Lecture 8
Energy Methods in Elasticity
Problem 8-1: Consider an elastic cantilever beam loaded at its tip.
P
l
x
EI
a) Specify the boundary conditions.
b) Derive the load-tip displacement relation using four methods presented in class.
Method I Solving uncoupled problems
Method II Solving coupled problem (direct integration)
Method III Castigliano Theorem
Method IV Ritz Method
Problem 8-1 Solution:
(a) Boundary conditions
w(0) 0
w'(0) 0
(b) Derive the load-tip displacement relation using four methods
1. Solving second order uncoupled problems
We will use
moment-curvature relationship
M
curvature-displacement relationship
N
EI
w ''
to solve the problem.
1
These relationships combine to give:
EIw ''
M
We need the moment.
Find reactions
¦ Fy
Ry
¦ Mr
Ry P 0
P
(M r
Pl ) 0
Pl
Mr
Find moment in beam base on force balance and moment balance
¦ Fy
V
P V
P
¦ M0
M
0
(Pl Vx M )
0
Pl Vx
Combing the above results from force balance and moment balance
M
Pl Px
Integrate moment once
2
EIw ''
M
1
Pl Px
EI
1
Px 2
Plx
C1
2
EI
w ' dx
w'
Use B.C.
w'(0) 0
C1 0
Integrate once
w
1
Plx
EI
Px 2
2
w
1 Plx 2
EI
2
Px 3
6
C2
Use B.C.
w( x)
w(0)
0
C2
0
1 Plx 2
EI
2
Px3
6
Pl 3
3EI
w(l )
2. Solve coupled problems using 4th order direct integration
We use the 4th order differentiated equation:
EIwIV
q
0
And we need four boundary conditions:
w(0)
0
w '(0)
0
M (0)
M (l )
Pl
0
EIw ''
EIw ''
Integrate and use boundary conditions to determine the constants
3
EIwIV dx 0
EIw''' C1
EIw'' C1 x C2
Use B.C.
M (0)
Pl
M (l ) 0
We get
C1
P
C2
Pl
Then
EIw''
Px Pl
EIw '
Px 2
2
Plx C3
Use B.C.
w'(0) 0
We get
C3
0
Px3
6
Plx 2
2
Then
EIw
C4
Use B.C.
w(0) 0
We get
C4
0
Finally
w( x)
1 Plx 2
EI
2
w(l )
Px3
6
Pl 3
3EI
4
3. Castigliano Theorem
Find moment in beam
M
Pl
Px
Calculate strain energy
U
1 M2
dx
2 EI
1 Px Pl
2
EI
2 3
Pl
6EI
2
dx
Use Castigliano Theorem
w(l )
U
P
w(l )
Pl 3
3EI
4. Ritz method
Assume shape function
w( x ) C1 x 3 C2 x 2 C3 x C4
Use boundary conditions
w(0) 0
w'(0) 0
We get
C3
C4
0
w( x ) C1 x 3 C2 x 2
Also
w(l ) C1l 3 C2l 2
wo
Find total potential energy
5
1 l
EI ( w '') 2 dx Pwo
2 0
1 l
EI (6C1 x 2C2 ) 2 dx Pwo
2 0
EI
12C12l 3 12C1C2l 2 4C2 2l
2
Pwo
Minimize
C1
0 and
0
C2
C1
EI
24C1l 3 12C2l 2
2
C2
EI
12C1l 2 8C2l
2
Pl 3
Pl 2
0
0
Solve for C1 and C2 using the above two equations
C1
C2
P
6 EI
Pl
2 EI
Finally
w( x)
1 Plx 2
EI
2
w(l )
Px3
6
Pl 3
3EI
Note: we obtain the exact solution with
w x ~O 3
If we were to assume shape function of lower order, we can only obtain an
approximate solution.
6
Problem 8-2: Consider a beam of length L and bending rigidity EI which is fully clamped
on both ends shown in Figure 1. The beam is subjected to a point force P at the midspan.
Solve the problem (find the expression between the load and the deflection under the load)
using:
Figure 1
a)
The direct integration of beam equation with the suitable boundary conditions (exact
solution).
b)
The Ritz Method (approximate solution).
c)
Compare the results and calculate the relative error of the Ritz Method.
Problem 8-2 Solution:
T
Solve for half the beam. Boundary conditions are:
V
a)
w(0) 0
(1)
w'(0) 0
(2)
l
w '( ) 0
2
(3)
EIw'''(0)
P
2
(4)
Direct integration
We use the 4th order differentiated equation:
EIwIV
where q
q
0
7
Integrate once
w ''' C1
Use B.C. (4), we have
w ''' 0
P
C1
2 EI
P
w '''
2 EI
By integration
P
x C2
2EI
P 2
x C2 x C3
4 EI
w ''
w'
Use B.C. (2), we have
w' 0
C3
0
P 2
x C2 x
4EI
w'
Use B.C. (3),
w'
l
2
C2
w'
0
Pl
8EI
P 2
x
4 EI
Pl
x
8EI
Integrate once
w
P 3
Pl 2
x
x C4
12 EI
16 EI
Use B.C. (1),
w 0
C4
0
0
8
w
P 3
Pl 2
x
x ,0
12EI
16EI
x
l
2
Mid-span deflection
b)
Pl 3
192EI
l
2
w
Ritz method
Assume shape function
y x
C1 x 3 C2 x 2 C3 x C4
Boundary conditions are:
y(0) 0
(5)
y '(0) 0
(6)
l
y '( )
2
0
(7)
l
y( )
2
w0
(8)
Use B.C. (5) and (6),
C3
y x
C4
0
C1 x3 C2 x 2
9
Apply B.C. (7) and (8),
l
y '( ) 0
2
2
l
l
3C1
2C2
0
4
2
l
y ( ) w0
2
3
l
l2
C1
C2
w0
8
4
C1 and C2
Combine the above two equations, we can solve for
C1
C2
Calculate
16
w0
l3
12
w0
l2
U W
24w0
l2
y '' x
U
1
EI
2
l
0
24 w0
l2
4x
1
l
4x
1
l
2
w0 2
96EI 3
l
dx
Minimize
w0
96EI
w0
2w0
l3
P
0
Pl 3
192EI
Finally
w
P 3
Pl 2
x
x ,0
12EI
16EI
x
l
2
10
c)
Compare Ritz method and exact solution
w0 exact
w0 Ritz
Pl 3
192 EI
Pl 3
192 EI
Because we guess the correct order of the shape function, we got the exact solution.
%error=0
11
Problem 8-3:
Figure
Use Castigliano’s Theorem to calculate the horizontal deflection at point D in
Problem 8-3 Solution:
Find reaction
Rx
P
MR
2 PL
where Rx and M R are reaction force and reaction moment
Find internal moment distribution at each section
Section AB
M1
2 PL
Section BC
M2
P 2 L r sin
12
Section CD
M3
Px2
Calculate strain energy
U
U1 U 2 U 3
1
2
L
0
1
2 EI
2PL
2
EI
2 PL
P 2 20 L3
2 EI
3
dx1
2
L rP
2
P 2 L r sin
1
2
rd
EI
0
2
2
0
4L
4 L2 r 8 Lr 2
1
2
2 2Lr sin
2
2L
0
r sin
2
Px2
EI
d
2
dx2
8 P 2 L3
3
r3
2
Use Castigliano
w0
U
P
2 P 20L3
2 EI
3
2
4 L r 8Lr
2
r3
2
13
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2.080J / 1.573J Structural Mechanics
Fall 2013
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