# Chap4 - CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES...

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1 CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES FINITE ELEMENT ANALYSIS AND DESIGN Nam-Ho Kim and Bhavani Sankar Instructor: Subrata Roy

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2 INTRODUCTION • We learned Direct Stiffness Method in Chapter 2 – Limited to simple elements such as 1D bars • In Chapter 3, Galerkin Method and Principle of Minimum Potential Energy can be applied to more complex elements • we will learn Energy Method to build beam finite element – Structure is in equilibrium when the potential energy is minimum Potential energy : Sum of strain energy and potential of applied loads • Interpolation scheme: () {} vx x =⋅ ⎢⎥ ⎣⎦ Nq Beam deflection Interpolation function Nodal DOF Potential of applied loads Strain energy UV Π =+
3 BEAM THEORY • Euler-Bernoulli Beam Theory – can carry the transverse load – slope can change along the span (x-axis) – Cross-section is symmetric w.r.t. xy-plane – The y-axis passes through the centroid – Loads are applied in xy-plane (plane of loading) L F x y F Plane of loading y z Neutral axis A

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4 BEAM THEORY cont . • Euler-Bernoulli Beam Theory cont. – Plane sections normal to the beam axis remain plane and normal to the axis after deformation (no shear stress) – Transverse deflection (deflection curve) is function of x only: v ( x ) – Displacement in x-dir is function of x and y: u ( x , y ) y y ( dv / dx ) θ = dv / dx v ( x ) L F x y Neutral axis 0 (, ) () dv uxy u x y dx =− 2 0 2 xx du ud v y xd x d x ε == θ=
5 BEAM THEORY cont . • Euler-Bernoulli Beam Theory cont. – Strain along the beam axis: – Strain ε xx varies linearly w.r.t. y; Strain yy = 0 –C u r v a t u r e : – Can assume plane stress in z-dir basically uniaxial status • Axial force resultant and bending moment 2 0 2 xx du ud v y xd x d x == 00 / du dx ε= 22 / dv d x 2 0 2 xx xx dv EE E y dx σ εε 2 0 2 2 2 0 2 xx AA A xx A P dA E dA E ydA dx M yd A E y d A E y d A dx σε =− + ∫∫ Moment of inertia I ( x ) 0 2 2 PE A ME I dx = = EA : axial rigidity EI : flexural rigidity

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6 BEAM THEORY cont . • Beam constitutive relation – We assume P = 0 (We will consider non-zero P in the frame element) – Moment-curvature relation: • Sign convention – Positive directions for applied loads 2 2 dv ME I dx = Moment and curvature is linearly dependent + P + P + M + M + V y + V y y x p ( x ) F 1 F 2 F 3 C 1 C 2 C 3 y x
7 GOVERNING EQUATIONS • Beam equilibrium equations – Combining three equations together: – Fourth-order differential equation y y dV Vd x dx + dM Md x + y V M dx p 0( ) 0 y yy y dV fp x d x V d x V dx ⎛⎞ =⇒ + + = ⎜⎟ ⎝⎠ () y px =− 0 2 y dM dx M x p d x V d x dx −+ + + = 4 4 dv E Ip x dx = y V

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8 STRESS AND STRAIN • Bending stress – This is only non-zero stress component for Euler-Bernoulli beam • Transverse shear strain – Euler beam predicts zero shear strain (approximation) – Traditional beam theory says the transverse shear stress is – However, this shear stress is in general small compared to the bending stress 2 2 xx dv Ey dx σ =− 2 2 ME I dx = () (, ) xx M xy I 0 xy uv vv yx xx γ ∂∂ =+= += 0
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## This note was uploaded on 08/22/2011 for the course EML 4500 taught by Professor Staff during the Fall '08 term at University of Florida.

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Chap4 - CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES...

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