cylindrical plates on foundation

1. 1. STRUCT. MECH., 7(3), 231-246 (1979)
The Interaction between a Uniformly Loaded
Circular Plate and an Isotropic Elastic
Halfspace: A Variational Approach
A. P. S
. Selvedurai
ABSTRACT
The axially symmetric flexural interaction of a uniformly loaded circular
plate resting in smooth contact with an isotropic elastic halfspace is examined
by using an energy method. In this development the deflected shape of
the plate is represented in the form of a power series expansion which satisfies
the kinematic constraints of the plate deformation. The flexural behavior of
the plate is described by the classical Poisson-KirchhofT thin plate theory.
Using the energy formulation, analytical solutions are obtained for the
maximum deflection, the relative deflection, and the maximum flexural
moment in the circular plate. The results derived from the energy method
are compared with equivalent results derived from numerical techniques.
The solution based on the energy method yields accurate results for a wide
range of relative rigidities of practical interest.
Copyright 0 1979by Marcel Dekker, Inc. All Rights Reserved. Neither this work nor any
Dart may be revroduced or transmittedin any form or by any means, electronicor mechan-
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2. I. INTRODUCTION
The flexural behavior of finite plates resting on deformable elastic media
is of interest to several branches of engineering. Solutions to such problems
are of particular importance in the analysis and design of structural founda-
tions resting on soil and rock media [I]. The classical problem relating to the
axisymmetric flexure of a circular plate resting on an isotropic elastic half-
space was examined by Borowicka [2] by using a power series expansion
technique. This analytical procedure was subsequently modified by lshkova
1'31and Brown [4] who incorporated the efiect of a singular term in the ap-
proximation of the contact stress distribution. Other numerical methods,
such as discretization procedures, finite difference techniques, and finite
element methods, have also found useful application in the treatment of the
circular plate problem (see, e.g., Refs. 5-9). A complete account of the
axisymmetric interaction problem related to the elastic halfspace is given by
Selvadurai [I].
The present paper examines the application of an energy method for the
analysis of axisymmetric flexure of a uniformly loaded circular plate resting
in smooth contact with an isotropic elastic halfspace. The analysis assumes
that the deflected shape of the circular plate can be represented in the form
of a power series in terms of the radial coordinate r. This deflected shape is
specified to within a set of undetermined constants and satisfies the kinematic
constraints of the axisymmetric deformation. The assumption of continuous
contact between the elastic plate and the isotropic elastic halfspace ensures
that the deflected shape of the plate is compatible with the surface deflections
of the halfspace within the plate region. The energy method of analysis
requires the development of the total potential energy functional for the plate-
elastic halfspace system which consists of (1) the strain energy of the half-
space region, (2) the strain energy of the circular plate, and (3) the work
component of the external loads. The strain energy of the isotropic elastic
halfspace can be developed by computing the work component of the surface
tractions which compose the contact stresses at the plate-elastic medium
interface. The contact stresses associated with the imposed displacement field
can be determined by making use of the integral equation methods developed
for mixed boundary value problems in classical theory of elasticity (see, e.g.,
Refs. 10-12). The strain energy of the plate region is composed of the flexural
and membrane energies corresponding to the prescribed deflected shape.
The total potential energy functional thus developed is defined in terms of the
constants characterizing the deflected shape of the plate. We may, however,
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3. UNIFORMLY LOADED CIRCULAR PLATE 233
eliminate two of these constants by invoking the Kirchhoff boundary con-
ditions applicable to the free edge of the plate. The remaining constants are
uniquely determined from the linearly independent algebraic equations
generated from the minimization of the total potential energy functional.
The general procedure outlined above is used to analyze the flexural
interaction of a uniformly loaded circular foundation, with a free edge, resting
on an isotropic elastic halfspace. The deflected shape of the plate is repre-
sented by an even order polynomial in r up to the sixth-order. This particular
deflected shape is assumed to represent, approximately, the flexural behavior
of a moderately flexible circular foundation. Using the energy method,
analytical solutions are derived for the central deflection, the differential
deflection, and the central flexural moment. Also, Boussinesq's [I31 solution
for the problem of a rigid circular plate resting on an isotropic elastic half-
space occurs as a limiting case of the energy solution (i.e., the relative rigidity
R, defined by Eq. (20), tends to infinity). Similarly as R -0, the energy solu-
tions for the central deflection and the differential deflection compare fa-
vorably with the corresponding solutions developed for the problem related
to a uniformly loaded circular area. A comparison with an existing solution
[4] indicates that the accuracy of the energy approximation is greatly im-
proved at the range of relative rigidities 0 to co. Numerical results presented
in this paper illustrate the manner in which the central deflection, the dif-
ferential deflection, and the central moment of the uniformly loaded circular
plate resting in frictionless contact with an isotropic elastic halfspace are
influenced by the relative rigidity of the plate-elastic medium system.
II. ANALYSIS
We examine the axisymmetric interaction of a thin elastic plate resting in
smooth contact with an isotropic elastic halfspace. The thin plate, of thickness
h and radius a, is subjected to a uniform load of stress intensity p, over its
entire surface area (Fig. 1). Due to the axisymmetric interaction, displace-
ments are induced at the plate-elastic halfspace interface. We assume that
there is no loss of contact at the interface. As such, the surface displacements
of the contact region r < a in the z-direction also represent the deflected
shape of the circular plate, w(r). An expression for the total potential energy
functional appropriate to the elastic plate-elastic halfspace system can be
developed by making use of the function w(r).
When considering the small deflection Poisson-Kirchhoff thin plate theory,
the elastic strain energy of the circular plate subjected to the axisymmetric
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4. k
20
rl
flexible
'
. 1111111
hf
0
elastic
halfspace
plate
Fig. 1 Uniformly loaded circular plate on an isotropic elastic halfspace.
deflection w(r) is composed of only the flexural energy of the plate U,, given
by
where
and Eband vb are, respectively, the elastic modulus and Poisson's ratio for
the plate material, and S corresponds to the plate region.
The second component of the total potential energy functional corresponds
to the elastic strain energy U, of the isotropic elastic halfspace which is sub-
jected to the displacement field w(r) in the region r < a. The elastic strain
energy can be developed by computing the work component of the surface
tractions which compose the interface contact stresses. Since the interface
is assumed to be smooth, only the normal surface tractions contribute to the
strain energy. These normal tractions can be uniquely determined by making
use of the integral equation methods developed by Sneddon (10, 1I], Green
[14], and Green and Zerna [I51for the analysis of mixed boundary value pro-
blems in classical elasticity theory. We consider the problem of an isotropic
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5. UNIFORMLY LOADED CIRCULAR PLATE U S
elastic halfspace which is subjected to the axisymmetric displacement field
u, = w(r) for z = 0, 0 Q r 6 a (2)
where u, is the component of the displacement vector in the z-direction. The
surface of the halfspace is subjected to the traction boundary conditions
where a
:
, and a,,are the normal and shear stress components of the Cauchy
stress tensor utj referred to the cylindrical polar coordinate system (r, 6.2).
By employing the integral equation formulation it can be shown that the
contact stress at the interface is given by
where
2 d ' rw(r) dr
g(t) = -
~ ~ d f [ ~ , I t ~ - r ~
- (for 0 6 r 6 a)
and G, and v, are, respectively, the linear elastic shear modulus and Poisson's
ratio of the elastic medium. From the above results, the elastic strain energy
of the isotropic elastic halfspace is given by
The total energy of the uniform external load (pa)applied to the plate is given
by
Up= -po 1I,w(r)rdr dfJ (7)
By combining (I), (6). and (7) we obtain the following expression for the
total potential energy functional:
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6. 236 SELVADURAI
For the total potential energy functional to satisfy the principle of stationary
potential energy, we require
where 6U is the variation in the total potential energy. Alternatively we
note that when the prescribed deflected shape w(r) satisfies the condition
6U=0 (for all 6w(r)), it can be shown that tv(r) is the solution of the associated
elasticity problem. In order to apply the principle of total potential energy to
the circular foundation, we assume that the deflected shape w(r) can be
represented in the form
where C, are arbitrary constants and 4,(r) are arbitrary functions which
render the plate deflections kinematically admissible. Of the n + I arbitrary
constants, two can be eliminated by invoking the Kirchhof boundary con-
ditions applicable for the free edge of the circular plate, i.e.,
The total potential energy functional can now be represented in terms of
(n - 1) arbitrary constants. The principle of total potential energy requires
that U be an extremum with respect to the kinematically admissible deflec-
tion field characterized by C, (see, e.g., Refs. 16-18). Hence
The above minimization procedure yields (n - 1) linear equations for the
undetermined constants Ci (i = 0, I, 2, ...,n - I).
Ill. THE UNIFORMLY LOADED CIRCULAR PLATE
RESTING ON AN ISOTROPIC ELASTIC HALFSPACE
To apply the formal procedure developed in the preceding section to the
analysis of the circular plate problem, it is assumed that the deflected shape
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7. UNIFORMLY LOADED CIRCULAR PLATE 237
of the plate can he approximated by the power series
where CZiare arbitrary constants. In (13), the particular choice of functions
corresponding to (bi(r) give a kinematically admissible deflection and finite
flexural moments and shearing forces in the plate region 0 $ r $ o. By
invoking the free edge boundary conditions (I]), this assumed expression for
the plate deflection can be reduced to the form
where
1 - -3(1+vb), A - (I + ~ b )
- 4(2 + v,) ' '- 6(2 + v,)
The contact stress distribution corresponding to the imposed displacement
field (14) can he determined by making use of the relationships (4) and (5);
we have
16 32
d r , 0)= n(l - v,)
Gp
Ja2 - r 2k 0 + ~ 2 { - 4 - F ~ , - - 1 2
25
Similarly, by making use of w(r) as defined by (14). the total potential energy
functional (8) reduces to the form
where the constants X. (n = 1, 2, 3, 4) are given by
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8. and
The constants C, and C2can be determined from the equations which are
generated from the minimization conditions,
The deflected shape of the uniformly loaded circular foundation correspond-
ing to (13) is given by
where R is a relative rigidity parameter of the circular plate-elastic halfspace
system defined by
and K, is a similar parameter defined by Brown 141. The relative rigidity pa-
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9. UNIFORMLY LOADED CIRCULAR PLATE 239
rameter R (or K,) can be used to investigate the accuracy and the limits of
applicability of the energy approximation for the plate deflection, (19).
A. LimitingCases
We note that as R -w, the circular foundation becomes infinitely rigid;
as such, the displacement (w,) of the uniformly loaded rigid circular plate
resting in smooth contact with an isotropic elastic halfspace is given by
Result (21) is in agreement with the classical solution obtained by Boussinesq
[I31 and Harding and Sneddon [I91 for the rigid displacement of a circular
punch resting on an isotropic elastic halfspace derived by considering results
of potential theory and integral equation methods, respectively.
As R -
.0, the problem reduces to that of the axisymmetric loading of an
isotropic elastic halfspace by a uniform flexible load. The two results of
particular importance in geotechnical engineering are the maximum surface
deflection (w(0)) and the differential deflection (w(0) - w(a)) within the uni-
formly loaded area. Considering the energy method, expression (19) yields
The exact result corresponding to the central surface displacement of the
uniformly loaded area is given by (see, e.g., Ref. 20)
The energy estimate for the central deflection of the uniformly loaded circular
area overpredicts the exact solution by approximately 4.5%.
The result for the differential deflection obtained from the energy method
is
The corresponding exact solution for the uniformly loaded circular area is
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10. In this case the energy estimate for the differential deflection of the uniformly
loaded area overpredicts the exact solution by approximately 0.3 %.
B. Flexural Moments inthe Circular Plate
The flexural moments induced in the uniformly loaded circular foundation
due to its interaction with the isotropic elastic halfspace can, in principle,
be calculated by making use of the expression for the foundation deflection
(19) and the relationships
Although the energy method provides an accurate estimate of the deflec-
tions of the plate, the accuracy with which w(r) is able to predict the flexural
moments in the foundation is, in general, considerably less (see, e.g., Refs.
21 and 22). Any inaccuracies that may be present in the energy estimate for
w(r) are greatly magnified in the computation of M, and M, owing to the
presence of derivatives of w(r) up to the second order. It can he shown that
the flexural moments in the circular foundation as determined from (19)
and (24) are somewhat lower than those predicted by Brown 14) using the
numerical method involving a power series technique. A more accurate
estimate of the flexural moments in the circular foundation can be obtained
by considering the flexural response of the circular plate under the combined
action of the uniform external load and the contact stresses a,,(r, 0
)
. The
maximum flexural moment [M,(O) = M,(O)] at the center of the circular
foundation can be computed by using the solutions developed for the flexure
of a circular plate simply supported along irs boundary.
(a) The maximum flexural moment at the center of an edge-supported
plate due to the external load p, is given by
(b) The maximum flexural moment due to the contact stress o,,(r, 0)
acting on an edge-supported plate is given by
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11. UNIFORMLY LOADED CIRCULAR PLATE 241
where a
,
.
(
[
, 0) is defined by (15). The expression (25b) is, in part,
composed of integrals which take the form
For the evaluation of (26), the following recurrence relationships have been
developed :
and
Alternatively, I, can be expressed in the form of the double series
Since the flexural moments derived from (25a) and (25b) are referred to the
same total load, their combined result will eliminate the free edge support
reaction. Using these results we obtain the following expression for the
central flexural moment (M,) in the uniformly loaded plate resting on an iso-
tropic elastic halfspace:
where
and
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12. IV. CONCLUSIONS
The classical problem relating to the axisymmetric flexure of a uniformly
loaded circular plate resting in smooth contact with an isotropic elastic
halfspace is examined by using an energy method. The energy method pro-
posed here utilizes an even order power series representation, up to the sixth
order in r, to approximate the deflected shape of the flexible plate. This power
series is specified to within a set of arbitrary constants. These constants are
evaluated by making use of the free edge KirchhoK boundary conditions and
from the minimization of the total potential energy functional. Using such a
technique, analytical results can be developed for the deflection of the circular
----- Brown(l969)
-Energy Method
-
I I I I I
Fig. 2 The variation of central displacernenl or the uniformly loaded circular plate:
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13. UNIFORMLY LOADED CIRCULAR PLATE 243
plate and the contact stresses at the plate-elastic medium interface. The latter
result is employed in the calculation of the maximum flexural moment in the
circular plate. The energy result for the plate deflection converges to the
exact result when the relative rigidity of the plate-elastic medium system
(K,) tends to infinity. Also, when the relative rigidity K, tends to zero, the
energy result yields total and differential surface displacements for the loaded
area of the halfspace which compare very accurately with existing exact
solutions.
In Figs. 2 to 4, the results derived from the energy method for the central
deflection, the differential deflection, and the central flexural moment in the
circular plate are compared with equivalent results derived from the power
series method [4]. The results of the energy method compare very accurately
with the exact solutions for relative rigidities, K,, in the range 0 to oo.[The
practical applicability of this range of relativo rigidities can be illustrated in
the following manner. Consider a uniformly loaded circular reinforced con-
----- Brown(1969)
Enorgy Method
0 1.0 2.0 3.0
Fig. 3 . The variation of the differential deflectionof the uniformly loaded circular plate:
op.(l -
W(0) - w(0) =
E.
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14. 214 SELVADURAI
------ Brown (19691
Energy Method
0.04 I I I I I
Fig. 4 The variation of the centralflexural moment in the uniformly loaded circular plate:
M,(O)= MdO) = P ~ O ' ~ ~
Crete raft (diameter 10 m; thickness I m; Eb = 1.5 x IO7 kN/m2; vb = 113)
which is founded on a saturated cohesive soil medium (v, = 112). In the case
of a soft clayey soil with E, = 4 x lo3 kN/m2, we have K, = 22.5. Similarly
for a stiff clayey soil with E, = 8 x lo4kN/m2, we have K, = 1.125.1
The energy method described in this paper can be effectivelyemployed to
provide approximate solutions to interaction problems where exact or
numerical solution is either mathematically complex or intractable. Interac-
tion problems which involve external loads of nonuniform variation or of
limited extent, edge boundary conditions of complete or partial fixity, etc.,
can be examined by using the general approach outlined in this paper. The
accuracy of the energy method depends on the choice of functions that are
used to represent the deflected shape of the plate. For example, the assumed
form of the deflected shape (13) is clearly inappropriate for situations involv-
ing localized loading of highly flexible (K, -t 0) circular plates. In this in-
stance, higher order and logarithmic terms in r have to be incorporated in the
assumed deflection functions to accurately model the flexural moments and
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15. UNIFORMLY LOADED CIRCULAR PLATE 245
deflections in the plate. Alternatively, the flexural response of a highly
flexible locally loaded circular plate can be examined by analyzing the ap-
propriate problem related t o the infinite plate resting o n a n isotropic elastic
halfspace (see, e.g., Refs. 1, 23, and 24).
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. .
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..
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16. 246 SELVADURAl
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Received August 29, 1978
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