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Analysis of moderately thick restrained circular plates resting on an isotropic elastic half space
Analysis of moderately thick restrained circular plates resting on an isotropic elastic half space M. Musharraf Zaman, M. Omar Faruque, and A. Mahmood School of Civil Engineering and Environmental Science, University of Oklahoma, Norman, Oklahoma, U S A Circular raft foundations for cylindrical structures such as nuclear reactors, tower silos, and storage tanks are often analyzed as circular plates resting on an elastic half space. Two important features that are commonly ignored in the analysis are the restraint at the plate boundary due to the rigidity of the superstructure and the effect of transverse shear deformation on the response of the plate. A mixed-variational formulation is developed herein to analyze a moderately thick restrained circular plate resting on an isotropic elastic half space. In this formulation the transverse plate deflection, the bending moments, and the shear force are treated as independent functions and are approximated by power series in terms of the radial coordinate and a set of undetermined parameters. These parameters are evaluated by utilizing the stationary property of the mixed-variational functional. Numerical results arepresentedfor various plate rigidities and edge restraints. The effect of transverse shear deformation on the plate deflection and bending moments is investigated. Keywords:
restrained circular plate, mixed-variational principle, shear deformation, plate-foundation
interaction Introduction Analysis of circular plates with restrained boundaries is important in the design of foundations for cylindrical structures such as nuclear reactors, tower silos, and storage tanks. In conventional techniques, generally circular plates (or foundations) are idealized as an isolated system subjected to loads from the superstructure. Such an idealization does not account for the edge restraint provided by the superstructure, and it essentially ignores the interaction due to the bending moments induced in the foundation by the wall (Figure I(a)). The effects of such interactions can be quite significant depending upon the relative stiffness and the connection of the foundation-superstructure system. The classical problem involving the axisymmetric interaction between a circular plate and an isotropic elastic half space was investigated by Borowicka’ using a power series expansion technique. This technique was subsequently extended by Brown2 and Ishkova,3 who incorporated the effects of singularity in the approximation of the contact stress distribution at the interface. Many other analytical and numerical methods have been introduced to analyze this interaction
Address reprint requests to Dr. Zaman at the School of Civil Engineering and Environmental Science, University of Oklahoma, Norman, OK 73019, USA. Received 22 August 1989; accepted 25 January 1990
352 Appl. Math. Modelling, 1990, Vol. 14, July
problem (see, for example, Refs. 4-10). An extensive account of various approaches to the solution of circular plate-half space interaction problems is presented in the review article by Popov.” More recently, Faruque and Zaman I2 have developed a mixed-variational formulation for analysis of the axisymmetric interaction of a thin circular plate resting in smooth contact with an isotropic elastic half space. Unlike unrestrained circular plates, the study of restrained circular plates has been limited. Booker and SmallI examined the problem of a cylindrical tank resting on an elastic half space. They developed an analytical procedure based on the flexibility method to analyze problems in which the tank wall and the base plate have the same radii. Selvadurai14 used the principle of minimum potential energy to analyze circular raft foundations with a restrained boundary. In this method the plate deflections were predicted with good accuracy, but no results were reported for the distribution of flexural moments, although knowledge of flexural moments is essential for the design of such restrained foundation plates. Issa and Zamani5 also employed the (potential) energy method to analyze the cylindrical tank-foundation-half space interaction problem by considering that the tank wall is rigidly connected to the foundation and by adding the strain energy of the tank wall to the strain energy of the circular plate-elastic half space system. For many circular plate foundations the ratio of the thickness to the radius of the plate is large,enough to have significant effect of shear deformation on the flexural response of the plate. Selvaduraii6 has analyzed
0 1990 Butterworth-Heinemann
Circular plates resting on an isotropic elastic half space: M. M. Zaman et al.
Cylindrical Tank Vail
Isotropic 53stCc
AZ&
of
Figure l(a).
Moderately thick circular plate-cylindrical tank wall resting on an isotropic elastic half space
elastic half space. The term “moderately thick” plate is used here to refer to a plate in which the effect of transverse shear deformation is important. In this approach the plate deflection, the flexural moments, and the transverse shear force are approximated independently by power series of different orders. The undetermined constants in the power series approximations are evaluated by invoking the stationary property of the derived variational functional at equilibrium. The proposed method is found to be more efficient and accurate than the potential energy-based formulations,‘,is particularly in terms of flexural moment evaluation. In this method the plate deflection as well as the flexural moments can be predicted with the same degree of accuracy because they are treated as independent variables and are approximated with different functions.
Proposed mixed-variational formulation
Isotropic Eiastic
Figure l(b). Idealization of the circular plate-cylindrical tank wall system
the interaction of an unrestrained circular plate resting on an elastic half space by including the effect of transverse shear deformation of the plate. An analytical method based on a mixed-variational principle” is presented in this paper to analyze the axisymmetric flexural interaction between a moderately thick restrained circular plate and an isotropic
Consider a circular plate of thickness h and radius c1 resting in smooth and continuous contact with an isotropic elastic half space as shown in Figure I(a). The plate is subjected to a uniformly distributed load of intensity PO acting over a circular area of radius ha and is restrained at the boundary owing to the superstructure. For convenience the interaction between the plate and the superstructure is idealized by providing rotational spring at the plate edge as shown in Figure l(b). Note that the self-weight of the superstructure can be represented by an equivalent ring load and its effect in the analysis can be incorporated by modifying the potential energy due to external loads. However, the effect of the self-weight may not be significant in comparison to other transverse loads. The horizontal shear force at the boundary can be accounted for by considering the stretching of the plate in the radial direction. For foundations of practical importance, radial stretching is usually negligible. As such, the effects of the self-weight of the superstructure and the horizontal shear force at the plate boundary are not taken into account. The total potential energy of the idealized superstructure-plate-half space system can be expressed in the form
(1) where
sb=;{(*.r+$)2-
2(1 - zJ)$$
} + F ($4 + +J*
c+,,(r) is the normal contact stress at the plate-elastic half space interface, w(r) is the transverse plate de-
(2)
flection, K is the rotational spring constant related to the geometry and elastic properties of the superstruc-
Appl.
Math. Modelling, 1990, Vol. 14, July 353
Circular plates resting on an isotropic elastic half space: M. M. Zaman et al. ture, and @l(r) is the rotation of a plane section originally normal to the middle surface of the plate. In equation (2), D denotes the flexural ridigity of the plate [D = Eh3/{12(1 - v2)}], E, G, and v are the elastic constants, and r and 8 represent the polar coordinates. For an elastic system in equilibrium the principle of minimum potential energy can be written as
s7Tp = 0
(3)
2lr ho
-J-I
0 0
where S7rP represents the first variation of the potential energy with respect to the independent functions w(r) and $(r). Following the steps outlined by Washizu,17 the variational principle in equation (3) can be modified to obtain a mixed-variational principle of the form 6%-R = 0
(4)
where
277 0
P,w(r)r dr d% + i
II 0 0
v;_(r)w(r)r dr d%
2Tr
(5)
where the subscript ,r represents the derivative with respect to r, Qr is the transverse shear force in the plate, and M, and MH are the radial and the circumferential moments, respectively. In equation (4) the independent functions subjected to variation are w(r), M,.(r), Me(r), and QJr) and hence can be called a mixedvariational principle. Since w(r), M,.(r), M,,(r), and Ql(r) are treated independently, solutions based on such a principle can provide the same order of accuracy for these quantities by representing them in terms of power series of same order. Note that equation (5) includes shear force QJr), which allows one to study the effect of transverse shear on the flexural response of the plate. In the formulation based on the principle of minimum potential energy, only the displacement w(r) is treated as an independent function. Although such a formulation can adequately predict the plate deflection, it cannot predict bending moments [M,.(r), Me(r)] and shear force [Qr(r)] with the desired accuracy. From this consideration the proposed formulation is superior to the potential energy-based formulations. Assuming that the superstructure can be considered as a cylindrical thin shell of semi-infinite length and thickness t, it can be shown that the stiffness factor K in equation (1) takes the following form:ih K = Eo6(,
T v6,{3(’ - 4(;)‘}“’
(6)
where E. and u. represent the elastic constants of the shell material. 354 Appl. Math. Modelling, 1990, Vol. 14, July
General application To employ the mixed-variational principle for the analysis of the interaction problem related to the moderately thick circular plate with restrained boundary, the plate deflection, the flexural moments, and the shear force are approximated by the following power series: 2i
w(r) =,a 5Aj t i=o 0
(7) Zi
M,.(r) = Poa2 5 Bj X j
=
O
0 Y
MA(r) = Poa2 2 Cj X j
=
O
Q,.(r) = Pea 5 Dk
(9) 0
0
6 2k-’
k=l
(10)
where A;, Bj, C;, and Dk represent the undetermined parameters. Note that the function 4(r) can be expressed as $(r) = -w(r),, + c $$
(11)
and therefore does not require an additional representation. The constant c in equation (11) is related to the nonuniformity of the shearing strain over the crosssection of the plate. It can vary between 1.0 and 1.5,
Circular plates resting on an isotropic elastic half space: M. M. Zaman et al.
depending upon the degree of nonuniformity of shearing strain, and is usually taken to be 615.” In equations (7)-( 10) there are a total of (n + 2m + s + 3) unknown parameters, of which four can be eliminated by invoking the boundary conditions that are applicable to the edge of the plate. These boundary conditions are
KWlr=. = K[W)lr=,
(12)
MW” = M&)lr=”
(13)
Qr(r)Ir=
-D
= 0
I =
(14)
Note that in equation (12) the radial moment M,.(v) at the edge of the plate is not zero because the edge of the plate is restrained owing to the presence of the superstructure. The magnitude of the moments induced by the superstructure at the plate boundary depends upon the rigidity of the joint. Owing to elimination of four constants from equations (7)-(lo), the total number of unknowns in the system reduces to (n + 2m + s - 1). These undetermined parameters are evaluated from a set of linear algebraic equations that are obtained by using the stationary conditions of the functional rR, which can be written as anR -=
ifA;
0,
2I = 0,
(15)
.[A, + A,(;)= + Az{(;)~ - ($)($}] M,(r) = f’~~~[Bo{ 1 - (b)“} + B,{ (i)* - (;)‘}I
Analysis of the restrained circular plate The mixed-variational principle outlined in the preceding section is used to analyze the interaction problem of restrained circular plates resting in smooth contact with an isotropic elastic half space. The following approximation functions are used:
(16) M,(r) = P,n=[B, + B,(i)’ + B2(;)4]
(17)
Me(r) = P0a2[ C0 + C,(b)* + C=(i)“]
(18)
Q,(r) = p”a[Dl(b) + D$)‘]
(19)
where Au, . . . , Ax, Bo, . . . , B2, Co, . . . , C2, D,, and D2 are the undetermined parameters. It should be noted that the approximation function for w(r) includes terms up to an order of six, whereas the approximation functions for M,(r) and Me(r) contain terms up to the fourth order only. This particular choice is motivated by the fact that M,(r) and MB(r) can be expressed in terms of derivatives of w(r) up to the second order. However, the order of approximation functions for M,(r) and M&r) can be selected independent of the order of w(r) for general application of the mixed-variational principle. By applying the boundary conditions given by equations (12)-(14) for the restrained circular plate, four unknown coefficients, Ax, B2, Co, and D2 are eliminated from equations (16)-( 19). The resulting moditied approximation functions are given by
w(r) =
(20) - +A, + ($42](3’
(21)
M&) = p,a*[B, + C,(i)* + Cz($“l
(22)
QJrI = $DI{(~) - (;)‘}I
(23)
+(r) =
--~[AI(:) + W{ (i)3 - (i)(k)5]] + (f’of$)[Dl((i) - (:)‘>1
rCr(rb, = - (i) [ AI + A2{6(:)’ - (y) (:)“)I + (PO&) [D,{l - 3(i)*}]
(24) (25)
The distribution of normal contact stress, u,,(r), at the plate-elastic half space interface due to displacement field W(T) can be expressed in an integral form as’* fl&) =
1
for
OSrSa
(26)
Appl. Math. Modelling, 1990, Vol. 14, July 355
Circular plates resting on an isotropic elastic half space: M. M. Zaman et al. where ES and u, are the elastic constants of the half space and (27) By substituting equation (20) into equations (26) and (27) the following expression for u,,(r) is obtained:
a,,(r) =
Esa
[ 7T(l - VZ)_
]{A,+A,[-2+4(;)2]
(28)
+Az[ -(E) - (Z)(5)‘+ (E)(L)‘- (%)($I) By using the expressions in equations (20)-(25) and (28) the functional nR in equation (5) can be written in terms of the unknown parameters AO, A,, A*, &, B,, C,, C2, and D, as
[cm = {Yi
??-R = X, + X2 + X, + x4 + x, + x, + x, + x, + x, + X,”
appendix. The first variation of rR with respect to the unknown parameters AO, A,, A2, &, B1, C,, C2, and D, will result in a set of linear simultaneous equations of the form
(29)
The expressions for X,, X2, . . . , Xl,, are given in the
(30) where [C] is the symmetric coefficient matrix of size (8 x 8) given in the appendix and the vectors {X} and {Y} are given by
(31) (32)
J (I XX) If we know the parameters AO, A,, AZ, &,, B,, C,, C2, and D,, then the plate deflection, w(r), the flexural moments, MJr) and M&r), and the contact stress, us(r), can be obtained from equations (20), (21), (22), and (28), respectively.
If we represent t/h and h/a by 0 and p, respectively, and assume that the cylindrical tank wall and the circular plate are of the same material (that is, E0 = E and v. = v), equation (34) reduces to the form l/2
Numerical results To facilitate the presentation of numerical results for the moderately thick restrained circular plate-elastic half space interaction, attention is restricted to the particular case in which the uniformly distributed load is acting over the entire plate surface (that is, A = 1). A relative rigidity parameter K, is defined in terms of the geometric and material properties of the plate-elastic half space system as follows:16
K=
(;)s($)(i>’
(33)
To demonstrate the influence of the edge restrainment on the flexural behavior of the plate, a nondimensional factor K, is defined as K,=K;
K, = 2[3(1 - ~*)]l’~0* $ 0
(35)
Since K, incorporates the geometric characteristics of both the plate (h and a) and the tank-wall (t), the effect of edge restrainment can be conveniently investigated by changing the values of 0 and p. For each selected value of 8 and p, K, will provide a numerical measure of edge restrainment. In the numerical results presented here the effects of both 8 and /1 are investigated. For convenience of isolating the effects of material properties and geometry the relative rigidity parameter is expressed in the form K, = K,p3
(36)
where K, is a modular ratio defined by (37)
= 2=($)(;)‘(;)[3(1 - v;)(;)*]“’ (34) 356 Appl. Math. Modelling, 1990, Vol. 14, July
For convenience of presentation of numerical results the plate deflection, the flexural moments, and
Circular plates resting on an isotropic elastic half space: M. M. Zaman
the contact stress are nondimensionalized w(r) =
41-4) 7rP(jA2 E.7 [
I
w(r)
et al.
as (38)
c+Z,W= ~oA2~z,,W
(39)
M,(r) = PoA2a2M,(r)
(40)
Me(r) = POh2a2MO(r)
(41)
where all quantities with an overbar are nondimensional variables. The numerical results presented in this section are valid for a Poisson ratio of 0.3 for the
1
Figure 4.
Variation of the normalized plate deflection versus radial distance for various modular ratios (for A = 1.0, p = 0.1, 0 = 2.0)
Figure 2.
Variation of the normalized plate deflection versus radial distance for various modular ratios (for A = 1.0, p = 0.1, 0 = 0) (Note: W(r) = [trp0h2a(l - v:)lE,IW,(r), Km = (~/6)[(1 - v:Ml - v’)] (E/E,))
Figure 5. (A = 1.0,
E I3
Plate central
Flexible plate:
Variation of the normalized plate deflection versus radial distance for various modular ratios (for A = 1.0, p = 0.1, 8 = 1.0)
versus h/a
for various
K,,,
plate material. For the limiting cases of completely flexible and rigid plates the central plate deflections are compared with the corresponding exact solutions’ as follows: Rigid plate:
Figure 3.
deflection
t/a = 0.5) (W(O) = [~~0A2a(l- v%E,IW(O))
W(0){exact:present method} = w(0){0.50:0.50} W(O){exact:present method} = W(O){O.6366:0.6398}
It is evident that the solutions for the limiting cases obtained from the proposed variational formulation compare accurately with the known exact results. Figure 2 shows the variation of plate deflection for 13= 0 (that is, the unrestrained case) and for various relative rigidities, K,.
Appl.
Math.
Modelling,
1990,
Vol.
14, July
357
Circular plates resting on an isotropic elastic half space: M. M. Zaman et al. Figures 3 and 4 show the variation of normalized plate deflection with the nondimensional -radial distance (r/a) for 8 = 1.0 and 2.0. It is observed that for plates of intermediate rigidities (K, = 100) the central plate deflection decreases with increasing 0, while the edge deflection increases. As a result, the differential deflection decreases as 8 increases. The deflection of a plate of high rigidity remains unaffected by the edge restrainment, as expected. The effect of p (the ratio of thickness to radius of the plate) on the central plate deflection, W(O), for different values of K,,, is shown in Figure 5. It is observed that when the relative rigidity (K,) is significantly large, the central deflection remains virtually unaffected by the change in the param-
Figure 8. Variation of the normal contact stress versus radial distance for various modular ratios (for A = 1.0, /* = 0.2, 0 =
Figure 6. Variation of the normal contact stress versus radial distance for various modular ratios (for A = 1.0, p = 0.1, 0 = 0) ( N o t e : uzz(r) = p0A2~zz(r), K,,, = (~/6)[(1 - &/(I - v*)](E/EJ)
Figure 9. Variation of the normalized plate moment versus radial distance for various modular ratios (for A = 1.0, r~ = 0 . 1 , 0 = 0) (Note: M,(r) = b&,(r) = poA2a2~,(r), K,,, = (46][(1 - v:)/(l - v2]]fE/E,)]
Figure 7. Variation of the normal contact stress versus radial distance for various modular ratios (for A = 1.0, /.L = 0.1, 8 = 2.0)
358 Appl. Math. Modelling, 1990, Vol. 14, July
eter p. For plates of moderate rigidity (e.g., 100 < K, < lo), W(0) is significantly affected by the change in I_L. For example, W(0) decreases by approximately 20% owing to the change in /J from 0.1 to 0.5 for K, = 100. The distribution of normalized contact stress, CZZ(r), for various K, and f3 is presented in Figures 6 and 7. It is evident that for plates of moderate rigidity the magnitude of the contact stress at the center of the plate is significantly influenced by the change in 0. Also, the shape of 7i,,(r) curves is substantially affected by the extent of edge restraint. Figure 8 shows the distribution of a,,(r) for various values of K, and for
Circular plates resting on an isotropic elastic half space: M. M. Zaman
Figure 10. Variation of the normalized plate moment versus radial distance for various modular ratios ( for A = 1 .O, p = 0.1, I9 = 1.0)
003
Figure 12. Plate central moment versus h/a for various (A = 1.0, t/a = 0.5) (M,(O) = M,(O) = poA2aZM,(0))
p = 0.2 and 0 = 2.0. Comparison of Figures 7 and 8 reveals that for K, = 100 the value of cZZ(0) decreases with increasing I_L.For Km = 100 this decrease in LYZZ(0) is on the order of 32% for an increase in p from 0.1 to 0.2. Distribution of flexural moments is important from the viewpoint of design of circular plate foundations. Figures 9, 10, and II show the variation of the normalized radial moment, a,(r), with the nondimensional radial distance, da, for various values of K, and 8. It is evident that the edge restraint (0) significantly influences the values as well as the distribution of MY(r). A point of contraflexure is noticed for the case of restrained plates (Figure 10). The normalized central
K,,,
moment, M,(O), significantly reduces, owing to the increase in the edge restrsint. As is evident, changing 8 from 0 to 2.0 decreases M,(O) from approximately 0.079 to 0.022 for K,,, = IOh. With the decrease in K,,, this effect decreases, as expected. For unrestrained plates ((3 = 0), the edge moment &‘,.(a) becomes zero (Figure 9). Edge restraint (0) is seen to induce significant edge moments M,.(a), as is evident from Figures IO and II. Such information is important for design of circular foundations. Figure 12 shows the effect of p on the central moment, M,.(O), for various values of K,,,. For the limiting cases of flexible and rigid plates the effect of p on M,.(O) is found to be insignificant, whereas for plates of moderate relative rigidity this effect is significant. Concluding
Figure 11. Variation of the normalized plate moment versus radial distance for various modular ratios (for A = 1 .O, p = 0.1, I9 = 2.0)
et al.
remarks
A formulation based on a mixed-variational principle is developed to analyze the axisymmetric flexural interaction between a moderately thick circular plate with a restrained edge and an isotropic elastic half space. In this formulation the plate deflection, the flexural moments, and the shear force are treated as independent functions and are approximated by power series in terms of the radial coordinate and a set of unknown parameters. The effect of the restrained boundary is included in the mixed-variational functional, rrR, by using an appropriate term that is applicable for the semi-infinite cylindrical shell connected monolithically to the circular plate at the edge. The unknown parameters associated with the approximation functions are evaluated by utilizing the stationary condition of rrR. Numerical results are presented to demonstrate the effects of edge restrainment and transverse shear deformation on the plate deflection, the contact stress distribution, and the flexural moments. It is found that both edge restraint and transverse shear deformation can significantly influence these quantities.
Appl.
Math. Modelling,
1990, Vol. 14, July
359
Circular plates resting on an isotropic elastic half space: M. M. Zaman et al.
The proposed mixed formulation can predict the plate deflection and the flexural moments with very good accuracy even when lower-order polynomials are used as approximation functions. In this paper, sixth-order and fourth-order polynomials are used to approximate the plate deflection and the flexural moments, respectively. The mixed-variational approach presented here can be used effectively to analyze flexural behavior of restrained and unrestrained circular plates resting on an elastic half space.
6 7 8 9
Palmov, V. A. The contact problem of a plate on an elastic foundation. J. Appl. Math. Mech. 1960, 24, 609-618 Selvadurai, A. P. S. The interaction between a uniformly loaded circular plate and an isotropic elastic half-space: A variational approach. .I. Struct. Mech. 1979. 7. 231-246 Tseitlin, A. I. On the flexure of a circular plate lying on a linearly deformable foundation. Izv. Akad. Nauk SSSR Mekh. Tverd. Tela 1969, 1, 99-112 (in Russian) Zaman, M. M., Kukreti, A. R. and lssa, A. Analysis ofcircular plate-elastic half space interaction using an energy approach.
Appl. Math. Modelling 1988, 12, 285-292
IO I1 12
References 13
Borowicka, H. Influence of rigidity of a circular foundation slab on the distribution of pressure over the contact surface.
Proceedings of the First Internutional Conference on Soil Mechanics Found Engineering. Vol. 2. DD. 144-149, 1936
14
Brown, P. T. Numerical analyses of uniformly loaded circular rafts on deep elastic foundations. Geotechnique 1969,19,301-306 Ishkova, A. G. Exact solution of the problem of a circular plate in bending on the elastic half space under the action of a uniformly distributed anti-symmetrical load. Dokl. Akud. Nauk
15
Zemochkin, B. N. Analysis of circular plates on elastic foundation. Moskov. Izd. Voenno. Inzh. Akad. 1939 (in Russian) Popov, G. Ya. Plates on a linearly elastic foundation (a survey). Soviet Appl. Mech. 1971, 8, 3-17 Faruque, M. 0. and Zaman, M. M. A mixed-variational approach for the analysis of circular plate-elastic half space interaction. Comput. Methods Appl. Mech. Engrg., 1990, in press Booker, J. R. and Small, J. C. The analysis of liquid storage tanks on deep elastic foundations. Internat. J. Numer. Anal. Methods Geomech. 1983, 7, 187-207 Selvadurai, A. P. S. Circular raft foundation with a restrained boundary. Geotech. Engrg. 1984, 15, 171-192 Issa, A. and Zaman, M. M. A cylindrical tank-foundation-half space interaction using an energy approach. Comput. Methods Appl. Mech. Engrg. 1986, 56, 47-60
16
USSR 1947, 4, 129-132
Cheung, Y. K. and Zienkiewicz, 0. C. Plates and tanks on elastic foundations-An application of finite element method. Internat. J. Solids Structures 1965, 1, 451-461 Gladwell, G. M. L. Unbended contact between acircular plate and an elastic foundation. The Mechanics of the Contact Between Deformable Bodies, ed. A. D. de Pater and J. J. Kalker. Delft University Press, The Netherlands, 1975, pp. 99-109
17 18
Sklvadurai, A. P. S. A contact problem for a Reissner plate and an isotrooic elastic half soace. J. M&an. Theor. Appl. 1984, 3, 181-i96 Washizu, K. Variational Methods in Elasticity and Plasticity, 2nd ed. Pergamon, Oxford, England, 1982 Harding, J. W. and Sneddon, 1. N. Elastic stresses produced by the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch. Proc. Cambridge Philos. Sot. 1945,41, 16-26
Appendix The expressions for XI, X2, . . . , Xl0 in equation (33) are as follows:
12rP;&{ ($), + (&))BT + (&)B”B, + (;)B&, + ($B”G + ($c: + (;)Gc2 + ($)ci - (+B; - &BOB, - ($VB”C, - ($B,C, - ($B,C2 - ($B,C2} x2 = 41 + 4Pk${ (&)D? + ($)B& - (;)B,D, + (+,a + (;)c*D,} X3 = -rPoa3 { (;)Mo + ($W, + A,C, + ($%G + (;)&‘o + (f)&B, + AZCI + (+bG} x, = -
X4 = -iiP,ai{12Aa + (i)A4A, + [ (:)A6 - (i)A8]A2} X, = [+$I {A:, + (;)AoA, + ($)AoAz + (;)A: + (g)A,A2 + (=)A:} X6 =
&rKa((t)A: + (:),,A2 + ($)A$}
X, = - [ ($K2g]{A: + (;)A,A2 + (;)A;} x8
=
[24&K&]{ (+Wo
+
(&%& +
360 Appl. Math. Modelling, 1990, Vol. 14, July
($Wo
+
(+M4}
Circular plates resting on an isotropic elastic half space: M. M. Zaman et al.
X, = [48rruP,K$1{ ($WO + (;)A&, + ($%C, + (;)A,& + (;)&C, + (+z} X,0 = [‘Ml + u)cP,K~]{ ($W, + (;)A#, j The matrix [C] in equation (34) is an (8
x
8) symmetric square matrix. The elements of [C] are given below:
C(l,l) = 2x,
C(4,7) = [ -4 + (;)v]xz
4 C(1,2) = 5 X1 0
C(4,8) = ;
272 C(l,3) = 15 XI ( ) C(1,4) = C(1,5) = C(1,6) = C(1,7) = C(1,8) = 0
x 3
0
C(5,5) = - ; x2 0 C(5,6) = vx2
C(5,8) = - t
x3
0
C(2,4) = -(y) + (+ + (;)x, C(2,5) = -(i) + (&)X6 C(2,6) = - 1 + i
x7
0
C(2,7)
=
-(;)
C(6,6) =
-4X2
C(6,7) = -3X2 1 C(6,8) = 5 0
x3
C(7,7) = - F x2 0 +
($X7
C(7,8) = ;
x3
0
C(2.8) = $ xs 0
C(8,8) = ;
x3
0
where C(3,4) = - (7) + (-$)X6 + ($)A? C(3,5) = -($ + (A)“,
XI =
x2 = POEh3
x3 =
C(3,6)= -l+ ; x, 0
ES ?I-P”(1 - vf) i
(1 + v)cP,,; K Pea’
x4=4C(3,7) = -({) + (&)“’ C(3,8) = - xg 0
24 K2 x5 = -0 7 PoaEh3 x6 = 24K;
C(4,4) = ,! (y) + 16v]x, C(4,5) = [ -@ + z,],, C(4,6) = [-6 + 3v]x2
x,= -48vKa Eh3 K xg = 4(1 + V)“Z
restrained circular plate, mixed-variational principle, shear deformation, plate-foundation
interaction Introduction Analysis of circular plates with restrained boundaries is important in the design of foundations for cylindrical structures such as nuclear reactors, tower silos, and storage tanks. In conventional techniques, generally circular plates (or foundations) are idealized as an isolated system subjected to loads from the superstructure. Such an idealization does not account for the edge restraint provided by the superstructure, and it essentially ignores the interaction due to the bending moments induced in the foundation by the wall (Figure I(a)). The effects of such interactions can be quite significant depending upon the relative stiffness and the connection of the foundation-superstructure system. The classical problem involving the axisymmetric interaction between a circular plate and an isotropic elastic half space was investigated by Borowicka’ using a power series expansion technique. This technique was subsequently extended by Brown2 and Ishkova,3 who incorporated the effects of singularity in the approximation of the contact stress distribution at the interface. Many other analytical and numerical methods have been introduced to analyze this interaction
Address reprint requests to Dr. Zaman at the School of Civil Engineering and Environmental Science, University of Oklahoma, Norman, OK 73019, USA. Received 22 August 1989; accepted 25 January 1990
352 Appl. Math. Modelling, 1990, Vol. 14, July
problem (see, for example, Refs. 4-10). An extensive account of various approaches to the solution of circular plate-half space interaction problems is presented in the review article by Popov.” More recently, Faruque and Zaman I2 have developed a mixed-variational formulation for analysis of the axisymmetric interaction of a thin circular plate resting in smooth contact with an isotropic elastic half space. Unlike unrestrained circular plates, the study of restrained circular plates has been limited. Booker and SmallI examined the problem of a cylindrical tank resting on an elastic half space. They developed an analytical procedure based on the flexibility method to analyze problems in which the tank wall and the base plate have the same radii. Selvadurai14 used the principle of minimum potential energy to analyze circular raft foundations with a restrained boundary. In this method the plate deflections were predicted with good accuracy, but no results were reported for the distribution of flexural moments, although knowledge of flexural moments is essential for the design of such restrained foundation plates. Issa and Zamani5 also employed the (potential) energy method to analyze the cylindrical tank-foundation-half space interaction problem by considering that the tank wall is rigidly connected to the foundation and by adding the strain energy of the tank wall to the strain energy of the circular plate-elastic half space system. For many circular plate foundations the ratio of the thickness to the radius of the plate is large,enough to have significant effect of shear deformation on the flexural response of the plate. Selvaduraii6 has analyzed
0 1990 Butterworth-Heinemann
Circular plates resting on an isotropic elastic half space: M. M. Zaman et al.
Cylindrical Tank Vail
Isotropic 53stCc
AZ&
of
Figure l(a).
Moderately thick circular plate-cylindrical tank wall resting on an isotropic elastic half space
elastic half space. The term “moderately thick” plate is used here to refer to a plate in which the effect of transverse shear deformation is important. In this approach the plate deflection, the flexural moments, and the transverse shear force are approximated independently by power series of different orders. The undetermined constants in the power series approximations are evaluated by invoking the stationary property of the derived variational functional at equilibrium. The proposed method is found to be more efficient and accurate than the potential energy-based formulations,‘,is particularly in terms of flexural moment evaluation. In this method the plate deflection as well as the flexural moments can be predicted with the same degree of accuracy because they are treated as independent variables and are approximated with different functions.
Proposed mixed-variational formulation
Isotropic Eiastic
Figure l(b). Idealization of the circular plate-cylindrical tank wall system
the interaction of an unrestrained circular plate resting on an elastic half space by including the effect of transverse shear deformation of the plate. An analytical method based on a mixed-variational principle” is presented in this paper to analyze the axisymmetric flexural interaction between a moderately thick restrained circular plate and an isotropic
Consider a circular plate of thickness h and radius c1 resting in smooth and continuous contact with an isotropic elastic half space as shown in Figure I(a). The plate is subjected to a uniformly distributed load of intensity PO acting over a circular area of radius ha and is restrained at the boundary owing to the superstructure. For convenience the interaction between the plate and the superstructure is idealized by providing rotational spring at the plate edge as shown in Figure l(b). Note that the self-weight of the superstructure can be represented by an equivalent ring load and its effect in the analysis can be incorporated by modifying the potential energy due to external loads. However, the effect of the self-weight may not be significant in comparison to other transverse loads. The horizontal shear force at the boundary can be accounted for by considering the stretching of the plate in the radial direction. For foundations of practical importance, radial stretching is usually negligible. As such, the effects of the self-weight of the superstructure and the horizontal shear force at the plate boundary are not taken into account. The total potential energy of the idealized superstructure-plate-half space system can be expressed in the form
(1) where
sb=;{(*.r+$)2-
2(1 - zJ)$$
} + F ($4 + +J*
c+,,(r) is the normal contact stress at the plate-elastic half space interface, w(r) is the transverse plate de-
(2)
flection, K is the rotational spring constant related to the geometry and elastic properties of the superstruc-
Appl.
Math. Modelling, 1990, Vol. 14, July 353
Circular plates resting on an isotropic elastic half space: M. M. Zaman et al. ture, and @l(r) is the rotation of a plane section originally normal to the middle surface of the plate. In equation (2), D denotes the flexural ridigity of the plate [D = Eh3/{12(1 - v2)}], E, G, and v are the elastic constants, and r and 8 represent the polar coordinates. For an elastic system in equilibrium the principle of minimum potential energy can be written as
s7Tp = 0
(3)
2lr ho
-J-I
0 0
where S7rP represents the first variation of the potential energy with respect to the independent functions w(r) and $(r). Following the steps outlined by Washizu,17 the variational principle in equation (3) can be modified to obtain a mixed-variational principle of the form 6%-R = 0
(4)
where
277 0
P,w(r)r dr d% + i
II 0 0
v;_(r)w(r)r dr d%
2Tr
(5)
where the subscript ,r represents the derivative with respect to r, Qr is the transverse shear force in the plate, and M, and MH are the radial and the circumferential moments, respectively. In equation (4) the independent functions subjected to variation are w(r), M,.(r), Me(r), and QJr) and hence can be called a mixedvariational principle. Since w(r), M,.(r), M,,(r), and Ql(r) are treated independently, solutions based on such a principle can provide the same order of accuracy for these quantities by representing them in terms of power series of same order. Note that equation (5) includes shear force QJr), which allows one to study the effect of transverse shear on the flexural response of the plate. In the formulation based on the principle of minimum potential energy, only the displacement w(r) is treated as an independent function. Although such a formulation can adequately predict the plate deflection, it cannot predict bending moments [M,.(r), Me(r)] and shear force [Qr(r)] with the desired accuracy. From this consideration the proposed formulation is superior to the potential energy-based formulations. Assuming that the superstructure can be considered as a cylindrical thin shell of semi-infinite length and thickness t, it can be shown that the stiffness factor K in equation (1) takes the following form:ih K = Eo6(,
T v6,{3(’ - 4(;)‘}“’
(6)
where E. and u. represent the elastic constants of the shell material. 354 Appl. Math. Modelling, 1990, Vol. 14, July
General application To employ the mixed-variational principle for the analysis of the interaction problem related to the moderately thick circular plate with restrained boundary, the plate deflection, the flexural moments, and the shear force are approximated by the following power series: 2i
w(r) =,a 5Aj t i=o 0
(7) Zi
M,.(r) = Poa2 5 Bj X j
=
O
0 Y
MA(r) = Poa2 2 Cj X j
=
O
Q,.(r) = Pea 5 Dk
(9) 0
0
6 2k-’
k=l
(10)
where A;, Bj, C;, and Dk represent the undetermined parameters. Note that the function 4(r) can be expressed as $(r) = -w(r),, + c $$
(11)
and therefore does not require an additional representation. The constant c in equation (11) is related to the nonuniformity of the shearing strain over the crosssection of the plate. It can vary between 1.0 and 1.5,
Circular plates resting on an isotropic elastic half space: M. M. Zaman et al.
depending upon the degree of nonuniformity of shearing strain, and is usually taken to be 615.” In equations (7)-( 10) there are a total of (n + 2m + s + 3) unknown parameters, of which four can be eliminated by invoking the boundary conditions that are applicable to the edge of the plate. These boundary conditions are
KWlr=. = K[W)lr=,
(12)
MW” = M&)lr=”
(13)
Qr(r)Ir=
-D
= 0
I =
(14)
Note that in equation (12) the radial moment M,.(v) at the edge of the plate is not zero because the edge of the plate is restrained owing to the presence of the superstructure. The magnitude of the moments induced by the superstructure at the plate boundary depends upon the rigidity of the joint. Owing to elimination of four constants from equations (7)-(lo), the total number of unknowns in the system reduces to (n + 2m + s - 1). These undetermined parameters are evaluated from a set of linear algebraic equations that are obtained by using the stationary conditions of the functional rR, which can be written as anR -=
ifA;
0,
2I = 0,
(15)
.[A, + A,(;)= + Az{(;)~ - ($)($}] M,(r) = f’~~~[Bo{ 1 - (b)“} + B,{ (i)* - (;)‘}I
Analysis of the restrained circular plate The mixed-variational principle outlined in the preceding section is used to analyze the interaction problem of restrained circular plates resting in smooth contact with an isotropic elastic half space. The following approximation functions are used:
(16) M,(r) = P,n=[B, + B,(i)’ + B2(;)4]
(17)
Me(r) = P0a2[ C0 + C,(b)* + C=(i)“]
(18)
Q,(r) = p”a[Dl(b) + D$)‘]
(19)
where Au, . . . , Ax, Bo, . . . , B2, Co, . . . , C2, D,, and D2 are the undetermined parameters. It should be noted that the approximation function for w(r) includes terms up to an order of six, whereas the approximation functions for M,(r) and Me(r) contain terms up to the fourth order only. This particular choice is motivated by the fact that M,(r) and MB(r) can be expressed in terms of derivatives of w(r) up to the second order. However, the order of approximation functions for M,(r) and M&r) can be selected independent of the order of w(r) for general application of the mixed-variational principle. By applying the boundary conditions given by equations (12)-(14) for the restrained circular plate, four unknown coefficients, Ax, B2, Co, and D2 are eliminated from equations (16)-( 19). The resulting moditied approximation functions are given by
w(r) =
(20) - +A, + ($42](3’
(21)
M&) = p,a*[B, + C,(i)* + Cz($“l
(22)
QJrI = $DI{(~) - (;)‘}I
(23)
+(r) =
--~[AI(:) + W{ (i)3 - (i)(k)5]] + (f’of$)[Dl((i) - (:)‘>1
rCr(rb, = - (i) [ AI + A2{6(:)’ - (y) (:)“)I + (PO&) [D,{l - 3(i)*}]
(24) (25)
The distribution of normal contact stress, u,,(r), at the plate-elastic half space interface due to displacement field W(T) can be expressed in an integral form as’* fl&) =
1
for
OSrSa
(26)
Appl. Math. Modelling, 1990, Vol. 14, July 355
Circular plates resting on an isotropic elastic half space: M. M. Zaman et al. where ES and u, are the elastic constants of the half space and (27) By substituting equation (20) into equations (26) and (27) the following expression for u,,(r) is obtained:
a,,(r) =
Esa
[ 7T(l - VZ)_
]{A,+A,[-2+4(;)2]
(28)
+Az[ -(E) - (Z)(5)‘+ (E)(L)‘- (%)($I) By using the expressions in equations (20)-(25) and (28) the functional nR in equation (5) can be written in terms of the unknown parameters AO, A,, A*, &, B,, C,, C2, and D, as
[cm = {Yi
??-R = X, + X2 + X, + x4 + x, + x, + x, + x, + x, + X,”
appendix. The first variation of rR with respect to the unknown parameters AO, A,, A2, &, B1, C,, C2, and D, will result in a set of linear simultaneous equations of the form
(29)
The expressions for X,, X2, . . . , Xl,, are given in the
(30) where [C] is the symmetric coefficient matrix of size (8 x 8) given in the appendix and the vectors {X} and {Y} are given by
(31) (32)
J (I XX) If we know the parameters AO, A,, AZ, &,, B,, C,, C2, and D,, then the plate deflection, w(r), the flexural moments, MJr) and M&r), and the contact stress, us(r), can be obtained from equations (20), (21), (22), and (28), respectively.
If we represent t/h and h/a by 0 and p, respectively, and assume that the cylindrical tank wall and the circular plate are of the same material (that is, E0 = E and v. = v), equation (34) reduces to the form l/2
Numerical results To facilitate the presentation of numerical results for the moderately thick restrained circular plate-elastic half space interaction, attention is restricted to the particular case in which the uniformly distributed load is acting over the entire plate surface (that is, A = 1). A relative rigidity parameter K, is defined in terms of the geometric and material properties of the plate-elastic half space system as follows:16
K=
(;)s($)(i>’
(33)
To demonstrate the influence of the edge restrainment on the flexural behavior of the plate, a nondimensional factor K, is defined as K,=K;
K, = 2[3(1 - ~*)]l’~0* $ 0
(35)
Since K, incorporates the geometric characteristics of both the plate (h and a) and the tank-wall (t), the effect of edge restrainment can be conveniently investigated by changing the values of 0 and p. For each selected value of 8 and p, K, will provide a numerical measure of edge restrainment. In the numerical results presented here the effects of both 8 and /1 are investigated. For convenience of isolating the effects of material properties and geometry the relative rigidity parameter is expressed in the form K, = K,p3
(36)
where K, is a modular ratio defined by (37)
= 2=($)(;)‘(;)[3(1 - v;)(;)*]“’ (34) 356 Appl. Math. Modelling, 1990, Vol. 14, July
For convenience of presentation of numerical results the plate deflection, the flexural moments, and
Circular plates resting on an isotropic elastic half space: M. M. Zaman
the contact stress are nondimensionalized w(r) =
41-4) 7rP(jA2 E.7 [
I
w(r)
et al.
as (38)
c+Z,W= ~oA2~z,,W
(39)
M,(r) = PoA2a2M,(r)
(40)
Me(r) = POh2a2MO(r)
(41)
where all quantities with an overbar are nondimensional variables. The numerical results presented in this section are valid for a Poisson ratio of 0.3 for the
1
Figure 4.
Variation of the normalized plate deflection versus radial distance for various modular ratios (for A = 1.0, p = 0.1, 0 = 2.0)
Figure 2.
Variation of the normalized plate deflection versus radial distance for various modular ratios (for A = 1.0, p = 0.1, 0 = 0) (Note: W(r) = [trp0h2a(l - v:)lE,IW,(r), Km = (~/6)[(1 - v:Ml - v’)] (E/E,))
Figure 5. (A = 1.0,
E I3
Plate central
Flexible plate:
Variation of the normalized plate deflection versus radial distance for various modular ratios (for A = 1.0, p = 0.1, 8 = 1.0)
versus h/a
for various
K,,,
plate material. For the limiting cases of completely flexible and rigid plates the central plate deflections are compared with the corresponding exact solutions’ as follows: Rigid plate:
Figure 3.
deflection
t/a = 0.5) (W(O) = [~~0A2a(l- v%E,IW(O))
W(0){exact:present method} = w(0){0.50:0.50} W(O){exact:present method} = W(O){O.6366:0.6398}
It is evident that the solutions for the limiting cases obtained from the proposed variational formulation compare accurately with the known exact results. Figure 2 shows the variation of plate deflection for 13= 0 (that is, the unrestrained case) and for various relative rigidities, K,.
Appl.
Math.
Modelling,
1990,
Vol.
14, July
357
Circular plates resting on an isotropic elastic half space: M. M. Zaman et al. Figures 3 and 4 show the variation of normalized plate deflection with the nondimensional -radial distance (r/a) for 8 = 1.0 and 2.0. It is observed that for plates of intermediate rigidities (K, = 100) the central plate deflection decreases with increasing 0, while the edge deflection increases. As a result, the differential deflection decreases as 8 increases. The deflection of a plate of high rigidity remains unaffected by the edge restrainment, as expected. The effect of p (the ratio of thickness to radius of the plate) on the central plate deflection, W(O), for different values of K,,, is shown in Figure 5. It is observed that when the relative rigidity (K,) is significantly large, the central deflection remains virtually unaffected by the change in the param-
Figure 8. Variation of the normal contact stress versus radial distance for various modular ratios (for A = 1.0, /* = 0.2, 0 =
Figure 6. Variation of the normal contact stress versus radial distance for various modular ratios (for A = 1.0, p = 0.1, 0 = 0) ( N o t e : uzz(r) = p0A2~zz(r), K,,, = (~/6)[(1 - &/(I - v*)](E/EJ)
Figure 9. Variation of the normalized plate moment versus radial distance for various modular ratios (for A = 1.0, r~ = 0 . 1 , 0 = 0) (Note: M,(r) = b&,(r) = poA2a2~,(r), K,,, = (46][(1 - v:)/(l - v2]]fE/E,)]
Figure 7. Variation of the normal contact stress versus radial distance for various modular ratios (for A = 1.0, /.L = 0.1, 8 = 2.0)
358 Appl. Math. Modelling, 1990, Vol. 14, July
eter p. For plates of moderate rigidity (e.g., 100 < K, < lo), W(0) is significantly affected by the change in I_L. For example, W(0) decreases by approximately 20% owing to the change in /J from 0.1 to 0.5 for K, = 100. The distribution of normalized contact stress, CZZ(r), for various K, and f3 is presented in Figures 6 and 7. It is evident that for plates of moderate rigidity the magnitude of the contact stress at the center of the plate is significantly influenced by the change in 0. Also, the shape of 7i,,(r) curves is substantially affected by the extent of edge restraint. Figure 8 shows the distribution of a,,(r) for various values of K, and for
Circular plates resting on an isotropic elastic half space: M. M. Zaman
Figure 10. Variation of the normalized plate moment versus radial distance for various modular ratios ( for A = 1 .O, p = 0.1, I9 = 1.0)
003
Figure 12. Plate central moment versus h/a for various (A = 1.0, t/a = 0.5) (M,(O) = M,(O) = poA2aZM,(0))
p = 0.2 and 0 = 2.0. Comparison of Figures 7 and 8 reveals that for K, = 100 the value of cZZ(0) decreases with increasing I_L.For Km = 100 this decrease in LYZZ(0) is on the order of 32% for an increase in p from 0.1 to 0.2. Distribution of flexural moments is important from the viewpoint of design of circular plate foundations. Figures 9, 10, and II show the variation of the normalized radial moment, a,(r), with the nondimensional radial distance, da, for various values of K, and 8. It is evident that the edge restraint (0) significantly influences the values as well as the distribution of MY(r). A point of contraflexure is noticed for the case of restrained plates (Figure 10). The normalized central
K,,,
moment, M,(O), significantly reduces, owing to the increase in the edge restrsint. As is evident, changing 8 from 0 to 2.0 decreases M,(O) from approximately 0.079 to 0.022 for K,,, = IOh. With the decrease in K,,, this effect decreases, as expected. For unrestrained plates ((3 = 0), the edge moment &‘,.(a) becomes zero (Figure 9). Edge restraint (0) is seen to induce significant edge moments M,.(a), as is evident from Figures IO and II. Such information is important for design of circular foundations. Figure 12 shows the effect of p on the central moment, M,.(O), for various values of K,,,. For the limiting cases of flexible and rigid plates the effect of p on M,.(O) is found to be insignificant, whereas for plates of moderate relative rigidity this effect is significant. Concluding
Figure 11. Variation of the normalized plate moment versus radial distance for various modular ratios (for A = 1 .O, p = 0.1, I9 = 2.0)
et al.
remarks
A formulation based on a mixed-variational principle is developed to analyze the axisymmetric flexural interaction between a moderately thick circular plate with a restrained edge and an isotropic elastic half space. In this formulation the plate deflection, the flexural moments, and the shear force are treated as independent functions and are approximated by power series in terms of the radial coordinate and a set of unknown parameters. The effect of the restrained boundary is included in the mixed-variational functional, rrR, by using an appropriate term that is applicable for the semi-infinite cylindrical shell connected monolithically to the circular plate at the edge. The unknown parameters associated with the approximation functions are evaluated by utilizing the stationary condition of rrR. Numerical results are presented to demonstrate the effects of edge restrainment and transverse shear deformation on the plate deflection, the contact stress distribution, and the flexural moments. It is found that both edge restraint and transverse shear deformation can significantly influence these quantities.
Appl.
Math. Modelling,
1990, Vol. 14, July
359
Circular plates resting on an isotropic elastic half space: M. M. Zaman et al.
The proposed mixed formulation can predict the plate deflection and the flexural moments with very good accuracy even when lower-order polynomials are used as approximation functions. In this paper, sixth-order and fourth-order polynomials are used to approximate the plate deflection and the flexural moments, respectively. The mixed-variational approach presented here can be used effectively to analyze flexural behavior of restrained and unrestrained circular plates resting on an elastic half space.
6 7 8 9
Palmov, V. A. The contact problem of a plate on an elastic foundation. J. Appl. Math. Mech. 1960, 24, 609-618 Selvadurai, A. P. S. The interaction between a uniformly loaded circular plate and an isotropic elastic half-space: A variational approach. .I. Struct. Mech. 1979. 7. 231-246 Tseitlin, A. I. On the flexure of a circular plate lying on a linearly deformable foundation. Izv. Akad. Nauk SSSR Mekh. Tverd. Tela 1969, 1, 99-112 (in Russian) Zaman, M. M., Kukreti, A. R. and lssa, A. Analysis ofcircular plate-elastic half space interaction using an energy approach.
Appl. Math. Modelling 1988, 12, 285-292
IO I1 12
References 13
Borowicka, H. Influence of rigidity of a circular foundation slab on the distribution of pressure over the contact surface.
Proceedings of the First Internutional Conference on Soil Mechanics Found Engineering. Vol. 2. DD. 144-149, 1936
14
Brown, P. T. Numerical analyses of uniformly loaded circular rafts on deep elastic foundations. Geotechnique 1969,19,301-306 Ishkova, A. G. Exact solution of the problem of a circular plate in bending on the elastic half space under the action of a uniformly distributed anti-symmetrical load. Dokl. Akud. Nauk
15
Zemochkin, B. N. Analysis of circular plates on elastic foundation. Moskov. Izd. Voenno. Inzh. Akad. 1939 (in Russian) Popov, G. Ya. Plates on a linearly elastic foundation (a survey). Soviet Appl. Mech. 1971, 8, 3-17 Faruque, M. 0. and Zaman, M. M. A mixed-variational approach for the analysis of circular plate-elastic half space interaction. Comput. Methods Appl. Mech. Engrg., 1990, in press Booker, J. R. and Small, J. C. The analysis of liquid storage tanks on deep elastic foundations. Internat. J. Numer. Anal. Methods Geomech. 1983, 7, 187-207 Selvadurai, A. P. S. Circular raft foundation with a restrained boundary. Geotech. Engrg. 1984, 15, 171-192 Issa, A. and Zaman, M. M. A cylindrical tank-foundation-half space interaction using an energy approach. Comput. Methods Appl. Mech. Engrg. 1986, 56, 47-60
16
USSR 1947, 4, 129-132
Cheung, Y. K. and Zienkiewicz, 0. C. Plates and tanks on elastic foundations-An application of finite element method. Internat. J. Solids Structures 1965, 1, 451-461 Gladwell, G. M. L. Unbended contact between acircular plate and an elastic foundation. The Mechanics of the Contact Between Deformable Bodies, ed. A. D. de Pater and J. J. Kalker. Delft University Press, The Netherlands, 1975, pp. 99-109
17 18
Sklvadurai, A. P. S. A contact problem for a Reissner plate and an isotrooic elastic half soace. J. M&an. Theor. Appl. 1984, 3, 181-i96 Washizu, K. Variational Methods in Elasticity and Plasticity, 2nd ed. Pergamon, Oxford, England, 1982 Harding, J. W. and Sneddon, 1. N. Elastic stresses produced by the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch. Proc. Cambridge Philos. Sot. 1945,41, 16-26
Appendix The expressions for XI, X2, . . . , Xl0 in equation (33) are as follows:
12rP;&{ ($), + (&))BT + (&)B”B, + (;)B&, + ($B”G + ($c: + (;)Gc2 + ($)ci - (+B; - &BOB, - ($VB”C, - ($B,C, - ($B,C2 - ($B,C2} x2 = 41 + 4Pk${ (&)D? + ($)B& - (;)B,D, + (+,a + (;)c*D,} X3 = -rPoa3 { (;)Mo + ($W, + A,C, + ($%G + (;)&‘o + (f)&B, + AZCI + (+bG} x, = -
X4 = -iiP,ai{12Aa + (i)A4A, + [ (:)A6 - (i)A8]A2} X, = [+$I {A:, + (;)AoA, + ($)AoAz + (;)A: + (g)A,A2 + (=)A:} X6 =
&rKa((t)A: + (:),,A2 + ($)A$}
X, = - [ ($K2g]{A: + (;)A,A2 + (;)A;} x8
=
[24&K&]{ (+Wo
+
(&%& +
360 Appl. Math. Modelling, 1990, Vol. 14, July
($Wo
+
(+M4}
Circular plates resting on an isotropic elastic half space: M. M. Zaman et al.
X, = [48rruP,K$1{ ($WO + (;)A&, + ($%C, + (;)A,& + (;)&C, + (+z} X,0 = [‘Ml + u)cP,K~]{ ($W, + (;)A#, j The matrix [C] in equation (34) is an (8
x
8) symmetric square matrix. The elements of [C] are given below:
C(l,l) = 2x,
C(4,7) = [ -4 + (;)v]xz
4 C(1,2) = 5 X1 0
C(4,8) = ;
272 C(l,3) = 15 XI ( ) C(1,4) = C(1,5) = C(1,6) = C(1,7) = C(1,8) = 0
x 3
0
C(5,5) = - ; x2 0 C(5,6) = vx2
C(5,8) = - t
x3
0
C(2,4) = -(y) + (+ + (;)x, C(2,5) = -(i) + (&)X6 C(2,6) = - 1 + i
x7
0
C(2,7)
=
-(;)
C(6,6) =
-4X2
C(6,7) = -3X2 1 C(6,8) = 5 0
x3
C(7,7) = - F x2 0 +
($X7
C(7,8) = ;
x3
0
C(2.8) = $ xs 0
C(8,8) = ;
x3
0
where C(3,4) = - (7) + (-$)X6 + ($)A? C(3,5) = -($ + (A)“,
XI =
x2 = POEh3
x3 =
C(3,6)= -l+ ; x, 0
ES ?I-P”(1 - vf) i
(1 + v)cP,,; K Pea’
x4=4C(3,7) = -({) + (&)“’ C(3,8) = - xg 0
24 K2 x5 = -0 7 PoaEh3 x6 = 24K;
C(4,4) = ,! (y) + 16v]x, C(4,5) = [ -@ + z,],, C(4,6) = [-6 + 3v]x2
x,= -48vKa Eh3 K xg = 4(1 + V)“Z