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# L22 - Chapter 10 Columns Stability of Structures In the...

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Chapter 10 Columns

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Stability of Structures In the design of columns: - allowable stress is not exceeded all A P σ σ = - deformation falls within specifications spec AE PL δ δ =
column is unstable under loading: suddenly becomes sharply curved or buckles. Unstable column

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Stability of Structures: 2 rods & torsional spring ( ) θ Δ 2 : moment restoring K θ θ Δ Δ 2 sin 2 : moment ing destabiliz L P L P
Column is stable (tends to return to aligned orientation) if destabilizing moment < restoring moment: ( ) L K P P K L P cr 4 2 2 = < Δ < Δ θ θ

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Stability of Structures After a perturbation, the system settles to a new equilibrium configuration at a finite deflection angle. ( ) θ θ θ θ sin 4 2 sin 2 = = = cr P P K PL K L P : the assumed configuration is only possible if P > P cr . P has to exceed P cr to obtain a new equilibrium. Steps to find P cr - draw FBD for the deflected configuration of the system - use statics to solve for P cr sin θ < θ
Euler’s Formula for Pin-Ended Beams 2 2 2 L EI n P π = 2 2

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