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RBF-based meshless method for the free vibration of beams on elastic foundations
Applied Mathematics and Computation 249 (2014) 198–208
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
RBF-based meshless method for the free vibration of beams on elastic foundations Husain J. Al-Gahtani ⇑, Faisal M. Mukhtar Department of Civil and Environmental Engineering, King Fahd University of Petroleum & Minerals, 31261 Dhahran, Saudi Arabia
a r t i c l e
i n f o
Keywords: Meshless method Radial basis function Free vibration analysis Beam on elastic foundation
a b s t r a c t In this paper, solution is obtained for the free vibration differential equations of motion of an axially loaded beam on elastic foundation using a meshless method. Use is made of the multiquadrics radial basis function (RBF) in obtaining the numerical solution for four different cases: (1) one end clamped, the other end simply supported; (2) both ends clamped; (3) both ends simply supported; and (4) a simple beam on elastic foundation with end rotational springs. The approach is easier to implement and program as compared to grid/mesh-based methods such as the finite difference method (FDM) and the finite element method (FEM). Accuracy of the results obtained using the proposed method was verified using the analytical results available in the literature for the first three cases considered. Numerical results of the fourth case were aimed at justifying the use of the numerical scheme for a problem whose analytical solution is not readily available and to show the high accuracy of the RBF method. The results prove that the method require much less number of nodes to converge to the correct solution as compared to FDM. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Many classical engineering problems whose behaviors are governed by differential equations require a smart choice of solution techniques that are fast, easy to program in a computer, and yet produce accurate and reliable results with minimum possible effort. The behavior of an axially loaded beam on elastic foundation is one such phenomenon whose free vibration is described by a fourth-order partial differential equation. As a result, various studies have been conducted to come up with reliable solutions to this problem using different approaches. A weak form quadrature element method (QEM) has been applied to analyze vibrations of beams on a nonlinear elastic foundation considering the effects of transverse shear deformation and the rotational inertia of beams by Yihua et al. [1]. Using Green’s functions, Wang et al. [2] presented a unified formulation for bending, buckling, and vibration problems of uniform Timoshenko and Euler–Bernoulli beams resting on various models of elastic foundations. The differential transform method (DTM) has been used to solve the free vibration differential equations of motion of one end clamped, the other simply supported axially loaded beams on elastic foundation by Çatal [3]. Örztuk [4] applied the variational iteration method (VIM) to analyze the free vibration of a beam on an elastic foundation with different end conditions. Akour [5] used Hamilton’s principle to derive the governing equation and investigated the behavior of a simply supported nonlinear beam resting on linear elastic foundation subjected to harmonic loading using Runge–Kutta technique. Eigenvalues for free vibration of beam-columns on elastic supports have been obtained by West and Mafi [6]. Three main parameters were investigated including the damping coefficient, the natural frequency, and ⇑ Corresponding author. http://dx.doi.org/10.1016/j.amc.2014.09.097 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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the coefficient of the nonlinear term. Hsu [7] applied the spline collocation method and solved for the natural frequencies of non-uniform beams resting on elastic foundations. Kacar et al. [8] studied the free vibration of an Euler–Bernoulli beam resting on a variable Winkler foundation using the differential transformation method. The method of discrete singular convolution (DSC) has been used by Civalek and Özturk [9] to model free vibration of tapered piles embedded in two-parameter elastic foundations. Another competitive type of solution is the grid-based or mesh-based methods such as FDM and FEM. Ansari et al. [10] developed a non-classical model for the free vibrations of nanobeams accounting for the surface stress effects. The influence of surface stress was incorporated into the Euler–Bernoulli beam theory based on Gurtin–Murdoch elasticity theory. In order to obtain the natural frequencies of the beams subject to different boundary conditions, discretization of the governing equation was achieved using a compact finite difference method (CFDM) of sixth order. Effects of beam thickness, surface density, surface residual stress, surface elastic constants and boundary conditions on the natural frequencies of the beams were investigated. Vo et al. [11] presented an FEM model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Hamilton’s principle was used to derive the governing equations of motion and boundary conditions of the sandwich beam whose core is fully metal or ceramic and functionally graded skin. Effects of several factors on the natural frequencies, critical buckling loads and load–frequency curves the beams were analyzed. Finite element equations for a variationally consistent higher order beam theory were presented by Heyliger and Reddy [12] for the static and dynamic behavior of rectangular beams. The stress-free conditions on the upper and lower surfaces of the beam were accounted for by the higher order beam theory. This theory overcomes the limitations of the elementary beam theory that disregards the transverse shear deformation effect and the limitation of Timoshenko beam theory (first order shear deformation theory) that assumes a constant transverse shear strain distribution; The higher order beam theory models, more accurately, the parabolic shear stress distribution without the need for shear correction coefficient needed in the Timoshenko theory. The recovery of the Kirchoff constraint for thin beams was resulted due to the Full integration of the shear stiffness terms. Accuracy of the model was demonstrated through several numerical examples. To [13] presented explicit expressions for mass and stiffness matrices of two higher order tapered beam finite elements for vibration analysis. The loss of computer time and round-off-errors associated with extensive matrix operations and numerical evaluation of the expressions were eliminated due to the explicit element mass and stiffness matrices. It was verified that the higher order tapered beam elements presented are superior to the lower order one as they offer more realistic representations of the curvature and loading history of the beam element. In all the aforementioned references, the methods involved are either based on analytical approaches that are mathematically involved or based on numerical solutions that require mesh for the solution interpolation. Nevertheless, the possibility of obtaining accurate numerical solutions without resorting to the mesh-based techniques mentioned above, has been the goal of many researchers throughout the computational mechanics community for the past two decades or so. There are several types of meshless methods as briefly discussed here. Based on the improved element-free Galerkin method, Zhang et al. [14] performed a numerical study for degenerate parabolic equations arising from the spatial diffusion of biological populations. Galerkin procedure was used to derive the discrete equation system and the essential boundary conditions imposed using the penalty method. Convergence study was carried out for the time dependent problem. A number of numerical examples were used to establish the applicability of the method. Based on the moving Kriging interpolation technique Zhu et al. [15] developed a meshless local Petrov–Galerkin approach for geometrically nonlinear thermoelastic analysis of functionally graded plates in thermal environments. The constructed shape functions possess Kronecker delta function property due to the Kriging interpolation method. This eliminates the need for special techniques for enforcing essential boundary conditions. Zhang et al. [16] studied the mechanical and thermal behaviors of ceramic–metal grade plates (FGPs) using Kriging meshless method. The local meshless method was developed based on the local Petrov–Galerkin weak-form formulation combined with shape functions having Kronecker delta function property. Postbuckling analysis of carbon nanotubereinforced functionally graded (CNTR-FG) cylindrical panels under axial compression was reported by Liew et al. [17]. Ritz method was employed, based on kernel particle approximations for the field variables, to obtain the discretized governing equations. Mesh-free kp-Ritz was employed by Zhang et al. [18] for analysis of flexural strength and free vibration of functionally graded carbon nanotube-reinforced composite cylindrical panels. Formulations were based on the first-order shear deformation shell theory and the 2-D transverse displacement field was approximated by the mesh-free kernel particles estimate. Similar mesh-free kp-Ritz method was adopted by Zhang et al. [19] for large deflection geometry nonlinear analysis of CNTR-FG cylindrical panel and by Cheng et al. [20] to analyze a degenerate parabolic equation arising in the spatial diffusion of biological population. RBF-based collocation method, as one of the most recently developed meshless methods, has attracted attention in recent years especially in the area of computational mechanics [21–33]. Due to its simplicity to implement, it represents an attractive alternative to FDM and FEM. The objective of this paper is to offer a simple yet accurate meshless method for the solution of free vibration equations of beam on elastic foundation. To the knowledge of the authors, so far, RBF collocation methods have not been applied to this specific free vibration problem. The proposed RBF method can be considered as a competitive candidate due to its simplicity, robustness and accuracy. Four different support conditions are considered: One end clamped and other end simply supported; both ends clamped; both ends simply supported; and a simple beam on elastic foundation with end rotational springs. For verification of the results obtained, use is made of the analytical solutions given by Çatal [3] for the case of one end clamped and other end simply supported beam on elastic foundation. For the cases of both ends clamped and both ends simply supported, the analytical results reported by Ozturk [4] are used for the verification purpose. The efficiency of the method is demonstrated for the first three cases by comparing the minimum number of RBF nodes needed for
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convergence to the exact solution to that needed by FDM. Numerical results of the fourth case were aimed at justifying the use of the numerical scheme for a problem whose analytical solution is not readily available and to show the higher accuracy of the RBF method over the mesh-based FDM scheme. 2. Governing differential equations Consider a beam of length L and cross-sectional area A resting on elastic foundation as shown in Fig. 1 (a). Let G, E and I be its shear modulus, elastic modulus and moment of inertia respectively. Fig. 1 (b) shows a differential segment of length dx out of this beam with the internal forces and deformations indicated. Idealizing the elastic foundation by Winkler model, the relation between displacement function y(x, t) of the beam and the distributed load q(x, t) acting beneath it due to the reaction within the elastic foundation can be written as q(x, t) = Csy(x, t). For a modulus of subgrade reaction C0, and beam width b, the parameter Cs can be written as Cs = C0b. Neglecting the second order terms, the equilibrium equations of the internal forces shown in Fig. 1 (b) are used to derive the motion equation (upon simplification) of the beam [34,35] as given by Eqs. (2.1) and (2.2).
N
N
x
Elastic foundation
Csy(x,t) y L
(a) M ( x, t )
M ( x, t )
N N
T ( x, t )
T ( x, t )
M ( x, t ) dx x ( x, t ) dx x
T ( x, t ) dx x
Csy(x,t) dx dx
(b) Fig. 1. (a) A beam on elastic foundation (b) internal forces and deformations of the beam on elastic foundation.
Fig. 2. Domain and boundary nodal distribution.
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Table 1 Some Commonly used RBF’s. Function
Definition pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /ðrÞ ¼ r 2 þ c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /ðrÞ ¼ ð r2 þ c2 Þ 2 /ðrÞ ¼ Rn
Multiquadrics (MQ) Inverse multiquadrics Nth Polynomial splines Gaussian
/ðrÞ ¼ eðcr
Thin plate (polyharmonic) splines (TPS)
" /i1v ðzÞ þ
/i2v ðzÞ þ
p2 Nr þ
Þ
/ðrÞ ¼ ð1Þkþ1 r 2k logðrÞ
# 2 x2 C s ÞkL x2 ÞL4 ðm ðC s m /1 ðzÞ ¼ 0; /001 ðzÞ þ AG EI
x2 ÞL4 ðC s m /2 ðzÞ ¼ 0; EI
2
0 6 z 6 1;
0 6 z 6 1:
ð2:1Þ
ð2:2Þ
In the above equations, the position variable x is substituted with a dimensionless parameter z = x/L. The dimensionless displacement function of the beam taking into consideration and neglecting the effects of axial and shear forces are, respec2
4
tively, given as /1(z) and /2(z). /001 ðzÞ ¼ d d/z12ðzÞ and /i1v ðzÞ ¼ d d/z14ðzÞ are, respectively, the second and fourth derivatives of the dimensionless displacement function with respect to the dimensionless position variable z. The time variable is denoted 2 as t and the ratio of axial load N to the Euler buckling load as N r ¼ NL pEI : m, x and k represent the distributed mass of the beam, beam circular frequency and shape factor due to shape of the beam section, respectively. Eq. (2.1) applies to the case in which the axial and shear forces are taken into consideration, and the coefficient of the second derivative represents the sum of the axial force and shear effects whereas the coefficient of the dimensionless displacement function represents the effect due to the foundation. Neglecting the effects of axial and shear forces results in Eq. (2.2).
N
N Cs L
(a) N
N Cs L
(b) N
N Cs L
(c) Ks
Ks
Cs L
(d) Fig. 3. (a) One end clamped and other simply supported (CS) (b) Both ends clamped (CC) (c) Both ends simply supported (SS) (d) Simple beam on elastic foundation with end rotational springs (RS).
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Table 2 Boundary conditions for the beams considered.* Beam Type
z=0
CS CC SS RS
z=1
/ðzÞ
/0 ðzÞ
/00 ðzÞ
EI /00 ðzÞ K s /0 ðzÞ
/ðzÞ
/0 ðzÞ
/00 ðzÞ
EI /00 ðzÞ K s /0 ðzÞ
0 0 0 0
0 0 – –
– – 0 –
– – – 0
0 0 0 0
– 0 – –
0 – 0 –
– – – 0
The dash symbol (–) means not applicable. * /ðzÞ ¼ /1 ðzÞ and /2 ðzÞ for Eqs. (2.1) and (2.2) respectively.
1/(Nd+1)
z
0.1
1 Domain and boundary nodes Dummy nodes Fig. 4. Domain and boundary discretization.
3. RBF method formulation Meshless methods have attracted the attention of many researchers during the last two decades. Though, uniquely simple, these methods provide solution accuracies that compete with those of finite elements and boundary elements without the difficult and time-consuming task for mesh connectivity [36,37]. 3.1. Radial basis function (RBF) Radial basis function is one of the most popular types of meshless methods, and its general principle of formulation is explained here using an elliptic type PDE with the domain L and boundary B operators on the dependent variable u (Eqs. (3.1) and (3.2)). F and g are the continuous functions of the position. The same principle applies when using this technique to solve other types of PDE’s.
Lu ¼ f ;
ð3:1Þ
Bu ¼ g:
ð3:2Þ
To apply RBF, we model the geometry of the problem by randomly distributed nodes Nd and Nb, respectively, on the domain X and on the boundary C as shown in Fig. 2. This forms a set of Np points (Np = Nd + Nb).Assuming a suitable RBF centered at a given number of points, the solution to Eq. (3.1) can be approximated using the direct collocation method as
uðxÞ ¼
Np X
aj /ðrj Þ;
ð3:3Þ
j¼1
where, / is the radial basis function centred at xj, rj ¼ jx xj j is the radial distance between the point x and the centre of RBF xj , and aj are coefficients to be determined. Some commonly used types of radial basis functions include those shown in Table 1. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi However, in the formulation of the problems solved in this study, use is made of the multiquadrics / ¼ jx xj j2 þ c2 , where c – 0 is an adjustable constant known as the shape parameter. The unknown coefficients (aj) are determined by solving the system of Np linear equations formed by applying the operators L and B at the domain nodes, Nd, and the boundary nodes, Nb, respectively.
Luðxi Þ ¼
Nd X
aj Lð/jxi xj jÞ;
i ¼ 1; Nd ;
ð3:4Þ
i ¼ 1; Nb :
ð3:5Þ
j¼1
Buðxi Þ ¼
Np X
aj Bð/jxi xj jÞ;
j¼N d þ1
The resulting collocation Eqs. ((3.4) and (3.5)) can be written in a matrix form thus,
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Table 3 = 3.705, C = 50.0 kN/m2, Frequency factors’ prediction for a one end clamped the other simply supported beam (CS) on elastic foundation Nr = 0.25, k s AG = 1,935,900 kN. Method
G
N
k 1
FDM/%Err.
G1
RBF/%Err. Analytical FDM/%Err.
G2
RBF/%Err. Analytical FDM/%Err.
G3
RBF/%Err. Analytical
10
100
1000
10000
4 8 12 11 N/A
2.6905 2.7846 2.8021 2.8219 2.8157
4.45 1.10 0.48 0.22
3.2064 3.3312 3.3552 3.3809 3.3745
4.98 1.28 0.57 0.19
3.8991 4.0102 4.0327 4.0561 4.0511
3.75 1.01 0.45 0.12
5.8213 5.8629 5.8718 5.8745 5.8794
0.99 0.28 0.13 0.08
10.0385 10.0472 10.0491 10.0493 10.0511
0.13 0.04 0.02 0.02
4 8 12 11 N/A
3.5968 3.8819 3.9349 3.9856 3.9766
9.55 2.38 1.05 0.23
4.5059 4.9088 4.9857 5.0598 5.0472
10.72 2.74 1.22 0.25
5.3798 5.9007 6.0063 6.1083 6.0925
11.70 3.15 1.41 0.26
6.6921 7.1361 7.2367 7.3499 7.3222
8.61 2.54 1.17 0.38
10.2728 10.4326 10.4734 10.5205 10.5099
2.26 0.74 0.35 0.10
4 8 12 11 N/A
4.0001 4.6118 4.7276 4.8236 4.8192
17.00 4.30 1.90 0.09
5.1180 5.9630 6.1270 6.2652 6.2581
18.22 4.72 2.09 0.11
6.2570 7.4053 7.6407 7.8710 7.8318
20.11 5.45 2.44 0.50
7.5378 8.8080 9.1000 9.4075 9.3458
19.35 5.75 2.63 0.66
10.6169 11.3378 11.5428 11.7737 11.7296
9.49 3.34 1.59 0.38
Table 4 ¼ 3:705, C = 50.0 kN/m2, Frequency factors’ prediction for a one end clamped the other simply supported beam (CS) on elastic foundation Nr = 0.5, k s AG = 1,935,900 kN. G
Method
N
FDM/%Err.
4 8 12 11 N/A
2.5776 2.6803 2.6993 2.7211 2.7141
5.03 1.25 0.55 0.26
3.0838 3.2159 3.2411 3.2685 3.2613
8.61 4.70 3.95 3.14
3.8084 3.9198 3.9423 3.966 3.9607
5.99 3.24 2.69 2.10
5.7914 5.8308 5.8392 5.8479 5.8463
1.50 0.83 0.68 0.54
10.0324 10.0406 10.0424 10.0424 10.0436
0.19 0.10 0.09 0.09
4 8 12 11 N/A
3.5398 3.8353 3.8901 3.9422 3.933
10.00 2.48 1.09 0.23
4.4349 4.8502 4.929 5.0049 4.992
11.16 2.84 1.26 0.26
5.3024 5.8344 5.9417 6.0453 6.0219
11.95 3.11 1.33 0.39
6.6374 7.0826 7.1833 7.2791 7.2685
8.68 2.56 1.17 0.15
10.2559 10.4129 10.453 10.4994 10.4887
2.22 0.72 0.34 0.10
4 8 12 11 N/A
3.9564 4.5826 4.7004 4.7979 4.7934
17.46 4.40 1.94 0.09
5.062 5.9251 6.0916 6.2318 6.2245
18.68 4.81 2.14 0.12
6.1915 7.359 7.5971 7.8153 7.7903
20.52 5.54 2.48 0.32
7.4794 8.7603 9.0541 9.3641 9.3023
19.60 5.83 2.67 0.66
10.5916 11.3088 11.5132 11.7435 11.6996
9.47 3.34 1.59 0.38
k 1
G1
RBF/%Err. Analytical FDM/%Err.
G2
RBF/%Err. Analytical FDM/%Err.
G3
RBF/%Err. Analytical
Lð/Þ Bð/Þ
fag ¼
10
100
1000
f : g
10000
ð3:6Þ
Eq. (3.6) makes it convenient for the values of the unknowns (a’s) to be determined. 3.2. Application of RBF to free vibration of beam on elastic foundation Fig. 3 shows the four types of beams resting on an elastic foundation each having different end conditions considered and solved in this study: One end clamped and other simply supported (CS); both ends clamped (CC); both ends simply supported (SS); and a simple beam on elastic foundation (no axial force) with end rotational springs (RS). The boundary conditions for these beam types are given in Table 2. The additional term K s /0 ðzÞ appearing in the slope boundary condition of the last case, RS, reflects the contribution of the rotational spring of spring constant Ks. The exact solution for this last case is not readily available for a finite value of Ks – 0. The domain of interest was discretized using a finite number of equally spaced domain nodes, Nd and two boundary nodes at the two ends of the beam (z ¼ 0 and z ¼ 1). However, since there exist four boundary conditions that need to be satisfied (Table 2), we end up with Nd + 4 equations to solve for Nd + 2 unknowns. This poses a problem to the scheme due to the fact that same number of equations as the number of unknowns is necessary for the complete solution of such linear algebraic equations. In the proposed RBF method, we created two additional ‘dummy’ nodes one each at positions z ¼ 0:1 and z ¼ 1:1 thereby satisfying the requirement (Nd + 4equations to solve for Nd + 4 unknowns by applying each in turn to Nd + 4 nodes). The final discretized beam with the two created ‘dummy’ nodes is shown in Fig. 4. With this, the
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problem is solved by satisfying the boundary conditions and the governing differential equation over selected points over the boundary and inside the domain, respectively. qffiffiffiffiffiffiffiffiffiffiffi 2 4 4 Recognizing that c ¼ m xEI L , the frequency factor, in (2.1) and (2.2) the general motion equations were solved using the RBF method in which the variable /(z) (i.e. /1(z) or /2(z)) was approximated with an equation of the form given in (3.3). Since the domain and boundary operators, L and B operate over /(z) to define (2.1) and/or (2.2) as the case may be and the relevant boundary conditions given in Table 1, respectively, we generated the equations and write them in the form described in (3.6). However, since the target of the formulation presented in this study is obtaining the values of the frequency factors c, we by-passed the step of obtaining a0 s and argued that for a non-trivial solution, determinant of coefficient matrix must be zero.
Lð/Þ Bð/Þ ¼ 0:
ð3:7Þ
This determinant yields a characteristic equation in terms of c. The positive real roots of the equation are the free vibration frequency factors (hence frequencies, x) of the beams on elastic foundation shown in Fig. 3.
Table 5 ¼ 3:705, C = 50.0 kN/m2, Frequency factors’ prediction for a one end clamped the other simply supported beam (CS) on elastic foundation Nr = 0.75, k s AG = 1,935,900 kN. G
Method
N
FDM/%Err.
4 8 12 11 N/A
2.4474 2.5621 2.5832 2.6076 2.5996
5.85 1.44 0.63 0.31
2.944 3.0863 3.1131 3.1429 3.1346
6.08 1.54 0.69 0.26
3.7102 3.8223 3.8447 3.869 3.863
3.96 1.05 0.47 0.16
5.7609 5.7979 5.8058 5.8141 5.8124
0.89 0.25 0.11 0.03
10.0264 10.0339 10.0355 10.0424 10.0367
0.10 0.03 0.01 0.06
4 8 12 11 N/A
3.4799 3.7871 3.8436 3.8974 3.8879
11.52 3.71 2.27 0.91
4.3603 4.7894 4.8703 4.9481 4.9349
12.65 4.06 2.44 0.88
5.2216 5.7658 5.8749 5.9806 5.9634
13.29 4.25 2.44 0.69
6.5815 7.0279 7.1286 7.2255 7.2141
9.45 3.31 1.92 0.59
10.2389 10.3931 10.4325 10.5189 10.4673
2.38 0.91 0.54 0.29
4 8 12 11 N/A
3.9111 4.5529 4.6727 4.7717 4.7672
17.96 4.50 1.98 0.09
5.0041 5.8865 6.0556 6.1955 6.1904
19.16 4.91 2.18 0.08
6.1238 7.3118 7.5528 7.7469 7.748
20.96 5.63 2.52 0.01
7.4196 8.7117 9.0075 9.3576 9.257
19.85 5.89 2.70 1.09
10.5662 11.2795 11.4833 11.8277 11.6665
9.43 3.32 1.57 1.38
k 1
G1
RBF/%Err. Analytical G2
FDM/%Err.
RBF/%Err. Analytical G3
FDM/%Err.
RBF/%Err. Analytical
10
100
1000
10000
Table 6 Frequency factors’ prediction for a one end clamped the other simply supported beam (CS) on elastic foundation with shear and axial force effects neglected. Method
G
N
k 1
G1
FDM/%Err.
RBF/%Err. Analytical G2
FDM/%Err.
RBF/%Err. Analytical G3
FDM/%Err.
RBF/%Err. Analytical a b
10
100
4 8 12 11 N/A
3.6813 3.8633 3.9003 3.9372 3.9307
6.34 1.71 0.77 0.17
3.7256 3.9018 3.9377 3.9736 3.9673
6.09 1.65 0.75 0.16
4.1003 4.2353 4.2635 4.2918 5.9315a
4 8 12 11 N/A
5.9922 6.7605 6.9277 7.0927 7.0693
15.24 4.37 2.00 0.33
6.0026 6.7677 6.9344 7.099 7.0757
15.17 4.35 2.00 0.33
6.104 6.8392 7.0009 7.1613 7.1384
4 8 12 11 N/A
7.4846 9.3897 9.8284 10.1976 10.2104
26.70 8.04 3.74 0.13
7.4899 9.3924 9.8308 10.2505 10.212
26.66 8.03 3.73 0.38
7.5429 9.4195 9.8544 10.2521 10.2336
1000 b
10000
5.8643 5.9122 5.9226 5.9331 5.9315
1.13 0.33 0.15 0.03
10.0454 10.055 10.0571 10.0575 10.0591
0.14 0.04 0.02 0.02
14.49 4.19 1.93 0.32
6.9163 7.4544 7.5806 7.7139 7.6896
10.06 3.06 1.42 0.32
10.3076 10.4855 10.5317 10.585 10.5729
2.51 0.83 0.39 0.11
26.29 7.96 3.71 0.18
8.02 9.6779 10.0815 10.4932 10.4733
23.42 7.59 3.74 0.19
10.7059 11.5461 11.7912 12.0703 12.0194
10.93 3.94 1.90 0.42
b b b
There seems to be a typographical error in this particular value of analytical solution reported in [3]. Not reported because the likely error in ‘a’ will affect these values.
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205
(a)
(b)
(c) Fig. 5. Variation of frequency factors, G due to relative stiffness, k for both ends clamped beam (CC) on elastic foundation for; (a) Nr = 0.25 (b) Nr = 0.5 (c) Nr = 0.75.
4. Numerical example Free vibration equations for four different types of beam on elastic foundation (Fig. 3) are solved using the RBF approach presented in this study. The analytical results of the same problem reported by Çatal [3] are used for verification of the results obtained for the case of one end being clamped and the other simply supported (CS), whereas the analytical results presented by Ozturk [4] are used for the verification of the next two cases where both ends are clamped (CC) and both ends are simply supported (SS). The Numerical results of the fourth case were aimed at justifying the use of the numerical scheme for a problem whose analytical solution is not readily available. The efficiency of RBF is demonstrated by comparing its results with those of FDM for the cases considered. The FDM scheme employed in all examples is based on the secondorder-accurate centered difference form. Same values of the parameters and beam profile are chosen as those used in [3] and [4]. Hence, the beam is made up of an IPB 500 steel profile resting on a Winkler type foundation of modulus of subgrade reaction of 50,000 kN/m2. Other parameters are:
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(a)
(b)
(c) Fig. 6. Variation of frequency factors, G due to relative stiffness, k for simply supported beam (SS) on elastic foundation for; (a) Nr = 0.25 (b) Nr = 0.5 (c) Nr = 0.75.
I ¼ 107:2 105 m4 ;
A ¼ 2:39 102 m2 ;
¼ 3:705; ¼ 0:19 kNs2 =m; k m
E ¼ 2:1 108 kN=m2 ;
G ¼ 8:1 107 kN=m2 : The number of domain nodes Nd used for the RBF scheme in this study is 7 in addition to the two boundary nodes and two ‘dummy’ nodes created, resulting into Np = 11. It has been noticed through numerical experiment that the optimum value of the parameter c depends on the number of nodes. In order to have a single global value of c for all cases of loading and boundary condition, the number of nodes has been fixed to 9. This number has been found sufficient to achieve a reasonable level of accuracy for the frequency factor values. The optimum value of c corresponding to this level of discretization has been found to be 0.6. Tables 3–5 show comparisons, for the first three modes of vibration of the CS beam with shear and axial force effects, between the frequency factors ðc1 ; c2 and c3 Þ obtained using RBF, FDM and the analytical method reported in [3] for N r ¼ 0:25; 0:5; and 0:75: Table 6 shows the RBF and analytical results for the same beam type for the case in which the shear and axial forces are neglected. Frequency factor values for this case are greater than the values when the axial
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Table 7 ¼ 3:705, C = 50.0 kN/m2, AG = 1,935,900 kN. Frequency factors’ prediction for a simple beam on elastic foundation with end rotational springs (RS) k = 10, k s G
G1
Method
FDM
RBF G2
FDM RBF
G3
FDM RBF
N
Ks 0
½
1
100
12 24 36 11 12
2.8829 2.8872 2.8880 2.8975 4.5751
2.7430 2.7459 2.7464 2.7503 4.5177
2.5571 2.5586 2.5588 2.5524 4.4530
4.1441 4.1745 4.1802 4.1843 5.5381
24 36 11 12
4.5998 4.6044 4.6281 5.7837
4.5402 4.5444 4.5672 5.7538
4.4731 4.4769 4.4981 5.7221
5.6133 5.6272 5.6378 6.6059
24 36 11
5.8482 5.8601 5.8885
5.8155 5.8269 5.8562
5.7809 5.7918 5.8220
6.7455 6.7713 6.7835
and shear effects are taken into consideration. Increase in the frequency factors for the first three modes of the beams with decrease in Nr was realized. The increase is more appreciable for lower modes. The results in all the three tables indicate that the proposed solution using RBF approach is in excellent agreement with the analytical solution. Furthermore, the comparison between the three approaches shows that RBF solution is more accurate than that obtained by FDM. This is clearly noticeable in the results tabulated which show that more FDM segments will be needed to converge to the exact solution as compared to those employed by the RBF. Taking into consideration the effects of bending moment, shear and axial forces, a comparison between the RBF results and the analytical ones [4] for the relationship between relative stiffness, k ¼ C s L4 =EI and the frequency factors ðc1 ; c2 and c3 Þ for the remaining two beam types (CC and SS) are presented in Figs. 5 and 6 for N r ¼ 0:25; 0:5; and 0:75: Both figures show a remarkable agreement between the RBF results and those obtained using the analytical solution. It should be noted that the replotting of the graphs of Ref. [4] has been achieved in this study using the WebPlotDigitizer [38], an online tool to extract numerical data from plot images. Results of the fourth case (RS) are presented in Table 7 which compares RBF with FDM for the first three frequencies and four levels of Ks ranging from a zero value corresponding to SS case to a high value of 100 that approximates CC case. High accuracy of RBF with less number of nodes as compared to FDM is evident in the results. Since the analytical solution for this last case is not readily available the problem is presented and solved in order to justify the use of the numerical scheme and to show the high accuracy of the RBF method over the FDM; as many as 36 FDM segments are needed to converge to the RBF solution obtained using just 8 segments (9 nodes). 5. Conclusion The study presented herein proposes the use of RBF-based meshless method for the solution of free vibration equations of beam on elastic foundation. Solved for different ratios of axial load to Euler buckling load and different boundary conditions, this work succeeded in obtaining results that are in excellent agreement with those of obtained by analytical methods as available in the literature. It is seen from the results that the RBF results agree very well with the analytical ones. Increase in the frequency factors for the first three modes of the beams with decrease in Nr, the ratio of axial load to the Euler buckling load, was realized. Two most noticeable advantages of the RBF method utilized here is the ease of programming in a computer and that no iteration (as in the case of DTM [3]) of any sort is needed to obtain solutions with high accuracy. Furthermore, far less number of nodes is needed to converge to the analytical solution as compared to the well-known FDM. Acknowledgment The authors would like to express their appreciation to King Fahd University of Petroleum & Minerals for supporting this study. References [1] M. Yihua, O. Li, Z. Hongzhi, Vibration analysis of Timoshenko Beams on a nonlinear elastic foundation, Tsinghua Sci. Technol. 14 (2009) 322–326. [2] C.M. Wang, K.Y. Lam, X.Q. He, Exact solutions for Timoshenko beams on elastic foundations using Green’s function, Mech. Based Des. Struct. Mach. 26 (1998) 101. [3] S. Çatal, Solution of free vibration equations of beam on elastic soil by using differential transform method, Appl. Math. Modell. 32 (2008) 1744–1757. [4] B. Ozturk, Free vibration analysis of beam on elastic foundation by the variational iteration method, Int. J. Nonlinear Sci. Numer. Simul. 10 (2009) 1255–1262. [5] S. N. Akour, Dynamics of nonlinear beam on elastic foundation, in: Proceedings of the World Congress on Engineering, WCE 2010, London, U.K. II: 2010.
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[6] H.H. West, M. Mafi, Eigenvalues for beam-columns on elastic supports, J. Struct. Eng. ASCE 110 (1984) 1305–1319. [7] M. Hsu, Vibration analysis of non-uniform beams resting on elastic foundations using the spline collocation method, Tamkang J. Sci. Eng. 12 (2009) 113–122. [8] A. Kacar, H.T. Tan, M.O. Kaya, Free vibration analysis of beams on variable Winkler elastic foundation by using the differential transform method, Math. Comput. Appl. 16 (2011) 773–783. [9] Ö. Civalek, B. Özturk, Free vibration analysis of tapered beam-column with pinned ends embedded in Winkler-Pasternak elastic foundation, Geomech. Eng. 2 (2010) 45–56. [10] R. Ansari, K. Hosseini, A. Darvizeh, B. Daneshian, A sixth-order compact finite difference method for non-classical vibration analysis of nanobeams including surface stress effects, Appl. Math. Comput. 219 (2013) 4977–4991. [11] T.P. Vo, H.-T. Thai, T.-K. Nguyen, A. Maheri, J. Lee, Finite element model for vibration and buckling of functionally graded beams based on a refined shear deformation theory, Eng. Struct. 64 (2014) 12–22. [12] P.R. Heyliger, J.N. Reddy, A higher order beam finite element for bending and vibration problems, J. Sound Vib. 126 (1988) 309–326. [13] C.W.S. To, Higher order tapered beam finite elements for vibration analysis, J. Sound Vib. 63 (1979) 33–50. [14] L.W. Zhang, Y.J. Deng, K.M. Liew, An improved element-free Galerkin method for numerical modeling of the biological population problems, Eng. Anal. Boundary Elem. 40 (2014) 181–188. [15] P. Zhu, L.W. Zhang, K.M. Liew, Geometrically nonlinear thermomechanical analysis of moderately thick functionally graded plates using a local Petrov– Galerkin approach with moving Kriging interpolation, Compos. Struct. 107 (2014) 298–314. [16] L.W. Zhang, P. Zhu, K.M. Liew, Thermal buckling of functionally graded plates using a local Kriging meshless method, Compos. Struct. 108 (2014) 472–492. [17] K.M. Liew, Z.X. Lei, J.L. Yu, L.W. Zhang, Postbuckling of carbon nanotube-reinforced functionally graded cylindrical panels under axial compression using a meshless approach, Comput. Methods Appl. Mech. Eng. 268 (2014) 1–17. [18] L.W. Zhang, Z.X. Lei, K.M. Liew, J.L. Yu, Static and dynamic of carbon nanotube reinforced functionally graded cylindrical panels, Compos. Struct. 111 (2014) 205–212. [19] L.W. Zhang, Z.X. Lei, K.M. Liew, J.L. Yu, Large deflection geometrically nonlinear analysis of carbon nanotube-reinforced functionally graded cylindrical panels, Comput. Methods Appl. Mechan. Eng. 273 (2014) 1–18. [20] R.J. Cheng, L.W. Zhang, K.M. Liew, Modeling of biological population problems using the element-free kp-Ritz method, Appl. Math. Comput. 227 (2014) 274–290. [21] X. Zhang, K.Z. Song, M.W. Lu, X. Liu, Meshless methods based on collocation with radial basis functions, Computational Mechanics 26 (2000) 333–343. [22] V.M.A. Leitao, A meshless method for Kirchhoff plate bending problems, Int. J. Numer. Methods Eng. 52 (2001) 1107–1130. [23] G.R. Liu, Y.T. Gu, A meshfree method: meshfree weak-strong (MWS) form method, for 2-D solids, Comput. Mechan. 33 (2003) 2–14. [24] J.T. Chen et al, A meshless method for free vibration analysis of circular and rectangular clamped plates using radial basis function, Eng. Anal. Boundary Elem. 28 (5) (2004) 535–545. [25] V.M.A. Leitao, RBF-based meshless methods for 2D elastostatic problems, Eng. Anal. Boundary Elem. 28 (2004) 1271–1281. [26] A.J.M. Ferreira, C.M.C. Roque, P.A.L.S. Martins, Radial basis functions and higher-order shear deformation theories in the analysis of laminated composite beams and plates, Compos. Struct. 66 (2004) 287–293. [27] A.J.M. Ferreira, Free vibration analysis of Timoshenko beams and Mindlin plates by radial basis functions, Int. J. Comput. Methods 2 (2005) 15–31. [28] H.J. Al-Gahtani, A meshless method for non-linear Poisson problems with high gradients, Comput. Assisted Mech. Eng. Sci. 13 (2006) 367–377. [29] C.M. Tiago, V.M.A. Leitao, Radial basis functions to linear and nonlinear structural analysis problems, Comput. Mech. Appl. 51 (2006) 1311–1334. [30] M. Naffa, H.J. Al-Gahtani, RBF-based meshless method for large deflection of thin plates, Eng. Anal. Boundary Elem. 31 (2007) 311–317. [31] H. Wang, Q.-H. Qin, Meshless approach for thermo-mechanical analysis of functionally graded materials, Eng. Anal. Boundary Elem. 32 (2008) 704–712. [32] H.J. Al-Gahtani, M. Naffa, RBF meshless method for large deflection of thin plates with immovable edges, Eng. Anal. Boundary Elem. 33 (2009) 176–183. [33] S. Xiang, G.-W. Kang, A nth-order shear deformation theory for the bending analysis on the functionally graded plates, Eur. J. Mech. A. Solids 37 (2013) 336–343. [34] H.H. Çatal, Free vibration of partially supported piles with the effects of bending moment, axial and shear force, Eng. Struct. 24 (2002) 1615–1622. [35] J.J. Tuma, Y.F. Cheng, Theory and Problems of Dynamic Structural Analysis, Schaum’s Outline Series, McGraw-Hill Inc., New York, 1983. pp. 234. [36] T. Belytschko, A. Huerta, S. Fernández-Méndez, T. Rabczuk, Meshless methods, Encyclopedia of computational mechanics, vol. 1, John Wiley & Sons, 2004, ISBN 0-470-84699-2. Chapter 10. [37] G.R. Liu, Mesh Free Methods, 1st ed., CRC Press, 2002, ISBN 0-8493-1238-8. [38] Rohatgi, A. (2013), WebPlotDigitalizer: HTML5 based online tool to extract numerical data from plot images. Version 3. [WWW document] URL http:// arohatgi.info/WebPlotDigitizer/?page_id=21 (accessed on July 2013).
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
RBF-based meshless method for the free vibration of beams on elastic foundations Husain J. Al-Gahtani ⇑, Faisal M. Mukhtar Department of Civil and Environmental Engineering, King Fahd University of Petroleum & Minerals, 31261 Dhahran, Saudi Arabia
a r t i c l e
i n f o
Keywords: Meshless method Radial basis function Free vibration analysis Beam on elastic foundation
a b s t r a c t In this paper, solution is obtained for the free vibration differential equations of motion of an axially loaded beam on elastic foundation using a meshless method. Use is made of the multiquadrics radial basis function (RBF) in obtaining the numerical solution for four different cases: (1) one end clamped, the other end simply supported; (2) both ends clamped; (3) both ends simply supported; and (4) a simple beam on elastic foundation with end rotational springs. The approach is easier to implement and program as compared to grid/mesh-based methods such as the finite difference method (FDM) and the finite element method (FEM). Accuracy of the results obtained using the proposed method was verified using the analytical results available in the literature for the first three cases considered. Numerical results of the fourth case were aimed at justifying the use of the numerical scheme for a problem whose analytical solution is not readily available and to show the high accuracy of the RBF method. The results prove that the method require much less number of nodes to converge to the correct solution as compared to FDM. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Many classical engineering problems whose behaviors are governed by differential equations require a smart choice of solution techniques that are fast, easy to program in a computer, and yet produce accurate and reliable results with minimum possible effort. The behavior of an axially loaded beam on elastic foundation is one such phenomenon whose free vibration is described by a fourth-order partial differential equation. As a result, various studies have been conducted to come up with reliable solutions to this problem using different approaches. A weak form quadrature element method (QEM) has been applied to analyze vibrations of beams on a nonlinear elastic foundation considering the effects of transverse shear deformation and the rotational inertia of beams by Yihua et al. [1]. Using Green’s functions, Wang et al. [2] presented a unified formulation for bending, buckling, and vibration problems of uniform Timoshenko and Euler–Bernoulli beams resting on various models of elastic foundations. The differential transform method (DTM) has been used to solve the free vibration differential equations of motion of one end clamped, the other simply supported axially loaded beams on elastic foundation by Çatal [3]. Örztuk [4] applied the variational iteration method (VIM) to analyze the free vibration of a beam on an elastic foundation with different end conditions. Akour [5] used Hamilton’s principle to derive the governing equation and investigated the behavior of a simply supported nonlinear beam resting on linear elastic foundation subjected to harmonic loading using Runge–Kutta technique. Eigenvalues for free vibration of beam-columns on elastic supports have been obtained by West and Mafi [6]. Three main parameters were investigated including the damping coefficient, the natural frequency, and ⇑ Corresponding author. http://dx.doi.org/10.1016/j.amc.2014.09.097 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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the coefficient of the nonlinear term. Hsu [7] applied the spline collocation method and solved for the natural frequencies of non-uniform beams resting on elastic foundations. Kacar et al. [8] studied the free vibration of an Euler–Bernoulli beam resting on a variable Winkler foundation using the differential transformation method. The method of discrete singular convolution (DSC) has been used by Civalek and Özturk [9] to model free vibration of tapered piles embedded in two-parameter elastic foundations. Another competitive type of solution is the grid-based or mesh-based methods such as FDM and FEM. Ansari et al. [10] developed a non-classical model for the free vibrations of nanobeams accounting for the surface stress effects. The influence of surface stress was incorporated into the Euler–Bernoulli beam theory based on Gurtin–Murdoch elasticity theory. In order to obtain the natural frequencies of the beams subject to different boundary conditions, discretization of the governing equation was achieved using a compact finite difference method (CFDM) of sixth order. Effects of beam thickness, surface density, surface residual stress, surface elastic constants and boundary conditions on the natural frequencies of the beams were investigated. Vo et al. [11] presented an FEM model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Hamilton’s principle was used to derive the governing equations of motion and boundary conditions of the sandwich beam whose core is fully metal or ceramic and functionally graded skin. Effects of several factors on the natural frequencies, critical buckling loads and load–frequency curves the beams were analyzed. Finite element equations for a variationally consistent higher order beam theory were presented by Heyliger and Reddy [12] for the static and dynamic behavior of rectangular beams. The stress-free conditions on the upper and lower surfaces of the beam were accounted for by the higher order beam theory. This theory overcomes the limitations of the elementary beam theory that disregards the transverse shear deformation effect and the limitation of Timoshenko beam theory (first order shear deformation theory) that assumes a constant transverse shear strain distribution; The higher order beam theory models, more accurately, the parabolic shear stress distribution without the need for shear correction coefficient needed in the Timoshenko theory. The recovery of the Kirchoff constraint for thin beams was resulted due to the Full integration of the shear stiffness terms. Accuracy of the model was demonstrated through several numerical examples. To [13] presented explicit expressions for mass and stiffness matrices of two higher order tapered beam finite elements for vibration analysis. The loss of computer time and round-off-errors associated with extensive matrix operations and numerical evaluation of the expressions were eliminated due to the explicit element mass and stiffness matrices. It was verified that the higher order tapered beam elements presented are superior to the lower order one as they offer more realistic representations of the curvature and loading history of the beam element. In all the aforementioned references, the methods involved are either based on analytical approaches that are mathematically involved or based on numerical solutions that require mesh for the solution interpolation. Nevertheless, the possibility of obtaining accurate numerical solutions without resorting to the mesh-based techniques mentioned above, has been the goal of many researchers throughout the computational mechanics community for the past two decades or so. There are several types of meshless methods as briefly discussed here. Based on the improved element-free Galerkin method, Zhang et al. [14] performed a numerical study for degenerate parabolic equations arising from the spatial diffusion of biological populations. Galerkin procedure was used to derive the discrete equation system and the essential boundary conditions imposed using the penalty method. Convergence study was carried out for the time dependent problem. A number of numerical examples were used to establish the applicability of the method. Based on the moving Kriging interpolation technique Zhu et al. [15] developed a meshless local Petrov–Galerkin approach for geometrically nonlinear thermoelastic analysis of functionally graded plates in thermal environments. The constructed shape functions possess Kronecker delta function property due to the Kriging interpolation method. This eliminates the need for special techniques for enforcing essential boundary conditions. Zhang et al. [16] studied the mechanical and thermal behaviors of ceramic–metal grade plates (FGPs) using Kriging meshless method. The local meshless method was developed based on the local Petrov–Galerkin weak-form formulation combined with shape functions having Kronecker delta function property. Postbuckling analysis of carbon nanotubereinforced functionally graded (CNTR-FG) cylindrical panels under axial compression was reported by Liew et al. [17]. Ritz method was employed, based on kernel particle approximations for the field variables, to obtain the discretized governing equations. Mesh-free kp-Ritz was employed by Zhang et al. [18] for analysis of flexural strength and free vibration of functionally graded carbon nanotube-reinforced composite cylindrical panels. Formulations were based on the first-order shear deformation shell theory and the 2-D transverse displacement field was approximated by the mesh-free kernel particles estimate. Similar mesh-free kp-Ritz method was adopted by Zhang et al. [19] for large deflection geometry nonlinear analysis of CNTR-FG cylindrical panel and by Cheng et al. [20] to analyze a degenerate parabolic equation arising in the spatial diffusion of biological population. RBF-based collocation method, as one of the most recently developed meshless methods, has attracted attention in recent years especially in the area of computational mechanics [21–33]. Due to its simplicity to implement, it represents an attractive alternative to FDM and FEM. The objective of this paper is to offer a simple yet accurate meshless method for the solution of free vibration equations of beam on elastic foundation. To the knowledge of the authors, so far, RBF collocation methods have not been applied to this specific free vibration problem. The proposed RBF method can be considered as a competitive candidate due to its simplicity, robustness and accuracy. Four different support conditions are considered: One end clamped and other end simply supported; both ends clamped; both ends simply supported; and a simple beam on elastic foundation with end rotational springs. For verification of the results obtained, use is made of the analytical solutions given by Çatal [3] for the case of one end clamped and other end simply supported beam on elastic foundation. For the cases of both ends clamped and both ends simply supported, the analytical results reported by Ozturk [4] are used for the verification purpose. The efficiency of the method is demonstrated for the first three cases by comparing the minimum number of RBF nodes needed for
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convergence to the exact solution to that needed by FDM. Numerical results of the fourth case were aimed at justifying the use of the numerical scheme for a problem whose analytical solution is not readily available and to show the higher accuracy of the RBF method over the mesh-based FDM scheme. 2. Governing differential equations Consider a beam of length L and cross-sectional area A resting on elastic foundation as shown in Fig. 1 (a). Let G, E and I be its shear modulus, elastic modulus and moment of inertia respectively. Fig. 1 (b) shows a differential segment of length dx out of this beam with the internal forces and deformations indicated. Idealizing the elastic foundation by Winkler model, the relation between displacement function y(x, t) of the beam and the distributed load q(x, t) acting beneath it due to the reaction within the elastic foundation can be written as q(x, t) = Csy(x, t). For a modulus of subgrade reaction C0, and beam width b, the parameter Cs can be written as Cs = C0b. Neglecting the second order terms, the equilibrium equations of the internal forces shown in Fig. 1 (b) are used to derive the motion equation (upon simplification) of the beam [34,35] as given by Eqs. (2.1) and (2.2).
N
N
x
Elastic foundation
Csy(x,t) y L
(a) M ( x, t )
M ( x, t )
N N
T ( x, t )
T ( x, t )
M ( x, t ) dx x ( x, t ) dx x
T ( x, t ) dx x
Csy(x,t) dx dx
(b) Fig. 1. (a) A beam on elastic foundation (b) internal forces and deformations of the beam on elastic foundation.
Fig. 2. Domain and boundary nodal distribution.
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Table 1 Some Commonly used RBF’s. Function
Definition pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /ðrÞ ¼ r 2 þ c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 /ðrÞ ¼ ð r2 þ c2 Þ 2 /ðrÞ ¼ Rn
Multiquadrics (MQ) Inverse multiquadrics Nth Polynomial splines Gaussian
/ðrÞ ¼ eðcr
Thin plate (polyharmonic) splines (TPS)
" /i1v ðzÞ þ
/i2v ðzÞ þ
p2 Nr þ
Þ
/ðrÞ ¼ ð1Þkþ1 r 2k logðrÞ
# 2 x2 C s ÞkL x2 ÞL4 ðm ðC s m /1 ðzÞ ¼ 0; /001 ðzÞ þ AG EI
x2 ÞL4 ðC s m /2 ðzÞ ¼ 0; EI
2
0 6 z 6 1;
0 6 z 6 1:
ð2:1Þ
ð2:2Þ
In the above equations, the position variable x is substituted with a dimensionless parameter z = x/L. The dimensionless displacement function of the beam taking into consideration and neglecting the effects of axial and shear forces are, respec2
4
tively, given as /1(z) and /2(z). /001 ðzÞ ¼ d d/z12ðzÞ and /i1v ðzÞ ¼ d d/z14ðzÞ are, respectively, the second and fourth derivatives of the dimensionless displacement function with respect to the dimensionless position variable z. The time variable is denoted 2 as t and the ratio of axial load N to the Euler buckling load as N r ¼ NL pEI : m, x and k represent the distributed mass of the beam, beam circular frequency and shape factor due to shape of the beam section, respectively. Eq. (2.1) applies to the case in which the axial and shear forces are taken into consideration, and the coefficient of the second derivative represents the sum of the axial force and shear effects whereas the coefficient of the dimensionless displacement function represents the effect due to the foundation. Neglecting the effects of axial and shear forces results in Eq. (2.2).
N
N Cs L
(a) N
N Cs L
(b) N
N Cs L
(c) Ks
Ks
Cs L
(d) Fig. 3. (a) One end clamped and other simply supported (CS) (b) Both ends clamped (CC) (c) Both ends simply supported (SS) (d) Simple beam on elastic foundation with end rotational springs (RS).
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Table 2 Boundary conditions for the beams considered.* Beam Type
z=0
CS CC SS RS
z=1
/ðzÞ
/0 ðzÞ
/00 ðzÞ
EI /00 ðzÞ K s /0 ðzÞ
/ðzÞ
/0 ðzÞ
/00 ðzÞ
EI /00 ðzÞ K s /0 ðzÞ
0 0 0 0
0 0 – –
– – 0 –
– – – 0
0 0 0 0
– 0 – –
0 – 0 –
– – – 0
The dash symbol (–) means not applicable. * /ðzÞ ¼ /1 ðzÞ and /2 ðzÞ for Eqs. (2.1) and (2.2) respectively.
1/(Nd+1)
z
0.1
1 Domain and boundary nodes Dummy nodes Fig. 4. Domain and boundary discretization.
3. RBF method formulation Meshless methods have attracted the attention of many researchers during the last two decades. Though, uniquely simple, these methods provide solution accuracies that compete with those of finite elements and boundary elements without the difficult and time-consuming task for mesh connectivity [36,37]. 3.1. Radial basis function (RBF) Radial basis function is one of the most popular types of meshless methods, and its general principle of formulation is explained here using an elliptic type PDE with the domain L and boundary B operators on the dependent variable u (Eqs. (3.1) and (3.2)). F and g are the continuous functions of the position. The same principle applies when using this technique to solve other types of PDE’s.
Lu ¼ f ;
ð3:1Þ
Bu ¼ g:
ð3:2Þ
To apply RBF, we model the geometry of the problem by randomly distributed nodes Nd and Nb, respectively, on the domain X and on the boundary C as shown in Fig. 2. This forms a set of Np points (Np = Nd + Nb).Assuming a suitable RBF centered at a given number of points, the solution to Eq. (3.1) can be approximated using the direct collocation method as
uðxÞ ¼
Np X
aj /ðrj Þ;
ð3:3Þ
j¼1
where, / is the radial basis function centred at xj, rj ¼ jx xj j is the radial distance between the point x and the centre of RBF xj , and aj are coefficients to be determined. Some commonly used types of radial basis functions include those shown in Table 1. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi However, in the formulation of the problems solved in this study, use is made of the multiquadrics / ¼ jx xj j2 þ c2 , where c – 0 is an adjustable constant known as the shape parameter. The unknown coefficients (aj) are determined by solving the system of Np linear equations formed by applying the operators L and B at the domain nodes, Nd, and the boundary nodes, Nb, respectively.
Luðxi Þ ¼
Nd X
aj Lð/jxi xj jÞ;
i ¼ 1; Nd ;
ð3:4Þ
i ¼ 1; Nb :
ð3:5Þ
j¼1
Buðxi Þ ¼
Np X
aj Bð/jxi xj jÞ;
j¼N d þ1
The resulting collocation Eqs. ((3.4) and (3.5)) can be written in a matrix form thus,
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Table 3 = 3.705, C = 50.0 kN/m2, Frequency factors’ prediction for a one end clamped the other simply supported beam (CS) on elastic foundation Nr = 0.25, k s AG = 1,935,900 kN. Method
G
N
k 1
FDM/%Err.
G1
RBF/%Err. Analytical FDM/%Err.
G2
RBF/%Err. Analytical FDM/%Err.
G3
RBF/%Err. Analytical
10
100
1000
10000
4 8 12 11 N/A
2.6905 2.7846 2.8021 2.8219 2.8157
4.45 1.10 0.48 0.22
3.2064 3.3312 3.3552 3.3809 3.3745
4.98 1.28 0.57 0.19
3.8991 4.0102 4.0327 4.0561 4.0511
3.75 1.01 0.45 0.12
5.8213 5.8629 5.8718 5.8745 5.8794
0.99 0.28 0.13 0.08
10.0385 10.0472 10.0491 10.0493 10.0511
0.13 0.04 0.02 0.02
4 8 12 11 N/A
3.5968 3.8819 3.9349 3.9856 3.9766
9.55 2.38 1.05 0.23
4.5059 4.9088 4.9857 5.0598 5.0472
10.72 2.74 1.22 0.25
5.3798 5.9007 6.0063 6.1083 6.0925
11.70 3.15 1.41 0.26
6.6921 7.1361 7.2367 7.3499 7.3222
8.61 2.54 1.17 0.38
10.2728 10.4326 10.4734 10.5205 10.5099
2.26 0.74 0.35 0.10
4 8 12 11 N/A
4.0001 4.6118 4.7276 4.8236 4.8192
17.00 4.30 1.90 0.09
5.1180 5.9630 6.1270 6.2652 6.2581
18.22 4.72 2.09 0.11
6.2570 7.4053 7.6407 7.8710 7.8318
20.11 5.45 2.44 0.50
7.5378 8.8080 9.1000 9.4075 9.3458
19.35 5.75 2.63 0.66
10.6169 11.3378 11.5428 11.7737 11.7296
9.49 3.34 1.59 0.38
Table 4 ¼ 3:705, C = 50.0 kN/m2, Frequency factors’ prediction for a one end clamped the other simply supported beam (CS) on elastic foundation Nr = 0.5, k s AG = 1,935,900 kN. G
Method
N
FDM/%Err.
4 8 12 11 N/A
2.5776 2.6803 2.6993 2.7211 2.7141
5.03 1.25 0.55 0.26
3.0838 3.2159 3.2411 3.2685 3.2613
8.61 4.70 3.95 3.14
3.8084 3.9198 3.9423 3.966 3.9607
5.99 3.24 2.69 2.10
5.7914 5.8308 5.8392 5.8479 5.8463
1.50 0.83 0.68 0.54
10.0324 10.0406 10.0424 10.0424 10.0436
0.19 0.10 0.09 0.09
4 8 12 11 N/A
3.5398 3.8353 3.8901 3.9422 3.933
10.00 2.48 1.09 0.23
4.4349 4.8502 4.929 5.0049 4.992
11.16 2.84 1.26 0.26
5.3024 5.8344 5.9417 6.0453 6.0219
11.95 3.11 1.33 0.39
6.6374 7.0826 7.1833 7.2791 7.2685
8.68 2.56 1.17 0.15
10.2559 10.4129 10.453 10.4994 10.4887
2.22 0.72 0.34 0.10
4 8 12 11 N/A
3.9564 4.5826 4.7004 4.7979 4.7934
17.46 4.40 1.94 0.09
5.062 5.9251 6.0916 6.2318 6.2245
18.68 4.81 2.14 0.12
6.1915 7.359 7.5971 7.8153 7.7903
20.52 5.54 2.48 0.32
7.4794 8.7603 9.0541 9.3641 9.3023
19.60 5.83 2.67 0.66
10.5916 11.3088 11.5132 11.7435 11.6996
9.47 3.34 1.59 0.38
k 1
G1
RBF/%Err. Analytical FDM/%Err.
G2
RBF/%Err. Analytical FDM/%Err.
G3
RBF/%Err. Analytical
Lð/Þ Bð/Þ
fag ¼
10
100
1000
f : g
10000
ð3:6Þ
Eq. (3.6) makes it convenient for the values of the unknowns (a’s) to be determined. 3.2. Application of RBF to free vibration of beam on elastic foundation Fig. 3 shows the four types of beams resting on an elastic foundation each having different end conditions considered and solved in this study: One end clamped and other simply supported (CS); both ends clamped (CC); both ends simply supported (SS); and a simple beam on elastic foundation (no axial force) with end rotational springs (RS). The boundary conditions for these beam types are given in Table 2. The additional term K s /0 ðzÞ appearing in the slope boundary condition of the last case, RS, reflects the contribution of the rotational spring of spring constant Ks. The exact solution for this last case is not readily available for a finite value of Ks – 0. The domain of interest was discretized using a finite number of equally spaced domain nodes, Nd and two boundary nodes at the two ends of the beam (z ¼ 0 and z ¼ 1). However, since there exist four boundary conditions that need to be satisfied (Table 2), we end up with Nd + 4 equations to solve for Nd + 2 unknowns. This poses a problem to the scheme due to the fact that same number of equations as the number of unknowns is necessary for the complete solution of such linear algebraic equations. In the proposed RBF method, we created two additional ‘dummy’ nodes one each at positions z ¼ 0:1 and z ¼ 1:1 thereby satisfying the requirement (Nd + 4equations to solve for Nd + 4 unknowns by applying each in turn to Nd + 4 nodes). The final discretized beam with the two created ‘dummy’ nodes is shown in Fig. 4. With this, the
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problem is solved by satisfying the boundary conditions and the governing differential equation over selected points over the boundary and inside the domain, respectively. qffiffiffiffiffiffiffiffiffiffiffi 2 4 4 Recognizing that c ¼ m xEI L , the frequency factor, in (2.1) and (2.2) the general motion equations were solved using the RBF method in which the variable /(z) (i.e. /1(z) or /2(z)) was approximated with an equation of the form given in (3.3). Since the domain and boundary operators, L and B operate over /(z) to define (2.1) and/or (2.2) as the case may be and the relevant boundary conditions given in Table 1, respectively, we generated the equations and write them in the form described in (3.6). However, since the target of the formulation presented in this study is obtaining the values of the frequency factors c, we by-passed the step of obtaining a0 s and argued that for a non-trivial solution, determinant of coefficient matrix must be zero.
Lð/Þ Bð/Þ ¼ 0:
ð3:7Þ
This determinant yields a characteristic equation in terms of c. The positive real roots of the equation are the free vibration frequency factors (hence frequencies, x) of the beams on elastic foundation shown in Fig. 3.
Table 5 ¼ 3:705, C = 50.0 kN/m2, Frequency factors’ prediction for a one end clamped the other simply supported beam (CS) on elastic foundation Nr = 0.75, k s AG = 1,935,900 kN. G
Method
N
FDM/%Err.
4 8 12 11 N/A
2.4474 2.5621 2.5832 2.6076 2.5996
5.85 1.44 0.63 0.31
2.944 3.0863 3.1131 3.1429 3.1346
6.08 1.54 0.69 0.26
3.7102 3.8223 3.8447 3.869 3.863
3.96 1.05 0.47 0.16
5.7609 5.7979 5.8058 5.8141 5.8124
0.89 0.25 0.11 0.03
10.0264 10.0339 10.0355 10.0424 10.0367
0.10 0.03 0.01 0.06
4 8 12 11 N/A
3.4799 3.7871 3.8436 3.8974 3.8879
11.52 3.71 2.27 0.91
4.3603 4.7894 4.8703 4.9481 4.9349
12.65 4.06 2.44 0.88
5.2216 5.7658 5.8749 5.9806 5.9634
13.29 4.25 2.44 0.69
6.5815 7.0279 7.1286 7.2255 7.2141
9.45 3.31 1.92 0.59
10.2389 10.3931 10.4325 10.5189 10.4673
2.38 0.91 0.54 0.29
4 8 12 11 N/A
3.9111 4.5529 4.6727 4.7717 4.7672
17.96 4.50 1.98 0.09
5.0041 5.8865 6.0556 6.1955 6.1904
19.16 4.91 2.18 0.08
6.1238 7.3118 7.5528 7.7469 7.748
20.96 5.63 2.52 0.01
7.4196 8.7117 9.0075 9.3576 9.257
19.85 5.89 2.70 1.09
10.5662 11.2795 11.4833 11.8277 11.6665
9.43 3.32 1.57 1.38
k 1
G1
RBF/%Err. Analytical G2
FDM/%Err.
RBF/%Err. Analytical G3
FDM/%Err.
RBF/%Err. Analytical
10
100
1000
10000
Table 6 Frequency factors’ prediction for a one end clamped the other simply supported beam (CS) on elastic foundation with shear and axial force effects neglected. Method
G
N
k 1
G1
FDM/%Err.
RBF/%Err. Analytical G2
FDM/%Err.
RBF/%Err. Analytical G3
FDM/%Err.
RBF/%Err. Analytical a b
10
100
4 8 12 11 N/A
3.6813 3.8633 3.9003 3.9372 3.9307
6.34 1.71 0.77 0.17
3.7256 3.9018 3.9377 3.9736 3.9673
6.09 1.65 0.75 0.16
4.1003 4.2353 4.2635 4.2918 5.9315a
4 8 12 11 N/A
5.9922 6.7605 6.9277 7.0927 7.0693
15.24 4.37 2.00 0.33
6.0026 6.7677 6.9344 7.099 7.0757
15.17 4.35 2.00 0.33
6.104 6.8392 7.0009 7.1613 7.1384
4 8 12 11 N/A
7.4846 9.3897 9.8284 10.1976 10.2104
26.70 8.04 3.74 0.13
7.4899 9.3924 9.8308 10.2505 10.212
26.66 8.03 3.73 0.38
7.5429 9.4195 9.8544 10.2521 10.2336
1000 b
10000
5.8643 5.9122 5.9226 5.9331 5.9315
1.13 0.33 0.15 0.03
10.0454 10.055 10.0571 10.0575 10.0591
0.14 0.04 0.02 0.02
14.49 4.19 1.93 0.32
6.9163 7.4544 7.5806 7.7139 7.6896
10.06 3.06 1.42 0.32
10.3076 10.4855 10.5317 10.585 10.5729
2.51 0.83 0.39 0.11
26.29 7.96 3.71 0.18
8.02 9.6779 10.0815 10.4932 10.4733
23.42 7.59 3.74 0.19
10.7059 11.5461 11.7912 12.0703 12.0194
10.93 3.94 1.90 0.42
b b b
There seems to be a typographical error in this particular value of analytical solution reported in [3]. Not reported because the likely error in ‘a’ will affect these values.
H.J. Al-Gahtani, F.M. Mukhtar / Applied Mathematics and Computation 249 (2014) 198–208
205
(a)
(b)
(c) Fig. 5. Variation of frequency factors, G due to relative stiffness, k for both ends clamped beam (CC) on elastic foundation for; (a) Nr = 0.25 (b) Nr = 0.5 (c) Nr = 0.75.
4. Numerical example Free vibration equations for four different types of beam on elastic foundation (Fig. 3) are solved using the RBF approach presented in this study. The analytical results of the same problem reported by Çatal [3] are used for verification of the results obtained for the case of one end being clamped and the other simply supported (CS), whereas the analytical results presented by Ozturk [4] are used for the verification of the next two cases where both ends are clamped (CC) and both ends are simply supported (SS). The Numerical results of the fourth case were aimed at justifying the use of the numerical scheme for a problem whose analytical solution is not readily available. The efficiency of RBF is demonstrated by comparing its results with those of FDM for the cases considered. The FDM scheme employed in all examples is based on the secondorder-accurate centered difference form. Same values of the parameters and beam profile are chosen as those used in [3] and [4]. Hence, the beam is made up of an IPB 500 steel profile resting on a Winkler type foundation of modulus of subgrade reaction of 50,000 kN/m2. Other parameters are:
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H.J. Al-Gahtani, F.M. Mukhtar / Applied Mathematics and Computation 249 (2014) 198–208
(a)
(b)
(c) Fig. 6. Variation of frequency factors, G due to relative stiffness, k for simply supported beam (SS) on elastic foundation for; (a) Nr = 0.25 (b) Nr = 0.5 (c) Nr = 0.75.
I ¼ 107:2 105 m4 ;
A ¼ 2:39 102 m2 ;
¼ 3:705; ¼ 0:19 kNs2 =m; k m
E ¼ 2:1 108 kN=m2 ;
G ¼ 8:1 107 kN=m2 : The number of domain nodes Nd used for the RBF scheme in this study is 7 in addition to the two boundary nodes and two ‘dummy’ nodes created, resulting into Np = 11. It has been noticed through numerical experiment that the optimum value of the parameter c depends on the number of nodes. In order to have a single global value of c for all cases of loading and boundary condition, the number of nodes has been fixed to 9. This number has been found sufficient to achieve a reasonable level of accuracy for the frequency factor values. The optimum value of c corresponding to this level of discretization has been found to be 0.6. Tables 3–5 show comparisons, for the first three modes of vibration of the CS beam with shear and axial force effects, between the frequency factors ðc1 ; c2 and c3 Þ obtained using RBF, FDM and the analytical method reported in [3] for N r ¼ 0:25; 0:5; and 0:75: Table 6 shows the RBF and analytical results for the same beam type for the case in which the shear and axial forces are neglected. Frequency factor values for this case are greater than the values when the axial
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Table 7 ¼ 3:705, C = 50.0 kN/m2, AG = 1,935,900 kN. Frequency factors’ prediction for a simple beam on elastic foundation with end rotational springs (RS) k = 10, k s G
G1
Method
FDM
RBF G2
FDM RBF
G3
FDM RBF
N
Ks 0
½
1
100
12 24 36 11 12
2.8829 2.8872 2.8880 2.8975 4.5751
2.7430 2.7459 2.7464 2.7503 4.5177
2.5571 2.5586 2.5588 2.5524 4.4530
4.1441 4.1745 4.1802 4.1843 5.5381
24 36 11 12
4.5998 4.6044 4.6281 5.7837
4.5402 4.5444 4.5672 5.7538
4.4731 4.4769 4.4981 5.7221
5.6133 5.6272 5.6378 6.6059
24 36 11
5.8482 5.8601 5.8885
5.8155 5.8269 5.8562
5.7809 5.7918 5.8220
6.7455 6.7713 6.7835
and shear effects are taken into consideration. Increase in the frequency factors for the first three modes of the beams with decrease in Nr was realized. The increase is more appreciable for lower modes. The results in all the three tables indicate that the proposed solution using RBF approach is in excellent agreement with the analytical solution. Furthermore, the comparison between the three approaches shows that RBF solution is more accurate than that obtained by FDM. This is clearly noticeable in the results tabulated which show that more FDM segments will be needed to converge to the exact solution as compared to those employed by the RBF. Taking into consideration the effects of bending moment, shear and axial forces, a comparison between the RBF results and the analytical ones [4] for the relationship between relative stiffness, k ¼ C s L4 =EI and the frequency factors ðc1 ; c2 and c3 Þ for the remaining two beam types (CC and SS) are presented in Figs. 5 and 6 for N r ¼ 0:25; 0:5; and 0:75: Both figures show a remarkable agreement between the RBF results and those obtained using the analytical solution. It should be noted that the replotting of the graphs of Ref. [4] has been achieved in this study using the WebPlotDigitizer [38], an online tool to extract numerical data from plot images. Results of the fourth case (RS) are presented in Table 7 which compares RBF with FDM for the first three frequencies and four levels of Ks ranging from a zero value corresponding to SS case to a high value of 100 that approximates CC case. High accuracy of RBF with less number of nodes as compared to FDM is evident in the results. Since the analytical solution for this last case is not readily available the problem is presented and solved in order to justify the use of the numerical scheme and to show the high accuracy of the RBF method over the FDM; as many as 36 FDM segments are needed to converge to the RBF solution obtained using just 8 segments (9 nodes). 5. Conclusion The study presented herein proposes the use of RBF-based meshless method for the solution of free vibration equations of beam on elastic foundation. Solved for different ratios of axial load to Euler buckling load and different boundary conditions, this work succeeded in obtaining results that are in excellent agreement with those of obtained by analytical methods as available in the literature. It is seen from the results that the RBF results agree very well with the analytical ones. Increase in the frequency factors for the first three modes of the beams with decrease in Nr, the ratio of axial load to the Euler buckling load, was realized. Two most noticeable advantages of the RBF method utilized here is the ease of programming in a computer and that no iteration (as in the case of DTM [3]) of any sort is needed to obtain solutions with high accuracy. Furthermore, far less number of nodes is needed to converge to the analytical solution as compared to the well-known FDM. Acknowledgment The authors would like to express their appreciation to King Fahd University of Petroleum & Minerals for supporting this study. References [1] M. Yihua, O. Li, Z. Hongzhi, Vibration analysis of Timoshenko Beams on a nonlinear elastic foundation, Tsinghua Sci. Technol. 14 (2009) 322–326. [2] C.M. Wang, K.Y. Lam, X.Q. He, Exact solutions for Timoshenko beams on elastic foundations using Green’s function, Mech. Based Des. Struct. Mach. 26 (1998) 101. [3] S. Çatal, Solution of free vibration equations of beam on elastic soil by using differential transform method, Appl. Math. Modell. 32 (2008) 1744–1757. [4] B. Ozturk, Free vibration analysis of beam on elastic foundation by the variational iteration method, Int. J. Nonlinear Sci. Numer. Simul. 10 (2009) 1255–1262. [5] S. N. Akour, Dynamics of nonlinear beam on elastic foundation, in: Proceedings of the World Congress on Engineering, WCE 2010, London, U.K. II: 2010.
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