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 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 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 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 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 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 , 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 mn11 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) m1 n  1 m1 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  mn11 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 0x  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 0x  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 mnab 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 mnab 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     mnab 00

2 2 m22 n m x n y 22 ()2wC 2   2  sin sin (22) 42m n a b 22mn IC1    mn (31) mnab ab 22 mnab 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 mnab 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) mnab 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  m12 wxy w xx w yy    Cmn cos cos sin sin  qC ( 1)ab 2  ab a b a b 0 mn mn    Ve   (53) amn11 m n 2 I2 () wxy w xx w yy dxdy (39)  m1 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 mn11  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   m1 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 D4 ab m2 n 2 mn UI   C 2 (43) b 1 22mn 2 2 4 mnab 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 UC2 (44) b 22mn  8 mnab  0 (57) Cmn The potential energy functional due to the applied distributed transverse load q(x, y) is ()UV  eeU 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)m1 (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   m1   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  m1 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 m1 m1 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  mn2 Hence, 8qa4 ( 1)m1 ( 1) 2 0 (72) m1 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 Dmn22 For m = 1, n = 1, and square plate, 1 ab 44 m x n y m1 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 mn22 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 m1 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:

m1 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 m1 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 m1 2 2 2 24sin 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)m1 m1 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 xyab, m   n   m  x n  y   ( 1)m1   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)m1 sin sin ab8  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

m1 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  m1 2 2 2 m x n y 42( 1) (nm    )sin sin 8qb0  ab (92) Myy  6 2  2 2 2 2 m1 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)m1 (nm 2   2  2 )sin sin 8qb2   0 ab (93) m1 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, xyab, 22 m1 2 2 2 mn 8qb2 ( 1) (nm    )sin sin (94) M xab, y   0 22 xx  22 4  2 2 2 2  mn mn() m  n m1 2 2 2 mn 8qb2 ( 1) (mn    )sin sin (85) M xab, y   0 22 xx  22 4  2 2 2 2  mn mn() m  n mn 1  m1 2 2 2 2 8 ( 1) (  1) (nm    ) 2 (95) mn  qb 1 4  2 2 2 2 0  m1 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  m1 2 2 2 2 mn 8 ( 1) (  1) (nm    ) 1  (97)  m1 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 m1 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 wq10 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 (0x  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 . 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

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