.97   8.97   10.93   10.93   10.93  

-­‐0.001   -­‐0.001   -­‐0.001   0.001   0.001   0.001   0.003   0.003   0.003   0.002   0.002   0.002  

0.884   0.202   -­‐0.48   1.917   0.332   1.256   14.692   2.313   10.076   17.899   2.813   -­‐12.273  

58.02   17.094   -­‐23.833   51.189   14.928   21.431   5.227   1.468   2.297   3.709   1.097   -­‐1.516  

Results of Seismic load combination (SEISMIC 1-8)

Select all the seismic load combinations Sort out the highest values of P, V and M

Page 122

Table 9.4: Analysis result of design values of V and M based on worst case seismic design combination Story  

Column  

Load  

Loc  

P  

V2  

V3  

T  

M2  

M3  

STORY1   STORY1   STORY1  

C2   C2   C2  

SEISMIC1  MIN   SEISMIC1  MIN   SEISMIC1  MIN  

0   1.38   2.76  

-­‐279.84   -­‐278.8   -­‐277.77  

-­‐27.4   -­‐27.4   -­‐27.4  

-­‐4.11   -­‐4.11   -­‐4.11  

-­‐0.002   -­‐0.002   -­‐0.002  

-­‐6.755   -­‐1.081   -­‐3.958  

-­‐52.756   -­‐14.979   -­‐21.344  

Table 9.5: Summarize of design values required to carry out the design of steel member Results

Design value

Symbol

Design axial force for gravity load combination (G+0.3Q)

NEd.G

245

Design axial force for the design seismic action alone

NEd.E

39

Design moment at y-y at end 1 (seismic load combination)

MEd.SC.y1

52.8

Design moment at y-y at end 2 (seismic load combination)

MEd.SC.y2

21.3

Design moment at z-z at end 1 (seismic load combination)

MEd.SC.z1

6.8

Design moment at z-z at end 2 (seismic load combination)

MEd.SC.z2

4.0

Shear forces at y-y at end (seismic load combination)

VEd.SC.y

27.4

Shear force at z-z at end 1 (seismic load combination)

VEd.SC.z

4.1

(kN/kNm)

Page 123

9.4.1 Design of steel column (Seismic design situation – Hand calculation) Detail design of steel column using Eurocode 3 and Eurocode 8 1. Rolled I - section 2. Limit to class 1 and 2 section 3. Column not susceptible to torsional deformations Design data Length of column

hc := 3m

Overstrength factor (EN1998-1-1,cl.6.1.3(2))

γ ov := 1.25

Omega factor of bracing members at storey 1

Ω := 2.5

Behavior factor q

q := 4

Ductlity class

Ductility_class := "DCM"

Total axial force due to the non-seismic actionsNEd.G := 245.17kN (G+ψ EiQ) Total axial force due to the non-seismic actionsNEd.E := 39kN (seismic) Design shear force due to Eurocode requirement (EN1998-1-1,cl.6.7.4(1)) Design shear force due to Eurocode requirement (EN1998-1-1,cl.6.7.4(1)) Design moment y-y axis (seismic combination) Design moment y-y axis (seismic combination) Design moment y-y axis (seismic combination) Design moment y-y axis (seismic combination) Maximum moment

MEd.y := max MEd.y1, MEd.y2 = 52.76kN ⋅ ⋅m

Maximum moment

MEd.z := max MEd.z1, MEd.z2 = 6.75⋅ kN⋅ m

VEd.y := 4.11kN

VEd.z := 27.4kN MEd.y1 := 52.76kN⋅ m MEd.y2 := 21.34kN⋅ m MEd.z1 := 6.75kN⋅ m MEd.z2 := 3.96kN⋅ m

(

(

)

)

Page 124

Design shear force due to Eurocode requirement (EN1998-1-1,cl.6.7.4(1)) Section properties:

NEd := NEd.G + 1.1⋅ γ ov⋅ Ω⋅ NEd.E = 379.233kN ⋅

Depth of section,h:

h := 270mm

Width of section,b:

b := 280mm

Thickness of web, tw:

tw := 8mm

Thickness of flange, tf :

tf := 13mm

Thickness of element

t := max tw, tf = 13⋅ mm

Second moment of area z-z:

Izz := 47630000mm

Second moment of area y-y:

Iyy := 1.367⋅ 10 mm

Cross section area, A:

A := 9730mm

Radius of section,r:

r := 24mm

Heigth of web, hw

hw := h − 2tf − 2r = 196⋅ mm

Area of the web

A w := h w⋅ tw = 1.568 × 10 ⋅ mm

Warping Constant, Iw:

Iw := 753.7⋅ 10 ⋅ mm

Torsional Constant, IT:

It := 635000mm

Plastic Modulus, Wply

W pl.y := 1112000mm

Plastic Modulus, Wplz

W pl.z := 518000mm

Elastic modulus, E:

Es := 210kN⋅ mm

Yield strength of steel , fy:

fy := 275N ⋅ mm

Ultimate strength, fu:

fu := 430N ⋅ mm

Shear modulus

G := 81kN⋅ mm

(

)

4

8

4

2

3

9

2

6

4 3

3

−2

−2 −2

−2

Page 125

fy :=

fy if t ≤ 16mm −2

fy − 10N ⋅ mm

−2

fy − 20N ⋅ mm

Reduction of yield and ultimate strenght of sections EN10025-2 fu :=

−2

if 16mm < t ≤ 40mm

fy = 275⋅ N ⋅ mm

if 40mm < t ≤ 80mm

fu if t ≤ 16mm −2

fu − 10N ⋅ mm

−2

fu − 20N ⋅ mm

−2

if 16mm < t ≤ 40mm

fu = 430⋅ N ⋅ mm

if 40mm < t ≤ 80mm

Partial safety factor Resistance of cross-sections whatever the class (CYS EN1993-1-1,cl 6.1(1)) Resistance of members to instability (CYS EN1993-1-1,cl 6.1(1)) Resistance of cross-section in tension (CYS EN1993-1-1,cl.6.1(1))

γ M0 := 1 γ M1 := 1 γ M2 := 1.25

Section classification ε :=

For section classification the coefficient ε is:

"CLASS 1" if

fy

= 0.924

−2

N⋅ mm

For a flange element:

Class_type flange :=

235

cf tf

cf := ≤ 9⋅ ε

"CLASS 2" if 9⋅ ε <

cf

"CLASS 3" if 10⋅ ε <

tf cf tf

(b − tw − 2.r ) 2

= 112⋅ mm

≤ 10⋅ ε

Class_type flange = "CLASS 2" ≤ 14⋅ ε

Page 126

For a web element:

Class_type web :=

cw := h − 2⋅ tf − 2⋅ r = 196⋅ mm

"CLASS 1" if

cw

≤ 72⋅ ε

tw

"CLASS 2" if 84⋅ ε <

cw

"CLASS 3" if 105⋅ ε <

(

Class_type := if Class_type flange

tw cw tw

Class_type web = "CLASS 1" ≤ 83⋅ ε

≤ 124⋅ ε

Class_type web , Class_type flange , "ADD MANUALY"

)

Class_type = "ADD MANUALY"

Requirements on cross-sectional class of dissipative elements depending on Ductility class (EN1998-1-1,cl.6.5.3(2)) Section classification rule for EC8 (EN1998-1-1,cl.6.5.3(2)) Class_type_req :=

"CLASS 1 , 2 or 3" if 1.5 < q ≤ 2 ∧ Ductility_class "CLASS 1 or 2" if 2 < q ≤ 4 ∧ Ductility_class "CLASS 1" if q > 4 ∧ Ductility_class

"DCM"

"DCM"

"DCH"

Class_type_req = "CLASS 1 or 2"

Note: The column now has to be check using the resistance verification checks of Eurocode 3 as shown in section 9.3 of this document.

Page 127

9.5 Design of composite beams - Hand calculations ETABS: Define > Wall/Slab/Deck sections Figure 9.9: Define deck section Comflor60 -Corus

Figure 9.10: Modified “Stiffness Modifiers” (crack-sections)

Page 128

ETABS: Analyze > Run analysis ETABS: Display > Show Tables >

Select all combinations

Page 129

Assumptions - Design and analysis This design process is envisaging a analyzed to determine the forces and moments in the individual structural members. Simple design approach: This method applies to structures in which the connections between members will not develop any significant restraint moments. Members forces and moments are calculated on the basic of the following assumptions: 1. Simply supported beam. 2. The steel sheeting with ribs is placed transverse to the beam. 3. Limited only to I abd H rolled sections with equal flanges 4. Ignored any contribution of steel sheeting to the transverse reinforcements Length of beam

Le := 5m

Spacing of the secondary beams (LHS)

L1 := 5m

Spacing of the secondary beams (RHS)

L2 := 5m

Loading length

L :=

L1 2

+

L2 2

= 5m

Slab design data Comfloor 60

Overall depth of slab

h := 150mm

Steel sheeting deck profile (Comflor 60)

hp := 60mm

Depth of concrete slab above steel sheeting

hc := h − hp = 90⋅ mm

Rib width at top

b1 := 131mm

Rib width at bottom

b2 := 180mm

Page 130

b1 + b2

Distance between shear connector (Assume single shear connector) Space of each troughs

e := 300mm

Thickness of steel sheeting

ts := 1mm

bo :=

= 155.5⋅ mm

2

Structural steel properties Depth of section, h:

ha := 240mm

Width of section,b:

b := 120mm

Thickness of web, tw:

tw := 6.2mm

Thickness of flange, tf :

tf := 9.8mm

Thickness of element

t := max tw, tf = 9.8⋅ mm

Radius of section,r:

r := 15mm

Heigth of web, hw

hw := ha − 2tf − 2r = 190.4mm ⋅

Area of the web

A w := h w⋅ tw = 1.18 × 10 ⋅ mm

Radious of gyration

iz := 26.9507mm

Second moment of area z-z:

Izz := 2840000mm

Second moment of area y-y:

Iyy := 38920000mm

Cross section area, A:

A := 3910mm

Torsional Constant, IT:

It := 130000mm

Warping Constant, Iw:

Iw := 753.7⋅ 10 ⋅ mm

Plastic Modulus, Wply

W pl.y := 367000mm

Plastic Modulus, Wplz

W pl.z := 73900mm

Yield strength

fy := 275N ⋅ mm

Ultimate strength

fu := 430N ⋅ mm

Modulus of Elasticity

Es := 210kN⋅ mm

Shear modulus

G := 81kN⋅ mm

(

)

3

2

4 4

2 4

9

6 3

3

−2 −2 −2

−2

Page 131

Concrete properties −2

Yield strength of reinforcement

fyk := 500N ⋅ mm

Cylinder strength

fck := 25N ⋅ mm

Modulus of Elasticity

Ecm := 31kN⋅ mm

−2 −2

Shear connector properties Diameter

d := 19mm

Overall height before welding

hsc := 95mm

Ultimate strength of shear connector

fus := 450N ⋅ mm

Number of stud per in one rib

n r := 1

−2

Material partial factors for resistance Resistance of cross-sections whatever the class (CYS EN1993-1-1,cl 6.1(1)) Resistance of members to instability (CYS EN1993-1-1,cl 6.1(1))

γ M0 := 1.0 γ M1 := 1.0

Partial factor for concrete (EN 1992 1-1 Table 2.1N)

γ c := 1.5

Partial factor for reinforcing steel (EN 1992 1-1 Table 2.1N)

γ s := 1.15

Partial factor for design shear resistance of a headed stud (CYS EN1994-1-1,cl.2.4.1.2(5)P) Partial factor for design shear resistance of a composite slab (CYS EN1994-1-1,cl.2.4.1.2(6)P)

γ v := 1.25

γ vs := 1.25

Partial factor for permanent action

γ G := 1.35

Partial factor for variable action

γ Q := 1.5

Design value of the cylinder compressive strength of concrete (EN1992-1-1,cl.

fck −2 fcd := = 16.667⋅ N ⋅ mm γc

Page 132

fyk −2 fyd := = 434.783N ⋅ ⋅ mm γs

Design value of the yield strength of structural steel Loading at construction stage Dead load

−2

Weight of steel deck (Comfloor 60)

g k.deck := 0.114kN⋅ m

Weight of wet concrete

gk.c.wet := 2.79kN⋅ m

Weight of steel beam (IPE240)

g k.b := 0.8kN⋅ m

−2

−1

Live load −2

Construction live load

q k := 0.75kN⋅ m

Total load at construction stage

(

)

−1

FEd := γ G⋅ gk.deck ⋅ L + gk.c.wet⋅ L + g k.b + γ Q⋅ q k⋅ L = 26.307⋅ kN⋅ m

2

Moment at construction stage

MEd.c :=

Shear force at construction stage

VEd.c :=

FEd⋅ L 8

FEd⋅ L 2

= 82.209⋅ kN⋅ m

= 65.767kN ⋅

Design moments and shear forces Shear force at composite stage

VEd.c = 65.767kN ⋅

Design moment at composite stage

MEd.c = 82.209kN ⋅ ⋅m

Shear force at composite stage

VEd := 55.5kN

Design moment at composite stage

MEd := 132kN⋅ m

Page 133

Ultimate limit state verification Construction stage Section classification (EN19931-1,cl.5.6(6)) fy :=

fy if t ≤ 16mm −2

fy − 10N ⋅ mm

−2

fy − 20N ⋅ mm

Reduction of yield and ultimate strength of sections EN10025-2 fu :=

−2

if 16mm < t ≤ 40mm

fy = 275⋅ N ⋅ mm

if 40mm < t ≤ 80mm

fu if t ≤ 16mm −2

fu − 10N ⋅ mm

−2

fu − 20N ⋅ mm

For section classification the coefficient ε is:

−2

if 16mm < t ≤ 40mm

fu = 430⋅ N ⋅ mm

if 40mm < t ≤ 80mm

ε :=

235 fy

= 0.924

−2

N⋅ mm

For a flange element:

Class_type flange :=

"CLASS 1" if

cf := cf tf

(b − tw − 2.r ) 2

= 41.9⋅ mm

≤ 9⋅ ε

"CLASS 2" if 9⋅ ε <

cf

"CLASS 3" if 10⋅ ε <

tf cf tf

≤ 10⋅ ε

Class_type flange = "CLASS 1"

≤ 14⋅ ε

Page 134

For a web element:

Class_type web :=

"CLASS 1" if

cw := ha − 2⋅ tf − 2⋅ r = 190.4mm ⋅ cw tw

≤ 72⋅ ε

"CLASS 2" if 84⋅ ε <

cw

"CLASS 3" if 105⋅ ε <

(

Class_type := if Class_type flange

tw cw tw

Class_type web = "CLASS 1" ≤ 83⋅ ε

≤ 124⋅ ε

Class_type web , Class_type flange , "ADD MANUALY"

)

Class_type = "CLASS 1"

Bending Resistance of the steel section (EN1993-1-1,cl.6.2.5) Design resistance for bending (EN1993-1-1,cl.6.2.5(2)) Bending resistance check checks (EN1993-1-1,cl.6.2.5(1))

Ma.pl.Rd :=

W pl.y ⋅ fy γ M0

= 100.925⋅ kN⋅ m

(

Check_1 := if MEd.c ≤ Ma.pl.Rd , "OK" , "NOT OK"

)

Check_1 = "OK"

Vertical Shear resistance of the steel section (cl.6.2.2) & (EN1993-1-1,cl.6.2.6) Factor for shear area (EN1993-1-1,cl.6.2.6(g)) Shear area of steel section (EN1993-1-1,cl.6.2.6(3)) A v :=

A v1 if A v1 > η ⋅ tw⋅ h w η ⋅ tw⋅ h w if A v1 < η ⋅ tw⋅ h w

Shear resistance of steel section Y-Y (EN1993-1-1,cl.6.2.6(2))

η := 1

(

)

Av1 := A − 2⋅ b ⋅ tf + tw + 2r ⋅ tf 3

2

A v = 1.913 × 10 ⋅ mm

Vpl.Rd := A v ⋅

fy ⋅ ( 3)

−1

γ M0

= 303.691kN ⋅

Page 135

Design of shear resistance check (EN1993-1-1,cl.6.2.6(1)P)

(

Check_2 := if VEd ≤ Vpl.Rd , "OK" , "NOT OK"

)

Check_2 = "OK"

Check if the verification of shear buckling resistance required or not (EN1993-1-1,cl.6.2.6(6)) ⎛ hw ⎞ ε Check_3 := if ⎜ < 72⋅ , "Not required shear buckling resistance" , "Required shear buckling resistance" ⎟ η ⎝ tw ⎠ Check_3 = "Not required shear buckling resistance"

Bending and shear interaction check (cl.6.2.2.4) Strong axis Y-Y VEd

Interaction check 1

vy :=

Reduced yield strength

⎛ 2VEd ⎞ ρ := ⎜ − 1⎟ = 0.403 ⎝ Vpl.Rd ⎠

Vpl.Rd

= 0.183 2

Reduced design plastic resistance moment (EN1993-1-1,cl.6.2.8(5))

2 ⎛ ρ⋅ A w ⎞⎟ ⎜ ⎜ W pl.y − 4t ⎟ ⋅ fy w ⎠ ⎝ M a.pl.Rd. := if v y > 0.5 γ

M0

M a.pl.Rd if v y < 0.5

Ma.pl.Rd = 100.925kN ⋅ ⋅m

Lateral torsional buckling resistance of steel beam (EN1993-1-1,cl.6.3.2) Status of effective length

Effective_Length := "Pinned Pinned"

Effective length factor (Guidance of EC3)

k :=

0.7 if Effective_Length

"Fixed Fixed"

0.85 if Effective_Length

"Partial restraint"

0.85 if Effective_Length

" Pinned Fixed"

1 if Effective_Length

=1

"Pinned Pinned"

Page 136

Effective length (pinned)

Lcr := k⋅ Le = 5m

Factor for end warping, kw

kw := 1.0

(SN003a-EN-EU) Coefficient factor C1

C1 := 1.348

(SN003a-EN-EU) Coefficient factor C2

C2 := 0.454

(SN003a-EN-EU) Distance between the point of load application and the shear centre

zg := 0m

Elastic critical moment for lateral torsional buckling (SN003a-EN-EU) 2

Mcr := C1⋅

π ⋅ Es ⋅ Izz

(Lcr)2

2 I

2

( cr) t k w 2 ⋅ ⎛⎜ ⎞⎟ ⋅ + + (C2⋅ zg) − C2⋅ zg = 176.744kN ⋅ ⋅m kw Izz 2 ⎝ ⎠ π Es ⋅ Izz L

G⋅ I

Lateral torsional buckling curve (EN1993-1-1,table 6.4)

Buckling_curve_Z :=

"b" if "c" if

h b h b

≤2 >2

Buckling_curve_Z = "b"

Imperfection factor for lateral torsional (EN1993-1-1,table 6.3)

αLT :=

0.21 if Buckling_curve_Z

"a"

0.34 if Buckling_curve_Z

"b"

0.49 if Buckling_curve_Z

"c"

0.76 if Buckling_curve_Z

"d"

αLT = 0.34

Non dimensional slenderness (EN1993-1-1,cl.6.3.2.2(1)) Parameter introducing the effect of biaxial bending (EN1994-1-1,cl.6.3.2.3(1)) Parameter λ LTO (EN1993-1-1,cl.6.3.2.3(1))

λ LT :=

W pl.y ⋅ fy Mcr

= 0.756

β := 0.75

λ LTO := 0.4

Page 137

Value to determine the reduction factor (EN1993-1-1,cl.6.3.2.2(1))

φ LT := 0.5⋅ ⎡1 + αLT⋅ λ LT − λ LTO + ⎛ β ⋅ λ LT ⎣ ⎝

Reduction factor for lateral-torsional buckling (EN1993-1-1,cl.6.3.2.2(1))

χ LT :=

(

1 2

φ LT +

2⎞⎤

)

2

⎠⎦ = 0.775

= 0.841

φ LT − β λ LT

1

Check_5 := if ⎛⎜ χ LT ≤ 1 ∧ χ LT ≤

2

⎜ ⎝

, "OK" , "NOT OK" ⎞⎟

⎟ ⎠

λ LT

Check_5 = "OK"

Design plastic resistance (EN1993-1-1,cl.6.3.2.1)

Mb.Rd := χ LT⋅

W pl.y⋅ fy γ M1

= 84.882⋅ kN⋅ m

Section verification for lateral torsional ⎛ MEd.c ⎞ Check_6 := if < 1, "OK" , "NOT OK" ⎟ ⎜ M buckling ⎝ b.Rd ⎠ (EN1993-1-1,cl.6.3.2.1(1)) Check_6 = "OK"

Composite stage Effective width of composite beam (cl.5.4.1.2(5)) Total effective width at mid-span (EN1994-1-1cl. 5.4.1.2(5))

⎛ ⎛ L1 L2 Le ⎞ ⎞ beff := bo + 2⎜ min⎜ + , ⎟ ⎟ 2 8 ⎠ ⎠ ⎝ ⎝ 2

Plastic resistance moment of composite section with full shear connection (cl.6.2) Tensile resistance of steel section (EN1993-1-1,cl.6.2.3(2))

fy ⋅ A 3 Npl.a := = 1.075 × 10 ⋅ kN γ M0

Compression resistance of concrete slab (EN1994-1-1,cl.6.2.1.2(1d)

Nc.f := 0.85⋅ fcd ⋅ b eff ⋅ h c = 1.792 × 10 ⋅ kN

Tensile resistance in web of steel section

Npl.w := fy ⋅ tw⋅ ha − 2⋅ tf

3

(

)

Page 138

Location of neutral axis (EN1994-1-1,cl.6.2.1.2(1)) Location_neutral axis :=

"Lies in the concrete slab"

if Nc.f > Npl.a

"Lies in the top flange of the beam"

if Nc.f ≤ Npl.a

"Lies in the web of the beam" if Nc.f < Npl.w

Location_neutral axis = "Lies in the concrete slab"

Bending resistance with full shear connection (EN1994-1-1,cl.6.1.2) M pl.Rd :=

Npl.a h c ⎞ ⎛ h a Npl.a ⋅ ⎜ +h − ⋅ ⎟ if Location_neutral axis Nc.f 2 ⎝ 2 ⎠

"Lies in the concrete slab"

ha ⎛ h c ⎞ Npl.a ⋅ + Nc.f ⋅ ⎜ + h p ⎟ if Location_neutral axis 2 ⎝ 2 ⎠

"Lies in the top flange of the beam"

2 ⎛ h c + h a + 2h p ⎞ Nc.f h a M a.pl.Rd + Nc.f ⋅ ⎜ ⎟ − ⋅ if Location_neutral axis "Lies in the top flange of the beam" 2 ⎝ ⎠ Npl.w 4

Mpl.Rd = 261.285kN ⋅ ⋅m

Bending resistance check checks (EN1993-1-1,cl.6.2.5(1))

(

Check_7 := if MEd ≤ Mpl.Rd , "OK" , "NOT OK"

)

Check_7 = "OK"

Vertical Sheat resistance of the composite steel section (cl.6.2.2) & (EN1993-1-1,cl.6.2.6) Shear resistance of steel section Y-Y (EN1993-1-1,cl.6.2.6(2)) Design of shear resistance check (EN1993-1-1,cl.6.2.6(1)P)

Vpl.Rd = 303.691kN ⋅

(

Check_8 := if VEd ≤ Vpl.Rd , "OK" , "NOT OK"

)

Check_8 = "OK"

Page 139

Check if the verification of shear buckling resistance required or not (EN1993-1-1,cl.6.2.6(6)) ⎛ hw ⎞ ε Check_9 := if ⎜ < 72⋅ , "Not required shear buckling resistance" , "Required shear buckling resistance" ⎟ η ⎝ tw ⎠ Check_9 = "Not required shear buckling resistance"

Design resistance of shear stud connector (cl.6.6.3.1(1)) For sheeting with ribs transverse to the beam For sheeting parallel to the beam see Equation 6.22 of EC4 Upper limit of reduction factor kt kt.max := 0.85 if n r 1 ∧ 1mm ≥ ts ∧ d < 20mm (EN1994-1-1,Table:6.2) 1.0 if n r 1 ∧ 1mm < ts ∧ d < 20mm 0.75 if n r

1 ∧ 1mm ≥ ts ∧ 19mm ≤ d < 22mm

0.75 if n r

1 ∧ 1mm < ts ∧ 19mm ≤ d < 22mm

0.70 if n r

2 ∧ 1mm ≥ ts ∧ d < 20mm

0.80 if n r

2 ∧ 1mm < ts ∧ d < 20mm

0.60 if n r

2 ∧ 1mm ≥ ts ∧ 19mm ≤ d < 22mm

0.60 if n r

2 ∧ 1mm < ts ∧ 19mm ≤ d < 22mm

kt.max = 0.75

(EN1994-1-1,cl.6.6.4.2)

bo ⎛ hsc ⎞ kt := 0.6⋅ ⋅ ⎜ − 1⎟ hp hp ⎝ ⎠

Limitation of kt

kt :=

Reduction factor kt

(EN1994-1-1,cl.6.6.4.2(2)) Minimum height of shear stud (EN1994-1-1,cl.6.6.1.2(1))

kt if kt < kt.max

= 0.75

kt.max otherwise

(

hmin := if hsc ≥ 4d , "Ductile" , "Not Ductile"

)

hmin = "Ductile"

Page 140

Limitation of stud diameter (EN1994-1-1,cl.6.6.1.2(1))

dlim:= if ( 16mm < d < 25mm, "Ductile" , "Not ductile" ) dlim = "Ductile"

Factor α (EN1994-1-1,cl.6.6.3.1(1))

hsc ⎛ hsc ⎞ + 1⎟ if 3 ≤ ≤4 =1 d ⎝ d ⎠

0.2⋅ ⎜

α :=

h sc

1 if

d

>4

Design shear resistance of a headed stud (EN1994-1-1,cl.6.6.3.1(1))

2 ⎛ ⎞ ⎜ 0.8⋅ f ⋅ π ⋅ d ⎟ 2 us ⎜ 4 0.29⋅ α⋅ d ⋅ fck ⋅ Ecm ⎟ PRd := kt⋅ min , ⎜ ⎟ = 55.298⋅ kN γv γv ⎝ ⎠

Degree of shear connection (cl.6.6.1.2(1)) Nc.f

= 1.667 Ratio of the degree shear connection η := Npl.a (EN1994-1-1,cl.6.2.1.3(3))

Minimum degree of shear connection ηmin := 1 − ⎛⎜ ⎜ for equal flange ⎜ (EN1994-1-1,cl.6.6.1.2(1)) ⎝

355 fy −2

N⋅ mm

⎞⎟ ⋅ ⎛⎜ 0.75 − 0.03⋅ Le ⎞⎟ if L < 25m e m ⎠ ⎟ ⎝ ⎟ ⎠

1.0 if Le > 25m

ηmin = 0.225

Check the degree of shear interaction within the limits Check_11 := if (η > ηmin ∧ η ≥ 0.4, "OK" , "NOT OK" ) (EN1994-1-1,cl.6.6.1.2(1)) Check_11 = "OK" 2⋅ Npl.a

Number of shear connector required

n :=

Numper of stud provided

Nstud := 40

Stud spacing

s prov :=

PRd

Le Nstud

= 38.889

= 0.125m

Page 141

Check the minimum spacing of studs (EN1994-1-1,cl.6.6.5.7(4))

s lim := if s prov ≥ 5⋅ d ∧ s prov < 6⋅ h , "OK" , "NOT OK"

Adequacy of the shear connection (EN1994-1-1,cl.6.6.1.3(3))

Check_12 := if Mpl.Rd < 2.5⋅ Ma.pl.Rd , "Uniform spacing" , "Not uniform spacing"

(

)

s lim = "OK"

(

Check_12 = "Not uniform spacing"

Design of transverse reinforcement (cl.6.6.6.2) & (EN1992-1-1,cl.6.2.4) Le

Length under consideration

Δ x :=

Longitudinal shear stress (EN1992-1-1,cl.6.2.4(3))

v Ed :=

Strength reduction factor (EN1992-1-1,Eq.6.6N)

v := 0.6⋅ ⎜ 1 −

Angle between the diagonal strut (EN1992-1-1,cl.6.2.4(4))

θf := 45deg

Assume spacing of the bars

s f := 200mm

Area of transverse reinforcement required (EN1992-1-1,cl.6.2.4(4))

A s.req :=

= 2.5m

2

Npl.a 2⋅ h c⋅ Δ x

⎛ ⎜ ⎝

fck

⎞⎟ − 2 ⎟ 250⋅ N ⋅ mm ⎠

v Ed⋅ h c⋅ s f

( ) ( )

sin θf fyd ⋅ cos θf 2

Area of transverse reinforcement provided As.prov := 193mm

(

Check_13 := if As.req ≤ As.prov , "OK" , "NOT OK"

)

Check_13 = "OK"

Check the crushing compression in the flange Check_14 := if (vEd ≤ v⋅ fcd ⋅ sin(θf )⋅ cos (cos (θf )) , "OK" , "NOT OK" ) (EN1992-1-1cl.6.2.4(4)) Check_14 = "OK"

Page 142

Serviceability limit state verification Construction stage −1

Dead load at composite stage

Gk := 10.88kN⋅ m

Live load at composite stage

Qk := 5.0kN⋅ m

−1

(

)

5⋅ Gk + Qk ⋅ Le

Maximum deflection at construction stage

δcon :=

Vertical deflection limit (CYS NA EN1993-1-1,table NA.1)

Check_15 := if ⎜ δcon <

4

= 15.812⋅ mm

384⋅ Es ⋅ Iyy

⎛

Le

⎝

250

⎞

, "OK" , "NOT OK" ⎟

⎠

Check_15 = "OK"

Short term elastic modular ration (EN1994-1-1,cl.7.2.1)

n o :=

r :=

Second moment of area of the composite section, Ic

Deflection with full shear connection Vertical deflection limit (CYS NA EN1993-1-1,table NA.1)

Iy :=

Es Ecm

A beff ⋅ hc

(

A ⋅ h + 2⋅ h p + h c

δcom :=

)2

+

(

)

(

) ( )4

4⋅ 1 + n o ⋅ r

5⋅ Gk + Qk ⋅ Le 384⋅ Es ⋅ Iy

⎛

Le

⎝

200

Check_16 := if ⎜ δcom <

b eff ⋅ h c 12⋅ n o

3 −4 4

+ Iyy = 1.563 × 10

= 3.938⋅ mm

⎞

, "OK" , "NOT OK" ⎟

⎠

Check_16 = "OK"

Page 143

m

Vibration (Simplified analysis): Loading: −1

Permanent load

Gk = 10.88⋅ kN⋅ m

Imposed load

Qk = 5⋅ kN⋅ m

For category B building

ψ1 := 0.5

Total weigth floor, Fv

Fv := Gk + ψ1⋅ Qk

−1

Increase the inertia, Ic by 10% to allow for the increased dynamic stiffness of the composite beam, Icl

(

)

Icl := Iy + Iy⋅ 0.1

Instantaneous deflection caused by re-application of the self weight of the floor and the beam to the composite beam, δ α δα :=

Natural frequncy, f

Check beam vibration (SCI-P-076)

(

)

5⋅ Fv ⋅ Le ⋅ Le 384⋅ Es ⋅ Icl

3

= 3.016⋅ mm

18 ⎞ ⋅ ⎜ δ ⎟ Hz = 10.364Hz α ⎜ ⎟ ⎝ mm ⎠

f := ⎛

Check_17 := if (f > 4⋅ Hz, "OK" , "NOT OK" ) Check_17 = "OK"

Page 144

9.5 Design of steel bracing

9.5.1 Main configuration of design of steel bracing Basic theory: Tension only, utilises two members at each storey but only the tension element is assumed to resist wind load and seismic load, the compression element is assumed to buckle and offer no resistance to lateral movement. Eurocode 8 requirement: The diagonals shall be taken into account as follows in an elastic analysis of the structure for the seismic action: a) in frames with diagonal bracings, only the tension diagonals shall be taken into account, b) in frames with V bracings, both the tension and compression diagonals shall be taken into account (EN1998-1-1,cl6.7.2(2). Taking into account of both tension and compression diagonals in the analysis of any type of concentric bracing is allowed provided that all of the following conditions are satisfied: a) a non-linear static (pushover) global analysis or non-linear time history analysis is used, b) both pre-buckling and post-buckling situations are taken into account in the modeling of the behavior of diagonals and, c) background information justifying the model used to represent the behavior of diagonals is provided (EN1998-1-1,cl6.7.2(3).

Page 145

Figure 9.11: Method of design bracing in this manual

Ignore compression members

Compression members Tension members Direction of shear

Steps for designing steel bracing member: 1. Delete the compression member. 2. Leave the tension members only. 3. Run the design of steel frame. 4. Find the suitable section and ensure that the section pass all the checks. 5. Ensure that the compression member has been placed at the construction drawings.

Page 146

9.5.2 Simplified design of frames with X bracing (Extract from Design guidance to EC8) In a standard design, the following simplified approach may be used:

•

The analysis of the structure is realized considering that only one diagonal in each X bracing is present, the other diagonal being considered as already buckled and unable to provide strength. This corresponds to an underestimation of both the stiffness and the strength of the structural system at the initial (pre-buckling) stage, but to a safe-side estimate at the post-buckling stage.

•

The beams and columns are capacity designed according to the real yield strength of the diagonals, for bending with increased axial force and bending moment from the analysis for the combination of the design seismic action with gravity loads.

However, this simplified approach could be dangerous for the stability of the structure, if it does not take into account that action effects of compression in columns and beams at the pre-buckling stage are higher than in the post-buckling stage envisaged in the analysis. Indeed, if the buckling loads of the diagonal are closed to their yield load in tension, the initial shear resistance Vinit of the X bracing is underestimated by a model where only one diagonal is considered present. If low-slenderness diagonals are used, Vinit can be close to double the value of Vpl.Rd computed with the hypothesis of one active yielded diagonal. The only way to prevent this unsafe situation is to design slender diagonal having their buckling load at most around 0.5Npl.Rd. This condition is behind the prescribed lower bound limit value of 1.3 for the slenderness λ. The prescribed upper bound limit max λ=2, is justified by the aim to avoid shock effects during the load reversal in diagonals.

Page 147

9.5.3 Model in ETABS Figure 9.12: Amendment model

Assume that the steel bracing resist the lateral force at the +X direction

Assume that the steel bracing resist the lateral force at the -X direction

Page 148

Assume that the steel bracing resist the lateral force at the +Y direction

Assume that the steel bracing resist the lateral force at the -Y direction

Page 149

STEP 2: Design > Steel frame design > Select design combo…

Figure 9.13: Lateral/gravity load combination at ULS

Figure 9.14: Gravity load combination at SLS

Page 150

Ultimate limit state (ULS)

Static load combination STATIC 11. 1.35DL + 1.5LL + 0.75WINDX STATIC 12. 1.35DL + 1.5LL - 0.75WINDX STATIC 13. 1.35DL + 1.5LL + 0.75WINDY STATIC 14. 1.35DL + 1.5LL - 0.75WINDY STATIC 15. 1.35DL + 1.5WINDX + 1.05LL STATIC 16. 1.35DL - 1.5WINDX – 1.05LL STATIC 17. 1.35DL + 1.5WINDY + 1.05LL STATIC 18. 1.35DL - 1.5WINDY – 1.05LL Seismic load combination for “Modal Analysis” SEISMIC 9. DL + 0.3LL + EQX + 0.3EQY SEISMIC 10. DL + 0.3LL + EQX – 0.3EQY SEISMIC 11. DL + 0.3LL - EQX + 0.3EQY SEISMIC 12. DL + 0.3LL - EQX – 0.3EQY SEISMIC 13. DL + 0.3LL + EQY + 0.3EQX SEISMIC 14. DL + 0.3LL + EQY – 0.3EQX SEISMIC 15. DL + 0.3LL - EQY + 0.3EQX SEISMIC 16. DL + 0.3LL - EQY – 0.3EQX Serviceability limit state (SLS)

DSTLD 3. DL + LL

Page 151

Figure 9.15: Design steel bracing member

Write click on member Brace name: D3 Storey level: Storey 1

Page 152

Table 9.6: Design value of brace D3

Story  

Brace  

Load  

Loc  

P  

V2  

V3  

T  

M2  

M3  

STORY1   STORY1   STORY1   STORY1   STORY1   STORY1   STORY1   STORY1  

D3   D3   D3   D3   D3   D3   D3   D3  

SEISMIC1  MIN   SEISMIC2  MIN   SEISMIC3  MIN   SEISMIC4  MIN   SEISMIC1  MIN   SEISMIC2  MIN   SEISMIC3  MIN   SEISMIC4  MIN  

0   0   0   0   2.915   2.915   2.915   2.915  

-­‐361.83   -­‐361.83   -­‐361.83   -­‐361.83   -­‐361.06   -­‐361.06   -­‐361.06   -­‐361.06  

-­‐1.41   -­‐1.41   -­‐1.41   -­‐1.41   -­‐0.13   -­‐0.13   -­‐0.13   -­‐0.13  

-­‐0.05   -­‐0.05   -­‐0.05   -­‐0.05   -­‐0.05   -­‐0.05   -­‐0.05   -­‐0.05  

-­‐0.044   -­‐0.044   -­‐0.044   -­‐0.044   -­‐0.044   -­‐0.044   -­‐0.044   -­‐0.044  

-­‐0.173   -­‐0.173   -­‐0.173   -­‐0.173   -­‐0.054   -­‐0.054   -­‐0.054   -­‐0.054  

-­‐1.792   -­‐1.792   -­‐1.792   -­‐1.792   0.443   0.443   0.443   0.443  

Page 153

Worst case combination

Modify the default steel design data if needed

Page 154

Modify the effective length factor if needed

Modify the omega factors if needed

Table 9.7: Summarize of design values required to carry out the design of steel member Results

Design value

Symbol

Design axial force for the worse case design load combination

NEd

361.83

Design moment at y-y at end 1 (worse case combination)

MEd.y1

-1.792

Design moment at y-y at end 2 (worse case combination)

MEd.y2

0.443

Design moment at z-z at end 1 (worse case combination)

MEd.z1

-0.173

Design moment at z-z at end 2 (worse case combination)

MEd.z2

-0.054

Shear forces at y-y at end (worse case combination)

VEd.y

-0.05

Shear force at z-z at end 1 (worse case combination)

VEd.z

-1.41

(kN/kNm)

Page 155

9.5.4 Design of steel bracing (Gravity/Seismic design situation) – Hand calculation

1. Rolled I - section 2. Limit to class 1 and 2 section Design data Overstrength factor (EN1998-1-1,cl.6.1.3(2))

γ ov := 1.25

Behavior factor q

q := 4

Ductlity class

Ductility_class := "DCM"

Number of storeys

Ns := 3

Length of bracing

hc := 5.831m

Total axial load on column, NEd Shear force y-y axis

NEd := 361.83kN VEd.y := 0.05kN

Shear force z-z axis

VEd.z := 1.41kN

Design moment y-y axis

MEd.y1 := 1.792kN⋅ m

Design moment y-y axis

MEd.y2 := 0.443kN⋅ m

Maximum moment

MEd.y := max MEd.y1, MEd.y2 = 1.792kN ⋅ ⋅m

Design moment z-z axis

MEd.z1 := −0.173kN⋅ m

Design moment z-z axis

MEd.z2 := −0.054kN⋅ m

Maximum moment

MEd.z := max MEd.z1, MEd.z2 = −0.054⋅ kN⋅ m

(

(

)

)

Section properties: Depth of section,d:

d := 120mm

Width of section,b:

b := 120mm

Thickness of web, tw: Thickness of flange, tf : Thickness of element

tw := 16mm tf := 16mm

(

)

t := max tw, tf = 16⋅ mm

Page 156

4

Second moment of area z-z:

Izz := 12280000mm

Second moment of area y-y:

Iyy := 12280000mm

4

2

A := 6656mm

Cross section area, A:

6

Warping Constant, Iw:

Iw := 0⋅ mm

Torsional Constant, IT:

It := 18000000mm

Plastic Modulus, Wply

W pl.y := 261600mm

Plastic Modulus, Wplz

W pl.z := 261600mm

Elastic modulus, E:

Es := 210kN⋅ mm

Yield strength of steel , fy:

fy := 275N ⋅ mm

Ultimate strength, fu:

fu := 430N ⋅ mm

Shear modulus

G := 81kN⋅ mm

4 3

3

−2

−2 −2

−2

fy :=

fy if t ≤ 16mm −2

fy − 10N ⋅ mm

Reduction of yield and ultimate strenght of sections EN10025-2 fu :=

−2

fy − 20N ⋅ mm

−2

fy = 275⋅ N ⋅ mm

if 40mm < t ≤ 80mm

fu if t ≤ 16mm −2

fu − 10N ⋅ mm

Partial safety factor

if 16mm < t ≤ 40mm

−2

fu − 20N ⋅ mm

if 16mm < t ≤ 40mm

−2

fu = 430⋅ N ⋅ mm

if 40mm < t ≤ 80mm

Resistance of cross-sections whatever the class (CYS EN1993-1-1,cl 6.1(1)) Resistance of members to instability (CYS EN1993-1-1,cl 6.1(1)) Resistance of cross-section in tension (CYS EN1993-1-1,cl.6.1(1))

γ M0 := 1 γ M1 := 1 γ M2 := 1.25

Page 157

Section classification ε :=

For section classification the coefficient ε is:

235 fy

= 0.924

−2

N⋅ mm

cf := d − 2tf = 88⋅ mm

Class_type_flange :=

"CLASS 1" if

cf t

≤ 33⋅ ε

"CLASS 2" if 33⋅ ε < "CLASS 3" if 38⋅ ε <

cf t cf t

≤ 38⋅ ε ≤ 42⋅ ε

Class_type_flange = "CLASS 1"

cw := d − 2tw = 88⋅ mm Class_type_web :=

"CLASS 1" if

cw t

≤ 72⋅ ε

"CLASS 2" if 72⋅ ε < "CLASS 3" if 83⋅ ε <

Class_type := if (Class_type_flange

cw t cw t

≤ 83⋅ ε ≤ 124⋅ ε

Class_type_web = "CLASS 1"

Class_type_web , Class_type_flange , "ADD MANUALY" ) Class_type = "CLASS 1"

Requirements on cross-sectional class of dissipative elements depending on Ductility class (EN1998-1-1,cl.6.5.3(2)) Section classification rule for EC8 (EN1998-1-1,cl.6.5.3(2)) Class_type_req :=

"CLASS 1 , 2 or 3" if 1.5 < q ≤ 2 ∧ Ductility_class "CLASS 1 or 2" if 2 < q ≤ 4 ∧ Ductility_class "CLASS 1" if q > 4 ∧ Ductility_class

"DCM" = "CLASS 1 or 2"

"DCM"

"DCH"

Class_type_req = "CLASS 1 or 2"

Page 158

Tension resistance (cl.6.2.2) Design plastic resistance of the cross section (EN1993-1-1,cl.6.2.3(2a))

A ⋅ fy 3 Npl.Rd := = 1.83 × 10 ⋅ kN γ M0

Modified plastic resistance of cross section as described in "Design Guidance to EC8" (cl.6.10.2) Design ultimate resistance (EN1993-1-1,cl.6.2.3(2b))

Npl.Rd := 0.5⋅ Npl.Rd = 915.2kN ⋅

Design tension resistance (EN1993-1-1,cl.6.2.3(2))

Nt.Rd := min Nu.Rd , Npl.Rd = 915.2kN ⋅

Check tension capacity (EN1993-1-1,cl.6.2.3(1)P)

Check_1 := if ⎜

0.9A ⋅ fy 3 Nu.Rd := = 1.318 × 10 ⋅ kN γ M2

(

)

⎛ NEd

⎞

⎝ Nt.Rd

≤ 1.0, "OK" , "NOT OK" ⎟

⎠

Check_1 = "OK"

Compression resistance (cl.6.2.3) Compression resistance of steel section (EN1993-1-1,cl.6.2.4(1)) Check compression capacity (EN1993-1-1,cl.6.2.4(1)P)

Nc.Rd := Npl.Rd = 915.2kN ⋅

⎛ NEd

⎞

Check_2 := if ⎜

⎝ Nc.Rd

≤ 1.0, "OK" , "NOT OK" ⎟

⎠

Check_2 = "OK"

Bending resistance (cl.6.2.5) Moment resistance of steel section at Y-Y (EN1993-1-1,cl.6.2.5(2)

Mc.Rd.y :=

Moment resistance of steel section at Z-Z (EN1993-1-1,cl.6.2.5(2)

Mc.Rd.z :=

W pl.y⋅ fy γ M0 W pl.z⋅ fy γ M0

= 71.94⋅ kN⋅ m

= 71.94⋅ kN⋅ m

Sheat resistance (cl.6.2.6) Factor for shear area (EN1993-1-1,cl.6.2.6(g))

η := 1

Shear area of steel section (EN1993-1-1,cl.6.2.6(3))

Avy :=

Shear area of steel section (EN1993-1-1,cl.6.2.6(3))

Avz :=

Shear resistance of steel section Y-Y (EN1993-1-1,cl.6.2.6(2))

A⋅ b b+d A⋅ d b+d

3

2

3

2

= 3.328× 10 ⋅ mm

= 3.328× 10 ⋅ mm

Vpl.Rd.y := A vy ⋅

fy ⋅ ( 3) γ M0

−1

= 528.391⋅ kN

Page 159

Shear resistance of steel section Z-Z (EN1993-1-1,cl.6.2.6(2))

Vpl.Rd.z := A vz⋅

fy ⋅ ( 3)

−1

= 528.391kN ⋅

γ M0

Check if the verification of shear buckling resistance required or not (EN1993-1-1,cl.6.2.6(6)) d ε Check_3 := if ⎛⎜ < 72⋅ , "Not required shear buckling resistance" , "Required shear buckling resistance" ⎞⎟ η ⎝ t ⎠ Check_3 = "Not required shear buckling resistance"

Bending and shear interaction check (cl.6.2.8) Strong axis Y-Y VEd.y

−5

Interaction check 1

vy :=

Reduced yield strength

⎛ 2VEd.y ⎞ ρ := ⎜ − 1⎟ = 1 ⎝ Vpl.Rd.y ⎠

Vpl.Rd.y

= 9.463× 10 2

2 ⎞ ⎛ ⎜ W pl.y − ρ⋅ A ⎟ ⋅ fy 4t ⎠ Reduced design plastic resistance moment ⎝ Mc.Rd.y := if v y > 0.5 γ M0 (EN1993-1-1,cl.6.2.8(5))

Mc.Rd.y if v y < 0.5

Mc.Rd.y = 71.94kN ⋅ ⋅m

Weak axis Z-Z VEd.z

−3

Interaction check 1

vz :=

Reduced yield strength

⎛ 2VEd.z ⎞ ρ := ⎜ − 1⎟ = 0.989 ⎝ Vpl.Rd.z ⎠

Vpl.Rd.z

= 2.668 × 10 2

Page 160

2 ⎞ ⎛ ⎜ W pl.z − ρ⋅ A ⎟ ⋅ fy 4t ⎠ Reduced design plastic resistance moment ⎝ M c.Rd.z := if v z > 0.5 γ M0 (EN1993-1-1,cl.6.2.8(5))

M c.Rd.z if v z < 0.5

Mc.Rd.z = 71.94kN ⋅ ⋅m

Check combination of axial and bending (EN1993-1-1,cl.6.2.1(7))

⎛ NEd

Check_4 := if ⎜

⎝ Npl.Rd NEd

Unity factor Npl.Rd

+

+

MEd.y Mc.Rd.y

+

MEd.y Mc.Rd.y

+

MEd.z Mc.Rd.z

⎞

≤ 1.0, "OK" , "NOT OK" ⎟

MEd.z Mc.Rd.z

⎠

= 0.42

Check_4 = "OK"

Bending and axial force interaction check (cl.6.2.9)

⎡ (A − 2b ⋅ tw)

Factor a

aw := min⎢

Factor a

af := min⎢

Factor n

n :=

Factor β

β :=

A

⎣

⎦

⎡ (A − 2d ⋅ tf ) A

⎣

NEd Npl.Rd

⎤

, 0.5⎥ = 0.423

⎤

, 0.5⎥ = 0.423

⎦

= 0.395

1.66 1 − 1.13n

2

if

1.66 1 − 1.13n

2

≤6

= 2.016

6 otherwise

Factor α

a := β = 2.016

Page 161

Strong axis Y-Y Mc.Rd.y⋅ ( 1 − n ) M := N.y.Rd Reduced design value of the resistance to 1 − 0.5aw bending moments making allowance for the MN.y.Rd := MN.y.Rd if MN.y.Rd ≤ Mc.Rd.y presence of axial forces Mc.Rd.y if MN.y.Rd > Mc.Rd.y (EN1993-1-1,cl.6.2.9.1(5)) MN.y.Rd = 55.168kN ⋅ ⋅m Mc.Rd.z⋅ ( 1 − n ) Weak axis Z-Z MN.z.Rd := 1 − 0.5af Reduced design value of the resistance to bending moments making allowance for the M N.z.Rd := MN.z.Rd if MN.z.Rd ≤ Mc.Rd.z presence of axial forces Mc.Rd.z if MN.z.Rd > Mc.Rd.z (EN1993-1-1,cl.6.2.9.1(5))

MN.z.Rd = 55.168kN ⋅ ⋅m

Check combination of axial and bending (EN1993-1-1,cl.6.2.1(7))

⎛ NEd

Check_5 := if ⎜

⎝ Npl.Rd NEd

Unity factor Npl.Rd

+

+

MEd.y Mc.Rd.y

+

MEd.y Mc.Rd.y

+

MEd.z Mc.Rd.z

⎞

≤ 1.0, "OK" , "NOT OK" ⎟

MEd.z Mc.Rd.z

⎠

= 0.42

Check_5 = "OK"

Bucking interaction check (cl.6.3) Strong axis Y-Y Status of effective length

Effective_Length := "Pinned Pinned"

Effective length factor (Guidance of EC3)

ky :=

0.7 if Effective_Length

"Fixed Fixed"

0.85 if Effective_Length

"Partial restraint"

0.85 if Effective_Length

" Pinned Fixed"

1 if Effective_Length

=1

"Pinned Pinned"

Page 162

Buckling length of column (fixed end)

Lcry := ky hc = 5.831m

Euler Buckling at y-y axis (EN1993-1-1,cl.6.3.1.2(1)

Ncry :=

Slenderness parameter at y-y axis (for class 1,2 and 3 cross-section) (EN1993-1-1,cl.6.3.1.3(1) Check for X bracing (EN1998-1-1,cl.6.7.3(4))

λ y :=

Es ⋅ Iyy ⋅ π

2

= 748.568⋅ kN

2

Lcry

A ⋅ fy

= 1.564

Ncry

(

Check_6 := if Ns ≥ 3, "Consider limitation (As EC8)" , "Ignore limitation (As EC3)" Check_6 = "Consider limitation (As EC8)"

Check for X bracing (EN1998-1-1,cl.6.7.3(1))

(

Check_7 := if 1.3 < λ y < 2, "OK" , "NOT OK"

)

Check_7 = "OK"

Type of the section

Section := "Hot finished"

Buckling curve (EN1993-1-1,table 6.2)

Buckling_curve :=

"a" if Section

"Hot finished"

"c" if Section

"Cold formed"

Buckling_curve = "a" αy :=

Imperfection factor (EN1993-1-1,table 6.1)

0.21 if Buckling_curve

"a"

0.49 if Buckling_curve

"c"

αy = 0.21 2⎤

Value to determine the reduction factor χ (EN1993-1-1,cl.6.3.1.2(1))

φ y := 0.5⋅ ⎡1 + αy ⋅ λ y − 0.2 + λ y ⎣

Reduction factor χ (EN1993-1-1,cl.6.3.1.2(1))

χ y :=

Reduction factor χ check

Check_8 := if χ y ≤ 1.0, "OK" , "NOT OK"

(

)

1 2

φy +

φ y − λy

(

2

⎦ = 1.866

= 0.347

)

Check_8 = "OK"

Page 163

)

Design buklcing resistance (EN1993-1-1,cl.6.3.1.1(3))

Nb.Rd.y :=

χ y⋅ A ⋅ fy

= 634.758⋅ kN

γ M1

⎛ NEd

Buckling resistance of compression member check (EN1993-1-1,cl.6.3.1.1(1))

Check_9 := if ⎜

⎝ Nb.Rd.y

⎞

, "OK" , "NOT OK" ⎟

⎠

Check_9 = "OK"

Weak axis Z-Z Status of effective length

Effective_Length := "Pinned Pinned"

Effective length factor (Guidance of EC3)

kz :=

0.7 if Effective_Length 0.85 if Effective_Length

"Partial restraint"

0.85 if Effective_Length

" Pinned Fixed"

1 if Effective_Length

Buckling length of column (fixed end)

=1

"Pinned Pinned"

Lcrz := kz hc = 5.831m

Euler Buckling at y-y axis (EN1993-1-1,cl.6.3.1.2(1)

Ncrz :=

Slenderness parameter at y-y axis (for class 1,2 and 3 cross-section) (EN1993-1-1,cl.6.3.1.3(1) Check for X bracing (EN1998-1-1,cl.6.7.3(4))

"Fixed Fixed"

λ z :=

Es ⋅ Izz⋅ π 2

2

= 748.568kN ⋅

Lcrz

A ⋅ fy Ncrz

= 1.564

(

Check_10 := if Ns ≥ 3, "Consider limitation (As EC8)" , "Ignore limitation (As EC3)" Check_10 = "Consider limitation (As EC8)"

Check for X bracing (EN1998-1-1,cl.6.7.3(1))

(

Check_11 := if 1.3 < λ z < 2, "OK" , "NOT OK"

)

Check_11 = "OK"

Type of the section

Section := "Hot finished"

Page 164

)

Buckling curve (EN1993-1-1,table 6.2)

Buckling_curve :=

"a" if Section

"Hot finished"

"c" if Section

"Cold formed"

Buckling_curve = "a" αz :=

0.21 if Buckling_curve

"a"

0.49 if Buckling_curve

"c"

αz = 0.21 2

Value to determine the reduction factor χ (EN1993-1-1,cl.6.3.1.2(1))

φ z := 0.5⋅ ⎡1 + αz⋅ λ z − 0.2 + λ z ⎤ = 1.866 ⎣ ⎦

Reduction factor χ (EN1993-1-1,cl.6.3.1.2(1))

χ z :=

Reduction factor χ check

Check_12 := if χ z ≤ 1.0, "OK" , "NOT OK"

(

)

1 2

φz +

2

= 0.347

φ z − λz

(

)

Check_12 = "OK"

Design buklcing resistance (EN1993-1-1,cl.6.3.1.1(3)) Buckling resistance of compression member check (EN1993-1-1,cl.6.3.1.1(1))

Nb.Rd.z :=

χ z⋅ A ⋅ fy γ M1

= 634.758⋅ kN

⎛ NEd

Check_13 := if ⎜

⎝ Nb.Rd.z

⎞

, "OK" , "NOT OK" ⎟

⎠

Check_13 = "OK"

Lateral torsional buckling check (cl.6.3.2) Effective length factor, k (SN003a-EN-EU) Factor for end warping, kw (SN003a-EN-EU)

kz = 1 kw := 1.0 MEd.y2

= 0.247

Ratio of the smaller and larger moment

ψ :=

Coefficient factor C1

C1 := 1.88 − 1.40ψ + 0.52ψ = 1.566

MEd.y1

2

(SN003a-EN-EU)

Page 165

(

Coefficient factor C1 check

Check_14 := if C1 ≤ 2.7, "OK" , "NOT OK"

)

(SN003a-EN-EU) Check_14 = "OK"

Coefficient factor C2

C2 := 1.554

(SN003a-EN-EU) Distance between the point of load application and the shear centre

zg := 0m

Elastic critical moment for lateral torsional buckling (SN003a-EN-EU) 2

Mcr := C1⋅

π ⋅ Es ⋅ Izz

(Lcrz)2

2 2 ⎛ kz ⎞ Iw (Lcrz) G⋅ It 2 3 ⋅ ⎜ ⎟ ⋅ + + (C2⋅ zg) − C2⋅ zg = 1.636× 10 ⋅ kN⋅ m kw Izz 2 ⎝ ⎠ π Es ⋅ Izz

Imperfection factor for lateral torsional CHS sections (EN1993-1-1,table 6.3)

αLT := 0.76

Non dimensional slenderness (EN1993-1-1,cl.6.3.2.2(1))

W pl.y ⋅ fy

λ LT :=

Mcr

= 0.21

Value to determine the reduction factor (EN1993-1-1,cl.6.3.2.2(1))

φ LT := 0.5⋅ ⎡1 + αLT⋅ λ LT − 0.2 + λ LT ⎤ = 0.526 ⎣ ⎦

Reduction factor for lateral-torsional buckling (EN1993-1-1,cl.6.3.2.2(1))

χ LT :=

(

1 φ LT +

2

)

2

2

= 0.992

φ LT − λ LT

Check_15 := if ⎛⎜ χ LT ≤ 1 ∧ χ LT ≤

⎜ ⎝

1 2

, "OK" , "NOT OK" ⎞⎟

⎟ ⎠

λ LT

Check_15 = "OK"

Parameter λ LTO

λ LTO := 0.4

(EN1993-1-1,cl.6.3.2.3(1)) Design buckling resistance moment (EN1993-1-1,cl.6.3.2.1(3))

Mb.Rd := χ LT⋅ W pl.y⋅ γ

⎛ MEd.y

Check_16 := if ⎜

⎝ Mb.Rd

fy

= 71.389⋅ kN⋅ m

M1

⎞

≤ 1, "OK" , "NOT OK" ⎟

⎠

Page 166

Check_16 = "OK"

Check if the lateral torsional buckling be ignored (EN1993-1-1,cl.6.3.2.2(4)) MEd.y ⎛ ⎞ 2 Check_17 := if ⎜ λ LT < λ LTO ∧ < λ LTO , "Ignored torsional buckling effects" , "Consider torsional buckling effects" ⎟ Mcr ⎝ ⎠ Check_17 = "Ignored torsional buckling effects"

Combine bending and axial compression cl.6.3.3 Moments due to the shift of the centroidal axis for class sections 1,2 & 3 (EN1993-1-1,cl.6.3.3(4)/table 6.7)

ΔM Ed.z := 0 ΔM Ed.y := 0

Characteristic resistance to normal force of the critical cross-section (EN1993-1-1,cl.6.3.3(4)/table 6.7)

3

NRk := fy ⋅ A = 1.83 × 10 ⋅ kN

Characteristic moment resistance of the critical cross-section (EN1993-1-1,cl.6.3.3(4)/table 6.7)

My.Rk := fy ⋅ Wpl.y = 71.94kN ⋅ ⋅m Mz.Rk := fy ⋅ Wpl.z = 71.94kN ⋅ ⋅m

ψy :=

MEd.y1 MEd.y2 MEd.y2 MEd.y1

Ratio of end moments (EN1993-1-1,Table B2) ψz :=

M Ed.z1 M Ed.z2 M Ed.z2 M Ed.z1

if −1 ≤

if −1 ≤

if −1 ≤

if −1 ≤

MEd.y1 MEd.y2 MEd.y2 MEd.y1 M Ed.z1 M Ed.z2 M Ed.z2 M Ed.z1

≤1

≤1

≤1

≤1

Equivalent uniform moment factor

Cmy := 0.6 + 0.4⋅ ψy = 0.699

Equivalent uniform moment factor

Cmz := 0.6 + 0.4⋅ ψz = 0.725

⎡⎡

⎡

⎢⎢ ⎢⎢ ⎣⎣

⎢ ⎢ ⎣

(

)

kyy := min⎢⎢Cmy⋅ ⎢1 + λ y − 0.2 ⋅

NEd

N ⎤⎤ ⎛ ⎞⎤ ⎥⎥ , Cmy⋅ ⎜ 1 + 0.8⋅ Ed ⎟⎥ = 1.018 NRk ⎥⎥ NRk ⎟⎥ ⎜ χ y⋅ χ ⋅ y γ ⎜ ⎟⎥ γ M1 ⎥⎥ M1 ⎠⎦ ⎦⎦ ⎝

Page 167

Interaction factors NEd ⎤⎤ N ⎡⎡ ⎡ ⎛ ⎞⎤ (EN1993-1-1,table ⎥⎥ , Cmz⋅ ⎜ 1 + 1.4⋅ Ed ⎟⎥ = 1.303 kzz := min⎢⎢Cmz⋅ ⎢1 + ( 2λ z − 0.6) ⋅ B.1&B.2) NRk ⎥⎥ NRk ⎟⎥ ⎢⎢ ⎢ ⎜ χ ⋅ χ ⋅ zγ zγ ⎢⎢ ⎢ ⎥⎥ ⎜ ⎟⎥ M1 ⎦⎦ M1 ⎠⎦ ⎣⎣ ⎣ ⎝ kyz := 0.6kzz = 0.782 kzy := 0.6kyy = 0.611

EN1993-1-1,Equation 6.61

⎛ NEd

Check_18 := if ⎜

⎜ ⎜ ⎝

χ y ⋅ NRk

+ kyy ⋅

χ LT⋅

γ M1

NEd

Unity factor

MEd.y + ΔM Ed.y

+ kyy ⋅

χ y ⋅ NRk

My.Rk

+ kyz⋅

γ M1

χ LT⋅

⎞

≤ 1.0, "OK" , "NOT OK" ⎟

M z.Rk

⎟ ⎟ ⎠

γ M1

MEd.y + ΔM Ed.y

γ M1

MEd.z + ΔM Ed.z

My.Rk

+ kyz⋅

γ M1

MEd.z + ΔM Ed.z M z.Rk

= 0.595

γ M1

Check_18 = "OK"

EN1993-1-1,Equation 6.62

⎛ NEd

Check_19 := if ⎜

⎜ ⎜ ⎝

χ z ⋅ NRk

+ kzy⋅

χ LT⋅

γ M1

NEd

Unity factor

MEd.y + ΔM Ed.y

χ z ⋅ NRk γ M1

+ kzy⋅

My.Rk

+ kzz⋅

MEd.z + ΔM Ed.z

γ M1

MEd.y + ΔM Ed.y χ LT⋅

My.Rk γ M1

M z.Rk

⎞

≤ 1.0, "OK" , "NOT OK" ⎟

⎟ ⎟ ⎠

γ M1

+ kzz⋅

MEd.z + ΔM Ed.z M z.Rk

= 0.584

γ M1

Check_19 = "OK"

Page 168

Eurocode 8 requirements Yield resistance (EN1998-1-1,cl.6.7.3(5)) Yield resistance check (EN1998-1-1,cl.6.7.3(5))

(

Check_20 := if NEd ≤ Npl.Rd , "OK" , "NOT OK"

)

Check_20 = "OK"

Check omega factor (EN1998-1-1,cl.6.7.3(8)) Axial force at storey 3

NEd.3 := 162.34kN

Axial force at storey 2

NEd.2 := 317.56kN

Area of steel section (RHS 100X100X10)

A := 3600mm

Design plastic resistance of the cross section Storey 3: RHS 100X100X10 (EN1993-1-1,cl.6.2.3(2a))

0.5A ⋅ fy Npl.Rd.3 := = 495⋅ kN γ M0

2

Omega factor

Npl.Rd Ωstorey1 := = 2.529 NEd

Omega factor

Npl.Rd Ωstorey2 := = 2.882 NEd.2

Omega factor

Npl.Rd.3 Ωstorey3 := = 3.049 NEd.3

Minimum omega

Ωmin := min Ωstorey1 , Ωstorey2 , Ωstorey3

(

)

Ωmin = 2.529

Minimum omega

(

Ωmax := max Ωstorey1 , Ωstorey2 , Ωstorey3

)

Ωmax = 3.049

Check Ω factor (EN1998-1-1,cl.6.7.3(8))

(

Check_21 := if Ωmax ≤ 1.25Ωmin, "OK" , "NOT OK"

)

Check_21 = "OK"

Page 169

10.0 Modal response spectrum analysis

10.1 Set the analysis options

1.

ETABS: Analyze > Set analysis Options

Calculate the number of modes:

Figure 10.1: Set the modal analysis parameters

Page 170

10.2 Evaluate the analysis results of the structure according to the modal analysis requirements

2.

ETABS: Display > Show Tables

Figure 10.2: Modal response spectrum results

Page 171

10.2.1 Assess the modal analysis results based on the EN1998

The requirements of the sum of effective modal masses for the modes taken into account amounts to at least 90% of the total mass of the structure is satisfied (EN1998-11,cl.4.3.3.3.1(3)). Page 172

Effective mass of mode 6 = 97% > 90% “OK”

11.0 Second order effects (P – Δ effects) according to EN1998-1-1,cl.4.4.2.2

The criterion for taking into account the second order effect is based on the interstorey drift sensitivity coefficient θ, which is define with equation (EN 1998-1-1,cl.4.4.2.2(2)).

Θ=

P!"! ∙ d! V!"! ∙ h

dr:

is the interstorey drift

h:

is the storey height.

Vtot:

is the total seismic storey shear.

Ptot:

is the total gravity load at and above storey considered in the seismic design situation (G+0.3Q).

Table 11.1: Consequences of value of P-Δ coefficient θ on the analysis θ≤0,1

No need to consider P-Δ effects P-Δ effects may be taken into account approximately by

0,1≤θ≤0,2

0,2≤θ≤0,3 θ≥0,3

!

amplifying the effects of the seismic actions by !!! P-Δ effects must be accounted for by an analysis including second order effects explicity Not permitted

Important note: If the above expression is not satisfied, second order effects, should be enable in ETABS.

ETABS: Analyze > Set analysis option >

> Set the parameters Page 173

11.1 Displacement calculation according to EN1998-1-1,cl.4.4.2.2

d! = q ∗ d! ds :

is the displacement of a point of the structural system induced by the design seismic action.

qd :

is the displacement behaviour factor, assumed equal to q unless otherwise specified.

de :

is the displacement of the same point of the structural system, as determined by a linear analysis based on the design response spectrum.

11.2 Interstorey drift Interstorey drift is the design interstorey drift, evaluated as the difference of the average lateral displacements ds at the top and bottom of the storey under consideration and calculated in accordance with EN1993-1-1,cl.4.3.4.

d! =

d!.!"# − d!.!"# 2

Page 174

11.3 Calculation of second order effect using ETABS

3.

ETABS: Run the model

4.

ETABS: Display > Show tables

Select the design combinations

Static load case combination (include wind load) STATIC 2. STATIC 3. STATIC 4. STATIC 5. STATIC 6. STATIC 7. STATIC 8. STATIC 9.

1.35DL + 1.5LL + 0.75WINDX 1.35DL + 1.5LL - 0.75WINDX 1.35DL + 1.5LL + 0.75WINDY 1.35DL + 1.5LL - 0.75WINDY 1.35DL + 1.5WINDX + 1.05LL 1.35DL - 1.5WINDX – 1.05LL 1.35DL + 1.5WINDY + 1.05LL 1.35DL - 1.5WINDY – 1.05LL

Seismic load case combination SEISMIC 1. SEISMIC 2. SEISMIC 3. SEISMIC 4. SEISMIC 5. SEISMIC 6. SEISMIC 7. SEISMIC 8.

DL + 0.3LL + EQX + 0.3EQY DL + 0.3LL + EQX – 0.3EQY DL + 0.3LL - EQX + 0.3EQY DL + 0.3LL - EQX – 0.3EQY DL + 0.3LL + EQY + 0.3EQX DL + 0.3LL + EQY – 0.3EQX DL + 0.3LL - EQY + 0.3EQX DL + 0.3LL - EQY – 0.3EQX Page 175

Figure 11.1: Displacement due to lateral load

For floor with the non use of diaphragm, the maximum displacement can be found in this table

For floor with the use of diaphragm, the maximum displacement can be found in this table

11.3.1 Interstorey drift displacement

Page 176

Sort smallest to largest in order to find the maximum displacement or Sort largest to smallest in order to find the maximum displacement Consider the maximum value Do this process for all storeys separately as showing below

Table 11.2: Displacement due to lateral load Storey no.

Max Displacement at X

Max Displacement at Y

Storey 3

Storey 2

Storey 1

Page 177

Table 11.3: Drift displacement

Storey  

Displacement   Displacement   Displacement   Displacement   Interstorey   Interstorey   Direction  x   Direction  y   Behaviour   dsx                                   dsy                                   drift                           drift                           dx.e                             dy.e                               factor  q   (mm)       (mm)     drx                       dry                       (mm)   (mm)   cl.4.4.2.2   cl.4.4.2.2   (mm)   (mm)  

Storey  3  

11.742  

11.7452  

4  

46.968  

46.9808  

6.7754  

6.7776  

Storey  2  

8.3543  

8.3564  

4  

33.4172  

33.4256  

9.0274  

9.0296  

Storey  1  

3.8406  

3.8416  

4  

15.3624  

15.3664  

7.6812  

7.6832  

d!" = q ∗ d!"

d!" =

d!".!"# − d!".!"# 2

d!" =

d!".!"# − d!".!"# 2

d!" = q ∗ d!"

11.3.2 Total gravity load Ptot ETABS: Display > Show tables

Select the design combinations

Static load case combination STATIC 10. DL + 0.3LL

Page 178

Export the results in Excel sheet

Filter the value of the bottom storey

Page 179

Story  

Load  

Loc  

P  

Record the total gravity load (G+ψEiQ) of each storey

STORY3   STATIC10   Bottom   1402.76   STORY2   STATIC10   Bottom   2804.93   STORY1   STATIC10   Bottom   4207.11  

11.3.2 Total seismic storey shear Vtot ETABS: Display > Show tables

Select the design combinations

Seismic load case combination SEISMIC 1. SEISMIC 2. SEISMIC 3. SEISMIC 4. SEISMIC 5. SEISMIC 6. SEISMIC 7. SEISMIC 8.

DL + 0.3LL + EQX + 0.3EQY DL + 0.3LL + EQX – 0.3EQY DL + 0.3LL - EQX + 0.3EQY DL + 0.3LL - EQX – 0.3EQY DL + 0.3LL + EQY + 0.3EQX DL + 0.3LL + EQY – 0.3EQX DL + 0.3LL - EQY + 0.3EQX DL + 0.3LL - EQY – 0.3EQX

Page 180

Export the results in Excel sheet

Sort smallest to largest in order to find the maximum shear force or Sort largest to smallest in order to find the maximum shear force Consider the worst load combination Do this process for all storeys separately as showing below

Page 181

Filter the values using the worst case combination Filter the value of the bottom storey

Story  

Load  

STORY1   STORY2   STORY3  

SEISMIC1  MAX   SEISMIC1  MAX   SEISMIC1  MAX  

Loc  

P  

Bottom   4207.11   Bottom   2804.93   Bottom   1402.76  

VX   663.91   550.8   330  

Repeat the above procedure in order to obtain the Vtot at Y-direction

VY   Story  

Load  

STORY1   STORY2   STORY3  

SEISMIC5  MAX   SEISMIC5  MAX   SEISMIC5  MAX  

Loc  

Record the total seismic shear of each storey for Vtot at X-direction

P  

Bottom   4207.11   Bottom   2804.93   Bottom   1402.76  

663.91   550.8   330  

Page 182

Table 11.4: Second order effects check (EN1993-1-1,cl.4.4.2.2(2)) Displacement   Displacement   Displacement   Displacement   Interstorey   Interstorey   Direction  x   Direction  y   Behaviour   dsx                                   dsy                                   drift                           drift                           dx.e                             dy.e                               factor  q   (mm)       (mm)     drx                       dry                       (mm)   (mm)   cl.4.4.2.2   cl.4.4.2.2   (mm)   (mm)  

Storey  

Storey  3  

11.742  

11.7452  

4  

46.968  

46.9808  

6.7754  

6.7776  

Storey  2  

8.3543  

8.3564  

4  

33.4172  

33.4256  

9.0274  

9.0296  

Storey  1  

3.8406  

3.8416  

4  

15.3624  

15.3664  

7.6812  

7.6832  

Total   Total   Total   Height  of   gravity  load               seismic   seismic   each   Ptot                     storey  shear   storey  shear   storey   (kN)   Vtotx  (kN)   Vtoty  (kN)   (mm)  

Interstorey  drift   Interstorey  drift   sensitivity  coefficient  θ   sensitivity  coefficient  θ   at                                                                                          at                                                                             X  direction   Y  direction  

663.91  

663.91  

663.91  

3000  

OK  

OK  

550.8  

550.8  

550.8  

3000  

OK  

OK  

330  

330  

330  

3000  

OK  

OK  

   

 

 

Θ=

P!"! ∙ d!" ≤ 0.10 V!"!# ∙ h

Θ=

P!"! ∙ d!" ≤ 0.10 V!"!# ∙ h

Page 183

12.0 Damage limitation according to EN1998-1-1,cl.4.4.3 The “damage limitation requirement” is considered to have been satisfied, if, under a seismic action having a larger probability of occurrence than the design seismic action corresponding to the “no-collapse requirement” in accordance with 2.1(1)P and 3.2.1(3), the interstorey drifts are limited in accordance with 4.4.3.2. The damage limitation requirements should be verified in terms of the interstorey drift (dr) (EN 1998-1-1,cl.4.4.3.2) using the equation below:

d! ∙ v ≤ 0.005 ∙ h     dr: is the difference of the average lateral displacement ds in CM at the top and bottom of storey. v: is the reduction factor which takes into account the lower return period of the seismic action. h: is the storey height Table 12.1: Damage limitation (EN1998-1-1,cl.4.4.3) For non-structural elements of brittle material attached to the structure

drv≤0.005h

drv≤0.0075h

For building having ductile non structural elements For building having non-structural elements fixed in a way so as not to

drv≤0.010h

interfere with structural deformation

Table 12.2: Reduction factor of limitation to interstorey drift (CYA NA EN1998-11,cl.NA.2.15)

Importance class

Reduction factor v

I

0.5

II

0.5

III

0.4

IV

0.4

Page 184

12.1 Calculation of damage limitation

Table 12.3: Interstorey drift (see table 11.3)

Storey  

Displacement   Displacement   Displacement   Displacement   Interstorey   Interstorey   Direction  x   Direction  y   Behaviour   dsx                                   dsy                                   drift                           drift                           dx.e                             dy.e                               factor  q   (mm)       (mm)     drx                       dry                       (mm)   (mm)   cl.4.4.2.2   cl.4.4.2.2   (mm)   (mm)  

Storey  3  

11.742  

11.7452  

4  

46.968  

46.9808  

6.7754  

6.7776  

Storey  2  

8.3543  

8.3564  

4  

33.4172  

33.4256  

9.0274  

9.0296  

Storey  1  

3.8406  

3.8416  

4  

15.3624  

15.3664  

7.6812  

7.6832  

Reduction   Heigh  of   factor                             each   v                                    storey   cl.4.4.3.2(2)   (mm)  

Damage  limitation   Damage  limitation   check                                                   check                                                   X-­‐direction   Y-­‐direction  

0.4  

3000  

OK  

OK  

0.4  

3000  

OK  

OK  

0.4  

3000  

OK  

OK  

 

 

d! ∙ v ≤ 0.005 ∙ h   d! ∙ v ≤ 0.005 ∙ h  

Page 185

ANNEX - A

ANNEX A.1 - Assumptions made in the design algorithm (Manual of ETABS – EC3 & EC8)

1. Load combination

•

The automated load combinations are based on the STR ultimate limit states and the characteristic serviceability limit states.

2. Axial force check

•

Tubular sections are assumed to be hot finished for selecting the appropriate buckling curve from EC3 Table 6.2. This is non conservative if cold formed sections are used.

3. Bending moment check

•

The load is assumed to be applied at the shear center for the calculation of the elastic critical moment.

•

Any eccentric moment due to load applied at other locations is not automatically accounted for.

4. Shear Force Check

•

Plastic design is assumed such that Vc,Rd is calculated in accordance with EC3 6.2.6(2).

•

The shear area, Av is taken from the input frame section property, rather than using the equations defined in EC3 6.2.6(3).

•

Transverse stiffeners exist only at the supports and create a non-rigid end post for the shear buckling check. No intermediate stiffeners are considered. Page 186

•

The contribution from the flanges is conservatively ignored for the shear buckling capacity.

5. Combined Forces Check

•

The interaction of bending and axial force is checked in accordance with EC3 6.2.1(7), which may be conservative compared to EC3 6.2.9.

•

The calculation of the equivalent uniform moment factors, Cm, assumes uniform loading, which is conservative.

A1.1:Limitation made in the design algorithm (Manual of ETABS – EC3&EC8) 6. General

•

Class 4 sections are not designed (EC3 5.5) and should be considered using other methods.

•

The effects of torsion are not considered in the design (EC3 6.2.7) and should be considered using other methods.

7. Axial Force Check

•

The net area is not determined automatically. This can be specified on a member-bymember basis using the Net Area to Total Area Ratio overwrite.

•

The axial buckling check does not consider torsional or torsional-flexural buckling.

8. Combined Forces Check

•

The effect of high shear is checked only for Class 1 or 2 I-sections when combined with bending. Other section shapes and classes require independent checks to be carried out. Page 187

ANNEX –B: Steel design flowcharts

BASIS OF STRUCTURAL DESIGN (EN1990:2002) Vertical deflection (EN1993-1-1,cl.7.2.1) w1 = Initial part of the deflection under permanent loads wc = Precamber in the unloaded structural member w2 = due to Permanent load w3 = due to Variable load STEEL MEMBERS (CYS NA EN1993-1-1,table NA.1) Vertical deflection Cantilevers Beams carrying plaster or other brittle finish Other beams (except purlin and sheeting rails) Purlins and sheeting rails General use

Limits wmax L/180 L/360 L/250 To suit cladding L/300

Horizontal deflection (EN1993-1-1,cl.7.2.2)

u = Overall horizontal displacement over the building height H ui = Horizontal displacement over height Hi STEEL MEMBERS (CYS NA EN1993-1-1,table NA.2) Horizontal deflection Top of columns in single storey buildings, exept portal frames Columns in portal frame buildings, not supporting crane runways In each storey of the building with more than one storey On the multi-storey building as a whole

Limits wmax H/300 To suit cladding Storey height/300 Building height/500

Page 188

Dynamic effects (vibration of floors) (EN1993-1-1,cl.7.2.3) STEEL MEMBERS (CYS NA EN1993-1-1,table NA.3) Design situation

Limits natural frequency 5Hz 9Hz

Floors over which people walk regularly Floor which is jumped or danced on in a rhythmical manner

Effective length (Design Guidance of EC3)

Figure 1: Effective length columns (Design Guidance of EC3)

End restraints Effective length factor, ky,z

Fixed/Fixed 0.7L

Partial restrain in direction 0.85L

Pined/Fixed

Pinned/Pined

0.85L

1.0L

Free in position/Fixed 1.2L

Free/Fixed 2.0L

Page 189

Compression resistance (EN1993-1-1,cl. 6.2.4)

Class 1 or 2and3

𝛮!.!" =

𝛢𝑓! 𝛾!!

𝑵𝑬𝒅 ≤ 𝑵𝒄,𝑹𝒅

Bending resistance (EN1993-1-1,cl. 6.2.5)

Class 1 or 2

𝑀!.!" =

Class 3

𝑊!",! 𝑓! 𝛾!!

𝑀!.!" =

𝑊!",!"# 𝑓! 𝛾!!

𝑴𝑬𝒅 ≤ 𝑴𝒄.𝑹𝒅

Fastener holes in tension flange may be ignored if:

𝑨𝒇,𝒏𝒆𝒕 𝟎. 𝟗𝒇𝒖 /𝜸𝑴𝟐 ≥ 𝑨𝒇 𝒇𝒚 /𝜸𝑴𝟎

Page 190

Shear resistance (EN1993-1-1,cl. 6.2.6)

Plastic design

Elastic design

𝐴! = ℎ! ∙ 𝑡!

𝐴! /𝐴! ≥ 0.6

𝜏!" =

𝑉!,!" =

𝑉!" 𝐴!

Rolled I and H sections (load parallel to web)

Rolled C channel sections (load parallel to web)

CHS

𝐴! = 2𝐴/𝜋

RHS

𝐴! = 𝐴ℎ/(𝑏 + ℎ)Load parallel to depth

𝜂= 1.0 (conservative value)

𝐴! = 𝐴𝑏/(𝑏 + ℎ)Load parallel to

𝜏!" 𝑓! /( 3𝛾!! )

width

𝐴! = 𝐴 − 2𝑏𝑡! + 𝑡! + 2𝑟 𝑡!

𝑽𝑬𝒅 ≤ 𝟏. 𝟎 𝑽𝒄.𝑹𝒅 but

≥𝜂ℎ! 𝑡!

𝑉!".!" =

𝐴! (𝑓! / 3) 𝛾!!

𝑽𝑬𝒅 ≤ 𝑽𝒄,𝑹𝒅

Ignore Shear buckling resistance for webs without intermediate stiffeners

𝒉𝒘 𝜺 > 72 𝒕𝒘 𝜼

Page 191

Combine Bending and shear (EN1993-1-1,cl. 6.2.8)

NO Reduction of resistances (effect on Mc,Rd)

𝑉!".!" =

𝜌= 1−

Shear design resistance

YES

𝑉!" ≤ 0.5 ∙ 𝑉!".!"

NO Reduction of resistances (no effect on Mc,Rd)

𝐴! (𝑓! / 3) 𝛾!!

2𝑉!" −1 𝑉!",!"

!

If torsion present:

2𝑉!" 𝜌= 1− −1 𝑉!",!,!"

!

For an I and H sections

𝑓!" = 1 − 𝜌 𝑓!

𝑉!",!,!" =

1−

𝜏!,!" 1.25 𝑓! / 3 /𝛾!!

Reduced design plastic resistance moment

𝐴! = ℎ! 𝑡!

𝑴𝒚.𝑽,𝑹𝒅 =

(𝑾𝒑𝒍,𝒚 −

𝝆𝑨𝒘 𝟐 𝟒𝒕𝒘

𝜸𝑴𝟎

)𝒇𝒚

      ≤ 𝑴𝒚,𝒄,𝑹𝒅

Page 192

𝑉!",!"

Bending & Axial force (EN1993-1-1,cl. 6.2.9) Class 1 or 2

Doubly symmetrical I and H sections Z-Z axis

Doubly symmetrical I and H sections Y-Y axis

𝑁!" ≤

0.5 ∙ ℎ! ∙ 𝑡! ∙ 𝑓! 𝛾!!

𝑁!" ≤

ℎ! ∙ 𝑡! ∙ 𝑓! 𝛾!!

𝑁!" ≤ 0.25𝑁!".!" NO NO

YES

YES Consider axial force

Consider axial force

Ignored axial force 𝑎=

𝑎=

𝐴 − 2𝑏𝑡! ≤ 0,5 𝐴

𝑛=

𝑁!" 𝑁!",!"

Ignored axial force

𝐴 − 2𝑏𝑡! ≤ 0,5 𝐴

𝑛=

𝑁!" 𝑁!",!"

𝑛𝑎

𝑀!,!,!" = 𝑀!",!,!" (1 − 𝑛)/(1 − 0,5𝑎) 𝑀!,!,!" = 𝑀!",!,!" 1 − MN,y,Rd≤ Mpl,y,Rd

𝑛−𝑎 1−𝑎

!

𝑀!,!,!" = 𝑀!",!,!"

𝑵𝑬𝒅 𝑴𝒚,𝑬𝒅 𝑴𝒛,𝑬𝒅 + + ≤ 𝟏. 𝟎 𝑵𝑹𝒅 𝑴𝒚,𝑹𝒅 𝑴𝒛,𝑹𝒅

Page 193

Bending & Axial force (EN1993-1-1,cl. 6.2.9) Class 1 or 2

For RHS Y-Y axis Z-Z axis

𝑁!" ≤

ℎ! ∙ 𝑡! ∙ 𝑓! 𝛾!!

NO

YES

Consider axial force

Ignored axial force

Hollow section

Welded box section

𝑎! = (𝐴 − 2𝑏𝑡)/𝐴) ≤ 0.5

𝑎! = (𝐴 − 2𝑏𝑡! )/𝐴) ≤ 0.5

𝑎! = (𝐴 − 2ℎ𝑡)/𝐴) ≤ 0.5

𝑎! = (𝐴 − 2ℎ𝑡! )/𝐴) ≤ 0.5

𝑀!,!,!" =

𝑀!",!,!" 1 − 𝑛 ≤ 𝑀!",!,!" 1 − 0.5𝑎!

𝑀!,!,!" =

𝑀!",!,!" 1 − 𝑛 ≤ 𝑀!",!,!" 1 − 0.5𝑎!

I and H section

CHS

𝑎=2 𝛽 = 5𝑛   ≥ 1 𝑛 = 𝑁!" /𝑁!",!"

𝑎=2 𝛽 = 5𝑛   ≥ 1 𝑛 = 𝑁!" /𝑁!",!"

𝑴𝒚,𝑬𝒅 𝑴𝑵,𝒚,𝑹𝒅

𝒂

+

𝑴𝒛,𝑬𝒅 𝑴𝑵,𝒛,𝑹𝒅

RHS 1.66 1 − 1.13𝑛! but𝑎 = 𝛽 ≤ 6

𝑎=𝛽=

𝜷

≤ 𝟏. 𝟎

Page 194

Buckling resistance in compression (EN1993-1-1,cl. 6.3.1.1) Class 1 or 2and3

Slenderness for flexural buckling

𝑁!" =

!! !" !!

λ=

for ideal strut

𝐴𝑓! 𝑁!"

𝜆 ≤ 0.2 𝑁!" /𝑁!" ≤ 0.04

NO (consider buckling effects)

Cross-section

YES (ignored buckling effects)

Limits tf≤40mm h/b>1.2 40mm 100mm

U-T and solid section L-sections Hollow sections

Hot finished Cold formed

Buckling curve Imperfection factor a

ao 0,13

a 0,21

Buckling about axis y-y z-z y-y z-z y-y z-z y-y z-z any any any any

b 0,34

c 0,49

Buckling curve a b b c b c d d C b a c

d 0,76

Φ = 0,5 1 + 𝑎 𝜆 − 0,2 + 𝜆!

χ=

1 Φ + Φ ! − λ!

𝑁!,!" =

≤ 𝜒 ≤ 1,0

𝜒𝐴𝑓! 𝛾!! )

𝑵𝑬𝒅 ≤ 𝑵𝒃,𝑹𝒅

Page 195

Buckling resistance in bending (EN1993-1-1,cl. 6.3.2)

Class 1 or 2and3

Slenderness for flexural buckling

𝜀=

235 𝑓!

λ! = 𝜋

𝐸 = 93,9𝜀 𝑓!

λ!" =

𝑊! 𝑓! 𝑀!"

1

χ!" = Φ!" +

!

See following pages for calculation of Mcr and λL

Φ!" − λ!"

!

≤ 𝜒!" ≤ 1,0

Φ!" = 0,5 1 + 𝑎!" 𝜆!" − 0,2 + 𝜆!" !

Cross-section

Limits

Rolled I-sections

h/b≤2 h/b>2 h/b≤2 h/b>2 -

Welded I-sections Other cross-sections Buckling curve Imperfection factor aLT

a 0,21

b 0,34

Buckling curve a b c d d c 0,49

d 0,76

Class 1 or 2

Class 3

Wy=Wpl,y

Wy=Wel,y

𝑀!,!" =

𝜒!" 𝑊! 𝑓! 𝛾!!

𝑴𝑬𝒅 ≤ 𝟏. 𝟎 𝑴𝒃.𝑹𝒅

Page 196

Calculation process of Mcr (www.access-steel.com - Document SN003a&b)

Term L E G Iz It Iw k kw zg

Step 1: Define the properties of member Description Values Distance between point of Lcr=kl lateral restraint Young’s modulus 210000 N/mm2 Shear modulus 80770 N/mm2 Second moment of area about the weak axis From section table Torsion constant Warping constant Effective length factor 1.0 unless justified otherwise Factor for end warping 1.0 unless justified otherwise Distance between the point of +/-(h/2) or 0 if the load is load application and the shear applied through the shear centre centre

Step 2: Calculate the coefficient C1 and C2 Loading and C2 Ψ=M1/M2 support conditions Pinned UDL 0,454 1.00 Fixed UDL 1,554 0.75 Pinned central P 0,630 0.50 Fixed central P 1,645 0.25 0 -0.25 -0.50 -0.75 -1.00 Pinned UDL Pinned, central P

C1 1,00 1.14 1,31 1,62 1,77 2,05 2,33 2,57 2,55 1,127 1,348

Point of application of the load is through the shear centre

YES zg=0

𝛭!"

𝜋 ! 𝐸𝐼! 𝐼! 𝐿!" ! 𝐺𝐼! = ! + ! 𝐿!" 𝐼! 𝜋 𝐸𝐼!

NO zg

!.!

𝛭!" = 𝐶!

𝜋 ! 𝐸𝐼! (𝑘𝐿!" )!

𝑘 𝑘!

!

𝐼! (𝑘𝐿!" )! 𝐺𝐼! + + 𝐶! 𝑧! 𝐼! 𝜋 ! 𝐸𝐼!

!

−   𝐶! 𝑧!

Page 197

Alternative method to calculate the Mcr and λLT

Non-dimensional slenderness

! !!

= 1.0(conservative value)

𝑈 = 0.9(conservative value)

𝑉 = 1.0

(conservative value)

𝜆! =

𝑘𝐿 𝑖!

K=1.0 for beams k=1.0 for free cantilever k=0.9 for lateral restraint to top flange k=0.8 for torsional restraint k=0.7 for lateral and torsional restraint

Simply supported rolled I, H and C section

𝝀𝑳𝑻 =

𝟏 𝑪𝟏

𝑼𝑽𝝀𝒛 𝜷𝒘

βw = 1.0 (conservative value)

Page 198

Member combined bending and axial compression (EN1993-1-1,cl. 6.3.3)

Class 1 and 2

𝑵𝑬𝒅 𝝌𝒚 𝑵𝑹𝒌

+ 𝒌𝒚𝒚

𝜸𝑴𝟏

𝑵𝑬𝒅 𝝌𝒛 𝑵𝑹𝒌 𝜸𝑴𝟏

+ 𝒌𝒛𝒚

𝑴𝒚,𝑬𝒅 𝝌𝑳𝑻

𝑴𝒚,𝑹𝒌

𝑴𝒚,𝑹𝒌 𝜸𝑴𝟏

𝑴𝒛,𝑬𝒅 𝑴𝒛,𝑹𝒌

≤ 𝟏. 𝟎

𝜸𝑴𝟏

𝜸𝑴𝟏

𝑴𝒚,𝑬𝒅 𝝌𝑳𝑻

+ 𝒌𝒚𝒛

Class 3

+ 𝒌𝒛𝒛

𝑴𝒛,𝑬𝒅 𝑴𝒛,𝑹𝒌

≤ 𝟏. 𝟎

𝜸𝑴𝟏

Method 2:Interaction factor kij for members not susceptible to torsional deformations (Recommended by CYS NA EN 1993-1-1,cl.NA2.20 – Table B.1) Interaction Plastic cross-sectional properties Elastic cross-sectional properties Type of sections factors Class 1 and 2 Class 3 𝑵𝑬𝒅 𝑵𝑬𝒅 𝑪𝒎𝒚 𝟏 + 𝝀𝒚 − 𝟎. 𝟐 𝑪𝒎𝒚 𝟏 + 𝟎. 𝟔𝝀𝒚 𝝌𝒚 𝑵𝑹𝒌 /𝜸𝑴𝟏 𝝌𝒚 𝑵𝑹𝒌 /𝜸𝑴𝟏 I-sections kyy RHS-sections 𝑵𝑬𝒅 𝑵𝑬𝒅 ≤ 𝑪𝒎𝒚 𝟏 + 𝟎. 𝟖 ≤ 𝑪𝒎𝒚 𝟏 + 𝟎. 𝟔 𝝌𝒚 𝑵𝑹𝒌 /𝜸𝑴𝟏 𝝌𝒚 𝑵𝑹𝒌 /𝜸𝑴𝟏 kyz kzy

I-sections RHS-sections I-sections RHS-sections

0.6kzz

kzz

0.6kyy

0.8kyy

𝑪𝒎𝒛 𝟏 + 𝟐𝝀𝒛 − 𝟎. 𝟔 I-sections ≤ 𝑪𝒎𝒚 𝟏 + 𝟏. 𝟏𝟒

𝑵𝑬𝒅 𝝌𝒛 𝑵𝑹𝒌 /𝜸𝑴𝟏

kzz 𝑪𝒎𝒛 𝟏 + 𝝀𝒛 − 𝟎. 𝟐 RHS-sections ≤ 𝑪𝒎𝒛 𝟏 + 𝟎. 𝟖

𝑵𝑬𝒅 𝝌𝒛 𝑵𝑹𝒌 /𝜸𝑴𝟏

𝑵𝑬𝒅 𝝌𝒛 𝑵𝑹𝒌 /𝜸𝑴𝟏

𝑪𝒎𝒛 𝟏 + 𝟎. 𝟔𝝀𝒛

𝑵𝑬𝒅 𝝌𝒛 𝑵𝑹𝒌 /𝜸𝑴𝟏

≤ 𝑪𝒎𝒚 𝟏 + 𝟎. 𝟔

𝑵𝑬𝒅 𝝌𝒛 𝑵𝑹𝒌 /𝜸𝑴𝟏

𝑵𝑬𝒅 𝝌𝒛 𝑵𝑹𝒌 /𝜸𝑴𝟏

Page 199

Method 2:Interaction factor kij for members susceptible to torsional deformations (Recommended by CYS NA EN 1993-1-1,cl.NA2.20 – Table B.2) Interaction Plastic cross-sectional properties Elastic cross-sectional properties factors Class 1 and 2 Class 3 kyy Kyy from Table B.1 Kyy from Table B.1 kyz Kyz from Table B.1 Kyz from Table B.1 𝟏−

𝟎. 𝟏𝝀𝒛 𝑵𝑬𝒅 𝑪𝒎𝑳𝑻 − 𝟎. 𝟐𝟓 𝝌𝒛 𝑵𝑹𝒌 /𝜸𝑴𝟏

≥ 𝟏− kzz

𝟎. 𝟏 𝑵𝑬𝒅 𝑪𝒎𝑳𝑻 − 𝟎. 𝟐𝟓 𝝌𝒛 𝑵𝑹𝒌 /𝜸𝑴𝟏 for𝜆! < 0.4:

𝟏−

𝟎. 𝟎𝟓𝝀𝒛 𝑵𝑬𝒅 𝑪𝒎𝑳𝑻 − 𝟎. 𝟐𝟓 𝝌𝒛 𝑵𝑹𝒌 /𝜸𝑴𝟏

≥ 𝟏−

𝟎. 𝟎𝟓 𝑵𝑬𝒅 𝑪𝒎𝑳𝑻 − 𝟎. 𝟐𝟓 𝝌𝒛 𝑵𝑹𝒌 /𝜸𝑴𝟏

𝒌𝒛𝒚 = 𝟎. 𝟔 + 𝝀𝒛 𝟎. 𝟏𝝀𝒛 𝑵𝑬𝒅 ≤𝟏− 𝑪𝒎𝑳𝑻 − 𝟎. 𝟐𝟓 𝝌𝒛 𝑵𝑹𝒌 /𝜸𝑴𝟏

Page 200

Summary design of steel member in bending

Design step

Choose yield strength of section, fy from table 3.1 in EN 1993-1-1

Get starinε from table 5.2 in EN 1993-1-1

Results

fy

S275 S355

ε

Substitute the value of εinto the class limits in table 5.2 to work out the class of the flange and web

Flange Class

Take the latest favourable class from the flange outstand, web in bending and web in compression results

Overall Section Class

Use the required value of W for the defined class to work out Mc,Rd

Steel grade

Web class

Mc,Rd

𝜀=

fy (N/mm2) Nominal thickness of element t (mm) t≤16 16≤t≤40 40≤t≤63 63≤t≤80 275 265 255 245 355 345 335 325

235 𝑓!

fy ε

235

275

355

420

1.00

0.92

0.81

0.75

Flange under compression: Web under pure bending:

c=(b-tw-2r)/2 c/tf c=(h-2tf-2r) c/tw

Class 1 or 2 Class 3 Class 4

Mc,Rd = Mpl,Rd = Wpl,yfy/γM0

Class 1 & 2

Mc,Rd = Mel,Rd = Wel,minfy/γM0

Class 3

Mc,Rd = Weff,minfy/γM0

Class 4

Cross-section Resistance check

Page 201

Summary design of steel member in shear

Design step

Calculate the shear area of the section, Av

Results

Av

Steel grade S275 S355

Calculate the design plastic shear resistance, Vpl,Rd

Vpl,Rd

Shear resistance check

VEd≤Vc,Rd

fy (N/mm2) Nominal thickness of element t (mm) t≤16 16≤t≤40 40≤t≤63 63≤t≤80 275 265 255 245 355 345 335 325

𝑉!".!" =

𝐴! (𝑓! / 3) 𝛾!!

Page 202

Summary of buckling resistance in bending

Design step

Calculate the design bending moment and shear

Section classification

Calculate critical length

Calculate Critical moment

Results

MEd &VEd

Wy&fy

Lcr

Mcr

Calculate non-dimensional slenderness λLT

λLT

Calculate imperfection factor αLT

αLT

Calculate reduction factor φLT

φLT

Calculate modified/reduction factor for lateral-torsional buckling χLTorχLT,mod

Calculate buckling resistance Mb,Rd

Buckling resistance check

χLTχLT,mod

Mb,Rd

𝑴𝑬𝒅 ≤ 𝟏. 𝟎 𝑴𝒃,𝑹𝒅

Page 203