Mathematical Modelling of Engineering Problems Vol. 5, No. 1, March, 2018, pp. 1-10
Journal homepage:http://iieta.org/Journals/MMEP
Ritz variational method for bending of rectangular kirchhoff plate under transverse hydrostatic load distribution
Clifford U. Nwoji1, Hyginus N. Onah1, Benjamin O. Mama1, Charles C. Ike2*
1 Dept of Civil Engineering,University of Nigeria, Nsukka, Enugu State, Nigeria 2Dept of Civil Engineering,Enugu State University of Science & Technology, Enugu State, Nigeria
Corresponding Author Email: [email protected]
https://doi.org/10.18280/mmep.050101 ABSTRACT
Received: 12 Feburary 2018 In this study, the Ritz variational method was used to analyze and solve the bending Accepted: 15 March 2018 problem of simply supported rectangular Kirchhoff plate subject to transverse hydrostatic load distribution over the entire plate domain. The deflection function was chosen based Keywords: on double series of infinite terms as coordinate function that satisfy the geometric and Ritz variational method, Kirchhoff force boundary conditions and unknown generalized displacement parameters. Upon substitution into the total potential energy functional for homogeneous, isotropic plate, hydrostatic load distribution Kirchhoff plates, and evaluation of the integrals, the total potential energy functional was obtained in terms of the unknown generalized displacement parameters. The principle of minimization of the total potential energy was then applied to determine the unknown displacement parameters. Moment curvature relations were used to find the bending moments. It was found that the deflection functions and the bending moment functions obtained for the plate domain, and the values at the plate center were exactly identical as the solutions obtained by Timoshenko and Woinowsky-Krieger using the Navier series method.
1. INTRODUCTION (i) to determine the total potential energy functional for the simply supported Kirchhoff plate under transverse Plates are commonly applied in many engineering fields as hydrostatic load distribution for a suitable choice of roof slabs, building floor slabs, bridge deck slabs, foundation displacement basis function footings, water tanks, bulkheads, retaining walls, turbine disks, (ii) to apply the variational principle to the total potential aerospacecraft panels, and ship hulls [1, 2, 3]. Their energy functional determined and hence find the unknown applications cut across the various branches of engineering displacement parameters which minimize the total potential and plates are used in civil, structural, marine, naval, aerospace energy functional and mechanical engineering. (iii) to determine the bending moment distributions from Plates are subjected to transversely applied loads, causing the bending moment curvature relations, and find the bending flexure and the development of flexural deformation and moments at the center of the plate stresses. Plates are defined by their shapes as rectangular, (iv) to find the deflection at the center of the plate. square, circular, elliptical, skew, quadrilateral plates, etc. They are also defined by: their constituent materials as homogeneous, non homogeneous, isotropic, anisotropic, and 2. LITERATURE REVIEW orthotropic [1, 2, 3]. They are also classified as membranes, thin plates, moderately thick plates, and thick plates, Review of plate theories depending on the ratio (h/a) of their thickness, h to the least inplane dimension, a. Sophie Germain used the method of variational calculus to Plates can be subjected to static loads or dynamic loads, and derive from the first principles, the differential equation can also be subjected to inplane compressive loads that cause governing thin plates subject to transverse distributed loads. buckling. Thus, plates can respond to applied loads by static Her differential equation was however defective, as she flexural response, dynamic flexural response or by buckling; neglected the contributions to the strain energy due to the depending on whether the loads are static or dynamic or twisting deformations of the plate middle surface. Lagrange compressive. later obtained the corrected governing differential equation for transversely loaded thin plates that accounted for the warping Research aim and objectives (twisting) of the plate middle surface. Subsequently, Navier, [4] using the theory of elasticity, obtained the differential The research aim is to apply the Ritz variational method to equation of bending of rectangular plates under distributed the flexural analysis of simply supported Kirchhoff plate under transverse loads. Navier [4] also used the Fourier double transverse hydrostatic load distribution over the entire plate trigonometric series method to solve and obtain exact domain. The objectives include:
1 solutions for the deflection of simply supported rectangular problem of Kirchhoff plates with clamped and simply thin plates under transverse loads. supported edges. Other researchers who have worked on the Poisson [5] [6] extended Navier’s research to the problem plate problem include Mama et al [28], Ezeh et al [29] and of circular plates. Kirchhoff [7] presented an extended theory Aginam et al [30]. of plates which considered both the transverse distributed loads and the in-plane loads on plates. Kirchhoff also applied the method of virtual displacements in solving plate problems. 3. RITZ VARIATIONAL METHOD Hencky [8, 9] and Reissner [10, 11] have presented improvements to the Kirchhoff plate theory which considered The Ritz variational method for the flexural analysis of the the effect of transverse shear deformation on the behaviour of Kirchhoff plate under given applied loads and edge support plates, and thus allow their application to the problem of conditions is based on the application of the principle of moderately thick and thick plates. Their theories are first order stationary total potential energy. The principle of minimum shear deformation plate theories. total potential energy states that the displacement field Mindlin [12] used a displacement based formulation to corresponding to the minimum total potential energy obtain the governing differential equation for first order shear functional of the structure under applied loads represents a deformable plates, and incorporated the effect of rotary inertia. state of equilibrium provided the displacement field satisfies Mindlin’s theory, however, requires a shear correction factor the prescribed boundary conditions of the structure. Hence, for to account for the error introduced by the assumption of a the plate flexure problem, the Ritz variational method seeks to constant shear strain through the plate thickness. determine the displacement field w(x, y) defined over the plate Other plate models and theories that have been developed domain, where x and y are the plate domain coordinates and to account for the limitations and imperfections of the classical w(x, y) is the displacement function or the displacement field, Kirchhoff-Love plate theory include: Shimpi refined plate such that the geometric and force boundary conditions are theory [13], Higher Order Plate Deformation theory [14], satisfied and the total potential energy functional, of the Reddy’s third order plate theory [15], Leung’s plate theory Kirchhoff plate, which is the sum of the stain energy in [16], Osadebe plate model [17], and modified plate theories bending and the potential of the external distributed load is [18]. minimized. The plate problem is generally, a boundary value problem Ritz assumed the displacement field w(x, y) in terms of a which is a system of differential equations which are required linear combination of products of basis (shape or coordinate) to be satisfied in the plate domain and the associated boundary functions of the plate in the x and y coordinate directions conditions to be satisfied at the plate boundaries [19]. The chosen such that these shape (basis or coordinate) functions plate problem has been successfully solved in the technical satisfy apriori the end conditions along the x and y coordinate literature using two fundamental methods; namely: analytical directions. The Ritz deflection field w(x, y) for plates is closed form methods and numerical or approximate methods. generally given by the double series of infinite terms: Ladeveze [20] introduced a new approach for the analysis and solution of homogeneous, isotropic elastic plates with constant thickness under arbitrary distributed flexural loads. w(,) x y wmn X m()() x Y n y (1) His theory was described as exact because it lead to exact mn11 values of the generalized two dimensional quantities. Contrary to classical plate theories, Ladeveze’s approach was not where Xm(x) and Yn(y) are the basis (coordinate or shape) limited to thin plates. Analytical closed form methods that functions in the x and y coordinate directions, respectively; and have been used in solving the plate problem include the double wmn are unknown parameters of the displacement field, which trigonometric series method by Navier [4], the single are sought. wmn are called generalized displacement trigonometric series method by Levy [21]; closed form parameters. solutions have also been obtained by Nadai [22]. For Kirchhoff plates under transversely applied static loads, Other researchers that have presented closed form the total potential energy functional for small deformation mathematical solutions to the plate problem include Mama et assumptions and linear elastic, isotropic and homogeneous, al [23] who used the finite Fourier sine transform method to material properties, becomes in terms of the displacement field: solve the problem of simply supported rectangular Kirchhoff plates under uniformly distributed transverse loads over the 2 D entire plate surface, uniformly distributed patch load over a wmn X m Y n wmn X m Y n 2(1 ) wmn X m Y n 2 m n m n m n given area of the plate, and point load on the plate. Numerical R methods have also been used to find approximate solutions to the plate bending problem. Nwoji et al [24] used the wmn X m Y n wmn X m Y n dxdy Kantorovich-Vlasov method to solve the simply supported m n m n plate problem for the case of uniformly distributed transverse load applied over the entire plate surface. Osadebe et al [25] p(,) x y wmn X m Y n dxdy (2) applied the Galerkin-Vlasov method to the flexural analysis of R mn simply supported Kirchhoff plates under uniformly distributed transverse loads applied on the entire plate under region. where the primes denote space derivatives of the basis Nwoji et al [26] applied the Galerkin-Vlasov method to the functions. flexural analysis of rectangular Kirchhoff plates with clamped Applying the principle of stationary total potential energy, and simply supported edges for the case of uniformly Ritz variational method requires that for equilibrium, the total distributed transverse loads. Ike [27] used the Kantorovich- Euler-Lagrange-Galerkin’s method to solve the flexural potential energy functional should be minimized with respect to the unknown displacement field. Hence, for an
2 extremum of with respect to wmn, the following condition is imposed:
0 (3) wmn
This yields a system of algebraic equations: given generally by:
kmn w mn Fmn (4) m1 n 1 m1 n 1
where kmn are stiffness and Fmn are force terms Figure 1. Rectangular Kirchhoff plate with simply supported and edges carrying hydrostatic load in the x-coordinate direction
kmn XYXY mnkl XXYY mknl XXYY mknl XXYY mknl A suitable displacement field that satisfies all the geometric R and force boundary conditions of simple supports at the four 2(1 )XYXY XXYY dxdy (5) edges is given by: m n k l m k n l
m x n y 1 w(,) x y Cmn sin sin (9) Fmn p(,) x y Xm Y n dxdy (6) ab D mn11 R where m = 1, 2, 3, … 22 kmn ( Xm Y n )( X k Y l ) (1 ) Xm X k Y n Y l n = 1, 2, 3, …
X X Y Y2 X X Y Y dxdy (7) m k n l m k n l Cmn are the generalized displacement coordinates of the displacement function which are unknowns, and which we An approximation to the displacement field is achieved in seek to determine in order to find the deflection function w(x, the Ritz variational method by a truncation of the Ritz y). The total potential energy functional for a Kirchhoff displacement field such that m and n are finite values. Thus, m plate made with homogeneous isotropic material is given by = 1, 2, … M; n = 1, 2, ... N. Consequently, the matrix equation becomes a finite system UVb (10) of algebraic equations in terms of the unknown displacement parameters wmn, which can be solved by matrix inversion, and where Ub is the strain energy functional for bending of the other methods of matrix algebra. Kirchhoff plate and V is the potential energy functional due to the applied transverse load q(x, y).
4. APPLICATION OF THE RITZ VARIATIONAL D METHOD U ( 2w ) 2 2(1 )(w2 w w) dxdy (11) b 2 xy xx yy Rxy Consider a rectangular Kirchhoff plate with isotropic, 2 homogeneous material properties and all edges simply 2 where supported. The plate is subjected to a hydrostatic load xy22 distribution of intensity q(x); where q() x q x where z 0 a 2w w q0 is the intensity at x = a, and q(x) acts over the entire area of xy xy the plate surface. The plate is defined by the Cartesian coordinate system shown in Figure 1, with the origin chosen 2w at a corner of the plate. The plate domain is defined by wxx x2 0x a , 0 y b 2 where a, and b are the length and width of the plate, w wyy respectively. The geometric and force boundary conditions are y2
w( x 0, y ) w ( x a , y ) 0 D is the plate flexural rigidity, and is the Poisson’s ratio w( x , y 0) w ( x , y b ) 0 of the plate material. (8) w( x 0, y ) w ( x a , y ) 0 Rxy is the two dimensional domain of the plate defined by xx xx 0x a , 0 y b , z 0 wyy ( x , y 0) wyy ( x , y b ) 0
3 ab 222 V qxywxydxdy(,)(,) (12) m n m x n y 4 2 2 2 Rxy IC1 mn sin sin ab22 ab 00 mn DD22 (13) Ub () wxx w yy dxdy 2(1 )(wxy w xx w yy ) dxdy 22
D (14) U( w w )22 dxdy D (1 ) ( w w w) dxdy b 2 xx yy xy xx yy m x n y m x n y RRxy xy sin sin sin sin dxdy (25) a b a b D 2 2 2 (15) U ( w ) dxdy D (1 ) ( w w w) dxdy b 2 xy xx yy mx ny RRxy xy The sinusoidal functions sin and sin are a b D orthogonal functions, and from the properties of orthogonal UIDIb 12 (1 ) (16) functions we obtain 2 a 22 m x m x where I1 () w dxdy (17) sin sindx 0 for mm (26) aa Rxy 0
22 b I() w w w dxdy (18) n x n y 2 xy xx yy sin sindy 0 for nn (27) R bb xy 0 where w2 w w is called the Gaussian curvature of the The orthogonality properties of the sinusoidal basis xy xx yy functions of the plate lead to the vanishing of the second term plate. in the expression for the integral I1 and the simplification of I1 to become: 2 2 2 2 m x n y 2 ()wC mn sin sin (19) ab m22 n m x n y ab I 4 C 2sin 2 sin 2 dxdy (28) mn 1 22mn mnab ab 00 2 2 2 2 2 m x n y 22 ab ()wC sin sin (20) 4 m n 2 2m x 2 n y (29) mn I1 Cmn sin sin dxdy ab 22 ab mn mnab 00
2 2 22 22 ab 22 m n m x n y (21) 4 m n 2 2 m x2 n y (30) ()wC mn sin sin I1 Cmn sindx sin dy a b a b 22 ab mn mnab 00
2 2 m22 n m x n y 22 ()2wC 2 2 sin sin (22) 42m n a b 22mn IC1 mn (31) mnab ab 22 mnab 22
2 22 2 2 2 4 m n m x n y 4 2 2 ()wC sin sin (23) ab m n 22mn ab 2 mnab IC1 mn (32) 4 ab22 mn 2 22 2 2 4 m n 2 2m x 2 n y ()wC mn sin sin Similarly, ab22 ab mn 2 m2 n 2 m 2 n 2 I() w w w dxdy (33) CC 2 xy xx yy 2 2 2 2 mn m n m n m n a b a b 2 m x n y m x n y mn m x n y sin sin sin sin (24) wC22 cos cos (34) a b a b xy mn mnab a b
Hence,
4 2 mn m x n y ab wC2 4 2cos 2 cos2 (35) m x n y xy mn where I3 x sin sin dxdy (49) mnab a b ab 00 ab 2 m x n y m m x n y w w C sin sin l I3 xsin dx sin dy (50) xx yy mn ab mn a a b 00
2 n m x n y 2 a b C sin sin (36) a m x ax m x b n y (51) mn I sin cos cos b a b 3 mn m a m a n b 0 0
22 m n m x n y 2 2 4C 2sin 2 sin 2 (37) a a b (52) 22 mn I3 sinm cos m (cosn 1) ab m m n mn ab
2 (38) 2 4 mn 2 2m x 2 n y2 m x 2 n y m12 wxy w xx w yy Cmn cos cos sin sin qC ( 1)ab 2 ab a b a b 0 mn mn Ve (53) amn11 m n 2 I2 () wxy w xx w yy dxdy (39) m1 2q abC ( 1) 1 V 0 mn (54) 2 (40) e 2 4 mn 2 2m x 2 n y2 m x 2 n y mn I2 Cmn cos cos sin sin dxdy mn11 ab a b a b mn
where m = 1, 2, 3, 4, 5, … 2 ab 4 mn 2 2 m x2 n y n = 1, 3, 5, 7, 9, … I2 Cmn cos cos dxdy ab a b The total potential energy functional is given in Equation mn 00 (10) ab 22m x n y sin sin dxdy (41) 2 4 2 2 m1 ab ab D m n 2 2q0 abCmn ( 1) (55) 00 Cmn 8 ab2 2 2 mn m n m n m = 1, 2, 3, …;n = 1, 3, 5, 7, … I2 = 0 (42)
2 fC() (56) D D4 ab m2 n 2 mn UI C 2 (43) b 1 22mn 2 2 4 mnab For extremum of with respect to the generalized coordinates, the principle of minimization of the total potential energy functional with respect to Cmn requires that: 4 2 2 2 ab D m n UC2 (44) b 22mn 8 mnab 0 (57) Cmn The potential energy functional due to the applied distributed transverse load q(x, y) is ()UV eeU V 0 (58) ab CCCmn mn mn V q(,)(,) x y w x y dxdy (45) e 2 4ab D m2 n 2 2q ab ( 1)m1 (59) 00 20C 0 C 8 2 2 mn 2 mn mn m n ab m n ab qx m x n y 2 0 C sin sin dxdy (46) 4 2 2 m1 mn ab D m n 2q0 ab ( 1) (60) a a b Cmn 00 mn 2 2 2 4 m n ab m n mn
ab m1 q0 m x n y 2q ab ( 1) Cmn x sin sin dxdy (47) 0 2 a mn a b mn 00 (61) Cmn 2 q 4abD m 2 n 2 0 CI (48) mn 3 22 a mn 4 ab
5 m1 m1 mn 8q0 ( 1) 8qa4 ( 1) sin sin (71) C (62) w w x ab, y 0 22 mn 2 c 22 62 22 D mn mn m22 n 6 mn D mn ab22 mn2 Hence, 8qa4 ( 1)m1 ( 1) 2 0 (72) m1 m x n y wc 62 8q ( 1) sin sin D 22 0 ab mn mn m n w(,) x y (63) 222 mn 6 mn Dmn22 For m = 1, n = 1, and square plate, 1 ab 44 m x n y m1 2 q a q a sin sin ( 1) 00 8q wc 0.002038 (73) 0 ab (64) 6 DD w(,) x y 62 D mn mn22 mn22 For square plates, summing up to m = n = 3, the center ab 4 qa0 Let b deflection is obtained as wc 0.002024 which is a D not significantly different from the center deflection for a one m1 m x n y 44 ( 1) sin sin term Ritz variational solution. 8qa w(,) x y 0 ab 62 Bending moment distribution D mn mn m2 2 n 2 (65) The bending moment distribution Mxx and Myy are given by:
m1 m x n y ( 1) sin sin 22 8 ww ab4 MD (74) w(,) x y q0 b xx 22 622 2 2 xy D mn mn m n 22 ww (66) MD (75) yy 22 a yx Let 1 b m1 m x n y From Equation (66), ( 1) sin sin 8 ab4 w(,) x y q a 2 62 0 m m x n y m1 2 2 2 24sin sin ( 1) D mn mn m n w8 qb a a b (76) 1 w xx 2 6 2 xDmn mn m2 2 n 2 (67) For square plates, 1, 2 n m x n y sin sin ( 1)m1 m1 m x n y 24 (77) 4 ( 1) sin sin w8 qb b a b wyy 8qa0 ab 2 6 2 2 2 2 w(,) x y yDmn mn m n 62D 22 mn mn m n Thus, (68) 22 Deflection at the plate center xyab, m n m x n y ( 1)m1 sin sin 22 4 8qb a b a b (78) The deflection at the plate center is given by Mxx 622 2 2 mn mn mn m n ( 1)m1 sin sin ab8 w x, y 22q a4 62 0 2222 22 D mn mn m2 n 2 2 m n m n 1 2 (79) 22 (69) ab ab
m1 mn ( 1) sin sin 8 22 4 22 qb0 (70) 2 mn 622 2 2 (80) D mn mn m n 22 bb 2 For square plates, the center deflection is given by
6 2 2 2 222 2 2 2 2 2 n m n m2 n m (90) 2 mn (81) ba2 2 2 2 22 b a b a bb nm2 2 2 2 2 nm2 2 2 (91) 2 2 2 2 2 2 b b b 2 mn (82) b2 m1 2 2 2 m x n y 42( 1) (nm )sin sin 8qb0 ab (92) Myy 6 2 2 2 2 2 m1 2 2 2 m x n y b mn mn() m n 42( 1) (mn )sin sin 8qb ab (83) Mxx 6 2 2 2 2 2 b mn mn() m n m x n y ( 1)m1 (nm 2 2 2 )sin sin 8qb2 0 ab (93) m1 2 2 2 m x n y 4 2 2 2 2 2 ( 1) (mn )sin sin mn mn() m n 8qb ab (84) 4 2 2 2 2 mn mn() m n At the center, At the center, xyab, 22 m1 2 2 2 mn 8qb2 ( 1) (nm )sin sin (94) M xab, y 0 22 xx 22 4 2 2 2 2 mn mn() m n m1 2 2 2 mn 8qb2 ( 1) (mn )sin sin (85) M xab, y 0 22 xx 22 4 2 2 2 2 mn mn() m n mn 1 m1 2 2 2 2 8 ( 1) ( 1) (nm ) 2 (95) mn qb 1 4 2 2 2 2 0 m1 2 2 2 2 mn mn() m n 8 ( 1) (mn )( 1) 2 (86) 4 2 2 2 2 qb0 mn mn() m n 2 Myy xx q0 b (96) 2 Mxx xx q0 b (87) yy is the bending moment coefficient for Myy and is given xx is the bending moment coefficient for Mxx and is given by: by: mn 1 m1 2 2 2 2 mn 8 ( 1) ( 1) (nm ) 1 (97) m1 2 2 2 2 yy 4 2 2 2 2 8 ( 1) ( 1) (mn ) (88) mn mn() m n xx 4 2 2 2 2 mn mn() m n The deflection coefficients for the center of the plate for m Similarly, = 1, n = 1 in the Ritz variational solution are tabulated in Table 1, for various aspect ratios of the rectangular plate. 22 The deflection coefficient for the center deflection of the m1 n m m x n y ( 1) sin sin plate m = 1, 3; n = 1, 3 in the Ritz variational solution are 4 b a a b (89) 8qb0 Myy 62 tabulated in Table 2, for various plate aspect ratios. D mn mn m2 2 n 2 Table 1. Deflection coefficients at center of simply supported Kirchhoff plate under hydrostatic load (for m = 1, n = 1) in the Ritz variational method
b qb4 wF()0 a c 1 D 3 1 2.0803 10 3 1.1 2.4945 10 3 1.2 2.8983 10 3 1.3 3.2844 10 3 1.4 3.6485 10 3 1.5 3.9883 10 3 1.6 4.3030 10 3 1.7 4.5929 10
7 3 1.8 4.8590 10 3 1.9 5.1027 10 3 2 5.3256 10 3 3 6.7402 10 3 4 7.3711 10 3 5 7.6935 10 3 8.3213 10
Table 2. Deflection coefficients at center of plate simply supported under hydrostatic load (for m = 1, 3; n = 1, 3 in the Ritz variational method)
Timoshenko and Relative error (%) b F1() a Woinowsky- Krieger 1.0 3 0.00203 0 2.0277 10 0.00203 1.1 3 0.00243 0 2.42963 10 0.00243 1.2 3 0.00282 0 2.82002 10 0.00282 1.3 3 0.00319 0 3.18955 10 0.00319 1.4 3 0.00353 0 3.53307 10 0.00353 1.5 3 0.00386 –0.26 3.8516 10 0.00385 1.6 3 0.00415 –0.24 4.1412 10 0.00414 1.7 3 0.00441 –0.23 4.40325 10 0.00440 1.8 3 0.00465 –0.22 4.6389 10 0.00464 1.9 3 0.00487 –0.41 4.84972 10 0.00485 2 3 0.00506 –0.40 5.03791 10 0.00504 3 3 0.00612 –1.63 6.02264 10 0.00602 4 3 0.00641 –3.12 6.21131 10 0.00621 5 0.00648 0.00651
For m = 1, n = 1 in the Ritz variational solution for , for 2 xx yy 0.02399 0.0240 where Myy yy q0 a . 1, xx 0.02669 0.0267, and for The results are tabulated in Table 3.
1.5 ba / , 0.0446 xx Table 4. Deflection coefficients for hydrostatic load on For m = 1, 3; n = 1, 3, in the Ritz variational solution for 1 simply supported Kirchhoff plate for and for xx 0.02346 0.0235, 1.5, xx 0.0399499 0.03995 where a4 b wq10 3 2 Et Mxx xx q0 a a x = 0.25a x = 0.5a x = 0.75a Similarly, for 1,mn 1,3; 1,3, 1 0.0143 0.0221 0.0177 yy 0.02346 and for 1.5,mn 1,3; 1,3, 1.2 0.0203 0.0308 0.0241 1.4 0.0257 0.0385 0.0298 Table 3. Bending moment coefficients for m = 1, 3; n = 1, 3 1.6 0.0303 0.0453 0.0346 in the Ritz variational method 1.8 0.0342 0.0508 0.0385 2 0.0373 0.0553 0.0417 3 0.0454 0.0668 0.0498 Exact Relative error yy (%) 4 0.0477 0.0700 0.0521 1.0 0.0235 0.0239 –1.67 0.0484 0.0711 0.0529 1.5 0.0240 0.0249 –3.61
8 Convergence to exact solution for the bending moment using two terms each of the double series for deflection yielded a relative error of less than 1% for plates with aspect coefficients xx and yy is achieved using m = 1, 3, 5, 7, 9; ratios less than 2. Convergence to the exact solution obtained n = 1, 3, 5, 7, 9 in the solution obtained by the Ritz variational by Timoshenko and Woinowsky-Krieger, who used Levy’s method. The converged results for deflection and bending single Fourier series method, was obtained by using three moment coefficients for various values of the ratio b/a are terms of the double series for displacement. Similarly, presented for x = 0.25a, x = 0.5a and x = 0.75a, for the convergence to the exact solution obtained by Timoshenko deflection coefficients and for x = 0.5a for the bending and Woinowsky-Krieger was obtained by using five terms of moment coefficients and tabulated in Tables 4 and 5. the double series for bending moments. However, reasonably accurate solutions were obtained for both deflection and Table 5. Bending moment coefficients for hydrostatic load bending moments by using a few terms each, of the double x series obtained using the Ritz variational method. q0 on simply supported Kirchhoff plate a
2 2 6. CONCLUSIONS M q a Myy yy q0 a b xx xx 0 a xx (xa 0.5 ) yy (xa 0.5 ) The following conclusions can be made: 1.0 0.0239 0.0239 (i) the Ritz variational method can be successfully 1.2 0.0313 0.0250 applied to the solution of rectangular Kirchhoff plate bending 1.4 0.0376 0.0253 problems for simply supported ends, and transverse 1.6 0.0431 0.0246 hydrostatic loading distribution over the entire plate area. 1.8 0.0474 0.0239 (ii) the solutions obtained for the deflection and bending 2 0.0508 0.0232 moment distributions are double sine series with rapidly 3 0.0594 0.0202 convergent properties, thus making the solutions easily 4 0.0617 0.0192 amenable to mathematical analysis 0.0625 0.0187 (iii) the solution obtained for the center deflections is a double series, with rapidly convergent properties; and convergence to the exact solution is obtained using three terms 5. DISCUSSIONS each of the double series. (iv) the solutions obtained for the bending moments Mxx The Ritz variational method has been successfully applied and Myy at the plate center are double series with good to the flexural problems of simply supported rectangular convergence properties and convergence to the exact solution Kirchhoff plates subjected to hydrostatic load distribution of is obtained using five terms each of the double series (v) the Ritz variational method yielded exactly identical intensity q() x q x over the entire area 0 a solutions as the exact solutions obtained by Timoshenko and (0x a , 0 y b ) of the plate surface. A Woinowsky-Krieger who used the Navier double Fourier sine displacement field was chosen as products of linear series method. Hence, the Ritz variational method presented in combination of displacement shape or coordinate functions this work yielded the exact solution for the plate flexure that apriori satisfied boundary conditions at the plate contours problem analyzed. This is attributable to the fact that the exact deflection shape function was used in the Ritz method and unknown generalized displacement parameters, Cmn, as Equation (9). The total potential energy functional was then presented in this study. determined using the chosen displacement field as Equation (55). It was observed that the total potential energy functional obtained is a double series, and is a function of the generalized REFERENCES displacement coordinates. The principle of minimum total potential energy functional was then applied to obtain the [1] Chandrashekhara K. (2011). Theory of Plates. unknown generalized displacement coordinates as Equation University Press, Hyderabad, India. (64). Thus, the displacement field was obtained as Equation [2] Timoshenko S, Woinowsky-Krieger S. (1959). Theory (64). The plate deflection was expressed in terms of the plate of Plates and Shells, 2nd Edition. McGraw Hill Book Co. New York. aspect ratios and 1 as Equations (66) and (67) [3] Szilard R. (2014). Theories and Applications of Plates respectively. The plate deflection was also obtained for square Analysis: Classical, Numerical and Engineering Kirchhoff plates as Equation (68). The deflection was found at Methods. John Wiley and Sons Inc. the center of the plate as Equations (69) and (70), for [4] Navier CLMN. (1823). Bulletin des science de la rectangular Kirchhoff plates, and as Equation (72) for square societe philomarhique de Paris. Kirchhoff plates. Bending moment-displacement relationships [5] Poisson SD. (1829). Sur les Equations generales de Equations (74) and (75) were used to obtain the bending l’Equilibre et du Mouvement des corps solides moment distribution on the plate as Equations (84) and (89). elastiques et des fluids. Journal de l’Ecole The bending moments at the plate center were found as Polytechnique, 13(20): 1-174. Equations (86) and (95). Equations for the deflections and [6] Poisson SD. (1829). Memoire sur l’Equilibre et le bending moment distributions were found as double series mouvement de corps elastiques. Mem. de l’Acad. Sci 8: with rapidly convergent properties. The equations for 357. deflections were more rapidly convergent that the equations for bending moments. The Ritz variational solutions obtained
9 [7] Kirchhoff G. (1850). Uber die Schwingnngen eigner [22] Nadai A. (1925). Die Elastichen Platten. Springer- kreisformigen elastischen sheibe annalen der physic. Verlag, Berlin. 157(10): 258-264. [23] Mama BO, Nwoji CU, Ike CC, Onah HN. (2017). [8] Hencky H. (1913). Der Spannungszustand in Analysis of simply supported rectangular kirchhoff Rechteckigen Plattern R. Oldernbourg, Munich, Berlin. plates by the finite fourier sine transform method. [9] Hencky H. (1947). Uber die Berucksichti gnng der International Journal of Advanced Engineering schubuerzerrungen in ebenen plattern Ing Arch 16. Research and Science (IJAERS) 4(3): 285-291. [10] Reissner E. (1945). The effect of transverse shear [24] Nwoji CU, Mama BO, Onah HN, Ike CC. (2017). deformation on the bending of elastic plates. Journal of Kantorovich-vlasov method for simply supported Applied Mechanics 12: 69-77. rectangular plates under uniformly distributed [11] Reissner E. (1944). On the theory of bending of elastic transverse loads. International Journal of Civil, plates. Journal of Mathematics and Physics 23: 184-191. Mechanical and Energy Science (IJCMES) 3(2): 69-77. [12] Mindlin RD. (1951). Influence of rotary inertia and [25] Osadebe NN, Ike CC, Onah HN, Nwoji CU, Okafor FO. shear on flexural motions of isotropic elastic plates. J. (2016). Application of Galerkin-Vlasov Method to the Appl. Mech. 18(1): 31-38. flexural analysis of simply supported rectangular [13] Shimpi RP. (2002). Refined Plate Theory and its Kirchhoff plates under uniform loads. Nigerian Journal variants. AIAA Journal 40: 137-146. of Technology (NIJOTECH) 35(4): 732-738. [14] Levinson M. (1980). An accurate simple theory of [26] Nwoji CU, Mama BO, Ike CC, Onah HN. (2017). statics and dynamics of elastic plates. Mechanics Galerkin-Vlasov method for the flexural analysis of Research Communications 7: 343-350. rectangular Kirchhoff plates with clamped and simply [15] Reddy JN. (1984). A refined non linear theory of plates supported edges. IOSR Journal of Mechanical and Civil with transverse shear deformation. International Engineering (IOSRJMCE) 14(2): 61-74. Journal of Solids and Structures 20: 881-896. [27] Ike CC. (2017). Kantorovich-Euler-Lagrange- [16] Leung AYT. (1991). An unconstrained third order Galerkin’s method for bending analysis of thin plates. theory. Computers and Structures 40(4): 871-874. Nigerian Journal of Technology 36(2): 351-360. [17] Osadebe NN. (1997). Differential equations for small [28] Mama BO, Onah HN, Ike CC, Osadebe NN. (2017). deflection analysis of thin elastic plates possessing Solution of free harmonic vibration equation of simply extensible middle surface. University of Nigeria Virtual supported Kirchhoff plate by Galerkin-Vlasov method. Library. Nigerian Journal of Technology 36(2): 361-365. [18] Suetake Y. (2006). Plate bending analysis by using a [29] Ezeh JC, Ibearugbulem OM, Onyechere CI. (2013). modified plate theory. CMES 2(3): 103-110. Pure bending analysis of thin rectangular flat plates [19] Sebastian VK. (1983). An elastic solution for simply using ordinary finite difference method. International supported rectangular plates. Nigerian Journal of Journal of Emerging Technology and Advanced Technology (NIJOTECH) 7(1): 11-16. Engineering (IJETAE) 3(3). [20] Ladeveze P. (2002). The exact theory of plate bending. [30] Aginam CH, Chidolue CA, Ezeagu CA. (2012). Journal of Elasticity 68(1): 37-71. Application of direct variational method in the analysis [21] Levy M. (1899). Memoire sur la theorie des plaque of isotropic thin rectangular plates ARPN. Journal of elastiques planes. Journal de Mathematiques Pures et Engineering and Applied Sciences 7(9). Appliquees, 3: 219.
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