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Beams_EBT_Ch5_2

# Beams_EBT_Ch5_2 - Euler-Bernoulli Beam Finite Element F0...

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Euler-Bernoulli Beam Finite Element Forces and their interrelationships at a point in the beam + M V q ( x ) V M c f x q ( x ) F 0 L z, w M 0 z y Beam crosssection c f De fi nitions of Stress Resultants M = Z A z · σ xx dA, V = Z A σ xz dA Equilibrium Equations dV dx + c f w = q, dM dx V = 0 d 2 M dx 2 + c f w = q Kinematic Relations u ( x, z ) = z dw dx , v = 0 , w ( x, z ) = w ( x ) ε xx = z d 2 w dx 2 , γ xz = 0 Constitutive Relations σ xx = E ε xx = Ez d 2 w dx 2 , σ xz = G γ xz = 0

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Moment-De fl ection and Shear Force-De fl ection Relations M = Z A z · σ xx dA = EI d 2 w dx 2 , V = dM dx = d dx μ EI d 2 w dx 2 Governing Equilibrium Equation in terms of w d 2 dx 2 μ EI d 2 w dx 2 + c f w = q ( x ) for 0 < x < L Weak Form 0 = Z x b x a μ EI d 2 v dx 2 d 2 w dx 2 + c f vw vq dx v ( x a ) Q e 1 μ dv dx ¶ ¯ ¯ ¯ ¯ x a Q e 2 v ( x b ) Q e 3 μ dv dx ¶ ¯ ¯ ¯ ¯ x b Q e 4 Q e 1 d dx μ EI d 2 w dx 2 ¶¸ ¯ ¯ ¯ ¯ x a = V ( x a ) Q e 2 μ EI d 2 w dx 2 ¶ ¯ ¯ ¯ ¯ x e = M ( x a ) Q e 3 ≡ − d dx μ EI d 2 w dx 2 ¶¸ ¯ ¯ ¯ ¯ x b = V ( x b ) Q e 4 ≡ − μ EI d 2 w dx 2 ¶ ¯ ¯ ¯ ¯ x b = M ( x b )