11.1 Introduction
* column  axial compressive load (usually concentric or sometimes eccentric)
* classification of axial compression member
 short column or post
- relatively larger section compared to the length
- crushing = strength-based failure: compressive stress  yield strength or ultimate strength
 long column
- slender and long column
- initial bending(lateral displacement) & increase of compressive loading
(i) restoration to straight line due to elasticity: stable equilibrium
(ii) equilibrium in bent state: neutral equilibrium (critical load or buckling load)
(iii) severe bending and lost column function: buckling = stability-based failure(elastic buckling)
 intermediate column
- between post and long column
- stability-based failure(inelastic buckling): (elastic limit of material) < (compressive stress) < (yield stress)
* function of structural members --- failure criteria
 strength --- yielding
 stiffness --- excessive deformation
 stability --- buckling
* objective of the chapter
- compressive stress of post
- buckling stress of long column
- experimental/empirical formulae of intermediate column
11.2 Short Columns with Eccentric Load
11.2.1 Axial Stress
* post under eccentric load (e = eccentricity)
- decomposition

P Pex

x
A Iy
- max/min stress
P Pex

c1
A Iy
P Pe
  B   x c2
A Iy
 max   D 
 min
- neutral axis (   0 )
x0  
Iy
Aex
* general eccentricity: K ( ex , e y )
- compressive stress

Pe
P Pex

x y y
A Iy
Ix
- neutral axis (   0 ): 1 
 intersects: x0  
Iy
Aex
,
Ae
Aex
x y y 0
Iy
Ix
y0  
Ix
Ae y
- tensile stress at shadow area(outside of NA)
11.2.2 Core of Section
* core = area of eccentric-load application without occurrence of tensile stress in a section
 core = location of K ( ex , e y ) with   0
1
e
ex
x  2y y  0 ;
2
ry
rx
radius of gyration: rx 
Ix
,
A
ry 
Iy
A
[<Example 11.2.1>] Core of circular section
- Assuming K (ex , 0) , then x0  r
r  
Iy
Aex

r 4 / 4
r 2 e x
 ex  e0  14 r
[<Example 11.2.1>] Core of rectangular section
- radius of gyration
I
I
h2
b2
, ry2  y 
rx2  x 
A 12
A 12
- Assuming K ( ex , e y ) :
1
- extreme points :  12 b  x  12 b ,
A(  12 b,  12 h)
B( 12 b,  12 h)
C( 12 b, 12 h)
D(  12 b, 12 h)
6
6
1  ex  e y
b
h
6
6
1  ex  e y
b
h
6
6
1  ex  e y
b
h
6
6
1  ex  e y
b
h
 12 h  y  12 h
0
(under the line mn)
0
(under the line pn)
0
(above the line pq)
0
(under the line qm)
12 x
12 y
ex  2 e y  0
2
b
h
1
e
ex
x  2y y  0
2
ry
rx
11.3 Long Columns with Concentric Load : Euler’s Formula
* 1757, leonhard Euler(Switzerland, 1707-1783)
* ideal column: no bending or lateral deformation
- prismatic(constant section), homogeneous, straight line, axial compressive loading
* real column – bending or lateral deformation due to initial imperfection or eccentricity
- defect of material, error in manufacturing, initial crookedness of axis, mistake
* buckling of real column
- buckling load or critical load: Pcr
Pcr
A
- possibility of buckling: (column length, L) > 10(minimum dimension of section)
or, (column length, L) > 30(minimum radius of gyration of section, r)
L
I
- slenderness ratio: k  ; r  min
r
A
- buckling stress or critical stress:  cr 
* Euler’s formula for the buckling load(stress) of ideal long column under concentric axial load
L
- slenderness ratio [ k  ]
r
- material property [ E ]
- end conditions [ free – hinge – fix ]
Hinged-Hinged, Free-Fixed, Fixed-Fixed, Hinged-Fixed
** Lateral deformation occurs in plane of minimum flexural stiffness (EI)min, or
the bending deformation occurs around the axis of minimum inertia moment Imin.
11.3.1 Hinged-Hinged Columns
Fig. (c): bending moment: M  Py
- DE for elastic curve: EIy    M   Py
P
 y   2 y  0 ,  2 
EI
- BC for DE: y (0)  0 , y ( L)  0
- general solution: y  C1 sin x  C2 cosx
- BC: C2  0 , C1 sin L  0
non-trivial solution  sin L  0
n 2 2 EI
 L  n , or P 
; n  1, 2,
L2
- buckling mode: y n  C1 sin
* smallest critical load and fundamental mode (n = 1 ): Pcr 
 2 EI
2
L
, y1  C1 sin
nx
; n  1, 2,
L
x
L
* behaviour of column and critical load
 P  Pcr  state of straight line and stable equilibrium
 P  Pcr  state of straight line or infinitesimal bending and neutral equilibrium
 P  Pcr
 state of straight line and unstable equilibrium, or buckling
* modes of buckling
Pcr  2 E I  2 Er 2  2 E
 2

 2
A
L A
L2
k
- generally,  cr   pl : less than proportional limit of material  elastic buckling
- Elastic buckling occurs only for very large slenderness ratio (k), or long column.
* buckling stress:  cr 
11.3.2 Free-Fixed Columns
- Let  = lateral displacement at free end
- bending moment: M  P ( y   )
- DE : EIy    M  P(  y )
 y   2 y   2
- BC: y (0)  0 , y (0)  0 , y (L)  
- general solution:
y  C1 sin x  C2 cosx  
- BC @ x = 0: C1  0 , C2  
 y   (1  cos x )
- BC @ x = L & non-trivial solution  cosL  0
n
n 2 2 EI
 L 
, or P 
; n  1, 3, 5,
2
4L2
nx 

- buckling mode: yn   1  cos
 ; n  1, 3, 5,
2L 

* smallest critical load and fundamental mode (n = 1 ): Pcr 
 2 EI
x 

, y1   1  cos 
2L 
( 2 L)

2
* effective length, Le
= equivalent length of column with various end conditions
corresponding to the hinged-hinged column
= the distance between inflection points
in the fundamental mode
* free-fixed column: Le  2 L
11.3.3 Fixed-Fixed Columns
- symmetric w.r.t. x  12 L
 zero horizontal reaction at both
ends
- bending moment: M  Py  M A
- DE : EIy    M   Py  M A
M 
 y    2 y   2  A 
 P 
- general solution:
y  C1 sin x  C2 cos x 
- BC: y (0)  y (0)  0
M
 y  A (1  cos x )
P
- BC:
y (0)  y (0)  0 &
non-trivial solution
MA
P
 cosL  1
4n  EI
; n  1, 2,
L2
M 
2nx 
- buckling mode: yn  A 1  cos
 ; n  1, 3, 5,
P 
L 
 L  2n , or P 
2
2
* smallest critical load and fundamental mode (n = 1 ): Pcr 
 2 EI
(0.5L)
2
, y1 
MA 
2x 
1  cos

P 
L 
* fixed-fixed column: Le  0.5L
11.3.4 Hinged-Fixed Columns
- H = horizontal reaction at supports  M A  HL
- bending moment: M  Py  M A  Hx
- DE : EIy    M   Py  H ( L  x )
H
 y    2 y   2  ( L  x )
P
- general solution:
H
( L  x)
P
- BC: y (0)  y (0)  y ( L)  0
HL
H
 C2  
, C1 
; C1 tan L  C2  0
P
P
L  4.4934
y  C1 sin x  C2 cos x 
- non-trivial solution: L  tan L

* smallest critical load and effective length: Pcr 
20.19 EI
 2 EI
,

L2
(0.7 L) 2
Le  0.7 L
11.3.5 Effective Length Factors
* dependency of buckling load of long column on end conditions  effective length of column
* effective length factor (K): Le  KL
End conditions
K
Fixed-Fixed
0.5
Fixed-Hinged
0.7
Hinged-Hinged
1
Fixed-Free
2
* general form of Euler’s formula
 2 EI
- buckling load: Pcr 
(KL) 2
- buckling stress:  cr 
Pcr
 2E
 2E


A ( KL / r ) 2 ( Kk ) 2
* Euler’s buckling = elastic buckling
 effective when  cr   pl
 Kk  
E
 pl
- sometimes,  pl  12  y
- Effective slenderness ratio (Kk) for the long column depends on material (E and pl).
 structural steel: Kk  100
 wood: Kk  80
cf: short column [crushing failure, compression failure]: Kk < 2060
[Example<11.3.1>] Allowable load of long column
proportional limit  pl  2.1 t / cm2 , yield stress  y  2.8 t / cm2 , elastic modulus E  2.1 103 t / cm 2 ,
length L  10 m , rectangular section 30  20 cm , fixed-hinged ends, FS = 2.0
I min 

(30)( 20)3
 20,000 cm 4 ,
12
E
 pl
r
I min
20,000

 5.774 cm ,
A
600
 99.3  Kk (long column) Euler’ formula:  cr 
 1,000 
Kk  0.7
  121.2
 5.774 
 2E
( Kk ) 2
 allowable stress and allowable axial load:  a   cr / FS  0.705 t / cm 2 ,
 1.410 t / cm 2
Pa   a A  423 t
11.4 Long Columns with Eccentric Load : Secant Formula
- Hinged-Hinged column
- e = eccentricity
- bending moment: M  P( e  y )
- DE : EIy    M   P(e  y )  y   2 y   2 e
- general solution:
y  C1 sin x  C2 cosx  e
- BC: y (0)  y ( L)  0
L


sin x  cos x  1
 y  e tan
2


L 

 1
- max deflection at x  12 L : y max    e sec
2


* discussions
 non-linear relations between P (or ) and   no superposition for several eccentric loads
 linear relations between   e

 2 EI 
  P  Pcr  2   L      ym a x   
L 

* maximum compressive stress in the column
 max moment at x  12 L : M max  P( e   )  Pe sec
 max compressive stress:  max
* secant formula:  max 
L
2
P M max
P Pec
L
 
c 
sec
A
I
A
I
2
P  ec  L
1  sec
A  r 2
 2r
P 

EA 
 The maximum stress,  max , is a function of the mean stress,
and two non-dimensional factors; the eccentricity ratio,
 r  0  short column under eccentric load
 e  0  P  Pcr  , Euler’s buckling load
* secant formula considering end conditions:
P

A
P
,
A
ec
L
, and the slenderness ratio,
.
2
r
r
 max
1
ec  KL P 

sec

r2
2
r
EA


 The secant formula can be applied in post, intermediate column, and long column.
 analysis procedure of eccentrically-loaded column [p340]
11.5 Empirical and Design Formulae
* estimation of allowable load
 Kk  160 ~ 200  Euler’s formula
 intermediate column or short column  empirical formulas
* empirical formulae: material, slenderness ratio, resulting load(allowable or critical?)
* failure criteria
 short column: yielding of material (  y )
 long column: effective slenderness ratio ( Kk ), modulus of elasticity ( E )
 intermediate column: all (  y , Kk , E )
11.5.1 Concentrically-Loaded Cases
*  cr - Kk relations for perfect column
 D-G:  cr   y (crushing failure)
 G-B-C: Euler’s stress (elastic buckling failure)
 D-G-B: empirical formula used
 dotted line D-B
** linear and parabolic approximation for the dotted line D-B
 B = point of tangency
between Euler’s curve
and approximation curve
 linear approximation
-- straight-line formula
(Johnson, Tetmajer)
 parabolic approximation
-- parabolic formula(Johnson)
 others : Rankine, codes, …
(1) Johnson’s straight-line formula: line D-B of Fig. 11.5.2(a)
* D(0,  y ) - B ( xB , yB )
Line D-B: tangent line of Euler’s curve A-B-C
1
 3 2 E  2

 , yB  y ;
 xB  
  
3
 y 
1
2 y   y  2
a


3  3E 
 2   2

P
* formula:  cr  cr   y 1   2y  ( Kk ) ,
A
 3  3 E 

1
1
 3 2 E  2

Kk  
  
y


(2) Tetmajer’s formula: straight line
* experimental results for various materials:  cr 
 y (kg / cm2 )
Material
Mild steel
3030
High-strength steel
3350
Wood
293
Pcr
  y 1  a ( Kk )
A
a
Kk
10~112 0.00426
0.00185
 89
2~100 0.00662
(3) Johnson’s parabolic formula: parabola D-B of Fig. 11.5.2(b)
* D(0,  y ) - B ( xB , yB )
Line D-B: tangent line of Euler’s curve A-B-C
1
 2 2 E  2

 , yB  y ;
 xB  
  
2
 y 
 y2
b
4 2 E
   

P
* formula:  cr  cr   y 1   2y ( Kk ) 2  ,
A
  4 E 

1
 2 2 E  2

Kk  
  
 y 
(4) Rankine’s formula
* experimental results for hinged-hinged columns by W.J.M. Rankine(England, 1820-1872)
* general form of the formula for all columns(short, long, and intermediate)
y
P
 cr  cr 
A 1  c( Kk ) 2
* consistency with Euler’s formula: As Kk   ,  cr   Euler 
  cr 
Pcr

A
y
  
1   2 y ( Kk ) 2
 E 
 2E
k2
 c
y
 2E
--- Gordon-Rankine’s formula
(5) Korean highway bridge design specifications
* design formula for concentrically-loaded column: Straight-line formula + Rankine’s formula
* allowable stress ( kg / cm 2 ) for structural steel(SS400, SWS400, etc) with  y  2400 kg / cm2
Pa
 1,400
A
P
 a  a  1,400  8.4( Kk  20)
A
P
12,000,000
a  a 
A 6,700  ( Kk ) 2
a 
; Kk  20
; 20  Kk  93
; Kk  93
(6) AISC specifications
(7) Design formulae for other materials
11.5.2 Eccentrically-Loaded Cases
* Practical design formula: allowable stress method, interaction formula
(1) Allowable stress method
fa
a

fb
a
 1 .0
(2) Interaction formula
fa
a

fb
b
 1.0