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

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

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

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

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

6. Virtual Work Principle F F (Complementary) virtual work principle Perturb F by dF fi di Thecomplementary virtual work done byan external virtual force system under the actual deformation of a structure is equal to the complementary strain energy done bythe 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 F F fi di Superposition Virtual system Actual system

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

9. Virtual Work Principle F F dfi fi

10. Virtual Work Principle F Temperature effect dfi DT

11. Virtual Work Principle F Misfit effect dfi DL

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

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

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

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

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

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

18. Betti’s Law of Reciprocal Deflections P P

19. Betti’s Law of Reciprocal Deflections P M

20. Betti’s Law and Flexibility Coefficients 1 Location j Location i 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