ho5 - MIT - 16.20 Fall, 2002 16.20 HANDOUT #5 Fall, 2002...

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MIT - 16.20 Fall, 2002 16.20 HANDOUT #5 Fall, 2002 Stability and Buckling Bifurcation Buckling and Snap-Through Buckling bifurcation point BIFURCATION BUCKLING Perfect Column: EI = constant Z X 4 2 • Governing Equation: EI dw + P = 0 dx 4 dx 2 • Solution: w = A sin P EI x + B cos P x + C + Dx Paul A. Lagace © 2002 Handout 5-1
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MIT - 16.20 Fall, 2002 2 π n 2 EI nx • Simply supported: P cr = l 2 mode shape: w = A sin l 2 EI Euler buckling load : P = cr l 2 cE I 2 General Case: P = c = coefficient of edge fixity cr l 2 • Various Boundary Conditions – Simply-supported (pinned) w = 0 2 M = E I dw = 0 dx 2 – Fixed end (clamped) w = 0 = 0 dx – Free end 2 M = E I = 0 dx 2 2 S = d = 0 dx dx 2 – Sliding 2 S = d = 0 dx dx 2 = 0 Paul A. Lagace © 2002 Handout 5-2
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MIT - 16.20 Fall, 2002 – Free end with axial load 2 M = E I dw = 0 dx 2 2 S = d EI = P 0 dx dx 2 dx – Vertical spring M 2 = E I = 0 dx 2 2 d S = dx dx 2 = k f w – Torsional spring w = 0 2 M = E I = k T dx 2 • Various Configurations
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ho5 - MIT - 16.20 Fall, 2002 16.20 HANDOUT #5 Fall, 2002...

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