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

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

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

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

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

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

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

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

Virtual Work Principle dfi F fi F

Virtual Work Principle Temperature effect dfi DT F

Virtual Work Principle Misfit effect dfi DL F

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

Virtual Work Principle DF 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

Virtual Work Principle For Frames w DF dF P Negligible

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

Castigliano’s Principle For Trusses For Beams For Frames

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

Betti’s Law of Reciprocal Deflections

Betti’s Law of Reciprocal Deflections M

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

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