Acta Mechanica Sinica (2013) 29(5):718–727 DOI 10.1007/s10409-013-0069-3
RESEARCH PAPER
Dynamic response to a moving load of a Timoshenko beam resting on a nonlinear viscoelastic foundation Yan Yang · Hu Ding · Li-Qun Chen
Received: 3 June 2013 / Revised: 1 August 2013 / Accepted: 18 August 2013 ©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2013
Abstract The present paper investigates the dynamic response of finite Timoshenko beams resting on a sixparameter foundation subjected to a moving load. It is for the first time that the Galerkin method and its convergence are studied for the response of a Timoshenko beam supported by a nonlinear foundation. The nonlinear Pasternak foundation is assumed to be cubic. Therefore, the effects of the shear deformable beams and the shear deformation of foundations are considered at the same time. The Galerkin method is utilized for discretizing the nonlinear partial differential governing equations of the forced vibration. The dynamic responses of Timoshenko beams are determined via the fourth-order Runge–Kutta method. Moreover, the effects of different truncation terms on the dynamic responses of a Timoshenko beam resting on a complex foundation are discussed. The numerical investigations shows that the dynamic response of Timoshenko beams supported by elastic foundations needs super high-order modes. Furthermore, the system parameters are compared to determine the dependence of the convergences of the Galerkin method. Keywords Nonlinear · Timoshenko beam · Pasternak foundation · Galerkin method · Convergence The project was supported by the State Key Program of National Natural Science Foundation of China (10932006 and 11232009), and Innovation Program of Shanghai Municipal Education Commission (12YZ028). Y. Yang · H. Ding (¬) · L.-Q. Chen Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, 200072 Shanghai, China email:
[email protected] L.-Q. Chen Department of Mechanics, Shanghai University, 200444 Shanghai, China
1 Introduction With the increase of road traffic and vehicle loads, earlier damage of asphalt pavement on highways has become more and more serious, which has greatly shortened the service life of the pavement. Many researches have shown that one of the most important reasons for road damage is the vehicle load. These pavement-vehicle systems can be theoretically modeled as beams supported by foundations subjected to moving forces. Therefore, the studies of the dynamic behavior of structures under moving loads have received enormous attention in the literatures [1, 2]. Winkler foundation model is mostly considered to represent the elastic foundation for mathematical simplicity [3]. This kind of model is considered as a system of mutually independent linear springs, in which it is assumed that the deflection of foundation at any point on the surface is directly proportional to the stress. However, it does not accurately represent the continuous characteristics of practical foundations because the interaction between the lateral springs is not taken into account in this model. To find a foundation model that is both closer to physical reality and mathematically simple, researchers proposed various two-parameter foundations [4, 5], including the Pasternak model, which is introduced to account for the interaction among the linear elastic springs. Uzzal et al. [6] presented the dynamic response of an Euler–Bernoulli beam lying on the Pasternak foundation under a moving load as well as a moving mass. Cao and Zhong [7] investigated the dynamic response of an Euler–Bernoulli beam on the Pasternak foundation subjected to moving loads. They found that the maximum deflection of a beam resting on the two-parameter foundation is much smaller than that of a beam on the Winkler foundation. With the development of researches on the dynamic response of beams resting on linear foundations, researchers began to pay attention to beams on nonlinear foundations, even though there are complexity and difficulties in dealing
Dynamic response to a moving load of a Timoshenko beam resting on a nonlinear viscoelastic foundation
with nonlinearity. Moreover, it is no wonder that the main factor, which strongly affects the dynamic of soil-structure problem, is the validity of the selected nonlinear foundation models. With linear-plus-cubic stiffness, Tsiatas [8] studied finite beams on a nonlinear foundation. He found that the linear analysis is inadequate to predict the real response of the beam even for small nonlinearity in the foundation, and the use of nonlinear foundation is essential. Senalp et al. [9] focused on the dynamic response of a simply-supported finite length Euler–Bernoulli beam resting on a linear and nonlinear viscoelastic foundation subjected to a moving concentrated force. They found that the dynamic responses of the beams on the nonlinear foundation model are always greater as compared to the linear foundation model. That is to say, the dynamic response for the nonlinear foundation model provides more conservative estimate in track or pavement design. Nguyen and Duhamel [10] solved both linear and nonlinear problems for all values of velocity and frequency of loads. They revealed significant dynamic effects when the nonlinearities of the foundation were taken into account. Dahlberg [11] obtained some results arising from moving load and found that the nonlinear model simulated the beam deflection fairly well as compared to measurements, whereas the linear model did not. That is to say, the influence of foundation’s nonlinearity can not be omitted. Wu and Thompson [12] compared the results from both the linear and nonlinear track models. They found that both the impact force and the track vibration level arising from nonlinear foundations are shown to be noticeably higher than those from linear foundations. Therefore, it is concluded that linear models are not appropriate for analyzing wheel/track impact because the foundation stiffness varies dramatically under the impact force. Kargarnovin et al. [13] investigated the nonlinearity effects on the beam response and compared the results for a nonlinear and an equivalent linear model. Sapountzakis and Kampitsis [14] studied the nonlinear response of beams resting on a nonlinear three-parameter foundation. They found the discrepancy between the results from the linear and the nonlinear analyses is remarkable. Based on the above-mentioned, a conclusion can be drawn that the nonlinear foundation is a very practical model for dynamic loading analyses. It should be remarked that the literatures on beams on nonlinear Pasternak foundations are rather limited. In the analysis of vibration of beams resting on foundations under a moving load, the beam has been modeled as an Euler–Bernoulli beam [15–17], or a Rayleigh beam [18] or a Timoshenko beam [13, 14, 19–24]. Ruge and Birk [25] compared the results of Timoshenko and Euler–Bernoulli beams on Winkler foundation in the frequency-domain. In this case, an additional equation governing the rotation of the beam cross-section is considered. They found that the physically more realistic Timoshenko beam model offers additional numerical advantages in unbounded domains. Zehsaz et al. [26] investigated the linear dynamic response of finite
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Timoshenko beam lying on a viscoelastic bed and subjected to a moving load. Ding et al. [27] investigated the dynamic response of infinite Timoshenko beams lying on nonlinear foundations subjected to a moving concentrated force. They found that the shear modulus of the beams have a significant influence on the dynamic response. Similar conclusions are obtained in Ref. [13]. Although the beam on a certain kind of foundation is extensively studied, it should be noted that the effects of the shear deformable beams and the shear modulus of foundations are rarely considered at the same time. Moreover, to our knowledge, there is no published work on the dynamic response of finite Timoshenko beams supported by nonlinear foundations. The dynamic response of elastic beams resting on nonlinear viscoelastic Pasternak foundation displays nonlinear and viscous characters, and its solution becomes difficult to achieve. To deal with dynamical problems for such cases, the Galerkin truncation method is the most common used tool. Zehsaz et al. [26] evaluated a Timoshenko beam on the linear Pasternak viscoelastic bed subjected to moving load via the 10-term Galerkin method. Ding et al. [15] studied the convergence of the Galerkin method for the dynamic response of an Euler–Bernoulli beam resting on a nonlinear Winkler foundation subjected to a moving concentrated load. They found that the investigation on dynamic responses of a vehicle-pavement-foundation system needs large truncation terms. However, there is no literature on the application of Galerkin method to the Timoshenko beam with nonlinear foundations. In the present paper, the shear modulus, the viscoelasticity, and the nonlinearity of foundations are considered, as well as the effects of the shear deformable beams. For the first time, the dynamic response of a finite Timoshenko beam on a nonlinear cubic Pasternak foundation with viscous damping are numerically determined via the Galerkin method. The parametric sensitivity analysis is carried out. Furthermore, the convergence of the Galerkin truncation is investigated for a Timoshenko beam resting on foundations. 2 Equation of motion The system under investigation is a finite elastic Timoshenko beam resting on nonlinear viscoelastic foundation subjected to a moving load, as shown in Fig. 1. F0 and V represent respectively the magnitude of the load and the load speed. Moreover, V is assumed to be constant. X and U are the spatial coordinate along the axis of the beam and the vertical displacement function, respectively. Consider a homogeneous beam with a constant cross-section A, a moment of inertial I, a length L, a density ρ, a modulus of elasticity E, a shear modulus G and a effective shear area k A. The foundation is taken as a nonlinear Pasternak foundation with linear-pluscubic stiffness and viscous damping as follows P(x, t) = k1 U(X, T ) + k3 U 3 (X, T )
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+μ
∂U(X, T ) ∂2 U(X, T ) , − Gp ∂T ∂X 2
(1)
where P represents the force induced by the foundation per unit length of the beam, k1 and k3 are the linear and nonlinear foundation parameters, respectively. Furthermore, Gp and μ are the shear deformation coefficient and the damping coefficient of the foundation, respectively, and T is the time.
∂ψ ∂2 u ∂u ∂2 u ∂2 u 3 + k + α u + k u + μ − − G 1 3 p ∂x ∂x2 ∂t ∂t2 ∂x2 = F0 δ(x − vt), ∂ψ ∂2 ψ ∂2 ψ ∂u = 0. − + β ψ − + kf ψ + cf 2 2 ∂x ∂t ∂t ∂x 3 Normal modes The harmonic solution can be assumed in the form ∞ ∞ ϕk (x)eiωk t , ψ(x, t) = υk (x)eiωk t , u(x, t)= k=1
Fig. 1 The model of a finite Timoshenko beam on a nonlinear viscoelastic Pasternak foundation
Using the Hamilton principle and considering the Timoshenko beam theory, one can develop the governing differential equations of motion for the beam as [27] ∂ψ ∂2 U ∂2 U − ρA 2 + k AG + k1 U + k3 U 3 ∂X ∂X 2 ∂T ∂U ∂2 U − Gp 2 = F0 δ(X − VT ), +μ ∂T ∂X 2 2 ∂ψ ∂ ψ ∂ ψ ∂U = 0, ρI 2 − EI 2 + k AG ψ − + kf ψ + cf ∂X ∂T ∂T ∂X
(2)
kf ↔ kf
L2 , EI
1 , EA L2 cf cf ↔ , I ρE
Gp ↔ Gp
k3 L4 k3 ↔ , EA F0 ↔
where ωk are the natural frequencies, and ϕk (x) and υk (x) are the corresponding mode functions of the beam resting on linear Pasternak foundation, which can be derived from Eq. (4) as follows d2 υk c dυk dϕk d2 ϕk = 0, = 0, (6) + aϕ − + bυk + β k β dx dx dx2 dx2 where a = (ω2k − k1 )/(α + Gp ), b = ω2k − β − kf and c = αβ/(α + Gp ). By eliminating function υk from Eq. (6), the Timoshenko beam equation of free vibration is obtained in the form d2 ϕk d4 ϕk + p + qϕk = 0, p = a + b + c, q = ab. dx4 dx2 Thus the characteristic equation has the form of r4 + pr2 + q = 0.
(7)
(8)
Replacing z = r , Eq. (8) is rewritten in the form z2 + pz + q = 0.
Introducing the dimensionless variables and parameters as follows U T E X u= , t= , x= , L L ρ L k G ρ ψ = ψ, v=V , α= , E E k1 L2 k1 ↔ , EA
(5)
k=1
2
where kf and cf are foundation rocking stiffness and damping coefficients, ψ(X, T ) is the slope function due to bending of the beam, δ(X − VT ) is the Dirac delta function used to deal with the moving concentrated load.
L2 β = k AG , EI μ L2 μ↔ , A ρE
(4)
(3)
F0 , EA
where x is the dimensionless spatial coordinate and tis the dimensionless time. Equation (2) can be transformed into the following dimensionless equation
Its roots are √ 1 z1 = − p+ Δ , 2
(9)
z2 =
√ 1 − p− Δ . 2
(10)
where Δ = (−c + a − b)2 + 4ac. Now one should discuss the sign of the roots z1 and z2 : ∀ωk ⇔ z2 < 0; z1 > 0 for q < 0; z1 < 0 for q > 0. Two possible solutions to Eq. (7) can be obtained: (1) For q < 0, ϕ(x) = C1 cosh λ1 x + C2 sinh λ1 x +C3 cos λ2 x + C4 sin λ2 x, υ(x) = C1 sinh λ1 x + C2 cosh λ1 x
(11)
+C3 sin λ2 x + C4 cos λ2 x. (2) For q > 0, ϕ(x) = H1 cos λ1 x + H2 sin λ1 x +H3 cos λ2 x + H4 sin λ2 x, υ(x) = H1 sin λ1 x + H2 cos λ1 x
(12)
+H3 sin λ2 x + H4 cos λ2 x, where the integration constants Ci , Ci , Hi , and Hi depend on the boundary conditions.
Dynamic response to a moving load of a Timoshenko beam resting on a nonlinear viscoelastic foundation
The simply supported boundary condition analyzed in this paper is the most frequently encountered one for the present problem. The boundary conditions for a simply supported beam are ϕ(0) = ϕ(1) = 0,
C1 + C3 = 0, λ21C1 − λ22C3 = 0.
(14)
The system of equation is satisfied when C1 = C3 = 0, which corresponds to the solution for an Euler–Bernoulli beam. The boundary conditions at x = 1 are expressed by the matrix equation of ⎤⎡ ⎤ ⎡ ⎤ ⎡ sin λ2 ⎥⎥⎥ ⎢⎢⎢ C2 ⎥⎥⎥ ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢ sinh λ1 ⎥⎥⎦ ⎢⎢⎣ ⎥⎥⎦ = ⎢⎢⎣ ⎥⎥⎦ . ⎢⎢⎣ (15) λ21 sinh λ1 −λ22 sin λ2 C4 0 The nontrivial solution to Eq. (15) is obtained from the condition that the main matrix determinant is equal to zero. Thus one can obtain the frequency equation sin λ2 = 0.
Consequently, the corresponding i-th normal modes with simply supported boundary condition can be obtained as υ(x) = C cos λ2 x.
(17)
(2) For q > 0, the solution to Eq. (7) takes form Eq. (12). From the boundary conditions at x = 0, the following equations are obtained λ21 H1 + λ22 H3 = 0.
(18)
The system of equation is satisfied when H1 = H3 = 0. The boundary conditions at x = 1 are expressed by the matrix equation of ⎤⎡ ⎤ ⎡ ⎤ ⎡ sin λ2 ⎥⎥⎥ ⎢⎢⎢ H2 ⎥⎥⎥ ⎢⎢⎢ 0 ⎥⎥⎥ ⎢⎢⎢ sin λ1 ⎥⎥⎦ ⎢⎢⎣ ⎥⎥⎦ = ⎢⎢⎣ ⎥⎥⎦ . ⎢⎢⎣ (19) −λ21 sin λ1 −λ22 sin λ2 H4 0 The nontrivial solution to Eq. (19) is obtained from the condition that the main matrix determinant is equal to zero. Thus one can obtain the frequency equation sin λ2 = 0.
(20)
Consequently, the corresponding i-th normal modes with simply supported boundary condition can be obtained as ϕ(x) = H sin λ2 x,
υ(x) = H cos λ2 x.
(21)
4 Galerkin discretization The Galerkin truncation method is used to discretize the system and the series expansion forms for u(x, t) and ψ(x, t) with simply supported condition are assumed as
ϕk (x) = sin(kπx),
qk (t)ϕk (x),
k=1
ψ(x, t) =
∞
(22) ζk (t)υk (x),
υk (x) = cos(kπx),
k=1
where ϕk (x) and υk (x) are the trial functions, qk (t) and ζk (t) are sets of generalized displacements. The first n terms of Eq. (22) is considered in this paper. Substituting Eq. (22) into Eq. (4) leads to n k=1
q¨ k (t) + μq˙ k (t)
+[k1 + Gp (kπ)2 + α(kπ)2 ]qk (t) ϕk (x) n n 3 −α (kπ)ζk (t)ϕk (x) + k3 qk (t)ϕk (x) k=1
k=1
(23)
= F0 δ(x − vt), n
ζ¨k (t) + cf ζ˙k (t) + [kf + (kπ)2 + β]ζk (t) υk (x)
k=1 n −β (kπ)qk (t)υk (x) = 0.
(16)
ϕ(x) = C sin λ2 x,
∞
ϕ (0) + aϕ(0) = ϕ (1) + aϕ(1) = 0. (13)
(1) For q < 0, the solution to Eq. (7) takes form Eq. (11). From the boundary conditions at x = 0, the following equations are obtained
H1 + H3 = 0,
u(x, t)=
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k=1
It should be remarked that weight function is taken as trial function itself for the Galerkin method. Multiplying Eq. (23) by the weight functions wi (x) and integrating it over the interval of 0 and 1, the Galerkin procedure leads to the following set of 2n second-order ordinary differential equations n q¨ k (t) + μq˙ k (t) + [k1 + Gp (kπ)2 + α(kπ)2 ]qk (t) k=1
×
1
ϕk (x)wi (x)dx
0
n − α(kπ)ζk (t)
0
k=1
+k3
1
1
n
0
ϕk (x)wi (x)dx
3 qk (t)ϕk (x) wi (x)dx = F0 wi (vt),
k=1
n
ζ¨k (t) + cf ζ˙k (t) + [kf + (kπ)2 + β]ζk (t)
k=1
×
1
0
υk (x)wi (x)dx
n − β(kπ)qk (t) k=1
(24)
0
1
υk (x)wi (x)dx = 0,
i = 1, 2, · · · n. With the normal ortho-normal condition, the simplysupported boundary conditions lead to
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ϕk (x)wi (x)dx = υk (x)wi (x)dx =
0, 1/2,
k i, k = i,
0, 1/2,
k i, k = i.
The above mentioned ordinary differential equations (ODES) can be solved via the fourth-order Runge–Kutta method [28]. In the numerical computation here, the initial conditions are ˙ = 0. ˙ = ζ(t) = ζ(t) (27) q(t) = q(t)
(25)
Substitution of Eq. (25) into Eq. (24) yields 2
t=0
q¨ i (t) + [k1 + Gp (iπ) + α(iπ) ]qi (t) 1 n 3 qk (t)ϕk (x) wi (x)dx +2k3 0
t=0
t=0
t=0
2
5 Numerical results In this part, numerical examples are given for investigating model truncation convergence and parametric dependence. The physical and geometrical properties of the Timoshenko beam, foundation and the moving load are listed in Table 1 [15, 27].
k=1
+μq˙ i (t) − α(iπ)ζi (t) = 2F0 wi (vt), ¨ζi (t) + cf ζ˙i (t) + [kf + (iπ)2 + β]ζi (t) − β(iπ)qi (t) = 0,
(26)
i = 1, 2, · · · n.
Table 1 Properties of the beam, foundation and load
Beam
Item
Value
Dimensionless value
Young’s modulus E/GPa
6.998
—
Shear modulus G/GPa
77
Mass density ρ/(kg·m−3 )
2 373
—
Height of pavement h/m
0.3
—
Width of pavement b/m
1.0
—
Length L/m
160
—
0.4
—
α
—
4.401
β
—
1.502 × 107
8
97.552
Nonlinear stiffness k3 /(MN·m )
8
2.497 × 106
Viscous damping μ/(MN·s·m−2 )
0.3
Shear coefficients k
Linear stiffness k1 /MPa −4
Foundation
Moving load
Shear deformation coefficient Gp /N
39.263 7
6.669 × 10 8
Rocking stiffness kf /N
10
Rocking damping coefficient cf /(N·s)
1.5 × 106
0.031 8 1.626 × 105 2.618 × 104
Load F0 /N
2.126 × 10
1.01 × 10−4
Speed V/(m·s−1 )
20
0.011 65
The vertical deflection of a beam when the load moves to the mid-point of the beam is shown in Fig. 2a, while the vertical deflection of beam center with time is illustrated in Fig. 2b. The Galerkin truncation term is set to four cases, namely, 50-term Galerkin truncation, 75-term Galerkin truncation, 150-term Galerkin truncation and 200-term Galerkin truncation. As shown in the two plots, the transverse deflection increases for x < 80 or t < 4, and the biggest deflection appears at x = 80 (with ±0.2% variations) or t = 4 (with ±0.15% variations). After reaching the peak values, the transverse deflection decreases and tends to zero. And the growth speed of the transverse deflection is almost the same as the reduced speed. The numerical results also demon-
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strated that there are large differences between the 50-term Galerkin truncation results with the 150-term ones and the 200-term ones. There is no doubt that 50-term Galerkin method is not accurate enough for analyzing the dynamical response of Timoshenko beams on nonlinear viscoelastic foundations subjected to a moving concentrated load, and there are discernible differences between the results of 75term and 150-term Galerkin truncation. Moreover, the results of 150-term and 200-term Galerkin truncation are almost the same. Therefore, the 150-term Galerkin truncation yields rather accurate results. The maximum relative differences between the maximum mid-point vertical deflections is 8.49 × 10−2 in 50-term Galerkin truncation and 200-term
Dynamic response to a moving load of a Timoshenko beam resting on a nonlinear viscoelastic foundation
Galerkin truncation, 4.06 × 10−2 in 75-term Galerkin truncation and 200-term Galerkin truncation, and 0.74 × 10−2 in 150-term Galerkin truncation and 200-term Galerkin truncation. The results are in good agreement with that of Ding et al. [15].
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Figure 3 shows the dependence of the convergence in terms of the vertical deflections of the beam on system parameters. The abscissa represents truncation terms n while the the ordinate represents the vertical displacements of the beam’s midpoint for X = L/2 and T = L/(2V). As seen in
Fig. 2 Effects of the Galerkin truncation terms. a The effects on the vertical deflection of the beam; b The effects on the vertical deflection of the beam’s midpoint
Fig. 3 Effects of parameters on the vertical displacements of the beam’s midpoint versus truncation terms. a Effects of the span length of the beam; b Effects of the flexural stiffness of the beam; c Effects of the shear modulus of the beam; d Effects of the shear deformation coefficient of the foundation; e Effects of the height of the beam; f Effects of the width of the beam
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these figures, the vertical deflections of the beam increase initially with the truncation terms, but gradually approach to a stable value. The numerical results illustrate that the convergence rate of the Galerkin truncation increases with the increase of the modulus of elasticity of the beam, the shear modulus of the beam, the shear coefficient of the foundation, the height and the width of the beam, but decreases with the growing length of the beam. That is to say, the longer the length of the beam, the more the truncation terms are required to achieve convergence. From the obtained results, it is concluded that the vertical displacements of the beam decrease with the growth of these parameters except the length of the beam. For various values of L, there is little difference among the vertical deflections of the beam when Un (L/2, L/(2V)) has been a stable value, as shown in Fig. 3a. Moreover, the numerical results also depict that there is little difference among the vertical deflections of the beam obtained from not rather large truncation terms for different values of the modulus of elasticity, the height and width of the beam. It should be noted that the effects of the above mentioned parameters have been investigated except the shear modulus of the beam and the shear coefficient of the foundation in Ref. [15]. In this paper, similar conclusions are drawn from Figs. 3b, 3e and 3f. However, the influence of the length of the beam on the vertical deflection of a Timoshenko beam on the Pasternak foundation can be neglected, which is different from Ref. [15], arising probably from model difference. The above discussions demonstrate that the above mentioned parameters have appreciable influences on the conver-
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gence of the Galerkin truncation. However, some parameters do not, including the linear foundation parameters, the rocking stiffness of the foundation, the damping coefficient of the foundation, and the nonlinear foundation parameters. On the other hand, these parameters have greater effects on the vertical displacements than the convergence of the Galerkin truncation. Consequently, it is difficult to demonstrate the dependence of the convergence on these parameters. In order to further study the effects of these parameters on the convergence of the Galerkin truncation, δn is introduced and is described by δn =
Un (L/2, L/2V) − Un−1 (L/2, L/2V) × 100%. Un−1 (L/2, L/2V)
(28)
The effects of other system parameters on the convergence of the Galerkin truncation are investigated in Fig. 4. By comparison with Fig. 3, on the contrary, the numerical results demonstrate that δn decrease with the truncation terms until they are zero. From Figs. 4a and 4b, it can be seen that the convergence of the Galerkin truncation decreases with greater linear foundation parameters, but increases with greater rocking stiffness of the foundation. Furthermore, 100-term Galerkin truncation does not have a convergent numerical solution. This conclusion coincides with the result of Ref. [15]. Figures 4c and 4d shows a more complicated phenomenon that the convergence of the Galerkin truncation concerns with not only the system parameters but also the number of truncation terms. As seen in Fig. 4c, when the truncation term is less than 37, the bigger the damping coe-
Fig. 4 Effects of parameters on δn versus truncation terms. a Effects of the linear foundation parameter; b Effects of the rocking stiffness of the foundation; c Effects of the damping coefficient of the foundation; d Effects of the nonlinear foundation parameter
Dynamic response to a moving load of a Timoshenko beam resting on a nonlinear viscoelastic foundation
fficient of the foundation, the faster the convergence. Otherwise, the smallest damping coefficient of the foundation leads to the fastest convergence. A similar conclusion can be drawn from Fig. 4d, however, the nonlinear foundation parameter has less influence on the convergence of the Galerkin truncation than the damping coefficient of the foundation. The time history diagrams of the vertical dynamic deflections of the mid-span of the beam are shown in Fig. 5 for different values of system parameters. As seen in these figures, the maximum value of the dynamic deflection occurs almost at the mid-span of the beam (with ±0.15% variations), as illustrated in Fig. 2b. However, it is interesting to note that the peak value occurs at a farther vicinity of the mid-span (with ±0.86% variations) when the damping coefficient of the foundation is a greater one, as seen from Fig. 5a. Thus the growth speed of the transverse deflection is far greater than the reduced speed. That is to say, the damping coefficient of the foundation is a reason of time-delay. Moreover, as the damping coefficient of the foundation increases, the deflection of the Timoshenko beams decreases accordingly.
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Similar results can be found in Ref. [27]. Above all, the numerical results show that the damping coefficient of the foundation has significant influence on the dynamic response of Timoshenko beams. In other words, the damping coefficient of the foundation can not be neglected when the dynamic response of finite Timoshenko beams resting on nonlinear viscoelastic Pasternak foundations is studied. The effects of the shear modulus of a beam and the shear deformation coefficient of its viscoelastic nonlinear foundation on the deflections of the beam are illustrated in Figs. 5b and 5c, respectively. From the simulation results, one can see that the biggest deflections decrease with the increase of shear modulus of beams and the increase of shear deformation coefficient of foundations. It should be noted that a Pasternak foundation turns into a Winkler foundation when Gp = 0. That is to say, the maximum deflection of a Timoshenko beam on a Pasternak foundation is much smaller than that of a beam on a Winkler foundation. It is noted that this conclusion corresponds with the result of Ref. [7].
Fig. 5 Effects of parameters on the deflection of the beam. a Effects of the damping coefficient of the foundation; b Effects of the shear modulus of the beam; c Effects of the shear deformation coefficient of the foundation; d Effects of the rocking stiffness of the foundation; e Effects of the linear foundation parameter; f Effects of the nonlinear foundation parameter
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From Fig. 5d, one can observe that the maximum deflection decreases with the increase of rocking stiffness of the foundation. In Ref. [27], Ding et al. has got a similar conclusion. Figures 5e and 5f display the effects of the linear elasticity parameter and nonlinear elasticity parameter of the foundation on the deflection of Timoshenko beams supported by viscoelastic nonlinear foundations. The numerical results indicate that the biggest deflections decrease with increasing linear and nonlinear elasticity parameter of the foundation. And the whole form of the deflection has little change for different linear elasticity parameters and nonlinear elasticity parameters of the foundation. Figure 6 depicts the influence of the nonlinear elasticity parameter on the vertical displacements of the beam at the mid-span when the load moves to the mid-point of the beam versus the magnitude of the moving load. The figure predicts that the difference between the results of k3 = 8.0 ×106 N/m4 and k3 = 8.0 × 1011 N/m4 is very small. But comparing the results of k3 = 8.0 × 106 N/m4 with k3 = 8.0 × 1014 N/m4 , there are big differences and the deflection decreases with increasing nonlinear elasticity parameter. Furthermore, the deflection of Timoshenko beams increases with increasing magnitude of the moving load.
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fficient can be neglected when the dynamic response of finite Timoshenko beams supported by nonlinear viscoelastic Pasternak foundations is studied. 6 Conclusions This paper is devoted to the Galerkin method and its’ convergence for the dynamic response of a finite Timoshenko beam resting on a nonlinear Pasternak foundation subjected to a moving concentrated load. The dynamic response of the system is obtained using the high-order Galerkin truncation method in conjunction with the fourth-order Runge–Kutta method. The convergence of the Galerkin truncation is studied for the case of a Timoshenko beam vibrating on a foundation. The numerical investigation shows that the dynamical response of Timoshenko beams supported by nonlinear foundations needs super high-order modes. Furthermore, the dependences of the convergence of Galerkin method on the system parameters are numerically studied. The present paper proves that some parameters have appreciable effects on the convergence of the Galerkin truncation. However, some parameters do not. On the other hand, these parameters have greater effects on the vertical displacements of the beams than on the convergence of the Galerkin truncation. It should be noted that the convergence of the Galerkin truncation increases with the increase of the shear modulus of the beam, the shear deformation coefficient of the foundation and the rocking stiffness of the foundation. On the contrary, the vertical displacements of the beams decrease with the growth of of these parameters. References
Fig. 6 Effects of the nonlinear elasticity parameter on the vertical displacement versus the magnitude of the moving load
As far as the effects of the rocking damping coefficient on the dynamic response of finite beams are concerned, one can conclude that the rocking damping coefficient could neither affect greatly on the deflection of the beam nor make a contribution to the convergence of the Galerkin truncation, as seen from Fig. 7. That is to say, the rocking damping coe-
Fig. 7 Effects of the rocking damping coefficient on the vertical deflection versus truncation terms
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