1. CIVL3310 STRUCTURAL ANALYSIS
Chapter 9:
Deflections Using Energy Methods
CIVL3310 STRUCTURAL ANALYSIS
Professor CC Chang

2. Work and Energy Principles
Work done by external forces F F F F F

3. Work and Energy Principles
Where does the work go?
Saved in the beam in terms of
“Strain energy”
Member
force
i-th component
Member
deformation F F
Work - Energy
Total strain energy

4. Virtual Work Principle
Under equilibrium, perturb the structure
Work - Energy F F F
Perturb DF by d DF
Perturb F by dF

5. Virtual Work PrincipleF F
(Complementary) virtual work principle
Perturb F by dF
Virtual work principle
Perturb DF by d DF

6. Virtual Work Principle
(Complementary) virtual work principle
Perturb F by dF
fi di F F
The complementary virtual work done by an external virtual force system under the actual deformation of a structure is equal to
the complementary strain energy done by the virtual stresses
under the actual strains
The complementary virtual work done by an external virtual force system
under the actual deformation of a structure is equal to
the complementary strain energy done by the virtual stresses
under the actual strains

7. Virtual Work Principle
fi di F F
Superposition
Virtual system
Actual system

8. Virtual Work Principle
Looking for DF 1
Virtual system
Actual system
fiLi
EiAi
fi di F F 1

9. Virtual Work Principle
dfi F fi F

10. Virtual Work Principle
Temperature effect
dfi DT F

11. Virtual Work Principle
Misfit effect
dfi DL F

12. Virtual Work Principle
For Beams
M=0, q≠0, k=0 dF
dM(x)
M(x)
M ≠ 0, q≠0, k ≠ 0 DF F x dq
Perturbation
M(x)
Integrate for the whole beam

13. Virtual Work PrincipleDF
dMA qA
M(x) dF
dM(x)
Moment due to
actual loads
Deformation due to
actual loads
Virtual force corresponds
to actual deformation
Virtual moment induced by
The virtual force

14. Virtual Work Principle
For Frames w DF dF P
Negligible

15. Castigliano’s Principle
Work-Energy Fi Fi Fi 1 k

16. Castigliano’s Principle
For Trusses
For Beams
For Frames

17. Betti’s Law of Reciprocal Deflections
Location 1
Location 2
Virtual work principle P Q P Q
Betti’s Law

18. Betti’s Law of Reciprocal Deflections

19. Betti’s Law of Reciprocal DeflectionsM

20. Betti’s Law and Flexibility Coefficients
Location j
Location i 1
Flexibility coefficient
Deflection at j due to a unit load at i 1
Deflection at i due to a unit load at j

21. 9. Deflections Using Energy Methods
Work-energy principle
Virtual work principle
Castigliano’s principle
Betti’s law