First Principles Derivation of Differential Equations of Equilibrium of Anisotropic Rectangular Thin Plates on Elastic Foundations

Transcription

1 Journal of Geotechnical and Tranportation Engineering Volume 4 Iue 1 Firt Principle Derivation of Differential Equation of Equilibrium of Aniotropic Rectangular Thin Plate... Ike Received 5/12/2018 Accepted 6/1/2018 Publihed 7/1/2018 Correponding Author: Charle Chinuba Ike; charleike@eut.edu.ng.jgtte.com Firt Principle Derivation of Differential Equation of Equilibrium of Aniotropic Rectangular Thin Plate on Elatic Foundation Charle Chinuba Ike Department of Civil Engineering, Enugu State Univerity of Science & Technology, Enugu State, Nigeria. ikecc2007@yahoo.com, charleike@eut.edu.ng Abtract In thi paper, the partial differential equation of equilibrium for thin rectangular aniotropic plate reting on elatic foundation are derived from fundamental principle of the theory of elaticity. The derivation ere done by imultaneou firt principle conideration of the requirement of kinematic relation, material contitutive relation, and the differential equation of tatic equilibrium. The derivation conidered tranvere ditributed load on the plate domain, and inplane load. Orthotropic plate on elatic foundation equation and iotropic plate on elatic foundation equation ere obtained a pecial cae of aniotropic plate on elatic foundation. Keyord: Aniotropic plate, orthotropic plate, iotropic plate, elatic foundation. 1. Introduction The interaction beteen tructure and upporting oil media i of fundamental importance in foundation analyi and deign and ha attracted the attention of reearcher [1]. The tructure are modeled a beam and/or plate, and the oil media a elatic foundation. The tructure oil interaction problem reduce to the analyi of beam and plate reting on elatic foundation, and uch problem find important application in the deign and analyi of foundation uch a individual footing, combined footing and mat (raft) foundation. Plate are claified a thin plate ith mall deflection, moderately thick plate and thick plate. Thin plate theorie are due to Kirchhoff, Love, Lagrange and Germain. Theorie for moderately thick plate ere due to Mindlin [2], Reiner [3, 4]. The mathematical theory of elaticity i ued to decribe the behaviour of thick plate. Shear deformation theorie of plate have alo been formulated by Shimpi [5], Reddy [6]. Elatic foundation model that have been ued to model the interaction of the oil medium on the tructure are: dicrete foundation parameter model and elatic continuum model. The elatic continuum model idealie the oil medium a an elatic half pace, and ue the olution of the claical theory of elaticity for half pace region [7, 8]. The claical Bouineq problem i ued to obtain analytical expreion for the reactive preure of the oil medium on the tructure. Dicrete parameter model are baed on replacing the oil medium ith a et of dicrete elatic pring, here each pring i connected to one another ith hear coupling link, in order to timulate the continuity of the dicrete model. The implet dicrete foundation model a formulated by Winkler [9] ho aumed the foundation i replaced ith dicrete independent linear elatic pring that are cloely paced a hon in Figure 1. In the Winkler model, the oil reaction preure at any point on the plate i directly proportional to the deflection of the plate at the point, and the proportionality contant, k, i called the modulu of ubgrade reaction or the Winkler foundation parameter or modulu. The mathematical equation for the Winkler model i: p (x, y) k(x, y) (1) Depite the obviou advantage of implicity in mathematical term, the diadvantage of the Winkler model include: (i) the foundation doe not deform beyond the loaded region and thi contravene practical reality. (ii) the Winkler model uffer from a lack of continuity in the upporting oil medium. (iii) there i difficulty in determining the tiffne of the elatic pring to characterize the oil medium. Figure 1: Winkler foundation model Other dicrete foundation model include Filonenko-Borodich foundation model, Paternak [10] foundation model, generalied to parameter foundation model, Hetenyi [11] foundation model, and Kerr foundation model. In the Filonenko-Borodich [12] foundation, the interaction and continuity beteen the dicrete individual pring element i introduced by connecting the top of the pring to a thin tretched elatic membrane hich i under a contant tenile force, T a hon in Figure 2. 10

2 Ike Figure 2: Filonenko-Borodich foundation model The oil reaction preure, p(x, y) of interaction ith the foundation for thi model i expreed mathematically a: p x y k x y T x y (2) i the biharmonic operator, and D i the flexural rigidity of the elatic plate. In the Kerr [13, 14] model, the oil medium i replaced by a et of dicrete elatic pring eparated by a hear layer ith hear modulu G, ith the pring contant in the firt and econd layer having different value a hon in Figure 5. here k and T are the to parameter of the Filonenko-Borodich model and x y (3) i the Laplace operator. The intenity of the tenile force T in the tretched elatic membrane characterize the interaction of the dicrete pring element. In the Paternak [10] foundation model, the interaction of the dicrete pring element i introduced by connecting their end to a plate that undergoe only tranvere hear deformation a hon in Figure 3. The oil preure of interaction on the tructure i given mathematically a Figure 5: Kerr foundation model The Kerr foundation model equation i given by: k G p p k G k k (8) p x y k x y G x y here G i the hear modulu of the hear layer, and k and G are the to dicrete foundation parameter of the Paternak foundation. Figure 3: Paternak foundation model The continuity in the Paternak foundation i characterized by the hear modulu G. Hetenyi foundation introduce inter-pring element continuity by incorporation of an elatic plate capable of flexural deformation at the top of the pring a hon in Figure 4. The model equation i: Figure 4: Hetenyi Foundation model p x y k x y D x y p x y k x y D x y here (7) x x y y (4) (5) (6) here k 1 i the elatic pring contant for the firt oil layer, k 2 i the elatic pring contant for the econd oil layer and (x, y) i the deflection. 1.1 Application of the plate on elatic foundation problem Plate on elatic foundation problem have extenive application in the deign of highay and airport runay pavement and machine foundation. Many partial differential equation in engineering are of the ame form a the plate on elatic foundation equation, and hence are analogue of the plate on elatic foundation problem. The governing equation of plate trip ubjected to tranvere line load are analogou to the equation of plate on elatic foundation. The reearch aim i to derive the differential equation of equilibrium of thin rectangular aniotropic plate reting on elatic foundation. The objective include: (i) to determine the internal tre reultant in aniotropic thin rectangular plate reting on elatic foundation. (ii) to obtain partial differential equation of equilibrium of aniotropic thin rectangular plate on elatic foundation for the general cae of tranvere ditributed load and inplane load. (iii) to obtain partial differential equation of equilibrium for pecial type of aniotropic plate on elatic foundation called orthotropic plate on elatic foundation and iotropic plate on elatic foundation. 2. Theoretical Frameork and Methodology 2.1 Baic aumption Thi ork i baed on the folloing aumption of the mall deformation, thin plate theory: (i) the plate material i linear elatic, homogeneou and aniotropic 11

3 Journal of Geotechnical and Tranportation Engineering vol. 4 (1) (ii) the plate urface i initially flat before the application of load (iii) the tranvere deflection of the middle urface i mall compared ith the plate thickne (iv) the traight line initially normal to the middle urface of the plate before bending remain traight and normal to the middle urface during deformation, and are unchanged in length. Thi aumption called the normality condition, implie that the hear train in the tranvere direction and ) are negligible, and the normal train in the tranvere direction i alo negligible (v) the tre normal to the middle plane i mall a compared ith the other tre component, and can be diregarded ithout introducing ignificant error. (vi) the middle plane i free of train, and i neutral during bending. 2.2 Equation of elaticity theory Thi ork applie the equilibrium approach in the theory of elaticity. Hence the equation are built on a imultaneou conideration of the three requirement of (i) kinematic relation, (ii) material contitutive la, and (iii) differential equation of equilibrium. For mall deformation aumption, the kinematic relation are: u x v y z u v y x u xz x z v y z (9) (10) (11) (12) (13) (14) here u, v, are the diplacement component in the x, y, and z Carteian coordinate direction, repectively,,, are the normal train in the x, y, and z coordinate direction xz and are the hear train. ( xz The material contitutive relation for aniotropic plate material i given in matrix form by: a a... a a a... a (15) zx zx a a a, The differential equation of equilibrium (tatic equilibrium) of an infiniteimal part of the plate element i given by the ytem of equation: yx zx f x x y z zy f y x y z xz f z x y z (16) (17) (18) here f x, f y and f z are the body force component on the x, y and z coordinate direction. 3. Derivation of Equation of Aniotropic Thin Plate on Elatic Foundation 3.1 Diplacement field The derivation proceed on the aumption that the diplacement at any point on the plate can be determined from the tranvere diplacement of the plate middle urface denoted by (x, y); and that the in-plane diplacement component are given in term of (x, y) by: u z x v z y here z i the ditance from the middle plane. 3.2 Strain field (19) (20) Applying Equation (19) and (20) in the kinematic relation, and uing the foundational aumption, the train field are obtained a: z x z y z x y xz 3.3 Stre-train relation (21) (22) (23) (24) (25) For aniotropic thin plate, the material contitutive relation given in general a Equation (15) implify to become: here,, are the normal tree in the x, y, and z direction,,, and zx are the hear tree and a ij, i 1,, 6, j 1,, 6 are the elatic contant. g g g g g g g g g here g 11, g 12, g 16 g 66 are the elatic contant. (26) 12

4 Ike Uing Equation (21) (23), the tre field are obtained a: z g x y z g x y z g x y (27) (28) (29) The hear tree and are obtained uing the differential zx equation of equilibrium Equation (16-18) a follo: 3.4 Stre reultant (32) (33) The tre reultant are given by the bending moment M, M, the titing moment M and the hear force Q x, Q y a follo: Subtitution of Equation (27-29), (32) and (33) yield: zy z x x y y (30) zx z g ) z x x y y (31) Integration ith repect to z, here h z h, give: zy z g ) h zx z g x x y ( g g ) y h zy z g x M M ( g ) x y y zdz zdz M M Q Q yx x y xz dz dz zdz M z g dz x y (34) (35) (36) (37) (38) (39) M z g dz x y M z g dz x y (40) (41) h z dz g ) x x y y (42) h z dz g ) x x y y (43) Evaluation of the integral and implification yield: M D x y M D x y M D x y (44) (45) (46) D ( D ) x x y y (47) D ( D ) x x y y (48) here in general, zgdz ij gijh D ij (49) D 11, D 12 are bending rigiditie, D 66 i the titing rigidity, D 16, D 26 are coupling rigiditie or econdary rigiditie. 3.5 Differential equation of equilibrium The differential equation of equilibrium in term of the tre reultant are: Q x x M x M x q( x, y) p( x, y) y M y M y (50) (51) (52) Equation (50) (52) can be expreed a the ingle equation: M M M M q x x y y x y M M M q x y (53) (54) Then by ubtitution of Equation (44-46), e obtain upon implification, 13

5 Journal of Geotechnical and Tranportation Engineering vol. 4 (1) D ( D ) x x y x y (55) Equation (55) i the governing partial differential equation of equilibrium of aniotropic rectangular thin plate on elatic foundation under ditributed tranvere load q(x, y). For orthotropic plate, D 16 D 26 0 and the differential equation of equilibrium implifie to become: D ( D ) p x y q x y x x y y (56) or, p x y q x y y D H p x y q x y x x y y (57) here H D 12 2D 66 (58) D D 11 (59) D D 22 hex D (60) (61) hey D (62) H D 2G (63) he D (64) hg G (65) E x, E y, and G are the orthotropic plate moduli, D, D, D, and G are the flexural rigiditie and the torional rigidity of an orthotropic thin plate. For orthotropic plate, the internal force become: M D D (66) x y x y M D D D D x y x y M D G D ( D ) x D ( D ) G x D H x ( D G) x y y H x y y (67) (68) (69) (70) (71) (72) (73) 3.6 Combined tranvere and inplane loading of aniotropic plate on elatic foundation For vertical equilibrium in the z direction of an aniotropic thin plate element reting on an elatic foundation and under combined tranvere and in-plane loading, Equation (50) change to account for the inplane load N, N, and N a follo: Q Q x y q( x, y) p x y x y x y (74) The governing partial differential equation of equilibrium for rectangular thin aniotropic plate on elatic foundation under combined ditributed tranvere and inplane load become: D ( D ) p x y x x y x y y q( x, y) x y (75) For orthotropic plate on elatic foundation under combined tranvere and inplane loading, e obtain: D H p x y x x y y q( x, y) x y 3.7 Aniotropic plate on Winkler foundation (76) For Winkler foundation, the ubgrade reaction preure p(x, y) at any point (x, y) under the plate i directly proportional to the deflection at that point (x, y), here the proportionality contant i the Winkler coefficient of ubgrade modulu or the Winkler foundation parameter, k, and mathematically, ubtitution of the Winkler expreion for the oil reaction into Equation (75) yield: D D ( D D ) x x y x y k( x, y) q( x, y) y x y (77) For orthotropic plate on Winkler foundation, ubtitution of the Winkler expreion for the oil reaction ith Equation (76) yield the governing differential equation of equilibrium a: D H k( x, y) x x y y q( x, y) x y (78) For iotropic thin plate on Winkler foundation, the governing differential equation of equilibrium i obtained from Equation (76) by ubtitution of the expreion for the Winkler oil reaction, and noting that for iotropic, homogeneou thin plate, D 11 D 22 H D, and e obtain: D k( x, y) x x y y 14

6 Ike q( x, y) x y here D i the flexural rigidity of the iotropic plate. 3.8 Aniotropic plate on Paternak foundation (79) The governing differential equation of equilibrium for aniotropic plate on Paternak foundation are obtained repectively a: ( ) D x x y x y y k( x, y) G ( x, y) q( x, y) x y (80) for aniotropic plate on Paternak foundation, and D H k( x, y) G( x, y) x x y y q( x, y) x y for orthotropic plate on Paternak foundation, and D k( x, y) G ( x, y) x x y y q( x, y) x y for iotropic plate on Paternak foundation. (81) (82) 3.9 Aniotropic thin rectangular plate on Filonenko- Borodich foundation The governing partial differential equation of equilibrium for aniotropic thin rectangular plate on Filonenko-Borodich foundation are obtained, repectively a follo: ( ) D x x y x y y k( x, y) T ( x, y) q( x, y) x y (83) and for the pecial cae of aniotropy called orthotropy, D 16 D 26 0, e obtain D H k( x, y) T( x, y) x x y y q( x, y) (84) x y Iotropic plate are pecial aniotropic plate, here D 11 D 22 H D, and e obtain, D k( x, y) T ( x, y) x x y y q( x, y) x y (85) 4. Dicuion In thi paper, the partial differential equation of equilibrium of aniotropic thin rectangular plate on elatic foundation ha been uccefully derived. The derivation applied the equilibrium approach in the mathematical theory of elaticity. Small deformation aumption of the elaticity theory ere ued to expre the Cauchy equation relating train to diplacement a Equation (9-14). Generalied tre-train la for the general cae of aniotropy a ued a Equation (15). The differential equation of tatic equilibrium ere alo ued a Equation (16-18). The diplacement field ued in the derivation ere conitent ith thin plate theory and are given by Equation (19) and (20). Internal tre reultant ere obtained for the general cae of aniotropic thin plate on elatic foundation a Equation (44-48). The partial differential equation of equilibrium of aniotropic thin plate on elatic foundation a obtained a Equation (55), for tranvere ditributed load. The equation a obtained a Equation (75) for aniotropic plate on elatic foundation under combined tranvere and in-plane load. The equilibrium equation for aniotropic thin plate on Winkler foundation a obtained a Equation (77) for combined tranvere ditributed load and inplane load. The differential equation for aniotropic thin plate on Paternak and Filonenko-Borodich foundation ere obtained, repectively a Equation (80) and (83) for combined tranvere ditributed load and inplane load. It a alo found that orthotropic thin plate on elatic foundation equation and iotropic thin plate on elatic foundation equation could be obtained a pecial cae of the aniotropic thin plate on elatic foundation equation. 4.1 Limitation of the governing equation derived The baic method/approache continuum and modeling method are ued in deriving equation of plate on elatic foundation. In the continuum approach, the oil i conidered a elatic, elatoplatic, hypoplatic, non homogeneou, aniotropic and /or layered. Thi demand complex mathematical method and analyi. The modeling approach i le complicated, and involve le complicated mathematical analyi. Hoever, the determination of the foundation model parameter() i a huge problem. 4.2 Limitation of the claical plate theory The tudy ued the claical thin plate theory (CPT) to model the plate. The claical thin plate theory (Kirchhoff plate theory) diregard the tranvere hear deformation of the plate, and can hence only correctly predict the flexural behaviour of thin plate having a ratio of thickne, h to idth, a, of le than approximately 0.05 (i.e. a < 0.05). 5. Concluion The folloing concluion are made: (i) The differential equation of equilibrium of rectangular thin aniotropic plate reting on elatic foundation can be obtained uing the equilibrium approach in the mathematical theory of elaticity. (ii) Orthotropic thin plate on elatic foundation problem are pecial cae of aniotropic thin plate on elatic foundation problem here D 16 D 26 0, and the equation for orthotropic thin plate on elatic foundation are obtained 15

7 Journal of Geotechnical and Tranportation Engineering vol. 4 (1) from the aniotropic thin plate on elatic foundation here D 16 D (iii) Iotropic thin rectangular plate on elatic foundation problem are particular type of the general aniotropic thin rectangular plate on elatic foundation here D 16 D 26 0, and D 11 H D 22 D; and the equation are obtained from the aniotropic thin rectangular plate on elatic foundation equation. Reference [1] Wang Y.H., Tham L.G. and Cheung Y.K. (2005) Beam and plate on elatic foundation: a revie, Progre in Structural Engineering and Material Structural Analyi and CAD, Volume 7, Iue 4, Wiley, October/December 2005, pp DOI: /pe.202. [2] Mindlin, R.D. (1951) Influence of rotary inertia on flexural motion of iotropic, elatic plate. Journal of Applied Mechanic, Vol. 18, No 1, pp [3] Reiner E. (1944) On the theory of bending of elatic plate. Journal of Mathematic and Phyic, Vol. 23, pp [4] Reiner E. (1945) The effect of hear deformation on the bending of elatic plate. Journal of Applied Mechanic, Vol. 12, pp [5] Shimpi R.P. (2002). Refined plate theory and it variant. AIAA Journal, Vol. 40, pp [6] Reddy J.N. (1984) A refined non-linear theory of plate ith tranvere hear deformation. International Journal of Solid and Structure, Vol. 20 pp , [7] Seeng-Min K. (2004) Bending and vibration of a plate on elatic foundation ubject to inplane compreion and moving load. International Journal of Solid and Structure, Vol. 41, Iue 20, October 2004, pp , Elevier Ltd. [8] Mylecki K. (2004) Approximate fundamental olution of equilibrium for thin plate on elatic foundation. Archive of Civil and Mechanical Engineering, Vol IV, No 1, pp [9] Winkler E. (1867) Die Lebre vonder elatizitat und fetigkei, Dominicu, Prague. [10] Paternak P.L. (1954) On a ne method of analyi of an elatic foundation by mean of to foundation contant (in Ruian) Goud Izd Lit po Stroiteltvu Arkhitekture Moco. [11] Hetenyi M. (1946) Beam on elatic foundation. The Univerity of Michigan Pre, Ann Arbour, Michigan. [12] Filonenko-Borodich M.M. (1945) A very imple model of an elatic foundation capable of preading load, Sbornik Mokovkovo Elecktro Intituta. [13] Kerr A.D. (1964) Elatic and vico-elatic foundation model. Journal of Applied Mechanic, ASME, 1964, 25(80), pp [14] Kerr A.D. (1965) A tudy of a ne foundation model. Acta Mechanica, Vol. 1, pp