1. Pure Elastic 2D Beam Bending Using Numerical Methods
Scott M. Steffan
12/9/99
Term Project
MEAE 4960HO2
Rensselaer Polytechnic Institute Hartford
Assumptions:
1. Beam Subject to Pure Bending
2. Material is Isotropic and Homogeneous
3. Deflections are Small
4. Beam Bending is Pure Elastic

2. Loadcase 1 - Simply Supported w/ Uniform Distributed Load
Pure Elastic Beam Bending - Using Numerical Methods
Two Different Loading Conditions Studied
w = 1e5
L = 10
E = 1e11
width = 1
height = 1
I = 1/12 y
Loadcase 2 - Simply Supported w/Linearly Varying, Distributed Load x z -w L
-wx

3. Pure Elastic Beam Bending - Using Numerical Methods
Closed Form Solutions Can Be Generated Through Integration
Loadcase 1:
Loadcase 2:
Simply Supported Boundary Conditions:

4. Pure Elastic Beam Bending - Using Numerical Methods
Finite Element Method - Loadcase 1 Mesh Density Study
- Better Approximation of True Beam Deformation w/ Higher # Elements
2 Element Mesh
40 Element Mesh
Studied 2, 4, 5, 8, 10, 20, and 40 Element Meshes
2-noded Linear Beam Elements
Monitored Deflection Every 0.25 units in length
Lower # Element Meshes Too Stiff - Underestimate Deflection & Stress
2 Elements

5. Pure Elastic Beam Bending - Using Numerical Methods
Other Numerical Methods Used to Approximate 2nd Order DE
Linear “Shooting” Method
Finite Difference Method
Raleigh-Ritz Method
h= Loadcase 1
h=0.5 - Loadcase 2
Each Method Produced Good Deflection Approximations for both Loadcases (within 0.3%)
Finite Difference Method Recorded Highest % Error for both Loadcases
2 Choices for Increased FDM Accuracy:
1.) Increase # of Stations (lower h)
2.) Approximate derivatives w/ Higher Order Taylor Series
Finite Difference Method - Loadcase 2

6. Pure Elastic Beam Bending - Using Numerical Methods
Each Method Has +’s and -’s
Finite Element Analysis
+ Very Accurate w/ well shaped elements and dense mesh
+ Versatile - can handle irregular shapes and difficult boundary conditions easily
- Longer Solution Time
Linear “Shooting” Method
+ Short Solution Time
+ Accurate
- Can present instability problems
Finite Difference Method
+ Better Stability than “Shooting”
- Lower Accuracy
Raleigh-Ritz Method
+ Method Similar to FEM
- Not as Accurate as FEM