1. CIVL3310 STRUCTURAL ANALYSIS
Chapter 8:
Deflections
CIVL3310 STRUCTURAL ANALYSIS
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
Inflection point
tensile side
Straight line
(why?)
Straight line
(why?)

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

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

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
-slope-deflection
Load-shear-moment

17. Conjugate-Beam Method
Mathematical equivalence
-slope-deflection
Load-shear-moment
Actual beam
Conjugate beam

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

19. Conjugate-Beam Methodor

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. Conjugate-Beam 5 kN/m 3 kN/m A F B C D E 20m 20m 50m 20m 20m M EI M EIV F M B C D E
= 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?