VM5

VM5
Laterally Loaded Tapered Support Structure

Overview

Reference:

S. H. Crandall, N. C. Dahl, An Introduction to the Mechanics of Solids, McGraw-Hill Book Co., Inc., New York, NY, 1959, pg. 342, problem 7.18.

Analysis Type(s):

Static Analysis (ANTYPE = 0)

Element Type(s):

2D 4-Node Sructural Solid Elements (PLANE182)

2D 8-Node Structural Solid Elements (PLANE183)

Input Listing:

vm5.dat

Test Case

A cantilever beam of thickness t and length has a depth which tapers uniformly from d at the tip to 3d at the wall. It is loaded by a force F at the tip, as shown. Find the maximum bending stress at the mid-length (X = ) and the fixed end of the beam.

Figure 6: Cantilever Beam Problem Sketch

Material Properties

Geometric Properties

E = 30 x 10⁶ psi

υ = 0.0

= 50 in

d = 3 in

t = 2 in

F= 4000 lb

The PLANE183 model uses the same node numbering at the element corners as the PLANE182 model and has additional midside nodes added to all elements.

Analysis Assumptions and Modeling Notes

Two different solutions are obtained. The first solution uses lower order PLANE182 elements and the second solution uses higher order PLANE82 elements. The 2 inch thickness is incorporated by using the plane stress with thickness option. Poisson's ratio is set to 0.0 to agree with beam theory.

Results Comparison

		Target	Mechanical APDL	Ratio
PLANE182	at mid-length stress, psi	8,333.	8,163.656	.980
PLANE182	at fixed end stress, psi	7,407.	7,151.096	.965
PLANE183	at mid-length stress, psi	8,333.	8,363.709	1.004
PLANE183	at fixed end stress, psi	7,407.	7,408.980	1.000

Figure 7: Displacement Display