Chapter 9
Deflections using Energy Methods
Reading Assignment:
Chapters 9.1 - 9.5
External Work
•Used for more complicated loadings or for
structures such as trusses and frames.
•Conservation of Energy Principle
Ue = Ui
External Work – Force/Moment
X

U e   Fdx
U e   Md
0
 P  - Linear Elastic Response
F   x

2


P
P
x
 
 
U e   Fdx    xdx    

   2 
0
0
X
P
Ue 
2
X
0
1
U e  M
2
Strain Energy – Axial Force
• Hooke’s Law
 σ = EЄ
• Normal Stress
 σ = N/A
 A=Constant Cross-Sectional Area
 Final Strain
 Є=Δ/L
 L=Length
 Δ = NL/AE
 Ui = N2L/2AE
Strain Energy – Bending
• Strain Energy
 dUi =M2dx/2EI
• Strain Energy over Beam Length
L
M 2dx
Ue  
2 EI
0
Example #1
Truss Example
Real Work Method
Principle of Work and Energy
P
Ue 
2
M 2dx  Px dx 1 P 2 L3
Ui  


2 EI
2 EI
6 EI
0
L
2
P P 2 L3
PL3
Ue  Ui 


2
6 EI
3EI
•Application of this method is limited
•Only one load may be applied to the structure
•Only the displacement under the force can be obtained.
Principle of Virtual Work
• Also known as the “unit load method”
• Used to determine displacement and slope at
a specific location on a structure.
• See Figure 9-6 in textbook.
Principle of Virtual Work
• External Virtual Work
 P’Δ
• Internal Virtual Work
 Σu*dL
• ΣPΔ = Σuδ
 P’ * Δ = Σu*dL
• Moment
 P’*θ = Σuθ *dL
Method of Virtual Work: Trusses
• External Loading
nNL
P'  
AE
• Temperature
P'   nTL
• Fabrication Error and Camber
P'  nL
Method of Virtual Work Procedure
1) Place unit load on the truss at the joint where the
desired displacement is to be determined and
remove all real loads.
NOTE: Load should be in the same direction of
displacement.
2) Calculate the internal forces in each truss member.
(+ = tension, - = compression)
3) Remove unit load and apply real loads to truss.
4) Calculate the internal forces in each truss member.
(+ = tension, - = compression)
Method of Virtual Work Important Notes
1) You must retain the algebraic sign for each of the n and N
values when placing them into the virtual work equation.
2) If the virtual work equation is positive, the displacement is
in the same direction as the unit load. If it is negative it is
in the opposite direction.
3) For Temperature Equation:
1)
ΔT = Positive with temperature increase, Negative with
temperature decrease.
4) For Fabrication Equation:
1)
ΔL = Positive with increase in length, Negative with decrease in
length.
5) MAKE SURE YOU PAY ATTENTION TO YOUR UNITS.
Example #2
Truss Example
Virtual Work Method
Example #3
Truss Example
Virtual Work Method