Research Article INTEGRATED FORCE METHOD FOR FIBER REINFORCED COMPOSITE PLATE BENDING PROBLEMS Doiphode G. S., Patodi S. C.* Address for Correspondence Assistant Professor, Applied Mechanics Department, Faculty of Tech. & Engg., M. S. University of Baroda, Vadodara - 390 001, Gujarat, India Professor,Department of Civil Engineering, Parul Institute of Engg. and Tech., Limda, Vadodara - 391 760, Gujarat, India ABSTRACT Solution to fiber reinforced composite plate problem under different types of loading and boundary conditions is attempted using Integrated Force Method (IFM). Internal moments developed due to external loading are calculated by imposing simultaneously the equilibrium equations and compatibility conditions. Various closed form matrices are developed by discretizing the potential strain energy for a four noded rectangular element having 9 force and 1 displacement degrees of freedom; which is named as RECT_9F_1D element. Using symmetry, the influence of aspect ratio on internal moments and deflections is studied. Matlab is exploited for plotting the moment contours and deflected shape of transversely loaded plates. The numerical results obtained for twelve Glass Fiber Reinforced Plastic (GFRP) plate problems are compared with the available small deflection theory results to validate the proposed formulation and its computer implementation; a good agreement is found. KEY WORDS: IFM, Orthotropic plate, Matlab, Moment contours. INTRODUCTION In many practical applications of orthotropic plates, rather than following hypothetical assumptions, it is important to consider actual directional dependent bending characteristics, behavior and stiffness. Generally speaking anisotropy can be introduced in plate either by modifying material constituents or by providing ribs, stiffeners or corrugations for which an exact analytical solution in terms of deflection and internal moments exists only for some simple boundary conditions [1]. The complete solution of plates with clamped boundary conditions is possible only through numerical or approximate method such as Rayleigh Ritz method []. Other methods such as Differential Quadrature Method, Modified Spline Function Method, Superposition Technique and Modified Fourier Series Method consist of very complex and tedious mathematical formulation [3]. Thus, there is a need of a method, which can be easily applied to any type of boundary condition and can easily handle any type of loading condition. Direct Moment Calculation (DMC) approach, which is the key aspect of Integrated Force Method (IFM) is well established since last few decades [4, 5]. The present work includes basic equations of IFM for a four noded rectangular plate bending element. Different closed-form matrices such as Global Equilibrium Matrix [S], Global Flexibility Matrix [G] and Auto generated Matlab based Global Compatibility Matrix [C] are derived by discretizing the expressions of potential and complimentary strain energies. The familiar process of differentiation used in popular stiffness method to evaluate stresses from displacements is avoided in the integrated force method. These matrices are assembled using the traditional procedure. Total twelve problems of orthotropic plate bending with variation in geometrical dimensions, boundary conditions and loading are considered. Using symmetry, quarter plate is discretized in 5 x 5 mesh. Nodal displacements and internal moments are calculated. Using Matlab software [6] moment contours and deflected shape for each case are plotted with maxima/minima values. Influence of aspect ratio for fixed and simply supported edge conditions under central point load and uniform lateral pressure are studied. Results obtained by the proposed method are compared with those based on classical small deflection theory [7]. FUNDAMENTAL THEORY OF IFM The IFM equations for a continuum discretized into finite number of elements with n and m force and displacement degrees of freedom respectively, are obtained by coupling the m` number of equilibrium equation and r =n m compatibility conditions. The m equilibrium equations (EE) are written as [4, 5] [B] {F} = {P} (1) and the r compatibility conditions are written as [C] [G] {F} = {δr} () These conditions are combined to obtain the IFM governing equations as [ B] [ ][ ] { } { P} { } F = Or [S] {F} = {P}. (3) C G δr From forces {F}, displacements {X} are calculated using the following equation. {X} = [J] {[G] {F} + {β 0 } (4) where, [J] = m rows of [[S] -1 ] T, [B] is of m x n size matrix, which is sparse and unsymmetrical, [G] is the symmetrical flexibility matrix; it is a block-diagonal matrix where each block represents a flexibility matrix for an element, [C] is the compatibility matrix of size r x n, {δr} = -[C] {β} 0 is the effective deformation vector with {β} 0 being the initial deformation vector of dimension n, [S] is the IFM
governing unsymmetrical matrix of size n x n and [J] is the m x n size deformation coefficient matrix which is calculated from [S] matrix. IFM has following 3 matrices that are equivalent to the operators of classical elasticity theory: 1. The equilibrium matrix [B] which links internal forces to external loads,. The compatibility matrix [C] which governs the deformations and 3. The flexibility matrix [G] which relates deformations to forces. Both the equilibrium and compatibility matrices of the IFM are unsymmetrical, whereas the material constitutive matrix and the flexibility matrix are symmetrical. FORMULATION FOR A RECTANGULAR ELEMENT Element Equilibrium Matrix [Be] represents the curvatures corresponding to each internal moment. Consider a 4-noded, 1 displacement degrees of freedom (ddof) (X 1 to X 1 ) rectangular element of thickness t with dimensions as a x b along the x and y axes as shown in Figs. 1 and. The force field is chosen in terms of nine independent forces as (7) Relations between internal moments and independent forces are written as Arranging Eqs. (8) to (10) in matrix form,... (8)... (9)... (10) Fig. 1 Rectangular Element or { } [ Y]{ } (11) M = F e... (1) where {F e } is the element force vector. The variation of above moments is considered bilinear along both the directions. The displacement field satisfies the continuity condition and the selected forces also satisfy the mandatory requirement. Polynomial function for lateral displacement can be written as U Fig. Nodal Displacements The elemental equilibrium matrix written in terms of forces at grid points represents the vectorial summation of n internal forces {F} and m external loads {P}. The nodal equilibrium equation in matrix notation can be stored as rectangular matrix [B e ] of size m x n. The variational functional is evaluated as a portion of IFM functional which yields the basic elemental equilibrium matrix [B e ] in explicit form as follows: p = D M x = { } D w + M x T y w + M y xy w dxdy x y (5) M { ε}ds (6) Where, {M} T =(M x, M y, M xy ) are the in-plane internal moments and {Є} T = (13) Or w(x,y) = [A]{α}.. (14) where [A] is a function of x and y matrix and {α} is a vector of constants. Substituting coordinates, one can find constants and finally the interpolation matrix [N] following the usual finite element procedure. Here each component of [N] is associated with nodal displacements X 1, X, -----------X 1 as shown in Fig.. Eq. 14 is now expressed in terms of nodal displacements as w(x,y) = [N]{X}... (15) By arranging all force and displacement functions properly, one can discretize the Eq. 5 to obtain the elemental equilibrium matrix as U e = {X} T [B e ]{F}... (16) where [B e ] = s [Z] T [Y] ds... (17) Here [Z] = [L][N] where [L] is the differential operator matrix, [N] is the displacement interpolation function matrix and [Y] is the force interpolation function matrix. The integration yields the following non symmetrical equilibrium matrix [B e ] for the element.
[B e ]= G e 77 =, G e 88 =, G e 99 = (18) Elemental Flexibility Matrix (Orthotropic) [G e ortho]: The element flexibility matrix is obtained by discretizing the complementary strain energy as [G e ortho]= s [Y] T [D ortho ][Y] dx dy (19) where [D ortho ] is the orthotropic plate material matrix of size ( 3 x 3 ) which is written as (0) where, D x = - E x h 3 /1(1- and D y = - E y h 3 /1(1- are the plate rigidities along x and y directions respectively, D xy = G xy h 3 /1 is torsional rigidity in xy plane, E x, E y and G xy are the Young s modulus of elasticity in x and y directions and shear modulus of the orthotropic material in shear plane respectively, are the Poisson s ratios along orthogonal axes and h is the thickness of plate. Substituting matrices given in Eqs. 11 and 0 in Eq. 19 and integrating within domain (a x b) yields the symmetrical flexibility matrix [G e ortho] of size (9 x 9). The nonzero components of the matrix are as follows: G e 11 =, G e 15 = G e 51 = - 11, G e =, G e 6 = G e 6 = - G e, G e 33 =, G e 37 = G e 73 = - G e 33, G e 44 =, G e 48 = G e 84 = - 44, G e 55 =, G e 66 =, Global Compatibility Matrix [C] The compatibility matrix is obtained from the deformation displacement relation ({β} = [B] T {X}). In DDR all the deformations are expressed in terms of all possible nodal displacements and the r compatibility conditions are developed in terms of internal forces i.e., F 1,------ F n, where n is the total number of internal forces in a given problem. The global compatibility matrix [C] can be evaluated by multiplying the coefficients of compatibility conditions developed by using Matlab and the global flexibility matrix [G]. NUMERICAL EXAMPLES Results of twelve different examples of laterally loaded glass fiber reinforced plastic plate are presented to validate the suggested formulation and computer implementation of integrated force method. For numerical solution, a plate of size m x m of thickness t = 3mm and having Young s modulus in x and y directions as 40kN/mm and 8kN/mm respectively and Poisson s ratio υ xy as 0.5 is considered. The shear modulus in xy plane is considered as 4kN/mm. Due to double symmetry, only quarter of the plate is discretized into 5 x 5 mesh. The following variations are considered in solving the rectangular plate problems. 1. Element aspect ratio R = 1.0, 1.5 and.0.. Central Point Load (CPL) of 10kN and Uniform Lateral Pressure (ULP) of 10kN/m. 3. Support conditions: Fully clamped and Simply supported. Following steps are used for solving the above problems: Step 1: A four-noded rectangular element (a x b) with 1 ddof and 9 fdof is used to discretize the problem using two- way symmetry. The elemental [B e ] matrix is obtained by substituting a = 0.1m, b = 0.1m for orthotropic case in Eq. 18 considering 5 x 5 discretization scheme. After assembling, the size of global equilibrium matrix will be 75 x 5. Step : The compatibility matrix for the twenty five elements is obtained from the displacement deformation relations (DDR) i.e. β = [B] T {X}. In the DDR, 5 deformations which correspond to 5 force variables are expressed in terms of 75 displacements (X 1, X, -------- X 75 ). The problem requires 150 compatibility conditions [C] that are obtained by eliminating the 75 displacements from the 5 DDR using auto-generated Matlab based computer program by giving input as upper part of the global equilibrium matrix. Step 3: The global flexibility matrix [G] for the problem is obtained by diagonal concatenation of the
twenty five elemental flexibility matrices calculated as per Eq. 19. Step 4: By multiplying compatibility matrix [C] and global flexibility matrix [G], bottom most part of the global equilibrium matrix is obtained. Assembling both the matrices gives complete [S] matrix of size 5 x 5, which comprises of EE and CC. The forces are obtained by using MatLab s Pseudo inverse command. Step 5: The displacements are calculated by using the relation ({X} = [J][G]{F}), where [J] = m rows of matrix [[S] -1 ] T. Linear static analysis is carried out for all the plate problems and nodal displacements and internal moments are calculated at each node of 5 x 5 discretization pattern. Tables 1 and show deflection and moments Mx, My and Mxy at the centre of plate for various aspect ratios. The values are compared with those given by classical plate theory [1,7]. Two dimensional moment contours as well as three dimensional deflected shapes are drawn for plate using Matlab software. Figures 3 and 4 show deformation patterns for simply supported plate whereas Figs. 5 and 6 show the same for clamped plate under central point loading and uniform lateral pressure. Figures 7 thru 9 show moment contours for Mx, My and Mxy respectively for simply supported plate subjected to central point loading. LOAD CPL ULP LOAD CPL ULP Table 1 Results obtained for Simply Supported GFRP Plate R w c (m) Mxc ( N-m) Myc ( N-m) Mxyc( N-m) IFM Exact 1 IFM Exact 1 IFM Exact 1 IFM Exact 1 1.0 0.0105 0.0105 -.09 -.1-0.418-0.439-0.00-0.01 1.5 0.014 0.01373 -.554 -.99-0.511-0.599-0.037-0.04.0 0.0141 0.01790-3.11-3.76-0.83-0.931-0.059-0.069 1.0 0.067 0.0679-5.340-5.60-1.063-1.095-0.056-0.059 1.5 0.078 0.03160-5.445-5.79 -.069 -.076-0.066-0.071.0 0.0395 0.04450-6.01-6.55 -.078 -.091-0.078-0.089 Table Results obtained for Fully Clamped GFRP Plate R w c (m) Mxc ( N-m) Myc ( N-m) Mxyc( N-m) IFM Exact 7 IFM Exact 7 IFM Exact 7 IFM Exact 7 1.0 0.001 0.00-0.163-0.17-0.041-0.043-0.089-0.09 1.5 0.006 0.008-0.181-0.1-0.05-0.058-0.091-0.095.0 0.0034 0.0039-0.17-0.9-0.094-0.104-0.098-0.11 1.0 0.0078 0.0079-0.591-0.61-0.147-0.15-0.044-0.046 1.5 0.0081 0.0087-0.633-0.69-0.38-0.5-0.056-0.069.0 0.009 0.0130-0.717-0.80-0.3-0.389-0.076-0.094 Fig. 3 Deformed Shape of a SS Plate under CPL
Fig. 4 Deformed Shape of a SS Plate under ULP Fig. 5 Deformed Shape of a Clamped Plate under CPL Fig. 6 Deformed Shape of a Clamped Plate under ULP
Fig. 7 Contours of Mxx for SS Orthotropic Plate under CPL along yy Axis along yy Axis Fig. 8 Contours of Myy for SS Orthotropic Plate under CPL
along yy Axis Fig. 9 Contours of Mxy for SS Orthotropic Plate under CPL CONCLUSIONS A method of analysis, which imposes simultaneously equilibrium equations and compatibility conditions to calculate internal moments, is presented for the analysis of transversely loaded fiber reinforced composite plates. Results obtained for a number of examples with change in aspect ratio, boundary condition and loading condition show good agreement with theoretical results. However, with the increase in aspect ratio, the difference in results is found to increase; which may be improved by finer discretization in the longer direction. The use of Matlab software in conjunction with the software developed based on integrated force method in Visual Basic makes the whole exercise quite attractive as it provides directly graphical output in the form of deflected shape and moment contours. The method has good potential for its application to reinforced concrete slab, stiffened plates and corrugated sheets. REFERENCES 1. Chandrashekhar K., Theory of Plates, Universities Press (India) Ltd., Hyderabad, 001.. Whitney J. M., Structural Analysis of Laminated Plates, Lancaster, PA, Technomic Publishing Co., 1987. 3. Gorman D. J., Free Vibration Analysis of Rectangular Plates using a set of Static Beam Function in Rayleigh-Ritz Method, Journal of Sound Vibration, Vol. 189, No.1, pp.81-87, 1996. 4. Patnaik S. N., Integrated Force Method for Discrete Analysis, International Journal of Numerical Methods and Engineering, Vol. 41, pp. 37-51, 1973. 5. Doiphode G. S., Kulkarni S. M. and Patodi S. C., Improving Plate Bending Solutions using Integrated Force Method, Proceedings of the Sixth Structural Engineering Convention (SEC 008), IIT Chennai, pp. 7 35, Dec. 008. 6. Matlab: Manual on Graphical Tool for D and 3D Plots, The Mathworks, Inc., Natick, MA 01760-098, USA. 7. Bairagi N. K., Plate Analysis, Khanna Publishers, New Delhi, 1986.