VM148

VM148
Bending of a Parabolic Beam

Overview

Reference: S. Timoshenko, Strength of Material, Part I, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1955, pg. 210, article 46.
Analysis Type(s): Static Analysis (ANTYPE = 0)
Element Type(s):
3D 20-Node Structural Solid Elements (SOLID95)
3D 20-Node Structural Solid Elements (SOLID186)
Input Listing: vm148.dat

Test Case

A beam having a parabolic depth-to-length variation is subjected to an end load as shown. The other end is supported at a wall. Determine the deflection δ at the tip of the beam.

Figure 204: Parabolic Beam Problem Sketch

Parabolic Beam Problem Sketch

Material PropertiesGeometric PropertiesLoading
E = 30 x 106 psi
G = 1.5 x 108 psi
υ = 0.0
= 4 in
ho = 2 in
b = 0.2 in
F = -1000 lb

Analysis Assumptions and Modeling Notes

The problem is solved first using 3D solid (SOLID95) and then using 3D solid SOLID186 elements.

A large shear modulus G is assumed (1.5 x 108) and the Poisson's ratio is taken as zero to match the theoretical assumptions. The six nodes at the top and bottom edges near the tip of element 1 are defined closer to the tip so that the two mid-edge nodes (11 and 71) are not improperly located. Other nodes along the parabolic edge are generated with parametric input, at uniform spacing along the axis, using the equation:

Results Comparison

TargetMechanical APDLRatio
SOLID95
Deflection, in-0.01067-0.01062[1]0.995
SOLID186
Deflection, in-0.01067-0.010761.009

  1. UY at node 11 or 71.