of smooth receding contact solutions in linear elastostatics is ... ASME, United Engineering Center, 345 East 47th Street, New York, N.Y.. 10017, and will be ...
Hui Li Research Assistant.
J. P. Dempsey Associate Professor. Mem. ASME Department of Civil and Environmental Engineering, Clarkson University, Potsdam, NY 13676
Unbonded Contact of a Square Plate on an Elastic Half-Space or a Winkler Foundation The unbonded frictionless receding contact problem of a thin plate placed under centrally symmetric vertical loading while resting on an elastic half-space or a Winkler foundation is solved in this paper. The problem is transformed into the solution of two-coupled integral-series equations over an unknown contact region. The problem is nonlinear by virtue of unilateral contact and therefore needs to be solved iteratively. Special attention is given to the edge and corner contact pressure singularities for the plate on the elastic half-space. Comparison is made with other relevant numerical results available.
Introduction The unbonded frictionless receding contact problem of a thin square plate placed under centrally symmetric vertical loading while resting on an elastic half-space or a Winkler foundation is solved in this paper. The combination of Navier solutions and Levy solutions is used to satisfy the governing differential equations, the equilibrium condition and the plate boundary conditions. Boussinesq's fundamental solution (1885) for the surface displacement of an elastic half-space due to a vertical point load is used to define the total surface displacement for an, as yet unknown, contact pressure distribution. Displacement compatibility between the plate and foundation, as well as the requirement that the foundation be tensionless, reduces the problem to the solution of twocoupled integral-series equations. The latter displacement and contact pressure inequalities make the problem nonlinear, and hence the contact region and pressure distribution are found by iteration. The corner and edge contact pressure singularities for a plate on an elastic half-space need special consideration. The contact treated in this paper can be characterized as receding, as identified by Dundurs (1975), since the contact surface within the loaded configuration is contained within the initial contact surface. A general feature of smooth receding contact solutions in linear elastostatics is for the extent of a receding contact to be independent of the level of loading. Moreover, if uplift is to occur, the change from initial contact (the flat plate initially has all of its surface in contact) to the contact in the loaded configuration takes place discontinuously.
Theoretical investigations into the contact behavior of plates resting on elastic foundations have been motivated to a considerable extent by the need in civil engineering to design mat and raft foundations, flexible column footings, and pavement slabs (Hooper, 1978). The earliest application of the finite element method to the analysis of rectangular plates on elastic foundations by Cheung and Zienkiewicz (1965) was followed by a great many related investigations most of which ignored nonlinear effects due to separation or uplifting between the plate and the supporting medium (Rajapakse and Selvadurai, 1986). The studies of these nonlinear effects using the finite element method include, among others, Cheung and Nag (1968), Svec (1974), Ascione and Grimaldi (1984), and Ascione and Olivito (1985). A theoretical treatment can be found in the paper by Villaggio (1983). Experimental work on the title problem is also scarce; Vint and Elgood (1935) loaded rectangular steel plates resting on a nest of springs and measured deflections, while Murphy (1937) loaded rectangular steel plates (resting on a layer of rubber) by corner, edge and central concentrated loads.
Basic Equations A thin square plate of side length a in unbonded frictionless contact with an elastic foundation is subjected to a centrally symmetric load q{x, y) and an unknown support reaction p(x, y), Fig. 1. The governing equation for the plate deflection, wp (w, y), is given by
Dv*wp(x,y)=q(x,y)-p(x,y), Contributed by the Applied Mechanics Division for presentation at the Winter Annual Meeting, Chicago, IL, November 28 to December 2, 1988, of The American Society of Mechanical Engineers. Discussion on this paper should be addressed to the Editorial Department, ASME, United Engineering Center, 345 East 47th Street, New York, N.Y. 10017, and will be accepted until two months after final publication of the paper itself in the JOURNAL OF APPLIED MECHANICS. Manuscript received by ASME Applied Mechanics Division, July 14, 1987; final revision, December 14, 1987. Paper No. 88-WA/APM-28.
(1) 3
where the flexural rigidity of the plate D equals E p /! /12(l vp) in which h is the thickness of the plate, while Ep and vp are the modulus of elasticity and the Poisson's ratio of the plate, respectively. For the symmetric square plate with four free edges, the boundary conditions of zero slope and zero vertical force at centerlines and zero moment and zero vertical force at edges have to be satisfied, viz. Transactions of the ASME
4 3 0 / V o l . 55, JUNE 1988
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W
P,X = 0
where Ay is the stiffness of the Winkler foundation. In addition to the foregoing equations, the compatibility conditions between the plate and the half-space must also be considered. Since the contact is unilateral, the support reaction is either compressive or zero. These can be written as, with Q denoting the contact region wf(x,y) = wp(x,y),
p(x,y)>0,
for
(x,y)£U;
(6a,b)
wf(x,y)>wp(x,y),
p(x,y)=0,
for
(x,y)£Q.
(6c,d)
Formulation The solution of the governing equation (1) can be expressed in the following as the combination of wx(x, y), w2(x, y), the particular solutions due to the support reaction and the load, respectively, w3(x, y), the complementary solution, and 8, the corner deflection wp (x,y) = wx(x,y) + w2 (x,y) + w^(x,y) + 5,
J
(11) p(u,v)dudv TrEf J Jo V ( x - « ) 2 + ( y - y ) 2
•if
+—
p(u,v)g(x,y;u,v)dudv
oo
-«
E
A,„{rm(x,y)+rm(y,x)}=w2(x,y)+&,
*D
m~?,3,
fk(u>v):
4sinaAr« ^ , •Vp-x
g(x,y;u,v) 4 •K
~
sina m x
m = l , 3 , 1=1,3:
n3 + (2-up)k2n (k2 + n2)2
sma„y smamu (m2 + n2)2
n^D VpTrv
E3_bk. „—
r,„(u,v) = {cosh(l3m-amu) ,(x,y)=a
YJ
A
m{rm(x,y)+rm(y,x)],
(9)
m=l,3,
in which 44 p a r» a = —r [p(x,y);g(x,y)) a1 Jo Jo
sina,„x
sina„.y
dxdy. (10a,&)
sina„y,
sina„t>
(13a)
(136)
-4£pk3m3 + - ^ — (tanh/3 m -f p f . m sech2/3„,)> (13c) ir(m2 + k2)2 tJCt
n~t,3, (m2 + n2)
(12)
where bmk is the Kronecker delta, fp = (1 - vp)/(i + vp),r\p = (1 - x„)(3 + *p/, v„), X„ (1 ~- vi pO)/2, / 2 , /3,„ -p -= (1 pm = m7r/2, and
/;Stf)=—