AN APPROXIMATE SOLUTION FOR PLATES RESTING ON WINKLER FOUNDATION

Transcription

1 INTERNATIONA JOURNA OF CIVI ENGINEERING AND TECHNOOGY (IJCIET) ISSN 97 8 (Prnt) ISSN 97 (Onlne) Volume 5, Issue, November (), pp. - IAEME: Journal Impact Factor (): 7.99 (Calculated by GISI) IJCIET IAEME AN APPROXIMATE SOUTION FOR PATES RESTING ON WINKER FOUNDATION Abdulhalm Karasn, Gulten Atas, Department of Cvl Engneerng, Dcle Unversty 8 Dyarbaır, Turey ABSTRACT In many engneerng structures transmsson of vertcal or horzontal forces to the foundaton s a maor challenge. The problem may be smplfed f an accurate fnte grd soluton s devsed for plates on elastc foundatons. The man propose of ths artcle s to develop an approxmate but computatonally manageable fnte grd soluton of plates as an applcaton of fnte element method. In ths method each dscrete element utlzed s equpped wth an exact soluton for a beam element on a Wnler foundaton. Conceptually, the soluton can be defned as an extenson of the dscrete parameter approach where the physcal doman s broen down nto dscrete sub-domans, each endowed wth a property sutable for the purpose of mmcng problem at hand. In another words t s based on the exact stffness, obtaned by exact shape functons for the beam element on Wnler foundaton to extend for solvng two-dmensonal plate problems. Some examples of rectangular plates on one parameter elastc foundaton ncludng bendng problems were solved. Comparson wth nown analytcal solutons and other numercal solutons yelds accurate results. Keywords: Wnler Foundaton, Plates, Bendng, Fnte Grd Soluton, Shape Functons.. INTRODUCTION Researches n the area of plates restng on elastc foundatons have receved consderable attenton due to ther wde applcablty n cvl engneerng. However t has been covered sporadcally n the lterature. In many engneerng structures assessment of stress condtons created by vertcal or horzontal forces to the foundaton s a frequent problem of desgn. For partcular plate problems, closed form solutons have been obtaned. However, even for conventonal plate analyss these solutons can usually be appled to the problems wth smple geometry, load and boundary condtons. Of course for elastc foundaton sol model underneath plate problems the soluton s more complex and there s analytcal soluton for elementary cases. Some numercal and approxmate methods, such as fnte element, fnte dfference, boundary element and framewor methods have been developed to overcome such problems. The obectve of the present study s to

2 present a numercal soluton for plates on elastc foundatons. In ths form, plates are dealzed as a grllage of beams of a gven geometry satsfyng gven boundary condtons. The exact stffness matrces of a beam element on Wnler foundaton s used to solve general plate bendng problems. In order to nclude behavour of foundaton properly nto the mathematcally smple representaton t s necessary to mae some assumptons. One of the most useful smplfed models nown as the Wnler model assumes the foundaton behaves elastcally, and that the vertcal dsplacement and pressure underneath t are lnearly related to each other. That s, t s assumed that the supportng medum s sotropc, homogeneous and lnearly elastc, provded that the dsplacements are small. Ths smplest smulaton of an elastc foundaton s consdered to provde vertcal reacton by a composton of closely spaced ndependent vertcal lnearly elastc sprngs. Snce the nteracton between structural foundatons and supportng sol has a great mportance n many engneerng applcatons, a consderable amount of research has been conducted on beams and plates on elastc foundatons. Much research has been conducted to deal wth bendng, buclng and vbraton problems of beam and plates on elastc foundaton. The am of most s to solve some real lfe engneerng problems such as structural foundaton analyss of buldngs, pavements of hghways, water tans, arport runways and bured ppelnes, etc. nowadays A broad range of the engneerng problems has been solved by computer-based methods such as fnte element and boundary element methods[-]. Closed form solutons of such complex plate problems have been publshed for a lmted number of cases. On the other hand n the case of the beam analyss, the formulatons based on nterpolaton (shape) functons have been used n soluton by fnte element method. In 98 s authors such as [5-7] have derved exact stffness matrces for beams on elastc foundatons. Razaqpur and Shah [8] derved a new fnte element to elmnate the lmtatons of the soluton, such as the necesstes of certan combnatons of beam and foundaton parameters, for beams on a two-parameter elastc foundaton. In Karasn [9] extended ths soluton to an analytcal soluton for the shape functons of a beam segment supported on a generalzed two-parameter elastc foundaton. In that study t s ponted out that the exact shape functons can be utlzed to derve exact analytc expressons for the coeffcents of the element stffness matrx. Ths paves the way for dervng wor equvalent nodal forces for arbtrary transverse loads and coeffcents of the consstent mass and geometrcal stffness matrces. Ths study let plates to be represented by a dscrete number of ntersectng beams. Thus, mechancal propertes of one-dmensonal beam elements are used for soluton of complex plate restng on elastc foundaton problems for varous loadng and boundary condtons.. THEORY AND FORMUATIONS As Wlson [] has ndcated the structural behavour of a beam resembles that of a strp n a plate. On the other hand Hrennof [] stated that the system cannot truly be equal to the contnuous structure. However t s shown that the ease n arrvng at results of engneerng accuracy outweghs the small errors that these results represent. Its errors are attrbutable to the torsonal constants of the grd members and the compromsed effects of dscretzng a contnuous problem [9]. Then the framewor method that replaces a contnuous surface by an dealzed dscrete system can represent a two-dmensonal plate conceptualzed n Fg.. By ths representaton, plate problems ncludng nonunform thcness and foundaton propertes, arbtrary boundary and loadng condtons and dscontnuous surfaces, can be solved n a general form. 5

3 Fgure : Grd representaton for a contnuous plate wth two smply supported opposte edges and others are free (SFSF) As a result of ths representaton, the theory of ths study let to replace the governng equaton for plates restng on Wnler foundaton problems expressed n Equaton () whch s qute a dauntng equaton to solve for general loads and boundary condtons by that of one-dmensonal beam element n Equaton (). For the governng dfferental equatons of a plate and a lne element supported by Wnler foundaton for transverse dsplacement w(x,y)and w(x) are gven as; w( x, y ) w( x, y ) w( x, y ) ( + + ) + w( x, y ) q( x, y ) () x x y y D d w( x ) EI + w( x ) q( x ) () d x Where s the Wnler parameter wth the unt of force per unt length/per unt length (force/length ), EI and D are the flexural rgdtes of the lne element and the plate element respectvely. From the dfferental equatons networs of beam elements that represent the plates has an obvous advantage to solve complcate plate problems. A representaton of the foundaton wth ndependent closely lnear sprngs underlyng for an ndvdual beam element s shown n Fg.. Fgure : Representaton of the Beam Element Restng on Wnler Foundaton - Dervaton of Exact Shape Functons Frstly to obtan homogeneous form of Equaton () let q(x) and after necessary smplfcatons; d w( dx x ) + w( x ) where λ EI λ ()

4 d And by operator method usng dx and the roots of the characterstc equaton wth magnary number () are; ( D + λ ) w( x) and λ + λ,d λ + λ,d λ λ and D λ λ n n D then the characterstc Equaton () can be wrtten n D () Then the closed form soluton of Equaton () wth hyperbolc functons and Insertng the angular dsplacements as Ø(x) a +a x due to torsonal effects obtaned as; w(x) c + c Sn c x + c Cos 5 [ λx] cosh[ λx] + csn[ λx] snh[ λx] + [ λx] cosh[ λx] + c Cos[ λx] snh[ λx] (5) Then, the closed form equaton can be expressed n matrx form as: T w B C () The arbtrary constant elements subscrpt of the vector C can be determned by relatng them to the end dsplacements whch forms boundary condtons shown n Fg.. Fgure : A fnte element of a beam (a) the dsplacements, (b) loads appled to nodes Vectors { d } and { F } represent the generalzed dsplacements and the loads appled to the nodes, respectvely. In order to relate the rotatonal elements of the dsplacement vector to the constant vector, t s necessary to dfferentate the bendng part of Equaton (5) and by the boundary condtons (for x and x values) n terms of the constants n matrx form as; [ d] [ H ] [ C] or [ C] [ H ] [ d] (7) where [H] s a x matrx from and substtute Equaton (7) nto Equaton (5) then the closed form soluton of the dfferental equaton by ntroducng matrx N that ncludes the shape functons and the generalzed dsplacements defned n Fg. can be wrtten n matrx form as; [ ] [ ] T [ ] [ ] [ ] T w N d where N B [ H ] (8) 7

5 After performng the necessary symbolc calculatons, the shape functons are obtaned. Each shape (nterpolaton) functon defnes the elastc curve equaton of the beam elements for a unt dsplacement appled to the element n one of the generalzed dsplacement drecton as the others are set equal to zero. The elements of the shape functons matrx N are; x sn cos cos cos x sn cos 5 cos cos [ λx] cosh[ λ( x) ] cosh[ λx] sn[ λx] [ λ( x) ] snh[ λx] cos[ λx] snh[ λx] λ( + cos[ λ] + cosh[ λ] [ λx] cosh[ λ( x )] + cosh[ λx] cos[ λ( x )] [ λx] cosh[ λx] + sn[ λx] snh[ λ( x )] snh[ λx] sn[ λ( x )] (cos[ λ] + cosh[ λ] ) + [ λ( ] cosh[ λ( ] cosh[ λ( + x] sn[ λ( ] [ λ( ] snh[ λ( ] cos[ λ( + x) ] snh[ λ( ] λ( + cos[ λ] + cosh[ λ] [ λ( x] cosh[ λ( x )] + cosh[ λ( x] cos[ λ( + x )] + [ λ( x] cosh[ λ( + x )] snh[ λ( x] sn[ λ( + x )] + sn[ λ( x] snh[ λ( + x )] (cos[ λ] + cosh[ λ] ) + (9 a) (9 b) (9 c) (9 d) (9 e) (9 f) The bendng shape functons are drectly affected by the foundaton parameter. It s possble to redefne them n non-dmensonal forms for comparng the functons wth the correspondng Hermtan polynomals. To have non-dmensonal forms, we nsert the followng relatons nto Equaton (). x ξ for x, and p λ () EI where s the length of the beam. Note that both p and ξ are non-dmensonal quanttes. Snce the torsonal shape functons are not affected ( ξ, -ξ ) then the non-dmensonal forms of the bendng shape functons can be consdered as follows; sn[ pξ ] cosh[ p( ξ )] cosh[ pξ ] sn[ pξ ] + cos[ p( ξ )] snh[ pξ ] cos[ pξ ] snh[ pξ ] (a) p( + cos[ p] + cosh[ p] 8

6 cos cos [ pξ ] cosh[ p( ξ )] + cosh[ pξ ] cos[ p( ξ )] [ pξ ] cosh[ pξ ] + sn[ pξ ] snh[ p( ξ )] snh[ pξ ] sn[ p( ξ )] (cos[ p] + cosh[ p] ) (b) sn[ p( ξ )] cosh[ p( ξ )] cosh[ p( + ξ )] sn[ p( ξ )] + 5 cos[ p( ξ )] snh[ p( ξ )] cos[ p( + ξ )] snh[ p( ξ )] (c) p( + cos[ p] + cosh[ p] cos cos [ p( ξ )] cosh[ p( ξ )] + cosh[ p( ξ )] cos[ p( + ξ )] + [ p( ξ )] cosh[ p( + ξ )] snh[ p( ξ )] sn[ p( + ξ )] + sn[ p( ξ )] snh[ p( + ξ )] (cos[ p] + cosh[ p] ) (d) On the other hand, the shape functons for flexure of unform beam element wthout any foundaton, whch s the lmts of Equaton () as tends to zero, t s possble to fnd out the nondmensonal forms of the shape functons as Hermtan polynomals. ξ ξ + ξ, ξ ξ, 5 ξ + ξ and ξ ξ () In order to observe the foundaton parameter effects, the expressons n Equatons () and () are portrayed graphcally n Fg. for comparson. Fgure : Effects of one-parameter foundaton on the shape functons. Dervaton of the Element Stffness Matrx The element stffness matrx relates the nodal forces to the nodal dsplacements. Once the dsplacement functon has been determned as n prevous secton for beam elements restng on one parameter elastc foundaton, t s possble to formulate the stffness matrx. The element stffness 9

7 matrx for the prsmatc beam element shown n Fgure can be obtaned from the mnmzaton of stran energy functonal U as follows: et Equaton () be multpled by a test or weghtng functon, ν(x) whch s a contnuous functon over the doman of the problem. The test functon ν(x) vewed as a varaton n w must be consstent wth the boundary condtons. The varaton n w as a vrtual change vanshes at ponts where w s specfed, and t s an arbtrary elsewhere. Frst step s to ntegrate the product over the doman, d w( x) ν ( x) EI + w( x) q( x) dx or dx ( x) e( x) dx ν () The purpose of the ν(x) s to mnmze the functon e(x), the resdual of the dfferental equaton, n weghted ntegral sense. Equaton () s the weghted resdual statement equvalent to the orgnal dfferental equaton. snce v(x) s the varaton n w(x), t has to satsfy homogeneous form of the essental boundary condton. Then, Equaton () taes the form of only twce dfferentable n contrast to Equaton (), whch s n fourth order dfferental equaton, as follows. ν( x ) EI + w( x ) q( x ) dx EI dx + ν( x )w( x )dx ν ( x )q( x )dx d w( x ) dx d ν( x ) d w( x ) dx dx () Equaton () s called the wea, generalzed or varatonal equaton assocated wth Equaton (). The varatonal soluton s not dfferentable enough to satsfy the orgnal dfferental equaton. However t s dfferentable enough to satsfy the varatonal equaton equvalent to Equaton (). In order to obtan the stffness matrx, the dsplacement felds can be defned as follows; w( x ) w and ν ( x ) (5) EI EI d d w dx + dx dx d d dx dx dx + w dx dx w q( x )dx q( x )dx and { K }{ w } { F } e e () The shape functons,,,,, 5 and, are already nown from Equaton (9). The w T φ, θ, w, φ,, θ w referrng to sgn conventon n Fg.. After nodal dsplacements are { } { }, performng the necessary symbolc calculatons, the terms of the stffness matrx are obtaned as; GJ (7a) EIλ(snh[ λ] sn[ λ] 55 + cosh[ λ] + cos[ λ] m EI EIλ (cos[ λ] cosh[ λ] + cosh[ λ] + cos[ λ] m EI (7b) (7c)

8 Internatonal Journal of Cvl Engneerng and Technology (IJCIET), ISSN 97 8 (Prnt), ISSN 97 (Onlne), Volume 5, Issue, November (), pp. - IAEME 5 5 EIλ(cosh [ λ] sn[ λ] cos[ λ] snh[ λ] + cosh[ λ] + cos[ λ] m EI (7d) EIλ (sn + cosh 8EIλ (cosh [ λ] + [ [ λ] + cos[ 5 snh λ λ [ λ] sn[ λ] cosh[ λ] 5 5 ] m ] + cos λ snh λ + cos λ 8EIλ snh λ sn λ + cosh λ cos λ 5 EI [ ] [ ] [ ] m [ ] [ ] [ ] + [ ] m EI EI (7e) (7f) (7g) (7h) It s obvous that when foundaton parameter tends to zero (or λ ), the terms n Equaton (7) reduces to the conventonal beam stffness terms obtaned by Hermtan functons. On the other hand n Fg. 5, the effect of the foundaton parameter on the stffness terms portrayed wth respect to correspondng terms of Hermtan functons. /(-EI/^) p Fgure : Influence of Wnler foundaton parameters on the normalzed stffness term. System Stffness Matrx for Grdwor In grdwor systems at edge nodes two or three, at nteror nodes four of the typcal dscrete ndvdual beam elements are ntersected. Matrx dsplacement method based on stffness-matrx approach s a useful tool to solve grdwors wth arbtrary load and boundary condtons. It can be defned as a horzontal frame structure wth rgd onts whose members and onts le n a common plane. The appled loads can be out of plane or normal to the plane of the structure as lmted by the degrees of freedom drectons.

9 Consder a typcal member from a structural grd as shown n Fgs. -. wth the ends of the member denoted by and. The local axes of the member as llustrated n Fg. 7. are x, y, z whle the global axes are X, Y, Z. The possble end deformatons of the element are a ont translaton n z- drecton and the torsonal and bendng rotatons respectvely about x- and y- axes. That s, the degrees of freedoms (possble end deformatons) of the element at are two rotatons, and, and one translaton,, at they are smlarly and 5 for rotatons and for translaton. Fgure 7: Transformaton of the degree of freedoms of a typcal plane element n local (x, y, z) coordnates to the global (X, Y, Z) coordnates and the transformaton matrx of an arbtrary plane element (C cos(θ) and S sn(θ)) By usng a proper numberng shame t s possble to collect all dsplacements for each nodal pont n a convenent sequence the stffness matrx of the system for rectangular grds can be generated as follow: sys NE T a a (8) where s the ndvdual element number, NE s the number of elements dependng on boundary condtons, a s the ndvdual rotaton element matrx, s the proper element stffness matrx for a beam element restng on one-parameter elastc foundaton or a conventonal beam Equaton (7) and sys s the stffness matrx of the total structure. In the method a plate edge s subdvded nto a number of strps and each strp s characterzed wth the lumped characterstcs of the correspondng wdth and plate depth.. CASE STUDY The valdty of the soluton technque s demonstrated through an example of a smply supported plate subected to a unformly dstrbuted load restng on a Wnler foundaton consdered []. In the reference the sde length of the square plate a, the flexural rgdty D and Posson raton υ were chosen as 8 m, Nm and. respectvely. The unformly dstrbuted load q was gven as N/mm. The smple supported plate on Wnler foundaton s consdered. The comparson of the Fnte Grd Method (FGM) results wth the ocal Boundary Integral Equaton method (BIE) on the centerlne of the plate for three dfferent Wnler coeffcents s gven n Table.

10 Table : The comparson of the deflectons at the centerlne for a smply supported plate restng on a Wnler foundaton wth the BIE method From the table one can see that the maxmum relatve error for deflectons of ponts located on the axs passng through the centre of the plate s about less than % whch reflects a hgh degree of accuracy.. CONCUSION Many studes have been done to fnd a convenent representaton of physcal behavour of a real structural component supported on a foundaton. The usual approach n formulatng problems of beams, plates, and shells contnuously supported by elastc meda s based on the ncluson of the foundaton reacton n the correspondng dfferental equaton of the beam, plate, or shell. In the oneparameter model the sol underneath beams or plates (the Wnler model) lead to a dscontnuty of the foundaton deformaton along the doman boundary. However ths method offers several attractve advantages. Frst, orthotropc plates whch have no analytcal solutons can be analyzed wth no addtonal effort. The analyses are not confned only to statc deflecton and nternal force calculatons for unform foundaton, but also cover non-unform foundaton problems as well. However, Plates of any geometry, not only of the evy type, can be analyzed. It s shown that the ease n arrvng at results of engneerng accuracy outweghs the small errors. Its errors are attrbutable to the torsonal constants of the grd members and the compromsed effects of dscretzng a contnuous problem. REFERENCES [] P. Gülan and B.N. Alemdar, An exact fnte element for a beam on a two-parameter elastc foundaton, Structural Engneerng and Mechancs, 7(), 999, [] J. Slade, V. Slade, H. A. Mang, Meshless local boundary ntegral equaton method for smply supported and clamped plates restng on elastc foundaton, Computer Methods n Appled Mechancs and Engneerng, 9(5-5),, [] M. A. A. Alsarraf and H. S. El Dn, The effectve wdth n mult-grder composte steel beams wth web openngs, Internatonal Journal of Cvl Engneerng and Technology, 5(9),, pp. -5. [] P. K. Snha and Roht, Analyss of complex composte beam by usng Tmosheno beam theory and fnte element method, Internatonal Journal of Desgn and Manufacturng Technology, (),, pp. -5.

11 [5] R.D. Coo and F. Zhaouha, Beam elements on two-parameter elastc foundaton, Journal of Engneerng Mechancs, 9(), 98, pp. 9-. [] M. Esenberger and J. Clastorn, Beams on varable two parameter elastc foundatons Journal of Engneerng Mechancs, ASCE, (), 987, pp.5-. [7] R. Bares and C. Massonnet, analyss of beam grds and orthotropc plates, Crosby ocwood & Son, td, ondon, 98. [8] A. G. Razaqpur and K. R. Shah, Exact analyss of beams on two-parameter elastc foundatons Internatonal Journal of Solds Structures, 7(), pp. 5-5, 99. [9] A. Karasn, An mproved fnte grd soluton for plates on generalzed foundatons, doctoral dss., Mddle East Techncal Unversty, Anara, Turey,. [] E.. Wlson, three dmensonal statc and dynamc analyss of structures, Computers and Structures Inc.,, Bereley. [] A. Hrenoff, Framewor method and ts technque for solvng plane stress problems, Internatonal Assocaton Brdge Structure Engneerng, 9, 99, 7-7.