1. Numerical Analysis of a Beam

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

3. 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

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

5. Governing ODE

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

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

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

9. Numerical Differentiation Results

10. Dormand and Prince (RK5)
ODE system

11. RK4 Results
ODE system

12. FEM Results

13. Exact Solution

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

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

16. 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
Kwon, Y., & Bang, H. (2000). The Finite Element Method. Florida.