Application of the Finite Element Method Using MARC and Mentat
31
Chapter 3: Tapered Bar
Keywords:
1D elasticity, 2D elasticity, plane stress, model symmetry, convergence
Modeling Procedures:
ruled surface, convert
3.1 Problem Statement and Objectives
A tapered bar subjected to an axial load will be analyzed in order to predict the distributions of
stress and displacement in the bar. The geometrical, material, and loading specifications for the
bar are given in Figure 3.1. The thickness of the bar is
2h
inches, where
h
is described by the
equation:
h
x
x
=

+
4
0 6
0 03
2
.
.
3.2 Analysis Assumptions
•
Because the bar is thin in the width (outofplane) direction, a state of plane stress can be
assumed.
•
Even though the load is exclusively axial, the taper in the bar may cause the state of stress to
be twodimensional in nature. The effect of taper on the stress state depends upon the degree
of the taper, and is difficult to assess
apriori
. Therefore, both a 2D plane stress elasticity
analysis and a 1D elasticity analysis will be performed.
Geometry:
Material:
Steel
Length: L=10”
Yield Strength: 36 ksi
Width: b=1” (uniform)
Modulus of Elasticity: 29 Msi
Thickness: 2h (a function of x)
Poisson’s Ratio: 0.3
Density = 0.0088 slugs/in
3
Loading:
Axial Load: P=10,000 lbs
Figure 3.1 Geometry, material, and loading specifications for a tapered bar.
P
2h
L
x
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32
3.3 Mathematical Idealization
Based on the assumptions above, two different models will be developed and compared. The first
model is a 1D elasticity analysis. In this model, the main axis of the bar is discretized using linear
twonoded 1D bar/truss finite elements having a uniform thickness within each element. Thus, the
geometry is idealized as having a piecewise constant crosssection, as shown in Figure 3.2. The
uniform thickness within each element is taken to be equal to the actual thickness of the tapered
bar at the xcoordinate corresponding to the centroid of that element.
The second model is a 2D plane stress model of the geometry as shown in Figure 3.1. The 2D
finite element model of this structure will be developed using 2D plane stress bilinear fournoded
quadrilateral finite elements. The present analysis can be greatly simplified by taking advantage of
the horizontal plane of symmetry in the bar. The geometry, material properties, and loading
conditions are all symmetric about this plane. Therefore, the response of the structure (i.e.,
displacements, strains, and stresses) will also be symmetric about this plane. Hence, it is necessary
to model only a onehalf of the bar, as shown in Figure 3.3. The boundary conditions on the
symmetry plane are those that occur naturally on this plane, as can be verified by obtaining a
solution using the entire bar domain. In particular, the vertical displacement and the shear traction
are zero along the symmetry plane. Taking advantage of symmetry reduces the modeling effort,
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 Spring '05
 AVERILL
 Partial Differential Equations, Finite Element Method, Stress, main menu, Finite element method in structural mechanics, Elliptic boundary value problem

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