高等材料力學
關百宸 老師
海洋科學科學(工程)與水下科技人才培育方案
磨課師 示範教學
Buckling Analysis of Beam Column
 Review
 Derive the critical loading for column structural members with Euler’s
Equation
 Practice the calculation and learn how to deal with beam column with
different boundary conditions
 Beam Column
 Find the effect due to distributed loading
Assumption of Ideal Column
 Ideal column:





Perfectly straight before loading
Made of homogeneous material
Load is applied through the centroid of the cross section
Linear-elastic material
Buckle and bend the beam in a single plane
Ideal Column with Pin Supports
 Theoretically, when the axial load P increases, the column fails due to :
 Material Yielding: Plastic deformation
 Fracture : Crack forming and propagating
P
P
P
fracture
yielding
Ideal Column with Pin Supports
 When the axial load P reaches the critical load Pcr, the column becomes
unstable.
 Any small lateral force F will cause the column in the deflected position,
even after the force F is removed.
P
Pcr
F
Pcr
Ideal Column with Pin Supports
 Free body diagram of Column under an axial load P
P
P
Momentum equilibrium at point A:
M A  F  0
where
x
A
ν
P
M
F
d 2
M  EI 2
dx
Axial force equilibrium:
F  P
d 2
EI 2  P  0
dx
Ideal Column with Pin Supports
 Ordinary differential equation of Column
d 2
EI 2  P  0
dx
Let
v  e x
EI 2e x  Pe x  0
2 
P
0
EI
  i
P
P
, i
EI
EI
Ideal Column with Pin Supports
 The general solution of buckling
  B1e
i
P
x
EI
 B2e
i
P
x
EI
 P 
 P 
  C1 sin 
x   C2 cos 
x
 EI 
 EI 
Euler's formula:
e i x  cos   x   i sin   x 
還記得吧…
Ideal Column with Pin Supports
 Apply the boundary condition of pin-pin ends
P
v
 P 
 P 
  C1 sin 
x   C2 cos 
x
 EI 
 EI 
x
L
EI
Boundary conditions:
when x = 0, v = 0
C2  0
Ideal Column with Pin Supports
 Apply the boundary condition of pin-pin ends
P
when x = L, v = 0
 P 
C1 sin 
L  0
 EI 
v
x
L
EI
Two possible solution
(a)
C1  0
(b)
 P 
sin 
L  0
EI


Ideal Column with Pin Supports
 when
 P 
sin 
L  0
 EI 
P
L  n
EI
Where n = 1, 2, 3, …
Function of sin(x)
EI  2 n 2
P
L2
When n = 1 the critical load
EI  2
Pcr  2
L
This critical load is sometimes
referred to as Euler load
Ideal Column with Pin Supports
 For different mode n we can get the corresponding buckled shape of pinpin support Beam-Column:
2
2
 n
v  C1 sin 
 L

x  0

EI 
Pcr  2
L
n=1
4 EI 
Pcr 
L2
n=2
9 EI  2
Pcr 
L2
n=3
16 EI  2
Pcr 
L2
n=4
P
P
但是如果樑-柱上有均
佈力怎麼辦？
q(x)
Linear Elastic Beam-Column
 Free body diagram of Beam-Column under a axial load P and distribution
load q(x)
Momentum equilibrium at point A:
P
M
V
∆x
M  V x  P   qdxx  0
q(x)
∆ν
A
V+ ∆ V
M+∆M
P
where α is a temporary variable
Dividing by ∆x
V 
M

P
  qdx
x
x
Linear Elastic Beam-Column
 Take the limits as x  0
V 
P
M
V
∆x
where
dM
d
P
dx
dx
d 2
M  EI 2
dx
q
∆ν
V+ ∆ V
M+∆M
P
d 3
d
V   EI 3  P
dx
dx
Linear Elastic Beam-Column
 Equilibrium in the v-direction gives
V
P
x
q
V  V  V    qdx  0
M
V
∆x
∆x
v
 V   qdx  0
V+ ∆ V
q
∆ν
x  0
V+ ∆ V
M+∆M
P
q
dV
dx
Linear Elastic Beam-Column
 The ODE of Beam-Column:
d 3
d
V   EI 3  P
dx
dx
P
dV
d 4
d 2
  EI 4  P 2  q
dx
dx
dx
d 4
d 2
EI 4  P 2  q
dx
dx
M
V
∆x
q
∆ν
V+ ∆ V
M+∆M
P
Linear Elastic Beam-Column
 The general solution of Beam-Column equation
d 4
d 2
EI 4  P 2  q
dx
dx
The general solution: v  vh  v p
vh is the homogeneous solution
vp is the particular solution
Linear Elastic Beam-Column
 Solving the homogeneous solution vh
The homogeneous equation:
Let
d 4 h
d 2 h
EI
P 2 0
4
dx
dx
vh  e x
  4 e x 
P 2 x
 e 0
EI
P 

  2  2 
0
EI 

  0, 0, i
P
EI
i
P
EI
Linear Elastic Beam-Column
 the homogeneous solution vh
vh  A1e0 x  A2 xe0 x  A3e
i
P
x
EI
 A4e
i
P
x
EI
vh  C1  C2 x  C3 cos  x  C4 sin  x
The general solution v:
v  C1  C2 x  C3 cos  x  C4 sin  x  v p
同學有沒有甚麼問題？
Example
 Find the equation for the elastic curve v(x) for the uniformly loaded
beam-columns shown
q0
(a)
P
x
EI
v
q0
(b)
P
x EI
v
Example
 Beam-Column with Side Sway Buckling
 Derive the critical loads for the classical Euler buckling problems in the side
sway problems
(a) Fix-Fix ends
(b) Pin-Fix ends
q0
q0
P
P
x EI
v
l
x EI
v l