analysis of plates on elastic foundations
DESCRIPTION
Analysis of Plates on Elastic FoundationsTRANSCRIPT
ANALYSIS OF PLATES ON ELASTIC FOUNDATIONS
by
WILLLIAM THOMAS STRAUGHAN. B.S. in Ind. Mgt., M.S. in Engr.
A DISSERTATION
IN
CIVIL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
Approved
May, 1990
?6l
© 1990, William Thomas Straughan
ACKNOWLEDGEMENTS
Botii professors, C V. G. Vallabhan and Y. C. Das, contributed significantiy to this
work through their counsel and giudance, and I would like especially to acknowledge the
encouragement and technical support of Dr. Vallabhan. I would also like to thank die other
members of my doctoral committee. Dr. Kishor Mehta, Dr. Ernst Kiesling, and Dr. Atila
Ertas, for their suggestions regarding this research.
A special thank you is in order for Dr. Kiesling for his persistent encouragement for
me to pursue my doctoral studies at Texas Tech University and for providing me the
opportunity to gain valuable teaching experience during this pursuit
Finally, I would like to dedicate this work to my wife for not only her encouragement
and support, but more importantiy, for her willingness to make the personal sacrifices
necessary for the completion of this work.
CONTENTS
ACKNOWLEDGEMENTS ii
ABSTRACT v
LIST OF TABLES vii
LISTOPHGURES vin
NOTATIONS X
CHAPTER
I. INTRODUCTION 1
Soil Behavior 1
Winkler Model 2
Two-Parameter Model 7
Objectives of the Research 11
n . DEVELOPMENT OF THE THEORY 15
Introduction 15
Application of die Potential Energy Principle 17
Minimization of tiie Potential Energy 20
Determination of Edge Shear Forces 23
Mathematical Model 30
Determination of the Y Parameter 30
Computational Technique 32
m . APPLICATION OF THE METHOD OF FINITE DIFFERENCES 34
General Procedure Explanation 34
Application of the Finite-Difference Equation for V w 37
m
Calculation ofEquivalent Nodal Forces 47
Development of die Coefficient Matrix 47
IV. RESULTS 49
Comparisons 49
Description of the Plate and Load Configurations Evaluated 53
Discussion of Results 58
V. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 78
Summary 78
Conclusions 79
Recommendaions 80
REFERENCES 83
^ ^ iv
ABSTRACT
Concrete slabs (plates) supported directiy by the soil continuum is a very common
construction form. The behavior of the slab when it carries external loads is influenced by
the soil, and the behavior of the soU is, in turn, influenced by the action of the slab under
load. Developing a realistic mathematical model for this complex soil-structure interaction
problem is essential in order to provide safe and economical designs. In the past, many
researchers have worked on this problem, which is referred to as "beams and plates on
elastic foundations." In many practical design problems of this type, the soil continuum is
layered and may be resting over rigid rock or a relatively stronger soiL
Most of die previous work began with the well known Winkler model, which was
originally developed for the analysis of railroad tracks. The use of the Winkler model
involves one major problem and one significant behavioral inconsistency. The problem
involves the necessity for determiiung the modulus of subgrade reaction, "k," and the
behavioral inconsistency is that an analysis of plates carrying a uniformly distributed load
will produce a rigid body displacement
Vlasov and Leont'ev (1966), recognizing die difficulty in determining values of "k"
for soils, as well as the behavioral inconsistency, postulated a two-parameter model
Vlasov's model provided for die effect of die neglected shear strain energy in the soil and
die subsequent shear forces on die plate edges as a result of ±e soil displacement Recent
woric by Vallabhan and Das (1987,1988,1989) strengdiened die Vlasov postulation for
beams on elastic foundations but stopped short of developing computational techniques for
plates.
This research develops a diice-parametcr madiematical model foe die analysis of plates
on elastic foundations. The involved equations are explained in a step-by-step manner.
v
The procediuBS developed are then used in a computer program to perform the analysis.
The necessity of determining die value of "k" and die soil shear parameter, "t," is avoided
through the computation of a third parameter, •'Y." which provides a deformation profile for
the soil continuiun.
VI
UST OF TABLES
Table
4.1 Comparative input data 51
4.2 Additional comparative output data 54
4.3 Tabulation of the diree parameter values for the uniformly distributed load 61
4.4 Tabulation of the three parameter values for the concentrated load at the plate center 65
4.5 Tabulation of the three parameter values for the concenttated load at the plate centerline on the right vertical edge 70
4.6 Tabulation of die diree parameter values for die concentrated load at the upper-right plate comer 74
4.7 Tabulation of the three parameter values for the uniformly distributed line load along die bottom, horizontal edge 77
vn
USTOFnOURES
Figures
1.1 Depiction of the deformation of a uniformly loaded plate, using die Winkler model 4
1.2 Illustration of the assumption of various methods for providing for interaction among the foundation spring elements 8
1.3 Depiction of the deformation of a uniformly loaded plate and the soil stratum using the three-parameter model 12
2.1 Depiction of a loaded plate resting on an elastic foundation, together with a deformation or mode shape function, <j) 16
2.2 Variation of (1> with Y 24
2.3 Depiction of the domain of the plate and the domain of the soil stratum, divided into eight regions 25
2.4 Illustration of die vertical displacements of the elastic-foundation surface beyond the plate edges 27
3.1 Outline of a typical plate showing equally spaced nodal grid pattern and specifically identifying the six nodes that are interior nodes in this example 36
3.2 Finite-difference molecules applicable to all nodes in the domain of the plate. Shown applied to node i 38
3.3 Hnite-difference molecules representing differential equations for plate bending moment and shear perpendicular to the plane of the plate when applied at node i on the boundary 40
3.4 Hiute-difference molecules on or near the plate boundary. Developed by VaUabhan and Wang (1981) 41
3.5 Development of the domain equation applicable to all interior nodes. Shown applied at node i 43
3.6 Development of the finite-difference molecules for the shear parameter, t, to be applied to boundary nodes odier than at the comers of die plate 44
viu
4.1 Comparison of displacement results firom the computer program for plates on an elastic foundation to die program BOEF for a beam on an elastic foundation by Vallabhan 52
4.2 niustrates load displacement and plotting directions for computer program output data for four special load cases. (Dashed lines indicate the position and direction chosen for plotting the computer program output data in subsequent figures.) 55
4.3 Common input data 57
4.4 Computer program results for a 30x40x0.5-ft plate with a 500-psf uniformly distributed load. (Results plotted in die direction of die "x" coordinate axis.) 59
4.5 Computer program results for a 30x40x0.5-ft. plate widi a 30-kip concentrated load at the center. (Results plotted in the direction of die "x" coordinate axis.) Es = 1,000 psi for all curves 63
4.6 Computer program results for a 30x40x0.5-fL plate widi a 30-kip load at the centerline on die right vertical edge (Results plotted in the direction of the "x" coordinate axis.) E^ = 1,000 psi for all curves 66
4.7 Computer program results for a 30x40x0.5-ft plate widi a 30-kip load at die centerline on die right, vertical edge. (Results plotted in die direction of the "y" coordinate axis.) E, = 1,000 psi for all curves 67
4.8 Computer program results for a 30x40x0.5-ft. plate widi a 30-kip load at die upper, right comer. (Results plotted in die direction of die "x" coordinate axis.) E, = 1,000 psi for all curves 72
4.9 Cwnputer program program results for a 30x40x0.5-ft plate with a 2-klf load along die bottom edge of die plate. (Results plotted in die direction of die "x" coordinate axis.) E, = 1,000 psi for all curves '^
IX
NOTATIONS
B = Beam width
D = Plate flexural rigidity
E = Modulus of elasticity of a beam
_ E,(l-v^ (l+v^(l-2vj
EI = Rexural rigidity of a beam
Ep = Modulus of elasticity of the plate
Es = Modulus of elasticity of the soil
G = Shear modulus of the soil
h = Spacing between nodes (used in the finite difference representation)
Hs = Depth of the soil stratum
I = McHnent of inertia of a beam
k = Modulus of subgrade reaction
Mb,Mf = Prescribed bending moments along the plate edge
Mx = Bending moment in the plate
p = Contact pressure
^ ,Qf = Net shear forces at die edge of the plate
Qx = Shear force in the plate
R = Prescribed plate comer reaction
T = Membrane tension
t = (no subscript) Soil shear parameter
tp = Thickness of die plate
u,v,w = Displacements at a point in the soil ccmtinuum in the x,y,z directions
.
Vb,Vf = Kirchoff shear forces along die edge of die plate in die x and y directions, respectively
w = Deflection in die direction of die "z" coordinate axis
Wb = Plate deflection along die edge where x = b
Wc = Plate deflection at die plate comer
Wf = Representation for a "fictitious" node off the plate boundary
Wr = Plate deflection along the edge where y = C
x,y,z = Coordinate axes
a = 1 ^ , a constant D
Y = (no subscript) Parameter that characterizes the vertical deformation profile within the soil con t inuum
Yx = Shear ing strain in the direction of the subscripted coordinate axis
Ex = Normal strain in the direction of the subscripted coordinate axis
X
Vj = Poisson's ratio of the soil
Vp = Poisson's ratio of the plate
Ox = Normal stress in the direction of die subscripted cowdinate axis
Xx = Shearing stress in the direction of die subscripted coordinate axis
<Kz) = Mode-shape function defining die variation of die deflection w(x ,y , z ) in the " z " coordinate direction
V = Laplace operator
.4. V^ = (V^ Biharmonic operator
2t
XI
:a.
CHAPTER!
INTRODUCnON
Soil Behavior
Successfiil applications of the principles of stmctural engineering arc intricately linked
to die ability of die engineer to model the stmcture and its support conditions in order to
perform an accurate analysis and a subsequendy "correct" design. Arriving at a realistic
model is complicated in foundation analysis by the extreme difficulty of modeling the soil-
structure interaction.
In particular, concrcte building slabs, supported directiy by die soil medium, is a very
common construction system. It is used in residential, commercial, industrial, and
instimtional stmctures. In some of diese structures, very heavy slab loads occur, such as
in libraries, grain storage buildings, warehouses, etc. A mat foundation, which is
commonly used in the support of multi-story building columns, is another example of a
heavily loaded concrete plate supported directiy by die soil medium. In all these stmctures,
it is very important to be able to compute plate displacements and consequent stresses widi
an acceptable degree of accuracy in order to ensure a safe and economical design.
Ultimately, all structure loads must be transferred to the soil continuum, and bodi the
soil and die stmcture act togedier to resist and support the loads. The integral nature of the
foundation and soil actions is further complicated by die complexity of die soil medium
itself. Soil is truly a nonhomogeneous and an anisotropic medium diat behaves in a non
linear manner, while concrete and steel stmctures can be adequately modeled and analyzed,
assuming isotropic and linear behavior. In addition, die properties of structural building
materials arc well known so diat die stiffness of die stmcture may be readily determined,
given member sizing and stmcture geometry.
1
2
On the other hand, soil properties are very difficult to determine because in addition to
die previously mentioned characteristics, it is a "soft" material, which makes it very
difficult to obtain samples for testing that will produce laboratory results paralleling its
acmal "in-ground" behavior. Among other problems, die type of soil affects the ability to
obtain representative samples (for example, stiff clay is more difficult to sample than soft
clay). Variations in sampling techniques among laboratories further complicate the
problem. Two additional complicating factors arc that soil material properties are sttess-
dependent, and the soil continuum will in practice consist of layers of materials with
different constitoitive relations and material properties. Because of these factors, the tme
properties and constituitive relations of the soil continuum are essentially unknown and
indeterminable. As a result, it is necessary to make a number of simplifying assumptions
to analyze the soil-stmcturc interaction.
Winkler Model
One very popular method for modeling the soil-stmcture interaction has its origins in
the work done by Winkler in 1867, where the vertical translation of the soil, w, at a point is
assumed to depend only upon the contact pressure, p, acting at that point in the idealized
elastic foundation and a propOTtionality constant, k.
p = kw. (1.1)
The proportionality constant, k, is commonly called die modulus of subgrade reaction
or die coefficient of subgrade reaction. This model was first used to analyze die deflecticHis
of and resultant stresses in railroad tracks. In die intervening years, it has been applied to
many different soil-structure interaction problems, and it is known as die Winkler model.
3
Description of the Model
Application of die Winkler model involves die solutiwi of a fourrii-otder differential
equation. For plates, die equation to be solved is of the form
DV^w + kw = q (1-2)
where D = the plate flexural rigidity, k = the modulus of subgrade reaction, and q = the uniformly distributed load on the plate.
The model consists of linearly elastic springs with a stiffiiess of "k," placed at discrete
intervals below the plate, where k is the modulus of subgrade reacticHi of the soil. The
model is also ftequentiy referred to as a "one-parameter model"
Inherent Problems with the Model
Determination of the Modulus of Subgrade Reaction
In addition to the relative conq}utational difficulties involved in die soluticxi of this
equation, it has not enjoyed widespread use by practicing engineers because erf the relative
difficulty involved in determining the value of the modulus of subgrade reaction, k.
Numerous researchers have worked on developing techniques for the determination of k;
however, one of the most definitive papers was published by Tcrzaghi (1955). His work
showed diat the modulus of subgrade reaction depends upon the dimensions of die area
acted upon by the subgrade reaction, and he incorporated size effects in his equations.
The Winkler model assumes that no interaction exists between adjacent points in die
soil continuum. In other words, as shown in Figure 1.1, die springs arc considered as
isolated foundation elements. In order to improve the physical representation and still
utilize a relatively simple madiematical model, researchers have proceeded in two
fundamentally different directions. Some, such as Cheung and Zienkiewicz (1965), have
considered the problem of plates on an elastic semi-infinite continuum by introducing
si ThinPlate^f i i J , i i i i l J , i
Soil Stratum
^^M^N^M^
Figure 1.1. Depiction of the deformation of a uniformly loaded plate, using the Winkler model.
• -4 i^f
5
simplifying assumptions widi respect to expected displacements and stresses. This woric is
not applicable in real situations where die soil is generally layeitsd. The use of Cheung's
and Zienkiewicz's approach will result in uncharacteristically large displacements for a fiilly
loaded plate or slab on a semi-infinite soil continuum. Otiier researchers have attempted to
make die Winkler model more realistic by assuming some kind of interaction between die
spring elements. The first approach is discussed in die next section.
Summary of die Research for die Modification of "k"
Biot (1937) solved the problem of an infinite beam with a concentrated load resting on
a three-dimensional subgrade by evaluating the maximum bending moment in the beam.
He found diat he could obtain a good coirelation witii die Winkler model for the maximum
moment case by setting:
, 0.95 E. k = — *
E,B -,0.108
4
(1.3) (1-v,^) L(l-Vs")EIJ
where E, = modidus of elasticity of the soil, v, = Poisson's ratio of the soU,
B = beam width,
E = modulus of elasticity of the beam, and
I = moment of inertia of the beara
Later work by Vesic (1961) showed diat k depends upon both die stiffness of the soil,
as well as the stiffness of the structure, so that similar size stmctures of different stif&esses
will yield different values of k for the same applied load. Vesic's work extended Biot's
solution by providing the disuibution of deflection, moment shear, and pressure along the
beam. When Vesic divided the pressure along the beam by deflection at the same point
along die beam, he found the ratio between die two to be neariy constant He found die
continuum solution correlated with die Winkler model by setting
k =
6
0.65 E, Q / E V
(l_v,2) V m (1.4)
where all terms were previously defined.
Even using these values of k found for die two special cases described, an exact
correlation with the Winkler model was not obtained for die corresponding values of all die
variables for the continuum model Using the continuum solution as a reference, Vesic
described dus lack of correlation as an "error" in die Winkler model which he found to be
a function of the ratio of the characteristic length to the widdi of the beam. In addition to
diese factors, the value of k is also dependent upon the load distribution, the depth of die
soil continuum and any layering effects present in die continuum. Qearly, there is no
unique value for k, and in many instances it may not be determinable, even considering
field testing.
Determination of Smicture Displacements
The second problem area in the utilization of the Winkler model for the analysis of
slabs or plates on elastic foundations is that a slab carrying a uniformly distributed load
with fiiee edges will produce a uniform vertical displacement of the slab rather than a "dish-
shaped" slab displacement that one would expect For a physical representation of this
behavior, refer to Hgure 1.1. Vallabhan and Das (1987) showed diat the use of the
Winkler model can lead to nonconservative design values. Vesic (1973) also performed
research in this area, and he found that the maximum moment in large slabs was scxnewhat
less than that calculated using the Winkler model, depending upon the value of k used.
Even Terzaghi (1955) recommends against die use of his dieories of subgrade reaction fw
estimating stmcture displacements.
7
Two-Parameter Mndd
Summary of Other Research
As mentioned earlier, several researchers, recogiuzing the inherent problems widi die
Winkler model, attenqited to make the model more realistic by assuming some form of
interaction among the spring elements that represent die soil continuum.
Filonenko-Borodich (1940) developed a model that improved upon the Winkler nxxlel
by connecting die top ends of die springs widi an elastic membrane stretched to a constant
tension, T. Refer to Figure 1.2. In this model die modulus of subgrade reaction is given
by:
p = kw-TV^w (1-5)
where V is the Laplace operator, and all other terms were previously defined; however, no
method is provided for the computation of k or T.
Hetenyi (1946 and 1950) created an interaction among the springs in the fovmdation
by imbedding an additional plate with flexural rigidity, D , in the Winkler foundation in a
manner shown in Hgure 1.2. According to this model, the modulus of subgrade reaction
is given by:
p = kw + D*vVw (1- )
where all terms have been previously defined; however, no method is provided for
determining die values of k and D*.
Pasternak (1954) inq)roved upon the Winkler model by connecting the ends of the
springs to a plate, or "shear layer," consisting of incompressible, vertical elements, which
can deform only by lateral shear. According to this model die modulus of subgrade
reaction is given by:
p = k w - G V V ( -" ^
Even though "G" represents the shear modulus of the elastic foundation, no unique
niethod is provided for the deterndnation of k.
IStretched membrane, - plate in bending,
or shear layer
Figure 1.2. Illustration of the assumption of various methods for providing for interaction among the foundation spring elements.
9
A more detailed description of all diese models is presented by Kerr in his p^ier,
"Elastic and Viscoelastic Foundation Models" (1964).
Recognizing die impracticalities involved in die application of diese modifications to
die Winkler model, Vlasov and Leont'ev (1966) used a new dieoretical approach to develop
a two-parameter model for plates. Recendy, Nogami and Lam (1987) used a similar
approach to develop a two-parameter model for beams on elastic foundations for the plane
strain case.
Vlasov's and Leont'ev's model (commonly known as the "Vlasov model") provided
for shear strains within the soil continuum and resulted in the domain equation,
DV\v - 2tV^w + kw = q (1-8)
where t = the soil shear parameter.
This equation, which considers shear interactions within the foundation and stmcture,
was developed using variational principles, i i addition, Vlasov and Leont'ev introduced
another parameter, which they identified as Y to characterize the vertical deformation profile
within the soil continuum. The real sttength of Vlasov's and Leont'ev's approach is in the
total elimination of the necessity to determine empirically the values of the modulus of
subgrade reaction, k, or even the shear parameter, t, as their values can be computed once
the value of Yis determined. This model has the disadvantage of requiring an estimate of
the Y parameter since no mechanism was developed for computing the value of Y-
Yang (1972) considered the analysis of rectangular plates on elastic foundations using
Vlasov's and Leont'ev's two-parameter model to represent the soil-stmcture interaction.
For his analysis, he used an iterative approach, which combines the finite-element method
for die plate widi die finite-diffaence technique for die boundary conditions. Like Vlasov
and Leont'ev, Yang did not provide a mediod for die computation of die important vertical
defwrnation profile parameter, Y- He used die same estimated values of Y and assumed a
semi-infinite continuum where die depdi of die soil is infinite, as Vlasov and Leont'ev did.
10
and dien computed the values of the other parameters, numerically. None of diese
researchers have solved the problem of analyzing plates on an elastic foundation with finite
soil layers.
Refinements of die Two-Parameter Model
Recognizing the importance of Yin die control of die stress distribution widiin the
foundation, Jones and Xenophontos (1977), using variational principles, enhanced the
Vlasov model by developing an expression for the computation of Y Their work
established a relationship between the Y parameter and the displacement of the structure
resting on the continuum, but it stopped short of actually computing the value of Y or of
even suggesting a computational method.
Vallabhan and Das (1988b) developed an iterative procedure for use in the calculation
of the Y parameter for beams on an elastic foimdation. They found that for a uniformly
loaded beam on an elastic foundation, the Y parameter is dependent upon the ratio of the
depth of the soil stratum to the length of the beam. They identified their model as a
modified Vlasov model or a "three-parameter model" because the elastic foundation can
be characterized by the duce interdependent parameters, k, 2t, and Y- All of these
parameters are influenced by the load distribution, as well as the material properties and the
geomedy of the stmcture and the foundation. The equation used to characterize beam
behavior is
E l i - S ^ - 2 t ^ + k w = q. (1.9) dx^ dx^
Vallabhan's and Das' study included displacement computations, using the mediod of
finite differences fca- du ee load cases involving beams on an elastic foundaticm. In general,
dus computational approach yielded results diat were in excellent agrecnacnt widi diose
determined for die same stmctural system and load case using the finite-dement mediod
11
In die case of a beam subjected to a uniformly distributed load, die shape of the
displacement curve was nearly parallel to, but with slightiy lower values dian, those
obtained by the finite-element method. A design using this analysis would be conservative,
since the maximum bending moment and shear values found using the finite-difference
method were slightiy higher than those found using the finite-element method. Vallabhan
and Das concluded that if the loads arc fairly evenly distributed on the beam, the results
from the Vlasov model arc sufficiendy accurate for practical designs.
Objectives of die Research
The objective of this research is to develop a workable approach for the analysis of
plates on elastic foundations that will provide the designer with realistic stress values for
use in the design of the plate or, more specifically, reinforced concrete slabs. In this
research, several types of plate loading arc considered, including die uniformly distributed
load, concentrated loads, and line loads. Combinations of these loading systems, as well
as other types of loads, such as applied, concentrated edge moments, may be easily
handled using the computational techniques presented.
Steps Involved in the Research
Selection of the Model
As can be determined from the prcvious emphasis, the selection of the model is die
most important step. The Vlasov and Leont'ev model as strcngthened by J<mes and
Xenophontos and subsequendy improved by Vallabhan and Das (three-parameter model),
is the one used in dus research. This model as mentioned earlier, considers shear strain in
the soil stratum and results in a "dish-shaped" plate deformation profile. Refer to
Figure 1.3. It also eliminates the necessity of attempting to determine values for the
modulus of subgrade reaction, k, or the shear parameter, t
12
Thm Plate
Figure 1.3. Depiction of the deformation of a uniformly loaded plate and the soil stratum using the three-parameter model.
13
While die minimization of die required amount of data concerning die soil material
properties and constituitive relationships is a definite goal of diis research, a more
overriding goal was die importance of constmcting a realistic model. Chapter U presents
die development of die dieories and principles involved in utilizing diis modeling approach.
Selection of a Compntarional Approach
One of the primary goals of the research is to develop not only a realistic procedure for
die analysis of plates on an elastic foundation, but a practical and easily applied procedure
as well Qearly, die solution of dus type of soil engineering problem, which involves
eqiulibrium equations together with constituitive relations, compatibility considerations,
and complex boundary conditions, would require such an effort diat a purely mathematical
approach is impractical An altemative approach is to use a numerical analysis technique
that will provide approximate solutions as close to the exact solutions as required for
practical engineering design problems.
Both the finite-element method and the method of finite differences were considered.
Each method will generate and require solutions for a set of simultaneous equations;
however, the use of the finite-element mediod will generate a coefficient matrix that is much
larger than that generated by the use of the finite-difference method. This means that there
are many more unknowns to be solved and that most problems involving the analysis of
plates on elastic foundations with the finite-element method are well beyond the capacity of
a programmable calculator or a microcomputer, both of which are prevalent in engineering
offices today.
In order to help ensure the acceptance of the method by practicing engineers in small,
medium, and large engineering offices, the selection of a method that is at least widiin the
Storage capacity of microcomputers is essential. The finite-difference method satisfies this
criteria. It makes direct use of the fourth-order differential equations through the use of
14
computational "molecules," which have been extensively developed for plates not only in
the domain of die plate, but for various boundary conditions. The more extensively used
references in diis research involved die work done by Crandall (1956), Tunoshenko and
Woinowsky-Krieger (1959), Bowles (1968), Ugural (1981), and Vallabhan and Wang
(1981).
The computational method used in this research is applicable to the analysis of "thin"
plates, which experience small deflections consistent with the definitions explained in many
textbooks on plates and shells (Timoshenko and Woinowsky-Krieger, 1959; and Ugural,
1981). Chapter HI presents a detailed explanation of the plication of the finite-difference
method to the analysis of plates on an elastic foundation as used in this research.
Application of die Mediod
The computer program developed in conjunction with this research was used to
analyze several realistic cases of uniformly loaded slabs to determine the vertical
displacement and internal strcss residtants at various points within the slabs. Comparisons
were made, where possible, to die results obtained by other investigators by eliminating or
simply bypassing those portions of the program with analysis aspects not considered by
them. Examples of these woidd be die edge shear terms, the computation of the modulus
of subgrade reaction, ete.
Results of some of the cases analyzed arc presented in Chapter IV, and the conclusions
pertinent to this research are presented in Ch^ter V.
CHAPTER n
DEVELOPMENT OF TEffi THEORY
Introduction
Plates on elastic foundations represent a complex soil-structure interaction problem.
The development of the equations for such problems and the associated solutions become a
challenge to die engineer. This chapter is devoted to die dieoretical development of die
equations, and Chapter HI deals with the numerical techruques employed to arrive at the
solutions. A majority of the steps during the theoretical development are omitted; however,
all the most important steps are shown so that readers familiar with the Kirchhoff theory of
plates and variational calculus may easily follow the derivations.
There are two fundamentally different mediods employed in solid mechanics to derive
the field equations and the boundary conditions for the analysis of complex stmctures using
the displacement formulation. One method commonly used involves die assumption of
displacement functions and the subsequent development of the equilibrium equations. The
other method involves the assumption of the displacement functions and the application of
the minimum potential energy principle, which is the method chosen for this research
because it is easier to apply for this complex soil-stmcturc interaction problem. The
principle of minimum potential energy is explained in various textbooks, such as Langhaar
(1962), Fung (1965), Timoshenko and Goodier (1971), ete., and it is not repeated here.
As mentioned in Chapter I die soil stratum considered has a finite depdi witii a rigid
boundary at the bottom. Figure 2.1 depicts a soil stramm depdi of Hj, a modulus of
elasticity, Es, and Poisson's ratio, v,. The elastic foundation supports a plate which is
subjected to a vertical load, q(x,y). The flexural rigidity of the plate is D.
15
16
<t>(Hs) = 0
*• x,u
Figure 2.1. Depiction of a loaded plate resting on an elastic foundation, together with a deformation or mode shape function, <|>.
D=3v 12(l-Vp2)
where Ep =
tp = Vp =
die modulus of elasticity, the thickness, and Poisson's ratio of die plate.
17
(2.1)
Application nf the Potential Energv Principle
The total potential energy in die plate-soil system may be written as TC = Upuue + Ujoa
+ V, where UpUte is die strain energy stored in die plate, Ujoa is the strain energy stored in
the soil, and V is the potential energy of die prescribed loads.
This expression may be written more explicidy as
•t^ + b
-C - b
^ = ^ \ \ j (VV-2( l -Vp a2 0 w
LI*''J ^2 o w a w
3x3y * dxdy
(2.2)
•ii + b
+jjj [« xex+ y^+ a,e,+ txyYxy+ txzYxz+ Wyz] dxdydz-f Jqw dxdy
where w = die vertical displacement of the plate in the z direction, and q = the applied uniformly distributed load; and a, t, E, Y represent normal stress, shear stress, normal strain, and shear strain in
the elastic foundation. The size of the plate is 2rx 2b x tp; X, y, z represent the coordinate system; and D and Vp were previously defined.
Constituitive Relationships
The stress-strain relationship in the soil may be represented in matrix form by
assuming that it is a linearly elastic, isotropic, and homogeneous material.
where
r^x
"xy
•xz
v ^ Es(l-Vs) r (1 Vs)(l-2v,)
y^hzj
c =
d =
1 c c 0 0 0
c 1 c 0 0 0
c c 1 0 0 0
0 0 0 d 0 0
0 0 0 0 d 0
0 0 0 0 0 d « J
,and
< Yxy
Yxz
>
l-v, 5
2(1-Vs)
Es(l-Vs)
18
(2.3)
(2.4)
Tlie term (i+vj(i_2vj ^ written as E in applicable subsequent expressions for
convemence.
Assumptions of the Displacement Functions
The soil-stmcture interaction problem posed here is a three-dimensional one. The
terms (u,v,w) represent displacements in the x, y, z directions, respectively, in the soil.
From a practical point of view, however, the lateral displacements in the soil are negligible
compared to the vertical displacement in the z direction, and, hence, it is assumed that
U(x,y,z) = 0
and (2.5)
v(x,y,z) = 0.
Following Vlasov and Leont'ev (1966), it is fimher assumed that
w(x,y,z) = w(x,y)(|»(z) (^.6)
where w is the vertical displacement of the plate, and <j»(z) is a mode shape defining the
variation of die deflection w(x,y,z) in die zdirectiwi. This function is depicted in
19
Figure 2.1, and since w(x,y) is die plate displacement, (j)(0) is equal to 1.0, and (|)(H,) i
equal to 0 (zero). is
Strain-Displacement Relationships
Using die strain-displacement equations of elasticity (Timoshenko and Goodier, 1971)
and Equations 2.5 and 2.6 the following expressions may be written:
and
e, = - g ^ = w ( x , y ) ^ .
' ay ax "'
y ay az ay ^^^^'
" az ax ax ^^•
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
Expansion of the Potential Energy Expression
Substituting the values for strain from Equations 2.7 through 2.12 into the potential
energy equation, die result is
•*t+h
-C-h
(vV-2(l-Vp) a w a w [ a w dxdy
i j j j h " ^ ^ ' ^ ^ *^ VIT^ <l»]dxdydz-j fqwdxdy. •rf + b (2.13)
-I -b
The corresponding stress components in the soil become a^ = E(cEx+cey+eJ = E e ^ = E w ^ (2.14)
r„ = Eiy„.^M^ •xz (l+vj(l-2v,) l-2v.
V = EdYyz = G ^ ( 0
2(1-V3)
aw ay
Y„ = G ^ *
where
G = shear modulus of the soil = 2(l+v,)
Now the potential energy may be expressed as
^ +b
^ = f j jj(VV-2(l-Vp) -c -b I
^2 o w ax2
^ 2 ^ a w
IVJ a w axayj dxdy
G(Vw)^ (j>
•rf +b
dxdydz- I I qw dxdy.
20
(2.15)
(2.16)
(2.17)
(2.18)
-c -b
Minimization of the Potential Ener^
Using variational principles and minimizing the total potential energy by taking
variations in w and (j>, yields
•*( + b
SIC "^^ d%
= f f (DV*w-2tV^w + kw-q)5wdxdy + J - m - ^ + n<|> 5(|>dz
-C - b 0
-I- boundary conditions (b.c.) = 0
where
•H^ dz.
H
m
2t=fG<|)^dz, 0
= f I Ew^ dxdy.
(2.19)
(2.20)
(2.21)
(2.22)
a 2f)
21 and
n = f fG(Vw)^dxdy. (2.23)
Since 5w and &J) by definition are not equal to 0 (zero), die terms widun die
parendieses and die boundary conditions must each equal 0. The first term in die
parcndieses in Equation 2.19 may be written as
DV*w-2tV^w + kw =q, (2.24)
which gives the main field equation for die plate on an elastic foundation problem. The
boundary conditions for this equation arc:
1. for die plate edge X = b,
Vb + 2 t | ^ = Q , a n d M , = M, (2.25)
2. for the plate edge y = r,
ay
3. for the plate comer,
V 2 t | ^ = 0 , and My = N (2.26)
2M^ = R. (2.27)
The terms used in the above equations are defined as:
Vb,Vf = Kirchhoff shear forces along the edge of die plate in the x and y
directions, respectively;
Qb'^/ = net shear forces alcmg the plate edge, including the prescribed shear
forces;
^b'^i = prescribed bending moments along the plate edge; and
R = net comer forces, including die prescribed comer forces.
Similar boundary conditions exist on edges x = -b and y = -£
Outside the domain of die plate, die field equation ±at controls ±e surface
displacement of the soil stratum is 2 ^ (2^8)
-2tV w + kw = 0.
22
The boundary conditions for this case are
Equivalent edge shear force at x = ±b = Qb and w = 0 at x = «, (2.29)
Equivalent edge shear force at y = ± r= Q/ and w = 0 at y = «. (2.30)
The second equation widun die parendieses in Equation 2.19 is die field equation
representing the deformation characteristics of the soil continuum. It is,
- m ^ + n . l , = 0 (2.31)
inside die soil domain where die boundary conditions are at z = 0, (0) = 1.0 and at z = Hj,
<j>(Hs) = 0.
Therefore, for the plate on an elastic foundation problem, there are three differential
equations diat must be solved as indicated by Equations 2.24,2.28, and 2.31. The first
equation involves the displacement of the plate; the second equation involves the
displacement of the soil surface outside the domain of the plate; and the diird equation
involves the displacement of the soil continuum. The first step in the overall technique
employed in this research is to solve Equation 2.31 explicidy to obtain the deformation
mode (|)(z). The second step is to use die residts of diis solution to solve Equation 2.28,
which will yield the boundary forces necessary for the solution of Equation 2.24, which is,
in turn, solved by numerical methods.
Deformation Mode ^z) in Soil
The solution of die differential Equation 2.31 applying die boundary conditions yields
sinhY
• = • (2.32)
sinhY
23
where Hg = the depth of the soil stratum and
J J (Vw) dxdy f Y ''
H. = -S = - | ^ = ^ ; = • (2.33)
J J w^ dxdy
It is important to note that the coefficient, k, which is usually represented as the
modulus of subgrade reaction among engineers and the coefficient, t, which represents the
shear deformation in the soil, are bodi dependent upon the vertical deformation mode-shape
function, <|), and the depth of the soil stratum, Hj. (Refer to Equations 2.20 and 2.21.)
But the mode-shape function, ((>, is dependent upon the value of Y-
The value of Y, in tum, varies with the displacement of the plate and die depth of the
soil stratum as shown by Equation 2.33. In odier words, for a given plate on an elastic
foundation problem, the variables w, k, t, and Y are all interconnected in a very complex
manner. Vallabhan and Das (1988b) published die results of a smdy of a modified Vlasov
model for beams on elastic foundations in which they confuted and plotted a family of
curves showing the variation of the (j) function with depth in the soil stratum for various
values of Y- This portion of their work is included as Figure 2.2.
Determination of Edge Shear Forces
The boundary forces on die plate arc evaluated by solving Equation 2.28, repeated
below. This equation,
-2tV^w + kw = 0,
has to be solved in a domain outside die plate boundaries widi z = 0. The cxart solution of
dus equation is too compUcated, and Vlasov and Leont'ev (1966) assumed an approximate
solution for die displacement fimction w(x,y). The domain outside die plate is divided into
eight regions as shown in Figure 2.3. Symmetry of diesc regions can be assumed for die
24
Hs
><|)(z)
Figure 2.2. Variation of (j) with Y
Plate
25
IV
m IV
n + - • • X n
IV m IV
Figure 2.3. Depiction of die domain of die plate and the domain of the soil stratum, divided into eight regions.
26
solution of die equation in tiiese eight regions, which facilitates die generation of solutions
in Regions H, DI, and IV. Even diough symmetry of the regions is assumed, die overall
problem is not necessarily symmetric widi respect to die x and y axes because die plate
loading may not be symmetrical.
RegionU: b < x < o o a n d - r < y < / '
In this region, an approximate solution of the type
w(x,y) = Wj e"^^""^ (2.34)
where Wb is die vertical displacement of the plate at x = b. Substituting Equation 2.34 and
assuming w(oo,y) = 0, one gets
X = 7 | . (2.35)
The shape of this fimction w(x,y) is illustrated in Figure 2.4. Equivalent boundary
forces from the soil continuum acting on the edge of the plate arc computed using the
principle of virtual displacement This solution approach is utilized because of the lack of
an exact solution for the displacement in this region. The virtual work done by the edge
shear force on the plate equals die virtual strain energy in die soil continuum undergoing
displacement in this region.
J Q ^ 5 W dy = J J J (OzSez + XXZSYXZ + V^yz) dxdydz (2.36)
-t 0 - f b
where
^^ = E^ ^ f txz&yxz=G^08
dz'
rdwi IdyJ
<1), and (2.37)
27
Hgure 2.4. Illustration of die vertical displacements of die elastic-foundation surface beyond die plate edges.
28
By using Equations 2.20 and 2.21, Equation 2.36 can be rewritten as
jQ,5W,dy = j J --c -c b
kW 5W^ + 2t aw 55 ' ' 2 "»
O W
\^J
_)_ aw gTaw' ay \dy^
'dxdy
r
i ; 2 H W , - 1 ^ "" ^ dy2
\ (2.38)
5W,dy.
By comparing die integrals, die equivalent edge shear force, Qb, is obtained
d vV
Using Equation 2.25, die Kirchhoff shear force at x = b becomes
as
(2.39)
r V x = - y2Ew. + 2t |^ , -
^ dV ^
ax •x=b ^ dy (2-40)
It is important to note the difference between Wb and w in Equation 2.40, where - ^ is the
slope of displacement of the plate in die x direction at x = b, while Wb is die displacement
at X = b. Since all displacement terms are referenced to the plate at x = b, Equation 2.40
can be rewritten as
I y2Ew + 2 t ^ - f i-5L
ax X ay >• • ' T =
(2.41) 'x=b ^ ^x=b
Equations for Vj at x = -b can be derived using die same logic, and die equation is die
same except for a negative sign in front of Vx.
RegionlH: r<y<«>and-b<x<+b
Following die logic presented in die previous region, die displacement fiinction w(x,y)
in Region III is assumed as
w(x,y) = W^ e -X(y-f) (2.42)
29
where Wf is die vertical displacement of die plate aty = /;andA.= M-,as obtained
earlier. Following similar derivations, die Kirchhoff edge shear force for die plate is
y=r
y2Ew + 2 t ^ - L ^ 9y X 3x2 J
y=f (2.43)
The equation for Vy at y = -/^ can be obtained by changing die sign in die above equation.
Region IV
The displacement function for this region is developed by combining Equations 2.34
and 2.42, such that
w(x,y) = W, e"^"-"^ e""^-'^ (2.44)
where Wc is the vertical displacement at die plate comers. Using the principle of virtual
work, the comer reaction is derived by equating the virtual strain energy in the soil to die
virtual work done by the edge shear forces and die equivalent comer reaction:
oo oo oo
R SWp = J J (-2tV^w + kw) W,.5w dxdy + J 2t | ^ | 5wdy [ h [ x=b
(2.45)
5wdx
Rc = ftw, (2.46)
All die odier comer forces arc now - t times die displacements of die plate at die respective
comers.
30
Madiematical MnHpl
The madiematical model for die plate on an elastic foundation is dius represented by
die following domain equation and die associated boundary conditions.
DV^w - 2tV^w + kw = q
in the domain of the plate with die boundary conditions that
at x = ±b, M3 = 0 and V,= - y2Ew + 2 t | ^ - J - ^ w ax 2X ayZ
(2.47)
x=b
and
at y = ± / ' , My = 0 and Vy = - / 2Ew + 2 t ^ - ^ ^ w ax 2X ay2 (2.48)
where M;j and My are the prescribed moments in the x and y directions, respectively, and
Vx and Vy are the Kirchhoff shear force in the x and y directions, respectively.
In order to solve this equation consistent with the deformation of the soil, a value of Y
must be determined. Vlasov assumed an arbitrary value of Y ; however, this research
presents a definitive approach for the determination of Y-
Determination of die Y Parameter
The Y parameter is computed from the equation shown below
r Y V l-2Vs H.
J J (Vw)^ dxdy
2(l-v^ J J w^ dxdy
(2.49)
over the entire domain of die surface, including the plate.
To understand die approach better for solving die first set of terms in die numerator
and in die denominator, die soil domain is divided into regions, as shown in Figure 2.3.
31
In Region n
, . , , , -X^x-b)
w(x,y) = W^ e
-C b
4
-I
4
j j (Vw) dxdy = ^jw^^dy -c b -[
(2.50)
(2.51)
(2.52)
In Region HI
w(x,y) = W^ e -X<y-0
Jjw^dxdy=i)w> l-b
,+b
-b
+b
jJ(Vw)^dxdy=|jw^^dx
f -b -b
(2.53)
(2.54)
(2.55)
In Region IV
w(x,y) = Wc e e
w; oo oo
r j w^dxdy =
t b
oo oo - j ^ . 2
j J ( V w ) ^ d x d y = ^
C b
(2.56)
(2.57)
(2.58)
32
In Region I
The two terms, •t +b 4 +b
J J w2 dxdy and J J (Vw)^ dxdy, (2.59) -^ -b -c-b
arc evaluated by numerical integration.
Y Computational Summary
Expressions have been developed and appropriately expanded for all terms required
for the computation of the soil deformation parameter, Y. Computation of some of die
terms wiU require the use of numerical integration coupled with die finite-difference
method. A general explanation of the computational techniques involved in this research is
included in the last section of this chapter with additional details provided in Chapter EL
Computational Technique
A computer program was written that uses the method of finite differences to constmct
die full coefficient matrix for rectangular plates on elastic foundations. It was necessary to
work widi die full coefficient matrix using die finite-difference model because die matrix
was found to be unsymmetrical when all boundary conditions were incorporated.
As previously indicated, diese displacement computations arc dependent upon die
value of the Y parameter. The solution technique is an iterative process in which Y is
initially set equal to one (1.0); and die mode shape, (j), is calculated and used in die
computation of the values of die modulus of subgrade reaction, k, and die soil shear
parameter, L Widi diese parameter values computed, die vertical displacements at discrete
points in the plate are dien calculated.
Upon completion of die assemblage of die coefficient mattix, die load vector is
computed, and die set of simultaneous equations is solved.
33
Then Y is calculated from Equation 2.49 using die plate displacement values, and a
comparison between diis calculated value of Y and die initially assumed Y value or die
previously calculated Y value in subsequent calculations is made. If die difference between
die two successive Y values is widiin a prescribed tolerance limit, die displacement
computations cease. Otherwise, another iteration is performed, and the process is repeated
until the final value of Y does not deviate from die previous Y value by more dian die
prescribed tolerance.
The plate displacement values determined during the final iteration for the Y value are
used in the computation of internal plate-bending moments and shear forces at discrete
positions within die plate. These values are necessary for the design of die plate, and they
arc considered as the final program output
Numerical integration is used extensively in the programs, both in conjunction with
and in addition to, the finite-difference mediod in solving die differential equations. A
detailed explanation of the use of the finite-difference equations is provided in the next
chapter, while the results of die various computer program "runs" are presented and
discussed in the subsequent chapter.
CHAPTER m
APPUCATION OF THE METHOD OF FINITE
DIFFERENCES
General Procedure Rxplanarinn
Solution of the fourth-order differential equation,
DV^w - 2tV^w + kw = q, 0- D
in die domain of the plate,
where w = the displacement of the plate,
t = the soil shear parameter, k = the modulus of subgrade reaction, and q = the applied uniformly distributed load,
is too complex for arbitrary loads when die boundary conditions are considered, unless a
numerical approach is used.
The applicable boundary conditions are:
a t x = ±b, M j = - D a_w_ a w ax^'^'^Pay^
= 0
andV^ = -D d w + (2-Vp)- w ax^ ' "" '^ dxdy^_
aty = ±/;M^=-D
/2iSw + 2 t 4 ^ - - H ^ ax 2X ay^
r:i2 -.2 a w a w = 0
andV^=-D 9'w^(2-vJ^ '^ ay^ ax2ay.
n^r- ^ o» aw 1 a w ^ ^ • ' ^ ^ a 7 " 2 X - ^
(3.2)
(3.3)
(3.4)
(3.5)
34
35
The approach selected for diis research is die mediod of finite differences, which
allows die conversion of a set of linear differential equations into a set of linear algebraic
equations diat may be solved simultaneously on a digital computer by employing matrix
mediods. It is, of course, an approximate mediod widi die fundamental approximation
being die replacement of a continuous domain by a set of points or "nodes" widiin die
domain. Using die finite-difference mediod, die derivative of a function w = f(x,y), widi
respect to X or y where w is die deflected shape of a plate in flexure, may be expressed as
die difference of die values of w at die nodes of a plate grid. Refer to Figure 3.1.
Another method diat could have been used is the finite-element method; however, die
finite-difference method was selected because of its computational efficiencies. To achieve
results of comparable accuracy would have required a coefficient matrix that is many orders
of magnitude larger than that required using die method of finite differences widi its
consequent requirement of more computer memory and much longer "running" time.
Moreover, because the finite-element method is an energy method, it requires additional
derivations of the coefficient matrix for bending of the plate. Also, die coefficient matrices
for k and t must be individually developed and assembled into the overall stmcture
coefficient or stiffness matrix. Since k and t are bodi dependent upon die value of Y, dien
these two coefficient sub-matrices must be generated and again assembled into die stmcture
coefficient matrix widi each iteration. This is a tedious and time-consuming procedure.
Finally, die finite-element mediod operates on a reduced order of derivatives, while die
finite-difference mediod operates directiy on die fourdi-order differential equations, again
yielding more accurate solutions with a reduced number of unknowns.
Figure 3.1 shows a grid drawn on die surface of a rectangular plate. The numbers
represent equally spaced points or "nodes" on die plate surface. As shown, each node is
placed a distance "h" from its nearest "neighbor" node in bodi die x and y directions. This
particular plate example will be used to illustrate die use of die finite-difference "molecules"
36
37
31
25
38
32
26
K h ,
39
33
27
40
34
28
41 42
35 36
T
29 30
h T
19
13
%
20
14
22
16
10
23 24
17 18
11 12
1 h + h Jf h + h + h
Figure 3.1. Outiine of a typical plate showing equally spaced nodal grid pattem and specifically identifying die six nodes diat are interior nodes in dus example.
37
to determine die plate deflections and, later, die in-plane bending moments and die shear
forces perpendicular to the plane of die plate.
Application of die Finite-Difference Equation for V w
Interior Nodes
The finite-difference representation of v w and V w is shown in Figure 3.2. This
equation is valid for all points within the domain, including the plate edge nodes.
However, the application of diis equation at the "edge" or boundary nodes will result in
nodes that are off the plate. In addition, the application of the v w molecule at nodes that
arc located either on the first row or the first column of nodes inboard of the plate boundary
will also result in nodes that are off the plate. The crosshatched portion of the plate shown
in Figure 3.1 represents the only plate area containing nodes to which the v w molecule
'may be applied in this particular example without resulting in nodes that are off die plate.
These nodes, which are at least two nodes inboard of the plate edge nodes will be
designated "interior" nodes.
Boundary Nodes
To handle die application of die V w molecule to die edge nodes and neighbor edge
nodes and die application of die V^w molecule to die edge nodes, die known boundary
conditions are utilized.
Prescribed bending moment =*1VI| = -D
Prescribed bending moment =>]VI^=-D
ajv a j ^ ax •'ay'j
2^'^pax2 [dy
= 0, along X = ±b.
= 0, along y = ± £
Net shear =* (^ = -D ->3 a w ^ + ( 2 - V p ) , ^ 2
ax^ ^ axay^j , along X = ± b.
(3.6)
(3.7)
(3.8)
38
v^-*^a
Figure 3.2. Finite-difference molecules applicable to all nodes in the domain of the plate. Shown applied to node i.
39
Net shear ^ Q = _D a w , /^.,K a w i7^''-^''^j-'°"^'"*' '3-''
where D is die flexural rigidity of die plate, as given in Equation 2.1. The representation of
these equations in molecular form is illustrated in Figure 3.3.
Numerous researchers have adjusted die edge nodes and die neighbor edge nodes of
rectangidar plates using die known boundary conditions for moment and shear. The results
of the work of Vallabhan and Wang (1981) for the analysis of diin plates are shown in
moleciUe form in Figure 3.4, and these are the ones used in this research. This figure
represents the application of the fourth-order terms in finite-difference-molecule form to
nodes that are not interior nodes (i.e., nodes that are positioned less than three nodes in
from the boundary in both the x and y directions). Five cases are shown where the
molecide is applied at node i. These cases may be considered typical (e.g., the first
molecule shown in Figure 3.4(a) may be applied to all four comer nodes).
All Nodes Except Boundary Nodes
The domain equation,
DV^w - 2tV V + kw = q,
may be represented in finite-difference form as
[ h V - a h V + p]w.=qj (3.10)
where a = ^
R = kh! ^ D
"^ = D
40
r
• ^ ^ - ^ i [ ® H 3 H T ) ] +V
H = -T& < •V [(SHgHj)]
r
a = -2F3 { [ @ - ® - ® - © - 0 ] M2- v)
V.
<i=-*<
@
© j + (2-v)
©
Note: v represents Poisson's ratio for die plate (Vp in die text).
Figure 3.3. Finite-difference molecules representing differential equations for plate bending moment and shear perpendicular to the plane of the plate when applied at node i on the boundary.
Free Edge —>.
.« ni,3 >-%*%
Free Edge
(a) At a comer node
Free Edge .>.
> - ^ * ^
Free Edge (b) Edge node adjacent to comer
41
Free Edge
-Free Edge (c) Internal node 1 division in both
directions from a comer
^Free Edge
(d) Internal node 1 division from the edge in 1 direction and 3 or more divisions from the edge in the other direction.
Free Edge
>=S*¥
(e) Edge node 3 ot more divisions from a comer
a, = i (l-v^) b4 = (-4+2v+2v2) a2=(-3+2vfv2) ci =i(15-8v-5v^) aa=(2-2v) v = Poisson's ratio for the plate a4=(3-2v-v^) q = uniformly distributed plate load bi=(-6+2v) D = plate flexural rigidity b2=(2-v) Q = plate edge shear bj=(8-4v-3v^) h = spacing or divisions between the nodes
Note: V represents Poisson's ratio for the plate (Vp in the text).
Figure 3.4. Finite-difference molecules on or near die plate boundary. Developed by Vallabhan and Wang (1981).
42
The finite-difference molecule for all interior nodes is developed as shown in
Figure 3.5. The first molecule shown represents die molecule for V^w, which is added to
the molecide representing — — V w for diose nodes located one row and one column
inboard of die edge nodes (Cases (c) and (d) in Figure 3.4). Finally, die molecule
kh^ representing -=r- is added to all diagonal terms in the coefficient matrix. The end result of
all these steps is the constmction of the last finite-difference molecule shown at die bottom
of Figure 3.5, which is applied to all non-boundary nodes to develop the coefficient maoix.
Boundary Nodes
For the nodes on the boundary, the appropriate known boundary conditions are
utilized. Mb = 0 along die left and right vertical plate edges, and M^ = 0 along die bottom and top horizontal plate edges.
These two boundary conditions are depicted on die plate outline and in molecule form
in Figure 3.6. The known boundary conditions are utilized to eliminate die fictitious nodes
(wf), which lie outside die plate boundary.
2th 2 The technique used for die development of the molecule to be used for die — — V w
terms along die right, vertical boundary and die bottom, horizontal boundary (except for die
comer nodes) is also illustrated in Figure 3.6. TTie molecules for die left, vertical boundary
nodes and die top. horizontal boundary nodes (except for die comer nodes) are die same as
diose shown for die right, vertical boundary nodes and die bottom, horizontal boundary
nodes, respectively. TTiese molecules are dien added to die appropriate molecules along die
boimdary as depicted in Figure 3.4. /
43
- a where a = -2^
V.
+ P = ® . where P=-^*
( h V ^ - a h V +
Figure 3.5. Development of die domain equation applicable to all interior nodes. Shown applied at node i.
Mb=Oon diis edge\
M^= 0 on diis edge / <vwf (b)
- ^ 37 42
»
1 ^ Mf= 0 on diis edge ^ ^ (b)
Mb= 0 on ^ dils edge
(a)
W ^ = - D ( & - v ^ )
M.= -g<
44
= 0
^«r=-D& + v ^ )
-2t55v2w = a ^ Q K i v O ^c = -^< = 0
Note: v represents Poisson's ratio for die plate (Vp in die text).
Figure 3.6. Development of the finite-difference molecules for the shear parameter, t, to be applied to boundary nodes other than at the comers of the plate.
45
—^ ig^ to die domain equation —.
if— I
a w = a "s [^
Case(b) to die domain equation
Note: v represents Poisson's ratio for die plate (Vp in die text).
Figure 3.6. Continued
46
Since bodi Mj = 0 and Ni = 0 at die plate comer nodes, diere is no correction for die
comer nodes, that is.
% = 0 = -D
Therefore,
'"^2 ^2 ^
iw . a_w ax2 - 2 V 3yj
= 0; and M =-D a w ^ y a w ay ax2
= 0. (3.11)
7^ :i2
a w _ a w _ = 0 (3.12) ax2 ay2
at the comer nodes. To account for edge shear at a boundary node, Q,, and the distributed
load on the plate, Vallabhan and Wang (1981) found that the net equivalent discrete force,
Fj, at the plate edges is
i ~ 2D D ' (3.13)
where node i is on the edge at x=b, and
Q =-2t xw+4^--La^ ax IX ay2
for vertical boundary nodes other than comer nodes, and
(3.14)
0,= -2t X w + | ^ - - V ^ 9y 2X ax^^
for horizontal boundary nodes other than comer nodes
(3.15)
The expressions shown in Equations 3.14 and 3.15 are bodi converted to die finite-
difference foim in a manner similar to diat presented previously and dien incorporated into
die overall coefficient matrix.
47
Calculation of Equivalent Nodal FoTres
In order to develop a method for die computation of equivalent nodal forces, die load
influence area is considered. The load influence area for a uniformly distributed load in die
plate comers and at die odier boundary nodes is, respectively, only j and -i die influence
area of a uniformly distributed load at all odier nodes. The term "factor" is used to express
the adjustments to the load vector, a, so that die adjusted-load vector is a divided by factor,
where:
factor = 4 for the comer nodes
factor = 2 for the boundary nodes, and
factor = 1 for all otiier nodes.
Factor is also used when die modulus of subgrade reaction, k, from the domain
equation,
4 4 4
v V - 2 % V^w + % w = q ^ , (3.16) D D ^ D
expressed in finite-difference form is added to each nodal displacement That is,
k ^ - ^ factor (3.17)
is added to each nodal displacement. These factors are abeady incorporated in die
coefficient matrix as shown in Figure 3.4.
Development of die Coefficient Matrix
The process previously described in dus chapter leads to die constmction of die full
coefficient matrix. The correct development of diis matrix is essential in determining
accurate values for plate nodal displacements. The equation to be solved is Aw=a. where
A is die coefficient matrix, w represents die nodal displacements, and a is die qipUed
equivalent-load vector adjusted by factor, which is previously described. Once die plate
48
nodal displacements are determined, the values of intemal-nodal-bending moments, Mx and
My, are computed using the following expressions:
M, = -D a w ^^ a w
V ay ay /
My=-D :J2 :V2 ^
a w a w
(3.18)
(3.19)
In a similar manner, die values of die internal shear force at discrete points are also
determined using the expressions for Qx and Qy.
(3.20)
(3.21)
CHAPTER IV
RESULTS
Comparisons
As indicated in Chapter H, a computer program was developed in conjunction with this
research, using the computational techniques briefly outlined in the last section of that
chapter. When developing a computer program for stmctural analysis, the need always
exists to verify the program computations and final results (program output). Intuitively, it
can be proved that a plate carrying a uniformly distributed load, q, resting on a Winkler
foundation, undergoes a rigid body displacement without any bending moment and shear
force in die plate. In odier words, die computer program widi t = 0 and edge shear
forces = 0 should simulate the Winkler model,
D V w + kw = q,
and die result should be w = -^. This concept was used to verify die computer program, k
and die results were exacdy as predicted. Moreover, if die plate is subjected to a uniformly
varying load, such as q = qo + qix, die displacement solution, w, was found to be exacdy
equal to "^^^^. This represents a rigid body movement of die plate, which cleariy
demonstrates die validity of die program in representing die Winkler part of die model. At
die same time, it proves die failure of die Winkler model to represent a dish-shaped
displacement of die plate, as would be expected in a real situation. In addition, an
inexperienced researcher involved in die smdy of die elastic foundations might not consider
die importance of die effect of edge shear forces. For example, a solution to die field
equation as given in Equation 2.24,
DV' w - 2tV^w + kw =q,
49
50
widiout die edge shear forces, would also yield a rigid body displacement (w = \ where q
is a constant The computer program developed resulted in a rigid body displacement
solution for diis case as expected, which rcinforces die importance of die edge shear force
constraints in die model, as well as die performance of die program. The above smdy
illustrates clearly that die indx)duction of die term 2tV w in die field equation is not
sufficient; the edge shear forces must be included in die overall model for an accurate
reprcsentation of plate behavior.
Once these program checks were successfully concluded, the program results,
including the edge shear force effects from die soil continuum beyond the plate boundaries,
still needed to be verified. Unfortunately, tiiere is no evidence in the literature of a
computer program having been developed for a three-parameter model of plates on an
elastic foundation or even an example showing computational results. Therefore, it was
necessary to look at the work done by other researchers for beams on an elastic foundation.
Qearly, the most similar modeling exists in the computer program written by
Vallabhan for beams on elastic foundations and used in conjunction with the paper
"Parametric Smdy of Beams on Elastic Foundations" by Vallabhan and Das (1988b). This
program was mn with the input data for the plane strain case shown in Table 4.1. For this
case, die beam is an equivalent 1-foot-wide strip of a 24-foot-wide plate, supported by die
soil wherc the plane strain assumption was employed for both the plate and the soil. Also
shown in Table 4.1, and arranged for a side-by-side comparison, is die input data for die
plate program developed in conjimction with this research.
The displacements obtained as output firom bodi programs arc plotted in Figure 4.1 for
die half lengdi of die beam and die half-widdi of die plate. The plate displacements along
die plate centerline in die direction of die "x" axis are die ones plotted for die plate. If die
51
Table 4.1. Comparative input data.
Member Dimensions
Dimension in the "x" direction
Dimension in the "y" direction
Thickness
Number of nodes
Modulus of elasticity for die member
Modulus of elasticity for the soil
Poisson's ratio for the member
Poisson's ratio for the soil
Applied uniformly distributed load
Depth of the soil stratum
Eqiuvalent* Beam
24
1
0.5
7
432,000,000
144,000
0
0.2
500
20
ft.
ft.
ft.
psft
psf
psf
ft
Plate
24 ft.
48 ft.
0.5 ft
325
432,000,000 psf
144,000 psf
0
0.2
500 psf
20 ft.
*A 1-foot-wide strip of die plate on soil is used as an equivalent beam.
tpsf=pounds per square foot
Displacement (feet')
0.0-1
.010-
.020
.030-
.040-
.050
.060
52
Plate on an elastic foundation
Beam on an elastic foundation
Centerline 2.0 4.0 6.0 — I — 8.0 10.0
Half-length of beam and plate (ft)
' Hereafter, all measurement in feet will be expressed as "fL"
— I — 12.0
Figure 4.1. Comparison of displacement results firom die computer program for plates on an elastic foundation to die program BOEF for a beam on an elastic foundation by Vallabhan.
53
plate is infinitely long, it can be idealized as a plane strain problem, and die beam plane
strain formulation can be used. A plate, whose lengdi was twice its widdi, was chosen to
minimize die effects of die relatively high comer shear on die plate displacement function
along die plate centerline in die narrow direction. As illusdrated in Figure 4.1, excellent
agreement was obtained in bodi die shape of die displacement curve functions and in die
relative difference in die absolute displacement values as well. The maximum deviation of
0.0008 feet in terms of absolute values for displacements occurred at die plate center, while
at die free end, die deviation is 0.0007 feet The numbers represent a difference of 1.6%
and 2.1%, respectively, of the beam displacement at the two points. Additional program
results in the form of output data are presented in Table 4.2.
The general agreement among all respective comparative program output data is
remarkable, considering the one-dimensional nature of die beam behavior versus the two-
dimensional nature of the plate behavior. This successfid comparison verified the validity
of the plate analysis computer program, which was dien used to analyze five different plate
load cases.
Description of the Plate and Load Confiyurarions Evaluated
Five different plate-load configurations were evaluated using die computer program
developed in conjunction widi diis research. The first load configuration considered was
diat of a 500-psf uniformly distributed load acting over die entire plate surface area.
Figure 4.2 depicts die odier four special load cases considered. For each load case,
separate computer program "mns" were executed for four different depdis of die soil
stramm. In all five load cases, die four different depdis considered were 10 feet 20 feet
30 feet, and 50 feet
54
Table 4.2. Additional comparative output data.
Modulus of subgrade reaction, k
Foundation shear parameter, t
Deformation profile parameter, y
Bending moment at the center
Displacement at the center
Beam Unitst Plate
8,062
184,675
0.79219
1,199
0.05083
lbs./ft.3
lbs./ft
dimensionless
Ib.-ft.
ft.
8,141
177,285
0.9847
1,186
0.05003
tUnits shown in this table for each of the respective values are also used for the presentation of all subsequent computer program results in the remainder of this chapter.
55
20 ft
15 ft—»i
X 40 ft
20 ft.
(a) 30 kip concentrated load at the plate center.
H—*• X
(b) 30 kip concentrated load at the centerline on the right vertical edge of the plate.
20 ft
v 15ft-^<—15 ft—»»
^>x
20 ft.
40 ft. - • X
(c) 30 kip concentrated load at die upper right comer of the plate.
vrzrzrzrzrzrzrz-zrz (d) 2 klf line load along
the bottom horizontal edge of die plate.
Figure 4.2. Illustrates load placement and plotting directions for computer program output data for four special load cases. (Dashed lines indicate die position and direction chosen for plotting die computer program output data in subsequent figures.)
56 Input Data
With the exception of die previously mentioned variations in the soil stratum depths,
each of the five plate loads considered were evaluated using the same plate physical
dimensions and material properties and the same soil material properties. An additional
computer run was made for the 500-psf uniformly distributed load case to illustrate the
effect of a soil modidus of elasticity tiiat was ten times the value of the soil modulus of
elasticity of 1,000 psi that was used for the uniformly distributed load case and all other
load cases. The plate behavior using diis 10,00O-psi soil modulus was only evaluated for a
sod stramm depth of 10 feet Figure 4.3 presents an itemized listing of the common input
data used for all five load cases.
For each computer "mn," die data on die following three functions arc plotted.
1. Displacement function curve,
2. Bending moment curve,
3. Shear force curve.
In addition to die load-placement position. Figure 4.2 also illusttates by dashed lines
die position and direction on die plate chosen for plotting diese functions in all subsequent
figures. For die uniform load case, die plotting position and direction on die plate chosen
for plotting diese fimctions are die same as diose shown in Figure 4.2(a). These curves are
plotted for all four soil stratum depdis considered in die remaining figures of dus chapter.
Figure 4.2 has a plot of all functions for a soU modulus of elasticity of 10,000 psi
(1,440,000 psf) for soil stramm depdi of 10 feet
The significance of diese curves is discussed in die next section. Odier output data
consists of die final computed values of die following duee parameters:
1. Modulus of subgrade reaction, k
2. Foundation shear parameter, t
3. Soil deformation profile parameter, Y-
57
Plate widdi in die "x" direction
Plate widdi in die "y" dilution
Plate duckness
Number of divisions in the "x" direction
Number of divisions in die "y" direction
Poisson's ratio for die plate
Poisson's ratio for die soil
Plate modulus of elasticity
Soil modulus of elasticity
Depdi of the soil stramm Case A
B
C
D
30.00 ft.
40.00 ft.
0.50 ft
12.00
16.00
0.20
0.25
432,000,000.00 psf
144,000.00 psf
10.00 ft
and 20.00 ft
and 30.00 ft
and 50.00 ft
Figure 4.3. Common input data.
58
The significance of die computed values of each of diese for each load case is also
discussed in die next section.
Discussion nf Results
Uniformly Distributed Load
Figure 4.4 presents plots of die computer program results for displacement bending
moment, and shear force for the 500-psf uniformly disdibuted load case. This is die only
set of curves diat includes plots of diese diree values (displacement, bending moment and
shear) for a soil modulus of elasticity (Eg) of 10,000 psi (1,440,000 psf).
Observations from PlottfYJ Rp.«;n1f«;
A rcview of the curves reveals a number of significant observations.
Plate Displacement Curves
1. A stiffer soil (higher soil modulus of elasticity) significantiy reduces the plate
displacement
2. The maximum displacement for all results occurs at the center of the plate.
3. The plate displacement increases with an increasing depth of the soil stratum;
however, die effect is less consequential as die soil stratum depdi becomes greater.
From die curves it would appear diat diere will be a value of die soil stramm depdi
beyond which die change in displacement becomes insignificant and practically
imperceptible.
Bending Moment Curves
1. As expected from die plot of die displacement fiinction, die stiffer soil significantiy
reduces die maximum moment experienced in die plate.
59
H,= 30ft.
H, = 50ft.
(a) Displacements along the plate centerline for four diffeient soil stratum depths.
Bending Moment ab.-ft.)
1,500-
1.000-
500-
0 -
E,= 1,000 psi
H,= 30ft.
(b) Bending moments along the plate centerline for four different soil stratum depths.
H,= 20ft.
H,= 30ft.
H.= 50ft.
(c) Shear forces along the plate centerline for four diffeient soil stramm dq>ths.
Figure 4.4. Computer program results for a 30x40x0.5-ft. plate widi a 500-psf uniformly distributed load. (Results plotted in die direction of die "x" coordinate axis.)
60
2. The maximum value of the plate bending moment increases as the soil stratum
depth increases.
3. The position of the maximum plate bending moment moves toward the center of
die plate with increasing soil sd^tum depths, and as the soil stratum depdi
approaches infinity, die plate bending moment curve will approximate a parabolic
shape. This would be similar to the case of a simply supported beam experiencing
a uniformly distributed load.
Shear Force Curves
1. A higher modulus of elasticity results in reduced shear forces.
2. The variation in shear forces at die edge of die plate is significant for all values of
the soil stramm depth considered.
3. The plot of shear forces approaches a sought line as die depdi of die soil stramm
incrcases. This behavior is again similar to diat of a simply supported beam
experiencing a uniformly distributed load.
Observations from Parameter Values
The computer program generated die results shown in Table 4.3. A review of diis
table clearly reveals diat diere is no "unique" value for any of die duee parameters. Of
particular interest to die practicing engineer is die significant variation in die value of die
subgrade modulus, k, for a plate soU foundation system widi variations in die depdi of die
soil stramm when all material properties and die unifonnly distributed load is identified.
61
Table 4.3. Tabulation of the three parameter values for the unifomHy distributed load.
Es= 10.000 psi 1.000 psi
Hs= 10 ft. 10 ft 20 ft 30 ft 50 ft.
Modulus of subgrade reaction, k 173,199 17,317 8,749 5,980 3,992
Foundation shear parameter, t 919,363 92,094 173,361 240,785 331,892
SoU defonnation profile parameter, Y 0.5766 0.5647 0.9017 1.2235 1.8713
62 Concentrated Load at die Plate Center
Figure 4.5 presents plots of die computer program results for plate displacement
bending moment and shear for die load case depicted in Figure 4.2(a), 30-kip concentrated
load at the plate center.
Observations fiom Plotted Rff»;n1t«
Plate Displacement Curves
1. The shape of all plate displacement curves reveals diat the plate displacement
increases rapidly towards the center of the plate as would be expected for this type
of loading.
2. While the variation in plate displacement is significant for variations in depdis up to
30 feet, the difference becomes much less significant for greater depths.
Bending Moment Curves
1. The positive plate bending moment attains its maximum value at the plate center for
all values of soil stratum depdis considered.
2. The variation in plate bending moment with die depth of die soil stratum is only
significant at die plate center where die maximum moment is attained, and dien die
variations are only significant up to a soil stratum depth of 30 feet
3. The negative bending moment may require some "top" reinforcement in reinforced
concrete slabs, but it would not control die design.
Shear Force Curves
1. The maximum shear force, as expected, occurs in die center (at die location of die
concentrated load).
Displacement 15 (ft) -I — ^
0 -
63
0.005-
0.010-
0.015-
(a) Displacements along the plate centerline for four diffeient soil stratum depths.
M, Bending Moment
ab.-ft.)
+4,500-
+3.000-
+1,500-
0 -
-1 ,500-
V, Shear Force
Oba.) +3,000-1
+2.000
+1.000-
0 -
-1,000-
-2,000-
-3.000-1
(b) Bending moments along the plate centeiline for four di^crent soil stratum depths.
(c) Shear forces along the plate centerline for four diffeiem soil stratum depths.
Figure 4.5. Computer program results for a 30x40x0.5-ft. plate widi a 30-kip concentrated load at die center. (Results plotted in die direction of die "x" cooniinate axis.) E, = 1.000 psi for all curves.
64
2. The magnimde of die maximum shear force suggests diat "punching shear" may
well control die design for diis load case.
3. Shear forces diminish very rapidly away from die centerline.
4. Different values for die soil suamm depdis cause no significant effect on die shear
forces experienced by die plate for diis load case.
Observations from Parameter Valnps
The computer program generated die results shown in Table 4.4 When die difference
in the computed value of the subgrade modulus is significant as is the case widi the 20-
foot versus the 10-foot depdi of die soil stratum, die difference in bodi die plate
displacement and the bending moment at die plate center is likewise significant
Similarly, it can be seen diat die variation in the value of the foundation- shear
parameter is significant in a soil stramm depth of 10 feet versus 20 feet. On die other hand,
die variation in both of these parameters is less significant between a soil stramm depth of
20 feet versus 30 feet The difference is relatively insignificant between a 30-foot and a 50-
foot soil stramm depdi, even diough die value of the soil deformation profile parameter
does vary significantiy. This "tracks" well widi die earlier discussion regarding die plate
displacement and bending moment curves.
Concentrated Load at die Plate Centeriine on die Right Vertical Edge
Figures 4.6 and 4.7 present plots of die computer program results for plate displace
ment, bending moment and shear forces for die load case depicted in Figurc 4.2(b)~
30-kip concentrated load at die centerline on die right vertical edge of die plate. The values
plotted in Figure 4.6 arc die computer program results along die "x" coordinate axis at die
plate centerline (across die narrow-plate dimension). This is die location of die values
65
Table 4.4. Tabulation of the three parameter values for the concentrated load at the plate center.
H,= 10 ft. 20 ft. 30 ft 50 ft
Modulus of subgrade reaction, k 17,744 9,913
Foundation shear parameter, t 82,669 134,049
Soil deformation profile parameter, Y 1.1073 1.8401
7,903 7,121
160,684 174,905
2.5531 4.1008
Displacement i e ,„ (ft) -, —^ 1 iH ^
0-
0.005-
0.010-
0.015
66
(a) Displacements along the plate centerline for four different soQ stratum depths.
M, Bending Moment
ab.-ft.) OH — I 1 I I ' I — I 1 — — I 1 1 1 I
-250-
-500-
-750-
-1.000-
V, Shear Force
Obs.) +2,000-
+1400-
+1.000
+500-
0-
H,= 20ft
(b) Bendjng moments along die plate centeriine for four different soil stratum depths.
H.= 20,30,and50ft>
(c) Shear forces along the plate centerline for four different soil stratum depths.
Figure 4.6. Computer program results for a 30x40x0.5-ft plate widi a 30-kip load at the centerline on die right vertical edge. (Results plotted in the direction of die "x" cooniinate axis.) E, = 1,000 psi for all curves.
Displacement (ft) 20
0-
0.005-
0.010-
0.015-
M, Bending Moment
Qb.-ft.) 4,500-
3,000-
1400-
0-
V, Shear Force
Obs.) +2,000-
+1400-
+1,000-
+500-
0-
-500-
-1.000-
-1.500-
-2.000-
H,= 20.30. and 50 ft.
(a) Displacements along the plate centerline for four diflereni soil stratum depths.
(b) Bending moments along the plaie centeiiine for four different soil stratum depths.
H,-20,30, and 50 ft;
H.= 10ft.
I I n y
(c) Shear foiees along the plate centerline for four diflisrent soO stiatum depths.
Figure 4.7. Computer program results for a 30x40x0.5-ft. plate widi a 30-kip load at die centerime on die nght vertical edge. (Results plotted in die direction of die "y" coordinate axis.) E, = 1,000 psi for all curves.
68
plotted in all curves in Figures 4.4 and 4.5. The curves plotted in Figure 4.7 represent die
computer program output for plate displacements, bending momem, and shear forces along
die right, vertical edge of die plate (die long-plate dimension).
Observations from PInttpH Ti^^]]]\^
Plate Displacement Curves
F i^ re 4.6.
1. The displacement is a maximum at the point of the load and rapidly diminishes at
positions closer to the plate center.
2. The plate displacement is essentially zero at die plate center.
3. The depth of die soil stramm has littie effect on displacements with an almost
imperceptible difference after a depdi of 20 feet
Figure 4.7.
1. The displacement increases symmetrically from the top and bottom horizontal plate
edges until it reaches a maximum at die point of the applied concentrated load.
2. The depth of die soil stramm has littie effect on displacements with an almost
imperceptible difference after a depth of 20 feet
Bending Moment Curves
Figure 4.6.
1. The bending moment is zero at die plate edge, but it increases rapidly to reach a
maximum very close to die plate edge and dien diminishes rapidly at positions
closer to the plate center,
2. The only significant difference in plate bending moments widi different depdis of
die soil stramm occurs at die location of die point of maximum bending moment
69
Even at diis point, die variation in plate bending moment is not significant for a sod
stratum depdi greater dian 20 feet
Figure 4 7
1. The maximum positive plate bending moment attains its maximum value at die
plate centerline ("x" coordinate axis location) for all values of soU stramm depdis
considered.
2. The variation in plate bending moment widi die depdi of die soil sttamm is not
significant
3. The negative bending moment may require some "top" reinforcement in reinforced
concrete slabs, but it would not control die design.
Shear Force Curves
Figure 4.6.
1. The shear force is a maximum at die point of die applied load and rapidly
diminishes to zero at positions closer to die plate center.
2. The variation of shear force widi different soil stratum depths is not significant
Figure 4.7.
1. The maximum shear force occurs at the plate centerline (at the location of the
concentrated load).
2. Shear forces diminish very rapidly away from the plate centerline.
3. The variation of shear force witii different soil stratum depths is not significant
Observations from Parameter Values
The computer program generated die results shown in Table 4.5. The variation in bodi
die modulus of subgrade reaction and die foundation shear parameter is insignificant for
soil sttatum depdis greater dian 20 feet even diough die value of die soil deformation
70
Table 4.5. Tabulation of die three parameter values for the concentrated load at the plate centerline on die right vertical edge.
H,= 10 ft. 20 ft. 30 ft 50 ft
Modulus of subgrade reaction, k
Foundation shear parameter, t
Soil deformation profile parameter, y
20,005
66,161
1.8819
13,928
89,953
3.1387
13,242
93,992
4.5892
13,187
94,345
7.6323
71
profile parameter varies significantiy. This is consistent widi die earlier discussion of plate
displacements and bending moments for diis load case.
Concentrated Load at die Upper-Right Plate Comer
Figure 4.8 presents plots of die computer program results for plate displacement
bending moment, and shear for die load case depicted in Figure 4.2(c)~30-kip concentrated
load at die upper-right comer of die plate. All results plotted are diose diat occur along die
top horizontal edge of die plate (parallel to die "x" axis).
Observation from PlnttftH MPMU]^^
Plate Displacement Curves
1. The shape of die plate displacement curves reveals that the displacements are a
maximum at die upper-right comer (die location of the concentrated load), and they
rapidly decrease at positions closer to die plate centerline and actually approach
zero at the plate centerline.
2. The variation in displacement values for all soU stratum depths considered is
imperceptible.
Bending Moment Curves
1. The bending moment is zero at die plate edge, but it increases n^idly to reach a
maximum very close to die plate edge and dien diminishes rapidly at positions
closer to the plate centerline.
2. The variations in soil stramm depdis considered have an msignificant effect on
plate bending moment values.
Displacement IS in « (ft) -1 — • 1 1 1 i. i-
0 -
0.005
0.010
0.015 -
10 15 7 2 -•— 1 1 t—^x
H,= 10,20.30. and 50 ft.-
(a) Displacements along the top horizontal edge of the plate for four diHisrent soil stratum depths.
M, Bending Moment
Ob.-ft.)
-500-
-1.000-
- 1 4 0 0 -
Vx Shear Force
Obs.) + « » -
+500-
+400-
+300
+200-
+100-
H.= 10ft.-J (b) Bending moments along the top horizontal edge of the plate for four different soil stratum depths.
-100-
0 - I I 1 H i 1—•x
(c) Shear forces along the top horizontal edge of the plate for four different soil stratum d^ths.
Figure 4.8. Computer program results for a 30x40x0.5-ft plate widi a 30-kip load at die upper, right comer. (Results plotted in die direction of die "x" coordinate axis.) E, = 1.000 psi for all curves.
73
Shear Force Curves
1. As expected, die plate shear force attains its maximum value at die point of die
concentrated load, and it rapidly diminishes at positions closer to die plate
centerline.
2. The only variation in shear force widi different soil stramm depdis occurs at die
point of maximum negative shear, which is several orders of magnimde less dian
the maximum positive shear. Even in dus area of negative shear, the variation
becomes insignificant for soil stramm depdis greater tiian 20 feet
Observations from Parameter Values
The computer program generated the results shown in Table 4.6. Even though the
values of the soil-deformation-profile parameter are varying significantiy. the effect on die
other two parameters is relatively insignificant for a soil stramm depth up to 20 feet and
totally insignificant for soil stratom depths in excess of 20 feet
Uniformly Distributed Line Load Along the Bottom Plate Edge
Figure 4.9 presents plots of die computer program results for plate displacements.
bending moments, and shear forces for die load case depicted in Figurc 4.2(d)~2-kip per
linear-foot-line load along die bottom horizontal edge of die plate. All results plotted are
diose diat occur along die bottom, horizontal edge of die plate (parallel to die "x" axis).
Ohsenrarion5i fmm Plotted Results
Plate Displacement Curves
1. The maximum displacement regardless of soU stramm depdi. occurs at die center
of die plate.
74
Table 4.6. Tabulation of the three parameter values for the concentrated load at die upper-right plate comer.
Hs = 10 ft. 20 ft. 30 ft 50 ft
Modulus of subgrade reaction, k 24,920 22,115
Foundation shear parameter, t 50,668 56,267
Soil deformation profile parameter, y 2.7345 5.1158
22,076 22,078
56,358 56,354
7.6660 12.7772
75 Displacement i <
(ft.) -,
0 -
0.0005-
0.0010 -
0.0015 -
H,= 30 ft. and 50 ft.
(a) Displacements along the bonom edge of the plate for four soil stiatum dq)ths.
M, Bending Moment
ab.-ft.)
7 5 -
5 0 -
2 5 -
0 -
(b) Bending moments along the bottom edge of the plate for four soil stratum depths.
(c) Shear foices along the bottom edge of the plate for four soil stramm depths.
Figure 4.9. Computer program results for a 30x40x0.5-ft. plate widi a 2-klf load along die bottom edge of die plate. (Results plotted in die direction of die "x" cooniinate axis.) E, = 1.000 psi for all curve
76
2. The plate displacement increases widi increasing soil depdi until a sod stramm
depdi of 30 feet is reached, and dien die variation in plate displacement becomes
imperceptible.
Bending Moment Curves
1. As the depth of the soil sd^mm incrcases, the maximum bending moment increases
up to a soil stramm depdi of 20 feet After diis depdi is attained, die incrcases in
die maximum value of plate bending moment is imperceptible. However, die
moment at the plate centerline does continue to increase, but with a reduced
differential increase at ever greater soil stratum depths.
Shear Force Curves
1. Therc does exist a significant variation in the maximum shear force values at the
plate edge for all values of die soil stramm depth considered
2. The plot of the shear force data does not approach a straight line for any value of
the soil stramm depdi considered, and it is not likely that it will for any depth due
to the unsymmetrical nature of die loading relative to the "x" coordinate axis.
Observations from Parameter Values
The computer program generated die results shown in Table 4.7. Once again, dicre is
clearly no "unique" value for die modulus of subgrade reaction aldiough die variation in die
value of diis parameter does decrease widi increasing depdis of die soil stramm.
77
Table 4.7. Tabulation of die three parameter values for the unifonnly distributed line load along the bottom, horizontal edge.
H.= 10 ft. 20 ft. 30 ft 50 ft
Modulus of subgrade reaction, k
Foimdation shear parameter, t
Soil deformation profile parameter, y
19,213
70,312
1.6851
11,277
113,477
2.3690
9,082
138,108
3.0576
8,210
151.581
4.7447
CHAPTER V
SUMMARY, CONCLUSIONS, AND
RECOMMENDATIONS
Summary
Certainly a major part of this research has centered on the development of an accurate
engineering model through building upon and strengthening die work of many researchers,
whose analytical derivations were checked. The principles of solid mechanics, combined
with variational calculus, arc employed to develop the model. Derivations are presented in
detail in Chapter n. The final result culminates in the development of die duee-paramcter
model, which is presented in detail in Chapter II, and it is die model used for die
development of the computational techniques. The reasons for the selection of this model
arc presented in Chapter I, and die computational results presented in Chapter IV are
consistent widi die expected behavior for plates and even for analogous beams, where
appropriate. For diese reasons tiien, die model selected appears to be very realistic.
The mediod of finite-differences supplemented widi numerical integration and odicr
computational techniques was die primary approach chosen for use in a computer program
to perfonn die analysis of plates on elastic foundations. The finite-difference mediod
provides an excellent representation of differential equations widi a high degree of accuracy
relative to die computer memory Ttq^mcd. Its use also requires considerably less computer
operating time dian odier mediods. This aspect becomes especially important m die
utiUzation of die diree-parameter model because of die use of an iterative procedure for rhc
calculation of die foundation shear parameter, y. Essentially, diis results in the necessity
for repeating all calculations for each iteration.
78
79
Chapter II presents a detailed explanation of die utilization of die mediod of finite
differences used in die computational procedures and incorporated in die computer program
developed in conjunction widi diis research. All computer "mns" made for diis research,
as well as diose presented as results in Chapter IV. were made using a microcomputer.
The results of twenty-two computer mns for various load cases, plate sizes, soil
stramm depdis, and soil stiffnesses are presented in Chapter IV. From the results, it is
relatively easy to observe diat diere is no truly unique value for die modulus of subgrade
reaction, k, for a given type of sod. More specifically, it can be concluded that many
factors can influence die value of k. For certain load cases, the depdi of die soil stramm
has a very significant effect. In odier load cases, die depth of die soil stratum has litde or
no significant effect on the value of k.
As indicated in Chapter I, the accurate physical determination of the modulus of
subgrade reaction from soil samples is extremely difficult Therefore, if the need for this
physical determination can be eliminated, then a major goal of this research is accom
plished. The computational results presented in Chapter IV indicate the capability of
computing the value of k and die subsequent effect on plate displacements, bending
moments, and shear forces.
Conclusions
The following itemized list reprcsents brief summaries of die more important
conclusions reached as a result of this research.
1. A sound madiematical model is developed for determining displacements, bending
moments, and shear forces in plates supported by an elastic foundation. Principles
of soUd mechanics are used instead of die empirical or experimental evaluation of
die coefficient or modulus of subgrade reaction, k.
80
2. Botii die parameter, k, which represents die modulus of subgrade reaction and die
parameter, t which represents die shear deformation of die soil are uniquely
calculated utilizing die modulus of elasticity, Poisson's ratio, and die geometry and
die deformation profile of the soil or elastic foundation.
3. The arbitrary aspect of die Vlasov and Leont'ev model has been madiematically
removed through the development of a computational technique for deteraiining the
value of a diird parameter, y, which reflects die variation of die soil deformation.
This parameter is computed internally based on die soil properties and geomedy.
4. There is no unique value for the modulus of subgrade reaction, k. This was
repeatedly demonstrated by the research, which, in fact showed that the value of k
depends on the soil stifftiess, the plate stiffness, the depth of the soil stramm, the
size of the plate, and most important of all, the distribution of the load on the plate.
5. The model and the program incorporating the finite-difference method for the
evaluation of differential equations is relatively simple, and it can be easdy
programmed to operate on a microcomputer. When edge shear forces arc ignored
on a plate with a uniformly distributed load, the results are identical to diose
obtained using die classical Winkler model. In odier words, die model and the
computer program has been validated and checked, respectively.
6. The program incorporates practical and efficient computational techniques, and it is
easy to use.
7. This research clearly demonstrates die extteme importance of die role diat die edge
shear forces play in die analysis of plates supported by a soil continuum.
p .;-nmmftnHarions
There were certain assumptions made in conjunction widi diis research diat could be
eliminated or modified to strengdien diis work further. In addition, odier feamres could be
81
considered bodi to strengdien and enhance die model. Finally, odier loading types could be
considered to expand die application for this research.
The following represents a more specific itemized listing of diese categories of fumre
work diat could strengdien, enhance, and expand die application of dus research.
1. This research makes provisions for the inclusion of one value for soil modulus of
elasticity, Ej-die ability to include more than one value for this important soil
property to account for soil stratification or layering, even assuming a linear
variation between the values would strengthen the model.
2. This research considers a constant soil stratum depdi under die plate. The model
would be further strengthened if provisions were made for the inclusion of
different soil stramm depths for soil segments below a given plate.
3. The soil is considered to be a homogeneous, isotropic material in diis research, but
soils arc inherentiy anisotropic. Again, the model would be strengthened dirough
the provision for anisotropic soil behavior.
4. Techniques need to be developed to facilitate the analysis of slabs incorporating
"grade" beams and "ribbed" slabs.
5. A constant plate flexural rigidity has been assumed for this research. If provision
were made for die inclusion of varying plate flexural rigidities, die model would be
enhanced.
6. A constant spacing between "nodes" using die finite-difference mediod was
assumed for bodi die "x" and "y" dircction in dus research. The ability to select a
different node spacing in die "x" direction versus die "y" direction would enhance
the usefulness of the computational approach.
7. A finite-element program should be written incorporating die principles developed
in dus research to pennit die analysis of slabs containing holes or odier
irtcgularities.
82
8. Large-scale experiments should be conducted to verify diese computational results.
9. Dynamic loading of plates was not considered in diis research. The dircc-
parameter model could be utilized to analyze plates on elastic foundations subjcaed
to tills type of loading. This would represent a major expansion of applications for
this research.
10. Linear, elastic soil behavior has been assumed for this research. Consideration of
visco-elastic soil behavior widi its consequent non-linear behavior would
strengthen the engineering model.
11. The effect of lateral displacement of the soil was ignored as it is believed to have
minimal effects on the engineering model. Additional research should be
performed either to confirm this assumption or to develop procedures for its
inclusion in the model.
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h
85 Vesic. A. S.. 1973. "Slabs on elastic subgrade and Winkler's hypodiesis." Eighth
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