ECIV 320 Structural Analysis I
Deflections
Deflection Diagrams & Elastic Curve
ASSUMPTION
Linear Elastic Material Response
A structure subjected to a load will return to its original
undeformed position after load is removed
OBJECTIVE
Calculate Slope and Deflection due to Bending at any point on a Beam
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METHODS
Double Integration Method
Moment Area Theorems
Conjugate Beam
Energy Methods
Last Time - Conjugate Beam Method
Use to find slopes and deflection due to
bending of beams
Same amount of Computations as Moment-Area Theorems
Method is based on
Principles of Statics Only
Energy Methods
Calculate Slope and Deflection
Double Integration Method
Moment Area Theorems
Conjugate Beam
Beams
Simple Loadings
Energy Methods
Complicated Loadings
Trusses
Frames
External Work & Strain Energy
Conservation of Energy
(Elastic Material Behavior)
Ue=Ui
Work of External
Forces & Moments
Strain Energy
Internal Forces
External Work
Gradually Applied Force
x
U e   Fdx
0
1
U e  P
2
External Work
Due to Another Force
x
U e   Fdx
0
U e  P
'
External Work

Moment
Ue 
 Md
0
Gradually Applied
1
U e  M
2
Due to Another Moment
U e  M
'
Strain Energy- Axial Force
Hooke’s Law:
  E
Stress:
 N A
Strain:
  L
Final Deflection:
NL

AE
2
1
N L
U i  N 
2
2 AE
N
Strain Energy - Bending
d M

dx EI
1
M2
dU i  Md 
dx
2
2 EI
L
2
M
Ui  
dx
2 EI
0
Example
Principle of Virtual Work
Bernoulli 1717
External and Internal Loads are Related through
Equations of Equilibrium
External and Internal Displacements are Related through
Equations of Compatibility
Principle of Virtual Work
Principle of Virtual Work
Virtual Work
1     udL
Virtual
Loads
Real
Displ.
Virtual Work for Trusses - External Loads
1     udL
Virtual
Loads
Real
Displ.
dL  NL / AE
u=n
nNL
1   
AE
Virtual Work for Trusses - Temperature
1     udL
Virtual
Loads
Real
Displ.
dL  aTL
u=n
1     naT L
Virtual Work for Trusses - Fabrication Errors
1     udL
Virtual
Loads
Real
Displ.
dL  L
u=n
1     nL
Virtual Work for Trusses - Combined Effects
nNL
1   
  naT L   nL
AE
Example
Step 1
• Remove ALL External Loads
• Apply Virtual (Unit) Load in the Direction of
the Unknown Displacement
• Calculate Internal Forces n
Step 2
• Apply REAL Loads
• Calculate Internal Forces N
Step 3
• Virtual Work Equation
0
nNL
1   
  naT L   nL
AE
Solution
58,000
58,000
58,000
43,500
58,000
6
8
6
10
8
0
0
0.75
-1.25
1
0
80
120
-100
80
0
0
0.009
0.029
0.011
0.049
0
0
0
0
7.2e-4 7.2e-4
7.2e-4
Virtual Work for Beams and Frames - BENDING
1     udL
Virtual
Loads
Real
Displ.
M
dL  d  dx
EI
Virtual Work for Beams and Frames - BENDING
Virtual
Loads
1     udL
Real
Displ.
u=m
1   
L
0
m( x ) M ( x )
dx
EI ( x )
Virtual Work for Beams and Frames - BENDING
1     udL
Virtual
Loads
Real
Displ.
1   
L
0
M
dL  d  dx
EI
u=m
m( x ) M ( x )
dx
EI ( x )
Virtual Work for Beams and Frames - BENDING
1   u dL
M
dL  d  dx
EI
Virtual
Loads
To Calculate Slope
q
Real
Displ.
u=mq
1
1   
L
0
m ( x ) M ( x )
dx
EI ( x )