1. ECIV 320 Structural Analysis I Deflections

2. 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 METHODS • Double Integration Method • Moment Area Theorems • Conjugate Beam • Energy Methods

3. 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

4. Energy Methods Calculate Slope and Deflection Double Integration Method Moment Area Theorems Conjugate Beam Beams Simple Loadings Complicated Loadings Trusses Frames Energy Methods

5. External Work & Strain Energy Conservation of Energy (Elastic Material Behavior) Ue=Ui Work of External Forces & Moments Strain Energy Internal Forces

6. External Work Force Gradually Applied

7. External Work Due to Another Force

8. External Work Moment Gradually Applied Due to Another Moment

9. Strain Energy- Axial Force Hooke’s Law: Final Deflection: Stress: Strain: N

10. Strain Energy - Bending

11. Example

12. 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

13. Principle of Virtual Work

14. Principle of Virtual Work

15. Virtual Loads Real Displ. Virtual Work

16. Virtual Loads Real Displ. Virtual Work for Trusses - External Loads u=n

17. Virtual Loads Real Displ. Virtual Work for Trusses - Temperature u=n

18. Virtual Loads Real Displ. Virtual Work for Trusses - Fabrication Errors u=n

19. Virtual Work for Trusses - Combined Effects

20. Example

21. Step 1 • Remove ALL External Loads • Apply Virtual (Unit) Load in the Direction of the Unknown Displacement • Calculate Internal Forces n

22. Step 2 • Apply REAL Loads • Calculate Internal Forces N

23. Step 3 • Virtual Work Equation 0

24. Solution 58,000 6 0 0 0 0 58,000 0 80 0 0 8 58,000 6 0.75 0 120 0.009 43,500 -1.25 0.029 0 -100 10 58,000 8 1 0.011 7.2e-4 7.2e-4 80 0.049 7.2e-4

25. Virtual Work for Beams and Frames - BENDING Virtual Loads Real Displ.

26. Virtual Work for Beams and Frames - BENDING Virtual Loads Real Displ. u=m

27. Virtual Work for Beams and Frames - BENDING Virtual Loads Real Displ. u=m

28. Virtual Work for Beams and Frames - BENDING Virtual Loads To Calculate Slope q Real Displ. u=mq 1