RENSSELAER POLYTECHNIC INSTITUTE
Application of the Finite
Element Method to
Deflection in a Cantilever
Beam
MANE 4240 – Introduction to Finite Elements
José DeFaria
5/14/2015
DeFaria 2
Abstract
A cantilever beam was modeled as a 1D, 2D, and 3D model in Abaqus using various element types to
determine its deflection. A mesh refinement study was conducted and the results compared to the exact
values to compare and contrast the accuracy of the various element types. Overall, it was determined
that quadratic elements are more accurate than linear elements, and traditional integration elements
are more accurate than reduced integration elements. Despite these generalized findings, the most
accurate element type at convergence was a linear, reduced integration element in 2D.
DeFaria 3
Table of Contents
Abstract ..................................................................................................................................................... 2
Table of Contents ...................................................................................................................................... 3
Introduction .............................................................................................................................................. 4
Finite Element Theory -- The Importance of Basis Functions .................................................................. 5
Abaqus Element Types .............................................................................................................................. 6
Model Construction .................................................................................................................................. 7
1D .......................................................................................................................................................... 7
2D .......................................................................................................................................................... 7
3D .......................................................................................................................................................... 8
Results ....................................................................................................................................................... 9
1D .......................................................................................................................................................... 9
2D ........................................................................................................................................................ 10
3D ........................................................................................................................................................ 11
1D, 2D, 3D Comparison ....................................................................................................................... 12
Conclusion ............................................................................................................................................... 13
References .............................................................................................................................................. 14
Appendix – Result Tables ........................................................................................................................ 15
1D ........................................................................................................................................................ 15
2D ........................................................................................................................................................ 15
3D ........................................................................................................................................................ 16
DeFaria 4
Introduction
Finite element analysis is a method of numerical analysis to approximate solutions to problems which
would be difficult to obtain using traditional analysis methods. Although variational calculus has existed
for centuries, the concept of using finite elements to solve difficult problems has taken new meaning in
the era of computers.
As computational strength increases, the effective cost of finite element analysis decreases. This has
increased the adoption of finite element analysis throughout industry as a method to determine
acceptability of a design.
Although finite element analysis is widely used, one must remember it is a method of approximation.
Different methods and inputs will affect the accuracy of the approximation.
This study will examine one of the classic problems in engineering: the cantilever beam. The cantilever
beam is a well-defined problem with specific formulas to determine its deflection based on the length
and section of the beam and the magnitude and location of the load. Because the cantilever beam is
well-defined, the exact value of its deflection can be determined and compared to the analysis results.
Analysis will be conducted on a 1D, 2D, and 3D model of the cantilever beam, using different mesh sizes
and element types to determine the relationship between element type/geometry and accuracy.
In a cantilever beam with a concentrated load at the free end, the maximum deflection occurs at the
free end. The maximum deflection is found by:
𝛿=
𝑃𝑙 3
3𝐸𝐼
For the purposes of this study, a cantilever beam 12 inches in length, 2 inches wide, and 1 inch tall
(I=0.166667 in4), will be considered to be constructed of steel (E=30,000,000 psi, v=0.3). A load of
500 lbs will be applied to the free end. Using these values, the exact solution of the maximum deflection
is:
𝛿=
(500 𝑙𝑏𝑠)(12 𝑖𝑛)3
𝑃𝑙 3
=
= 0.0576 𝑖𝑛
3𝐸𝐼 3(30,000,000 𝑝𝑠𝑖)(0.166667 𝑖𝑛4 )
DeFaria 5
Finite Element Theory -- The Importance of Basis Functions
The success of finite element analysis depends heavily on the selection of a test function. This test
function is determined by the collection of the finite element basis functions. This study uses two
different orders of elements: first-order elements (P1 or linear) and second-order elements (P2 or
quadratic). The graphs of sample linear and quadratic basis functions are presented below for an
interval divided into two elements.
Linear Basis Function
Quadratic Basis Function
In the linear example above, the interval is divided into two elements. Each element has one node at
each end for a total of three nodes. Each node has its own basis functions, which are assembled into the
global basis function for each element. The global basis functions for both elements are plotted above.
In the quadratic basis function above, the interval is also divided into two elements, but each element
has a node in the center of the element in addition to the nodes at each end. This results in a total of
five nodes. Each node has its own basis functions, which are assembled into the global basis function for
each element. The global basis functions for both elements are plotted above. The quadratic basis
functions are much more detailed since they are of second-order and they introduce additional nodes;
therefore, quadratic elements are expected to be more accurate than linear elements.
DeFaria 6
Abaqus Element Types
Although it is unknown what particular test functions the Abaqus software uses, the concepts presented
above are the same. Elements can be linear or quadratic. Quadratic elements introduce additional
nodes compared to linear elements. This can be observed for 1D, 2D, and 3D elements below.
Linear
1D
2 nodes
2D
4 nodes
3D
8 nodes
Quadratic
3 nodes
8 nodes
20 nodes
In addition to the element types above, Abaqus also offers a “reduced integration” option which is on by
default in 2D and 3D analysis. When this method is off, the software uses Simpson’s rule, a method of
numerical analysis of definite integrals where the interval is broken into fixed intervals for evaluation.
When the reduced integration option is turned on, a different method called Gaussian Quadrature is
used. This does not use fixed intervals and can often reach the same approximation as Simpson’s rule
with less coordinates and therefore is less computationally expensive. This reduced integration method
can be prone to “hourglassing” in certain applications.
Geometry
Element Type
1D
B21
B22
S4
S4R
S8R
C3D8
C3D8R
C3D20
C3D20R
2D
3D
Order
Linear
Quadratic
Linear
Linear
Quadratic
Linear
Linear
Quadratic
Quadratic
Description
# of Nodes
2
3
4
4
8
8
8
20
20
Integration
Reduced
Reduced
Reduced
Reduced
DeFaria 7
Model Construction
1D
The 1D model was created by creating a wire part twelve inches long. The steel material properties were
defined in the property module and the cross-section was defined in the create profile tab. In the load
module, the left end of the beam was considered fixed (encastre) and a 500 pound load was placed on
the right end. The beam was meshed with an increasing number of elements. For B21 elements (linear),
the number of elements was increased from 1 through 20 and the deflection in the y-direction was
recorded using each mesh. For B22 elements (quadratic) the number of elements was increased from 1
through 10. At that point, it was determined the results had already converged, and the mesh
refinement was terminated. It should be noted that Abaqus automatically added additional nodes to the
1D model. This was not observed with the 2D or 3D models.
2D
The 2D model was created by creating a planar shell twelve inches long and 2 inches wide. The steel
material properties were defined in the property module and the thickness of 1 inch was applied in the
section properties. In the load module, the left end of the beam was considered fixed (encastre) and a
500 pound load was placed at the center of the beam on the right end. The beam was meshed with
quadrilateral elements. For S4, S4R, and S8R elements, the mesh was created by creating a single
lengthwise element. The model was solved and the deflection in the y-direction was recorded. The mesh
was then refined by doubling the number of elements and the new deflection was recorded. This
process was repeated up to 1024 lengthwise elements. Some hourglassing was observed with 1028 S8R
elements
DeFaria 8
3D
The 3D model was created by creating a solid extrusion twelve inches long, 2 inches wide, and 1 inch
deep. The steel material properties were defined in the property module. In the load module, the left
face of the beam was considered fixed (encastre) and a 500 pound load was placed in the center of the
right face. The beam was meshed with hexahedral elements. For C3D8, C3D8R, C3D20, and C3D20R
elements, the mesh was created by creating a single lengthwise element. The model was solves and the
deflection in the y-direction was recorded. The mesh was then refined by doubling the number of
elements and the new deflection was recorded. This process was repeated up to 512 lengthwise
elements. Some hourglassing was observed at 512 reduced integration elements.
DeFaria 9
Results
The recorded deflections were noted in Microsoft Excel, and graphed as a function of the number of
lengthwise elements. In each case below, the horizontal axis is logarithmic.
1D
1D Mesh Refinement
Exact
B21
B22
0.0585
0.058
0.0575
Deflection (inches)
0.057
0.0565
0.056
0.0555
0.055
0.0545
0.054
1
2
4
8
16
Number of (Lengthwise) Elements
As shown in the graph above, both the linear (B21) and quadratic (B22) elements converged to a value
which was within 0.5% of the actual value. Although better accuracy was noted with some of the other
models presented later, this accuracy is impressive considering the minimal resources necessary to
evaluate a simple one-dimensional system.
The graph above also highlights the accuracy difference between linear and quadratic elements. Using
only a single quadratic element, the model was within 0.5% of the exact value, whereas using a single
linear element the displacement was within 5.5% of the exact value. This is an order of magnitude in
precision.
DeFaria 10
2D
2D Mesh Refinement
Actual
S4
S4R
S8R
0.06
Deflection (inches)
0.055
0.05
0.045
0.04
1
2
4
8
16
32
64
128
256
512
1024
Number of (Lengthwise) Elements
Similar to what was observed in the 1D case, the quadratic elements (S8R) were far more accurate at
the beginning than the linear elements (S4, S4R). With a single quadratic element, the deflection was
found to be within 1.5% of the actual value, while a single linear element was only within 25% of the
actual value. As the number of S8R elements increased, the accuracy increased, until approximately 512
elements, when minor hourglassing was observed. It should be noted there is no option for a quadratic
element without reduced integration to compare these values with.
The linear elements can be used to compare the results of reduced integration. The reduced integration
elements (S4R) were more accurate than the traditional elements (S4) throughout all of the evaluated
meshes. Using 1024 elements, the traditional elements resulted in a deflection that was within 0.14% of
the actual value, and the reduced integration elements resulted in a deflection that was within 0.06%
the actual value. The reduced integration elements (S4R) resulted in the highest accuracy of all element
types evaluated in this study.
DeFaria 11
3D
3D Mesh Refinement
Actual
C3D8
C3D8R
C3D20
C3D20R
0.09
0.08
0.07
Deflection (inches)
0.06
0.05
0.04
0.03
0.02
0.01
0
1
2
4
8
16
32
64
128
256
512
Number of (Lengthwise) Elements
The three-dimensional model provided the only opportunity to compare both linear and quadratic
elements with both reduced and traditional integration.
The reduced integration elements were both more accurate than the traditional elements initially. The
linear reduced integration element (C3D8R) was within 0.1% of the actual value, while the quadratic
integration element (C3D20R) was within about 4%. These accuracies are much better than the
traditional linear element (C3D8, 98%) and the traditional quadratic (C3D20, 26%). As the mesh was
refined the C3D20R element became more accurate, until 512 elements were used and minor
hourglassing was experienced; however, the C3D8R elements continued to become more inaccurate,
and never truly converged compared to the other element types.
In comparison between linear and quadratic elements, the quadratic elements were much more
accurate. The values that the quadratic elements converged upon were within 0.5% and 1.5% of the
actual value. In this case, the traditional quadratic element was more accurate than the reduced
integration element due to the hourglassing discussed above.
DeFaria 12
1D, 2D, 3D Comparison
Linear Element Comparison by Geometry
Actual
B21
S4
C3D8
0.07
0.06
Deflection (inches)
0.05
0.04
0.03
0.02
0.01
0
1
2
4
8
16
32
64
128
256
512
1024
Number of (Lengthwise) Elements
The graph above compares traditional linear elements in each of the three geometries. Minor
differences in how the software interprets boundary conditions in loads in 1D/2D/3D may affect the
convergence value. However, the left-hand side of the graph, depicting the low-element count results
paints a clear picture: reducing the degrees of freedom increases the accuracy of the analysis. With only
a single element, the 1D, 2D, and 3D models were within 0.5%, 26%, and 98% of the actual value,
respectively. Mesh refinement greatly reduces these gaps, but a simpler model is solved more
accurately by the software.
DeFaria 13
Conclusion
This study evaluated a cantilever beam as a 1D, 2D, and 3D model and compared linear and quadratic
elements in all three model spaces, as well as reduced integration elements with traditional elements.
The table below lists all the element types evaluated and presents the final variance from the actual
value at the maximum number of elements (20 for 1D, 1028 for 2D, 512 for 3D).
Geometry
1D
2D
3D
Element
Type
B21
B22
S4
S4R
S8R
C3D8
C3D8R
C3D20
C3D20R
Order
Linear
Quadratic
Linear
Linear
Quadratic
Linear
Linear
Quadratic
Quadratic
Description
# of
Integration
Nodes
2
3
4
4
Reduced
8
Reduced
8
8
Reduced
20
20
Reduced
Variance from
Actual Value at
Max # of Elements
+ 00.5163 %
+ 00.5311 %
- 00.1358 %
+ 00.0623 %
+ 00.9789 %
+ 07.7670 %
+ 44.3150 %
- 00.4698 %
+ 01.6653 %
In respect to model geometry, generally speaking, reducing the number of nodes and degrees of
freedom also reduces the computational cost and increases the accuracy of the model. In practical
applications, engineers should simplify models if possible to generate and accurate, reliable, and costeffective solution.
In respect to linear versus quadratic elements, quadratic elements are nearly always more accurate
which agrees with the theory. Quadratic elements were more accurate in both 3D cases, and the values
were nearly identical in the 1D case. In the 2D case the linear element was more accurate than the
quadratic, but this is due to minor hourglassing due to the use of reduced integration elements. A
quadratic element without reduced integration is not an option in Abaqus for 2D models.
In respect to reduced integration elements, they are always more accurate than traditional elements
when a low number of elements is used; however, as the mesh is refined, they are nearly always less
accurate. This failure with a high number of elements is due to hourglassing. Use of reduced integration
elements is likely acceptable when a quick evaluation is needed, or for preliminary runs of the model to
determine baseline acceptance. When a higher degree of accuracy is required, reduced integration
elements should not be selected.
This study also highlighted the importance of mesh refinement. Without mesh refinement, an engineer
is simply accepting any random point along the line as the answer. However, in some situations, the
answer received with a course mesh is significantly different than that with a fine mesh. The S4R
element was the most accurate overall, but with using only 1 S4R element, the result was within 24% of
the actual value. Refining the mesh to ensure the values are heading towards convergence is an
important step in developing confidence in the results of finite element analysis.
DeFaria 14
References
American Forest & Paper Association. (2007). Beam Design Formulas with Shear and Moment Diagrams.
National Design Specification for Wood Construction. Washington, DC: American Wood Council.
American Institute of Steel Construction. (2005). Steel Construction Manual. American Institute of Steel
Construction.
Dassault Systemes Simulia Corp. (2013). Abaqus/CAE User's Guide. Abaqus 6.13. Providence, Rhode
Island.
Dunn, D. J. (n.d.). Mechanics of Solids - Beams Tutorial 3, The Deflection of Beams. Retrieved May 02,
2015, from FreeStudy: http://www.freestudy.co.uk/statics/beams/beam%20tut3.pdf
Gutierrez-Miravete, E. (2015, May 11). Index. Retrieved May 13, 2015, from Introduction to Finite
Elements: http://www.ewp.rpi.edu/hartford/~ernesto/S2015/IFEM/
The Welding Institute. (2015). FAQ: What is reduced integration in the context of finite element analysis?
Retrieved April 30, 2015, from The Welding Institute: http://www.twi-global.com/technicalknowledge/faqs/structural-integrity-faqs/faq-what-is-reduced-integration-in-the-context-offinite-element-analysis/
Timoshenko, S., & Goodier, J. N. (1951). Chapter 3, Two-Dimensional Problems in Rectangular
Coordinates. In Theory of Elasticity (pp. 29-44). McGraw-Hill Book Company.
DeFaria 15
Appendix – Result Tables
1D
# of
Elements
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
B21
Deflection (in)
0.05451765
0.05705882
0.05752941
0.05769412
0.05777035
0.05781176
0.05783673
0.05785294
0.05786405
0.05787200
0.05787788
0.05788235
0.05788583
0.05788860
0.05789082
0.05789265
0.05789416
0.05789543
0.05789649
0.05789741
% of actual
94.6487%
99.0605%
99.8774%
100.1634%
100.2957%
100.3676%
100.4110%
100.4391%
100.4584%
100.4722%
100.4824%
100.4902%
100.4962%
100.5010%
100.5049%
100.5081%
100.5107%
100.5129%
100.5147%
100.5163%
B22
Deflection (in)
0.05790616
0.05790595
0.05790591
0.05790590
0.05790589
0.05790589
0.05790589
0.05790589
0.05790589
0.05790589
0.05790589
0.05790589
0.05790589
0.05790589
0.05790589
0.05790589
0.05790589
0.05790589
0.05790589
0.05790589
% of actual
100.5315%
100.5312%
100.5311%
100.5311%
100.5311%
100.5311%
100.5311%
100.5311%
100.5311%
100.5311%
100.5311%
100.5311%
100.5311%
100.5311%
100.5311%
100.5311%
100.5311%
100.5311%
100.5311%
100.5311%
2D
# of
Elements
1
2
4
8
16
32
64
128
256
512
1024
S4
Deflection (in) % of actual
0.04257879
73.9215%
0.05227075
90.7478%
0.05585212
96.9655%
0.05699317
98.9465%
0.05736917
99.5993%
0.05743067
99.7060%
0.05751153
99.8464%
0.05751923
99.8598%
0.05752116
99.8631%
0.05752165
99.8640%
0.05752177
99.8642%
S4R
Deflection (in)
0.04358575
0.05369245
0.05652184
0.05729374
0.05753655
0.05748225
0.05762911
0.05763422
0.05763550
0.05763583
0.05763590
% of actual
75.6697%
93.2161%
98.1282%
99.4683%
99.8898%
99.7956%
100.0505%
100.0594%
100.0616%
100.0622%
100.0623%
S8R
Deflection (in)
% of actual
0.05678657
98.5878%
0.05705557
99.0548%
0.05729967
99.4786%
0.05739415
99.6426%
0.05743768
99.7182%
0.05747188
99.7776%
0.05751752
99.8568%
0.05757328
99.9536%
0.05766518
100.1132%
0.05783490
100.4078%
0.05816386
100.9789%
DeFaria 16
3D
# of
Elements
1
2
4
8
16
24
32
64
128
256
512
C3D8
Deflection (in)
0.00103100
0.00391180
0.01312751
0.03206815
0.05026453
0.05619560
0.05862415
0.06118198
0.06185892
0.06203071
0.06207382
% of actual
1.7899%
6.7913%
22.7908%
55.6739%
87.2648%
97.5618%
101.7780%
106.2187%
107.3940%
107.6922%
107.7670%
C3D8R
Deflection (in) % of actual
0.05748955
99.8082%
0.07096117
123.1965%
0.07508980
130.3642%
0.07627323
132.4188%
0.07682697
133.3802%
0.07713806
133.9202%
0.07740237
134.3791%
0.07835814
136.0384%
0.08021381
139.2601%
0.08398886
145.8140%
0.08312544
144.3150%
# of
Elements
1
2
4
8
16
24
32
64
128
256
512
C3D20
Deflection (in) % of actual
0.04245245
73.7022%
0.05230047
90.7994%
0.05560777
96.5413%
0.05666790
98.3818%
0.05705437
99.0527%
0.05716575
99.2461%
0.05721729
99.3356%
0.05728694
99.4565%
0.05731603
99.5070%
0.05732640
99.5250%
0.05732942
99.5302%
C3D20R
Deflection (in) % of actual
0.05521088
95.8522%
0.05609409
97.3856%
0.05670690
98.4495%
0.05700196
98.9617%
0.05718949
99.2873%
0.05727248
99.4314%
0.05732628
99.5248%
0.05746246
99.7612%
0.05764943
100.0858%
0.05796485
100.6334%
0.05855921
101.6653%