1. CIVL3310 STRUCTURAL ANALYSIS Chapter 8: Deflections Professor CC Chang

2. Deflection- Displacement & Rotation • Deflections maybe due to loads, temperature, fabrication errors or settlement • Linear elastic behavior: linear stress-strain relationship and small deflection • Need to: • Plot qualitative deflection shapes before/after the analysis • Calculate deflections at any location of a structure

3. Deflection Diagrams/Shapes curvature +M -M tensile side tensile side

4. Example 8.1 Draw the deflected shape of each of the beams. Need to show 1st order (slope) and 2nd order (curvature) information tensile side tensile side Inflection point Straight line (why?) Straight line (why?)

5. Example 8.2 negligible All members are axially inextensible! M≠0 M=0 No curvature

6. Calculation of Deflections • Direct integration • Moment-area method • Conjugate-beam method • Energy methods (in Chapter 9)

7. Calculation of Deflections • Note: Superposition can be used for linear structures =

8. Direct Integration + disp Bernoulli-Euler beam apply boundary conditions

9. Example 8.3 The cantilevered beam is subjected to a couple moment Mo at its end. Determine the eqn of the elastic curve. EI is constant. x at x = 0 dv/dx = 0 & v = 0 C1 = C2 =0. Max

10. 5 kN/m 3 kN/m A F 0 B C E D 20m 20m 20m 20m 50m 30 241 50 kN 149 91 90 50 V -30 -59 -150 M M1 M3 + + M2 _ y Direct Integration

11. Moment-Area Theorems qB qA + 1st moment-area theorem

12. Moment-Area Theorems • 2nd moment-area theorem Horizontal dis btw A & centroid of M/EIA-B M/EI area btw A & B

13. Moment-Area Theorems

14. Moment-Area Theorems 1st moment-area theorem 2nd moment-area theorem +

15. Example 8.5 Determine the slope at points B & C of the beam. Take E = 200GPa, I = 360(106)mm4

16. Conjugate-Beam Method • Mathematical analogy

17. Conjugate-Beam Method • Mathematical equivalence Actual beam Conjugate beam

18. P P Conjugate-Beam Method Conjugate beam Actual beam ?

19. Conjugate-Beam Method or

20. Conjugate-Beam Method Conjugate beams are mathematical/imaginative beams No need to worry about their stability Just use EQ concept to obtain V and M from w

21. Example 8.5 Determine the max deflection of the steel beam. The reactions have been computed. Take E = 200GPa, I = 60(106)mm4 Max disp Max M

22. Solution Conjugate beam: Max Moment Note: when 6.71

23. M EI M EI M V M Conjugate-Beam 5 kN/m 3 kN/m A F B C E D 20m 20m 20m 20m 50m Conjugate beam F B E C D = y = q

24. Summary • Can you plot a qualitative deflection shape for a structure under loads (need to show 1st & 2nd order information)? • Can you calculate structural deformation (displacement & rotation) using • Direct integration? • Moment-area principle? • Conjugate-beam method?