WINKLER PLATES BY THE BOUNDARY KNOT METHOD

Transcription

1 WINKLER PLATES BY THE BOUNARY KNOT ETHO Sofía Roble, Berardi Sensale, Facultad de Ingeniería, Julio Herrera y Reissig 565, ontevideo Abstract. Tis paper describes te application of te Boundary Knot etod (BK) in te solution of problems regarding Kircoff plates resting on a Winkler foundation. Tis is te first time te BK is applied to tis kind of problems. Te BK is a boundary, mesless and integration-free metod. To solve problems of plates resting on a Winkler foundation, te inomogeneous term is approximated considering knon particular solutions. Te Kelvin and te modified Kelvin non-singular functions are used to approximate te omogenous term. Even toug te approac of te BK is similar to te etod of Fundamental Solutions (FS), te BK only uses non-singular general solutions, terefore, te artificial boundary used in FS is not required. Te BK is validated troug its application to solve several problems it different boundary conditions: clamped, simply supported and free edges ere considered. Several cases are discussed and te solutions obtained it te BK are compared to te solutions obtained using te Finite Element etod (FE). Keyords: Winkler plate, Boundary Knot etod. INTROUCTION Boundary metods ave gained importance over te last decades as tey are an alternative to te classical Finite Element etod (FE). In te last decade, te mesless metods ave become relevant, specially te etod of Fundamental Solutions (FS) proposed by Kupradze and Aleksidze (96). Tis metod considers fundamental solutions, i.e., te solutions corresponding to a concentrated source in an infinite domain. Te FS approximates te solution of a omogeneous problem as te sum of several fundamental solutions considering several sources. Te sources are located on an artificial boundary outside te domain. Te FS is an integration-free and mesless metod. Hoever, te use of te artificial boundary outside te domain as some drabacks. Te artificial boundary must be determined. Te source points are usually placed on a omotetic boundary of te pysical one but te omotetic ratio as a significant influence on te final result. It as been proved teoretically, tat orse conditioning (Lyngby (98)) and better accuracy (aton and Jonston (977), Bogomolny (985)) are obtained en te sources are located farter. Te solution of te Kircoff rectangular plate on a Winkler s foundation using te FS as proposed by Wen (988, 989). Te Boundary Knot etod (BK) as developed by Cen and Tanaka (00) as an alternative tecnique. Te BK uses a base of non-singular general solutions to approximate te displacement field. Eac of tese general solutions is defined it respect to a reference point (erein also called source point) ic is placed in te pysical boundary of te problem. In addition, if a collocation sceme is used to satisfy te boundary conditions, te BK does not need integration. Terefore, te BK is an integration-free, mesless metod tat as te advantage to avoid te artificial boundary required by te FS. Using te BK, te solution is obtained after solving a linear system of equations tat is usually ill-conditioned (as ell as using te FS). Te ill-conditioned problem is overcome troug regularization tecniques. Te BK as applied to te solution of Kircoff plates resting on a Winkler foundation. Some examples of a uniformly distributed load considering several boundary conditions are studied. To te autor s knoledge, tis is te first time te BK as applied to solve tis kind of problems. Te obtained results suggest tat te BK is an accurate metod for solving tin plates resting on a Winkler foundation. Tis paper is organized as follos. Te basic formulae are presented in Section. In Section 3, some remarks about te linear system of equations of te BK are analyzed. In Section, te numerical results are discussed. Finally, in Section 5, conclusions are dran.. BASIC EQUATIONS Te BK is described for te classic Kircoff tin plate teory. Tis teory is suitable en te plate is tin and te displacements are small. In tis case, te deflection of an elastic, isotropic and omogeneous, tin plate resting on a Winkler foundation is governed by te equation P ( ) + k P ( ) = qp ( ) P Ω ()

2 ere Δ is te Laplace operator, Ω is an open domain in te to-dimensional Euclidean Space, (P) is te deflection 3 E at a generic point P, q(p) is te transversal load at P, k is te stiffness of te foundation and =, ere is ν te tickness of te plate, E is te Young s modulus, and ν is te Poisson s ratio. ( ) In order to determine te deflection field, boundary conditions must be imposed on eac edge of te plate. Te most frequent boundary conditions on an edge are son in Tab., ere n and t are te outard normal and te tangential directions, respectively. Table. Boundary conditions Clamped edge = 0 θn = = 0 Simply supported edge = 0 mn = + υ = 0 t Free edge 3 3 mn = + υ = 0 v ( ) n = + ν 3 t t = 0.. Application of te BK Consider a plate resting on a Winkler foundation, te plate as te folloing boundary condition P ( ) = f( P) P Γ Ω () ere Γ Ω is te boundary of te domain Ω. Calling λ = k, Eq. () can be reritten as qp ( ) P ( ) + λ P ( ) = (3) Te solution of Eq. (3) can be expressed in te form P ( ) = ( P) + ( P) () Were equation and p p are te omogenous and particular solutions, respectively. Te particular solution satisfies te qp ( ) p( P) + λ p( P) = (5) but not necessarily te boundary conditions. Te omogenous solution satisfies te equations ( P) + λ ( P) = 0 P Ω (6) ( P) = f( P) ( P) P Γ Ω (7) p and can be approximated by a base of general solutions given by W. Cen et al (005).

3 n i 0 i+ 0 i = ( λ λ ) # ( P) = A Ber ( r) + A Bei ( r) (8) ere r is te distance beteen P (te point ere te deflection is approximated) and P i (te source point). rpp (, ) = ( x x) + ( y y) i i i (9) Ber 0 and Bei 0 represent te Kelvin and te modified Kelvin functions of te first kind and zero order. Notice tat tere are no singular terms in te solution, tus, te source points can be placed directly on te boundary. Tis is a clear advantage over te FS because tere is no need for an artificial boundary. As in SF, in te BK te number of source points taken is critical to obtain te required accuracy it a reasonable computational effort. It is son tat an increase of te number of source points improves te accuracy of te approximated solution. Since Eq. (8) satisfies te field equation (Eq. (6)), to solve te problem, it only remains to impose te boundary conditions of Eq. (7). Troug collocation in te boundary, te metod determines te coefficients A i y A i +. Wen finding iger orders, singularities may appear in te general solution but tey can be easily solved. For example, te expression of te slope at point P, placed on te boundary is n # ( ) ( ( ( ) ( ) ) ( ( ) ( ) r r θ P = A i Bei r Ber r A i Bei r Ber r ) n λ λ + λ + + λ λ ) nx + ny (0) i = x y ere n= ( nx, ny) is te outard normal vector at P and, Ber and Bei represent te Kelvin and te modified Kelvin functions of te first kind and order one. Tese functions are not defined at r = 0, but tey ave a finite limit tat can be found troug Taylor series. Similar situation occurs for te bending moments and te sear forces. A uniformly distributed load is considered ere. For tis type of load, te particular solution tat satisfies Eq. (5) is te classical equation given by Timosenko (959) ( ) q p P = () k 3. NUERICAL IPLEENTATION Te implementation of te BK is similar to te implementation of te FS. Were, te source points (N) and field points () must be cosen on te boundary. Te field points can be te same as te source points, but not necessarily. Te only condition tat must be folloed is N. Tere are to coefficients to be determined for eac source point and to equations (boundary conditions) for eac field point. Tus, te linear system of equations to be solved as dimensions N. Te numerical examples so tat tis system is ill-conditioned and its condition number gros en te number of source and field points increase. Figure sos tat te condition number increases strongly en te number of nodes per size becomes larger. Te standard double precision floating point numbers ave a precision of about 6 decimal digits. In te analyzed cases, te condition number is given by Fig., ic sos tat for more tan 5 nodes per side, te solution of te systems can be affected by te precision of te computer. Tus, special care must be taken in te solution of te linear systems. Te truncated singular value decomposition (TSV) is used for tat purpose. Te TSV is a popular metod for computing regularized estimates in ill-posed inverse problems. For te solution of te system a metod proposed by Hansen (007) is used and better results are obtained even toug a larger number of nodes is taken.. RESULTS AN ISCUSSION A atlab computer program as developed. Te program is used to perform te numerical analysis of te proposed metod, applicability, reliability and accuracy are analyzed as ell. Te response of plates subjected to distributed loads and resting on a Winkler foundation are presented. Several boundary conditions are analyzed (clamped, simply supported and free edges). Te accuracy of te metod is evaluated by comparing te results obtained using BK it te ones obtained by te FE. In order to develop te FE analysis te commercial program SAP 000 as used.

4 Figure. Condition number against number of nodes per side.. Comparison studies Comparisons ere made for tree examples: a plate it all edges simply supported, a plate it all edges clamped and a plate it tree edged simply supported and one free as son in Fig.. Te side lengt of te plates is m. A uniformly distributed load it q = kpa is considered. E and ν are 3 0 pa and 0.5, respectively. For te FE analysis, te mes considered as 600 four-node quadrilateral elements. In te BK analysis, 6 nodes ere taken on eac side. For te presentation of te numerical results, te folloing dimensionless parameter is defined k = ka () 6 k takes values 50, 00 and 500 ic correspond it values, N ,, N and 3, 5 0 N for k. m m m Figure. Boundary conditions of te tree cases studied: simply supported, clamped and simply supported it a free edge Table. Comparison of te imum deflection, te imum bending moment and te imum sear force for a simply supported plate a k 3 0 FE 3 0 FE Nº Nº N º Nº Nº Nº Results of a simply supported plate are son in Tab.. eflections so an error of less tan % and in te case of te bending moments te error is around %. For te sear forces, te results so a sligtly larger error, taking values beteen % and 0%. Figures 3 and so te contour diagrams of deflections and bending moments in te example Nº. As it can be seen, te values obtained for te compared magnitudes are very similar over te ole domain. V V FE

5 x Figure 3. eflection contour diagram of example Nº, BK solution (left) and FE solution (rigt) Figure. Bending moment contour diagram of te example Nº, BK solution (left) and FE solution (rigt) Table sos te comparison for a clamped plate. Te results presented are te imum deflection and te imum and minimum bending moments. Table, sos te results for te plate it tree simply supported edges and one free edge. Te values compared are te same as in Tab.. Table 3. Comparison of te imum deflection and imum and minimum bending moments for a clamped plate + + a k 3 FE 3 0 FE FE Nº Nº Nº Nº Nº Nº a Table. Comparison of te imum deflection, imum bending moments and imum sear force for a plate it tree simply supported edges and one free k 3 0 FE 0 3 FE Nº Nº Nº Nº Nº Nº V V FE

6 Te results obtained ave a difference of less tan 5 %, except for te sear forces. Even toug sear values are not as accurate, en sear is imposed as a boundary condition (case analyzed in Tab. ) te results are acceptable because deflections and bending moments so an error of less tan,5%... Convergence studies Wen te deflection in te middle of te plate obtained by te BK is compared against te FE, te relative error is calculated as ε BK FE = (3) FE Te example Nº of te simply supported plate and Nº of te clamped plate are analyzed to so tat te TSV as effective dealing it te ill-conditioning of te linear systems in tese cases. Figure 5 sos te procedure tat solves te system does not become unstable and te accuracy of te solution increases en te number of nodes becomes larger. Te figure sos as ell tat an accurate solution is obtained for a relatively small number of nodes. 5. FINAL REARKS Figure 5. Relative error in te deflection at te midpoint against te number of nodes for a simply supported plate and a clamped plate Te difference beteen te BK and oter boundary collocation metods is tat te general solution used is nonsingular. As a consequence, te artificial boundary used by te FS is not needed. As tere is no need for integration, te implementation of te BK is straigtforard. Tis paper proposed a ne tecnique based on te BK for te solution of elastic plates resting on a Winkler foundation. Te drabacks of te proposed metod are te need for a particular solution and te ig condition number of te linear system obtained. Te presented numerical results so tat te ill-conditioning difficulties ere overcome using te TSV. Oter regularization metods sould be evaluated in future orks. Te results obtained it te proposed metod are very similar to tose obtained by te FE. Te sear forces so te largest differences, but te deflections and bending moments alays so ig accuracy, even en te boundary condition imposed is te sear force. 6. REFERENCES Bogomolny, A Fundamental solutions metod for elliptic boundary value problems, SIA Journal of Numerical Analysis, Vol., pp

7 Cen, W., Sen, A.J., Sen L.J. and Yuan, G.W., 005. General solutions and fundamental solutions of varied orders to te vibrational, tin, te Berger, and te Winkler plates, Engineering Analysis it Boundary Elements, Vol. 9, pp Cen, W. and Tanaka., 00. A esless, Integration-Free, and Boundary-Only RBF Tecnique, Computers and atematics it Applications, Vol. 3, pp Hansen, C Regularization Tools, A atlab Package for Analysis and Solution of iscrete Ill-Posed Problems, Numerical Algoritms, Vol. 6, pp Kupradze, V.. and Aleksidze,.A., 96. Te metod of te fundamental equations for te approximation of certain boundary value problems. USSR Comput. at. at. Pys. Vol., pp Lyngby, S.C, 98. Condition number of matrices derived from to classes of integral equations, atematical etods in te Applied Sciences, Vol. 3, pp aton, R. and Jonston, R.L, 977. Te approximate solution of elliptic boundary-value problems by fundamental solutions, SIA Journal on Numerical Analysis, Vol., pp Timosenko, S., and Woinosky-Krieger, S., Teory of Plates and Sells, cgra-hill, 959. Wen, P.H., 988. Te numerical metod for complex restrained problem of rectangular plate on elastic base, J. Cent.- Sout Inst. in. etall. Vol. 9 (3), pp Wen, P.H., 989. Boundary collocation metod for rectangular plate it free corners resting on te elastic foundation, Sangai ec. Vol. 0 (), pp RESPONSIBILITY NOTICE Te autors are te only responsible for te printed material included in tis paper.