Beams_EBT_Ch5_2

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

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
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 f 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 ,w ( w ( x ) ε xx = z d 2 w dx 2 , γ xz Constitutive Relations σ xx = E ε xx = Ez d 2 w dx 2 , σ xz = G γ xz
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Moment-De f ection and Shear Force-De f ection Relations M = Z A z · σ xx dA = EI d 2 w dx 2 ,V = dM dx = d dx μ d 2 w dx 2 Governing Equilibrium Equation in terms of w d 2 dx 2 μ d 2 w dx 2 + c f w = q ( x )f o r0 <x<L Weak Form 0= Z x b x a μ 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 d 2 w dx 2 ¶¸¯ ¯ ¯ ¯ x a = V ( x a ) Q e 2 μ d 2 w dx 2 ¯ ¯ ¯ x e = M ( x a ) Q e 3 ≡− d dx μ d 2 w dx 2 ¯ ¯ ¯ x b = V ( x b ) Q e 4 μ d 2 w dx 2 ¯ ¯ ¯ x b = M ( x b )
Background image of page 2
Weak Form 0= B ( v,w ) l ( v ) B ( )= Z x b x a μ EI d 2 v dx 2 d 2 w dx 2 + c f vw dx l ( v Z x b x a 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 Finite Element Approximation of w w ( x ) w e h ( x c e 1 + c e 2 x + c e 3 x 2 + c e 4 x 3 e 1 w e h ( x a ) , e 2 ≡− dw e h dx ¯ ¯ ¯ ¯ x = x a , e 3 w e h ( x b ) , e 4 dw e h dx ¯ ¯ ¯ ¯ x = x b Primary variables (generalized displacements) Secondary variables (generalized forces) 1 1 2 2 e e 2 1 Δ θ e e 4 2 Δ e e w 1 1 Δ e e w 3 2 Δ e e q , Q 2 2 e e q , Q 4 4 e e q , Q 1 1 e e q , Q 3 3 e h e h (In terms of the bilinear and linear form)
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
e 1 = w e h ( x a )= c e 1 + c e 2 x a + c e 3 x 2 a + c e 4 x 3 a e 2 = dw e h dx ¯ ¯ ¯ ¯ x = x e = c e 2 2 c e 3 x a 3 c e 4 x a e 3 = w e h ( x b c e 1 + c e 2 x b + c e 3 x 2 b + c e 4 x 3 b e 4 = dw e h dx ¯ ¯ ¯ ¯ x = x b = c e 2
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 11

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online