Journal of Theoretical and Applied Mechanics, Sofia, 2014, vol. 44, No. 2, pp. 57 70 RECURSIVE DIFFERENTIATION METHOD FOR BOUNDARY VALUE PROBLEMS: APPLICATION TO ANALYSIS OF A BEAM-COLUMN ON AN ELASTIC FOUNDATION Mohamed Taha Dept. of Eng. Math. and Physics, Faculty of Engineering, Cairo University, Giza, Egypt, e-mails: mtahah@eng.cu.edu.eg, mtaha@alfaconsult.org [Received 19 August 2013. Accepted 10 March 2014] Abstract. In the present work, the recursive differentiation method (RDM) is introduced and implemented to obtain analytical solutions for differential equations governing different types of boundary value problems (BVP). Then, the method is applied to investigate the static behaviour of a beam-column resting on a two parameter foundation subjected to different types of lateral loading. The analytical solutions obtained using RDM and Adomian decomposition method (ADM) are found similar but the RDM requires less mathematical effort. It is indicated that the RDM is reliable, straightforward and efficient for investigation of BVP in finite domains. Several examples are solved to describe the method and the obtained results reveal that the method is convenient for solving linear, nonlinear and higher order ordinary differential equations. However, it is indicated that, in the case of beam-columns resting on foundations, the beam-column may be buckled in a higher mode rather than a lower one, then the critical load in that case is that accompanies the higher mode. This result is very important to avoid static instability as it is widely common that the buckling load of the first buckling mode is always the smaller one, which is true only in the case of the beam-columns without foundations. Key words: Boundary value problems, recursive differentiation method, differential equations, beams on elastic foundation. 1. Introduction Most physical and engineering boundary value problems (BVP) can be modelled as functional equations. However, for most of these equations, exact solutions are very rare. Several analytical and numerical methods are being developed to obtain approximate solutions for such models. The commonly used analytical methods are: Adomian decomposition method (ADM) [1 4], variational iteration method (VIM) [5 6], homotopy perturbation method (HPM)
58 Mohamed Taha [7 8], deferential transform method (DTM) [9] and perturbation techniques [10 11]. On the other hand, numerical methods such as finite difference method [12], finite element method (FEM) [13], and differential quadrature method (DQM) [14 15], offer tractable alternative solutions for BVP that involve non-uniform characteristics or complicated boundaries. The analytical methods construct a solution for BVP as a polynomial such that the coefficients of these polynomials are obtained to satisfy both the governing differential equation and the boundary conditions. On the other hand, numerical techniques transform the differential equation into a system of algebraic equations either on the boundary of the BVP domain or at discrete points in the BVP domain and solve the new system. Two main issues face these methods, the mathematical manipulation in constructing the solution expressions and the accuracy (or convergence) of the results. Indeed, the degree of method success in overcoming such issues determines its efficiency and popularity. The present work introduces the recursive differentiation method (RDM) to solve differential equations governing different types of BVP s. The method constructs an analytical series solution for the differential equations but in a way different from the traditional higher order Taylor series method. However, in the RDM, the coefficients of the solution series are obtained through recursive differentiations of the governing differential equation. Investigations of the method reveal that the method yields exact solutions for linear differential equations. For complicated differential equations however, the method yields, with less effort, the same analytical expressions obtained from other techniques. The RDM is tested on linear, nonlinear and higher order differential equations which represent practical boundary value problems to illustrate its efficiency. Furthermore, both the ADM and the RDM method are applied to obtain analytical solutions for the axially loaded beams (beam-columns) on two parameter foundation subjected to different types of lateral loading. Exact expression for the critical loads of a beam-column resting on two parameter foundation required for stability analysis is obtained. The illustrated examples show that the method possesses the same accuracy of DTM and ADM but in relatively short mathematical manipulations. 2. Recursive differentiation method (RDM) Consider the nonlinear n-order boundary value problem of the form: (1) y (n) (x) = F(x,y,y (1),y (2),...,y (n 1) ), x 0 x x 1, with the boundary conditions: B i (y,y (1),...,y (n 1) ) = b i, i = 1,2,...,n,
Recursive Differentiation Method for Boundary Value Problems... 59 where y (n) is the n-th-derivative and b i are constants. In the RDM, the solution of Eqn (1) is assumed as a polynomial in the form: (2) y(x) = N m=0 (x x o ) m T m, m! where T m are coefficients obtained to satisfy both the governing equation and the boundary conditions, x 0 is the boundary coordinates, N is the truncation index selected to achieve the pre-assigned accuracy. The coefficients T m are related to the governing differential equation on the boundary as: (3) T m = y (m) x=x0. It is found that to enhance the accuracy in the RDM, the solution domain is to be transformed to [0, 1]. In addition, to decrease the mathematical effort required to obtain the coefficients T m, a recurrence formula to be derived from the recursive differentiations of the governing equation as it will be illustrated in the following numerical examples. Example 1. Consider the Riccati differential equation which appears in random process, optimal control and diffusion problems [3]; (4) y (1) (t) = y 2 + 1, 0 t 1. Subject to the initial condition: y(0) = 0. Let the solution of Eqn (4) be in the form: (5) y(t) = N t m T m m!. m=0 Recursive differentiation of Eqn (4) yields the recursive equations: (6) y (m) (t) = F(y,y (1),y (2),...,y (m 1) ), m 2. Using Eqn (3), the coefficients T m are obtained as: T 0 = T even = 0, T 1 = 1, T 3 = 2, T 5 = 16, T 7 = 272, T 9 = 7936. Substitution T m into Eqn (5), the solution for N = 9 is: (7) y(t) = t 2 3! t3 + 16 5! t5 272 7! t7 + 7936 t 9 + O(t 11 ). 9! The exact solution to this problem is [3]: y(t) = e2t 1 e 2t + 1. It is found that the numerical results obtained from the RDM expression are compatible with those calculated from the exact solution.
60 Mohamed Taha Example 2. Consider the linear fifth-order boundary value problem which arises in the mathematical modelling of viscoelastic flow [4]: (8) y (5) (x) = y(x) 15e x 10xe x, 0 x 1, with the boundary conditions: y(0) = 0, y (1) (0) = 1, y (2) (0) = 0, y(1) = 0 and y (1) (1) = e. Let the solution of Eqn (8) be in the form: (9) y(x) = N m=0 T m x m m!. The recursive differentiations of Eqn (8) yield the recurrence relation: (10) y (5+n) (x) = y (n) (x) (15 + 10n)e x 10xe x, n = 0,1,2,... Using Eqn (3), the coefficients T m for m 5 are obtained as: T 0 = 0, T 1 = 1, T 2 = 0, T 5 = 15. In addition, using Eqn (3) and Eqn (10), a recurrence formula for the calculations of the coefficients T m for m > 5 is obtained as: (11) T n+5 = T n (15 + 10n), n = 1,2,3,... Substitution into Eqn (9) yields the solution as: (12) y(x) = x + 1 3! T 3x 3 + 1 4! T 4x 4 15 N 5 x n+5 5! x5 + (n + 5)! (T n (15 + 10n)). n=1 The unknown coefficients T 3 and T 4 are obtained using the boundary conditions at x = 1 with Eqn (12), for different values of truncation index N as: T 3 = 2.9653, T 4 = 8.17213 for N = 7 T 3 = 3, T 4 = 8 for N = 12. Then, the solution of Eqn (8) is: (13) y(x) = x 3 3! x3 8 4! x4 15 5! x5 24 6! x6 35 7! x7 48 8! x8 63 9! x9 + which is the exact solution: y(x) = x(1 x)e x.
Recursive Differentiation Method for Boundary Value Problems... 61 Fig. 1. A beam-column on an elastic foundation 3. Analysis of a beam-column on an elastic foundation The equation of motion of an infinitesimal element of the axially loaded beam subjected to lateral load (beam-column) and resting on two-parameter foundation shown in Fig. 1 may be expressed as: (14) EI d4 y dx 4 + (p k 2) d2 y dx 2 + k 1y(x) = q(x), where EI is the flexural stiffness, p is the axial applied load, k 1 and k 2 are the linear and shear foundation stiffness per unit length, q(x) is the lateral load, E is the modulus of elasticity, I is the moment of inertia, y(x) is the lateral displacement, x is the coordinate along the beam. The applied lateral load is assumed in the nonlinear form: (15) q(x) = q 0 + q 1 x + q 2 x 2. Introducing the dimensionless variables; ξ = x/l and w = y/l, where L is the beam length, Eqn (14) may be rewritten as: (16) d 4 w dξ 4 + (P1 K2)d2 w dξ 2 + K1 w(ξ) = Q 0 + Q 1 ξ + Q 2 ξ 2, where the following dimensionless parameters are defined: (17) K1 = k 1L 4 EI, K2 = k 2L 2 pl2, P1 = EI EI, Q 0 = q 0L 3 EI, Q 1 = q 1L 4 EI and Q 2 = q 2L 5 EI. The boundary conditions at the beam ends for beam pinned at its ends (P-P) in dimensionless forms may be expressed as: (18) w(0) = w (0) = 0 and w(1) = w (1) = 0. In the following sections, the solution of non- homogeneous linear differential equation (Eqn 16) will be obtained using both the ADM and the RDM to illustrate the advantages of the proposed method.
62 Mohamed Taha 3.1. Adomian Decomposition Method (ADM) It is an analytical method proposed by Adomian [1] and has been used by many investigators [2, 3, 4] to obtain approximate analytical solution for numerous BVP s. To apply the ADM, Eqn (16) is rewritten in the operator form as: (19) lw = (P1 K2)w K1w(ξ) + Q 0 + Q 1 ξ + Q 2 ξ 2, where: l = d4 dξ 4. Appling the inverse operator l 1 to both sides of Eqn (19) and substituting of the boundary conditions at ξ=0 yields: (20) w(ξ) = ξw (0)+ ξ3 3! w (0) l 1 ( Pw K1w(ξ) ) ξ 4 +Q 0 4! +Q 1 ξ 5 ξ 6 5! +Q 2 6!, where: P = P1 K2, w (0) and w (0) are constants to be obtained. The ADM suggests the solution as a finite series in the form: (21) w(ξ) = N w n (ξ), where N is the truncation index that achieves the pre-assigned accuracy. Substitution of Eqn (21) into Eqn (20) and assuming: n=0 (22) w 0 (ξ) = ξw (0) + ξ3 ξ 4 3! w (0) + Q 0 4! + Q 1 ξ 5 ξ 6 5! + Q 2 6!. Then, the components w n (ξ), n > 0 may be elegantly derived from the recurrence formula: (23) w n (ξ) = P l 1 w n 1 K1l 1 w n 1. Using Eqn (21), Eqn (22) and Eqn (23), the lateral deflection of the beam in terms of the two unknowns w (0) and w (0) may be expressed as: (24) w(ξ) = w 0 (ξ) + w 1 (ξ) + w 2 (ξ) + The substitution of the boundary conditions at ξ = 1 yields the unknowns w (0) and w (0). 3.2. Application of Recursive Differentiation Method To use the RDM, the Eqn (16) is rewritten in the recursive form: (25) w (4) (ξ) = Pw (2) (ξ) K1w (0) (ξ) + Q 0 + Q 1 ξ + Q 2 ξ 2.
Recursive Differentiation Method for Boundary Value Problems... 63 Let the solution of Eqn (25) be in the form: (26) w(ξ) = N m=0 T m ξ m m!. The recursive differentiations of Eqn (25) yield the recurrence formula: (27) w (m) (ξ) = Pw (m 2) (ξ) K1w (m 4) (ξ) for m 7. Using Eqn (3) and the boundary conditions at ξ=0, T m for m<7 are obtained as: T 0 = 0, T 2 = 0, T 4 = Q 0, T 5 = P T 1 K1T 3 +Q 0 and T 6 = PQ 0 +2Q 2. as: The recurrence formula for coefficients T m, m 7 may be expressed (28) T m = P T m 2 K1T m 4 for m 7. is; Then, the lateral displacement in terms of the two unknowns T 1 and T 3 (29) w(ξ) = T 1 A 1 (ξ) + T 3 A 2 (ξ) + Q 0 A 3 (ξ) + Q 1 A 4 (ξ) + 2Q 2 A 5 (ξ), where the recursive functions A i (ξ), i = 1,2,...,5 are: A 1 (ξ) = ξ K1 ξ5 ξ7 + P K1 5! 7! (P 2 K1 K1 2 ) ξ9 9! A 3 (ξ) = +(P 3 K1 2PK1) ξ11 11! + ξ4 4! P ξ6 6! + (P 2 K1) ξ8 8! (P 3 2PK1) ξ10 10! +(P 4 3P 2 K1 + K1 2 ) ξ12 12! + ξ6 A 5 (ξ) = 6! P ξ8 8! + (P 2 K1) ξ10 10! (P 3 2PK1) ξ12 12! + A 2 (ξ) = A 3 (ξ) and A 4(ξ) = A 5 (ξ). The boundary conditions at ξ=1 yield the coefficients T 1 and T 3 as: (30 a) T 1 = (A 32A 21 A 31 A 22 )Q 0 + (A 42 A 21 A 41 A 22 )Q 1 + (A 52 A 21 A 51 A 22 )Q 2 A 11 A 22 A 12 A 21, (30 b) T 3 = (A 31A 12 A 32 A 11 )Q 0 + (A 41 A 12 A 42 A 11 )Q 1 + (A 51 A 12 A 52 A 11 )Q 2 A 11 A 22 A 12 A 21,
64 Mohamed Taha where: A i1 = A i (1) and A i2 = A i (1), i = 1,2,...,5. It is clear that the expressions obtained for the lateral displacement using the RDM are similar to those obtained from the ADM but in a straightforward procedure and with relatively short mathematical manipulations. The bending moment distribution M(ξ) and the shearing force distribution V (ξ) along the beam may be obtained as: (31) M(ξ) = EI w (ξ)/l and V (ξ) = EI w (ξ)/l 2. Furthermore, inspection of Eqn (30) indicates that the lateral displacement is unbounded as the denominator approaches zero. Actually, this situation represents the buckling condition and the axial load in this case is denoted as the buckling (or critical) loads P cr. Thus, the buckling loads P cr can be calculated from the condition: (32) A 11 A 22 A 12 A 21 = 0. 3.3. Verification Additional verification is presented in Table 1 where values of the stability parameter λ (λ = (P cr L 2 /EI)) are calculated using RDM and compared with those obtained from the FEM [13] and the DQM [14], though the proposed RDM is verified against the widely common ADM. It is clear that the RDM results for truncation index N = 12 are in close agreement with other techniques which validate the accuracy of the proposed method. Table 1. The stability parameter (λ) for P P beam on elastic foundation K2 K1 0 π 2 2.5π 2 Method λ 3.1415 4.4428 5.8774 FEM 0 3.1414 4.4425 5.8719 DQM 3.1415 4.4429 5.8773 RDM (N = 12) 4.4723 5.4654 6.6840 FEM 100 4.4642 5.4595 6.6799 DQM 4.4724 5.4654 6.6840 RDM (N = 12) 4. Numerical results The objectives of the present work are: 1) to implement the new proposed RDM; 2) to investigate the distributions of the straining actions along the beam (bending moments M(ξ) and shearing force V (ξ)) due to different
Recursive Differentiation Method for Boundary Value Problems... 65 types of lateral loading (uniform, linear or nonlinear); and 3) to investigate the influence of the different system parameters on the critical loads P cr. To achieve these objectives, the obtained expressions for the straining actions (Eqn 31) and for the critical load (Eqn 32) are to be used. However, investigation of Eqn (16) reveals that the influence of K2 on the beam-column behaviour is inversely proportional to the influence of P 1. Thus, the effect of K2 can be considered as an axial tension load ( P1) acting on the beam-column. The obtained expressions may be inserted in a short MATLAB code or even in an Excel spread sheet to draw the required results. Furthermore, although these expressions are obtained in dimensionless forms to be valid for any specific case, the properties of the beam considered in the present parametric study are: concrete beam, b = 0.2 m, h = 0.5 m, L = 5 m, E = 2.1E10 Pa, Poisson ratio µ is 0.15 and the total lateral load acting on the beam Q T = 250000 N. 4.1. Identification of the foundation stiffness parameters K1 and K2 There are two models have been widely used to simulate the foundation influence, namely, the linear elastic foundation model (Winkler model) and the two parameter model (Pasternak model). However, in the case of two parameter model, both the values of k 1 and k 2 increase with the increase in the foundation stiffness (weak, medium and stiff foundation etc.). Indeed, values of foundation parameters k 1 and k 2 depend on both the configurations of the foundation and the beam. The following expressions may be utilized for the determination of foundation parameters in the static analysis of rectangular beams on two parameter elastic foundations [16]. (33) k 1 = E 0 b γ 2(1 µ 2 0 ) χ and k 2 = E 0 b χ 4(1 µ) γ, where: χ = 3 2EI (1 µ 2 0 ) be 0 (1 µ 2 ), E 0 = E s and µ 0 = µ s 1 µ s, 1 µ 2 s E and µ are the elastic modulus and Poisson ratio of the beam respectively, E s and µ s for the foundation and γ is a parameter that accounts for the beam-foundation loading configuration (it is a common practice to assume γ = 1). Typical values of the elastic modulus and Poisson ratio for different types of foundation are given in Table 2. Investigations of Eqn (17) with Eqn (33) indicate that the values of K1 and K2 depend on the type of the foundation, the beam material and the
66 Mohamed Taha Table 2. Typical values for E s and µ s for foundation [17] Type of Soil E s N/m 2 µ s Type of Soil E s N/m 2 µ s Loose sand 5E6 0.42 Soft clay (saturated) 3E6 0.5 Medium sand 1E7 0.38 Medium clay 3E7 0.35 Dense sand 4E7 0.3 Hard clay 1E8 0.25 Sand and gravel 1.2E8 0.25 Fig. 2. Influence of foundation stiffness on the critical load (K2 = 0) slenderness ratio of the beam L/r (r = I/A, A is the beam cross sectional area). 4.2. Critical load Using Eqn (32), values of the critical loads for different values of foundation linear stiffness parameter K1 are calculated and presented in Fig. 2. It is observed, that the values of the critical loads parameter P cr may be correlated to the foundation stiffness parameters as: (34) P cr n = K2 + n 2 π 2 + K1 n 2 π 2 (for P P beams), where P cr is the critical load parameter and n is the buckling mode. Both, the critical load parameter and the buckling mode for a certain configurations depend on the foundation stiffness. Actually, the beam buckling mode depends on the minimum potential energy of the beam-foundation system which determines the stable equilibrium position in the buckled configuration. As the foundation stiffness increases, the system potential energy increases, but the rate of increase in the higher modes is less than the rate of increase in the lower modes. In other words, as the foundation stiffness increases to a certain value, the system minimum potential energy due to a buckling mode n approaches that one for the buckling mode n + 1. However, if the foundation stiffness ex-
Recursive Differentiation Method for Boundary Value Problems... 67 ceeds that value, the system minimum potential energy of the buckling mode n + 1 becomes smaller than that for buckling mode n and the beam buckles in the mode n + 1 instead of the mode n. The value of the foundation stiffness parameter K1 n at which the system minimum potential energy due to a buckling mode n is equal to that for a buckling mode n + 1 is given by: (35) K1 n = n 2 (n + 1) 2 π 4. If for a certain configuration of beam-foundation system, K1 exceeds K1 n the beam will buckle in the mode n+1. Further, for 0 < K1 < 4π 4, the beam will buckle in the first mode and for 4π 4 < K1 < 36π 4 the beam will buckle in the second mode and so on. This fact reveals that the buckling fundamental mode not always is the first mode and the critical load may be smaller than that calculated one using the first mode. Moreover, the investigations of the homogeneous solution of Eqn (16) lead to the same conclusions which validate the accuracy of the RDM. 4.3. Beam straining actions The distributions of M (ξ) and V (ξ) are shown in Fig. 3 and Fig. 4 respectively, for different loading types and different system parameters. The following dimensionless parameters are defined: (36) P = P1 π 2, M (ξ) = M(ξ) M U (max) and V (1) = V (ξ) V U (1), Fig. 3. Effect of loading type on the distribution of M (ξ) Fig. 4. Effect of loading type on V (ξ) where M U (max), V U (1) are the maximum values of the corresponding parameters for a free beam (beam without foundations) with the same properties carrying the same total load distributed uniformly along the beam. The loading types are denoted as U for uniform loading, L for linear and N for nonlinear loading. It is clear that the straining actions increase as the axial load increases and as the foundation stiffness decreases. The effects of the type of
68 Mohamed Taha loading; for the same total load; may be ignored in the calculations of the lateral displacement W, but it is more noticeable for both the bending moment distribution and the shearing force distribution. The maximum values of the bending moment for uniform loading and linear loading are the same and slightly greater than the maximum bending moment due to nonlinear loading. It is also observed that the maximum shearing forces for different loading types are relatively different and should be considered in the shear force calculations. The effect of the axial load parameter P on the maximum values of the straining actions (M and V ) for different loading types is shown in Fig. 5. It is found, that M and V increase as P increases. However, the influence of P on M and V decreases as the foundation stiffness increases. It should be noted that value of P for a beam resting on a foundation for a certain buckling mode is greater than value of P for the same beam but without foundation. Actually, P of a beam without foundation is used to produce the given figures. In addition, the effects of the loading type (U, L or N) on both W (max) and M (max) may be practically ignored but the effects of the loading type on V (1) should be taken into consideration. A. Bending moment M (max) B. Shearing force V (1) Fig. 5. Effect of axial load and foundation stiffness on the maximum straining actions (K2 = 0) 5. Conclusions The recursive differentiation method (RDM) is introduced and implemented to solve numerous examples of boundary value problems (BVP) in finite domain. It is illustrated that the method is simple and straightforward in constructing analytical solutions for the given differential equations. In addition, it is indicated that the obtained analytical expressions using the RDM for linear, nonlinear and higher order differential equations are compatible with
Recursive Differentiation Method for Boundary Value Problems... 69 those obtained from closed solutions or resulted from other analytical techniques. Further, it is found that, the accuracy of the obtained expressions using RDM is greatly enhanced when the solution domain is transformed to the domain [0, 1]. Moreover, a recurrence formula may be derived to decrease the mathematical effort in constructing the solution expressions. The RDM is used to investigate the static behaviour of beam-column subjected to different types of lateral loading and resting on two parameter foundation. Exact expressions for critical loads are obtained. However, it is found that if the foundation stiffness exceeds a certain level, the beam buckles in the second mode rather than the first mode and the critical load in such case may be much smaller than the critical load of the first mode. This result is very important for the stability analysis of beams resting on elastic foundation such as railway tracks, structural elements and piles. Further, the maximum values of straining actions affecting the beam cross section are obtained and illustrated against different types of loading types and different beam-foundation parameters. The results reveal that the method is versatile and efficient in dealing with beam BVP. R EFERENCES [1] Adomian, G. Solving Frontier Problems of Physics: the Decomposition Method, Boston, Vol. 60 of the Fundamental Theories of Physics, Kluwer Academic Publishers, 1994. [2] Taha, M. H., A. Omar, M. Nassar. Dynamics of Timoshenko Beam on Nonlinear Soil. Int. J. Civil Eng. Research, 3 (2012), No. 2, 93 103. [3] Bahnasawi, A. A., M. A. El-Tawil, A. Abdel-Naby. Solving Riccati Differential Equation using Adomian s Decomposition Method. Appl. Math. Comput., 157 (2004), 503 514. [4] Wazwas, A. M. The Numerical Solution of Fifth-order Boundary Value Problems by Decomposition Method. J. Comp. and Appl. Math., 136 (2001), 259 270. [5] Ji.Huan, He. Variational Iteration Method: Some Recent Results and New Interpretations. J. Comp. Appl. Math., 207 (2007), 3 17. [6] Noor, M. A., S. T. Mohyud-Din. Modified Variational Iteration Method for Heat and Wave-like Equations. Acta Applicandae Mathematicae, 104 (2008), No. 3, 257 269. [7] Tan, Y., S. Abbasbandy. Homotopy Analysis Method for Quadratic Riccati Differential Equation. Commun. Nonlin. Sci. Numer. Simul., 13 (2008), No. 3, 539 546.
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