Numerical Analysis of a Beam

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Presentation transcript:

Numerical Analysis of a Beam

The Problem Use several numerical analysis tools to calculate tip deflection and compare accuracies F L

Euler-Bernoulli beam theory Simplification of the linear theory of elasticity which relates loads to deflections in beams Beam theory governing equation Often, u=u(x), w=w(x), and EI is constant, yielding curve u(x) describes deflection u of the beam at some position x, and w is the distributed load Shear Force Bending Moment Slope Deflection

Boundary conditions x=o (fixed end) x=L (free end) F L

Governing ODE

Inputs Material properties F E=200x10^9 Pa I=6.7x10^-5 m^4 L=6m F=1000 N F L

Boundary conditions 0<=x<=L IVP v(x=0)=0 v’(x=L)=slope F L

Methods for comparison Numerical Differentiation Backward Euler Method RK4 FEM (NASTRAN/PATRAN) exact

Numerical Differentiation Results

Dormand and Prince (RK5) ODE system

RK4 Results ODE system

FEM Results

Exact Solution

Results Summary Method Max Tip Displacement % Error Numerical Differentiation .005379m 0.11 RK5 .005371m 0.037 RK4 FEM .005350m 0.428 Exact Solution .005373m 0.00

Animation of Deflection diving board\animation.m *courtesy of University of Wyoming Electrical and Computer Engineering Dept

References Riley,W.,Sturges,L.,& Morris, D. (1999). Mechanics of Materials. New York University of Wyoming Electrical and Computer Engineering Dept. (2008). Beam Deflection. From http://www.eng.uwyo.edu/classes/matlabanimate/StaticBeam/cantileverbeamoneload.m Kwon, Y., & Bang, H. (2000). The Finite Element Method. Florida.