DESIGN OF COMPRESSION MEMBERS SUBJECTED TO AXIAL FORCE AND MOMENT

1 Length of the Column L = 4

2 Type of Structural Member Hot Rolled Wide Flange section W 310 X 310 X 226

3 Axial Load P = 2500

4 Load Factor 1

5 Factored Axial Load = 2500

6 Moment about 'y' axis at Top = 100

7 Moment about 'y' axis at Bottom = 0

8 Moment about 'z' axis at Top = 350

9 Moment about 'z' axis at Bottom = -350

10 Factored Moment about 'y' axis at Top = 100

11 Factored Moment about 'y' axis at bottom = 0

12 Factored Moment about 'z' axis at Top = 350

13 Factored Moment about 'z' axis at Bottom = -350

14 Yield Stress of Steel = 250

15 Grade of Steel fe 410

16 Section Properties of Hot Rolled Wide Flange section W 310 X 310 X 226

= 317

Outstanding width of Flange b =bf/2 158.5

H = 348

= 35.6

= 22.1

Depth of Web d = 246.8

Moment of Inertia about 'zz' azis = 595600000

Moment of Inertia about 'yy' azis = 189300000

Radius at Root R = 15

Radius of gyration about 'z' = 143.6

Radius of gyration about 'y' = 81

Elastic/Yield Section Modulus about z axis = 3423000

Elastic/Yield Section Modulus about y axis = 1194000

Plastic Section Modulus about z axis = 3948812

Plastic Section Modulus about y axis = 1822502

Gross area of the section = 28880

Youngs Modulus E = 200000

Poisons Ratio µ = 0.3

Shear Modulus G =E/2(1+µ) 76923.077

Partial Factor of Safety of Material 1.1

16 Cross Section Classification

ε = 1

Flange Classification = 4.452

Pf

My

My

Mz

Mz

My

My

Mz

Mz

fy

bf

tf

tw

Iz

Iy

rz

ry

Zez

Zey

Zpz

Zpy

Ag

γmo

b/tf

Hence, the flanges are Class I (Plastic)

Web Classification = 11.167

Hence, the web is Class I (Plastic)

The overall cross section is classified as Class I (Plastic)

For Plastic & Compact Sections = 1.00

for semi compact sections =

17 Compression Resistance of the Cross Section

= 6563.64

> 2500

Hence, the design Compression Resistance is Alright

18 Bending Resistance of the Cross Section (As per Cl. 8.2.1.2 of IS 800-2007)

= 897.46

> 350

Hence, OK

= 414.21

> 100

Hence, OK

19 Shear Resistance of the Cross Section (As per Cl. 8.4.1 of IS 800-2007)

=

= 175

Area of web = 7690.8

= 1009.16

> 175

Hence OK

= 25

Area of web = 22570.4

= 2961.60

d/tw

βb

βb Ze/Zp

The design Compression resistance of the cross section Nd Ag.fy/γmo

a) The Design Major axis bending Resistance of the cross section about major 'z'z' axis

Mdz βb*Zpz*fy/γmo

b) The Design Major axis bending Resistance of the cross section about major 'z'z' axis

Mdy βb*Zpy*fy/γmo

The design bending resistance of the section is fine along both major z-z axis and minor y-y axis. Hence, the section is OK.

The design Plastic shear resistance of the cross section Vp (Av*(fyw/(3)0.5))/γmo

Maximum Shear Force (Load parellel to web i.e., about z-z) Vz

Av H*tw

For rolled sections load applied parallel to the web, the shear force is Vpz

Maximum Shear Force (Load parellel to flanges i.e., about y-y) Vy

Av 2*bf*tf

For rolled sections load applied parallel to the flanges, the shear force is Vpy

> 25

Hence OK

Hence, the Shear is alright

20 Shear Buckling of the Cross Section (As per Cl. 8.4.2 of IS 800-2007)

= 11.17

Hence, no shear buckling check is required

21 Cross Section resistance check (As per Cl. 9.3.1 of IS 800-2007)

= 616.75

> 350.00

n = 0.381

392.40

> 100.00

22

For I and H sections, =5n 1.904

= 2

0.396

Hence, OK

23 Member Buckling Resistance in compression (As per Cl. 7.1.2 of IS 800-2007)

Nd =

=

Φ =

λ =

=

for un-stiffned webs, the shear buckling need not be considered if d/tw is lessthan 67ε

d/tw

The Shear Force is less than 60% of the designed plastic shear resistance and hence, the cross section needs to be checked for bending and axial force only

Reduced Plastic Moment of Resistance of the section about 'z-z' axis for Rolled I or H sections

Mndz

1.11*Mdz(1-n)<Mdz

(if n > 0.2)

N/Nd

Reduced Plastic Moment of Resistance of the section about 'y-y' axis for Rolled I or H sections Mndy =1.56*Mdy(1-n)(n+0.6)

<Mdz

(if n > 0.2)

Hence, the Moment of Resistance of the section for major axis 'z-z' and minor axis'y-y' are alright

Cross Section check for biaxial bending with reduced moment resistances (As per Cl. 9.3.1.1 of IS 800-2007)

In the design members subjected to combined axial force (tension or compression) and Bending moment,should satisfy the relation = (My/Mndy)α1+(Mz/Mndz)α2 < 1.0

α1

α2

Ac*fcd

fcd (fy/γmo)/(Φ+(Φ2-λ2)0.5)

0.5(1+α(λ-0.2)+λ2)

(fy/fcr)0.5

fcr π2*E/(KL/r)2

=0.65*L 2.6

=1.0*L 4.0

= 6026.17

= 0.2037

= 810.08

= 0.5555

for Hot Rolled I & H sections = 1.10

= 35.60

Select Imperfection Factor (α) from table 7 of IS 800-2007

Imperfection Factor for curve 'b' = 0.34

Imperfection Factor for curve 'c' = 0.49

Buckling Resistance about 'z-z' axis

= 0.5214

= 0.9987

= 6555.08

> 2500

Hence, OK

Buckling Resistance about 'y-y' axis

= 0.7414

= 0.8114

= 5325.87

> 2500

Hence, the Buckling Resistance of the section about both major & minor axis is fine

24 Member Buckling Resistance in bending (As per Cl. 8.2.2 of IS 800-2007)

=

=

For buckling about the major axis "z-z', from Table 11 of IS 800-2007 KLz

Note: If stiffness of the beams are known, the effective length should be calculated by using Wood's curves.

For buckling about the major axis "y-y', from Table 11 of IS 800-2007 KLy

fcr,z π2*E/(KLz/rz)2

λz (fy/fcr,z)0.5

fcr,y π2*E/(KLy/ry)2

λy (fy/fcr,y)0.5

Selection of Buckling curve from table 10 of IS 800-2007

H/bf

tf

From Table 10, for buckling about zz axis, use curve 'b' & for buckling about yy axis, use curve 'c'

αb

αc

Pd,z

Φz 0.5(1+αb(λz-0.2)+λz2)

Xy (1/(Φz+(Φz2-λz

2)0.5)

Pd,z Xy*fy*Ae/γmo

Pd,y

Φy 0.5(1+αc(λy-0.2)+λy2)

Xz (1/(Φy+(Φy2-λy

2)0.5)

Pd,y Xz*fy*Ae/γmo

As the member is 4 m long and is unsupported along its length with no torsional or lateral restraint. Ewqual and opposite design end moments of 350 KN.M are applied about the major axis 'z'z'. Hence, the full length of the column must be checked for lateral torsional buckling.

Md βp*Zp*fbd

Mcr C1[(π2EIyh)/2(Kly)2)](1+(1/20)[(Kly/ry)/(h/tf)2]0.5

ψ = -1

K = 1

= 2.752

= 9.775

= 16884.83

= 0.2418

For rolled steel sections = 0.21

= 0.5336

= 0.9908

Lateral Torsional Buckling Resistance = 889.17

= 0.3936

Hence, OK

25

=

&

=

= 0.4694

= 0.3814

= 1.1669

1.3755

= 1.0014

1.3051

= 0.9243

0.6871

-1

0

For equal and opposite end moments, the values of ψ, K and C1 from table 42 of IS 800-2007 are

C1

H/tf

Mcr

Non Dimensional lateral torsional slenderness Ratio λLT (βp*Zpz*fy/Mcr)0.5

αLT

Reduction factor for lateral torsional buckling XLT

ΦLT 0.5(1+αLT(λLT-0.2)+λLT2)

XLT (1/(ΦLT+(ΦLT2-λLT

2)0.5)

Md XLT*βp*Zpz*fy/γmo

M/Md

Member Buckling Resistance in Combined bending and axial compression & Determination of moment amplification factors (As per Cl. 9.3.2.2 of IS 800-2007)

Members subjected to combined axial compression and biaxial bending shall satisfy the Interaction relationships given as

(P/Pdy)+(KyCmyMy/Mdy)+(KLTMz/Mdz)<1.0

(P/Pdz)+(0.6KyCmyMy/Mdy)+(KzCmzMz/Mdz)<1.0

Ratio of Applied axial force to the design axial strength for buckling about the y axis ny P/Pdy

Ratio of Applied axial force to the design axial strength for buckling about the z axis

nz P/Pdz

Ky 1+(λy-0.2)P/Pdy

< 1+0.8P/Pdy

Kz 1+(λz-0.2)P/Pdz

< 1+0.8P/Pdz

KLT 1-(0.1λLT ny)/(Cmlt-0.25)

> 1-(0.1ny)/(Cmlt-0.25)

Ratio of Moments at bottom & top about zz axis ψz =MZb /MZt or MZ2 /MZ1

Ratio of Moments at bottom & top about yy axis ψy =Myb /Myt or My2 /My1

0.6

0.2

Hence, the value is 0.4

0.4

= 0.9989

= 0.6390

Hence, the section is suitable to resist the design action affects.

Equivalent uniform moment factors obtained from Table 18 of IS 800-2007 which depends on the shape of the bending moment diagram between lateral bracing points in the appropriate plane of bending

Cmy =0.6+0.4ψy > 0.4

Cmz =0.6+0.4ψz > 0.4

Equivalent uniform moment factor for lateral torsional buckling as per Table 18 of IS 800-2007 corresponding to the actual moment gradient between lateral suports against torsional deformation in the critical region under consideration

Cmlt =

(P/Pdy)+(KyCmyMy/Mdy)+(KLTMz/Mdz)<1.0

(P/Pdz)+(0.6KyCmyMy/Mdy)+(KzCmzMz/Mdz)<1.0

DESIGN OF COMPRESSION MEMBERS SUBJECTED TO AXIAL FORCE AND MOMENT

m

Hot Rolled Wide Flange section W 310 X 310 X 226

KN

KN

KN.M

KN.M

KN.M

KN.M

KN.M

KN.M

KN.M

KN.M

mm

mm

mm

mm

mm

mm

mm

mm

mm

< 9.4ε

N/mm2

mm4

mm4

mm3

mm3

mm3

mm3

mm2

N/mm2

N/mm2

< 42ε

KN

KN

KN.M

KN.M

KN.M

KN.M

KN

KN

KN

KN.M

KN.M

The design bending resistance of the section is fine along both major z-z axis and minor y-y

mm2

mm2

KN.M

<67ε

KN.M

KN.M

>0.2

KN.M

KN.M

<1.0

Member Buckling Resistance in compression (As per Cl. 7.1.2 of IS 800-2007)

is lessthan 67ε

The Shear Force is less than 60% of the designed plastic shear resistance and hence, the

Hence, the Moment of Resistance of the section for major axis 'z-z' and minor axis'y-y' are

Cross Section check for biaxial bending with reduced moment resistances (As per

>1.0

m

m

<2

<100 mm

Select Imperfection Factor (α) from table 7 of IS 800-2007

KN

KN

KN

KN

Member Buckling Resistance in bending (As per Cl. 8.2.2 of IS 800-2007)

N/mm2

N/mm2

From Table 10, for buckling about zz axis, use curve 'b' & for buckling about yy axis, use

As the member is 4 m long and is unsupported along its length with no torsional or lateral restraint. Ewqual and opposite design end moments of 350 KN.M are applied about the major axis 'z'z'. Hence, the full length of the column must be checked for lateral torsional buckling.

)](1+(1/20)[(Kly/ry)/(h/tf)2]0.5

KN.M

KN.M

&

<1.00

Member Buckling Resistance in Combined bending and axial compression & Determination of moment amplification factors (As per Cl. 9.3.2.2 of IS 800-2007)

)+(KLTMz/Mdz)<1.0

)+(KzCmzMz/Mdz)<1.0

>0.4

<0.4

<1.0

<1.0

Hence, the section is suitable to resist the design action affects.