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

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## CIVL3310 STRUCTURAL ANALYSIS

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**CIVL3310 STRUCTURAL ANALYSIS**Chapter 8: Deflections Professor CC Chang**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**Deflection Diagrams/Shapes**curvature +M -M tensile side tensile side**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?)**Example 8.2**negligible All members are axially inextensible! M≠0 M=0 No curvature**Calculation of Deflections**• Direct integration • Moment-area method • Conjugate-beam method • Energy methods (in Chapter 9)**Calculation of Deflections**• Note: Superposition can be used for linear structures =**Direct Integration**+ disp Bernoulli-Euler beam apply boundary conditions**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**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**Moment-Area Theorems**qB qA + 1st moment-area theorem**Moment-Area Theorems**• 2nd moment-area theorem Horizontal dis btw A & centroid of M/EIA-B M/EI area btw A & B**Moment-Area Theorems**1st moment-area theorem 2nd moment-area theorem +**Example 8.5**Determine the slope at points B & C of the beam. Take E = 200GPa, I = 360(106)mm4**Conjugate-Beam Method**• Mathematical analogy**Conjugate-Beam Method**• Mathematical equivalence Actual beam Conjugate beam**P**P Conjugate-Beam Method Conjugate beam Actual beam ?**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**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**Solution**Conjugate beam: Max Moment Note: when 6.71**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**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?

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