Chapter10Columns

Chapter10Columns - Chapter 10 Columns 10.1 Introduction...

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Chapter 10 Columns
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10.1 Introduction Column = vertical prismatic members subjected to compressive forces Goals of this chapter: 1. Study the stability of elastic columns 2. Determine the critical load P cr 3. The effective length 4. Secant formula
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Previous chapters: -- concerning about (1) the strength and (2) excessive deformation (e.g. yielding) This chapter: -- concerning about (1) stability of the structure (e.g. bucking)
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10.2 Stability of Structures allow P A σ = < cr PL AE δ = < 2 2 ( )sin ( ) cr L P K θ = (10.1) sin ∆ ≈ ∆ (10.2) Concerns before: New concern: Stable? Unstable?
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2 2 ( ) ( ) cr L P K θ = sin ≈ ∆ 4 / cr P K L = (10.2) The system is stable, if The system is unstable if Since 4 / cr P K L < 4 / cr P K L A new equilibrium state may be established
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2 2 ( )sin ( ) L P M K θ = = 4 sin PL K = The new equilibrium position is: 4 sin PL K = (10.3) or
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After the load P is applied, there are three possibilities: 1. P < P cr – equilibrium & θ = 0 -- stable 2. P > P cr – equilibrium & θ = θ -- stable 3. P > P cr unstable – the structure collapses, θ = 90 o
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10.3 Euler’s Formula for Pin-Ended Columns Determination of P cr for the configuration in Fig. 10.1 ceases to be stable 2 2 d y M P y dx EI EI = = - 2 2 0 d y P y dx EI + = Assume it is a beam subjected to bending moment: (10.5) (10.4)
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Chapter10Columns - Chapter 10 Columns 10.1 Introduction...

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