NONLINEAR ANALYSIS OF AXISYMMETRIC CIRCULAR
PLATES INCLUDING SHEAR DEFORMATION
by
DEVANAND V.A.J. KONDUR, B.E.
A THESIS
IN
CIVIL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE
IN
CIVIL ENGINEERING
Approved
Accepted
August, 1992
ACKNOWLEDGMENTS
I would like to express my deepest gratitude to Dr. C. V. G. Vallabhan for his
advice and guidance during the course of this research. I would like to thank Dr. W. P.
V ann for his advice and comments which helped me to complete this thesis. I would also
like to thank Dr. Y. C. Das for his guidance during the initial stages of this research.
I finally thank my family for their support during course of my studies in the
United States.
11
TABLE OF CONTENTS
ACKNOWLEDGMENTS 11
LIST OF FIGURES v
LIST OF SYMBOLS Vll
I. INTRODUCTION 1
Shear Deformation 1
Review of Past Research 1
Linear Plate Theories 1
Nonlinear Plate Theories 3
Scope of Current Research 4
II. A MATHEMATICAL MODEL FOR AXISYMMETRIC CIRCULAR PLATES INCLUDING SHEAR DEFORMATION 5
Introduction 5
Nonlinear Von Karman Equations 5
New Nonlinear Equations Including Shear Deformation 7
Strain-Displacement Equations 8
Stress-Strain Relationship 9
Total Energy Function of the Plate 11
Application of the Principle of Minimum Potential Energy 12
Boundary Conditions 14
III. FINITE DIFFERENCE MODEL FOR THE FIELD AND BOUNDARY EQUATIONS 16
Introduction 16
Finite Difference Model 16
111
IV. SOLUTION OF THE FINITE DIFFERENCE EQUATIONS 25
Steps in the Iterative Procedure 27
Convergence of the Finite Difference Model 28
Example Problem 28
v. ANALYSIS IN NON-DIMENSIONAL PARAMETERS 39
Introduction 39
N on-dimensionalized Parameters 39
Comparison with Previous Results 40
Other Results from the Current Model 43
Radial Distribution of the Displacements and Stresses 44
VI. CONCLUSIONS AND RECOMMENDATIONS 65
Conclusions 65
Recommendations 66
LIST OF REFERENCES 67
iv
LIST OF FIGURES
2.1. Undeformed and deformed cross sections of the plate. 15
4.1. Alpha parameter curve. 30
4.2. Results of a convergence study for different aft ratios. 31
4.3. Maximum lateral displacement versus applied uniform lateral pressure. 32
4.4. Radial distribution of the lateral displacement of the plate for an applied uniform lateral pressure. 33
4.5. Radial distribution of the in-plane displacement of the plate for an applied uniform lateral pressure. 34
4.6. Radial distribution of the total rotation of the plate for an applied uniform lateral pressure. 35
4.7. Radial distribution of the radial stresses (bending, membrane and total) on the bottom surface of the plate for an applied uniform lateral pressure. 36
4.8. Radial distribution of the radial stresses (bending, membrane and total) on the top surface of the plate for an applied uniform lateral pressure. 37
4.9. Radial distribution of the shear stress of the plate for an applied uniform lateral pressure. 38
5.1. Comparison of maximum non-dimensional lateral displacement of the plate versus non-dimensional lateral pressure. 46
5.2. Non-dimensional radial stress at the center of the plate versus non-dimensional lateral pressure. 48
5.3. Maximum non-dimensional tangential stress at the edge of the plate versus non-dimensional lateral pressure. 50
5.4. Maximum non-dimensional in-plane displacement versus non-dimensional lateral pressure. 51
5.5. Maximum non-dimensional radial stress on the bottom surface of the plate versus non-dimensional lateral pressure. 52
5.6. Maximum non-dimensional radial stress on the top surface of the plate versus non-dimensional lateral pressure. 53
v
5.7. Maximum non-dimensional total rotation of the plate vs. non-dimensional lateral pressure. 54
5.8. Maximum non-dimensional shear stress of the plate vs. non-dimensional lateral pressure 55
5.9. Radial distribution of the non-dimensional lateral displacement for different values of non-dimensional lateral pressure Q. 56
5.10. Radial distribution of the non-dimensional in-plane displacement for different values of non-dimensional lateral pressure Q. 57
5.11. Radial distribution of the non-dimensional radial stress on the bottom surface of the plate for different values of non-dimensional lateral pressure Q. 58
5.12. Radial distribution of the non-dimensional radial stress on the top surface of the plate for different values of non-dimensional lateral pressure Q. 60
5.13. Radial distribution of the non-dimensional total rotation of the plate for different values of non-dimensional1ateral pressure Q. 62
5.14. Radial distribution of the non-dimensional shear stress of the plate for different values of non-dimensional lateral pressure Q. 63
vi
LIST OF SYMBOLS
a = Radius of the plate
[A] = Matrix containing linear terms of the in-plane
displacements u
[B] = Matrix containing linear terms of the lateral displacement w and the total rotation VI
D = Flexural rigidity of the plate
E = Young's modulus of the material of the plate
G = Shear modulus of the material of the plate
h = Size of the finite difference elements
k = Shear coefficient
Mr,M9 = Bending stress resultants of the plate
n = Number of divisions in the radial direction
Nr,N9 = Membrane stress resultants of the plate
q = Uniform lateral load applied on the plate
Q = Non-dimensional uniform applied lateral load on the plate
Q,. = Shear stress resultant of the plate
r = Coordinate of the plate in the radial direction
R = Non-dimensional coordinate of the plate in the radial direction
t = Thickness of the plate
u = In-plane displacement of the plate
{u} = In-plane displacement vector of the plate
U,W = Non-dimensional in-plane and lateral displacements of the plate
Vll
u = Total strain energy of the plate
um ,Ub ,Us = Membrane, bending and shear strain energies of the plate
w = Lateral displacement of the plate
{w} = Lateral displacement vector of the plate
a = Interpolating parameter
e,,e9, Yn = Strain components of the plate
v = Poisson's ratio of the material of the plate
Q = Work potential due to the applied load q
rr = Total potential energy of the system
"' = Total rotation of the plate
'I' = Non-dimensional total rotation of the plate
a,,a9, "n = Stress components of the plate
s = Non-dimensional normal stresses on the plate
T = Non-dimensional shear stress on the plate
vm
CHAPTER I
INTRODUCTION
Shear Deformation
In the classical plate theory the effect of transverse shear is neglected due to the
assumption that the thickness of the plate is negligible compared to the lateral dimensions
of the plate. Also, it is well known that the assumption that the normals to the mid
surface before deformation remain straight and normal to the mid surface after
deformation underpredicts the deflections. As the thickness of the plate increases
compared to its lateral dimensions, the effect of the transverse shear on the deflection of
the plate increases. All of these observations are good for small deflections of the plate.
But, as the lateral deflection of the plate becomes larger than its thickness, the load
deflection relationship becomes nonlinear. In case of metals like steel, the plate can
become geometrically nonlinear as well as elasto-plastic, while in the case of glass such
as that used in windows, decorative glass, etc., the plates remain elastic even when the
deflections are large. Such glass plates break by brittle fracture. There are researchers
who have developed solutions for nonlinear plate behavior, but a question among the
minds of new researchers has been whether there is any effect of shear deformation on
the overall behavior of the plates. The aim of this research is to explore the effect of
shear deformation on large deflections of axisymmetric circular plates.
Review of Past Research
Linear Plate Theories
Kirchoff developed a theory for the linear analysis of plates, extending the
classical Euler beam theory [1]. Refined linear plate theories which were an
improvement over the classical Kirchoff theory were developed as early as 1877 by Levy
1
2
[2]. From equilibrium conditions we can see that the transverse shear although small is
not zero. For linear theories it is observed that as the ratio of the radius of a circular
plate to the thickness of the plate decreases the effect of the transverse shear, on the
lateral deflection and the stresses increases. Further theories which include the effect of
transverse shear deformation have been developed by Reissner [3], Hencky [4], Mindlin
[5], Kromm [6] and Reddy [7].
In 1945 Reissner [3] published a paper on the effect of transverse shear defo
rmation on the bending of an elastic plate. He developed a system of equations for the
theory of bending of thin elastic plates which takes into account the transverse shear. He
compared the differences between his theory and the classical plate theories. Mindlin [5]
in 1951 developed a slightly different theory for the influence of rotary inertia and shear
on flexural motions of isotropic, elastic plates. His theory includes the effect of rotary
inertia and shear in the same manner as Timoshenko's one-dimensional theory of beams.
Both Reissner and Mindlin suggested in their theory a value of k, to be taken to account
for the non-uniformity of the shear across the thickness of the plate, but they
independently suggested different values of k.
Kromm [6] further developed a generalized plate theory which takes into account
the transversal strain. He solved the example of a simply supported plate with uniformly
distributed load. He examined the distribution of the shearing forces along the edges in
the elementary plate theory. Based on Mindlin's theory, Dym and Shames [7] developed
a set of equations for circular plates with shear deformation. But these equations are not
applicable to nonlinear analysis. Reddy [8] presented a solution for the maximum lateral
deflection of an axisymmetric circular plate including the effect of shear deflection. He
also presented a higher-order theory for rectangular plates. All of the theories mentioned
above are applicable only to thin plates with small lateral deflections.
3
Nonlinear Plate Theories
Having observed that the pressure-deflection data become nonlinear for large
deflections of plates, von Karman developed a new theory applicable to thin plates
undergoing very large deflections by introducing the effect of membrane stresses into the
model. For rectangular plates, Schmidt [9] developed a refined nonlinear theory of
plates with transverse shear deformation. His theory includes the effects of transverse
shear deformation and rotary inertia, and is a generalization of the nonlinear theory
developed by von Karman. Schmidt solved a numerical example in his paper but
simplified the non-linear equations to make them linear. Von Karman's nonlinear plate
equations were solved numerically by Vallabhan [ 10] for rectangular plates using the
fmite difference method. Federhofer and Egger [ 11] presented a solution for disp
lacements and stresses of the von Karman equations for axi-symmetric circular plates in a
graphical form. Timoshenko and Woinowsky-Krieger [12] presented approximate
equations for the maximum lateral deflections and stresses at the center and at the edge of
the plate. Chia [13] modified Timoshenko and Woinowsky-Krieger's approximate equ
ations and presented non-dimensional relationships between the lateral pressure, lateral
displacement and bending and membrane stresses at the center for axisymmetric circular
plates. Vallabhan and Das [14] solved axisymmetric circular plates with four different
boundary conditions, and they presented the results in non-dimensional form.
Unfortunately, this solution does not contain the effect of shear deformation. After
reviewing the literature the author has not been able to pinpoint the effect of shear stress
on nonlinear plate deformation.
4
Scope of Current Research
As discussed earlier all the theories presented above were either linear theories
including the effect of shear deformation or nonlinear theories excluding the effect of
shear deformation except the theory of Schmidt, who did not solve the problem. So, in
this research, using a displacement approach, a set on nonlinear equations in the sense of
von Karman is developed including the effect of shear deformation using Mindlin's
concept. These equations are derived using the principle of minimum potential energy
and solved by the finite difference method. The parameters are non-dimensionalized and
the results obtained using this model are compared with the results presented by previous
researchers.
Chapter II contains the derivation of the nonlinear equations including the effect
of shear deformation along with the associated boundary conditions. The finite diff
erence model applied to solve this system of equations is presented in Chapter III. The
iterative solution procedure used to solve these nonlinear algebraic equations is explained
in Chapter IV. A numerical example is also given in this chapter and the convergence
characteristics of the model are presented as well. In Chapter V the results are compared
with previously published results and some new results from this analysis are presented.
Conclusions and recommendations for further research are given in Chapter VI.
CHAPTER II
A MATHEMATICAL MODEL FOR AXISYMMETRIC
CIRCULAR PLATES INCLUDING SHEAR
DEFORMATION
Introduction
The objective of this research is to develop a model for the nonlinear analysis of
axisymmetric plates that includes the effect of shear deformation. Unlike the classical
Kirchoff plate theory, which is good for thin plates undergoing small lateral deflections,
this model is also applicable for the analysis of thicker plates. In this model, the effect of
transverse shear is taken into consideration using the concept developed by Mindlin.
The plate is circular and axially symmetric about the center with respect to the boundary
conditions and the applied loading. Therefore, the displacements vary only in the radial
direction. The plate is assumed to be simply supported and without restraint on the in
plane displacement at the edges. This boundary condition does not place any limitation
on the model; in other words, any other axisymmetric boundary conditions can be easily
incorporated into the model. For the purpose of comparison, nonlinear von Karman
equations for circular plates [ 12] are given first in this section.
Nonlinear Von Karman Equations
For rectangular plates von Karman considered the added effect of the membrane
stresses that stiffen the plate during large deformations. Like Kirchoff, von Karman used
w as the displacement function for developing the differential equations, but introduced
an Airy stress function, (/J(.x,y), for representing the in-plane stresses. For axisymmetric
5
6
circular plates Timoshenko [ 12] has shown the derivation of the von Karman equations
using a complete displacement approach, and these equations are presented below.
where
d2~ + _!_ du _ ~ = _ 1-v (dwJ2
_ _ dw __ d_2
w_ dr r dr r 2 2r dr dr dr 2
w = the lateral displacement of the plate,
u = the in-plane displacement of the plate,
a = the radius of the plate,
r = the radial distance,
v =the Poisson's ratio,
t = the thickness of the plate,
q = the applied radial distance, and
E = the modulus of elasticity of the plate.
(2.1)
(2.2)
The solution of these equations was obtained by Federhofer and Egger [11]. Vallabhan
and Das [14] have presented solutions for four different boundary conditions for
axisymmetric circular plates using w and ¢ functions, where the plate is subjected to a
uniform pressure. Ramasamudra [ 15] has solved the above equations using the finite
difference method, and presented the results in his master's thesis.
7
New Nonlinear Eguations Including Shear Deformation
Using the Mindlin concept of shear deformation, a new set of equations for the
analysis of axisymmetric circular plates are derived below. The assumptions used in this
theory are:
1. The material of the plate is linearly elastic, homogeneous, isotropic.
2. The radial strain e, and the circumferential strain e9 vary linearly in the lateral
direction.
3. The strain in the lateral direction £4
is assumed to be equal to zero.
4. The products of the derivatives of the in-plane displacements are assumed to be
negligible compared to the products of the derivatives of the lateral displacements
and hence are neglected.
The displacements at the mid-plane of the plate are taken as the fundamental variables.
Using the above assumptions the displacement functions u, v and w at any point (r,z) of
the plate can be written as follows
u(r,z) = u(r) + z 1f!(r)
v(r,z) = 0
w(r' z) = w(r) (2.3)
where u and w are displacements in the r and z-directions at any point on the middle
surface of the plate. Following Mindlin, in equation 2.3 a new function VJ(r) is
introduced. This function represents the rotation of a line element originally perpend
icular to the longitudinal plane of the plate, as shown in Fig. 2.1.
8
Strain - Displacement Equations
The nonlinear strains after removing all nonlinear terms according to von
Karman's assumptions are given by
e, dU +.!.( dW )' dr 2 dr
du + .!.( dw )' + /'1' dr 2 dr dr
e9 = u = u "' (2.4) -+z-r r r
Yrz dii dw dw -+- 1j1+-dz dr dr
where c.; and c.; represent the membrane strains and are independent of z, c.;, e! are the
bending strains and do contain z, and 1n is the shearing strain. Also, er9 = e9z = 0 due to
axisymmetry.
9
Stress - Strain Relationship
Using Hooke's law the strains for a linearly elastic and isotropic material are given in
terms of stresses by
and the relationship may be inverted to give
1 v 0 E
v 1 0 (1- v2) 0 0 (1- v)
(2.5)
2
Thus the stresses are given by
ar = ~[{du +.!(dw)2
+v ~} + z{d1fl +v VI}] 1-V dr 2 dr r dr r
(2.6)
a9 = _E_[v{du +_!_(dw)2
}+ u + z{v d1fl +VI}] 1 - v 2 dr 2 dr r dr r
(2.7)
(2.8)
where
G = the shear modulus of the plate.
From the above stresses we get the membrane stress resultants, Nr,N9 , the bending
moment resultants, Mr, M9 , and the shear stress resultant, Qr, as follows
10
[ ( )' ] N, Et du 1 dw u
= ( 1 - v 2 ) dr + 2 dr + v ;:-
(2.9)
[ { ( J}] No Et u du 1 dw
= ( 1 - v 2 } ;:- + v dr + 2 dr
(2.10)
M, = Et' ( If/ dlfl) 12 ( 1 - v 2 ) v 7 + dr
(2.11)
(2.12)
(2.13)
In the expression for the shear resultant, Q,., k is a constant factor used to take into
consideration the variation of the shear stress across the thickness of the plate. Since the
equations are based on Mindlin's theory, the value of k is obtained from the following
expression given by Mindlin[5]
where ; = (1-2v) 2(1+v)"
(2.14)
11
Total Energy Function of the Plate
The total strain energy stored in the plate, U , can be written as the sum of the
membrane, bending and shear strain energies. Then the potential energy function IT of
the plate can be written as the sum of U and the work potential, n, i.e.,
(2.15)
where
U m = membrane strain energy in the plate, -b U = bending strain energy in the plate,
Us = shear strain energy in the plate, and
n = work potential due to applied external loads.
The expression for the membrane strain energy in terms of the displacements is given by
um = J112 12n1aum rdrde dz -r/2 0 0
r/2 2n a E du 1 dw [{
2}2
= L J, J, 2( 1-v 2 ) dr + 2 C, )
+2v: { ~: + ~(~; J}+(: J}drdOdz.
The bending strain energy is given by
_ Jr/2 i2nia ub = ub rdrde dz -r/2 0 0
(2.16)
12
(2.17)
The shear strain energy is given by
_ Jr/212n 1a us = us rdrdo dz -r/2 0 o
( )
2 r/2 2n a G dw = J r r- '1/f+- rdrd(}dz. -r/2 Jo Jo 2 dr
(2.18)
Finally, the work potential due to the applied load ij is given by
- r2n ra n = - Jo Jo ij r dr d (} . (2.19)
Application of the Principle of Minimum Potential Energy
The principle of minimum potential energy states that of all the geometrically
possible configurations that a body can assume, the true one, corresponding to the
satisfaction of stable equilibrium, is identified by a minimum value of the potential
energy [16]. By applying this principle the field equations are developed. Taking the
variation of the potential energy function, Il, with respect to the variations of u, w and 1/f,
and collecting the terms, and setting each of the coefficients equal to zero we get
for 8u:
d2u 1 du u _ d2w dw ( 1 - v) ( dw )2
-+--------- -dr2 r dr r 2 dr2 dr 2r dr
(2.20)
13
for &v:
-kG [(d1fl + 1{/) +(d2w ~ dw)] __ + Et [{du + 1 (dw)2 + u} d 2
w t dr -; dr2 r dr - q ( 1 - v2 ) dr 2 dr v-; dr2
and for D1fl:
D ( d2
1f/ 1 d1fl 1f1) ( dw) - -+---- - 1{1+- - 0 kG t dr2 r dr r2 dr - ·
(2.22)
The above equations can also be expressed in terms of the membrane stress resultants, N,
and N9 , the bending resultants, M,and M9 , and the shear stress resultant, Q,., as
dN, + N9 = O dr r
dM, + M, - M 9 _ Q = O dr r r
dQ, Q, N d2w N9 dw - __ 0 -+-+ -+---q
dr r 'dr2 r dr
It may be noted that these are the equilibrium equations of the plate.
(2.23)
(2.24)
(2.25)
14
Boundary Conditions
Here, in this research the case of a simply supported circular plate with no in
plane edge restraints is considered as the boundary conditions. For this particular case
the boundary conditions are
at r=O
u = 0, (2.26)
dw = 0 dr '
(2.27)
"' = 0, (2.28)
and at r=a
du 1 dw u ( J dr + 2 dr +v; = 0 (2.29)
w = 0, (2.30)
Et' ( v yt + dyt) _ O 12( 1- v2 } r dr -
(2.31)
15
M r
Undefonned State
(x,y)
Defonned State
Fig. 2.1. Undefonned and defonned cross sections of the plate.
CHAPTER III
FINITE DIFFERENCE MODEL FOR THE FIELD AND
BOUNDARY EQUATIONS
Introduction
The three field equations, 2.20 through 2.22, are nonlinear differential equations
in terms of the in-plane displacement u, the lateral deflection w and the total rotation lfl.
As it is not possible to obtain a closed form solution for these equations, a numerical
procedure with iteration is necessary to solve the equations. The finite difference method
is used in this analysis.
Finite Difference Model
The classical finite difference approach is used to discretize the continuous
functions u, w and lfl at every node of the finite difference mesh. All the nonlinear terms
in equations 2.20 and 2.21 are transferred to the right-hand sides so that the left-hand
sides of the equations contain only linear terms and they are converted into linear
algebraic equations using the central difference formulation. The functions u, w and lfl
are then stored as discrete values at the nodes of the finite difference mesh. The
coefficients of the algebraic equations from equation 2.20 are stored in matrix [A], while
those generated from equations 2.21 and 2.22 are stored in matrix [B]. The matrices [A]
and [B] are given by
(A] { u} = {.ft ( w , u)} (3.1)
and
16
17
[B]{w, 'If} = {q} + {.t;(w,u)} (3.2)
where
{ u} = the in-plane displacement vector,
{ w} = the lateral displacement vector,
{'If} = the total rotation vector,
{ q} = applied pressure vector,
{.h ( w , u)} = vector representing the nonlinear terms in equation 2.19 at every
node, and
{.t; ( w , u)} = vector representing the nonlinear terms in equations 2.20 and 2.21
at every node.
The coefficient matrix [A] contains linear second-order differential operators and can be
developed as a tri-diagonal matrix with three coefficients. The radius of the plate is
divided into n number of divisions along the radial direction. The nodes at the center of
the plate and at the edge are also considered as unknowns, with a total of n+ 1 unknowns.
The terms in the [A] matrix at the i-th node are given by
From i = 1 to n+ 1
A u. I + B u. + c. u. I = F l J- I I I I+ J
(3.3)
where
1 1 ~=---
h2 2r; h
2 1 B.=----
1 h2 2 r;
1 1 C.=-+--
1 h2 2r; h
F _ ( 1 -vJ(dw)2
(dw d2wJ i -- ~ dr i - dr dr 2 ;.
18
The subscript i in the above equations represents that the quantity is evaluated at
that particular node. At the plate boundaries, equation 3.3 has to be modified to
incorporate the boundary conditions. Backward difference is used to evaluate the
differentials at the boundary since they cannot be evaluated using the central difference
formulation. The modified equations at the boundary are as follows
For i = 1:
F; = 0.0;
For i = n+l:
2 ~+I= 11
B = _2__ 2 0 _v __ (l+v) n+l 2 • 2
h r,.+ 1 h r,.+ 1
c,.+1 = o.o
19
The [B] matrix formed from equations 2.21 and 2.22 contains seven diagonal
terms and a special algorithm is developed to solve this matrix. These two equations are
clubbed together because the linear part on their left-hand sides contain both the w and ljl
terms. The storage space required for the [B] matrix is of the size 14(n+l), with two
unknowns w and ljl at every point in the mesh. Since the [B] matrix contains two
differential equations, it has two different sets of coefficients for consecutive rows in the
matrix. The two sets of coefficients in the [B] matrix are given by
aur. 3
+bw. 2
+c.ur. 2
+d.w. 1+e.ul.
1+ fw.+g.ur. =q. for i = 1,3,5, ... 2n-1 (3.4)
1T1- I 1- 1l'1- I 1- 1l'1- Ji I 1'1'1 I
where
a. = 0 I
b 1 1
--+--I
h2 2fi h
1 ci -
2h
and
and
where
2 d.=-
I h2
1 ei = -
r I
1 1 t=----1 h2 2r ih
1 gi = --
2h
1 [- Et [{(du) 1 (dw)2
u}(d2w) ~ = kG t q + ( 1 -v 2) dr i + 2 dr i + v t; dr 2 i
+ {;, + ~ ( ( ~: J + ~ ( ~; J)} ( ~; J ]]
20
aw. 3 +bw 3 +c.w. 2 +d.Hr. 2 +e.w. 1 + .f.11r. 1+g.w. =q. for i = 2,4,6, .. 2(n+1) (3.5) I 1- 1't'1- I 1- 1't'1- I 1- }j't'l- I I I
1 ai = 2h
D ( 1 1 ) bi = kG t /1- 2 r; h
21
c = 0 I
d. = _ _!!__(~+_!_)-1 I kGt h2 ,-2
I
D ( 1 1 ) h = kG t h2 + 2 r; h
and
At the boundaries, equations 3.4 and 3.5 have to be modified to incorporate the
boundary conditions. The modified equations at the boundaries are as follows.
Equation 3.4
fori= 1
~ = 0.0
bl = 0.0
c1 = 0.0
A 2.0 '; = IF
22
el = 0.0
h 2.0
=--h2
2 gl = --
h
1 { E t ( du) ( d2w) }
q1 = kG t q + ( 1-v) dr 1
dr2 1
;
and for i = 2n+1
llu,+l = 0.0
~n+l = 0.0
c2n+l = 0.0
£4n+1 = 1.0
e2n+l = 0.0
hn+l = 0.0
82n+l = 0.0
q2n+1 = 0.0.
Equation 3.5
fori= 2
23
~ = 0.0
h = 0.0
g2 = 0.0
q2 = 0.0;
and for i = 2(n+ 1)
~(n+J) = 0.0
b2(n+l)
2 = --
h2
c2(n+l) = 0.0
c4(n+l) = 2.0(-v-+..!_)+ (l+v) r,+l h h2 ,. 2
n+1
e2(n+1) = 0. 0
h(n+l) = 0.0
g2(n+l) = 0. 0
24
q2(n+l) == - q r,+I 2D.
CHAPTER IV
SOLUTION OF THE FINITE DIFFERENCE
EQUATIONS
The field equations 2.20, 2.21 and 2.22 are converted to algebraic equations using
the central difference formulation at all nodes. An iterative procedure is used to solve
the equations thus obtained. The equations resulting from the finite difference
formulation in Chapter III are again expressed below in load increment form. In these
equations k represents the k-th load increment and the i represents the i-th iteration.
and
where
{ w } = the interpolated lateral displacement vector,
{ u } = the interpolated in-plane displacement vector,
{ q} = the applied pressure vector,
{_h(w,u)} = a vector representing the nonlinear terms in equation 2.20, and
(4.1)
(4.2)
{.t;(w,u)} = a vector representing the nonlinear terms in equation 2.21 and 2.22
at every node.
In the above equations the load vector { q } remains constant during the iterations
for every load increment. Also, the [A] and [B] matrices are constant during the entire
iterative process. The { q } vector is updated at every load increment applied to the
system. The { _h} and { J;} functions are nonlinear and are updated for every iteration.
The initial values of w and u are taken as zero and hence the right hand side of equation
25
26
4.1 is equal to 0 in the beginning. During the first iteration, solving equation 4.1 gives a
value of { u} equal to 0, and hence no computation is necessary at this stage. Then the
values of { w} and { u} are used to calculate the right hand side of Equation 4.2. So the
right hand side of Equation 4.2 is just the applied load { q } . By solving this equation the
new values of { w} and { VI} are obtained. These values of { w} and { u} are then used to
calculate the { ft} and { f 2 } vectors in equations 4.1 and 4.2, which are then solved for
new values of { u}, { VI} and { w}. At every stage the {i-1) th values of { w} and { u} are
used to calculate the i-th value of the { h } and { f 2 } vectors during the iteration. This
process is repeated until convergence is obtained. For the k-th load increment, the
converged values of { w} and { u} from the (k-1) th load increment are used to start the
iteration for that load step.
This iterative process works only for small deflections. As the deflection
increases the number of iterations required for convergence increases. Sometimes the
procedure may even give a divergent solution. Thus an interpolation technique is adopted
to overcome this problem. This technique was used by Vallabhan and Das to solve
nonlinear von Karman equations [ 14]. They used a coefficient a based on the wit ratio to
interpolate the lateral displacement vector { w}. The alpha parameter is obtained by trial
for a particular problem. The alpha parameter curve for the present problem is given in
Fig. 4.1. The formula for the interpolation is given in the next section. The in-plane
displacement is interpolated by taking the average from the i-th and the {i-1) th iterations.
27
Steps in the Iterative Procedure
1. The initial values of { w} ~ and { u }~ are chosen as { 0} to calculate {It (u, w n: and
equation 4.1 is solved for {u}:. Note {u}: = {0} and hence is not calculated.
2. Using the updated values of {u} equation 4.2 is solved to obtain the value of {w}:.
3. The value of a is obtained using the alpha parameter curve based on the value of
(I) W max(l)
t
4. For the i-th iteration the lateral deflection w; is interpolated using the following
equation
5. The terms {It ( w, u)} ~ are calculated using w;. I
6. Equation 4.1 is solved to obtain the in-plane displacements { u} k. I
7. The average of the in-plane displacements is calculated.
{u;} = ~ {u; +u;_J
8. The terms {.t; { w , u)} k are calculated using u;. I
9. Equation 4.2 is solved to obtain the lateral displacements { w }~.
10. Convergence is checked based on w (max) using the following formula.
where
e = tolerance used for the convergence of the lateral deflection w.
11. If convergence is not obtained steps 3 through 10 are repeated until necessary
convergence is obtained.
28
The above iterative procedure is carried out using a computer program written in
FORTRAN and developed exclusively for this problem.
Convergence of the Finite Difference Model
The convergence of the model developed here varies for different aspect ratios
alt. The convergence of the model can be seen from Fig. 4.2. As the aspect ratio
increases we need fewer divisions in the finite difference mesh in the radial direction for
convergence of the model. In Fig. 4.1, the ratio of the converged value of the maximum
lateral displacement for n divisions in the finite difference mesh and the converged
maximum lateral displacement for 1000 divisions is plotted versus the number of
divisions in the radial direction of the mesh. For an aspect ratio of 150 we can see that
we need more divisions for convergence. But as the aspect ratio increases the number of
divisions required increases.
Example Problem
The results obtained for a particular problem using the computer program are
given in this section. A plate of 1.0 in. thickness and radius equal to 20 in. is chosen. A
load of 300 psi. is applied on the plate. The values of the maximum lateral deflection,
the lateral displacement w, the in-plane displacement w and the total rotation 1/f are
presented as a function of the radius in Fig. 4.3 through 4.6. The variations of the total
tensile, the total compressive, and shearing stresses across the radius are shown in Fig.
4.7 through 4.9. The parameters chosen for this particular case are given below
E =The modulus of Elasticity of the plate= 107 psi.
v =The Poisson's ratio of the plate= 0.25
29
The stresses are calculated by applying the finite difference formulation to
equations 2.6 through 2.8. The ratio of the maximum lateral displacement to the
thickness of the plate is usually taken as a measure of the nonlinearity of the behavior. In
this case this ratio approximately equal to 2.
0 . -. = 0
.
N 0
0 . 0
\
'\ ~
0.0 4.0 8.0 12.0 16.0
w ~x (maximum lateral displacement/thickness)
'--------~---
Fig. 4.1. Alpha parameter curve.
30
20.0
en = 0 ·~ V)
·~ > :.a 0 0 0 -
.. e ~
.......... V)
= 0 ·-V)
·~ > :.a
0 -co 0 .
co 0 .
• . 0
N 0
0 . 0
!d
J..-o- 1-1- 1- .1---·-:-.. :: ;: ;-: .-: .. -: ... ~ I/ l.......- ~~ .. .. . ..
· ' .. .. .· / , ..
v v / , v/ ,
/ , .
, , v· ~
.I i
I I
v I
I I
I I
i
I I
I I
I I
I
present model present model
-.- . - · present model
- ------ present model
... ........ ......... present model ---- present model
I I I '' I I I
n (number of divisions in the r-clirection)
Fig. 4.2. Results of a convergence study for different aft ratios.
31
~
aft= 5 all= 10 all= 20
1-1-
aft= 50 all= 100 a/1 = 150
I I I 'I 'I I
32
0 . C\l
.
cc - -.... c 0 E 0 (.) ~ -c. V) C\l
"0 --~
f;
/ J£ '~' lllllfiiJ l;llj;-r,u / -~
z.w v /
/ 1-o 0 .... ~ -E :::3 co E . ..... 0 >< ~
E '-'
~ 6 ~ •
v I .
0
.
. 0 . 0
0.0 64.0 128.0 192.0 256.0 320.0
q (applied uniform lateral pressure) psi.
Fig. 4.3. Maximum lateral displacement versus applied uniform lateral pressure.
...-.. ...... 0 (1)
a (1) (.) ~ -~ Vl
;a -~ ..... (1.) ...... ro -'-"
~
0 N
co -N . -CX)
0
0 . 0
0.0
I l {;
------~ JE 1:1 I I I I I f'J _'J Jll, __ _r, U
""' z.w
'\
\ : 1\
4.0 8.0 12.0 16.0
r (radial distance) in.
Fig. 4.4. Radial distribution of the lateral displacement of the plate for an applied uniform lateral pressure.
33
t--
20.0
-~ t::
8 0 (.) C1:S -0.. Vl :.a <1) c C1:S -0..
I
c ...... ......__,
:::s
"' N 0 0
0 0 0 0
"' N 0
0 I
0
"' 0
0 I
"' l' 0 0 I
0 0 -. 0 I
-------------~
~ .
~ \
1\
---- f
J£ 1:1 I Ill 'f'-1_11; I IS);;·_ r,ll z.w
I I
0.0 4.0 8.0 12.0 16.0
r (radial distance) in.
Fig. 4.5. Radial distribution of the in-plane displacement of the plate for an applied uniform lateral pressure.
34
\
20.0
..-. ~ ..... ~ -c.. ~ ..c .....
4-4 0 ~ 0 ·~ ..... ~ ..... 0 1-o -~ ..... 0 ..... '-' ~
0 co -0
CX) N -0
co C)
0 . 0
~ co 0 0
N M 0 0
0 0 0 . 0
-1--
.
.
0.0
35
v {;
/ Jl! •:'''''' fi_I_]IJIIJl};; __ r,u
z. w v /
v /
. /
4.0 8.0 12.0 16.0 20.0
r (radial distance) in.
Fig. 4.6. Radial distribution of the total rotation of the plate for an applied uniform lateral pressure.
36
f ,.-.. 0
t q Q.) . ~ 0 ~-·· · ··- · · - ·· -·- -·- -··-··- .. r,u
----< ~ -..- ~I a nJ,fn 0.. Q.) z,w -5 ~ 0 0 ----Q.) ~ (.)
] (I)
e 0 0 . ~ co 0
..0 0 -5 d 0 0
~ (I)
Q.)
.............
~ ---- ----- -------- ---------- .. '\ .. .. ' '
~ ' '
......................... membr. stress ',
-~ ------ - - bend. stress
total stress (I) (I)
Q.)
l:1 (I)
-.......,; 0 b C\l
0
················ ·· · ...... .... ···················
··. \ . ··········· ······ ... ..
········- ... . 0
0.0 4.0 8.0 12.0 16.0 20.0
r (radial distance) in.
Fig. 4. 7. Radial distribution of the radial stress (bending, membrane and total) on the bottom surface of the plate for an applied unifonn lateral pressure.
....-.. ~ ...... ~ ...... 0.. d)
..c ...... 4-1 0 ~ (.)
ct ::s V)
0.. 0 ...... ~
-s Q 0 tn d) V) V)
~ V) ~
b
0 . N
0 0
0 . N I
0 . • I
0 . co I
0 . CX)
I
.
····· ····· ···· ····· ···· ·· .. . . ···· · .... . ... ....
(
JE 'i'''''l'-",;''±Jf -r,u -~
z,w
...... .... ........... .... membr. stress -1-- ------ - -bend. stress
total stress
- r----------t---:
--------- --- ---- ----- -----
0.0 4.0 8.0
37
.. .... . .. ... ... ... ... ... ... ... ·· . ..
I /'
/ I
~,' ,
, , , , , , , , .. ..
12.0 16.0 20.0
"---~--------- r (raclial distance) in.
Fig. 4.8. Radial distribution of the radial stress (bending, membrane and total) on the top surface of the plate for an applied uniform lateral pressure.
.....-en en Q,)
l:: en ..... ~ Q,)
..c Vl _.. ....
0 . 0 0 0 -0 0
0 . 0 0 0 -0 0 0 0 N I
0 . 0 0 0 M I
0 0 0 0 ~ I
.
.
~ - ----~ ~
\ 1\
:
0.0 4.0 8.0 12.0 16.0
r (radial distance)
Fig. 4.9. Radial distribution of the shear stress of the plate for an applied uniform lateral pressure.
38
\ 20.0
CHAPTER V
ANALYSIS IN TERMS OF NON-DIMENSIONAL
PARAMETERS
Introduction
The results obtained from the computer program are discussed in this chapter.
Non-dimensionalized curves are presented for the maximum values of the lateral
displacement, the in-plane displacement, the total rotation, the stress at the bottom of the
plate, the stress at the top of the plate and the shear stress. Also, the variations of these
parameters with different values of the aspect ratio a/t are presented. The results are
compared with the solution by Federhofer and Egger using von Karman equations [11],
and the solution obtained by approximate equations as developed by Timoshenko and
Woinowsky-Krieger [12] and Chia [13]. Also, the results obtained by Ramasamudra
[15], using a displacement approach for solving the von Karman equations, are given.
Non-dimensionalized Parameters
The various parameters used in the analysis are non-dimensionalized as follows:
R _:_ - ' a
w = w. t
a U = u-,
t2
a '¥ = llf-,
t
s = ~(;)'.
39
and
T = ~(;)'.
Q = !(;J. ?,70 ? ,?
Using the above parameters the system equations~ 19 through }21 are non-
dimensionalized as given below:
d2U +_.!._ dU _!!_ = _ d
2W dW _ (1- v)(dW)
2
dR2 R dR R2 dR2 dR 2R dR
+ {!!_ + ~(dU + _!(dW)2
)} dW] ~ RdR 2dR dR
(d
2
'P 1 d'P '¥) (a)2
( dW) -+---- -6k(l-v) - '¥+- = 0. dR2 RdR R2 t dR
Comparison with Previous Results
40
(5.1)
(5.2)
(5.3)
Figure 5.l(a) shows the maximum lateral displacement, Wmax' the versus non-
dimensional pressure, Q, for the four different types of solutions discussed previously.
Federhofer and Egger [11] presented an exact solution for the non-dimensional lateral
displacement in a graphical form and Timoshenko and Woinowsky-Krieger [12] gave an
approximate solution for the lateral displacement in an equation form. These equations
were converted by the author to a non-dimensional form for the purpose of comparison;
and they are, for deflection:
W+0.262 W3 = o.696Q, (5.4)
for the total radial stress at the center of the plate:
Sr = 0.295W2 + 1.778W,
and for the total tangential stress at the edge of the plate:
Sr = -0.427W2 +0.755W.
41
(5.5)
(5.6)
Chia [9] modified these equations and presented approximate formulas for the
displacements and the stresses as follows:
W + 0.2872W3 = 0.6957Q,
and the total radial stress at the center of the plate is given by
Sr = 1.7790W + 0.2959W2- 0.1460W3
- 0.01451 W4•
(5.7)
(5.8)
Ramasamudra [15] presented the variation of the maximum lateral displacement
with Q in a graphical form and it is repeated in the Figure 5.1 (a). Timoshenko and
Woinowsky-Krieger [12], and Chia [13] used a Poisson's ratio of 0.3 in for their
approximate equations, while Ramasamudra used a Poisson's ratio of 0.22. The Poisson's
ratio used in the current model is 0.25. However, the effect of the change in Poisson's
ratio can be shown to be negligible.
From Fig. 5.1 (a) we can see that the solutions given by Federhofer and Egger and
by Ramasamudra almost matches the results from the present solution for an aspect ratio
aft of 10. The solution obtained from the formula by Chia gives a slightly lesser value of
the displacement for higher loads. The solution obtained from equation 5.4 matches the
solution from the present model. The maximum lateral displacements for different aspect
ratio's are given if Fig. 5.1(b). It can be observed from this figure that as the aspect ratio
increases to 40, the differences between the solutions increase but are negligible from a
practical point of view. This difference is due to the effect of the addition of shear
deformation in the plate.
Two comparisons of the total stress at the center of the plate for the four results
mentioned above are illustrated next, first for a value of Q up to 30, and then for a value
of Q up to 300. In Fig. 5.2(a), it can be seen that the solution obtained using the
42
approximate formula given in Timoshenko and Woinowsky-Krieger is higher than the
solution obtained by Ramasamudra and this author. The graphical solution of Federhofer
and Egger gives a lower value of stress than the present results for small non-dimensional
pressures, but matches with the current solution when the non-dimensional pressure Q is
between 25 and 60. The solution obtained by using Chia's approximate stress formula
gives a correct result only for values of Q up to 4. For higher values of Q it diverges and
the stress is too small. It is recommended that this formula not be used for values of Q
higher than 4.0.
Figure 5.2(b) shows the comparison of the total stress at the center of the plate
obtained with the present model with the results presented by Federhofer and Egger and
with Timoshenko's approximate formula. From this figure we can see that the graphical
results obtained by Federhofer and Egger overestimate the stress at the center of the plate
for values of Q greater than 60. Also, the solution obtained for the central stresses from
equation 5.5 yields higher values of stress than the graphical solution by Federhofer and
Egger. But the differences between the results obtained by the present model and that
obtained by Ramasamudra are negligible even for a value of the aspect ratio of 20. The
values for an aspect ratio of 40 are almost the same as for an aspect ratio of 20. From
these observations, it can be seen that the results given by Federhofer and Egger,
Ramasamudra, Timoshenko and Chia give good values for the non-dimensional
displacement, but the approximate solutions of Federhofer and Timoshenko and
Woinowsky-Krieger do not give good values of the central radial stress, even for lower
non-dimensional loads. In fact, results given by Chia are good only for values of non
dimensional load, Q, up to 4.
Figure 5.3 shows a comparison of results for the maximum tangential stress at the
edge of the plate. This stress is always compressive. The value of the maximum
tangential stress at the edge is not given by Chia. The three results obtained by
43
Federhofer and Egger in graphical form, by Timoshenko by equation 5.6, by
Ramasamudra in [15] and by the current model are fairly comparable for low aspect
ratios. As the aspect ratio increases the results obtained by this model differ from results
by Ramasamudra, as shown in the figure, and this effect is believed to be due to the shear
deformation.
Other Results from the Current Model
In this section, the maximum values of the in-plane displacement U max, the
maximum tensile stress, and the stress at the top face are plotted for various values of Q.
The results obtained by Ramasamudra are also presented in these graphs. The maximum
values of the total rotation 'I' max and the shear stress T max are also presented.
Figure 5.4 gives the maximum in-plane displacement U max for different aspect
ratios as obtained from the current model, with comparison to results from
Ramasamudra. The two compare well for an aspect ratio up to 5, but there is a
significant difference between these two as the aspect ratio increases to 20. For aspect
ratios higher than 20, the results do not vary significantly.
Figures 5.5 and 5.6 show the variations of the maximum stress at the bottom and
at the top surfaces of the plate, respectively with comparison to the results of
Ramasamudra. Values of the maximum stresses are not given by either Federhofer and
Egger, Timoshenko and Woinowsky-Krieger, or Chia. One can observe some difference
in the maximum stress for different values of the aspect ratio, alt. There is a significant
difference between the radial stress at the center and the maximum stress which does not
occur at the center. There is a substantial difference between the solution obtained by
Ramasamudra and the solution obtained by the current model, as shown in Fig. 5.5,
except for values of Q less than 25. As the aspect ratio decreases, the difference between
44
these two models further increases for both the bottom and the top stresses. None of
these results are published in the literature except those by Ramasamudra.
Figure 5.7 shows the maximum non-dimensional total rotation, '¥, as a function
of Q. There is some difference in the value of'¥ as the aspect ratio increases. Figure 5.8
shows the maximum shear stress at the edge of the plate and here the aspect ratio has
virtually no influence. These results are not available in any literature.
Radial Distribution of the Displacements and Stresses
In the above section, the radial stress at the center of the plate has been presented
as a function of Q. However, the radial stress at the center need not represent the
maximum stress in the plate as the deflection of the plate becomes larger. It would be
useful to observe the distributions of the displacements and the stresses in the radial
direction for different values of Q. The radial distribution of the lateral displacement, W,
is presented in Fig. 5.9. There is no significant difference visible in the distribution
patterns. In Fig. 5.10 the in-plane displacements, U, for different values of Q are given.
The maximum in-plane displacement occurs at the edge and is negative. The in-plane
displacement reduces from the edge to the center of the plate. The in-plane displacement
is zero at about a non-dimensional radial distance ria equal to 0.65, then becomes
positive, and then reduces to zero at the center of the plate. It is observed that the point
where the in-plane displacement becomes zero is independent of the applied load.
Stress distributions on the bottom surface of the plate are shown in Figs. 5.11 (a)
and (b). For values of Q less than 5 the maximum stress occurs at the center of the plate
but for higher values of Q the maximum stress occurs near the edge of the plate. Figure
5.ll(b) shows that as the non-dimensional pressure Q increases, the maximum stress
increases dramatically in the vicinity of ria equal to 0.8.
45
One can see the radial stress distribution on the top surface of the plate from Figs.
5.12(a) and 5.12(b). From Fig. 5.12(a}, it can be observed that the maximum
compressive stress occurs at the center of the plate for a non-dimensional pressure Q
approximately equal to 3. For higher pressures, the location of the maximum stress
moves toward the edges. Also, as the value of Q increases above 5, the compressive
stress at the center decreases. From Fig. 5 .12(b) it should be observed that the stress at
the center of the plate becomes positive, while the near the edge of the plate remains
compressive. The point where the stress changes from tensile to compressive is
dependent on the load applied to the plate. The radial stress is zero at the edge of the
plate.
Figure 5.13 shows the distribution of the total rotation of the plate. Figures
5.14(a) and 5.14(b) show the shear stress distribution. From Fig. 5.14(a) we can see the
maximum shear stress occurs at the edge of the plate. Also, for a low value of Q of 3,
there is a negative shear stress throughout the radius of the plate. But as the pressure
increases, we get positive shearing stresses near the center of the plate. In Fig. 5.14(b},
although the total rotaton is positive, along the radius, the shear stress changes from
positive to negative stress. This can be understood by looking at equation 2. 7, which
gives the shearing stress in terms of the total rotation and the lateral displacement. The
change of sign in the shear stress is due to the dwldr term in this equation, i.e., for lower
pressures the slope of the lateral displacement is very small, but as the pressure Q
increases the slope of the displacement becomes higher.
II
46
0 . ~~------~--------r--------.-------.--------~----~
{;
. CD
Ji! :[:I lllltU 11111,_ r,U t--t--~~+------1 0
z.w
0 ~1---~~t-------+-------~----~-------+------~
0.0
)( )( )(
0 0 0
50.0 100.0 150.0
present model aft = 20 Federbofer and Egger's solution Timosbenko and Woinowsky Krieger's formula(Eq. 5.4) Cbia's formula (Eq. 5.7) Ramasamudra's solution
200.0 250.0
Q =! (; J (non-dimensional lateral pressure)
(a)
300.0
Fig. 5.1. Maximum non-dimensional lateral displacement of the plate versus non-· dimensional lateral pressure.
47
0 . 0 ....
-~ c a 0 Q) . (..) co ~ -0.. en :.a -~ ..... QJ 0 ~
~ co --~ c 0 ...... ti'J c 8 0 :.a ~
I
('
~ . Jd •['''''' fu _uJ·~S);;---· r,u
_....,.....-:: ,... .... . ~
-t- ~~
~·· ~
7W
/ ~
A~ / '
/ v
c:: 0 c:: >< ~
8 0 ..._ . ~~1-
C\l
present model aft= 5
-.- . - · - present model aft= 10 f--
I --------- present model aft= 20
... ... ... .. .... ... ........ . present model aft= 40 II ..
':t·} 0 0
I I I
0.0 50.0 100.0 150.0 200.0 250.0 300.0
Q =! ( ~ r (non-dimensional lateral pressure)
(b)
Fig. 5.1. Continued.
-0 .... ~ -c.. 0 -s
'+-o 0 .... 0 .... c 0 u 0 ~ .... ....
48
0
~,---------r--------,---------r--------~------~--------~
0 C\l - )( )( )(
0 0 0
present model aft = 20 Federhofer and Egger's solution Timoshenko and Woinows Krieger's formula(Eq. 5.4) Chia's formula (Eq. 5.7) Ramasamudra's solution
f;
JE '~''''''fi!_IJJ!'fJ;-r,u zw
~ 0 ~ ~~--------+-------~--------~------~~-------4------~ ~
~ ~
E ·-~ I
c 0 c:: --
----------- --0 ~4---------~----~~--------~--~--~~~----~------~
0 . 0~~~~~~~~~4-~~-r-+~~--~r-~~~-r~~-r-1
0.0 5.0 10.0 15.0 20.0 25.0
Q =! (; J (non-dimensional lateral pressure)
(a)
30.0
Fig. 5.2. Non-dimensional radial stress at the center of the plate versus non-dimensional lateral pressure.
II
49
0 . ~~-------r-------r-------r------~------~----~~
£;
o Jf: 'f' '''''f'''-'J'~,_r,u ~lr~------r-~~~w----~------L-4-~~--~--------~------~
0 0 C\l
0 0 -
0 . 0
0.0
,4'
,...- ..
------ present model aft = 20 Federhofer and Egger's
t.t.!J.!J.!J.t. solution
~E ~E ~E Timosbenko and Woinows Krieger's formula(Eq . 5.4)
---- Ramasamudra's solution
50.0 100.0 150.0 200.0 250.0
Q =! (; J (non-dimensional lateral pressure)
(b)
-,
300.0
Fig. 5.2. Continued.
co = <U -e :a I = 0
5
50
Q . Q 2~----~----r-----r----.----~-----1
Q . Q CD I
f;
Jd II I II fll I ~I '$#;- r,U r--r-----~----+----_J w
Q . ~t-----~r------t------+-----~~~~~~--_j I
Q . ~j-------~-----+--~~~----~~----~----_j I
present model aft = 5 _ . _ . _ . present model aft = 10 _ _ _ _ _ _ present model aft = 20
..... ............... present model aft = 40 Fededlofer and Egger's
ll.ll.A.A.ll.ll. solution Ramasamudra's solution
Q . Q~~~~t-~~~r-r-~~~~~-4---T~-+~----J
2~0.0 300.0 0.0 ~0.0 100.0 1~0.0 200.0
Q =! (; J (non-dimensional lateral pressure)
Fig. 5.4. Maximum non-dimensional tangential stress at the edge of the plate versus nondimensionallateral pressure.
51
0 . 0 r-.. I 1
.-. ~
d 0 d) e 0 0 co (.) I t'::S -0.. V) 0 ;.a .
0
{; / -f-..- JE 1:1 I I I I I fiJ _IJ 11 ~,-- r, U vh' 7 \i / ~- .. ... . ... . .
d) lC')
d I ~ -0..
I 0 d ·- 0 / v~· .. w .... ... . . . ... - ~
~
= I 0 ·-V)
0 d d)
. e 0
M ;.a I
I d 0 0 d
>< 0 C\l
~ I a ..._, l::jl'""- 0 .. 0 e -~ I
II .. e 0 ~ 0
~ ::;..-d-' ... ... .~·
~/· II .
~ .
;·"·
~, f-..-
present model all = 5
~ ..... ... .. .... .. .... present model all= 10
/ ------- present model aft = 20
- . - . -· present model all= 40
/ f-..-
---- Ramasamudra's solution
I I I
0.0 50.0 100.0 150.0 200.0 250.0 300.0
Q =! (; )' (non-dimensional lateral pressure)
~-------------------
Fig. 5.4. Maximum non-dimensional in-plane displacement versus non-dimensional lateral pressure.
II
52
0 . ~~------r-----~-------r------r-----~------~
0 0 ~
0
f;
JE l:llllll f''-IJJ'-~+- --'· " t--t----t--~-1--/~
~~------t-------t-------~~--~~~~~~----~
0
~~-------r----~~~~~~------~--------~----~
0.0 50.0 100.0
present model aft = 5 . _ . _ . present model aft = 10
present model aft = 20 ... ................. present model aft= 40
--- Ramasamudra's solution
150.0 200.0 250.0
Q =! ( ~ r (non-dimensional lateral pressure)
300.0
Fig. 5.5. Maximum non-dimensional radial stress on the bottom surface of the plate versus non-dimensional lateral pressure.
..-.. ~ ..... ~ -0.. ~ .s
4-o 0
~ ;::::j V)
0.. 0 .....
e :.a I c
0 c ...._,
II
53
0
~~----~------~----~------,-----~----~ I
f
O J:dl[lllt[ _IJJI~w.:-r,U ~ ~~:~----~~~-----r-----L--~------~-------7~/ ______ j I
0 . ~1r--------~r---~~~~----~ I
/
. .. · ... _, ... _ ... ·~ ·
~ ...... ·. ~ ...
_, ... ·•.
present model aft = 5 _. _. _. present model aft= 10 _ _ _ _ _ _ _ present model aft = 20
.. .. .. .... .. .... .. .. present model aft= 40 ---- Ramasamudra's solutio
0
0~-r-r~-t~-,~~r-r-~~4-,-~-r~-r-r~-+~~-,~ 0.0 50.0 100.0 150.0 200.0 250.0 300.0
Q = ! (; )' (non-dimensi onallateral pressure)
~--------------------
, Fig. 5.6. Maximum non-dimensional radial stress on the top surface of the plate versus non-dimensional lateral pressure.
54
0 . I() N
-c 0 0 ·.::: . ~ 0 .... N 0 ~ -~ .... 0 .... -~ 0 c . 0 &n ·-tn -c 0 E
:.a I c
0 0 c . 0
f; ,-:;7 . "·
JE •:''''' 'f' IJJ;!CEJ_r,u
./~
~:,.;.; v -- , /
z.w ~ ...:
.~ . ,~ . / . v
.I :< -~
8 ......._,
~~ .... .
>< .. e ~ 0 II
. &n
><
'9--e
present model all = 5
1/ present model all= 10 ~ -·-·-·
------- present model all = 20 .......... .......... present model all= 40
0 . 0 I I
0.0 60.0 120.0 180.0 24-0.0 300.0
Q = ! ( ~ J (non-dimensional lateral pressure) ~-----------------
Fig. 5.7. Maximum non-dimensional total rotation of the plate versus non-dimensional lateral pressure.
.... ~ ~:::!I
'---/
~~I e,:, II ..
~:--.e
55
0 . 0
~~----T~-----r~-----r----~~----~--~
(
~ ~- Jf! 'i''''''fu.uJ~';J:- -'·" r--r----+----+--~/~....J ~ 7.W vv ~~----r----+----4---~~---~--_J
~ /v oj------i------,_-?L-~------~-----4----__j
:/ ~ +----+7"'?..~--~----....J present model a/t = 5 I _ . _ . _ . present model alt = 10
_ _ _ _ _ _ _ present model alt = 20
.. .. ... .... .. .. ..... present model a/t = 40
0.0 50.0 100.0 150.0
Q = ! ( ~ r (non-dimensional lateral pressure)
-
300.0
Fig. 5.8. Maximum non-dimensional shear stress of the plate versus non-dimensional lateral pressure.
-~ ~ -c.. 0 ..c ...
c...., 0 ... c 0 E d} (.) ctS -c.. Cl')
-~ "'" d} ... ctS --~ = 0 ·-
I
= 0 = "-"
0 . 0 -0
= .
0 cc
0 . ~
0 . C\l
0 . 0
56
T I ( ---~ ' J£ 1:11 II II ~lj_IIJII , __ r, U --·-- . ~----·~ · ~ ~ ~w
~------- ~-~-,""' -- .. "'"'·
.... .. .. .. .. .. ..
"" ... .. ... .. .. ... ······ ··· · ····· ·· ·· .. .. ...
'\ .... .... ... ' ... ... ... ... .. ... \ ' ', \ \\ ' ' ' ' ' ',\\ '
-- Q =50 ', \'
......................... '\\\ ------ - - Q = 100
-.- . - · - Q =200 ··-. \~ . ' Q= 300 '\ -
.. \
I I .. ·'
I I
0.0 0.2 0.4 0.6 0.8 1.0
R =!.. (non-dimensional radial distance) a
Fig. 5.9. Radial distribution of the non-dimensional lateral displacement fo r different values of non-dimensional lateral pressure Q.
-0 ..... C':S -0. Cl)
..c ..... 4-o 0 ..... c: 0
E 0 (.) ~ -0. CIJ ·-"0 Cl)
c: ~ -0..
I c: ·--~ c: 0 V'J c: Cl)
E ·-"0 I
c: 0 c:
'-" 1::3 , ........
:::t II ~
57
0
~~------~------~------r-----~------~
0 - -~ I
0 0 -f--I{)
I
0 0 cc I
0.0
I I
Q =50 Q = 100
Q =200
Q= 300
T I
0.2 0.4 0.6 0.8
R =!.. (non-dimensional radial distance) a
\ 1.0
Fig. 5.1 0. Radial distribution of the non-dimensional in-plane displacement for different values of non-dimensional lateral pressure Q.
0 . - ~ (1) ..... ~ -0.. (1)
..c 0 co . -.....
~ 0
~ :::s Cl)
..... 0
..c (1)
..c ..... = 0 Cl) Cl)
~ Cl)
E ·-
0 M
'"0 0 I ' = C\l
0 c ...._,
-
58
I ~
r-- JE 1:1 I I I I I tiJ_IJ ;, I SF-"· u
z.w
-~
- \ - - 1\ ~ -·----. ----- """'. ' l\\ -- -----
.... ""'"\ .... ......... ... ........ ············· ······ ················ ... .. .. .. .. .. . """
1---. '
Q=3 ··. '
-~\~ ·· ·············· ········· ·········· ... ~ Q=4 ------ - - --..:<\: -·-·-·- Q=5
II
0
0
-. 1--- ----- Q=6
Q=lO . -~\~ . ' . ·. ' · .. ' ~ ·.' . ·'
0.0 I I
0.2 0.4 0.6 0.8
R =!:.. (non-dimensional radial distance) a
(a)
Fig. 5.11. Radial distribution of the non-dimensional radial stress on the bottom surface of the plate for different values of non-dimensional lateral pressure Q.
1.0
0 . 0 ~
59
l I
. 0 C\l (")
-1--
f;
/ ~ JE •[•••• ''f'!-'!1'-'3};;--'·u
-0 .D d) 0 .c . - ~ c: C\l 0 en en
~ en
E ·-""0 I
0 cc ......
c: 0 c: ..._, 0 .
r;---..._ cc ~~
"------"" b'"l~
II ~-- 0
0
7.W
_.__........-·
--- ------------ ---
····· .. .. ········· ······ ·· ·· ··········
v
v ,.--.\
/ ' / .
/ v· ... --, ... .. , , ' , ' , , ' , , , ,
, , ... , ... ······ ·· ·· ... .. ...
... ... .. ·· ,•' , • ' ...
.. ........................ Q =50
------ - - Q = 100
-·-·-·- Q =200
Q= 300
I I I I
0.0 0.2 0.4 0.6 0.8
Fig. 5.11. Continued.
R = !._ (non-dimensional radial distance) a
(b)
.\ '·
\ \ ' \
\
\
\
\\ I .
I ' I I
I .
I I I
·.I ·.I
1.0
II
60
0 -~----r----,----~----~--~
0 M I
t q
--·-·-··-··-··-·-·- ··-··-·- r, u a
w
Q = 3 Q=4 Q=5 Q=6 Q= 10
.. -... ....
--.. , ,
t.-)... 0 . 11-~--r-~-r-,--~~-+--r-~~-+--r-~~-+--r-~~~
0.8 1.0 0.0 0.2 0.4 0.6
R = :_ (non-dimensional radial distance) a
(a)
Fig. 5.12. Radial distribution of the non-dimensional radial stress at the top surface of the plate for different values of non-dimensional lateral pressure Q.
0 0
...-. -0 -~ -c.. 0 0 . -= "' 4-o 0
ct ::::3 Vl
c.. 0 -C1.)
..c -c:: 0 Vl Vl C1.)
~ Vl
E I
c:: 0 c:: "-'
II
0 0 .
. 0
"' I
0 0 -I 0
"' -I
0 . "' C\l I
61
I I ---~ f ·-- ~J; ':'''''t'·''J~~~ _ r,u ----· ....... 1------- ~ --- - ·, --- ........... ··· ·· ··· ·· . .... . .. ' ··· ··. .. ....
>~~\ ···· · .... .
··· · ·~:-~ ,' I
.'I
I
'II
l\··. ; I : I
. .. I I ·· ..
,•' I li '• ...... · / ,, '' ....
\\'•,_ ';J . -- ,-..... ! . -~ \\ . ········· ········ ········ Q =50 \ '---'/ -------- Q = 100
-·-·-·- Q =200 -~ Q= 300 v
I I I I
0.0 0.2 0.4 0.6 0.8 1.0
R =!... (non-dimensional radial distance) a
(b)
Fig. 5.12. Continued.
62
0 .n N
{; . - -~ = 0 ·.;:: ~ ..... 8
0 0 N J2 {•lllllf"-'J]"±J-r,u V-/
z.w I -~ = 0 ......
V}
= s :.a I = 0 = '--"
0
"' . -
0 . 0 -0
"'
0 . 0
I ;' -~ Q=50 ......................... /;-' . Q = 100 --------
Q = 200 -·-·-·- , Q = 300
, , . II , , ,
~, '/ , .
~ , , ,
. , .· , .· , .. ·
~/ ~, .. ······ .. ··
p v , .. · , .. , .· , , ... ·· .· .. .. ... . . ~:.:.
-~ . ~.~··· · -=-···· ·
f"'lor"· ~· •• •
0.0 0.2 0.4 0.6 0.8
R = ~ (non-dimensional radial distance) a
, , ,
Fig. 5.13. Radial distribution of the non-dimensional total rotation of the plate for different values of non-dimensional lateral pressure Q.
, , ,
. . ····· ..
1.0
-0 -CIS -c.. 0 .c -
-CIS c 0 ·-tn c QJ
E ·-"'0 I
c 0 c
"-"
63
c ri~------~----~r-----~--~---r-------
c c -c 10 -c .
t q
a
z.w
·····················-·· ---------·-·-·------
Q=3 Q=4 Q=5 Q=6 Q= 10
c ~~~--~~-+--~~-T--~~-T~~~~~--~4--T~~~~
0.6 0.8 1.0 0.0 0.2 0.4
R =!:.. (non-dimensional radial distance) a
(a)
Fig. 5.14. Radial distribution of the non-dimensional shear stress of the plate for different values of non-dimensional lateral pressure Q.
-Cl.) .... ~ -0.. Cl.)
..c .... 4-o 0 til til
~ til
-~ c 0 til c Q.)
E ·-"'0 I c 0 c '-'
64
0
~~-------r------~------~--------~----~
t q
0 --·-·- -··-· _ ·-_ _ ___ r, u g __ ~l a • z.w -I
0 . 0 0 -1--
C\l I
0 . 0 l()
C\l I
0.0
························· Q =50
-------- Q = 100
- · - · - · - Q =200
Q= 300
I I I I
0.2 0.4 0.6 0.8
R = :_ (non-dimensional radial distance) a
(b)
1.0
Fig. 5.14. Continued.
CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
Conclusions
The following conclusions are drawn from the current research:
1. The effect of the shear deformation on the lateral displacement of the plate is
negligible and can be neglected for practical purposes.
2. The graphical solution given by Federhofer and Egger for the radial stress at the
center of the plate is only valid for a non-dimensional lateral pressure less than 60.
3. The approximate formula given by Timoshenko and Woinowsky-Krieger gives
higher values of stresses than the present results for values of Q greater than 4.
4. The graphical results of Federhofer and Egger, the approximate formula of
Timoshenko and Woinowsky-Krieger and the results of Ramasamudra give
comparable values of the tangential stress at the edge of the plate for aspect ratios
smaller than 10.
5. The results for the maximum radial stress given by Ramasamudra, in which he
shear deformation effects are ignored, are higher than those obtained from the
current model, in which shear deformations are incorporated.
6. The approximate formula given by Chia (Eq. 5.8) for calculating the radial stress at
center of the plate is valid only up to a non-dimensional lateral pressure of 4. For
values of Q larger than 4, it yields results that are too low.
65
66
Recommendations
The following recommendations are made for future research:
1. Study the effects of shear deformation on rectangular plates, especially since there
are significant differences in the values of the maximum stresses when shear
deformations are included.
2. Incorporate the effects of shear deformation into the analysis of circular laminated
glass plates.
3. Incorporate the effects of shear deformation into the analysis of laminated
rectangular glass plates.
LIST OF REFERENCES
[1] Ugural, A. C., Stresses in Plates and Shells, Me Graw-Hill Book Company, New York, 1981.
[2] Levy, M., "Memoire Sur La Theorie Des Plaques Elastiques Planes," J. Math. Pures Appl. Vol. 3, 1877.
[3] Reissner, E., "The Effect of Transverse Shear Deformation on the Bending of Elastic Plates," Journal of Applied Mechanics., Trans., ASME, Vol. 12, No.55, A69-A77, 1945.
[4] Hencky, H., "Uber die Berucksichtigung der Schubverzenung in ebenen Platten," Ing. Arch., Vol. 16, pp. 72-76, 1947.
[5] Mindlin, R. E., "Influence of Rotary Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates," Journal of Applied Mechanics., Vol. 18, pp. 31-38, 1956.
[6] Kromm, A., "Uber die Randquerkraftre bei gestutzten Platten," Z. Angew. Math. Mech., Vol. 35, pp. 231 - 242, 1955.
[7] Dym, L. C., and Shames, I. H., Solid Mechanics. A Variational Approach, McGraw-Hill Book Company, Inc., New York, 1973.
[8] Reddy, J. N., Energy and Variational Methods in Applied Mechanics, John Wiley & Sons, Inc., New York, 1984.
[9] Schmidt, R., "A Refined Nonlinear Theory of Plates with Transverse Shear Deformation," The Journal of Industrial Mathematics Society., Vol. 27, Part I, pp. 23-35, 1977.
[10] Vallabhan, C. V. G., "Iterative Analysis of Nonlinear Glass Plates," Journal of Structural Engineering, Vol. 109, No. 2, February, 1983.
[11] K. Federhofer and H. Egger, Sitzber. Akad. Wiss. Wien, IIa, vol. 155, p. 15, 1946.
[12] Timishenko, S. P., Woinowsky-Krieger, S., Theory of Plates and Shells, McGrawHill Book Company, Inc., New York, 1959.
[ 13] Chia, Chuen-Yuan, Nonlinear Analysis of Plates, Me Graw-Hill International Book Company, New York, 1980.
67
68
[14] Vallabhan, C. V. G., and Das, Y. C., "Nonlinear Stress Analysis of Circular Plates," Advances in Structrual Testing, Analysis and Design, ICST AD Proceedings, July 29-August 3, Bangalore, India, 1990.
[15] Ramasamudra, M., "A Mathematical Model for the Nonlinear Analysis of Circular Glass Plates by Finite Difference Method," Master's thesis, May, 1991.
[16] Langhaar, H. L., Energy Methods in Applied Mechanics, John Wiley and Sons, Inc., New York, 1962.
