Finite Element Analysis of a Cantilever
Beam in Bending
Chris Bickford
Introduction to Finite Elements: Term Project
Spring 2015
Introduction to Finite Elements: Term Project
ABSTRACT
This report evaluates the performance of five types of solid continuum elements in
Abaqus 6.14, Reference (1). Finite element analysis (FEA) was carried out to evaluate
the tip deflection of a cantilever beam for two separate loading conditions (end load and
uniform load) and each of the five element types was used with varying mesh densities.
The approximations produced by each element type/mesh density combination were
compared against the closed form solution, Reference (2). Overall, it was found that
both the fully integrated and reduced integration linear elements performed the worst,
while the quadratic elements were the most accurate. However, the linear elements with
incompatible modes performed almost as well as the quadratic elements, and are less
costly.
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TABLE OF CONTENTS
ABSTRACT ..................................................................................................................... 1
TABLE OF CONTENTS .................................................................................................. 2
LIST OF FIGURES .......................................................................................................... 2
LIST OF TABLES ............................................................................................................ 2
A
Purpose .................................................................................................................... 3
B
Introduction: Cantilever Beam in Bending ................................................................. 3
C
B.1
Cantilever Beam Subject to End Loading ........................................................... 3
B.2
Cantilever Beam Subject to Uniform Loading .................................................... 4
Finite Element Model Development .......................................................................... 5
C.1
Parametric Study: Mesh Density and Element Type .......................................... 6
D
Results and Discussion ............................................................................................ 8
E
Conclusion .............................................................................................................. 11
References .................................................................................................................... 12
LIST OF FIGURES
Figure 1 Overview of the Cantilever Beam and Material Properties ................................ 3
Figure 2 End Load Condition ........................................................................................... 4
Figure 3 Loading and Boundary Conditions for End Load ............................................... 6
Figure 4 Loading and Boundary Conditions for Uniform Loading .................................... 6
Figure 5 Mesh Densities.................................................................................................. 7
Figure 6 End Loaded Cantilever Beam: Tip Displacement vs. Mesh Density.................. 9
Figure 7 Uniformly Loaded Cantilever Beam: Tip Displacement vs. Mesh Density ....... 10
Figure 8 Von Mises Stress Distribution for End Loaded Cantilever Beam..................... 11
LIST OF TABLES
Table 1 End Loaded Cantilever Beam............................................................................. 8
Table 2 Uniformly Loaded Cantilever Beam .................................................................... 8
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A
Purpose
The purpose of this report is to assess the validity of a finite element analysis (FEA)
approximation using various solid continuum elements in Abaqus 6.14. It will be
determined which elements provide the best approximation when evaluating a
fundamental engineering problem such as a cantilever beam in bending.
B
Introduction: Cantilever Beam in Bending
To adequately assess the performance of FEA for different elements, it is important to
choose a problem which can be readily solved with a closed form solution. This
provides a benchmark by which to measure the adequacy of a numerical approximation.
A cantilever beam with a rectangular cross-section will be the subject of the analyses
detailed herein.
The conditions for a cantilever beam are that the beam is fixed at one end and free at
the other. The same beam dimensions with be used for both loading conditions
described in the following subsections. It also assumed that the beam is made of
common structural steel. Dimensions and parameters for beam are outlined in Figure 1.
L
Y
Fixed
Free
Z
Cross-section
Elastic Modulus: E = 200 Gpa
Poisson’s Ration: ν = 0.3
Density: ρ = 7850 kg/m2
a
b
Length: L = 1.0 m
Height: a = 0.1 m
Width: b = 0.1 m
Figure 1 Overview of the Cantilever Beam and Material Properties
B.1 Cantilever Beam Subject to End Loading
The first case that is evaluated is the condition where the beam is subjected to a load at
the free end of the beam. For this analysis, the end load force is arbitrarily chosen to be
1x106 N. The exact solution for this case is readily obtained using formulas for
Reference (2) and is simplified below.
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Figure 2 End Load Condition
By starting with the equation for the bending moment along the length of the beam, you
have a second order partial differential equation, EQ. 1, subject to boundary conditions.
𝐸𝐼
𝑑2𝑦
𝑑𝑧 2
= −𝐹𝑧
EQ. 1
At x=L, y=0 (no deflection)
At x=L, dy/dx=0 (gradient horizontal)
After integrating twice you are left with EQ. 2:
𝐸𝐼𝑦 = −
𝐹𝑧 3
6
+ 𝐴𝑧 + 𝐵
EQ. 2
By plugging in the boundary conditions and solving for the constants of integration you
get the formula for deflection at any z, EQ. 3:
𝐸𝐼𝑦 = −
𝐹𝑧 3
6
+
𝐹𝐿2 𝑧
2
−
𝐹𝐿3
3
EQ. 3
Which simplifies to EQ. 4 at the free end (z=0):
𝑦=−
𝐹𝐿2
2𝐸𝐼
EQ. 4
B.2 Cantilever Beam Subject to Uniform Loading
The second case evaluated is the condition where the beam is uniformly loaded along
its length. For this analysis, the load per unit length is arbitrarily chosen to be 10000
N/m. Like the end loading condition, the exact solution can be readily found using
formulas from Reference (2). The derivation of the exact solutions is simplified below.
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Again, starting with the equation for the bending moment along the length of the beam,
you have a second order partial differential equation, EQ. 5, subject to boundary
conditions.
𝐸𝐼
𝑑2𝑦
𝑑𝑧 2
= −𝑃
𝑧2
EQ. 5
2
At x=L, y=0 (no deflection)
At x=L, dy/dx=0 (gradient horizontal)
After integrating twice you are left with EQ. 6:
𝐸𝐼𝑦 = −
𝑃𝑧 4
24
+ 𝐴𝑧 + 𝐵
EQ. 6
By plugging in the boundary conditions and solving for the constants of integration you
get the formula for deflection at any z, EQ. 7:
𝐸𝐼𝑦 = −
𝑃𝑧 4
24
+
𝑃𝐿3 𝑧
6
−
𝑃𝐿4
8
EQ. 7
Which simplifies to EQ. 8 at the free end (z=0):
𝑦=−
C
𝑃𝐿4
8𝐸𝐼
EQ. 8
Finite Element Model Development
The finite element model was developed using Abaqus 6.14 for static linear-elastic
analysis. Defining this model was simple due to small amount of loading and boundary
conditions. For both loading conditions, an encastre boundary condition was applied at
the surface of the fixed end of the beam. To simulate an end load, a surface traction
load was applied to the surface at the free end of the beam. For uniform loading, a
uniform pressure was applied to the top surface of the entire beam.
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ENCASTRE BC:
Surface is constrained in all
DOF
Surface Traction Load:
A Shear load is applied to the
free end surface to simulate an
end load
Figure 3 Loading and Boundary Conditions for End Load
ENCASTRE BC:
Surface is constrained in all
DOF
Surface Pressure:
A Pressure is applied to the top
surface to simulate uniform
loading
Figure 4 Loading and Boundary Conditions for Uniform Loading
C.1 Parametric Study: Mesh Density and Element Type
As stated in previous sections, the main concern of this report is to evaluate the
performance of different element types. To do this effectively, a parametric analysis was
performed in which every element type was used to analyze both loading conditions for
five different mesh densities. For this report, I used the student version of Abaqus 6.14,
therefore, I was limited to 1000 nodes per analysis, which was adequate for this study.
All mesh densities used 4 elements on the cross-section face, which isolated the mesh
variance to one dimension (along the length of the beam). An overview of the different
mesh refinements is show in Figure 5.
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2 Elements in the
z-direction
8 Elements in the
z-direction
16 Elements in
the z-direction
24 Elements in
the z-direction
32 Elements in
the z-direction
Figure 5 Mesh Densities
An important consideration for FEA is the order of element that is used. Most
commonly, elements are either linear (1st order) or quadratic (2nd order) for structural
analysis. The type of element that is used can have a large impact on the outcome of
the analysis. The element order is determined by the order of its basis function, which is
used to calculate approximate values of the equations being evaluated at each node.
Quadratic elements are more accurate because the quadratic basis function allows it to
achieve contours between nodes that linear functions cannot.
Another factor for element selection is the integration scheme. Typically, elements are
either fully integrated or they use a reduced integration scheme, which simply uses less
integration points between nodes. In Abaqus, fully integrated linear elements utilize four
integration points between nodes, while reduced integration elements only use one
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integration point between nodes. The values for the function being analyzed are
calculated at the integration points then extrapolated to other points by means of the
elements basis functions.
The elements evaluated for this study are:
1) Fully integrated linear solid elements, (C3D8 in Abaqus)
2) Reduced integration linear solid elements, (C3D8R in Abaqus)
3) Fully integrated linear solid elements with incompatible modes, (C3D8I in
Abaqus)
4) Fully integrated quadratic solid elements, (C3D20 in Abaqus)
5) Reduced integration quadratic solid elements, (C3D20R in Abaqus)
The C3D8I elements are similar to the C3D8, except they have added internal degrees
of freedom that improve their behavior in bending, which is why they are considered
here. The disadvantage to C3D8I elements is they rapidly loose performance as
element skew is increased, which does not make them ideal for complex geometries.
D
Results and Discussion
Considering all of the combinations for the parametric study, a total of fifty finite element
analyses were carried out. Each individual approximation was compared to the exact
solution to evaluate its performance. The results are summarized in Table 1 and Table
2.
Table 1 End Loaded Cantilever Beam
Elements
along the
length of the
beam
2
8
16
24
32
Linear
(C3D8)
90.480%
34.354%
6.658%
-1.286%
-4.402%
Percent Error of Exact Solution
Linear with
Linear Reduced
Incompatible
Integration
modes
(C3D8R)
(C3D8I)
13.797%
-24.532%
2.099%
-32.500%
0.775%
-32.910%
0.395%
-33.009%
0.223%
-33.047%
Quadratic
(C3D20)
99.662%
1.017%
0.386%
0.219%
0.146%
Quadratic
Reduced
Integration
(C3D20R)
1.233%
0.443%
0.234%
0.152%
0.103%
Table 2 Uniformly Loaded Cantilever Beam
Elements
along the
length of the
beam
2
8
16
24
32
Linear
(C3D8)
89.668%
33.972%
6.466%
-1.436%
-4.540%
Percent Error of Exact Solution
Linear with
Linear Reduced
Incompatible
Integration
modes
(C3D8R)
(C3D8I)
8.642%
-33.038%
2.101%
-33.193%
0.796%
-33.219%
0.376%
-33.254%
0.176%
-33.271%
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Quadratic
(C3D20)
9.816%
1.235%
0.403%
0.180%
0.081%
Quadratic
Reduced
Integration
(C3D20R)
1.445%
0.483%
0.201%
0.091%
0.024%
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Overall, the performance of each element was similar between the two loading cases.
The only outlier is the fully integrated quadratic element with 2 elements the length of
the beam. In that case, the mesh is really too coarse to make a fair judgement. When
doing parametric mesh studies in the future, it would be best to avoid a mesh that
course. The following plots, show how the approximations for each element type
compare to the exact solution.
Cantilever Beam Subject to End Loading:
Performance of Element Type vs. Mesh Density
0.3000
0.2500
Predicted Tip Deflection (m)
Linear (C3D8)
0.2000
Linear with Incompatible Modes
(C3D8I)
0.1500
Linear Reduced Integration
(C3D8R)
Quadratic (C3D20)
0.1000
Quadratic Reduced Integration
(C3D20R)
0.0500
Exact Solution
0.0000
0
5
10
15
20
25
30
35
Number of Elements Along the Length of the Beam
Figure 6 End Loaded Cantilever Beam: Tip Displacement vs. Mesh Density
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Cantilever Beam Subject to Uniform Loading:
Performance of Element Type vs. Mesh Density
0.001200
0.001000
Predicted Tip Deflection (m)
Linear (C3D8)
0.000800
Linear with Incompatible Modes
(C3D8I)
Linear Reduced Integration
(C3D8R)
0.000600
Quadratic (C3D20)
0.000400
Quadratic Reduced Integration
(C3D20R)
Exact Solution
0.000200
0.000000
0
5
10
15
20
25
30
35
Number of Elements Along the Length of the Beam
Figure 7 Uniformly Loaded Cantilever Beam: Tip Displacement vs. Mesh Density
The plots in Figure 6 and Figure 7 are good illustrations of the performance of each
element. The quadratic elements are the most accurate, as one would expect. The
reduced integration linear elements have a tendency to over predict the deflection and
therefore under predict the geometries stiffness. The fully integrated linear elements
tend to do the opposite, and over predict the stiffness, which is evidence of shear
locking. The linear elements with incompatible modes eliminate that behavior and
produce a much better prediction overall. Figure 8 shows the stress distribution given by
both element types and there is a visible difference between the two. This is important,
because if your using fully integrated elements, you may be over predicting the stiffness
and under predicting the strain/stress, which is often the first variable looked at for
structural qualification.
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Fully integrated Linear
Elements (C3D8)
Max Stress: 5.4E+09
Fully integrated Linear
Elements with incompatible
modes (C3D8I)
Max Stress: 5.9E+09
Figure 8 Von Mises Stress Distribution for End Loaded Cantilever Beam
E
Conclusion
Overall, the parametric study was successful in assessing the different performance
characteristics of various element types. Clear conclusions can be made about which
elements are more accurate in the presence of a closed form solution. The study
showed that the quadratic elements were consistently the most accurate, especially the
reduced integration version (C3D20R). However, the drawback to these elements is
they can be costly. While the fully integrated and reduced integration elements are not
adequate replacements, the linear element with incompatible modes proved to be
nearly as accurate as the quadratic elements. Being that they are less costly, they
would be best suited element to model a simple beam in bending.
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References
1) Dassault Systemes Simulia Corp. (2014). Abaqus/CAE 6.14. Providence, RI
2) Timoshenko, S., & Goodier, J,N, (1951). Theory of Elasticity. McGraw-Hill Book
Company.
3) Dunn, D.J (n.d.) Mechanics of Solids – Beams Tutorial 3, The Deflection of
Beams. http://www.freestudy.co.uk/statics/beams/beam%20tut3.pdf
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