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Unilateral plate contact with the elastic-plastic winkler-type foundation
Com~~ers & S~wcrwes Vol. 39, No. 6, pi. 641-651, Printed in Great Britain.
0045.7949191 $3.00 + 0.00 Q 1991 Pergamon Press
199t
UNILATERAL
ELASTIC-PLASTIC
plc
PLATE CONTACT
WINKLER-TYPE
WITH THE
FOUNDATION
R. LEWANDOWSKIand R. SWITKA Technical University of Poznan, ul. Piotrowo 5, 60-965 Poznati, Poland (Received 11April 1990)
Abstract-In this work contact problems of a plate with the elastic-plastic Winkler-type foundation has been solved. An unilateral character of bonds between plate and foundation has been taken into consideration. The effect of friction forces in the contact plane has been neglected and an incremen~l approach has been applied. An incremental problem has been fo~ulat~ in a variational manner and solved by use of the linite element method. The results of numerical calculations for rectangular plates subjected to the simple and complex load processes have been presented.
1. INTRODUCTION
The contact problem of a construction with a deformable foundation is an essential analysis task in civil engineering. The ground on which a building structure is located exhibits the features of viscoelastic-plastic materials. Moreover it is possible loss of contact between structure and foundation and friction forces occur in the contact plane. Taking all ground features into consideration simultaneously and conditions of unilateral contact is a very difficult task and at present only the solutions of chosen problems are known. The analysis of behaviour of plates on the different types of elastic foundations, considered as a contact in problem with unilateral bonds, appears Refs [l-l I]. In these papers friction force in the contact plane has been neglected. For solutions the analytical methods [l-3], the finite element method [4-91 or boundary element method [10-l l] have been applied. The bending problem of the plates resting upon a visco-elastic foundation as a rule was considered as a contact problem with bilateral bonds [12-151. Recently work [16] has been published in which the conditions of unilateral contact of a circular plate subjected to bending and resting upon a visco-elastic Winkler-type foundation are taken into account. Also Ref. [17] considers the conditions of unilateral contact and gives a general variational formulation for plates resting upon a visco-elastic foundation. In Ref. [I81 a contact probiem of a beam with a fo~dation of nonlinear, physical properties was analysed. The solution was obtained with help of the finite difference method and the conditions of unilaterality of bonds were observed by the use of an iterative approach. The static problem of a beam on an elastic-plastic Winkler-type foundation was dealt with in Ref. [19]. An incremental approach enabling an analysis of the simple and complex load processes CAS 39,bE
was applied. For the solution the finite element method was used and the area of contact and plastic zone were determined by an iterative method. Moreover, the process of reaching the limit load by the Winkler-type foundation mating with a beam loaded with a concentrated force was investigated by analytical methods in Ref. [20]. Static analysis of an elastic-plastic foundation under a rigid plate loaded by a wheel of a passing vehicle was the theme of the work [21]. The problem was solved in an analytical manner. The foundation state was analysed in successive phases of wheel passing and adaptation of foundation subjected to multiple passages. The basis of the finite elements method used to analyse contact problems is built on the variational inequalities method. Lagrange multipliers, the penalty method or mathematical programing methods are employed as the fundamental numerical techniques. They are described in Refs [22-241. The literature honeying the contact problems of another kind are not discussed here. A comprehensive discussion of these problems as well as the review of corresponding literature is contained in Ref. [25] In this work a numerical method of static analysis of elastic plate resting upon an elastic-plastic foundation is presented. It has been assumed that the loading process is a complex one and consists of an optional number of the simple processes. Moreover, it is assumed that the load increments are uniparametric. To the description of plate behaviour both classical thin and thick plate theory has been applied. The uniparameter Winkler-type foundation, taking into account plastic defo~ation and reinforcement, has been adopted. Moreover, the unilaterai character of bonds between plate and foundation are considered and friction forces are neglected. The accepted Winkler model is the simplest one and it may arouse reservations as to its adequacy. However, it should be taken into account that while considering the elastic-plastic states of foundation it 641
642
R.
LEWANDOWSKIand
is impossible to make use of Green function and superposition principle. Such restrictions cause the application of more complex models, e.g. a biparameter model (Vlasov, Pasternak) or elastic-plastic half space to involve great difficulties and the enlargement of the task itself. For the foundation area discretization becomes necessary and that, in the course of the plastic analysis, requires the division into a large number of finite elements. The difficulties in the application of the general yield condition in foundation area are also of importance. The yield condition on the compressed edge area of a beam or a plate ceases to be univocal due to the peculiarities taking place in the stress state of the foundation. These serious and strenous problems do not appear in the Winkler model. Unilateral contact constitutes the domineering feature, especially for flexible plates exposed to the activity of strongly local loads. Unilateral bonds also may become evident in case of eccentrically loaded thick plates or in case of plastic deformations appearing in the elastic-plastic foundation due to variable loads. The present paper enables the analysis of all the above cases. For the solution of the problem the incremental approach has been applied. The incremental task has been formulated in a variational manner including unilateral constrains into functional. For a numerical techniques the finite elements method connected with iterative procedure of determination contact and plastic zones has been applied. The results of calculation of the plates subjected to simple as well as to complex load processes are presented.
2. PROBLEM FORMULATION
2.1. Model of foundation Figure 1 illustrates the behaviour of the discussed foundation model, where q, v, ki denote reaction of foundation, displacement and foundation coefficient, respectively. In the process of primary load the foundation coefficient is equal k, , and after reaching by the reaction of the value q. is equal k2, where k, > k, > 0. The foundation reaction q. hereafter will be called the yield limit. In the process of unloading, foundation deflection increments are purely elastic
R.
SWITKA
9
I
Fig. 1. Constitutive relation for foundation,
and the foundation
coefficient has the value k,, but
k, 2 k, .
In the proposed model the plastic deformations appear just from the beginning of the loading process if k, > k,. For k, = k, the foundation receives, in some interval, purely elastic deformations. At the repeated loading foundation deflection increments are also elastic, until reaching by the reaction of a maximum value which appeared at this point of foundation in the past. During further loading processes the foundation behaves in the same way as during primary loading one. The proposed foundation model describes approximately the behaviour of some kinds of nonirrigated grounds. One can here refer to the experimental results of Allen et al. [26] as well as Aljeksjejew [27] relating to dynamic characteristics of grounds at uniaxial compression. From these investigations it clearly results that the adopted foundation model approximates well enough the experimental curve for nonirrigated sands and argillaceous grounds in the range of stresses l-2 MPa, as well as the unloading modulus can be greater than the modulus of primary loading. On a base of the above experiments, in the work [28] a series of different idealizations of the relationship ot has been proposed and used for description of the propagation of waves in ground. A diagram of the relationship reactiondisplacement of foundation, which was already loaded in the past, can display the shape shown in Fig. 2a or the one shown in Fig. 2b.
tq
Fig. 2. Constitutive relation for foundation loaded in the past.
643
Unilateral plate contact The maximum reaction which in the past took place at a given point of foundation has been designated by q2, while v2 represents the displacement corresponding to this reaction. In the first case we have q2 < q. and the diagram consists of three straight segments and for its unique determination are the coordinates qi, vi, i = 1,2,3 of the three characteristic points on a plane q-v (compare with Fig. 2a). The coordinates of point 3 equal q3 = qo, vj = vg. If the foundation in the past was not loaded then points 1 and 2 coincided with the origin of the coordinate system. In the course of loading of foundation the point 2 shifts along the primary path of loading. The coordinate u, of the point 1 determines plastic displacement of foundation. In the second case q2 2 q,,, point 2 coincides with point 3 and their coordinates are equal to q2 = q3, v2 = vj. The diagram q-v consists now of two straight segments (compare with Fig. 2b). In the process of unloading, the position of the discussed points does not change. Let us assume that k, = k,. In such a case the plastic deformations of the foundation will accrue, if d >o,
The same equations expressed by the generalized displacements increments can be presented in the following manner:
D
F’d,+~(6&&2)] t +GhK(G*,-
i- Ghrc(ti,, - b2) = 0.
2.3. Unilateral condition in the contact plane The conditions of unilateral contact can be written in a form
40
for
q2 G cl0
q+o20,
q2
for
q2 >
g+gao,
40
where ( * ) denotes the increment of quantity over which it appears. In the case of foundation for which k, > k, the above relationships take the form ti >o,
B =q2-
An incremental constitutive law for the considered foundation model have the form 4 = k(v)d
(1)
where k(v) is temporary modulus of foundation (compare with Fig. 1). The temporary modulus of foundation is determined according to the concept of secant modulus what has been shown in Fig. I. 2.2. IncremePttal equations of the thin and thick plates Consideration has been limited to the isotropic plates subjected to loads inducing small displacements and strains. Plate material is linear and elastic. From the Kirchhoff-Love theory results the following incremental equations of a thin plate DV4+ = p - 4,
(2)
(3)
where G, h, & denote Kirchhoff’s modulus, the plate thickness and the rotation angles of a straight line normal to the plate central plane with regard to axis a and K = 5/6.
4=
i
4,)= 0,
(q+4)(s+&?)=O,
(4)
where g = v - w denotes the ‘gap’ between plate and foundation. This means that the foundation can be only compressed; plate and foundation cannot mutually penetrate and that passive pressure q and ‘gap’ cannot simultaneously differ from zero at one and the same point of foundation. A complete set of relations describing the discussed problem can be obtained if the boundary conditions of the plate are joined to relations (l)-(4) and the state of system at the beginning of loading process is given. 3. VARIATIONAL FORMULATION OF AN INCREMENTAL PROBLEM
It is very difficult to solve the discussed contact problem in a direct manner. Below, it will be formulated in a variational manner, more suitable for a numerical analysis. The variational problem can be written as follows: minimize the functional
8,)8,)
I(@ i], 8,, ti, A) = Z,(& r
where D, i, 9, denote the flexural rigidity of plate, the increment of deflection of the plate and the increment of external plate load, respectively. For the description of the behaviour of a thick plate the Mindlin plate theory has been applied. Brietly, the fundamental equations of this theory were derived in Ref. [29J.
-
/J
+@)(w
+*)dR
J
-
s
I&-l-d-i)dR
f-I
n [q + $(v)d Jti dC?.
+
s
(5)
R. LEWANDOWSKI
444
satisfying the conditions
and R.
SWITKA
for a thick plate
g+ti-i&O
Gh~[v*(w + 6) - (@,,,+ &) - (82.2+
=q+g--p-p,
in the area of the plate a. The above Z,(+, t$, &) denotes the strain energy of the plate, which can be determined from the relations: for a thin plate
I,($, 8,, 4,) = Z,(kiJ)= ;
d2,2,1
([V(w + tiJ)]2 s It
D V2(& + 4,) + ~(S,,,;+i [
L w - v)[(w + k),,,(w + i)J,
I.12- 8 2.11 -
+Gh~(w,+~,~-6,-8,)=0. - [(w + ~t>,,,l”t>do,
* 4.1,
1 (11)
(6) From the condition (8,) the boundary conditions for the plate are also obtained, but they are not expressed directly here. Taking into account that before load increment the equations of plate are also satisfied, we come to the incremental equations given by relations (2) and (3). Namely both formulations are equivalent.
for the Mindlin plate
4. SOLUTION METHOD
+(w,, + Gii - 0, -
ti,)‘]da.
(7)
The variational formulation is equivalent to the direct approach discussed above. In order to demonstrate it the stationary conditions of functional (5) will be given. From a generalized Kuhn-Tucker theorem [30] the following necessary conditions
(q + kti - A)& dsZ = 0
+
(8)
1n are obtained. From the second integral at (8,) it results that the function of Lagrange multipliers equals the function of passive pressure I=q+kt;=q+cj.
The condition
(9)
(82) is identical with the condition
(4,), and the condition (8,) is equivalent to condition
(43). Taking into account eqn (9) in (8,) and after calculation of H,(ti, 8,, t?,) we obtain the following equations: for a thin plate DV4(W+*)=p+@-q-4
(10)
Summarizing the successive solutions of incremental problems we obtain a solution for an assumed loading process. Such a procedure is necessary, because the description of bonds between plate and foundation as well as plastic properties of foundation is made with the help of nonlinear equations and inequalities. Moreover, the accretion process of the plastic deformations depends on load history and for the accurate description of this process is essential incremental analysis. It is assumed that in a typical incremental approach the changes of loading are proportional to one parameter. An approximate solution of an incremental problem has been obtained by application of the finite element method. Plate area n is being divided in finite elements and out of generalized displacements of nodes a vector U with dimension n x 1 is being formed. Numerical analysis of thin plates has been made using the finite element developed by Bogner et al. [31]. It is compatible, the four-nodal element having 16 degrees of freedom. For the analysis of thick plate the element described in Ref. [32] has been applied. This element is perfectly suitable for discretization of a plate resting on a foundation or elastic half space as it has been demonstrated in Ref. [8]. It is a nine-nodal efement with 26 degrees of freedom. In the foundation some characteristic points are chosen. At these points the contact conditions are verified and the foundation state is determined. The vectors V and Q with dimensions p x 1 created of displacements and foundation reaction and determined in characte~stic points are introduced. With the discussed points some foundation field located in their surroundings is associated. The area of the field associated with the point i of foundation is being marked by si. Typical division of a plate on
Unilateral plate contact
645
points of foundation, G = V - W is the ‘gap’ vector and S denotes the diagonal matrix, whose elements Sii are equal to the foundation area s, associated with the ith characteristic point. The above notation of functional (14) is possible if to the numerical calculation of integrals in relation (5) a complex rectangle method is applied and the characteristic points of foundations chosen as the points of numerical integration. Considering the equation of equilibrium of the plate before the increment of load Lxl2m
KpU + Q = P,
9.6m
Fig. 3. Example l-geometry of the plate, finite element division and distribution of characteristic points.
finite elements and typical choosing of characteristic points within one element has been shown in Fig. 3. Let us denote by W a vector with dimensions p x 1 of the displacements of points of the plate which coincide with the characteristic points of foundation. These displacements and their increments can be determined on the basis of displacements and displacements increments of the nodes using the relations W=AU,
W=AU,
of Lagrange
A=Q+Q,
(17)
the stationary condition of the functional (14) can be written as K&J + A’SK,,,V = P, G+V-W>O, Q+K,V>O, (G + V - W)‘S(Q + K,V) = 0.
(12)
where A is a matrix of dimension p x n whose elements can be determined on the basis of the shape functions applied for description of plate element. Relation (I) written in a matrix notation has the form
t) = K,V,
and taking into account interpretation multipliers
(16)
(18)
A characteristic feature of the unilateral conditions (1 8*) and (18X) is that they are a series of independent
inequalities. If conditions (18,)-(18,) can be satisfied then the following relation between increments of plate deflection and increments of foundation reaction holds
(13) V=HW,
where Q and V denote the vector of reaction increment and the vector of displacements increment in the characteristic points of foundation, K, is the diagonal matrix whose elements k, are coefficients of the foundation in those points. After introducing the above discretization, the variational formulation of the problem can be expressed in the following manner: minimize the functional
where H is the diagonal matrix whose elements depend on the state of the system after increment of load. Elements h, are equal h,= 1
when gi = 0 and gi = 0,
hii = 0
when qi = 0 and di = 0,
h,i = tii/tii
when g,=O and g,#O
I&J, v,A) = f (U + U)‘K,(U + U)
or
when qi = 0 and di # 0.
-(P+P)‘(U+U)-a’S(G+V-W) + Q’S%’+ ;V 'SK, V’,
(19)
(14)
Substituting eqn (19) into (18,) and taking into account (12) we obtain the nonlinear system of equations in a form
such that G+ir-VVZO.
(15)
Above, K, denotes the global plate stiffness matrix, ( )’ vector transposition, P is the vector of nodal plate loads, A is the vector whose elements are Lagrange - multinliers determined in the characteristic
K($J, V)U = P,
(20)
K$J, V) = Kp + A’SK,HA.
(21)
where
Equation (20) has a structure typical for the finite element method equations. Additionally the diagonal
646
R.
LEWANDOWSKIand
elements of matrices S, K,, H as well as elements of the matrix A have to be stored. The solution of eqn (20) has been obtained by the iterative procedure described below. It is assumed, that for a given load increment P, the approximations of vectors *, 0, * and matrix Km, denoted by vi, Uj, tij. K, where i means a number of iteration are known. As first approximation f7, = 0, = 0 were adopted and the elements of matrix Km, are determined on the basis of foundation state before load increment and the elements of matrix H, are to be taken hii = 1 when gi = 0 or h, = 0 when qi = 0. From assumption these approximations satisfy the unilateral conditions and relations (12) and (13). On their basis the elements of matrix Hi and Ki and from
where c1 and Q are the admissible errors of calculation and II - )I denotes an Euclidean norm. CALCULATIONS
Using the above method a computer program for the rectangular plates resting on an elastic-plastic foundation has been performed. Some results for chosen examples were presented below. Example 1. The thin, symmetricallyloaded plate Calculations have been done for a rectangular plate which have the following dimensions a = 9.6m,
Ip$MPcfJ
Fig. 4. Example b-deflection
aJ
.I-‘--‘_’
-I-‘--
‘-7
(22)
the new approximation of the vector U can be obtained. Now, one used the above model of foundation and the unilateral conditions the vector gj+l and the matrix Kmj+l can be determined. Some cases appear here and the detailed relations are given in Appendix 1. The iterative process are repeated until the following inequalities are satisfied
OF NUMERlCAL
SWITKA
bJ
I$+, =K,+,
5. RESULTS
R.
of the plate ccntre vs load.
Fig. 5 Example l-limits of contact and plastic zone for various values of load.
b = 7.2 m, h = 0.2 m. The remaining data characterizing the plate and foundation are E = 23 GPa. v = O.l7,k, = 20.0 MPa, k3 = 40.0 MPa, k2 = 0.1 MPa, g0 = 0.2 MPa. Plate division in elements and location of characteristic points within one element has been shown in Fig. 3. Because symmetry of plate and load, the calculations have been performed for a quarter of plate only. It has been assumed that the foundation is subjected to a load for a first time. The whole plate was loaded by a uniformly distributed load equal 0.004 MPa (a proper weight). Next, the centre of the plate (the dashed area in Fig. 3) has been loaded with an additional, unifo~Iy distributed load. The load increment has been chosen equal p = 0.01 MPa, and the calculation accuracy c, = 62= 0.01. Results of the calculation are presented in Figs 4 and 5. Figure 4 presents the relationship between the deflection of plate centre and load p. The passive pressure exceeded conventional yield limit q0 when the load at the centre reached the value pI = 0.37 MPa. Here a signifi~nt effect of unilaterality of bonds and plastic properties of foundation on the system behaviour is visible. The action of an additional load in the centre of the plate has caused a tear-off of a significant part of the plate from foundation, Next a zone appeared in which the foundation reaction was greater than the conventional yield point of the foundation. This zone expanded together with load increment, but the whole contact zone expanded insignificantly. In this
647
Unilateral plate contact
L
bxl.Zm
I
Fig. 8. Example 3-geometry of the plate.
Fig. 6. Example 2-deflection of the plate centre vs load. way a narrow zone was formed in which the foundation reaction was less than qQ.Further loading of the plate caused the shifting of contact boundary towards plate boundaries and further narrowing of zone in which q < qo. In Fig. 5 the limits of discussed zones for different values of p are shown. In the discussed example, a tearing-off zone remained until the end of the loading process, nevertheless the plate suffered large displacements. Since the description of the plate hehaviour was applied the geometrically linear theory, consequently the results for large loads probably have some errors. Example 2. Thick plate symmetricallyloaded The calculation for a plate with height h = 1.2m, with other data remaining the same as in example 1, have also been performed. At the beginning the plate was loaded with a load uniformly distributed on whole surface of the plate equal to 0.024 MPa and an additional load in the same manner for the thin plate was applied. During the whole process of loading the plate was in contact with foundation. In Fig. 6 the diagram of dependence of deflection of the plate centre on load acting in the centre is shown. The limits of the zone in which q < q. for different load values are shown in Fig. 7. In comparing these diagrams it becomes evident that, in this example, the foundation state ._.__-.-
--
’
‘_..-_‘-_. li”.~.-.-._. . I:1
very near to the limit is associated deflections of the plate.
with small
Example 3. Thin plate loaded asymmetrically Successive examples represent calculation results of a plate subjected to an asymmetrically distributed load in relation to one of the symmetry axes of the plate. The plate with dimensions in projection: a = 4.8 m, b = 3.6 m, h = 0.4 m has been used for calculations. The remaining data relating to plate and foundation are the same as in example 1, Projection of the examined plate is presented in Fig. 8. The ~l~iations were based on the thick plate theory. Upon loading the plate with a uniformly distributed load on the whole plate surface of intensity 0.008 MPa an additional Ioad has been applied on the surface corresponding to the dashed area shown in Fig. 8. Figure 9 presents a diagram between the deflection of the point 1w, and density of load acting on the dashed area p. In Fig. 10 the limit of the contact zone and the zone in which q r q. is shown for a load p = 1.008 MPa. Under influence of an applied load, in a significant part of foundation the reaction exceeded qo, a narrow zone in which q < q@has been formed, and the plate itself suffered &lPa)
0.6
f
I -t i I
I I I
1 ‘.
’ ‘..
I
I
I
j.
I
1 Fig. 7. Example 2-limits of plastic zone for various values of load.
8 5
I 10
Fig. 9. Example 3deflection
I IS
, w km)
20
of a point 1 vs load.
648
R. ~w~~ws~
and R.
~SWITKA
“__-~_+_f____y__
~~1~~Pa 1 ,f .“.II-_ confacf zone
I
Itt ii I
I
1 -
-I 1 ---
1 I t
---
t
3
Fig. 10. Example 3-limits
f i
I I
1
I
Fig. 13. Example 4Aimits
of contact and plastic zone, loaded element 3.
of contact and plastic zone for
p = !.OMPa.
--
j__l 1
---
-
-I - f
.
ut
_ITconf&f 1 :I 0 , \zone -, - ? _.fy.. - _ I I
Fig. 14. Example +--limits of contact and plastic zone, loaded element 4. Fig. Il. Example 4-limits of contact and pla&c zone, loaded element 1.
.--
_; ------
I
;--
1 I
.-- -c___;___&--I i
,c-+
A.,
I I .+--+,confucf gone
i
l
Fig. 15 Example 4-limits of contact zones, unloaded plate. Fig. 12. Example 4-limits of ccmtact and plastic zone, loaded element 2.
large displacements, which consist before all in the movement of plate as a rigid body. Example 4. A thin plate loaded by a slowiy moving load A plate of height h = 0.1 m was analysed. The remaining data concerning the plate and foundation were as in example 3. First of all the plate was loaded with a ~n~fo~~y distributed load of the intensity of 0.002 MPa (dead load of the plate). Next the finite element No. 1 was loaded. The additional load was increasing from zero to the value p = 0.5 MPa. The load increment was assumed to be $ = 0.01 MFa. Figure 11 presents the contact zone range and the range in which q > qafor the load p = 0.5 MPa. Next the load of the discussed element was decreased, increasing at the same time the load of the neighbour-
ing element designated by 2 in Fig. 12. This figure also shows the contact zone range and the yielding zone range (4 > go) which appear in the founda~o~ svhen the additional load reaches p = 0.5 MPa. The datted line, as in the previous examples, shows the plasticity zone boundary. This process was repeated twice, moving the load in succession on the elements designated by 3 and 4. The boundaries of the contact zone are shown in Figs 13 and 14, They refer to loading a corresponding plate element by additional load p = 0.5 MPa. Figure 15 shows contact zone boundaries after the additional load is completely eliminated. 6. FINAL REMARKS This paper constitutes one of the scarce attempts of taking into consideration plastic phenomena in the contact problem with unilateral bonds. The task concerning plates on the elastic-plastic foundation has been solved effectively with the application of a
649
Unilateral plate contact relatively simple incremental procedure. However, this refers to the simplest foundation model and the simplest model of the elastic-plastic body with linear hardening, it affords possibilities for the quality analysis of the behaviour of plates losing contact with the plastically deformed foundation. Moreover, the analysis of the behaviour of the foundation under a plate exposed to a slowly moving load has been conducted. Acknowledgement-This paper was supported by the Polish Academy of Sciences by the CPBP OZ.01Research Program. REFERENCES 1. Y. Weitsman, On the unbonded contact between plates and elastic half space. J. Appl. Mech., Trans. ASME 36,
198-202 (1969).
2. G. M. L. Gladwell and K. R. P. Iyer, Unbonded contact 3. 4. 5, 6. 7. 8.
9.
between a circular plate and a elastic half-space. J. Elast. 4, 115-130 (1974). H. Jung, Druckverteilung unter elastisch gelagerten Kreisplatten. Ing. A&iv 20, 8-12 (1952). Y. K. Cheung and D. K. Nag, Plates and beams on elastic foundations-linear and non-linear behaviour. Geotechnique 18, 250-260 (1968). 0, J. Svec, The unbonded contact problem of a plate on the elastic half space. Camp. method Appl. Mech. Engng 3, 105-I 13 (1974). L. A&one and A. Grimaldi, Unilateral contact between a plate and an elastic foundation. Meccnnicu 19, 223-233 (1984). G. Cyrok and R. Switka, A finite element analysis of plates resting upon elastic half-space, Arch. Ini &d. 29, 381-394 (1983) (in Polish). R. K. N. D. Rajapakse and A. P. S. Selvadurai, On the performance of Mindlin plate elements in modeling plate--elastic medium interaction: a comparative study. Int. J. Numer. Methods Engng 23, 1229-1244 (1986). H. Li and J. P. Dempsey, Unbonded contact of a square plate on an elastic half-space or a Winkler foundation. J. Appf. Mech., Trans. ASME 55,43&436 (1988).
10. J. Puttonen
and P. Varpasuo, Boundary element analysis of a plate on elastic foundations. Int. J. Numer. Methods Engng 23, 287-303 (1986).
11. G. Bezine, A new boundary
element method for bending of plates on elastic foundations. Int. J. Solids Struct.-24, 557-565 (1988). 12. K. Pister and M. L. Williams. Bendine of nlates on a viscoelastic foundation. J. Engng Mech. Diu., Proc. AXE 86, 3144 (1960). 13. E. L. Marvin, Viscoelastic plate on poroelastic foundation. J. Enana Mech. Div.. Proc. ASCE 98. 9I1-927 (1972). - 14. J. Mtjczka and G. Szefer, Plates on a consolidating space. Arch. In?. Lgd. 25, 21 l-229 (1979) (in Polish). 15. K. Sonda, H. Kobayashi and T. Ishio, Circular plates on linear viscoelastic foundations. .J. Engng Mech: Div., 1
Proc. AXE
18. F. W. Beau&t and P. W. Hoadley, Analysis of elastic beams on nonlinear foundations. Comput. Struct. 12, 669-676 (1980).
19. R. Lewandowski and R. Switka, Bending of a beam
on the elastic-plastic Winkler foundation with unilateral constrains. Arch. Ini. I&d. 34, 35-52 (1988) (in Polish). 20. G. Ganowicz, R. Lewandowski, R. Switka and M. k;adzihska-Depko, Analysis of a beam on an elastic-plastic foundation with unilateral constraints. Zest. Nauk. Polit. Pozn., s. Budownictwo 31, 23-32 (1988) (in Polish). Elastic-plastic response of 21. M. zadziriska-Depko, subgrade under a rigid slab with unilateral constraints and subjected to a moving load. Arch. Ini. I&Id. 32, 569-584 (1986) (in Polish). 22. T. R. J. Hughes, R. L. Taylor, J. L. Sackman, A. Cumier and W. Kanoknukulchai, A finite element method for a class of contact-impact problems. Comp. Meth. Applied Mech. Engng 8, 249-276 (1976). 23. N. Rikuchi and J. T. Oden, Contact problems in elastostatics. In Finite Elements; Special Problems in Solid Mechanics. Vol. 4 (Edited bv J. T. Oden and G. F. Carey). Prentice-Hall, Engiewood Cliffs, NJ (1984). Inequality Problems in 24. P. D. Panagiotopoulos, Mechanics and Applications. Birkhauser, Base1 (1985). 25. J. J. Telega, Variational methods in contact problems of mechani&. Adv. Mech. 10, 3-96 (1987) (in-Russian). 26. W. A. Allen. F. B. Mavfield and M. L. Morrison, Dynamics oi a project& penetrating sand. J. Appf. Phys. 3, (1957). 27. N. A. Aljeksjejew, Mietod oprjedelijenija dinamiczeskich charakteristik g~ntow pri bolszich dawlenijach. Dinamika gruntow, Sb. No. 44, Moskwa (1961) (in Russian). 28. E. Whdarczyk, Fale plastycze, w: Mechanika Techniczna. t. 3 Drgania i fale. PWN, Warszawa (1986). 29. T. J. R. Hughes, M. Cohen and M. Haroun, Reduced and selective integration techniques in the finite element analysis of plates. Nuclear Engng Design 46, 203-222 (1978).
30. D. G. Luenberger, Teoria optymalizacji. PWN, Warszawa (1974). 31. F. K. Bogner, F. L. Fox and L. A. Schmidt, The generation of inter element-compatible stiffness and mass matrices by the use of inte~olation formulae. Prof. Co& Matrix method in Struct. Mech. Air Force Institute of Technology, Wright Patterson A.F. Base, Ohio, October (1965). 32. T. J. R. Hughes and M. Cohen, The ‘heterosis’ finite element for plate bending. Comput. Struct. 9, 445-450 (1978).
1
104, 819-827 (1978).
16. R. Switka, Axially-symmetrical bending of a circular plate on a viscoelastic foundation under the conditions of an incomplete contact. Zest. Nuuk. Polit. Pozn., s. Budownictwo 31, 203-213 (1988) (in Polish). 17. M. Kuczma, Variational formulation of the contact problem a plate on a viscoelastic foundation. Zesr. Nauk. Polit. Pozn., s. Budownictwo 31, 73-84 (1988) (in Polish).
APPENDIX 1 Let the increment of deflection in the point i of the plate coinciding with the characteristic point i of the foundation be given. The foundation state satisfying the conditions of unilaterality and plastieization of foundation is to be determined. Here appear a series of cases which will be successively described. For each case the conditions of its identification, formulas for increments of variables of the foundation state, as well as formulas for increments of the coordinates of the characteristic points 1, 2 and 3 shown in Fig. 2 have been presented. The symbols d,, tip,k, h, t’, , d,, (lz,ti,, &, denote the elastic and plastic part of increment of the displacement of the foundation, the temporary foundation modulus, the diagonal element of matrix H associated with point i and increments of the coordinates of characteristic points 1,2,3 shown in Fig. 2.
650
R. LEWANDOWSKI and R. SWITKA
Fig. Al. Incremental constitutive relation-case
I.
Fig. A2. Incremental constitutive relation--case
2.
Fig. AS. Incremental constitutive relation--case
6.
Fig. A6. Incremental constitutive relation-case
7.
Case 2. Decay of ‘gap’ and foundation secondary loaded (Fig. A2) u, #O,
g= -g,
ti >g,
++< uz+g -u,,
d = dl = I+ + g,
c&=0,
g # 0,
Q = k,ii,
hk = 4/G,
c, = ir, = d, = & = Q, = 0.
v
(W
Case 3. Decay of ‘gap’ and foundation secondary loaded (Fig. A3) Fig. A3. Incremental constitutive relation-cases
3 and 4.
g # 0, g=-g,
g + VI -
u, c ti < g + Dj- 0,)
d=ti+g,
cj=k,(u,-u,)+k,(u,+ti-u2),
ri,=cj/k,, ti, = 4,
u, # 0,
$,=ti
d, = ii - 02 + u, ,
-d,,
hk =4/G,
&=q+cj-q2,
I&=&=0. (A3)
Case 4. Arise of ‘gap’ (Fig. A2) q # 0,
Fig. A4. Incremental constitutive relation-se
5.
4 = -4,
ti=ti,=q/k,,
u, + i < 0, $=O,
%=d-GJ,
t;, = 6, = d, = 4* = Q3= 0. Case 1. De-cay of ‘gap’ and foundation
hk=tj/G, (A4)
primary loaded
(Fig. Al)
Case 5. Loading of foundation (Fig. A4) u, = 0,
g=-d,
i > g, ti=G+g, . . up=u-ul.
. u, = up,
(j* = 4,
ti
g 20,
q#O,
u,>v+izoj+,
ti,=q/k,,
g=o,
ti=ti,
hk =cj/ti, ti, =ti,
d, = q, =o.
cj =k,&-u)+k,(u vp=u
(AlI
ti, = I$,
forj=2
.
-u,,
.
tij=u+ti-u,,
andu,#v,, +ti -u,),
h = 1,
110, tis=cj/kj,
k =4/G,
cj,=q+cj--q,.
(AS)
Unilateral plate contact
651
Cease 6. Loading of foundation (Fig. AS) qzo, g=O,
vz>v,
ir=G,
v+i>v,,
v,#v,,
tiJ>o,
q=k,(v,-v)+k,(v,-v,)+k,(v
d,=cj/k,,
dp=ti -d,,
+ti -vj),
h = 1,
k =4/C,,
1 -
ti, =d p,
d,=v
+ti -V*,
4*=q+4-q2,
d,=v
(‘46)
ti>O,
v,+i
g=O,
v=$,
v,=v-v,,
forj=2
Case 8. Secondary loaded foundation unloading (Fig. A7)
andv,#v,,
cj=k,ti,
t&=4/k,,
k=k,,
h=l,
V
I
Fig. A7. Incremental constitutive relation-+ase
Case 7. Primary loaded foundation (Fig. A6) q#O,
ti
g=O,
v=d,=i,
qzo, cj=k,d,
4, = ti, = 0
4 =
c,
4, = 4,
for j = 2.
(A7)
8.
and foundation
v+ti-v,>o, tip=O,
d, = v, = v, = 4* = 4, = 0.
ti, = tip,
’
A. /y&/J
+ti -v,,
9,=4+4-q,.
kJ
k=k,,
h=l, (A@
The foundation state after load increment is obtained by adding the increments of the values characterizing foundation to their former ones.
0045.7949191 $3.00 + 0.00 Q 1991 Pergamon Press
199t
UNILATERAL
ELASTIC-PLASTIC
plc
PLATE CONTACT
WINKLER-TYPE
WITH THE
FOUNDATION
R. LEWANDOWSKIand R. SWITKA Technical University of Poznan, ul. Piotrowo 5, 60-965 Poznati, Poland (Received 11April 1990)
Abstract-In this work contact problems of a plate with the elastic-plastic Winkler-type foundation has been solved. An unilateral character of bonds between plate and foundation has been taken into consideration. The effect of friction forces in the contact plane has been neglected and an incremen~l approach has been applied. An incremental problem has been fo~ulat~ in a variational manner and solved by use of the linite element method. The results of numerical calculations for rectangular plates subjected to the simple and complex load processes have been presented.
1. INTRODUCTION
The contact problem of a construction with a deformable foundation is an essential analysis task in civil engineering. The ground on which a building structure is located exhibits the features of viscoelastic-plastic materials. Moreover it is possible loss of contact between structure and foundation and friction forces occur in the contact plane. Taking all ground features into consideration simultaneously and conditions of unilateral contact is a very difficult task and at present only the solutions of chosen problems are known. The analysis of behaviour of plates on the different types of elastic foundations, considered as a contact in problem with unilateral bonds, appears Refs [l-l I]. In these papers friction force in the contact plane has been neglected. For solutions the analytical methods [l-3], the finite element method [4-91 or boundary element method [10-l l] have been applied. The bending problem of the plates resting upon a visco-elastic foundation as a rule was considered as a contact problem with bilateral bonds [12-151. Recently work [16] has been published in which the conditions of unilateral contact of a circular plate subjected to bending and resting upon a visco-elastic Winkler-type foundation are taken into account. Also Ref. [17] considers the conditions of unilateral contact and gives a general variational formulation for plates resting upon a visco-elastic foundation. In Ref. [I81 a contact probiem of a beam with a fo~dation of nonlinear, physical properties was analysed. The solution was obtained with help of the finite difference method and the conditions of unilaterality of bonds were observed by the use of an iterative approach. The static problem of a beam on an elastic-plastic Winkler-type foundation was dealt with in Ref. [19]. An incremental approach enabling an analysis of the simple and complex load processes CAS 39,bE
was applied. For the solution the finite element method was used and the area of contact and plastic zone were determined by an iterative method. Moreover, the process of reaching the limit load by the Winkler-type foundation mating with a beam loaded with a concentrated force was investigated by analytical methods in Ref. [20]. Static analysis of an elastic-plastic foundation under a rigid plate loaded by a wheel of a passing vehicle was the theme of the work [21]. The problem was solved in an analytical manner. The foundation state was analysed in successive phases of wheel passing and adaptation of foundation subjected to multiple passages. The basis of the finite elements method used to analyse contact problems is built on the variational inequalities method. Lagrange multipliers, the penalty method or mathematical programing methods are employed as the fundamental numerical techniques. They are described in Refs [22-241. The literature honeying the contact problems of another kind are not discussed here. A comprehensive discussion of these problems as well as the review of corresponding literature is contained in Ref. [25] In this work a numerical method of static analysis of elastic plate resting upon an elastic-plastic foundation is presented. It has been assumed that the loading process is a complex one and consists of an optional number of the simple processes. Moreover, it is assumed that the load increments are uniparametric. To the description of plate behaviour both classical thin and thick plate theory has been applied. The uniparameter Winkler-type foundation, taking into account plastic defo~ation and reinforcement, has been adopted. Moreover, the unilaterai character of bonds between plate and foundation are considered and friction forces are neglected. The accepted Winkler model is the simplest one and it may arouse reservations as to its adequacy. However, it should be taken into account that while considering the elastic-plastic states of foundation it 641
642
R.
LEWANDOWSKIand
is impossible to make use of Green function and superposition principle. Such restrictions cause the application of more complex models, e.g. a biparameter model (Vlasov, Pasternak) or elastic-plastic half space to involve great difficulties and the enlargement of the task itself. For the foundation area discretization becomes necessary and that, in the course of the plastic analysis, requires the division into a large number of finite elements. The difficulties in the application of the general yield condition in foundation area are also of importance. The yield condition on the compressed edge area of a beam or a plate ceases to be univocal due to the peculiarities taking place in the stress state of the foundation. These serious and strenous problems do not appear in the Winkler model. Unilateral contact constitutes the domineering feature, especially for flexible plates exposed to the activity of strongly local loads. Unilateral bonds also may become evident in case of eccentrically loaded thick plates or in case of plastic deformations appearing in the elastic-plastic foundation due to variable loads. The present paper enables the analysis of all the above cases. For the solution of the problem the incremental approach has been applied. The incremental task has been formulated in a variational manner including unilateral constrains into functional. For a numerical techniques the finite elements method connected with iterative procedure of determination contact and plastic zones has been applied. The results of calculation of the plates subjected to simple as well as to complex load processes are presented.
2. PROBLEM FORMULATION
2.1. Model of foundation Figure 1 illustrates the behaviour of the discussed foundation model, where q, v, ki denote reaction of foundation, displacement and foundation coefficient, respectively. In the process of primary load the foundation coefficient is equal k, , and after reaching by the reaction of the value q. is equal k2, where k, > k, > 0. The foundation reaction q. hereafter will be called the yield limit. In the process of unloading, foundation deflection increments are purely elastic
R.
SWITKA
9
I
Fig. 1. Constitutive relation for foundation,
and the foundation
coefficient has the value k,, but
k, 2 k, .
In the proposed model the plastic deformations appear just from the beginning of the loading process if k, > k,. For k, = k, the foundation receives, in some interval, purely elastic deformations. At the repeated loading foundation deflection increments are also elastic, until reaching by the reaction of a maximum value which appeared at this point of foundation in the past. During further loading processes the foundation behaves in the same way as during primary loading one. The proposed foundation model describes approximately the behaviour of some kinds of nonirrigated grounds. One can here refer to the experimental results of Allen et al. [26] as well as Aljeksjejew [27] relating to dynamic characteristics of grounds at uniaxial compression. From these investigations it clearly results that the adopted foundation model approximates well enough the experimental curve for nonirrigated sands and argillaceous grounds in the range of stresses l-2 MPa, as well as the unloading modulus can be greater than the modulus of primary loading. On a base of the above experiments, in the work [28] a series of different idealizations of the relationship ot has been proposed and used for description of the propagation of waves in ground. A diagram of the relationship reactiondisplacement of foundation, which was already loaded in the past, can display the shape shown in Fig. 2a or the one shown in Fig. 2b.
tq
Fig. 2. Constitutive relation for foundation loaded in the past.
643
Unilateral plate contact The maximum reaction which in the past took place at a given point of foundation has been designated by q2, while v2 represents the displacement corresponding to this reaction. In the first case we have q2 < q. and the diagram consists of three straight segments and for its unique determination are the coordinates qi, vi, i = 1,2,3 of the three characteristic points on a plane q-v (compare with Fig. 2a). The coordinates of point 3 equal q3 = qo, vj = vg. If the foundation in the past was not loaded then points 1 and 2 coincided with the origin of the coordinate system. In the course of loading of foundation the point 2 shifts along the primary path of loading. The coordinate u, of the point 1 determines plastic displacement of foundation. In the second case q2 2 q,,, point 2 coincides with point 3 and their coordinates are equal to q2 = q3, v2 = vj. The diagram q-v consists now of two straight segments (compare with Fig. 2b). In the process of unloading, the position of the discussed points does not change. Let us assume that k, = k,. In such a case the plastic deformations of the foundation will accrue, if d >o,
The same equations expressed by the generalized displacements increments can be presented in the following manner:
D
F’d,+~(6&&2)] t +GhK(G*,-
i- Ghrc(ti,, - b2) = 0.
2.3. Unilateral condition in the contact plane The conditions of unilateral contact can be written in a form
40
for
q2 G cl0
q+o20,
q2
for
q2 >
g+gao,
40
where ( * ) denotes the increment of quantity over which it appears. In the case of foundation for which k, > k, the above relationships take the form ti >o,
B =q2-
An incremental constitutive law for the considered foundation model have the form 4 = k(v)d
(1)
where k(v) is temporary modulus of foundation (compare with Fig. 1). The temporary modulus of foundation is determined according to the concept of secant modulus what has been shown in Fig. I. 2.2. IncremePttal equations of the thin and thick plates Consideration has been limited to the isotropic plates subjected to loads inducing small displacements and strains. Plate material is linear and elastic. From the Kirchhoff-Love theory results the following incremental equations of a thin plate DV4+ = p - 4,
(2)
(3)
where G, h, & denote Kirchhoff’s modulus, the plate thickness and the rotation angles of a straight line normal to the plate central plane with regard to axis a and K = 5/6.
4=
i
4,)= 0,
(q+4)(s+&?)=O,
(4)
where g = v - w denotes the ‘gap’ between plate and foundation. This means that the foundation can be only compressed; plate and foundation cannot mutually penetrate and that passive pressure q and ‘gap’ cannot simultaneously differ from zero at one and the same point of foundation. A complete set of relations describing the discussed problem can be obtained if the boundary conditions of the plate are joined to relations (l)-(4) and the state of system at the beginning of loading process is given. 3. VARIATIONAL FORMULATION OF AN INCREMENTAL PROBLEM
It is very difficult to solve the discussed contact problem in a direct manner. Below, it will be formulated in a variational manner, more suitable for a numerical analysis. The variational problem can be written as follows: minimize the functional
8,)8,)
I(@ i], 8,, ti, A) = Z,(& r
where D, i, 9, denote the flexural rigidity of plate, the increment of deflection of the plate and the increment of external plate load, respectively. For the description of the behaviour of a thick plate the Mindlin plate theory has been applied. Brietly, the fundamental equations of this theory were derived in Ref. [29J.
-
/J
+@)(w
+*)dR
J
-
s
I&-l-d-i)dR
f-I
n [q + $(v)d Jti dC?.
+
s
(5)
R. LEWANDOWSKI
444
satisfying the conditions
and R.
SWITKA
for a thick plate
g+ti-i&O
Gh~[v*(w + 6) - (@,,,+ &) - (82.2+
=q+g--p-p,
in the area of the plate a. The above Z,(+, t$, &) denotes the strain energy of the plate, which can be determined from the relations: for a thin plate
I,($, 8,, 4,) = Z,(kiJ)= ;
d2,2,1
([V(w + tiJ)]2 s It
D V2(& + 4,) + ~(S,,,;+i [
L w - v)[(w + k),,,(w + i)J,
I.12- 8 2.11 -
+Gh~(w,+~,~-6,-8,)=0. - [(w + ~t>,,,l”t>do,
* 4.1,
1 (11)
(6) From the condition (8,) the boundary conditions for the plate are also obtained, but they are not expressed directly here. Taking into account that before load increment the equations of plate are also satisfied, we come to the incremental equations given by relations (2) and (3). Namely both formulations are equivalent.
for the Mindlin plate
4. SOLUTION METHOD
+(w,, + Gii - 0, -
ti,)‘]da.
(7)
The variational formulation is equivalent to the direct approach discussed above. In order to demonstrate it the stationary conditions of functional (5) will be given. From a generalized Kuhn-Tucker theorem [30] the following necessary conditions
(q + kti - A)& dsZ = 0
+
(8)
1n are obtained. From the second integral at (8,) it results that the function of Lagrange multipliers equals the function of passive pressure I=q+kt;=q+cj.
The condition
(9)
(82) is identical with the condition
(4,), and the condition (8,) is equivalent to condition
(43). Taking into account eqn (9) in (8,) and after calculation of H,(ti, 8,, t?,) we obtain the following equations: for a thin plate DV4(W+*)=p+@-q-4
(10)
Summarizing the successive solutions of incremental problems we obtain a solution for an assumed loading process. Such a procedure is necessary, because the description of bonds between plate and foundation as well as plastic properties of foundation is made with the help of nonlinear equations and inequalities. Moreover, the accretion process of the plastic deformations depends on load history and for the accurate description of this process is essential incremental analysis. It is assumed that in a typical incremental approach the changes of loading are proportional to one parameter. An approximate solution of an incremental problem has been obtained by application of the finite element method. Plate area n is being divided in finite elements and out of generalized displacements of nodes a vector U with dimension n x 1 is being formed. Numerical analysis of thin plates has been made using the finite element developed by Bogner et al. [31]. It is compatible, the four-nodal element having 16 degrees of freedom. For the analysis of thick plate the element described in Ref. [32] has been applied. This element is perfectly suitable for discretization of a plate resting on a foundation or elastic half space as it has been demonstrated in Ref. [8]. It is a nine-nodal efement with 26 degrees of freedom. In the foundation some characteristic points are chosen. At these points the contact conditions are verified and the foundation state is determined. The vectors V and Q with dimensions p x 1 created of displacements and foundation reaction and determined in characte~stic points are introduced. With the discussed points some foundation field located in their surroundings is associated. The area of the field associated with the point i of foundation is being marked by si. Typical division of a plate on
Unilateral plate contact
645
points of foundation, G = V - W is the ‘gap’ vector and S denotes the diagonal matrix, whose elements Sii are equal to the foundation area s, associated with the ith characteristic point. The above notation of functional (14) is possible if to the numerical calculation of integrals in relation (5) a complex rectangle method is applied and the characteristic points of foundations chosen as the points of numerical integration. Considering the equation of equilibrium of the plate before the increment of load Lxl2m
KpU + Q = P,
9.6m
Fig. 3. Example l-geometry of the plate, finite element division and distribution of characteristic points.
finite elements and typical choosing of characteristic points within one element has been shown in Fig. 3. Let us denote by W a vector with dimensions p x 1 of the displacements of points of the plate which coincide with the characteristic points of foundation. These displacements and their increments can be determined on the basis of displacements and displacements increments of the nodes using the relations W=AU,
W=AU,
of Lagrange
A=Q+Q,
(17)
the stationary condition of the functional (14) can be written as K&J + A’SK,,,V = P, G+V-W>O, Q+K,V>O, (G + V - W)‘S(Q + K,V) = 0.
(12)
where A is a matrix of dimension p x n whose elements can be determined on the basis of the shape functions applied for description of plate element. Relation (I) written in a matrix notation has the form
t) = K,V,
and taking into account interpretation multipliers
(16)
(18)
A characteristic feature of the unilateral conditions (1 8*) and (18X) is that they are a series of independent
inequalities. If conditions (18,)-(18,) can be satisfied then the following relation between increments of plate deflection and increments of foundation reaction holds
(13) V=HW,
where Q and V denote the vector of reaction increment and the vector of displacements increment in the characteristic points of foundation, K, is the diagonal matrix whose elements k, are coefficients of the foundation in those points. After introducing the above discretization, the variational formulation of the problem can be expressed in the following manner: minimize the functional
where H is the diagonal matrix whose elements depend on the state of the system after increment of load. Elements h, are equal h,= 1
when gi = 0 and gi = 0,
hii = 0
when qi = 0 and di = 0,
h,i = tii/tii
when g,=O and g,#O
I&J, v,A) = f (U + U)‘K,(U + U)
or
when qi = 0 and di # 0.
-(P+P)‘(U+U)-a’S(G+V-W) + Q’S%’+ ;V 'SK, V’,
(19)
(14)
Substituting eqn (19) into (18,) and taking into account (12) we obtain the nonlinear system of equations in a form
such that G+ir-VVZO.
(15)
Above, K, denotes the global plate stiffness matrix, ( )’ vector transposition, P is the vector of nodal plate loads, A is the vector whose elements are Lagrange - multinliers determined in the characteristic
K($J, V)U = P,
(20)
K$J, V) = Kp + A’SK,HA.
(21)
where
Equation (20) has a structure typical for the finite element method equations. Additionally the diagonal
646
R.
LEWANDOWSKIand
elements of matrices S, K,, H as well as elements of the matrix A have to be stored. The solution of eqn (20) has been obtained by the iterative procedure described below. It is assumed, that for a given load increment P, the approximations of vectors *, 0, * and matrix Km, denoted by vi, Uj, tij. K, where i means a number of iteration are known. As first approximation f7, = 0, = 0 were adopted and the elements of matrix Km, are determined on the basis of foundation state before load increment and the elements of matrix H, are to be taken hii = 1 when gi = 0 or h, = 0 when qi = 0. From assumption these approximations satisfy the unilateral conditions and relations (12) and (13). On their basis the elements of matrix Hi and Ki and from
where c1 and Q are the admissible errors of calculation and II - )I denotes an Euclidean norm. CALCULATIONS
Using the above method a computer program for the rectangular plates resting on an elastic-plastic foundation has been performed. Some results for chosen examples were presented below. Example 1. The thin, symmetricallyloaded plate Calculations have been done for a rectangular plate which have the following dimensions a = 9.6m,
Ip$MPcfJ
Fig. 4. Example b-deflection
aJ
.I-‘--‘_’
-I-‘--
‘-7
(22)
the new approximation of the vector U can be obtained. Now, one used the above model of foundation and the unilateral conditions the vector gj+l and the matrix Kmj+l can be determined. Some cases appear here and the detailed relations are given in Appendix 1. The iterative process are repeated until the following inequalities are satisfied
OF NUMERlCAL
SWITKA
bJ
I$+, =K,+,
5. RESULTS
R.
of the plate ccntre vs load.
Fig. 5 Example l-limits of contact and plastic zone for various values of load.
b = 7.2 m, h = 0.2 m. The remaining data characterizing the plate and foundation are E = 23 GPa. v = O.l7,k, = 20.0 MPa, k3 = 40.0 MPa, k2 = 0.1 MPa, g0 = 0.2 MPa. Plate division in elements and location of characteristic points within one element has been shown in Fig. 3. Because symmetry of plate and load, the calculations have been performed for a quarter of plate only. It has been assumed that the foundation is subjected to a load for a first time. The whole plate was loaded by a uniformly distributed load equal 0.004 MPa (a proper weight). Next, the centre of the plate (the dashed area in Fig. 3) has been loaded with an additional, unifo~Iy distributed load. The load increment has been chosen equal p = 0.01 MPa, and the calculation accuracy c, = 62= 0.01. Results of the calculation are presented in Figs 4 and 5. Figure 4 presents the relationship between the deflection of plate centre and load p. The passive pressure exceeded conventional yield limit q0 when the load at the centre reached the value pI = 0.37 MPa. Here a signifi~nt effect of unilaterality of bonds and plastic properties of foundation on the system behaviour is visible. The action of an additional load in the centre of the plate has caused a tear-off of a significant part of the plate from foundation, Next a zone appeared in which the foundation reaction was greater than the conventional yield point of the foundation. This zone expanded together with load increment, but the whole contact zone expanded insignificantly. In this
647
Unilateral plate contact
L
bxl.Zm
I
Fig. 8. Example 3-geometry of the plate.
Fig. 6. Example 2-deflection of the plate centre vs load. way a narrow zone was formed in which the foundation reaction was less than qQ.Further loading of the plate caused the shifting of contact boundary towards plate boundaries and further narrowing of zone in which q < qo. In Fig. 5 the limits of discussed zones for different values of p are shown. In the discussed example, a tearing-off zone remained until the end of the loading process, nevertheless the plate suffered large displacements. Since the description of the plate hehaviour was applied the geometrically linear theory, consequently the results for large loads probably have some errors. Example 2. Thick plate symmetricallyloaded The calculation for a plate with height h = 1.2m, with other data remaining the same as in example 1, have also been performed. At the beginning the plate was loaded with a load uniformly distributed on whole surface of the plate equal to 0.024 MPa and an additional load in the same manner for the thin plate was applied. During the whole process of loading the plate was in contact with foundation. In Fig. 6 the diagram of dependence of deflection of the plate centre on load acting in the centre is shown. The limits of the zone in which q < q. for different load values are shown in Fig. 7. In comparing these diagrams it becomes evident that, in this example, the foundation state ._.__-.-
--
’
‘_..-_‘-_. li”.~.-.-._. . I:1
very near to the limit is associated deflections of the plate.
with small
Example 3. Thin plate loaded asymmetrically Successive examples represent calculation results of a plate subjected to an asymmetrically distributed load in relation to one of the symmetry axes of the plate. The plate with dimensions in projection: a = 4.8 m, b = 3.6 m, h = 0.4 m has been used for calculations. The remaining data relating to plate and foundation are the same as in example 1, Projection of the examined plate is presented in Fig. 8. The ~l~iations were based on the thick plate theory. Upon loading the plate with a uniformly distributed load on the whole plate surface of intensity 0.008 MPa an additional Ioad has been applied on the surface corresponding to the dashed area shown in Fig. 8. Figure 9 presents a diagram between the deflection of the point 1w, and density of load acting on the dashed area p. In Fig. 10 the limit of the contact zone and the zone in which q r q. is shown for a load p = 1.008 MPa. Under influence of an applied load, in a significant part of foundation the reaction exceeded qo, a narrow zone in which q < q@has been formed, and the plate itself suffered &lPa)
0.6
f
I -t i I
I I I
1 ‘.
’ ‘..
I
I
I
j.
I
1 Fig. 7. Example 2-limits of plastic zone for various values of load.
8 5
I 10
Fig. 9. Example 3deflection
I IS
, w km)
20
of a point 1 vs load.
648
R. ~w~~ws~
and R.
~SWITKA
“__-~_+_f____y__
~~1~~Pa 1 ,f .“.II-_ confacf zone
I
Itt ii I
I
1 -
-I 1 ---
1 I t
---
t
3
Fig. 10. Example 3-limits
f i
I I
1
I
Fig. 13. Example 4Aimits
of contact and plastic zone, loaded element 3.
of contact and plastic zone for
p = !.OMPa.
--
j__l 1
---
-
-I - f
.
ut
_ITconf&f 1 :I 0 , \zone -, - ? _.fy.. - _ I I
Fig. 14. Example +--limits of contact and plastic zone, loaded element 4. Fig. Il. Example 4-limits of contact and pla&c zone, loaded element 1.
.--
_; ------
I
;--
1 I
.-- -c___;___&--I i
,c-+
A.,
I I .+--+,confucf gone
i
l
Fig. 15 Example 4-limits of contact zones, unloaded plate. Fig. 12. Example 4-limits of ccmtact and plastic zone, loaded element 2.
large displacements, which consist before all in the movement of plate as a rigid body. Example 4. A thin plate loaded by a slowiy moving load A plate of height h = 0.1 m was analysed. The remaining data concerning the plate and foundation were as in example 3. First of all the plate was loaded with a ~n~fo~~y distributed load of the intensity of 0.002 MPa (dead load of the plate). Next the finite element No. 1 was loaded. The additional load was increasing from zero to the value p = 0.5 MPa. The load increment was assumed to be $ = 0.01 MFa. Figure 11 presents the contact zone range and the range in which q > qafor the load p = 0.5 MPa. Next the load of the discussed element was decreased, increasing at the same time the load of the neighbour-
ing element designated by 2 in Fig. 12. This figure also shows the contact zone range and the yielding zone range (4 > go) which appear in the founda~o~ svhen the additional load reaches p = 0.5 MPa. The datted line, as in the previous examples, shows the plasticity zone boundary. This process was repeated twice, moving the load in succession on the elements designated by 3 and 4. The boundaries of the contact zone are shown in Figs 13 and 14, They refer to loading a corresponding plate element by additional load p = 0.5 MPa. Figure 15 shows contact zone boundaries after the additional load is completely eliminated. 6. FINAL REMARKS This paper constitutes one of the scarce attempts of taking into consideration plastic phenomena in the contact problem with unilateral bonds. The task concerning plates on the elastic-plastic foundation has been solved effectively with the application of a
649
Unilateral plate contact relatively simple incremental procedure. However, this refers to the simplest foundation model and the simplest model of the elastic-plastic body with linear hardening, it affords possibilities for the quality analysis of the behaviour of plates losing contact with the plastically deformed foundation. Moreover, the analysis of the behaviour of the foundation under a plate exposed to a slowly moving load has been conducted. Acknowledgement-This paper was supported by the Polish Academy of Sciences by the CPBP OZ.01Research Program. REFERENCES 1. Y. Weitsman, On the unbonded contact between plates and elastic half space. J. Appl. Mech., Trans. ASME 36,
198-202 (1969).
2. G. M. L. Gladwell and K. R. P. Iyer, Unbonded contact 3. 4. 5, 6. 7. 8.
9.
between a circular plate and a elastic half-space. J. Elast. 4, 115-130 (1974). H. Jung, Druckverteilung unter elastisch gelagerten Kreisplatten. Ing. A&iv 20, 8-12 (1952). Y. K. Cheung and D. K. Nag, Plates and beams on elastic foundations-linear and non-linear behaviour. Geotechnique 18, 250-260 (1968). 0, J. Svec, The unbonded contact problem of a plate on the elastic half space. Camp. method Appl. Mech. Engng 3, 105-I 13 (1974). L. A&one and A. Grimaldi, Unilateral contact between a plate and an elastic foundation. Meccnnicu 19, 223-233 (1984). G. Cyrok and R. Switka, A finite element analysis of plates resting upon elastic half-space, Arch. Ini &d. 29, 381-394 (1983) (in Polish). R. K. N. D. Rajapakse and A. P. S. Selvadurai, On the performance of Mindlin plate elements in modeling plate--elastic medium interaction: a comparative study. Int. J. Numer. Methods Engng 23, 1229-1244 (1986). H. Li and J. P. Dempsey, Unbonded contact of a square plate on an elastic half-space or a Winkler foundation. J. Appf. Mech., Trans. ASME 55,43&436 (1988).
10. J. Puttonen
and P. Varpasuo, Boundary element analysis of a plate on elastic foundations. Int. J. Numer. Methods Engng 23, 287-303 (1986).
11. G. Bezine, A new boundary
element method for bending of plates on elastic foundations. Int. J. Solids Struct.-24, 557-565 (1988). 12. K. Pister and M. L. Williams. Bendine of nlates on a viscoelastic foundation. J. Engng Mech. Diu., Proc. AXE 86, 3144 (1960). 13. E. L. Marvin, Viscoelastic plate on poroelastic foundation. J. Enana Mech. Div.. Proc. ASCE 98. 9I1-927 (1972). - 14. J. Mtjczka and G. Szefer, Plates on a consolidating space. Arch. In?. Lgd. 25, 21 l-229 (1979) (in Polish). 15. K. Sonda, H. Kobayashi and T. Ishio, Circular plates on linear viscoelastic foundations. .J. Engng Mech: Div., 1
Proc. AXE
18. F. W. Beau&t and P. W. Hoadley, Analysis of elastic beams on nonlinear foundations. Comput. Struct. 12, 669-676 (1980).
19. R. Lewandowski and R. Switka, Bending of a beam
on the elastic-plastic Winkler foundation with unilateral constrains. Arch. Ini. I&d. 34, 35-52 (1988) (in Polish). 20. G. Ganowicz, R. Lewandowski, R. Switka and M. k;adzihska-Depko, Analysis of a beam on an elastic-plastic foundation with unilateral constraints. Zest. Nauk. Polit. Pozn., s. Budownictwo 31, 23-32 (1988) (in Polish). Elastic-plastic response of 21. M. zadziriska-Depko, subgrade under a rigid slab with unilateral constraints and subjected to a moving load. Arch. Ini. I&Id. 32, 569-584 (1986) (in Polish). 22. T. R. J. Hughes, R. L. Taylor, J. L. Sackman, A. Cumier and W. Kanoknukulchai, A finite element method for a class of contact-impact problems. Comp. Meth. Applied Mech. Engng 8, 249-276 (1976). 23. N. Rikuchi and J. T. Oden, Contact problems in elastostatics. In Finite Elements; Special Problems in Solid Mechanics. Vol. 4 (Edited bv J. T. Oden and G. F. Carey). Prentice-Hall, Engiewood Cliffs, NJ (1984). Inequality Problems in 24. P. D. Panagiotopoulos, Mechanics and Applications. Birkhauser, Base1 (1985). 25. J. J. Telega, Variational methods in contact problems of mechani&. Adv. Mech. 10, 3-96 (1987) (in-Russian). 26. W. A. Allen. F. B. Mavfield and M. L. Morrison, Dynamics oi a project& penetrating sand. J. Appf. Phys. 3, (1957). 27. N. A. Aljeksjejew, Mietod oprjedelijenija dinamiczeskich charakteristik g~ntow pri bolszich dawlenijach. Dinamika gruntow, Sb. No. 44, Moskwa (1961) (in Russian). 28. E. Whdarczyk, Fale plastycze, w: Mechanika Techniczna. t. 3 Drgania i fale. PWN, Warszawa (1986). 29. T. J. R. Hughes, M. Cohen and M. Haroun, Reduced and selective integration techniques in the finite element analysis of plates. Nuclear Engng Design 46, 203-222 (1978).
30. D. G. Luenberger, Teoria optymalizacji. PWN, Warszawa (1974). 31. F. K. Bogner, F. L. Fox and L. A. Schmidt, The generation of inter element-compatible stiffness and mass matrices by the use of inte~olation formulae. Prof. Co& Matrix method in Struct. Mech. Air Force Institute of Technology, Wright Patterson A.F. Base, Ohio, October (1965). 32. T. J. R. Hughes and M. Cohen, The ‘heterosis’ finite element for plate bending. Comput. Struct. 9, 445-450 (1978).
1
104, 819-827 (1978).
16. R. Switka, Axially-symmetrical bending of a circular plate on a viscoelastic foundation under the conditions of an incomplete contact. Zest. Nuuk. Polit. Pozn., s. Budownictwo 31, 203-213 (1988) (in Polish). 17. M. Kuczma, Variational formulation of the contact problem a plate on a viscoelastic foundation. Zesr. Nauk. Polit. Pozn., s. Budownictwo 31, 73-84 (1988) (in Polish).
APPENDIX 1 Let the increment of deflection in the point i of the plate coinciding with the characteristic point i of the foundation be given. The foundation state satisfying the conditions of unilaterality and plastieization of foundation is to be determined. Here appear a series of cases which will be successively described. For each case the conditions of its identification, formulas for increments of variables of the foundation state, as well as formulas for increments of the coordinates of the characteristic points 1, 2 and 3 shown in Fig. 2 have been presented. The symbols d,, tip,k, h, t’, , d,, (lz,ti,, &, denote the elastic and plastic part of increment of the displacement of the foundation, the temporary foundation modulus, the diagonal element of matrix H associated with point i and increments of the coordinates of characteristic points 1,2,3 shown in Fig. 2.
650
R. LEWANDOWSKI and R. SWITKA
Fig. Al. Incremental constitutive relation-case
I.
Fig. A2. Incremental constitutive relation--case
2.
Fig. AS. Incremental constitutive relation--case
6.
Fig. A6. Incremental constitutive relation-case
7.
Case 2. Decay of ‘gap’ and foundation secondary loaded (Fig. A2) u, #O,
g= -g,
ti >g,
++< uz+g -u,,
d = dl = I+ + g,
c&=0,
g # 0,
Q = k,ii,
hk = 4/G,
c, = ir, = d, = & = Q, = 0.
v
(W
Case 3. Decay of ‘gap’ and foundation secondary loaded (Fig. A3) Fig. A3. Incremental constitutive relation-cases
3 and 4.
g # 0, g=-g,
g + VI -
u, c ti < g + Dj- 0,)
d=ti+g,
cj=k,(u,-u,)+k,(u,+ti-u2),
ri,=cj/k,, ti, = 4,
u, # 0,
$,=ti
d, = ii - 02 + u, ,
-d,,
hk =4/G,
&=q+cj-q2,
I&=&=0. (A3)
Case 4. Arise of ‘gap’ (Fig. A2) q # 0,
Fig. A4. Incremental constitutive relation-se
5.
4 = -4,
ti=ti,=q/k,,
u, + i < 0, $=O,
%=d-GJ,
t;, = 6, = d, = 4* = Q3= 0. Case 1. De-cay of ‘gap’ and foundation
hk=tj/G, (A4)
primary loaded
(Fig. Al)
Case 5. Loading of foundation (Fig. A4) u, = 0,
g=-d,
i > g, ti=G+g, . . up=u-ul.
. u, = up,
(j* = 4,
ti
g 20,
q#O,
u,>v+izoj+,
ti,=q/k,,
g=o,
ti=ti,
hk =cj/ti, ti, =ti,
d, = q, =o.
cj =k,&-u)+k,(u vp=u
(AlI
ti, = I$,
forj=2
.
-u,,
.
tij=u+ti-u,,
andu,#v,, +ti -u,),
h = 1,
110, tis=cj/kj,
k =4/G,
cj,=q+cj--q,.
(AS)
Unilateral plate contact
651
Cease 6. Loading of foundation (Fig. AS) qzo, g=O,
vz>v,
ir=G,
v+i>v,,
v,#v,,
tiJ>o,
q=k,(v,-v)+k,(v,-v,)+k,(v
d,=cj/k,,
dp=ti -d,,
+ti -vj),
h = 1,
k =4/C,,
1 -
ti, =d p,
d,=v
+ti -V*,
4*=q+4-q2,
d,=v
(‘46)
ti>O,
v,+i
g=O,
v=$,
v,=v-v,,
forj=2
Case 8. Secondary loaded foundation unloading (Fig. A7)
andv,#v,,
cj=k,ti,
t&=4/k,,
k=k,,
h=l,
V
I
Fig. A7. Incremental constitutive relation-+ase
Case 7. Primary loaded foundation (Fig. A6) q#O,
ti
g=O,
v=d,=i,
qzo, cj=k,d,
4, = ti, = 0
4 =
c,
4, = 4,
for j = 2.
(A7)
8.
and foundation
v+ti-v,>o, tip=O,
d, = v, = v, = 4* = 4, = 0.
ti, = tip,
’
A. /y&/J
+ti -v,,
9,=4+4-q,.
kJ
k=k,,
h=l, (A@
The foundation state after load increment is obtained by adding the increments of the values characterizing foundation to their former ones.