Finite Element Analysis (FEA) method, and
its Application in evaluating Stress – Strain
characteristics
John William Mubeezi
A Seminar submitted to the Faculty of Rensselaer at Hartford in partial fulfillment of the
requirements of the Degree of MASTER of Science
Major Subject: Mechanical Engineering
The original of the Seminar is on file at the Rensselaer at Hartford Library
Approved by Seminar Advisor: Prof. Ernesto Gutierrez – Miravete
Clinical Associate Professor
Department of Engineering and Science
Rensselaer at Hartford, Hartford, CT
December 20 2004
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
2
Abstract:
This paper is concerned with reviewing the elastic and plastic solution of a thin uniformly
loaded rectangular plate with simply supported edges.
The Finite Element Method has been used extensively in solving elastic and plastic
solutions of this nature. In this paper ANSYS is employed to come up with numerical
solutions while altering criteria like mesh size, and type of elements. In the case of the
elastic solution, the results are compared with the exact solution, which in this paper is
derived from the Navier solution for thin plates a replica of the exact theory of plates that
is exclusively governed by the theory of elasticity.
This paper then goes on to solve for a plastic solution employing exclusively the plastic
modeling capabilities provided by ANSYS. The Stress – Strain behavioral trend for this
component is plotted and compared to similar trend for a perfectly elastic material.
Tables, diagrams and source code for the solutions in this research are attached in
appendices to the paper or simply appear within the paper text where deemed relevant.
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
Table of Contents
Abstract------------------------------------------------------------------------------- (2)
Table of contents........................................................................................... (3)
1. Introduction ………………………………………………………. (4)
2. Methodology.......................................................................................(5)
2.1 Classical deflection behavior of a thin plate …………………...(5)
2.2 The Governing Equations for deflection of plates………………(5)
2.3 The Boundary Conditions ………………………………………(6)
2.4 Navier Solution for Simply Supported Rectangular Plates……...(6)
2.5 The Finite Element Method …………………………………....(8)
3. Analysis..............................................................................................(10)
3.1 Solving for the Elastic exact (analytical) solution ……………...(10)
3.2 Solving for the Elastic numerical solution (ANSYS)…..…….....(10)
3.3 Solving for the Plastic numerical solution (ANSYS)…………... (10)
4. Results and Discussion....................................................................... (12)
4.1 Elastic solution results and discussion……………………….......(12)
4.2 Comment on stress distribution……………………………….....(14)
4.3 Plastic solution results and discussion.……………………...…...(15)
5. Conclusions .........................................................................................(17)
6. Table of Symbols ................................................................................(18)
7. References.......................................................................................... (19)
8. Appendix A .........................................................................................(20)
9. Appendix B........................................................................................ (22)
10. Appendix C .........................................................................................(24)
11. Appendix D .........................................................................................(26)
12. Appendix E .........................................................................................(29)
3
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
4
1. Introduction:
The first significant analytical treatment of the mechanics of plates occurred in the
1800s.Since then many great cases of plate bending problems have been worked out: the
fundamental theory (principally by Navier, Kirchhofff, and Levy) and numerical
approaches (by Glerkin, and Whal, and others). There is extensive literature relating to
plate and shell analysis.
The Navier method of solving simply supported rectangular plates using double
trigonometric series was first presented by Navier in a paper presented on this subject to
the French academy in 1820 [1].
In 1956 Turner, Clough, Martin, and Topp [2] introduced the finite element method,
which permits the numerical solution of complex plate and shell problems in an
economical way.
The general behavior of plates is often discussed in text books and journals of advanced
solid mechanics, and therefore this is not a paper introducing a new discovery but really
just expounding on the existing theories to establish a relation between analytical and
numerical solutions for a simple rectangular plate under a uniform pressure load. As
mentioned earlier, it is surprising to note that I could not find an analytical plastic
solution for a rectangular plate under any form of the common boundary conditions!
(This is a thought for a future research paper)
DISTRIBUTED LOAD
The specimen geometry as seen here is a basic
simply supported 1m X 1m X .015m plate
under a uniform pressure (.5e5 Pa). A
numerical elastic solution is obtained using
ANSYS and an analytical solution using the
Navier Solution for simply supported
PLATE
rectangular plates. Different meshes are
1m x 1m x .015m
applied for the numerical solution and similar
settings are replicated in the FORTRAN code for the analytical solution. Data is collected
and compared in form of graphs. Next ANSYS is set up to run the plastic solution by inputting the known Stress – Strain characteristics beyond  Y (the yield stress) for the
specific material. In this case we used mild Steel and applied the following material
characteristics:
Young’s Modulus E = 2.085 e 11 Pa
Poisson’s Ratio   .285
Yield Stress  Y  215Mpa
The tangential Modulus ET (3.0e10) is derived from an existing stress-strain curve for
mild steel, and this is the value that is used when solving for the plastic solution. (See
Figure 3.1)
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
5
2. Methodology:
2.1 Classical deflection behavior of a thin plate
The classical deflection behavior of thin plates is based on the small-deflection theory.
The fundamental assumptions of the small deflection theory of bending (classical theory
for isotropic, homogeneous, elastic- thin plate), is based upon the geometry of
deformations. These assumptions are:
a) The deflection of the midsurface is small compared with the thickness of the
plate. The slope of the deflected surface is therefore very small and its square
is therefore negligible.
b) The midplane remains unstrained (neutral) due to bending
c) Planar sections initially normal to the midsurface remain planer and normal to
the midsurface after bending. This implies that vertical shear strains are
negligible, and the deflection of the plate is therefore principally associated
with bending strains. It also follows that the normal strain from the transverse
loading can be omitted.
d) The stress normal to the midplane is small compared to the other stress
components and maybe neglected. This assumption is not very reliable in the
vicinity of a highly concentrated transverse load. (Employing the Von Mises
criteria in ANSYS for Yield stress –similar in expression to the equivalent
stress is therefore a good practice)
Assumptions a) through d) are known as Kirchhoff hypotheses which have been proven
valid by various tests. Because of these assumptions the complexity of the problem is
decreased, and hence a three – dimensional plate problem reduces to one involving only
two dimensions. The resulting governing plate equation can therefore be derived in a
concise and straightforward manner.
2.2The Governing Equations for deflection of plates.
Looking at twist of a plate element, and beginning with the known strain –displacement
relations the basic differential equation for the deflection of plates has been derived in
many text books of solid mechanics and is given as:
 2 k xy  2 k y
 2kx
p
(i)
2


2
2
xy
D
x
y
Where k x , k xy ,&k y are the curvatures at the mid-surface in planes parallel to the xz, yz,
and xy planes. And p is the lateral loading, D is the flexural rigidity given by:
Et 3
D
, where E is Young’s modulus, t is the thickness of the plate, and  is
12(1   2 )
Poisson’s ratio.
Now from the strain equations at any point in the plate:
2w
2w
2w
 x   z 2 ,  y   z 2 ,  xy  2 z
(ii)
xy
x
y
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
6
We can obtain the strain – curvature relations as:
 x   zk x ,  y   zk y ,  xy  2 zk xy
(iii)
Therefore by simple substitution of equations (ii) and (iii) into (i) yields
4w
4w
4w p
(iv)

2


x 4
x 2 y 2 y 4 D
This is the governing differential equation for deflection of thin plates (First derived by
Lagrange in 1811), and for more detailed derivations refer to [1], [3] & [4]
To determine w (the deflection), it is required to integrate this equation with the constants
of integration dependant upon the appropriate boundary conditions (in our case, those of
a simply supported plate).
2.3 The Boundary Conditions.
For a simply supported plate with dimensions as seen in Figure X below, the boundary
conditions are given as follows:
w  0 and
2w
0
x 2
w  0 and
2w
 0 at (y=0, y=b)
y 2
at (x=0, x=a)
Figure 2.1 Reference coordinate system for Navier’s method [3]
2.4 Navier Solution for Simply Supported Rectangular Plates.
The approach of solving the above differential equation using Fourier series was first
introduced by Navier in 1820. The deflection must satisfy the differential equation with
the relevant boundary conditions. In our case those of a simply supported plate indicated
above in Para. 2.3.
Taking the loading on the plate to be given by: q  f ( x, y ).
And representing the function f(x, y), in the form of a double trigonometric series.
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
7
mx
ny
sin
(v)
a
b
m 1 n 1
We can obtain any particular coefficient am ' n ' of this series (Fourier coefficients) by


f ( x, y )   sin
multiplying both sides of the equation (v) above by
n 'y
sin
dy , and integrating from 0 to b while noting that:
b
b
ny
n 'y
'
0 sin b sin b dy  0, when n  n ,

b
0
sin
ny
n 'y
b
sin
dy  ,
b
b
2
when
n  n'
The result is

b
0
f ( x, y) sin
n 'y
b 
mx
dy   amn' sin
b
2 m1
a
Now multiplying both sides of equation (vi) by sin
we obtain:
(vi)
m' x
dx , and integrating from 0 to a,
a
m' x
n' y
ab
sin
dxdy 
a m 'n ' , hence
a
b
4
4 a b
m' x
n' y
a m 'n ' 
f ( x, y ) sin
sin
dxdy
(vii)


ab 0 0
a
b
Performing the integration in expression (vii) for a given load distribution f(x, y), we find
the coefficients of the series (v), and represent in this way the given load as a sum of
partial sinusoidal loadings.
The total deflection is therefore given as a sum of the deflections produced by each
partial sinusoidal loading and is given as:
a mn
1  
mx
ny
(viii)
w  4 
sin
sin
2
2
2
a
b
 D m1 n1  m
n 
 2  2 
b 
a
Taking the case of a uniformly distributed load f ( x, y)  q0 , where q 0 is the intensity of
the uniformly distributed load; and substituting into equation (vii), we find that:
4q a b
16q
mx
ny
(ix)
amn  0   sin
sin
dxdy  2 0
ab 0 0
a
b
 mn
where m and n are odd integers since when m or n or both are even, a mn  0 .
a
b
0
0

f ( x, y ) sin
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
8
Now substituting (ix) into (viii) we get
mx
ny
sin
  sin
16q
a
b
w  6 0 
(x)
 D m 1 n 1  m 2 n 2  2
mn 2  2 
b 
a
We will use this equation to (x) solve for our analytical (exact) elastic solution
2.5 The Finite Element Method.
The powerful finite element method developed in the 1960’s together with the
widespread use of digital computers and the increasing emphasis upon numerical
methods. The solution is obtained without the use of the governing differential equations.
The finite element method is an approximate Ritz method combined with a variational
principle applied to continuum mechanics.
In the finite element method, the plate is discretized into a finite number of elements
(usually rectangular or triangular shape), connected at their nodes and along hypothetic
interelement boundaries. The application of fundamental concepts of the finite element
method has already been extended to practical problems in many engineering fields to
include: thick plates and shells, geometric and material nonlinearities, plasticity,
vibration, viscoelasticity and viscoplasticity, fracture, laminated plates and shells,
buckling, thermal stresses, dynamic response, aero- and hydroelastic analysis of
structural systems, and many more.
A convenient approach for derivation of the finite element governing expressions and
characteristics is based upon the principle of potential energy. The variation in the
potential energy  of the entire plate can be written as:
n
n
1
1
    ( M x  x  M y  y  2M xy  xy )dxdy    ( pw)dxdy  0
A
(xi)
A
where n, A, and p represent the number of uniform thickness elements comprising the
plate, surface area of an element, and the lateral load per unit surface area, respectively.
Expression (xi) can be rewritten as:
n
  (  M 
1
T
e
e
 pw)dxdy  0
(xii)
A
In which subscript T denotes the transpose of a matrix. Expression (xii) can further be
written as:
n
   (k     Q )  0
T
e
e
e
e
(xiii)
1
In which the stiffness matrix k e is given as:
k e   B DBdxdy
T
A
(xiv)
The element nodal force matrix Qe , due to initial strain and transverse load is:
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
Qe   B D 0 dxdy   P
T
T
A
pdxdy
9
(xv)
A
Since the changes in  e are independent and arbitrary, equation (xiii) leads to:
k e  e  Qe for the element nodal force equilibrium.
The governing equation therefore for the entire plate is given as:
(xvi)
K    Q
n
where
K    k e
1
n
and
Q   Qe
1
We observe that the plate stiffness matrix K  and the plate nodal force matrix Q are
determined by superposition of all element stiffness and nodal force matrices,
respectively.
The general procedure for solving a plate-bending problem by finite element method is as
follows:
n
1) Determine k e in terms of the given element properties. Generate K    k e
1
n
2) Determine Qe in terms of the applied loading. Generate Q   Qe
1
3) Determine the nodal displacements by satisfying the boundary conditions:
   K 1 Q
For this paper this procedure will be automated for us by employing the robust ANSYS
program
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
10
3. Analysis:
3.1 Solving for the Elastic exact (analytical) solution
Recalling equation (x) for the plate displacement at any point on the plate surface:
16q  
w  6 0 
 D m 1 n 1
mx
ny
sin
a
b
2
2 2
m
n 
mn 2  2 
b 
a
sin
Courtesy of Prof. Ernesto Gutierrez – Miravete, we will use the FORTRAN source code
attached in Appendix A to evaluate the displacements of the plate.
3.2 Solving for the Elastic numerical solution (ANSYS)
We will begin with the Ansys.log file (attached in Appendix B) to solve for the elastic
numerical solution applying the boundary conditions specified in Para.2.3. We will then
alter the element type and size within the log file, and run various solutions monitoring
the converging behavior of each set of characteristics.
Plate deflections for the different element types and mesh sizes (Degrees of freedom
DOF) are contained in Appendix D of this document.
3.3 Solving for the Plastic numerical solution (ANSYS)
Again we begin with a plastic. log file (attached in Appendix C), and input the stressstrain plastic characteristics for mild steel, solving with load sub steps beyond the yield
stress to initiate plastic behavior. A typical Stress- Strain curve for iron (containing
0.15%C) is used for inputs of stress and strain behavior beyond yield. See Figure 3.1
The four points shown on the graph are used to compute the tangential modulus ET that
is used in the plastic. log file to simulate plasticity for this material after yield. In this
model we employ rate – independent plasticity. Rate independent plasticity is
characterized by the irreversible straining that occurs in a material once a certain level of
stress is reached. The plastic strains are assumed to develop instantaneously, that is,
independent of time. Further still we characterize the material behavior as Classical
Bilinear Kinematic Hardening (BKIN) (a characteristic of the yield surface) [6].
(ANSYS offers seven options to characterize different types of material behaviors).
The rate –independent plasticity theory is characterized by three ingredients: a) the yield
criterion (von Mises in our case), b) the flow rule, and c) the hardening rule. More
detailed description and formulae derivations are outlined by Hill in [6].
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
1
2
3
4
Figure 3.1 Stress-strain curve for Iron (containing 0.15% C) [5]
11
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
12
4. Results and Discussion:
4.1 Elastic solution results and discussion
The results from the elastic solution are tabulated below (Table I ), and further illustrated
in graphs that follow (Figures 4.2, &4.3).
The location if Wmax is always in the center of the plate.
It is also observed that when applying the numerical solution, the mesh size significantly
influences the magnitude of Wmax. The element type is also a factor. Shell 63 is just a 4
noded elastic shell whereas Shell 181 is also 4 noded but is well suited for linear, large
rotation, and/or large strain nonlinear applications. Change in shell thickness is accounted
for in nonlinear analyses. In the element domain, both full and reduced integration
schemes are supported. SHELL181 accounts for follower (load stiffness) effects of
distributed pressures. This is evident in the results in that Shell 181 seems to converge
faster to the exact solution with finer meshing, but Shell 63 seems to maintain a result
closest to the exact solution at finer meshes.
Table I: Comparison of Exact solutions with the numerical solutions.
Wmax
No of
Elements
n
2
2X2
~3x3
~3x3
5x5
10x10
20x20
40x40
80x80
160x160
DOF (Degrees
of Freedom)
(n  1)  4n
2
1
4
4
16
81
361
1521
6241
25281
Exact Solution
1.074056E-02
1.074056E-02
1.074056E-02
1.074056E-02
1.074056E-02
1.074056E-02
1.074056E-02
1.074056E-02
1.074056E-02
ANSYS -Shell 63
0.008613
0.007327
0.010141
0.009425
0.010641
0.010716
0.010734
0.010739
0.01074
ANSYS - Shell 181
0.009657
0.007978
0.010778
0.009793
0.010762
0.010768
0.010784
0.010808
0.010828
It should also be noted that the mesh sizes 0.3 and 0.4 seem not to give coherent results,
and this is probably because they are not a multiple of the plate width and length. Graph
in Figure 4.4 is therefore plotted omitting mesh sizes 0.3 and 0.4; to give a more accurate
trend of convergence of the Shell 63 & Shell l81 solutions towards the exact solution.
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
Figure 4.1 A typical Gnu plot for the plate deflections from the exact solutions.
Mesh size Variation with Exact Solution
1.20E-02
1.10E-02
Wmax
1.00E-02
Exact Solution
Shell 63
Shell 181
9.00E-03
8.00E-03
7.00E-03
6.00E-03
0
0.2
0.4
0.6
Mesh Size
Figure 4.2 Plate deflection variations with Mesh size and type.
13
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
14
Neglecting Mesh size .4 and .3
1.20E-02
1.00E-02
Wmax
8.00E-03
Exact Solution
Shell 63
Shell 181
6.00E-03
4.00E-03
2.00E-03
0.00E+00
0
0.1
0.2
0.3
0.4
0.5
0.6
Mesh Size
Figure 4.3 Plate deflection variations with Mesh size and type.
4.2 A comment on the stress distribution
Using the 0.0125 mesh and Shell 63 element (combination that seems to give the
displacement solution closest to the exact solution), we re-run the ANSYS solution and
plot results for the Von Mises stress distribution along the plate entire surface(Figure
4.5). ANSYS normally uses the Von Mises Yield Criterion, which takes into account
both hydrostatic and deviatoric stresses as compared to the Tresca yield criterion that
only considers the maximum shear stresses. Tests suggest that von Mises yield criterion
provides a slightly better fit to experiment than Tresca. More theory on this can be found
in many solid mechanics textbooks and journals.
From Figure 4.5, we observe that with an applied UDL (uniformly distributed load) of
.5e5 Pa, the stress is greatest at the corners of the plate. This is due to the way in which
the plate is constrained i.e with a Ux, Uy, & Uz =0 placed directly on the line edges; the
corners tend to experience additional rigidity that causes the stresses to increase in this
region.
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
15
Figure 4.4 Von Mises stress distribution along plate surface.
4.3 Plastic solution results and discussion
Table II: Elastic and Plastic solution results comparison.
PERFECTLY ELASTIC AND ELASTIC-PLASTIC SOLUTION RESULTS
PRESSURE
LOAD
5.00E+04
6.00E+04
7.00E+04
9.00E+04
1.00E+05
PERFECTLY ELASTIC ELASTIC-PLASTIC
MATERIAL EQV
MATERIAL EQV
STRESS
STRESS
1.66E+08
1.66E+08
1.99E+08
1.99E+08
2.32E+08
2.17E+08
2.98E+08
2.27E+08
3.31E+08
2.33E+08
PERFECTLY ELASTIC ELASTIC-PLASTIC
MATERIAL DISP Wz MATERIAL DISP Wz
1.07E-02
0.010784
1.29E-02
0.012941
1.50E-02
0.015107
1.94E-02
1.99E-02
2.15E-02
0.022788
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
16
STRESS-DISPLACEMENT CURVE
3.50E+08
3.00E+08
STRESS (Pa)
2.50E+08
 Y  215Mpa
2.00E+08
PERFECTLY ELASTIC
ELASTIC-PLASTIC
1.50E+08
1.00E+08
5.00E+07
0.00E+00
0.010784
0.012941
0.015107
0.019931
0.022788
Wz(m)
Figure 4.5 Stress-Displacement curves for Elastic – Plastic and Perfectly Elastic
solutions.
From the Table II above and Graph 4.5, we observe that the ANSYS results depict typical
elastic –plastic behavior. It should be noted that once again the stresses used are by the
Von Mises equivalent stress calculations.
Figures showing the variation of stresses over the plate surface area are included in
Appendix E of this paper. From these figures we observe as the area of max stress
increases towards the center of the plate. Also the shape of the max stress region reflects
the classical yield lines [8] for a plate with similar boundary conditions.
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
17
Conclusions:
It is evident that the Finite Element Method is a pretty robust means of numerically
solving for vast majority structural problems. Of coarse the specimen used here was a
simple flat plate, but the observations made in this paper can apply to even complex
structures with minor changes like tailoring the right element type and boundary
conditions. The numerical results provided by ANSYS or any other numerical code
require careful scrutiny and interpretation to ensure that the known principles of applied
mechanics have been properly and relevantly applied to the solution at hand. This is why
starting with a simplified problem (like in this case a simply supported plate) before
moving to more complex ones can be profitable.
In this paper we have shown that the numerical elastic solution converges to the exact
(analytical) solution dependant on the mesh element type and size. Meshing, is therefore
an important aspect in running a numerical solution, and must always be treated with
caution. A different mesh type can yield different answers depending on how the element
code is set up.
In the case of the plastic solution, we observed that Shell 63, could not be used because it
is not set up to solve for plasticity. The plastic numerical solution also showed typical
plastic behavior of coarse dependant on how the plasticiy.log file was set up. This shows
that ANSYS really depends on the user to input the right information in order to produce
scientifically viable results.
It should also be noted however that we could not find a classical elastic-plastic solution
for a simply supported thin rectangular plate. This could be something to work on in the
future if such a solution is viable analytically.
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
List of symbols:
a
length of plate
b
width of plate
E
Young’s Modulus
ET
Tangent Modulus

Poisson’s ratio
Y
Yield Stress
Flexural Rigidity
Curvature at midsurface in plane xz
Curvature at midsurface in plane yz
Curvature at midsurface in plane xy
Thickness of plate
Strain in the x-direction
Strain in the y-direction
Shear-strain between x and y axes
Magnitude of plate displacement
Stiffness matrix
Nodal force matrix
Elasticity matrix (relates stress to strain)
Matrix that maps strains (function of element)
D
kx
ky
k xy
t
x
y
 xy
w
K 
Q
D
B
18
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
19
References:
[1] S.Timoshenko, Theory of Plates and Shell, McGraw-Hill, New- York, 1940.
[2] Stiffness and Deflection Analysis of Complex Structures, Journal of Aeronautical
Science, Vol.23, 1956, pp.805-823
Turner, Clough, Martin, and Topp
[3] A.C.Ugural, Stresses in Plates and Shells, McGraw-Hill, New York, 1999
[4] R.D.Cook, W.C. Young, Advanced Mechanics of Materials 2nd ed., Prentice-Hall,
Inc., New- Jersey, 1999.
[5] American Society for Metals – Metals Hand Book, 1961
[6] R.Hill, The Mathematical Theory of Plasticity, Oxford University Press Inc., NewYork, 2003.
[7] F.B.Hildebrand Advanced Calculus for Applications 2nd ed., Prentice-Hall, Inc., NewJersey, 1976.
[8] Yield-Elements: for elastic bending of plates and slabs, Journal of Engineering
Structures, Vol.17, Number 2, 1995, pp.87-94
Mitchell Gohnert and Alan R.Kemp
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
Appendix A
Fortran Source Code for deflection of a plate using equation (x)
C
C
Deflection of a plate
parameter(nx=21) (No of Elements)
parameter(ny=21) (No of Elements)
parameter(nmax=11)
parameter(mmax=11)
C
dimension w(nx,ny)
C
PI = 4*ATAN(1.0)
C
E = 2.085e11 (Young’s Modulus)
enu = 0.285 (Poisson’s Ratio)
a = 1.0
(Plate dimension)
b = 1.0
(Plate dimension)
h = 0.01
(Plate thickness)
q0 = .5e5
(Uniform load)
C
D = (E*h**3)/(12*(1-enu**2)) (Flexural Rigidity)
C = 16*q0/(D*PI**6)
dx = a/float(nx-1)
dy = b/float(ny-1)
C
x=0.0
do ix=1,nx
y=0.0
do jy=1,ny
sum=0.0
do j=1,mmax
EM=float(j)
do i=1,nmax
EN=float(i)
denom = EM*EN*((EM/a)**2+(EN/b)**2)**2
sum = sum + sin(EM*PI*x/a)*sin(EN*PI*y/b)/denom
enddo
enddo
C
w(ix,jy) = C*sum
C
y = y + dy
enddo
x = x + dx
enddo
c
c
x=0.0
do i=1,nx
y=0.0
do j=1,ny
write(6,*) x,y,w(i,j)
20
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
write(66,*) x,y,w(i,j)
y=y+dy
enddo
x=x+dx
enddo
C
stop
end
21
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
Appendix B
Ansys.log file for solving the numerical elastic solution
/BATCH
/COM,ANSYS RELEASE 7.11C1 UP20030709
04:50:44 11/11/2004
/PREP7
!*
ET,1,SHELL181 ! (ELEMENT TYPE)
!*
!*
R,1,.01, , , , , , ! (PLATE THICKNESS)
RMORE, , , ,
RMORE
RMORE, ,
!*
!*
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,1,,2.085e11 ! (YOUNG’S MODULUS)
MPDATA,PRXY,1,,.285 ! (POISSON’S RATIO)
ESIZE,0.025,0,
! (ELEMENT SIZE)
RECTNG,0,1,0,1,
! (PLATE DIMENSIONS)
!*
!*
/GO
FLST,2,4,4,ORDE,2
FITEM,2,1
FITEM,2,-4
!*
/GO
DL,P51X, ,UX,
! (BOUNDARY CONSTRAINT)
FLST,2,4,4,ORDE,2
FITEM,2,1
FITEM,2,-4
!*
/GO
DL,P51X, ,UY,
! (BOUNDARY CONSTRAINT)
FLST,2,4,4,ORDE,2
FITEM,2,1
FITEM,2,-4
!*
/GO
DL,P51X, ,UZ,
! (BOUNDARY CONSTRAINT)
FLST,2,1,5,ORDE,1
FITEM,2,1
/GO
!*
SFA,P51X,1,PRES,.5E5 ! (APPLIED PRESSURE LOAD)
!*
MSHKEY,0
CM,_Y,AREA
ASEL, , , ,
1
CM,_Y1,AREA
CHKMSH,'AREA'
CMSEL,S,_Y
!*
AMESH,_Y1
!*
CMDELE,_Y
CMDELE,_Y1
22
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
CMDELE,_Y2
!*
FINISH
/SOL
SOLVE
FINISH
23
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
Appendix C
Ansys_plastic.log file
/BATCH
/COM,ANSYS RELEASE 7.11C1 UP20030709
04:50:44 11/11/2004
/PREP7
!*
ET,1,SHELL181
!*
!*
R,1,.01, , , , , ,
RMORE, , , ,
RMORE
RMORE, ,
!*
!*
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,1,,2.085e11
TB,BKIN,1,1
! BILINEAR KINEMATIC HARDENING BEHAVIOR
TBDATA,1,215e6,3.0e10
! YIELD STRESSES AND TANGENT MODULUS
MPDATA,PRXY,1,,.285
ESIZE,0.025,0,
RECTNG,0,1,0,1,
!*
!*
/GO
FLST,2,4,4,ORDE,2
FITEM,2,1
FITEM,2,-4
!*
/GO
DL,P51X, ,UX,
FLST,2,4,4,ORDE,2
FITEM,2,1
FITEM,2,-4
!*
/GO
DL,P51X, ,UY,
FLST,2,4,4,ORDE,2
FITEM,2,1
FITEM,2,-4
!*
/GO
DL,P51X, ,UZ,
FLST,2,1,5,ORDE,1
FITEM,2,1
/GO
!*
SFA,P51X,1,PRES,.6E5
!*
MSHKEY,0
CM,_Y,AREA
ASEL, , , ,
1
CM,_Y1,AREA
CHKMSH,'AREA'
CMSEL,S,_Y
!*
AMESH,_Y1
!*
CMDELE,_Y
CMDELE,_Y1
24
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
CMDELE,_Y2
!*
FINISH
/SOL
NSUBST,10
! 10 SUBSTEPS FOR TIME STEP
OUTPR,BASIC,1
! PRINT BASIC SOLUTION FOR EACH SUBSTEP
OUTRES,NSOL,1
! STORE NODAL SOLUTION FOR EACH SUBSTEP
SOLVE
FINISH
25
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
Appendix D
Plate deflections for the different element types and mesh sizes
ELEMENT TYPE – SHELL 63
ELEMENT TYPE – SHELL 181
DOF=25281
wz  .01074
wz  .010828
wz  .010739
wz  .010808
DOF=6241
DOF=1521
26
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
wz  .010734
wz  .010784
wz  .010716
wz  .010768
wz  .010641
wz  .010762
wz  .009425
wz  .009793
DOF=361
DOF=81
DOF=16
27
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
DOF=4
wz  .010141
wz  .010778
wz  .007327
wz  .007978
DOF=4
DOF=1
wz  .008613
wz  .009657
Arrow direction shows increasing magnitude in mesh element size.
28
Engineering Seminar MANE6900HEG, Rensselaer Hartford, Fall 2004
Finite Element Analysis (FEA) method) and its application in evaluating Stress-Strain characteristics
- John Mubeezi
12-20-2004
Appendix E
Stress Distribution for various pressure loads
P = .5 E5 Pa
P = .6 E5 Pa
P = .7 E5 Pa
P = .9 E5 Pa
P =1.0 E5 Pa
29