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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
© 2010 The Steel Construction Institute
BS 5950 – simplified approach
2
© 2010 The Steel Construction Institute
BS 5950 – more exact approach
© 2010 The Steel Construction Institute
BS 5950 – more exact approach
© 2010 The Steel Construction Institute
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
© 2010 The Steel Construction Institute
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
© 2010 The Steel Construction Institute
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)
© 2010 The Steel Construction Institute
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
© 2010 The Steel Construction Institute
Columns in simple construction
From access-steel.com
© 2010 The Steel Construction Institute
Annex B, Cm factors
is the ratio of the bending moment at the ends
© 2010 The Steel Construction Institute
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
© 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
Example N-M
• Classify – Class 1
• Cross section - OK
(c) 2010 The Steel Construction Institute
Flexural resistance
Major axis
L/iy = 5000/88.2 = 56.7
1 = 76
Minor axis
75.0767.56
(c) 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
Flexural resistance
Major axis
L/iy = 5000/88.2 = 56.7
1 = 76.4
Curve ‘b’
Minor axis
74.04.767.56
(c) 2010 The Steel Construction Institute
(c) 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
(c) 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
Choice of curve
h/b = 203/203 = 1.0
(c) 2010 The Steel Construction Institute
(c) 2010 The Steel Construction Institute
Reduction factor,
86.0LT
LT
11
(c) 2010 The Steel Construction Institute
Correction factor, kc
752.033.1
1
033.033.1
1c
k
(c) 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
Equivalent moment factors
(c) 2010 The Steel Construction Institute
Equivalent moment factors
• Major axis
• = 0
• Cmy = CmLT = 0.6
• Minor axis
• = 0
• Cmz = 0.6
(c) 2010 The Steel Construction Institute
kyy factor
• Table B2 refers to Table B1
13
(c) 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
kyz factor
kyz = 0.6 kzz = 0.6 × 1.19 = 0.71
(c) 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
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
(c) 2010 The Steel Construction Institute
Comments
• At least its not Annex A!
• As a minimum, use a spreadsheet!
TM
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