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

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

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

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

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

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 -slope-deflection Load-shear-moment

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

Conjugate-Beam Method Actual beam Conjugate beam P ? P

Conjugate-Beam Method or

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

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