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Bending Deflection –
Method of Superposition
AE1108-II: Aerospace Mechanics of Materials
Dr. Calvin Rans
Dr. Sofia Teixeira De Freitas
Aerospace Structures
& Materials
Faculty of Aerospace Engineering
Deflection by Superposition
• If stress-strain behaviour of the beam material remains
linear elastic, principle of superposition applies
• Problem can be broken down into simple cases for
which solutions may be easily found, or obtained from
data handbooks (see Appendix C of the textbook)
The following slides show all the
standard solutions you will be
permitted to use on the exam!
Cantilever Beam Standard Solutions
xz
A
Deflection (at B)
q
xz
A
qL4
vB 
8 EI
qL3
B 
6 EI
EI
PL3
vB 
3EI
PL2
B 
2 EI
ML2
vB  
2 EI
ML
B  
EI
P
B
L
xz
A
EI
B
L
EI
L
M
B
Slope (at B)
Simply Supported Beam Standard Solutions
Deflection
q
xz
A
EI
5qL4
vC 
384 EI
EI
PL3
vC 
48EI
B
C
Slope
 A, B
qL3

24 EI
 A, B
PL2

16 EI
L
xz
P
A
L/2
L/2
xz
M
B
C
EI
A
B
C
L/2
M
L/2
ML2
vC 
8EI
C  0
Example
Example Problem
z
Find the deflection at B
We can break into two sub problems
z
z
Example Problem (cont)
Sub Problem (1)
z
vC 
v(1)  vC  C
4
7 wL
v(1) 
384 EI
 2
L
C 
Standard Solution 1:
z
wL4
v
8 EI
wL3

6 EI
 2
w L
4
8 EI
 2
w L
6 EI
wL4

128 EI
3
wL3

48 EI
Example Problem (cont)
Sub Problem (2)
z
 v(2)
PL3

3EI
Standard Solution 2:
PL3
z
v
3EI
PL2

2 EI
Example Problem (cont)
Superposition:
z
z
7 wL4
v(1) 
384 EI
v(2)
4
3
7 wL
PL
 vB 

384 EI 3EI
PL3

3EI