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4.Compression Members-1
4.0 AISC – Column Strength Curve
(ASD & LRFD)
1. ASD (F.S1.67)
For
KL
 Cc ,
r
Inelastic Buckling
 ( KL / r ) 2 
1  2C 2  Fy
c


Fa 
5 3( KL / r ) ( KL / r ) 3


3
8Cc
8Cc3
-(1)
Safety Factor
For
KL
 Cc ,
r
F .S 
Elastic Buckling
23
12 2 E
 1.92 Fa 
12
23( KL / r ) 2
-(2)
Where
Cc 
2 2 E
Fy
Corresponds to
Fcr
1
1
 2 
Fy C c 2
2. LRFD
Pn  Ag Fcr
For  c  1.5
Fcr  (0.658 )Fy
2
c
-
(3)
-
(4)
Inelastic Buckling
 0.877
Fcr   2  Fy
 c 
For  c  1.5
-
(5)
Elastic Buckling
Corresponds to
λc 
Fy
Fe

kl
rπ
Fy
E
c  1.5 
Fcr
1

 0.44
Fy 2.25
, slenderness parameter
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4.Compression Members-2
4.1 Euler’s Column Equation,Elastic Buckling
P
1
P
y
l
M=-EIy”
x
y
P
P
(a)
(b)
d2y
M
y"  2  
dx
El
 Ely"  Py
Ely" Py  0
2nd Order Homogeneous D.E.
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4.Compression Members-3
Let k 2 
P
EI
y" k 2 y  0
Assume y   e mx ,  y'   m e mx , y"   m 2 e mx
( m 2  k 2 ) e mx  0
 m   ki
y  1 e kix   2 e  kix
 A cos kx  B sin kx  two unknown constants
Boundary Condition
x  0, y  0 ; A  0
x  l , y  0 ; B sin kl  0
 kl  n (n 은 1,2,3,… )
( n ) 2
P
k 

2
l
EI
( n ) 2 EI
 Pcr 
l2
2
when n=1,
Pcr 
 2 EI
l2
 PE
Euler buckling load
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4.Compression Members-4
Mode shape
y  B sin kx  B sin(
1) n=1
Pcr  PE 
 2 EI
l2
Pcr
nx
 X )  B sin
EI
l
2)n=2
4 2 EI
Pcr 
 4 PE
l2
l
2
3)n=3
9 2 EI
Pcr 
 9 PE
l2
l
3
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4.Compression Members-5
Neutral Equilibrium.
P
P
N.E
U.E
Pcr  PE
S.E
x
Column with deficiencies
y
y
P
If there are some deficiencies such as
eccentricity, nonhomogeniety in material, we
can’t get bifurcation buckling load.
Pcr
 2 El  2 E
  cr 

A
Al 2
( l )2
r
cr ; critical stress
l/r ; slenderness ratio
cr
Invalid
Inverse Parabola
y
Euler Hyperbola
1/2y
0
Inelastic Bklg
Elastic Bklg
l/r
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4.Compression Members-6
Residual stress
residual comp. stress(50%)
residual tensile stress
Steel Industry
*Actually y is lower bound
of yielding strength of steel
ex)of
y=36 ksi
18 ksi may already exist as residual stress
Fcr
λ
Power function Fcr  (α c )Fy , α  const
2
Inverse Parabola
y = ax2+bx+c , in ASD
y
1/2y
kl/r
Inelastic Bklg
Cc 
Elastic Bklg
kl
1
kl
When σ cr  ; σ y  π 2 E/(kl/r)2   C c 
r
2
r
λc  1.5 means
λc 
Fy
Fe
2π 2 E
Fy
 1.5  Fe  0.44 F y
in practice, if a section is subject to
1/2y, it is assumed that some section is
subject to inelastic behavior.
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4.Compression Members-7
Effective Column Length
Euler’s equation is derived based on the
assumption that the ends of the column are pin
supported. When this condition is not met then
an equivalent length(effective length) is used.
KL denotes effective length
Where K is effective length factor and
L is actual length
P
P
KL=L
Theoretical
AISC
P
P
P
P
KL=0.7L
KL=0.5L
K=1
K=0.7
K=0.5
K=1
K=0.8
K=0.65
(Sidesway prevented, Braced)
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P
KL=2L
Theoretical K=2
AISC
K=2.1
4.Compression Members-8
P
KL=L
K=1
K=1.2
P
KL=2L
K=2
K=2.0
(Sidesway not prevented, Laterally unbraced)
No translation, free rotation
No translation, no rotation
Translation free, no rotation
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4.Compression Members-9
4.2 Inelastic Column Buckling Strength
without Any Imperfection.
 Tangent Modulus Theory, Pt 
 2 Et I
L2
 Reduced (Double) Modulus Theory, Pr 
Er 
 2 Er I
L2
E I 1  Et I 2
I
 Post Buckling Behavior (by Shanley, 1947)
Buckling commences at P  Pt
 After buckling, increase in stiffness due to
elastic unloading of some fibers in the sections.
 results in increase in load.
 without further yielding, P  Pr
 Further yielding
⇒ further decrease in stiffness
 P  Pm ax , Pt  Pmax  Pr
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4.Compression Members-10
B. Factors affecting column strength.
 Out-of-straightness.
 Eccentricity of axial load.
 Material nonlinearity.
 End restraints.
 Residual stress.
imperfections
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4.Compression Members-11
C. Design of Metal Columns as related
to Strength theories.
1. Emperical formulas based on column tests.
 not rational because they are entirely
dependent on test results.
 not rational in considering end restraint
(Earliest Column strength, since 1840s)
2. Formulas based on the yield limit state.
 The strength of a column is defined as the
load which will give an elastic stress for an
initially imperfect column, equal to the
yield stress. ( Large displ. theory )
 popular since 1850s upto the present.
(e.g.,British use of Perry-Robertson formula)
AASHTO ( 12th. Ed )
σ
σ all  max 
F.S.
σ y / F.S.

l
1   0.25 sec
2r

(σ a ) F.S. 


E

 lack of inelastic basis of column strength
can not rationally consider end restraint.
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4.Compression Members-12
3. Formulas based on the tangent–modulus theory.
 Two methods of treating the imperfections.
a. reduce
Pmax to Pt
emperically justified reduced column strength.
 basis for cold-formed columns and aluminum columns.
b. represented as flexural effects in the
interaction equation (SSRC).
 CRC-Column Strength Curve.(1960)
(CRC–Column
Research Council, former name of SSRC)
 cr
2
 1
y
4
and
 cr
1
 2
y

where
 
for  
for  
2
2
KL 1  y
r  E
 can accurately account for end restraints.
4. Formulas based on max. strength.
 Formulas based on a numerical fit of curves
obtained from max. strength analyses considering
imperfections and residual stresses.
 Modern trend.
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4.Compression Members-13
 SSRC Curves 1, 2, and 3 (1976)
(multiple column curves)
 Euro code 3 (ECS, 1994)
 Canadian Standard (CSA, 1994)
D. Effect of Residual Stresses.
1. Hot rolled shapes.
residual stress distribution depends on types of
sections, rolling temperature, cooling conditions,
straightening procedures, and metal properties.
t f  4.91
t w  3.07
heaviest
A  215 in2
d  22.42
b f  17.89
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4.Compression Members-14
※ Because of
this distribution
pattern of
residual stresses,
Pmax . ) minor  Pmax . ) major
for same
slenderness ratio.
Note when the
inelastic bklg.
begins depending
on the mag. of
flange tip
residual stresses.
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2. Welded Columns
4.Compression Members-15
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4.Compression Members-16
Poxigencut  Puniv. mill
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4.Compression Members-17
Pheavy  Plight
P stress  relieved  Poxygencut  Puniversalmill
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4.Compression Members-18
 Inelastic Behavior of Column.
 Column strength curves for H-shaped sections
having compressive residual stresses at
flange tips.
When
Max. comp.
residual stress
When
 0.3 Fy
F
 0.5 , it is
Fy
considered in most
design codes that some
parts of column are in
inelastic range
 inelastic buckling.
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4.Compression Members-19
․SSRC Multiple Column Curves
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4.Compression Members-20
Algebraic representations of the three column
strength curves were obtained by curve fitting,
and the resulting equations are given as Eqs. 3.13
through 3.15.
SSRC Curve 1
1. For
0    0.15
2. For
0.15    1.2
 u   y (0.9 9 0 0.1 2 2  0.3 6 72 )
3. For
1.2    1.8
 u   y (0.051  0.8012 )
4. For
1.8    2.8
 u   y (0.0 0 80.9 4 22 )
5. For
  2.8
u   y
 u   y 2 (  Euler curve)
(3.13)
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4.Compression Members-21
SSRC Curve 2
1.For
0    0.15
u   y
2.For
0.15    1.0
 u   y (1.035  0.202  0.222 2 )
3.For
1.0    2.0
 u   y (0.111  0.636 1  0.0872 )
4.For
2.0    3.6
 u   y (0.009 0.8772 )
5.For
  3 .6
 u   y 2 ( Euler curve)
( 3.14)
SSRC Curve 3
1.For
0    0.15
u   y
2.For
0.15    0.8
 u   y (1.093  0.622 )
3.For
0.8    2.2
 u   y (0.128  0.707 1  0.1022 )
4.For
2.2    5.0
 u   y (0.008 0.792 2 )
5.For
  5 .0
 u   y 2 ( Euler curve)
( 3.15)
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4.Compression Members-22
4.3 Theories of Inelastic Buckling
a. Double Modulus Theory (By Engesser, 1989)
ref. Text pp. 37~42 , Salmon’s pp. 303~306
(Reduced)
Inelastic Bklg. of straight Column
Assumptions
1. Small   material nonlinearity only
2. Plane sections remain plane
 Bernoullis’ Hypothesis
3. The relationship between stress-strain in any
longitudinal fiber is given by the stress strain diagram of the material.
(Comp. – Tension, Same Relationship)
4. The col. section is at least singly symmetric
and the plane of bending is a plane of
symmetry
5. The axial load remains constant as the member
moves from the straight to deformed position
※ This theory gives higher strength
than test results.(Salmon pp.304)
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4.Compression Members-23

Et
 cr
unloading path
P=PE
proportional
limit
E
E

permanent set
h2
h1
dx
R
d
ds≒dx
N.A
Z2
Z1
2
inside
cr
concave
S2
S1
composition
e
1
outside
concave
 1  E 1
 2  Et 2
E t ; the slope of stress-strain curve at    cr
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4.Compression Members-24
1 d 2 y d
small  theory 
Curvature   2 
R dx
dx
 1  Z1 y ,  2  Z 2 y
 1  Eh1 y ,  2  Et h2 y
or S1  EZ1 y , S2  Et Z 2 y

h1
0
S1 dA  
h2
0
S2 dA  0
--------- 
(pure bending only)

h1
0
S1 Z1  e  dA  
h2
0
S2 Z 2  e  dA  py
-------- 
( y measured from C.G.)
h1
from  Ey Z1 dA  Et y
0
h2
0
Z 2 dA  0 (C+T=0)
h1
let Q1   Z1dA
0
h2
Q2   Z 2 dA statical moment of area
0
about neutral axis
EQ1  E t Q2  0
-------- 
( Z 1  Z 2 both taken as +)
then the 2nd equil. eq. gives



y E 0 Z12 dA  Et 0 Z 22 dA  ey E 0 Z1 dA  Et 0 Z 2 dA  py
h1
h2
h1
h2
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4.Compression Members-25
EI1  Et I 2
I
Reduced modulus depends on the stress-strain
let E 
relationship of the material & the shape of the
cross-section.
E Iy  py  0 ; I : modulus of inertia about the C.G.
reduced pr 
 2 Er I
l2
introducing  r 
or  r
cr

 2 Er
l r 2
E
,   E t E  1.0
E
EI r y  py  0
r 
Et I 2 I 1
I
I
  2  1
E I
I
I
I
 r  pcr A 
 2 E r
l r 2
Procedures for determining
r
1) For    diagram  prepare    diagram.
2) From step 1) prepare  r   curve.
3) From step 2) prepare  r  l r curve.
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4.Compression Members-26
b. Tangent Modulus Theory
Assumption ; same as D.M. theory except #5.
the axial load increases during the
transition from straight to slight
bent position such that the increase
in average stress in compression is
greater than the decrease in stress
due to bending at the extreme fiber on
the convex side, i.e., no strain
reversal takes place on the convex
side. The compressible stress
increases at all point ; the tangent
modulus governs the entire crosssection.
P
bent

Et
P+ P
cr
E
P
1
r
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As P  P
4.Compression Members-27
P  P 
Et Iy  Py  0
Pt   2 Et I  l 2
 t  Pt A   2 E l r 
2
where   Et E
hence,  t  l r curve is not affected by crosssection shape.
 Procedure
1) From    diagram  Establish    curve
2) From step 1), prepare  t  l r curve
 This theory gives lower strength than tested
ultimate strength.
Note; Shanley Concept – True column behavior
PS
•
Pt
•
increment of load  ΔC  ΔT
Double Modulus
Neglect ps  pt , pt may be considered as p cr
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4.Compression Members-28
4.4 Design Specifications
AISC – Column Strength Curve (ASD
&
LRFD)
1. ASD (F.S1.67)
For
KL
 Cc ,
r
 ( KL / r ) 2 
1  2C 2  Fy
c


Fa 
5 3( KL / r ) ( KL / r ) 3


3
8Cc
8Cc3
-(1)
Safety Factor
For
KL
 Cc ,
r
23
12 2 E
F .S 
 1.92 Fa 
12
23( KL / r ) 2
-(2)
Where
Cc 
2 2 E
Fy
Corresponds to
Fcr
1 1
 2 
Fy c 2
2. LRFD
Pn  Ag Fcr
For  c  1.5
Fcr  (0.658 )Fy
For  c  1.5
 0.877
Fcr   2  Fy
 c 
2
c
Corresponds to c  1.5 
-
(3)
-
(4)
-
(5)
Fcr
1

 0.44
Fy 2.25
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4.Compression Members-29
● Comparison of AISC’s ASD
Strength
&
LRFD Column
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4.Compression Members-30
● Comparison of LRFD-equation(AISC) with
Test-results
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4.Compression Members-31
● Multiple Column Curves by G.Schulz
곡선 1 : 잔류응력
 r  0.2 y 와 편심량 f  l / 1000을 가정한 I 형강의
강축에 관한 곡선, 이음 없는 강관, 소둔한 상자형 단면에도 적
용.
곡선 2 :
 r  0.2 y 와
   곡선.
f  l / 1000을 가정한 I 형강의 약축에 관한
상자형 단면, 강축에 관한 각종 I 형 단면(압연,용
접)등 적용범위가 가장 넓다.
곡선 3 :
 r  0.4 y 와
f  l / 1000을 가정한 I 형강의 약축에 관한
   곡선.약축에
곡선 4 :
 r  0.5 y 와
관한 각종 I 형강, T 형강에도 적용.
f  l / 1000을 가정한 I 형강의 약축에 관한
   곡선.잔류응력이
큰 용접 I 형 단면(약축)에만 적용.
Steel Design I
Tel.(02) 3290-3317
Fax. (02) 921-5166
Prof. Kang,YoungJong
4.Compression Members-32
● 도로교시방서 허용압축응력(G.Schulz 하한치)
(1996)
Steel Design I
Tel.(02) 3290-3317
Fax. (02) 921-5166
Prof. Kang,YoungJong
● 도로교 설계기준(2000)
4.Compression Members-33
Steel Design I
Tel.(02) 3290-3317
Fax. (02) 921-5166
Prof. Kang,YoungJong
4.Compression Members-34
● 철도 설계기준 (철도교편, 1999)
Steel Design I
Tel.(02) 3290-3317
Fax. (02) 921-5166
Prof. Kang,YoungJong
● 도로교 설계기준(2005)
4.Compression Members-35
Steel Design I
Tel.(02) 3290-3317
Fax. (02) 921-5166
Prof. Kang,YoungJong
● 도로교 설계기준(2005)
4.Compression Members-36