Application of the Finite Element Method Using MARC and Mentat
41
Chapter 4: Tapered Beam
Keywords:
elastic beam, 2D elasticity, plane stress, convergence, deformed geometry
Modeling Procedures:
ruled surface, convert
4.1 Problem Statement and Objectives
A tapered beam subjected to a tip bending load will be analyzed in order to predict the
distributions of stress and displacement in the beam. The geometrical, material, and loading
specifications for the beam are given in Figure 4.1. The geometry of the beam is the same as the
structure in Chapter 3. The thickness of the beam is
2h
inches, where
h
is described by the
equation:
h
x
x
=

+
4
0 6
0 03
2
.
.
4.2 Analysis Assumptions
•
Because the beam is thin in the width (outofplane) direction, a state of plane stress can be
assumed.
•
The lengthtothickness ratio of the beam is difficult to assess due to the severe taper. By
almost any measure, however, the lengthtothickness ratio of the beam is less than eight.
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:
Tip Load: P=10,000 lbs
Figure 4.1 Geometry, material, and loading specifications for a tapered beam.
P
2h
L
x
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View Full DocumentApplication of the Finite Element Method Using MARC and Mentat
42
Hence, it is unclear whether thin beam theory will accurately predict the response of the beam.
Therefore, both a 2D plane stress elasticity analysis and a thin elastic beam analysis will be
performed.
4.3 Mathematical Idealization
Based on the assumptions above, two different models will be developed and compared. The first
model is a beam analysis. In this model, the main axis of the bar is discretized using straight two
noded 1D thin beam finite elements having a uniform crosssectional shape within each element.
Thus, the geometry is idealized as having a piecewise constant crosssection, as shown in Figure
4.2. The uniform thickness within each element is taken to be equal to the actual thickness of the
tapered beam 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 4.1. The 2D
finite element model of this structure will be developed using 2D plane stress bilinear fournoded
quadrilateral finite elements. In the present analysis, the geometry and material properties are
symmetric about the midplane of the beam. However, the loading is
not
symmetric about this
plane, so the response of the structure (i.e., displacements, strains, and stresses) will not be
symmetric about this plane. Hence, it is necessary to model the entire domain of the beam, as
shown in Figure 4.1.
4.4 Finite Element Model
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 Spring '05
 AVERILL
 Finite Element Method, Stress, Chemical element, main menu, Finite element method in structural mechanics, Elliptic boundary value problem

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