Abstract
A unified solution for the inplane vibration analysis of multispan curved Timoshenko beams with general elastic boundary and coupling conditions by combining with the improved Fourier series method and RayleighRitz technique is presented in this paper. Under the current framework, regardless of boundary conditions, each of displacements and rotations of the curved Timoshenko beams is represented by the modified Fourier series consisting of a standard Fourier cosine series and several closedform auxiliary functions introduced to ensure and accelerate the convergence of the series representation. All the expansion coefficients are determined by the RayleighRitz technique as the generalized coordinates. The convergence and accuracy of the present method are tested and validated by a lot of numerical examples for multispan curved Timoshenko beams with various boundary restraints and general elastic coupling conditions. In contrast to most existing methods, the current method can be universally applicable to general boundary conditions and elastic coupling conditions without the need of making any change to the solution procedure.
1. Introduction
As one of the important structural components, curved Timoshenko beams has abundant engineering applications such as bridges, aircraft structures, space vehicles, turbomachines and other industrial applications owing to their excellent engineering characteristics. Notably, these beams frequently work in complex environments and may suffer to arbitrary boundary restraints. Therefore, a good understanding of the vibration behavior of curved Timoshenko beams subjected to dynamic loads and general boundary conditions is of particular importance for satisfying the design requirements of strength and stiffness in practical designs.
In the last few decades, a number of computational techniques have been proposed and developed, such as Differential Quadrature method, the Galerkin method, meshless method, the Ritz method, Finite element method, dynamic stiffness method and discrete singular convolution method. An interesting review of the subject can be found in the review articles [13]. Culver and Oestel [4] presented a new method for determining the natural frequencies of multispan horizontally curved girders. Lin and Lee [5] used closedform solutions to analyze dynamic response of extensional circular Timoshenko beams with general elastic boundary conditions. Kang et al. [6] presented a systematic approach for the free inplane vibration analysis of a planar circular curved beam system. Issa et al. [7] extended the dynamic stiffness matrix method to analyze the vibration of continuous circular curved beams with the clampedclamped boundary condition. Chen [8, 9] developed an analytical technique to study the inplane vibration of continuous curved beams. Wang [10] investigated the effects of rotary inertia and shear on natural frequencies of continuous circular curved beams undergoing inplane vibrations by using the dynamic stiffness matrix method. Kawakami et al. [11] performed the inplane and outofplane free vibration of horizontally curved beams with arbitrary shapes and variable crosssections by an approximate method. Riedel and Kang [12] employed wave propagation techniques to study the free vibration of elastically coupled dualspan curved beams subject to classical boundary conditions. Lee [13] applied the pseudospectral method to analyze the free vibration of circularly curved multispan Timoshenko beams with classical boundary and rigid coupling conditions. Huang et al. [14, 15] derived the inplane and the outofplane transient response of a hingedhinged and a clampedclamped noncircular Timoshenko curved beam by using the dynamic stiffness matrix method and the numerical Laplace transform. Leung and Zhu [16] used finite element method to analyze the inplane vibration of thin and thick curved beams with classical boundary conditions. Krishnan and Suresh [17] utilized a simple cubic linear beam element to study static and free vibration analysis of curved beams using finite element method. Chen [18] applied the differential transform method to investigate the inplane vibration of arbitrarily curved beam structures. Yang et al. [19] studied free inplane vibration of uniform and nonuniform curved beams with variable curvatures, including the effects of the axis extensibility, shear deformation and the rotary inertia by using the Galerkin finite element method. Ozturk [20] introduced the reversion method and finite element method to predict inplane free vibration of a large deflected prestressed cantilever curved beam. Eisenberger and Efraim [21] presented an exact dynamic stiffness matrix for a circular beam with pinnedpinned and clampedclamped boundary conditions.
In view of the aforementioned issues and concerns, it should be emphasized that most of the existing contributions were restricted to a single or twospan curved beam subjected to a limited set of classical supports. Little research has been devoted to the inpane vibration problem of multispan curved Timoshenko beams with general elastic boundary and coupling conditions. However, in practical engineering applications, the boundary and coupling conditions of multispan curved Timoshenko beams may not always be classical boundary and rigid coupling conditions in nature, and there will always be some elastic boundary and coupling conditions. The inplane vibration behaviors of multispan curved Timoshenko beams with general boundary and coupling conditions have remained unsolved until now. Moreover, to the best of the authors’ knowledge, no unified, efficient and accurate solution is available in the literature for the inplane vibration analysis of multispan curved Timoshenko beams subjected to general elastic boundary and coupling conditions.
Recently, a modified Fourier series technique proposed by Li [22, 23] is widely used in the vibrations of plates and shells with general boundary constraints by Ritz method, e.g., [2432]. Therefore, the present work can be considered as an extension of the method and attempts to provide a unified solution method for the inplane vibration of multispan curved Timoshenko beams with general elastic boundary and coupling conditions. Under the current framework, the modified Fourier series method together with the RayleighRitz procedure and the artificial stiffnesslike spring technique is adopted to derive the theoretical formulation. The general elastic boundary and coupling constraints of the multispan curved Timoshenko beams are realized by applying the artificial stiffnesslike spring technique. Each of displacements and rotations of each curved Timoshenko beam is represented by the modified Fourier series consisting of a standard Fourier cosine series and several closedform auxiliary functions introduced to ensure and accelerate the convergence of the series representation. Thereby, all the Fourier expanded coefficients are treated equally and independently as the generalized coordinates and are solved directly by using the RayleighRitz procedure. The convergence and accuracy of the present formulation are checked by a considerable number of convergence tests and comparisons. A variety of numerical examples are presented for the inplane vibration of multispan curved Timoshenko beams with general elastic boundary and coupling conditions, which may serve as benchmark solutions for validating new computational techniques in future.
2. Theoretical formulations
2.1. Geometrical configuration
Fig. 1 shows a multispan curved Timoshenko beam system, which consists of multiple curved beams coupled together via a set of joints, which are modeled by two groups of linear springs and one group of rotational springs. The use of the coupling springs between two adjacent curved beams allows accounting for the effects of some nonrigid or resilient connectors. The conventional rigid connectors can be considered as a special case when the stiffnesses of these springs become substantially large with reference to the bending rigidities of the involved curved beams. Each curved beam may also be independently supported on a set of elastic restraints at both ends. All the traditional intermediate supports and classical boundary conditions (i.e., the combinations of the simply supported (S), free (F), and clamped end conditions (C)) can be readily obtained from these general boundary conditions by accordingly setting the stiffness constants of the restraining springs to be equal to zero or infinity.
Fig. 1A multispan curved Timoshenko beam subjected to general elastic boundary and coupling conditions
2.2. General boundary and coupling conditions
The Timoshenko model for a curved beam consists of three partial differential equations for the curved beam radial displacement $w$, the tangential displacement $u$ and the rotation $\theta $ due to the bending of a cross section. Thus, The differential equations for the vibration of the $i$th curved Timoshenko beam can expressed as[16]:
where $\omega $ is the angular frequency of the curved beam, ${k}_{i}$, ${G}_{i}$, ${A}_{i}$, ${I}_{i}$, ${\rho}_{i}$ and ${R}_{i}$ are the shear correction factor, the shear modulus, the crosssectional area, the second moment of the area, the density of the beams and radius of curvature, respectively. The extensional strain ${\epsilon}_{i}$, flexural strain ${\kappa}_{i}$, and shear strain component ${\gamma}_{i}$ in the coordinate system are expressed in:
According to the linearly elastic theory, the normal force ${N}_{i}$ is linearly related to ${\epsilon}_{i}$, while the bending moment is proportional to the change in curvature as in the technical theory of beams. The shear forceshear strain relation is the familiar one from curved Timoshenko beam theory. Thus:
From the previous reviews, in this study, the artificial stiffnesslike spring technique is adopted to simulate the arbitrary boundary conditions and continuity conditions. With this method, the boundary and continuity conditions can be expressed as follows.
At ${\theta}_{i}=$0:
At ${\theta}_{i}={\varphi}_{i}$:
At the left end (of the first curved beam):
At the right end (of the $N$th curved beam):
where, referring to Fig. 1, ${k}_{i,j}^{cu}$ and ${k}_{i,j}^{cw}$ denote the stiffnesses of the linear coupling springs in the ${\theta}_{i}$direction and ${z}_{i}$direction, and ${K}_{i,j}^{c\theta}$ denote the stiffness of the rotational coupling spring at the junction of beams $i$ and $j$, respectively; ${k}_{\theta 0}^{ui}$, ${k}_{\theta 0}^{wi}$, ${k}_{\theta 1}^{ui}$, ${k}_{\theta 1}^{wi}$ are the stiffnesses of linear boundary springs, and ${K}_{\theta 0}^{\theta i}$, ${K}_{\theta 1}^{\theta i}$ are the stiffnesses of the rotational boundary springs at the left and right ends of the curved Timoshenko beam $i$, respectively.
All the conventional (homogeneous) curved beam boundary conditions can be considered as the special cases of Eqs. (10)(21). For example, the simply supported end condition is easily modeled by simply setting the stiffnesses of the linear springs and rotational springs to be infinity and zero, respectively.
2.3. Admissible displacement functions
The admissible function is the essence of the weak formulation such as the RayleighRitz method to achieve an accurate, convergent and unified solution. The traditional Fourier series, a wellknown form of admissible function for its excellent convergence, is limited to some very simple boundary conditions and would result in the discontinuities of the displacements and their derivatives as well. For the titled problem, the admissible functions are required not only to be regular enough to be differentiable, but also satisfy the geometry boundary conditions and continuity conditions at the junction. Recently, a modified Fourier series technique proposed by Li [22, 23] is widely used in the vibrations of plates and shells with different boundary conditions by RayleighRitz method, e.g., [2432]. In this technique, each displacement of the structure under consideration is expressed as a conventional cosine Fourier series with the addition of several supplementary terms. The purpose of introducing the supplementary terms, taking the linear vibration of a classical beam for example, is explained here. Though an exact solution generally exists in the form of sine Fourier series when the beam is with the simply supported ends, it cannot be widely applicable for other boundary conditions. This is because that the original displacements and their derivatives of the edges exist potential discontinuities, in other words, the expanded expressions can’t be differentiated through termbyterm, which will make the solution not converge or converge slowly. The detail illustration is given in Ref. [22]. More information about the improved Fourier series can be seen in Refs. [2332]. In this formulation, the modified Fourier series technique is adopted and extended to investigate the inplane vibration of multispan curved Timoshenko beams with general elastic boundary and coupling conditions.
Combining Eqs. (1)(3) and (10)(21), it is obvious that each displacement/rotation component of a multispan curved Timoshenko beam is required to have up to the second derivative. Therefore, regardless of boundary and coupling conditions, each displacement/rotation component of the curved Timoshenko beams is assumed to be a onedimensional modified Fourier series as:
where ${j}^{2}=$–1 and ${\lambda}_{m}=m\pi /{\varphi}_{i}$. ${\xi}_{l}\left(x\right)$ denote the supplementary terms introduced to remove all the discontinuities potentially associated with the firstorder derivatives at the boundaries and then ensure and accelerate the convergence of the series expansion of the curved beam displacement. ${A}_{m}^{i}$, ${B}_{m}^{i}$ and ${C}_{m}^{i}$ are the expansion coefficients of standard cosine Fourier series. ${a}_{l}^{i}$, ${b}_{l}^{i}$ and ${c}_{l}^{i}$ represent the corresponding expansion coefficients of the supplementary terms ${\xi}_{l}\left(x\right)$. These two supplementary terms are defined as:
It is easy to verify that:
2.4. Energy expressions
For the multispan curved Timoshenko beams, the total strain energy ($V$) and kinetic energy ($T$) can be expressed as:
where ${V}_{b,i}$ and ${T}_{b,i}$ represent the strain energy and kinetic energy of the $i$th curved Timoshenko beams, and ${V}_{i,i+1}^{s}$ is the potential energy expression in the connective springs related to $i$th and $i+$1th beams. The detailed expression of the ${V}_{b,i}$, ${V}_{i,i+1}^{s}$ and ${T}_{b,i}$ can be written as:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+\frac{1}{2}{\left\{{k}_{\theta 0}^{ui}{u}_{i}^{2}+{k}_{\theta 0}^{wi}{u}_{i}^{2}+{K}_{\theta 0}^{\theta i}{\psi}_{i}^{2}\right\}}_{\theta =0}+\frac{1}{2}{\left\{{k}_{\theta 1}^{ui}{u}_{i}^{2}+{k}_{\theta 1}^{wi}{u}_{i}^{2}+{K}_{\theta 1}^{\theta i}{\psi}_{i}^{2}\right\}}_{\theta =\varphi},$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left.+{K}_{i,i+1}^{c\theta}{\left({\left({\psi}_{i}\right)}_{\theta =0}\mp {\left({\psi}_{i+1}\right)}_{\theta =\varphi}\right)}^{2}\right),$
2.5. Solution procedure
Having established the admissible displacement functions and energy functions of the multispan curved Timoshenko beams, next, the corresponding coefficients of the admissible functions should be determined. In a weak formulation such as the RayleighRitz technique, however, all the expansion coefficients are considered as the generalized coordinates independent of each other. The strong and weak solutions are mathematically equivalent if they are constructed with the same degree of smoothness over the solution domain. The RayleighRitz technique will be adopted in this study since the solution can be obtained more easily. More importantly, such a solution process is better suitable for the future modeling of builtup structures.
The Lagrangian for the multispan curved Timoshenko beams can be generally expressed as:
Substituting Eqs. (29) and (30) into the Lagrangian function Eq. (34), taking its derivatives with respect to each of the undetermined coefficients and making them equal to zero:
a total of $3\left(M+3\right)N$ equations can be obtained and they can be summed up in a matrix form as:
where $\mathbf{K}$ and $\mathbf{M}$ represent the stiffness matrix and the mass matrix of the beam, respectively. They are defined as:
The detail expressions for the substiffness and submass matrices are not shown here since they are easy to gain. According to the above formula, the general vibration characteristics of the multispan curved Timoshenko beam will be obtained. Specifically, the frequencies (or eigenvalues) can be obtained directly by solving the Eq. (36), and the mode shapes will be acquired by substituting the corresponding eigenvectors into the series representations of displacement and rotation components. It should also be noted that the current method is particularly advantageous in obtaining other variables of interest such as power flows. Since the displacements are constructed sufficiently smoothly as required in a strong formulation, postprocessing the solution can be done easily through appropriate mathematical operations, including termbyterm differentiations.
3. Numerical results and discussion
In this section, a comprehensive investigation concerning the inplane free vibration of multispan curved Timoshenko beams with various boundary and coupling conditions is given to demonstrate the accuracy and reliability of the present method. Throughout these examples, unless otherwise stated, the nondimensional $\mathrm{\Omega}=\omega {L}_{1}^{2}/{\left(12{\rho}_{1}/{E}_{1}^{{h}_{1}^{2}}\right)}^{1/2}$ is used in the presentation, and the material and geometry properties of all the curved beams under consideration are: ${\rho}_{i}=$7800 kg/m^{3} ($i=$1, 2,…, $N$), ${\mu}_{i}=$0.3 ($i=$1, 2,…, $N$), ${E}_{i}=$ 2.1×10^{11} Pa ($i=$ 1, 2,…, $N$), ${\varphi}_{i}=$ 120° ($i=$1, 2,…, $N$), ${R}_{i}=$ 1 m ($i=$1, 2,…, $N$), and ${a}_{i}\times {b}_{i}=$ 0.005 m×0.005 m ($i=$1, 2,…, $N$).
3.1. Determination of the boundary and coupling spring stiffness
In the present work, the general boundary and coupling conditions are implemented by the artificial stiffnesslike spring technique introduced to simulate the boundary forces and displacements, with the help of which, the general boundary and coupling conditions of the multispan curved Timoshenko beams can be achieved by assigning the proper stiffness to the boundary and coupling springs. Taking a clamped end boundary (C) and rigid coupling (R) conditions for example, it can be realized by simply setting the stiffness of the entire springs to be “infinitely large” which is instead of a sufficiently large number in the actual calculation. So, it’s of great significance to investigate the effects of the spring stiffness of the boundary and coupling spring on the modal characteristics.
Effects of elastic boundary and coupling stiffness parameters on the frequency parameters $\mathrm{\Omega}$ of twospan curved Timoshenko beams are studied. In Fig. 2, variation of the lowest three frequency parameters $\mathrm{\Delta}\mathrm{\Omega}$ versus the elastic boundary and coupling restraint parameters ${\mathrm{\Gamma}}_{\lambda}$ for twospan curved Timoshenko beams is shown. The elastic boundary restraint parameter ${\mathrm{\Gamma}}_{\lambda}$ refers to the situation that the beam is completely free at the left end boundary, rigid coupling restraint at the junction and elastically supported at the right end boundary, which is realized by only one group of spring component with the stiffness varying from 10^{2} to 10^{14}. According to Fig. 2, we can see that the frequency parameters exist the large change as the stiffness parameters increase in the certain range. In conclusion, the “infinitely large” in the actual calculations can be equal to 1×10^{14}.
Fig. 2Variation of the frequency parameters ΔΩ versus the elastic restraint and coupling parameters for Timoshenko beams
Then the vibration analysis will be conducted and the frequencies and modal shapes of multispan curved Timoshenko beams with different boundary conditions including the arbitrary classical, general elastic, general elastic coupling and their combinations will be presented. Here the left and first joint of the beam is taken as the example, considering three types of classical boundary conditions, three types of elastic boundary conditions and four types of coupling boundary conditions which are commonly encountered in engineering practices, the corresponding spring stiffness parameters are given as follows respectively:
Boundary conditions:
Coupling conditions:
As previously, the classical boundary conditions are defined in terms of the boundary spring parameter, the appropriateness of which deserves great attention and will be discussed and proved in later subsections. Notably, in this paper the boundary conditions of the multispan curved Timoshenko beam are represented by several simple letter strings introduced to make the expression succinct (seen in the Fig. 3), i.e., $CR{C}^{1}S$ identifies the threespan curved Timoshenko beams with $C$, $S$ boundary conditions at the left and right ends boundary of beam, and $R$, ${C}^{1}$ coupling conditions at joint 1 and joint 2, respectively.
3.2. Convergence study
Theoretically, there are infinite terms in the modified Fourier series solution. However, the series is numerically truncated and only finite terms are counted in actual calculations. The excellent convergence of the proposed method will be proved firstly. Considering the single curved Timoshenko beam as an element of the multispan curved Timoshenko beams, thus the convergence can be studied by just checking the single curved Timoshenko beam’s. In the Table 1, the first six frequency parameters $\mathrm{\Omega}$ for $CC$ and $FF$ curved Timoshenko beams with eleven truncation schemes (i.e. $M=$ 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18) are presented. The frequency parameters of the beams are calculated by MATAB on a notebook. The configuration of the computer is: Inter Core(TM) i74970 CPU and 8 GB RAM. It is obvious that the present method has an excellent convergence, and is sufficiently accurate even when only a small number of terms are included in the series expressions. The maximum difference between the $M=$12 and $M=$18 is less than 0.051 % for the worst case. Besides, from the table, we can see that although the series are truncated as much as 50, the computing time is less 0.09 s. Unless otherwise stated, the truncated number of the displacement expressions will be uniformly selected as $M=$12 in the following discussions.
Then the accuracy and reliability of the current formulation will be validated further by some more numerical examples. In each case, the convergence study is performed and for brevity purpose, only the converged results are presented here.
Table 1Convergence of the first six frequency parameters Ω for a single curved Timoshenko beam with CC and FF boundary conditions
$M$  $CC$  $FF$  
1  2  3  4  5  6  1  2  3  4  5  6  
8  51.48  100.3  183.1  245.8  317.8  381.1  19.87  56.80  114.4  190.8  285.2  397.0 
9  51.48  100.2  183.1  245.6  317.5  381.1  19.78  56.80  114.4  190.7  284.9  396.4 
10  51.44  100.2  183.0  245.6  317.4  380.8  19.78  56.80  114.4  190.7  284.8  395.9 
11  51.44  100.1  183.0  245.6  317.3  380.8  19.76  56.80  114.4  190.7  284.7  395.7 
12  51.43  100.1  182.9  245.6  317.3  380.7  19.76  56.80  114.4  190.6  284.7  395.6 
13  51.43  100.1  182.9  245.5  317.3  380.7  19.75  56.80  114.4  190.6  284.7  395.5 
14  51.42  100.1  182.9  245.5  317.3  380.7  19.75  56.80  114.4  190.6  284.7  395.5 
15  51.42  100.1  182.9  245.5  317.2  380.7  19.74  56.80  114.4  190.6  284.6  395.4 
16  51.42  100.1  182.9  245.5  317.2  380.7  19.74  56.80  114.4  190.6  284.6  395.4 
17  51.42  100.1  182.9  245.5  317.2  380.7  19.74  56.80  114.4  190.6  284.6  395.4 
18  51.42  100.1  182.9  245.5  317.2  380.7  19.74  56.80  114.4  190.6  284.6  395.4 
3.3. Multispan curved Timoshenko beams with general boundary and coupling restraints
In this subsection, multispan curved Timoshenko beams with general boundary and coupling restraints are investigated. Firstly, the accuracy and reliability of the present method is validated by a verification study about the classical boundary conditions. In Tables 24, the first eight frequency parameters $\mathrm{\Omega}$ with classical boundary and rigid coupling conditions for single curved Timoshenko beam, twospan curved Timoshenko beam and threespan curved Timoshenko beam are presented, respectively. The results obtained from the FEM (ABAQUS) are also listed in the table as the reference, and the two results match very well. The differences between the two results are very small, and do not exceed 0.92 % for the worst case. Next, the inplane vibration of multispan curved Timoshenko beams with general elastic restraints will be studied.
Table 2Frequency parameters Ω for a single curved Timoshenko beam with different classical boundary conditions
Mode  $CC$  $SS$  $FF$  $CF$  $CS$  
Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  
1  50.175  50.366  30.385  30.384  19.779  19.776  3.844  3.843  40.407  40.396 
2  103.27  103.55  76.737  76.740  57.139  57.133  16.068  16.070  89.665  89.797 
3  184.71  185.32  148.12  148.13  115.69  115.68  53.24  53.24  167.38  167.48 
4  280.60  281.13  234.56  234.51  194.28  194.25  111.77  111.77  257.99  257.39 
5  394.14  394.77  345.31  345.36  292.72  292.66  190.23  190.25  374.08  374.34 
6  537.97  537.33  471.24  471.20  410.94  410.84  288.65  288.68  504.16  503.84 
7  678.58  678.24  621.35  621.45  548.88  548.73  406.84  406.83  659.35  660.13 
8  871.79  871.95  786.73  786.58  706.52  706.35  544.72  544.76  829.13  828.82 
*The FEM results are form ABAQUS software 
Table 3Frequency parameters Ω for a twospan curved Timoshenko beam with different classical boundary conditions
Mode  $CRC$  $SRS$  $FRF$  $CRF$  $CRS$  
Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  
1  6.0095  6.0268  2.6165  2.6790  5.9400  5.9393  0.9826  0.9917  4.8761  4.8775 
2  21.271  21.355  15.224  15.228  10.775  10.774  5.583  5.605  17.572  17.619 
3  39.781  40.096  30.361  30.384  26.717  26.728  7.471  7.414  35.513  35.452 
4  63.996  63.565  54.465  54.546  42.689  42.715  24.178  24.054  60.169  59.783 
5  89.462  89.797  76.729  76.740  69.369  69.550  41.879  41.761  83.537  83.629 
6  128.51  128.56  112.13  112.27  96.454  96.480  67.750  67.816  119.86  119.86 
7  164.83  164.48  148.16  148.13  133.07  133.16  93.253  93.330  157.84  157.63 
8  206.60  207.20  786.73  786.58  170.17  170.16  130.83  130.83  201.78  201.98 
*The FEM results are form ABAQUS software 
Table 4Frequency parameters Ω for a threespan curved Timoshenko beam with different classical boundary conditions
Mode  $CRRC$  $SRRS$  $FRRF$  $CRRF$  $CRRS$  
Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  
1  3.7242  3.7150  1.2770  1.2829  2.4610  2.4609  0.5704  0.5730  2.0351  2.0198 
2  6.6241  6.6359  3.7794  3.7831  7.1872  7.1861  2.0940  2.0782  5.7446  5.7586 
3  12.917  12.963  11.519  11.524  7.9489  7.9423  5.0785  5.0696  12.107  12.078 
4  25.576  25.561  20.204  20.192  17.296  17.304  7.2865  7.3351  22.485  22.495 
5  37.095  37.055  30.397  30.385  28.778  28.758  15.235  15.241  34.004  33.939 
6  53.729  53.758  46.921  46.902  39.009  39.023  27.269  27.243  50.196  50.189 
7  71.283  71.390  62.487  62.495  54.833  54.815  38.089  38.092  66.462  66.550 
8  85.488  85.646  76.729  76.751  73.932  73.921  54.281  54.296  81.550  81.701 
*The FEM results are form ABAQUS software 
Tables 57 show the first eight frequency parameters $\mathrm{\Omega}$ of single curved Timoshenko beam, twospan curved Timoshenko beam and threespan curved Timoshenko beam subjected to classicalelastic restraints and elastic boundaries, respectively. Besides, due to the lack of the open reported reference results and to be used as the comparison, the contrast results obtained using an FEM (ABAQUS) model are also given in Tables 57. An excellent agreement is achieved between the current and the FEM solutions. Finally, the inplane vibrations of multispan curved Timoshenko beams with general elastic boundary and coupling restraints are presented.
Table 5Frequency parameters Ω for a single curved Timoshenko beam with different elastic boundary conditions
Mode  $C{E}^{1}$  $C{E}^{2}$  ${E}^{1}{E}^{1}$  ${E}^{2}{E}^{2}$  ${E}^{3}{E}^{3}$  
Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  
1  40.422  40.392  6.0279  6.0268  30.379  30.379  10.411  10.412  51.944  51.944 
2  89.646  89.757  21.352  21.355  76.685  76.689  34.648  34.649  103.37  103.38 
3  167.37  167.43  66.572  66.565  148.01  148.03  83.464  83.464  188.04  188.05 
4  257.27  257.10  128.58  128.56  234.05  234.08  151.77  151.78  279.93  279.95 
5  373.97  374.04  213.19  213.20  344.79  344.84  240.53  240.56  403.43  403.48 
6  502.77  502.62  315.79  315.76  469.28  469.36  348.80  348.84  532.41  534.42 
7  658.81  659.25  439.47  439.62  619.61  619.72  477.05  477.08  695.95  696.19 
8  824.37  824.33  581.93  581.93  778.92  779.14  624.80  624.91  849.83  850.06 
*The FEM results are form ABAQUS software 
Table 6Frequency parameters Ω for a twospan curved Timoshenko beam with different elastic boundary conditions
Mode  $CR{E}^{1}$  $CR{E}^{2}$  ${E}^{1}R{E}^{1}$  ${E}^{2}R{E}^{2}$  ${E}^{3}R{E}^{3}$  
Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  
1  4.9212  4.9475  0.5005  0.8976  2.6791  2.6790  2.6791  2.6790  6.0279  6.0265 
2  17.533  17.618  6.039  6.089  15.228  15.227  10.411  10.412  21.352  21.352 
3  35.505  35.450  10.746  10.845  30.397  30.381  15.235  20.407  40.370  40.389 
4  60.124  60.132  27.821  27.599  54.539  54.539  34.666  34.649  66.536  66.547 
5  83.395  83.615  48.025  47.940  76.729  76.715  54.539  54.546  89.720  89.742 
6  119.87  119.84  73.712  73.815  112.21  112.23  83.464  83.464  128.47  128.47 
7  157.84  157.83  101.57  101.58  148.09  148.09  112.24  112.27  167.33  167.36 
8  201.71  202.02  139.77  139.85  191.66  191.66  151.77  151.78  212.97  213.00 
*The FEM results are form ABAQUS software 
Table 7Frequency parameters Ω for a threespan curved Timoshenko beam with different elastic boundary conditions
Mode  $CRR{E}^{1}$  $CRR{E}^{2}$  ${E}^{1}RR{E}^{1}$  ${E}^{2}RR{E}^{2}$  ${E}^{3}RR{E}^{3}$  
Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  
1  2.0251  2.0198  0.8059  0.8075  1.2843  1.2830  1.2843  1.2829  3.7169  3.7147 
2  5.7442  5.7586  2.8042  2.7927  3.7831  3.7831  3.7831  3.7831  6.6352  6.6359 
3  12.053  12.078  5.2625  5.2651  11.519  11.524  10.415  10.411  12.954  12.962 
4  22.500  22.495  10.304  10.306  20.204  20.191  11.519  11.525  25.540  25.558 
5  33.976  33.938  16.707  16.710  30.397  30.383  20.204  20.191  37.058  37.051 
6  50.200  50.189  29.551  29.539  46.884  46.899  34.666  34.649  53.729  53.751 
7  66.460  66.543  42.468  42.453  62.487  62.487  46.884  46.902  71.356  71.371 
8  81.543  81.694  58.182  58.208  76.7 29  76.733  62.487  62.491  85.598  85.609 
*The FEM results are form ABAQUS software 
In Tables 8 and 9, the detailed comparisons between results obtained by the present method and those provided by FEM solutions (ABAQUS) are presented, in which two types of multispan curved Timoshenko beams (twospan curved Timoshenko beam and threespan curved Timoshenko beam) are included. It’s very clear that the current results have a great agreement with the reference data. In order to improve our comprehension of the effects of elastic boundary and coupling restraints on vibration characteristic of multispan curved Timoshenko beams. The first six mode shapes of the single curved Timoshenko beam, twospan curved Timoshenko beam and threespan curved Timoshenko beam with different boundary and coupling restraints are given in Figs. 46, respectively. It can be seen that the elastic boundary and coupling restrain have a quite significant effect on the vibration characteristics of the beam structures.
Fig. 3A simple letter string of a multispan curved Timoshenko beam
Fig. 4The lowest six mode shapes for single curved Timoshenko beams with different boundary conditions
According to the above analysis, it can be seen that the current method is reliable to make correct predictions of the modal characteristics for the multispan curved Timoshenko beam with the elastic restraint boundary and coupling conditions as well as the classical boundary and rigid coupling conditions. It should be noted that for sake of simplifying the research, only the threespan curved Timoshenko beam is studied in this paper, but it doesn’t mean the current method is restricted to the threespan. Through the theoretical formulations, it can be seen that when the number of the curved beams is added, which merely increases the dimensional of the stiffness matrix and mass stiffness, the corresponding analysis can be easily obtained.
Table 8Frequency parameters Ω for a twospan curved Timoshenko beam with different elastic coupling conditions
Mode  $C{C}^{1}C$  $C{C}^{2}C$  ${E}^{1}{C}^{1}{E}^{1}$  ${E}^{2}{C}^{2}{E}^{2}$  ${E}^{3}{C}^{3}{E}^{3}$  
Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  
1  3.7459  3.8427  3.7635  3.8427  0.0468  0.0471  0.0147  0.0152  6.0279  6.0261 
2  16.066  16.068  6.0004  6.0261  10.775  10.773  0.0147  0.0160  21.352  21.351 
3  40.316  40.392  16.149  16.068  30.379  30.380  0.0736  0.0573  40.370  40.385 
4  53.236  53.221  21.392  21.355  42.709  42.700  0.0736  0.0581  66.536  66.547 
5  89.610  89.786  53.002  53.221  76.699  76.707  5.9249  5.9382  89.720  89.727 
6  111.74  111.75  66.666  66.565  96.46  96.43  10.415  10.411  128.47  128.48 
7  167.54  167.48  111.78  111.75  148.05  148.07  26.717  26.724  167.33  167.35 
8  190.40  190.20  128.02  128.56  170.09  170.04  34.666  34.648  212.97  213.02 
*The FEM results are form ABAQUS software 
Table 9Frequency parameters Ω for a threespan curved Timoshenko beam with different elastic coupling conditions
Mode  $C{C}^{1}{C}^{1}C$  $C{C}^{2}{C}^{2}C$  ${E}^{1}{C}^{1}{C}^{1}{E}^{1}$  ${E}^{2}{C}^{2}{C}^{2}{E}^{2}$  ${E}^{3}{C}^{3}{C}^{3}{E}^{3}$  
Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  Present  FEM*  
1  2.9826  2.9699  3.5457  3.5304  7.1096  7.1025  3.9340  3.9321  3.7152  3.7147 
2  3.0338  3.0255  4.8516  4.8533  14.4988  14.4950  7.9195  7.9195  6.6360  6.6359 
3  9.8554  9.9093  7.5732  7.5835  30.380  30.382  10.415  10.411  12.962  12.962 
4  17.469  17.459  16.892  16.881  37.192  37.191  23.442  23.421  25.557  25.558 
5  36.253  36.227  19.151  19.127  48.616  48.599  30.250  30.253  37.045  37.047 
6  46.308  46.244  28.191  28.247  76.703  76.715  34.666  34.649  53.745  53.747 
7  54.380  54.373  54.312  54.480  87.846  87.843  63.628  63.588  71.362  71.367 
8  84.173  84.292  60.592  60.555  105.051  105.007  75.809  75.795  85.574  85.591 
*The FEM results are form ABAQUS software 
Fig. 5The lowest six mode shapes for twospan curved Timoshenko beams with different boundary conditions
Fig. 6The lowest six mode shapes for threespan curved Timoshenko beams with different boundary conditions
4. Conclusions
A unified method is presented for inplane vibration analysis of multispan curved Timoshenko beams with general elastic boundary and coupling conditions. Each of the displacements and rotations of every curved Timoshenko beam, is expressed as a modified Fourier series, which is constructed as the linear superposition of a standard onedimensional Fourier cosine series supplemented with auxiliary polynomial functions introduced to eliminate all the relevant discontinuities with the displacement and its derivatives at the ends and accelerate the convergence of series representations. All the expansion coefficients are determined by the RayleighRitz technique as the generalized coordinates. The excellent accuracy and reliability of the current solutions are confirmed by comparing the present results with FEM solution, and numerous new results for multispan curved Timoshenko beams with various classical cases, classicalelastic restraints, and elastic boundary and coupling conditions, are presented, which can be served as the benchmark solutions for other computational techniques in the future research.
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The authors would like to thank the anonymous reviewers for their useful and constructive comments.