Comparison of Finite
Element Analysis to
Analytical Models when
Bending a Thin Plate
MANE 4240 – INTRODUCTION TO FINITE ELEMENT ANALYSIS
THOMAS PROVENCHER
Table of Contents
Table of Contents
Abstract: .......................................................................................................................................... 1
Introduction: .................................................................................................................................... 2
Formulation and Solution: .............................................................................................................. 3
Empirical Solution ...................................................................................................................... 3
Finite Element Analysis Mesh Density Analysis........................................................................ 4
Empirical and FEA Reliabily as Thickness is Modified ........................................................... 10
Discussion: .................................................................................................................................... 12
Conclusions:.................................................................................................................................. 14
References: .................................................................................................................................... 14
Abstract:
This report will analyze the accuracy of several different types of Finite Element Analysis (FEA)
elements and mesh densities by comparing them to theoretical, empirical solutions. A
rectangular plate will have a pressure load placed on it and plate bending theory will be used to
calculate the maximum stress in the plate as well as its maximum deflection. The results from
the various FEA analysis methods will then be compared to the plate bending theory results to
determine which element type and mesh density provides the greatest degree of accuracy without
severely increasing computational time. The robustness of the plate bending theory will then be
tested by increasing the thickness of the plate and comparing its new stress and deflection values
to those calculated by an FEA analysis. A minimum recommended plate width to thickness ratio
will be calculated; below which any plate bending theory results should be scrutinized carefully.
1
Introduction:
Plate bending is a well understood mechanical property with hundreds of years of wellestablished practical research able to provide accurate analytical predictions for material stress
and deflection. The Roark’s Formulas for Stress and Strain, Reference (a), is one of many
resources available which can provide empirical solutions to nearly any plate loading case from
round plates to rectangular plates, from point or line loads to uniform pressure distributions, all
with practically any set of boundary conditions. Over the past 20-30 years, the numerical
calculation capacity of computers has allowed for the use of Finite Element Analysis (FEA) to
solve for nearly any mechanical system from simple plate bending to complex interaction models
between parts in an assembly. The extreme versatility of these FEA software packages does
inherently increase the complexity of properly using them as the many different element types
and mesh densities can have a dramatic impact on the accuracy of the solution. These
investigations will attempt to clarify which element types and mesh densities should be used for
plate bending.
For this analysis, a rectangular plate will be subjected to a uniform pressure load across the entire
top surface with simple supports along three of the bottom edges and a free, unsupported, edge
along one of the short sides. Reference (a) will be used to provide the benchmark maximum
stress and deflection values and will be used to compare the accuracy of the various FEA
element and mesh density options available within DASSAULT SYSTEMS ABACUS UNIFIED
FEA. The flexibility and robustness of the Reference (a) equations will then be analyzed by
increasing the thickness of the plate and comparing the deflection and stress results to those
provided by the FEA analysis using the element types and mesh density found to be the most
accurate in the first investigation.
2
Formulation and Solution:
Empirical Solution
For this investigation, a 6061-T6 aluminum plate, two meters long, one meter wide, and
nominally one cm thick, was subjected to a uniform pressure across its entire surface of
1000pascals. The material properties, shown in Table 1 were from Reference (b). Three of its
lower edges, both long ones and one of the short ones, were simply supported leaving the final
edge free to move. The equations provided by Reference (a), shows in Figure 1, were used to
determine the maximum deflection of the plate as well as the maximum stress. Both of these
values were located at the center of the free edge. Per Reference (a) the deflection values
calculated are negative which assumes the pressure load placed on the plate is in the negative
direction. The material properties, plate dimensions and pressure load was input into PTC
MATHCAD which then provided the final maximum deflection and stress using the Reference
(a) equations. The MATHCAD output shown in Figure 2 provided a maximum stress of 7.9MPa
and a maximum deflection of approximately 2.4mm in the downward direction.
Table 1: Material properties, plate dimensions, and pressure load
6061-T6 Aluminum Physical Properties
Density (kg/m3)
Modulus of Elasticity, E (Gpa)
Poisson's Ratio
2700
68.9
0.33
Plate Dimensions and Loads
Length, a (m)
Width, b (m)
Thickness, t (m)
Uniform Pressure, q (Pa)
2
1
0.01
1000
Figure 1: Reference (a) deflection and stress equations
3
Figure 2: MATHCAD input values and final maximum deflection and stress
Finite Element Analysis Mesh Density Analysis
The thin plate was modeled in ABACUS using the proper dimensions and material
characteristics from Table 1 in order to make the analysis comparable to the empirical results
calculated in Figure 2. Figure 3 shows the final shape of the plate. The mesh chosen for the
analysis consisted of simple rectangular cuboids throughout the entire plate. The initial mesh
density chosen created four solid elements throughout the thickness and 5,000 solid elements
across the surface, thus producing 20,000 solid elements in total. This mesh can be seen in
figures 4 and 5. The initial solid element type selected was linear, incompatible modes, which
are known to predict bending accurately.
Figure 3: ABACUS model of the plate
4
Figure 4: Initial mesh density chosen for the first FEA computation
Figure 5: The initial mesh density provided 4 elements through the thickness
A uniform pressure load of 1000pascals was placed across the entire top surface of the plate in
the negative “z” direction. The bottom edge of both the long sides as well as one of the short
side edges were restricted to only being able to move in the “x” and “y” directions by using
boundary conditions to lock the “z” direction. One corner was completely locked in all three
directions and another was locked in the “x” and “z” directions. These boundary conditions were
chosen to simulate all three of those edges being simply supported. The simply supported
condition prevents up and down movement but permits sliding. In order to prevent the whole
plate from sliding around during the FEA computations, the corner boundary conditions were
added to provide a solid reference location for the plate to remain locked to without preventing
the edges from moving as required. Figures 6 and 7 illustrate the boundary conditions.
5
Uniform pressure load in
the negative “z” direction
Figure 6: Pressure loading on the top surface
Locked in all directions
No boundary
conditions,
free to move
Locked in the “z”
direction only
Locked in the “x” and
‘z” directions, free to
slide in the “y” direction
Figure 7: Boundary conditions placed on the bottom edges of the plate
The ABACUS software was then instructed to run the analysis with the provided geometry,
mesh controls, elements types, loading, and boundary conditions. The maximum absolute
principle stress results of this initial solution can be seen in Figure 8. The mesh density was then
6
modified by decreasing or increasing the number of elements through the thickness as well as the
number of elements present along each edge. Table 2 summarizes the varying mesh densities
analyzed. The element type was also modified by selecting either linear, linear reduced
integration, or quadratic, in addition to the initially selected linear incompatible modes. The
maximum absolute principle stress and maximum deflection values for the three mesh densities
and four element types are tabulated in Tables 3 and 4, respectively
Figure 8: Max absolute principle stresses, 1 cm thick plate, 4 elements thick, 20,000 linear incompatible modes elements
Table 2: Three different mesh densities studied and the number of elements present
Number of
Number of
Number of
Number of
Total
Mesh
elements through elements on the elements on the elements per number of
Density the thickness
short edges
long edges
layer
elements
1
2
25
50
1250
2500
2
4
50
100
5000
20000
3
6
67
133
8911
53466
A fifth type of element was also analyzed, the shell element. Shell elements are theoretical
elements which have no physical thickness when viewed in the model. These elements are able
to accurately predict plate bending even with a low surface mesh density as their computations
are heavily based on plate bending theory. Shell elements also reduce computational time
because instead of the calculation complexity needed to solve 3D FEA problems, only 2D FEA
calculations are required. The thickness of the plate is added to the 2D calculations as a property
instead of forcing 3D equations to be used.
The mesh densities chosen for the various shell element solutions was the same as those used by
the solid elements except there was effectively only one element through the thickness.
7
Therefore, the final number of elements in each shell element analysis is equal to the number of
elements per layer shown in Table 2. The maximum absolute principle stress output model for
“mesh density 2” is shown in Figure 9. Tables 3 and 4 also include the maximum absolute
principle stress and maximum deflection values, respectively, of the shell element solutions.
Figure 9: Max absolute principle stresses, 1 cm thick plate, 5,000 shell elements
Table 3: Effect of element density on maximum stress
Maximum Absolute Principle Stress (N/m2) and Percent Error
Mesh Type
1
2
3
Roark's
Reduced
Incompatible
Equations
Linear
Integration modes
Quadratic
7.900E+06
2.133E+06
3.048E+06
7.916E+06 7.915E+06
73.00
61.42
0.20
0.19
% Error
7.900E+06
4.285E+06
5.354E+06
7.932E+06 7.918E+06
45.76
32.23
0.41
0.23
% Error
7.900E+06
5.325E+06
5.943E+06
7.926E+06 7.914E+06
32.59
24.77
0.33
0.18
% Error
Shell
7.829E+06
0.90
7.886E+06
0.18
7.893E+06
0.09
Table 4: Effect of element density on maximum deflection
Maximum Deflection (m) and Percent Error
Mesh Type
1
2
3
Roark's
Reduced
Incompatibl
Equations
Linear
Integration
e Modes
Quadratic Shell
-2.395E-03
-3.360E-04
-1.715E-03
-2.384E-03 -2.404E-03 -2.389E-03
85.97
28.39
0.46
0.38
0.25
% Error
-2.395E-03
-9.666E-04
-2.123E-03
-2.398E-03 -2.404E-03 -2.400E-03
59.64
11.36
0.13
0.38
0.21
% Error
-2.395E-03
-1.322E-03
-2.151E-03
-2.399E-03 -2.405E-03 -2.401E-03
44.79
10.19
0.17
0.42
0.25
% Error
8
Effect of Mesh Density Variation
on Stress
9.0E+06
80.00
8.0E+06
70.00
Effect of Mesh Density Variation
on Deflection
100.00
Roark's
90.00
Linear
Reduced
Integration
80.00
Reduced
Integration
Incompatible
modes
70.00
Incompatible
modes
0.0E+00
Roark's
Linear
6.0E+06
50.00
5.0E+06
40.00
4.0E+06
30.00
3.0E+06
20.00
2.0E+06
1.0E+06
10.00
0.0E+00
0.00
Quadratic
Shell
Linear %
Error
Reduced
Integration %
Error
Incompatible
Modes %
Error
Quadratic %
Error
-1.0E-03
-1.5E-03
50.00
40.00
-2.0E-03
30.00
20.00
-2.5E-03
10.00
Shell % Error
1
2
Mesh Density
3
Quadratic
60.00
Percent Error
60.00
Maximum Deflection (m)
7.0E+06
Percent Error
Maximum Absolute Principle Stress (N/m2)
-5.0E-04
Shell
Linear %
Error
Reduced
Integration
% Error
Incompatible
Modes %
Error
Quadratic %
Error
Shell % Error
-3.0E-03
0.00
1
2
3
Mesh Density
Figure 10: Mesh Density and element type maximum absolute principle stress and maximum deflection results
9
Empirical and FEA Reliabily as Thickness is Modified
Plate bending theory, used in the Reference (a) Roark’s equations, is based on the assumption
that the plate is sufficiently thin. This investigation is intended to learn how thick a plate can
become before plate theory fails. The above investigation found that incompatible modes
provides very accurate stress and deflection solutions when compared to plate theory for a plate
which is 100 times thinner than it is wide. This investigation will consider linear incompatible
modes elements to be the standard upon which the Reference (a) equations and shell elements
will be compared to as the thickness is modified. The number of elements will remain the same
throughout the investigation, “mesh density 2,” except for the final thickness where six elements
through the thickness will be used instead of four.
Figure 11 shows the result of increasing the thickness of the plate too much. The simply
supported boundary conditions do not work well for solid elements as the restricted nodes
become pulled apart beyond what is physically possible. In order to properly represent the
bending of a thick block like is shown in Figure 11 the whole edge of the block would have to be
secured by boundary conditions. Tables 5 and 6 as well as Figures 12 and 13 illustrate the effect
of modifying the thickness of the plate and its effect on the results of plate theory and the FEA
solid and shell element solutions.
Figure 11: Boundary condition and FEA failure as thickness grew; 20cm on the left and 30cm thick on the right
10
Table 5: Maximum absolute Principle stress output as thickness is varied
4 elements thick, 20,000 total
Thickness
0.005
0.01
0.05
0.1
0.2
0.3*
Maximum Absolute Principle Stress (N/m2)
Incompatible Modes
Roark's
Roark's % Error Shell Elements Shell % Error
3.175E+07 3.160E+07
0.475575572
3.154E+07
0.667695506
7.932E+06 7.900E+06
0.403429148
7.886E+06
0.5799294
3.154E+05 3.160E+05
0.1876934
3.144E+05
0.311659818
7.851E+04 7.900E+04
0.62796948
7.841E+04
0.119734546
2.887E+04 1.975E+04
31.58040601
1.953E+04
32.32869119
1.583E+04 8.778E+03
44.53080569
8.708E+03
44.97314376
* Note: This model had 6 elements through its thickness and thus had 30,000 elements total
1.0E+08
50
45
1.0E+07
40
35
1.0E+06
30
25
1.0E+05
20
15
1.0E+04
Percent Error
Maximum Absolute Principle Stress
(N/m2)
Effect of Thickness Variation on Stress Error
10
Incompatible
Modes
Roark's
Shell Elements
Roark's % Error
Shell % Error
5
1.0E+03
0
0
0.1
0.2
Plate Thickness (m)
0.3
Figure 12: Maximum absolute Principle stress output as thickness is varied
Table 6: Maximum Negative deflection as thickness is varied
4 elements thick, 20,000 total
Negative Maximum Deflection (m)*
Thickness Incompatible Modes
Roark's
Roark's % Error Shell Elements
Shell % Error
0.005
1.915E-02 1.916E-02
0.046994935
1.917E-02
0.078324892
0.01
2.398E-03
2.395E-03
0.125104254
2.400E-03
0.083402836
0.05
1.962E-05
1.916E-05
2.364451692
1.961E-05
0.061149613
0.1
2.577E-06
2.395E-06
7.062475747
2.540E-06
1.435778036
0.2
3.955E-07
2.993E-07
24.32440633
3.910E-07
1.128686436
0.3**
1.635E-07
8.870E-08
45.74492161
1.180E-07
27.83829907
* Note: The deflection was made positive to permit the use of a logarithmic scale in Figure 13
** Note: This model had 6 elements through its thickness and thus had 30,000 elements total
11
Effect of Thickness Variation on Deflection Error
50
45
1.0E-01
40
1.0E-02
35
1.0E-03
30
1.0E-04
25
1.0E-05
20
15
1.0E-06
Percent Error
Negative Maximum Deflection (m)
1.0E+00
Incompatible
Modes
Roark's
Shell Elements
Roark's % Error
10
1.0E-07
5
1.0E-08
Shell % Error
0
0
0.05
0.1
0.15
0.2
Plate Thickness (m)
0.25
0.3
Figure 13: Maximum deflection as thickness is varied
Discussion:
The accuracy of plate bending theory and the strength and versatility of Finite Element Analysis
were confirmed in these investigations. In the first investigation several different element types,
both solid and shell, were examined and compared to the standard set by the Reference (a) plate
bending Roark’s equations. Neither the linear nor the linear reduced integration elements were
able to provide accurate stress measurements. The reduced integration only provided marginally
accurate deflection values while the linear element’s deflection predictions were less accurate.
Increasing the mesh density did help these element types yield greater accuracy, but they never
provided the exactness provided by the other element types. This shows that both of these
elements, while potentially good at solving for other problem types, should not be used when
bending is the primary mode of deflection in an analysis. The linear incompatible modes,
quadratic, and shell elements were all able to provide extremely accurate stress and deflection
values at all the mesh densities analyzed. These results show that if bending is the primary mode
of deflection for a part, shell or linear incompatible modes are ideal candidates for analysis.
Quadratic elements could also be used, but their greater computational requirements would
necessitate a solid reason for their use.
12
The second investigation analyzed the impact of increasing the thickness of the plate. Plate
bending theory, Reference (a), and shell elements were selected to be compared to linear
incompatible modes elements which was set as a standard. The stress and deflection analysis
results were all very close to each other when the plate was 0.5cm, 1cm, and 5cm thick. The
Reference (a) Roark’s equations began to provide deflection predictions with greater and greater
deviation from the incompatible modes standard as the plate’s thickness increased beyond 5cm.
The shell elements’ deflection predictions began to diverge as the plate’s thickness increased to
more than 20cm, which is also when the incompatible modes elements began to stretch at the
boundary conditions. The stress values provided by both the Reference (a) equations and the
shell elements were following the incompatible modes values until the plate reached 20cm thick
as the pinched solid elements had higher stress levels than the center of the free edge where the
maximum stress levels had been.
This investigation has shown that if the plate’s width to thickness ratio is less than 50:1, any
results provided by plate bending theory or shell elements may be questionable and below 10:1,
they should be heavily scrutinized. The use of certain boundary conditions must also be
questioned if the solid elements are being pinched or bent excessively as is the case for the 20
and 30cm thick incompatible modes analyses.
13
Conclusions:
In the first investigation several different element types and mesh densities were analyzed. The
results from these analyses indicated that the linear and linear reduced integration elements were
unable to provide accurate solutions when compared to the standard set by the plate bending
theory Roark’s equations provided by Reference (a). The accuracy of these element types did
increase as the mesh density increased, but not enough to be comparable to the linear
incompatible modes, quadratic, or shell elements. The latter three element types provided stress
and deflection solutions which were extremely close to those provided by plate bending theory
regardless of the mesh density chosen. The second investigation researched the impact of
increasing the thickness of the plate on the plate bending theory equations as well as the shell
elements by setting the linear incompatible modes elements as the comparison standard. This
investigation showed that if the aspect ratio of plate with to thickness drops below 50:1, the plate
bending equations and shell elements should begin to be questioned. Once the ratio drops below
10:1, any results provided by those two methods should be heavily scrutinized.
References:
a) Young, Warren C., and Richard G. Budynas. "Chapter 11, Flat Plates." Roark's Formulas for
Stress and Strain: Warren C. Young; Richard G. Budynas. 7.th ed. New York: McGraw-Hill,
2002. 505. Print.
b) "ASM Material Data Sheet." ASM Material Data Sheet. ASM Aerospace Specification
Metals, Inc. Web. 1 May 2015.
14