Applied Mathematical Sciences, Vol. 9, 5, no. 6, 769-78 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.988/ams.5.575 Numerical Solution of Non-Linear Biharmonic Equation for Elasto-Plastic Bending Plate Feda Ilhan Department of Mathematics Abant Izzet Baysal University, Bolu, Turkey Zahir Muradoglu Department of Mathematics Kocaeli University, Kocaeli, Turkey Copyright 5 Feda Ilhan and Zahir Muradoglu. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A numerical solution for the non-linear boundary value problem of elasto-plastic bending plate by means of finite difference method is proposed. Test functions, as defined along this work, are employed for verifying the applicability of the computer program. Accuracy of the approximate solutions is demonstrated through the numerical examples. Keywords: Non-linear biharmonic equation, elasticity, plasticity, deformation, finite difference method, elasto-plastic plate Introduction Biharmonic equation plays an important role in different scientific disciplines, but it is difficult to solve due to the existing fourth order derivatives. They arise in several areas of mechanics such as two dimensional theory of elasticity and the deformation of elastic and elasto-plastic plates [, ]. Various approaches for the numerical solution of biharmonic equation have been considered in the literature over several decades. But these kind of problems are usually obtained through the use of finite element method which requires the use of nonphysical dissipation. However, the finite element solution is not as stable as the finite difference solution which has not only the ease of grid generation but also dissipative character.
77 Feda Ilhan and Zahir Muradoglu A general variational approach has been constructed by Hasanov [3] for solving the non-linear bending problem of elasto-plastic plate by using monotone operator theory. In this work existence of the weak solution of the non-linear problem in H (Ω) Sobolev space is given and by using finite difference method numerical solution for linear bending problems with various boundary conditions is obtained. Plate structures are most commonly encountered in analyzing engineering structures. In recent years, considerable attempts have been made in the development of numerical methods for the analysis of solid mechanics problems. The elasto-plastic behavior of plates has been analyzed by several numerical methods such as the finite element [, 5], boundary element [6, 7], finite strip [8, 9], meshless based methods [, ] and the others. The essential purpose of this study is to obtain a numerical solution for the non-linear biharmonic equation for elasto-plastic bending plate with different boundary conditions by using finite difference method. The paper is organized as follows. In section the mathematical model and governing equations for the non-linear bending problems are represented. In section 3 finite difference approximation of the problem is derived. In section by using a test function applicability of the finite difference method which has been carried out in Matlab is shown and the numerical algorithm and examples related to the non-linear problem are considered. In section 5 conclusions are given. The mathematical model and governing equations In this paper, bending problem of the plate which is tailored with an elasto-plastic and homogeneously isotropic incompressible features is studied. For simplicity, the plate is supposed to be rectangular. The basic plate equations contain no such shape restriction, but solutions are most easily determined for rectangular plates and circular plates. It is assumed that the plate with thickness is placed to the coordinate system Ox x x 3 such that the middle surface of the plate is located in Ox x plane. The plate is supposed to be in equilibrium under the action of the loads applied on the upper surface of the plate in the x 3 axis direction while its lower surface is free. It is known from the deformation theory of plasticity that [, 3] as ω = ω(x) is the deflection of a point x Ω on the middle surface of the plate, which is placed in the region Ω = {(x, x ) R : x α l α, α =,}, satisfies the following nonlinear biharmonic equation: Aω x [g(ξ (ω)) ( ω x + ω x )] + [g(ξ (ω)) ω ] x x x x + x [g(ξ (ω)) ( ω x + ω )] = F(x), x Ω x R (.) F(x) = 3q(x)/ h 3 and q = q(x) is the intensity of the loads applied on the plate. Boundary conditions which may also be called edge or support conditions include simply supported, clamped, free and a wide variety of conditions applied
Numerical solution of non-linear biharmonic equation 77 to certain applications. Though in practice simply supported condition and especially clamped boundary condition are difficult to enforce this study specifically exploits these two conditions []:. Clamped boundary condition ω(x) = ω (x) = (.) n. Simply supported boundary condition ω(x) = ω n = (.3) where n is the unit outward normal to the boundary Ω. Corresponding to Kachanov's model of elasto-plastic deformation theory for isotropic homogeneous deformable materials the relation between deviations of the stress σ D = {σ ij D } and the deformation ε D = {ε ij D } i, j =,, 3 is described by the Hencky relation [, 3]: σ ij D (u) = g (Γ )ε ij D (u) (.) Consequently, the relationship between the intensities of the shear stress T = (.5σ D ij σ D ij ) and the shear strain Γ = (ε D ij ε D ij ) is expressed as: T = g (Γ ) Γ (.5) where Γ = x 3 [( ω x ) + ( ω x ) + ( ω ) + ω ω x x x x ] (.6) With the aid of [5], the dependent variable ξ = ξ(ω) is expressed as: ξ (ω) = ( ω x ) + ( ω x ) + ( ω x x ) + ω x ω x (.7) Taking into account that g = g (Γ ) and Γ = x 3 ξ, instead of g (Γ ), a new function depending only on ξ is determined as in [5] g(ξ ) = h h 3 g (x 3 ξ ) x 3 dx 3 (.8) h where the function g = g(ξ ) describes the elasto-plastic behaviour of a deformable plate and is called modulus of plasticity. This function is defined as G, ξ ξ g(ξ ) = { G ( ξ κ ) ξ, ξ < ξ (.9)
77 Feda Ilhan and Zahir Muradoglu where ξ is the elasticity limit and κ (,) is the strength hardening parameter. For pure elastic bending, i.e. when ξ ξ, this is a constant function, g(ξ ) = G. Here G = E (( + ν)) is the modulus of rigidity. E > is Young's modulus and ν > is Poisson's ratio. According to the deformation theory of plasticity the coefficient g = g(ξ ) in the nonlinear bending equation (.) satisfies the following bounds [3,, 5, 6 ]: (i) c g(ξ ) c ; (ii) g(ξ ) + g (ξ ) ξ c ; (iii) g (ξ ), ξ [, ξ M ]; (.) (iv) ξ (, ξ M ), g(ξ ) = G, ξ [, ξ ], where c i are positive constants. Consequently, the given relation T = g (Γ ) Γ takes the form T = g(ξ )ξ (.) It is known from deformation theory that M, M are bending and M is twisting moments and they can be presented as []: M = h3 3 g(ξ ) ( ω x + ω x ) M = h3 3 g(ξ ) ( ω x + M = h3 6 g(ξ ) ω x x ω x ) The non-linear equation (.) is derived by substituting the above expressions in the moment equation: M x M M x x = q(x) (.) x 3 The Numerical Scheme To apply any numerical method gaining the solution of the nonlinear bending problem a linearization process is required to be performed. The iteration scheme given in [3] permits to solve the non-linear problem (.)-(.) (or.3) via a sequence of linearized problems. Exploiting this iteration, linearized bending equation can be presented as below: x [g(n ) (ξ ) ( ω (n) + ω (n) x )] + [g (n ) (ξ ) ω (n) ] x x x x x + x [g(n ) (ξ ) ( ω (n) + ω (n) x )] = F(x), x Ω x R (3.)
Numerical solution of non-linear biharmonic equation 773 By using a modification of Samarskii-Andreev finite difference scheme [7] we obtain the most appropriate approximation of the non-linear bending equation (.): (g (n ) h (ξ (n) h ) (y x x + y (n) x )) + (g h x x + {(g (n ) h (ξ (n) h ) y x )x + (g (n ) h (ξ h ) y x x x (n ) (ξ h ) ( y x (n) x )x } x (n) x + y (n) x )) x x + {(g (n ) h (ξ (n) h ) y x x )x + (g (n ) h (ξ (n) h ) y x x )x x x } = F(x ij ) (3.) Here F(x ij ) = 3q(x ij )/h 3 where x ij = (x (i), x (j) ), gh (n ) (ξ h ) = g(ξ h (y (n ) (x))), and ξ h (y) = y x x + y x x +.5(y x x + y x x ) + y x x y x x (3.3) is the finite difference approximation of the effective value of the plate curvature ξ(ω). In (3.) we use standard finite difference notations, so that y(x), x Ω h is a mesh function: y(x) = y(x ij ) = y ij, and y = ( y x ij y i j ) h, y x = ( y i+j y ij ) h, y x x = (y i+j y ij + y i j ) h. After repeating the process for corresponding boundary condition, the finite difference approximation of the linearized problem (3.) i.e. the discrete problem is obtained. Let ω(x) be the solution of the nonlinear problem (.)-(.) (or.3), ω (n) (x) be the solution of the linearized problem (3.)-(.) (or.3) and y (n) ij be the solution of the discrete problem. In order to find the order of approximation of the finite difference scheme (3.) for each n =,, 3, we have ω (n) y (n) = sup ω (n) (x) y (n) ij = O(h ). Numerical Results The first series of the numerical experiments is conducted to verify the accuracy of the finite difference scheme of the discrete equation (3.). For this purpose, two test functions ω(x, x ) = ( cos πx )( cos πx ) (.) and ω(x, x ) = sin πx sin πx (.) which are satisfying the clamped and simply supported boundary conditions respectively are used. For g(x, x ) = e x +x and g = g(ξ (ω)) the test functions are assumed to be analytical solutions of the nonlinear equation (.). The forcing term F(x, x ) is obtained by applying the biharmonic operator to the test functions. Results of the computational experiments on the uniform square meshes with different sizes are given in Table.
77 Feda Ilhan and Zahir Muradoglu The absolute error and the relative error are defined by ε y = ω(x ij ) y (n) ij and δ = (ω(x ij ) y (n) ij ) ω(x ij ) respectively. Table : Absolute and relative errors and approximation number of the finite difference scheme for g(x, x ) = e x +x and g = g(ξ (ω)) using the test function ω(x, x ) = ( cos πx )( cos πx ) Mesh Size N N Absolute Error g(x, x ) = e x +x g = g(ξ (ω)) ( ξ =.7 and κ = Relative Error n εy.5) Absolute Error Relative Error x.7..7.3 3 x 3.6.56.75..5.395 x.35.3.89.6.6.596 5 x 5.89..99. 9.78.768 6 x 6.63.6.997.3 8..7786 7 x 7.7..93.5 6.57.66.35.9.8.87 8 x 8.36 9.35 n εy Here n εy is called the approximation number and is calculated by the formula n εy = ln (ε y(n () )/ε y (N () )) where N (), N () are the size of consecutive meshes, ln (N () N () ) ε y (N () ) and ε y (N () ) are absolute errors for the corresponding meshes. Since the approximation error of the finite difference scheme is O(h ), n εy is expected to be and it is obtained roughly. The results indicate that the accuracy is well. The finite difference method is efficient to deal with the problem. When we analyze Table we see that for the test function (.) which is satisfying clamped boundary condition on the boundary, while the maximum relative error of the approximate solution for the case g(x, x ) = e x +x is found %., it is obtained %.3 for g = g(ξ (ω)) for the mesh of x. In Fig.(a)-(b) the graphics of the approximate solutions for the test function (.) are presented for g(x, x ) = e x +x and g = g(ξ (ω)) respectively.
Numerical solution of non-linear biharmonic equation 775 (a) (b) 7 6 3 5 3.5.8.5.6.. G G G 3.. G 3.6.8 Fig. : Approximate solution of the test function ω(x, x ) = ( cos πx )( cos πx ) for g(x, x ) = e x +x and g = g(ξ (ω)) in the figures (a) and (b) respectively Similarly, when the absolute and relative errors which are obtained by using the test function (.) are analyzed, for g(x, x ) = e x +x the relative error of the numerical solution is %.6 and it is obtained of %.6 for the nonlinear case. That means the finite difference scheme that is obtained for finding the approximate solution of the nonlinear problem is applicable. In addition, when the table given above is examined further, we deduce that as the mesh size increases the absolute and the relative errors decrease. Next, real implementation problems which satisfy various boundary conditions are discussed and the maximal deflections occurred on the surface of the bending plate are compared. The geometrical and physical parameters given in Table are used for all examples considered below. Table : The data used for the computational experiments. Geometric properties Side length of the plate l = l = [cm] Thickness of the plate h =.3 [cm] Mesh size N N = 5 5 Physical properties Elastic parameters E = kncm, ν =.5 κ =.5, ξ =. Example : A load is applied at the central and four symmetric neighboring nodes of the elasto-plastic plate which has the geometric and physical properties given in Table. For different initial approaches iteration number for obtaining numerical solution is derived. The results for clamped and simply supported boundary conditions are given in Table 3.
776 Feda Ilhan and Zahir Muradoglu Table 3: Iteration number for different initial approaches ω () (x, x ) where ω (k+) (x ij ) ω (k) (x ij ) < ε ε Clamped boundary condition (q = 35[kN]) Simply supported boundary condition (q = 9[kN])...... ω () Iteration number Iteration number e x +x 5 9 7 6 e x +x 5 3 3 6 3 sinπx sinπx 7 7 9 3 8 Example : It is assumed that a load F(x) is applied at the central and four symmetric neighbouring nodes of the elasto-plastic plate which has the geometric and physical properties given in Table. The intensity of the applied load is q = 3 [kn]. For the initial approach ω () (x, x ) = e x +x and with the accuracy ε =. the numerical solution of the nonlinear problem is obtained. For the case which clamped boundary condition (.) is satisfied on the whole boundary, the graph of the bending surface of the plate is given in Fig.(a) and when the simply supported boundary condition (.3) is satisfied the bending surface graph of the plate is given in Fig.(b). (a)numerical Solution (b)numerical Solution - - - - -3 5 G G 3 6 8-6 5 G G 3 6 8 Fig. : (a) Numerical solution for the clamped boundary condition.(b) Numerical solution for the simply supported boundary condition When the clamped boundary condition is satisfied, the approximate solution of the problem, corresponding to the given value of the load is reached after n = 5 iterations, while the simply supported boundary condition is satisfied the approximate solution is found for n = 8 iterations. For clamped and simply supported boundary conditions the maximal deflections occurred at the central point of the elasto-plastic plate are obtained as ω max =.59 [cm] and ω max = 5.986 [cm] respectively. The expected result that the deflection of the plate
Numerical solution of non-linear biharmonic equation 777 which satisfy simply supported boundary condition is more than the deflection of the plate satisfying clamped boundary condition is obtained. Example 3: An elasto-plastic plate which has the geometric and physical properties given in Table are taken and for the increasing values of the intensity of the load q k applied at the central point of the plate which has clamped and simply supported boundary respectively. The maximal value of deflection and ξ (ω max ) values corresponding to these deflections are observed (Table ). Table : Variation of maximal deflection ω max and ξ for increasing values of intensity of the load q k which is applied to the elasto-plastic plate satisfying clamped and simply supported boundary condition 3 5 Clamped Boundary Condition Simply Supported Boundary Condition k q k ω max [cm] ξ (ω max ) q k ω max [cm] ξ (ω max ) 6.95.83 3.79.39 9.777.7396 3 3.688.69 37..3835.9 6..8.5 3.59.85 9.695 3.937 35.6663 3.78 3 5.986 7.83 From Table one sees that for the plate satisfying clamped boundary condition the load q 3 = 37.[kN] corresponds to the last elastic state since the effective value of plate curvature is ξ (ω max ) = ξ =.. This means that for all q(x) > q 3 there will arise plastic deformations and for all q(x) < q 3 the deformations will be elastic. Similarly, when the same applications are made for a plate satisfying simply supported boundary condition, the load to be applied for the last elastic state is 6.[kN] and ξ (ω max ) =. = ξ.if the applied load is less than 6. [kn] elastic deformation occurs i.e. when the effect of the applied load is removed the bending plate returns to the its initial shape. If the intensity of the applied load q(x) > 6.[kN] then when the load is removed the trace of deformation remains on the plate. Relation between the intensity of the loads and maximal deflections for the plate satisfying clamped and simply supported boundary conditions is given in Fig. 3(a). Additionally, for this implementation problem it is observed that as the intensity of the applied load increase the deflection of the bending plate increase.
778 Feda Ilhan and Zahir Muradoglu 7 (a)deflections produced by the loads of different intensities 8 (b)deflections for different thickness 6 clamped boundary 7 Load 5 3 simply supported boundary Deflection [cm] 6 5 3 6 8 6 8 Deflection..5.3.35..5 Plate Thickness [cm] Fig. 3: (a) Relation between the intensity of the loads and maximal deflection for the plate satisfying clamped and simply supported boundary conditions. (b)the relation between deformation and thickness of the plate satisfying clamped boundary condition Example : An elasto-plastic plate which has the geometric and physical properties given in Table is taken and loads of different intensities are applied at the central and four symmetric neighbouring nodes of the plate. When the plate thickness is changed the change in deformation is observed for the clamped boundary condition (Table 5). Table 5: Maximal deflections for different thickness k h [cm] ω max [cm] q=6 [kn] q=38 [kn] q=35 [kn] 3 5 6 7 8.6.8.3.3.3.36.38. 3.939.6.95.687.3.99.967.837.97 3.97.389.9676.6.389.75.7 5.568 3.563.6666.656.855.5.93.88 When Table 5 is analyzed we see that, when thickness of the plate increases the maximal deflection decreases whatever the force is (Fig. 3(b)). Example 5: We take an elasto-plastic plate satisfying only the geometric properties given in Table. This plate is assumed to be made of rigid and soft materials respectively. For changing values of the strength hardening parameter κ, the maximal deflections on the surface of the bending plate for different boundary conditions are obtained and results are given in Table 6 and for rigid and soft materials respectively.
Numerical solution of non-linear biharmonic equation 779 Table 6: Deflections ω max [cm] on the surface of elasto-plastic plate made of rigid and soft material for different κ values E = [kncm ] E = [kncm ] κ ξ =. q = [3 kn] ξ =. q = [3 kn] Clamped Simply Supported Clamped Simply Supported.5.35.5.59.6.35 5.986 5.93 5.53.8855.875.8667.69.777 3.97 From Table 6 we deduce that the smaller strength hardening parameter κ, the less deflection on the surface of the plate. Because as κ decreases, the rigidity of the material increases. Numerical examples indicate that the finite difference method possesses no numerical difficulty in the analysis of the elasto-plastic problem of the plate. 5 Conclusions We obtained a numerical solution for the boundary value problem related to the fourth order nonlinear PDE for a bending plate by using finite difference method with various boundary conditions. The results given in the computational experiments have interesting aspects from both mathematical and engineering points of view. The presented numerical examples show the effectiveness of the given approach. References [] S. Timoshenko, A Course in the Theory of Elasticity, Naukova Dumka, Kiev, 97. [] S. Lurie, V. Vasiliev, The Biharmonic Problem of the Theory of Elasticity, Gordon and Breach Publishers, London, 995. [3] A. Hasanov, Variational approach to non-linear boundary value problems for elasto-plastic incompressible bending plate, Int. J. Nonl. Mech., (7), 7-7. http://dx.doi.org/.6/j.ijnonlinmec.7.. [] D. R. J Owen, E. Hinton, Finite Elements in Plasticity: Theory and Practice, Pineridge Press, UK, 98. [5] H Armen Jr., A. Pifko, H.S. Levine, A Finite Element Method for the Plastic Bending Analysis of Structure, Grumman Aircraft Engineering Corporation, Bethpage, New York, 998.
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