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

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

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

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

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=0.25 - 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

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