Combined Axial and Bending

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TM

1

Combined axial compression and

bending

© 2010 The Steel Construction Institute

Combined compression and bending

Two philosophies

1. Interaction method, 6.3.3

2. General method, 6.3.4


BS 5950 – simplified approach

2


BS 5950 – more exact approach


BS 5950 – more exact approach


EC3 Interaction method, 6.3.3

1

M1

Rkz,

Edz,Edz,yz

M1

Rky,LT

Edy,Edy,yy

M1

Rky

Ed

M

MMk

M

MMk

N

N(Eq 6.61)

(Eq 6.62)1

M1

Rkz,

Edz,Edz,zz

M1

Rky,LT

Edy,Edy,zy

M1

Rkz

Ed

M

MMk

M

MMk

N

N

Edz,Edy, ; MM Are the additional moments due to the shift of neutral axis for Class 4 sections

3


Some odd terms

Rky N

Rkz N

Rky,LTM

Rkz,M

yy fA Major axis flexural buckling resistance

yz fA Minor axis flexural buckling resistance

yyLT fW LTB resistance

yz fW Minor axis bending resistance


EC3 Interaction method, 6.3.3

1

M1

yz

Edz,yz

Rdb,

Edy,yy

Rdy,b,

Ed

fW

Mk

M

Mk

N

N(Eq 6.61)

1

M1

yz

Edz,zz

Rdb,

Edy,zy

Rdz,b,

Ed

fW

Mk

M

Mk

N

N(Eq 6.62)


The k coefficients

Taken from Annex A or B (NA may decide)

UK NA says…. use either

SCI says…..use Annex B• “It looks easier”

M1Rkc,y

Edmy 8.01

N

NC

4


Columns in simple construction

From access-steel.com


Annex B, Cm factors

is the ratio of the bending moment at the ends


A simplified approach

I Struct E Technical Note, November 2008

85.0Rkz,

Edz,mz

Rdb,

Edy,my

Rdy,b,

Ed M

MC

M

MC

N

N

78.078.0Rkz,

Edz,mz

Rdb,

Edy,

Rdz,b,

Ed M

MC

M

M

N

N

5


A much simplified approach

In the ISE Manual

78.0Rkz,

Edz,mz

yRd,b,

Edy,

zRd,b,

Ed M

MC

M

M

N

N

Combined Bending and Axial

Numerical demonstration

(c) 2010 The Steel Construction Institute

590 kN

590 kN30 kNm

Major axis

2 kNm

Minor axis

203 UKC 46 S3555 m long

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TM

6


Example N-M

• Classify – Class 1

• Cross section - OK


Flexural resistance

Major axis

L/iy = 5000/88.2 = 56.7

1 = 76

Minor axis

75.0767.56


Which curve?• Table 6.2, page 58

Buckling curve

Cross section Limits Buckling

about axis S 275

S 355

t f 40 mm y – y

z – z

a

b

h/b

1

,2

40 mm < t f 100 y – y

z – z

b

c

t f 100 mm y – y

z – z

b

c Ro

lled

sec

tio

ns

hy y

z

zb

tf

h/b

1

,2

t f > 100 mm y – y

z – z

d

d

7


Flexural resistance

Major axis

L/iy = 5000/88.2 = 56.7

1 = 76.4

Curve ‘b’

Minor axis

74.04.767.56



Flexural resistance

Major axis

L/iy = 5000/88.2 = 56.7

1 = 76.4

Curve ‘b’

= 0.75

Nb,y,Rd = 0.75 × 5870 × .355

Nb,y,Rd = 1563 kN

Minor axis

74.04.767.56

8


Flexural resistance

Major axis

L/iy = 5000/88.2 = 56.7

1 = 76.4

Curve ‘b’

= 0.75

Nb,y,Rd = 0.75 × 5870 × .355

Nb,y,Rd = 1563 kN

Minor axis

L/iz = 5000/51.3 = 97.5

1 = 76.4

74.04.767.56 28.14.76

5.97


Which curve?• Table 6.2, page 58

Buckling curve

Cross section Limits Buckling

about axis S 275

S 355

t f 40 mm y – y

z – z

a

b

h/b

1

,2

40 mm < t f 100 y – y

z – z

b

c

t f 100 mm y – y

z – z

b

c Ro

lled

sec

tio

ns

hy y

z

zb

tf

h/b

1

,2

t f > 100 mm y – y

z – z

d

d


Flexural resistance

Major axis

L/iy = 5000/88.2 = 56.7

1 = 76.4

Curve ‘b’

= 0.75

Nb,y,Rd = 0.75 × 5870 × .355

Nb,y,Rd = 1563 kN

Minor axis

L/iy = 5000/51.3 = 97.5

1 = 76.4

Curve ‘c’

74.04.767.56 28.14.76

5.97

9



Flexural resistance

Major axis

L/iy = 5000/88.2 = 56.7

1 = 76.4

Curve ‘b’

= 0.75

Nb,y,Rd = 0.75 × 5870 × .355

Nb,y,Rd = 1563 kN

Minor axis

L/iy = 5000/51.3 = 97.5

1 = 76.4

Curve ‘c’

= 0.40

Nb,z,Rd = 0.40 × 5870 × .355

Nb,z,Rd = 834 kN

74.04.767.56

y 28.14.765.97

z


LTB resistance

Triangular BMD

From NCCI, C1 = 1.77

kNm1.3465000

500077.1

z2

t2

z

w2z

2

cr EI

GI

I

IEIM

71.0101.346

355104976

3

cr

yyLT

M

fW

10


Choice of curve

h/b = 203/203 = 1.0



Reduction factor,

86.0LT

LT

11


Correction factor, kc

752.033.1

1

033.033.1

1c

k


a little more resistance

• 6.3.2.3(2)

fLT

modLT,

2LTc 8.00.2115.01 kf

88.08.071.00.21752.015.01 2 f


a little more resistance

• 6.3.2.3(2)

fLT

modLT,

97.088.0

86.0modLT,

kNm171100.1

3551049797.0 63

b.Rd M

12


Equivalent moment factors


Equivalent moment factors

• Major axis

• = 0

• Cmy = CmLT = 0.6

• Minor axis

• = 0

• Cmz = 0.6


kyy factor

• Table B2 refers to Table B1

13


kyy factor

M1Rky

Edmy

M1Rky

Edymyyy /

8.01/

)2.0(1

N

NC

N

NCk

72.01/)1563(

590)2.074.0(16.0

78.01/)1563(

5908.016.0

So kyy = 0.72


kzz factor

So kzz = 1.19

M1Rkz

Edmz

M1Rkz

Edzmzzz /

4.11/

6.021

N

NC

N

NCk

43.11/)834(

5906.0)28.12(16.0

19.11/)834(

5904.116.0


kzy factor

So kzy = 0.80

)/(25.0

1.01

)/(25.0

1.01

M1Rkz

Ed

mLT

M1Rkz

Ed

mLT

zzy

N

N

C

N

N

Ck

74.0)1/834(

590

25.06.0

28.11.01

80.0)1/834(

590

25.06.0

1.01

14


kyz factor

kyz = 0.6 kzz = 0.6 × 1.19 = 0.71


Verification (6.61)

0.1/)/(/ M1z.Rk

z.Edz.Edyz

M1y.RkLT

y.Edy.Edyy

M1Rky

Ed

M

MMk

M

MMk

N

N

52.01/9.81

271.0

)1/171(

3072.0

1/)1563(

590


Verification (6.62)

0.1/)/(/ M1z.Rk

z.Edz.Edzz

M1y.RkLT

y.Edy.Edzy

M1Rkz

Ed

M

MMk

M

MMk

N

N

88.01/9.81

219.1

)1/171(

3080.0

1/)834(

590

So the column is satisfactory!

15


Comments

• At least its not Annex A!

• As a minimum, use a spreadsheet!

TM

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Combined Axial and Bending

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