# Effects of Transverse Deformation on Free Vibration of 2D Curved Beams with General Restraints

Authors: Xueren Wang; Xuhong Miao; Di Jia; Shengyao Gao
Journal: Shock and Vibration
 (2017)

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## Abstract

An efficient modified Fourier series-based sampling surface approach is proposed for the analytical evaluation of the vibration characteristics of thick curved beams subjected to general restraints. The theoretical models of the beams are formulated by the theory of elasticity in two dimensions, which allows arbitrary thickness configurations to be tackled. As an innovation of this work, the approach is based upon the sampling surface method combined with the use of modified Fourier series approximation. In particular, the transverse beam domain is discretized by a set of sampling surfaces with unequal spaces, and the displacement components in beam domain coinciding with these surfaces are mathematically described as a set of modified Fourier series in which certain supplementary functions are included to remove all the relevant discontinuities with the displacements and their derivatives at the boundaries to form a mathematically complete set and guarantee the results convergent to the exact solutions. The final results are numerically solved using a modified variational principle by means of Lagrange multipliers and penalty method for the sake of arbitrary boundary conditions. The influences of transverse normal and shear deformation on the vibration characteristics with respect to the geometrical dimension and boundary conditions are systematically evaluated.


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## Body

## 1. Introduction

Beams are one of the most extensively used structural components in a variety of branches of engineering applications, such as aircraft, civil construction, automobile, and naval vessel. The analytical evaluation of the vibration characteristics of beams has attracted much attention in the past decades because this information is very important for the low-vibration design and safety validation of engineering structures.Strictly speaking, beams are three-dimensional (3D) blocks in physical sense for which the axial length is relatively larger than the other two dimensions. The 3D linear theory of elasticity may be applied in the theoretical modeling. However, such studies require high computing performance and lager storage capacities [1]. As a consequence, the beam problems are always simplified to a variety of one-dimensional (1D) representations by introducing several hypotheses in the kinetic relations and constitutive equations since the axial dimensions are relatively larger than the others. A variety of simplified 1D theories have been proposed so far, which are commonly divided into two aspects as follows: the classical beam theory (CBT) and the shear deformation beam theories (SDBTs). These specialties make them very attractive in the mechanics analysis of beams [2–10]. However, it is needed to be pointed out that the CBT is incapable of considering transverse deformation effect. The error of the calculating result is always great when dealing with moderately thick beams [11], since the shear effects on the cross section are more pronounced in moderately thick to thick beams and they are disregarded in the CBT. The FSDT overcomes this drawback and offers a more accuracy modeling theory since transverse deformation is further taken into account, even though the solutions based on the FSDTs are still not accurate due to the fact that the transverse normal components are still neglected. In addition, shear correction factors have to be incorporated in the FSDTs to adjust the transverse shear stiffness due to the fact that the transverse shear strains in the FSDTs are assumed to be constant in the thickness direction. The shear correction factors are difficult to determine because they depend not only on the geometric parameters, but also on the loading and boundary conditions. In order to obtain accurate solutions for thick beams, higher-order variation of axial displacement has been introduced into a wide variety of HSDTs. These theories are more accurate than the CBT and FSDTs without shear correction factors. But, unfortunately, the transverse normal effects are ignored in the conventional HSDTs. Thus, in order to analyze thick beams accurately, more advanced theories considering the through-thickness shear deformations are essentially required. Recently, Carrera [12, 13] developed the so-called Carrera Unified Formulation (CUF). According to the CUF, the obtained theories can have an order of expansion depending on the thickness functions that are used, which allows one to take into account the effects of the transverse normal effects.The static and dynamic analysis of thick beam has been extensively investigated by many researchers. Chen et al. [14] proposed a mixed approach for the bending and free vibration of arbitrarily thick beams. In their method, the state space method and the differential quadrature method are combined to solve the problems. The method was further applied to the calculation of the elasticity solution of FGM beams by Ying et al. [15]. Hasheminejad and Rafsanjani [16] obtained semianalytical results for the transient dynamic response of thick simply supported beams through a powerful state space technique and the Laplace transformation. Thermoelastic behavior of arbitrarily simply supported beams subjected to thermomechanical loads is studied by Xu and Zhou [17, 18]. Zenkour et al. [19] studied the influence of transverse deformations on fiber reinforced viscoelastic beams. Malekzadeha and Karami [20] developed a mixed differential quadrature (DQ) and finite element (FE) approach for free vibration and buckling analysis of thick beams. This method applies a finite element discretization technique along axial direction while the thickness direction is discretized using DQM. The developments of studies of static and dynamic analysis of beams can be found in several monographs by Qatu [1], Rosen [21], Chidamparam and Leissa [22], Hodges [23], and Hajianmaleki and Qatu [24].From the review of the literature, it is clear that although a lot of attention has been focused on static and dynamic analysis of thick beams, the extensive volume of literature on this subject was mainly limited to uniform straight beams with classical boundary conditions since their governing equation is much easier to be derived and tackled. The equations for a curved beam are more complicate and sophisticated because of curvilinear geometry. Inevitably, this introduces inherent complexity in finding their solutions [25]. In addition, the previous reviews showed that most beams are analyzed based on Euler-Bernoulli beam theory, Timoshenko beam theory, or the higher-order one-dimensional theory models which neglect the transverse normal deformation effect (thickness stretching). This appears quite inappropriate since the effect of transverse normal deformation on the static dynamic characteristics of thick beams is significant, especially at higher vibration modes of curved beams. Carrera et al. [26] and Koiter [27] recommended that a refinement of one-dimensional simplification theories is meaningless, unless the effects of transverse shear and normal deformations are all taken into account. Thus, seldom works are available that investigate the influences of transverse shear and normal deformations on the vibration characteristics of evident thick curved beams. The present work attempts to fill this gap.In this paper, the modified Fourier series-based sampling surface method is further extended to the evaluation of elasticity solution of thick curved beams. The method was developed by Ye and Jin [28] based on a modified Fourier series technique proposed by Li [29] and SaS approach originally proposed by Kulikov et al. [30, 31]. The method combines the advantages of both approaches. A comprehensive numerical analysis and discussions are conducted to investigate the influence of transverse normal and shear deformations on the vibration characteristics of curved beams. The article is organized as follows: the theoretical formulation including model description, plan stress assumption, application of sampling surface method, and modified Fourier series approximation is presented in Section 2; convergence studies, results verification, and transverse deformation investigation are given in Section 3 and the concluding remarks are summarized in Section 4.


## 2. Theoretical Formulations

### 2.1. Model Description

A thick circular beam shown in Figure1 is considered, in which b, h, and R represent the width, thickness, and inner radius of the beam. The beam is bounded along its edges by the boundaries θ=θ0 and θ=θ1. The bottom surface of the beam is selected as the reference surface with the three orthogonal curvilinear coordinates θ, y, and z; see Figure 1. In this paper, the beams are assumed to be isotropic and homogeneous and to vibrate freely in the θ-z plane. u, v, and w denote the three displacement components in the axial, lateral, and normal directions, respectively.Figure 1
Geometry and reference system for a curved beam and the diagrammatic sketch of sampling surface distribution.
(a)
Coordinate system
(b)
Sampling surface distribution


### 2.2. Plane Stress Assumption

As mentioned previously, the beam under consideration vibrates freely in theθ-z plane. Therefore, the plane stress hypothesis is adopted in the theoretical formulation for the purpose of improving the computational efficiency and maintaining the modeling precision synchronously.For a curved beam, the 3D strain-displacement relations for any point in the domain of the beam can be found as [32](1)εθ=1Rz∂u∂θ+w,γθz=∂u∂z+1Rz∂w∂θ-u,εy=∂v∂y,γyz=∂v∂z+∂w∂y,εz=∂w∂z,γθy=∂u∂y+1Rz∂v∂θ,where Rz=R+z, εθ, εy, and εz stand for the normal strains, and γyz, γθz, and γθy are the shear components. In the case of homogeneous materials and linear small deformation assumptions, the 3D stresses can be derived according to Hooke’s law:(2)σθσyσzτyzτθzτθy=C11′C12′C13′000C12′C22′C23′000C13′C23′C33′000000C44′000000C55′000000C66′εθεyεzγyzγθzγθy=C′ε,where σθ, σy, and σz stand for the normal stresses; τyz, τθz, and τθy are the shear component. C′ is the material stiffness matrix.(3)C11′=C22′=C23′=E1-ν1+ν1-2ν,C12′=C13′=C23′=vC11′1-ν,C44′=C55′=C66′=E21+νin which E means Young’s module of the material andν represents Poisson’s ratio.Furthermore, (2) can be rearranged in the form of matrix as(4)σiσo=Cii′Cio′Coi′Coo′εiεo,σi=σθσzτθz,σo=σyτyzτθy,εi=εθεzγθz,εo=εyγyzγθy,Cii′=C11′C13′0C13′C33′000C55′,Cio′=C12′00C23′00000=Coi′T,Coo′=C22′000C44′000C66′.Therefore, the final stress-strain relations for the beam under plane stress hypothesis can be obtained as(5)σi=Cεi,where C=Cii′-Cio′Coo′-1Cio′.


### 2.3. Application of Sampling Surface Technique

The sampling surface technique was originally proposed by Kulikov et al. [33–35]. A brief resume and application of this technique are included in this section.As shown in Figure1, S1,…,Sj,…,SJ stand for the chosen sampling surfaces inside the transverse domain of the beam to introduce the displacement components of these surfaces as basic beam variables. J is the total number of the sampling surfaces. These surfaces are selected to be nonequally spaced and paralleled to the beam’s middle surface. For application of the sampling surface technique, it was found that the distribution of the sampling surface has a great effect on the convergence and accuracy of the solutions. For the sake of the convergence, transverse coordinates of these surfaces are chosen as the roots of Chebyshev polynomial:(6)z1=0,zJ=h,zj=z1+zJ2-h2cos⁡π2j-32J-4;2≤j≤J-1.Therefore, the basic variables in the axial and normal directions of an arbitrary sampling surface can be given by(7)uθ,zj,t=ujθeiωt,wθ,zj,t=wjθeiωt;1≤j≤J,where uj(θ) and wj(θ) stand for the axial and transverse displacement components, respectively. t stand for the time variable, ω is the circular frequency. As a consequence, displacement field of the beam under vibration can be calculated by(8)uθ,z,t,wθ,z,t=∑jLjzujθ,Ljzwjθeiωt;z1≤z≤zJand Lj(z) is Lagrange’s interpolation of degree J-1:(9)Ljz=∏r≠jz-zrzj-zr;z1≤z≤zJ.According to (1), strains on the jth sampling surface can be found as(10)εθθ,zj,t,εzθ,zj,t,γθzθ,zj,t=εθjθ,εzjθ,γθzjθeiωt;1≤j≤J,εθjθ=1R+zj∂ujθ∂θ+wjθ,εzjθ=∑rMjrwrθ,γθzjθ=∑rMjrurθ+1R+zj∂wjθ∂θ-ujθ,where Mjr are determined by(11)Mjr=1zr-zj∏s≠r,jzj-zszr-zsforr≠j,Mjj=-∑r≠jMjrforr=j.Similarly, strain distribution in the whole space should be represented as a linear combination of their corresponding strain components of the entire sampling surfaces as (8). (12)εθ,εz,γθz=∑jLjzεθjθ,Ljzεzjθ,Ljzγθzjθeiωt;z1≤z≤zJ.The energy functional for the curved beam under the circumstance of free vibration is(13)Πs=Ts-Us,where Us and Ts denote the strain and kinetic energy function defined as follows:(14)Us=12∬S∫z1zJσθεθ+σzεz+τθzγθzR+zdθdydz,Ts=12∬S∫z1zJρu,t2+w,t2R+zdθdydz,where ρ stands for the material density. Substituting (5) and (12) into (14), the two energy functions can be further written as(15)Us=12∬S∫ζ1ζJC11∑i∑jLiLjRiRj∂ui∂θ∂uj∂θ+2∑i∑jLiLjRiRj∂ui∂θwj+∑i∑jLiLjRiRjwiwj+2C12∑i,r∑jMriLrLjRjwi∂uj∂θ+2C12∑i,r∑jMriLrLjRjwiwj+C22∑i,s∑j,rMsiMrjLsLjwiwj+C33∑i,s∑j,rMsiMrjLsLjuiuj+C33∑i,s∑jMsiLsLjRjui∂wj∂θ-2C33∑i,s∑jMsiLsLjRjuiuj+C33∑i∑j,rMrjLiLrRi∂wi∂θuj+C33∑i∑jLiLjRiRj∂wi∂θ∂wj∂θ-2C33∑i∑jLiLjRiRj∂wi∂θuj+C33∑i∑jLiLjRiRjuiujRzdθdydz,Ts=12∬S∫ζ1ζJ-ρω2∑i∑jLiLjuiuj+∑i∑jLiLjwiwjRzdθdydz.


### 2.4. Modified Fourier Series Approximation

The modified Fourier series approximation is introduced to represent the possible deformations of the curved beams. Particularly, each of the basic beam variables is mathematically described as a set of modified Fourier series including a standard cosine Fourier series as well as certain auxiliary functions [36–41]. The auxiliary terms are introduced for the purpose of removing the entire possible discontinuities with the basic beam variables and their derivatives at the edges to form a mathematically complete set and then ensure the convergence and speed up the calculation [39, 42–45]. In addition, the governing equations of the beams are derived and numerically solved by a modified variational principle for the sake of making arbitrary boundary conditions applicable.As mentioned previously, the displacement variables at an arbitrary sampling surface in the modified form of Fourier series are(16)ujθ=∑n=0N-2ujncos⁡nπθΔθ+ujN-1θθΔθ-12+ujNθ2θ/Δθ-1Δθ,wjθ=∑n=0N-2wjncos⁡nπθΔθ+wjN-1θθΔθ-12+wjNθ2θ/Δθ-1Δθ,where ujn and wjn(n=0,1,…,N) are the expansion coefficients; Δθ=θ1-θ0.N represents the truncation number.The boundary conditions of the curved beams are supposed to be of essential type. The necessary boundary equations can be stated in functional form as follows by applying the penalty technique and Lagrange multipliers [6, 46–49]:(17)Πb=Πb1+Πb2;Πb1=∫∫z1zJ∑l=12-1lηulσθu-u-l+ηwlτθzw-w-lθ=θldydz,Πb2=12∫∫z1zJ∑l=12ηulkulu-u-l2+ηulkwlw-w-l2θ=θldydz,where u-l and w-l denote the boundary values. kul and kwl represent the penalty parameters. ηul and ηwl are the parameters which define different restraint conditions. The boundary potential Πb1 is introduced by means of Lagrange multiplier technique while the boundary potential Πb2 is introduced by the aid of the penalty technique to ensure a uniform formulation to tackle general boundaries [6] and to ensure a computational stability in computational process. Taking the end of θ=θ0, for example, the values of the penalty parameters and boundary coefficients for different classical restraint conditions are shown in Table 1. For elastic boundary conditions, the boundary potentials Πb1 in (17) should be neglected and the penalty parameters will be determined at proper values [49].Table 1
Values ofηul, ηwl, kul, and kwl for different classical boundary conditions.
Boundary conditions
Boundary coefficients
Penalty parameters
η
u
0
η
w
0
k
u
0
k
w
0
F (free):σθ=0, τθz=0
0
0
0
0
S1 (simply supported):σθ=0, w=0
0
1
0
103E
S2 (simply supported):u=0, τθz=0
1
0
103E
0
C (clamped):u=0, w=0
1
1
103E
103ETherefore, the final variational functional for the curved beam with general boundaries is defined as(18)Πtotalujn,wjn=Πs+Πb;0≤n≤N,1≤j≤J.Finally, let the variation of theΠtotal with respect to each coefficient (ujn and wjn) equal zero; the governing equations can be derived in a matrix form as (19)K-ω2MG=0,whereK andM stand for the final stiffness and mass matrices of order 2(N+1)∗J.G denotes the vector of the unknown generalized displacements. Thus, solutions can be obtained directly by the eigenvalue decomposition of (19) and the roots of the decomposition are the square of eigenfrequency ω. The mode shape of the curved beam corresponding to each eigenfrequency can be constructed by substituting the corresponding eigenvector back into the displacement variables given in (16) and then substituting it in the displacement distribution formula given in (8).


## 2.1. Model Description

A thick circular beam shown in Figure1 is considered, in which b, h, and R represent the width, thickness, and inner radius of the beam. The beam is bounded along its edges by the boundaries θ=θ0 and θ=θ1. The bottom surface of the beam is selected as the reference surface with the three orthogonal curvilinear coordinates θ, y, and z; see Figure 1. In this paper, the beams are assumed to be isotropic and homogeneous and to vibrate freely in the θ-z plane. u, v, and w denote the three displacement components in the axial, lateral, and normal directions, respectively.Figure 1
Geometry and reference system for a curved beam and the diagrammatic sketch of sampling surface distribution.
(a)
Coordinate system
(b)
Sampling surface distribution


## 2.2. Plane Stress Assumption

As mentioned previously, the beam under consideration vibrates freely in theθ-z plane. Therefore, the plane stress hypothesis is adopted in the theoretical formulation for the purpose of improving the computational efficiency and maintaining the modeling precision synchronously.For a curved beam, the 3D strain-displacement relations for any point in the domain of the beam can be found as [32](1)εθ=1Rz∂u∂θ+w,γθz=∂u∂z+1Rz∂w∂θ-u,εy=∂v∂y,γyz=∂v∂z+∂w∂y,εz=∂w∂z,γθy=∂u∂y+1Rz∂v∂θ,where Rz=R+z, εθ, εy, and εz stand for the normal strains, and γyz, γθz, and γθy are the shear components. In the case of homogeneous materials and linear small deformation assumptions, the 3D stresses can be derived according to Hooke’s law:(2)σθσyσzτyzτθzτθy=C11′C12′C13′000C12′C22′C23′000C13′C23′C33′000000C44′000000C55′000000C66′εθεyεzγyzγθzγθy=C′ε,where σθ, σy, and σz stand for the normal stresses; τyz, τθz, and τθy are the shear component. C′ is the material stiffness matrix.(3)C11′=C22′=C23′=E1-ν1+ν1-2ν,C12′=C13′=C23′=vC11′1-ν,C44′=C55′=C66′=E21+νin which E means Young’s module of the material andν represents Poisson’s ratio.Furthermore, (2) can be rearranged in the form of matrix as(4)σiσo=Cii′Cio′Coi′Coo′εiεo,σi=σθσzτθz,σo=σyτyzτθy,εi=εθεzγθz,εo=εyγyzγθy,Cii′=C11′C13′0C13′C33′000C55′,Cio′=C12′00C23′00000=Coi′T,Coo′=C22′000C44′000C66′.Therefore, the final stress-strain relations for the beam under plane stress hypothesis can be obtained as(5)σi=Cεi,where C=Cii′-Cio′Coo′-1Cio′.


## 2.3. Application of Sampling Surface Technique

The sampling surface technique was originally proposed by Kulikov et al. [33–35]. A brief resume and application of this technique are included in this section.As shown in Figure1, S1,…,Sj,…,SJ stand for the chosen sampling surfaces inside the transverse domain of the beam to introduce the displacement components of these surfaces as basic beam variables. J is the total number of the sampling surfaces. These surfaces are selected to be nonequally spaced and paralleled to the beam’s middle surface. For application of the sampling surface technique, it was found that the distribution of the sampling surface has a great effect on the convergence and accuracy of the solutions. For the sake of the convergence, transverse coordinates of these surfaces are chosen as the roots of Chebyshev polynomial:(6)z1=0,zJ=h,zj=z1+zJ2-h2cos⁡π2j-32J-4;2≤j≤J-1.Therefore, the basic variables in the axial and normal directions of an arbitrary sampling surface can be given by(7)uθ,zj,t=ujθeiωt,wθ,zj,t=wjθeiωt;1≤j≤J,where uj(θ) and wj(θ) stand for the axial and transverse displacement components, respectively. t stand for the time variable, ω is the circular frequency. As a consequence, displacement field of the beam under vibration can be calculated by(8)uθ,z,t,wθ,z,t=∑jLjzujθ,Ljzwjθeiωt;z1≤z≤zJand Lj(z) is Lagrange’s interpolation of degree J-1:(9)Ljz=∏r≠jz-zrzj-zr;z1≤z≤zJ.According to (1), strains on the jth sampling surface can be found as(10)εθθ,zj,t,εzθ,zj,t,γθzθ,zj,t=εθjθ,εzjθ,γθzjθeiωt;1≤j≤J,εθjθ=1R+zj∂ujθ∂θ+wjθ,εzjθ=∑rMjrwrθ,γθzjθ=∑rMjrurθ+1R+zj∂wjθ∂θ-ujθ,where Mjr are determined by(11)Mjr=1zr-zj∏s≠r,jzj-zszr-zsforr≠j,Mjj=-∑r≠jMjrforr=j.Similarly, strain distribution in the whole space should be represented as a linear combination of their corresponding strain components of the entire sampling surfaces as (8). (12)εθ,εz,γθz=∑jLjzεθjθ,Ljzεzjθ,Ljzγθzjθeiωt;z1≤z≤zJ.The energy functional for the curved beam under the circumstance of free vibration is(13)Πs=Ts-Us,where Us and Ts denote the strain and kinetic energy function defined as follows:(14)Us=12∬S∫z1zJσθεθ+σzεz+τθzγθzR+zdθdydz,Ts=12∬S∫z1zJρu,t2+w,t2R+zdθdydz,where ρ stands for the material density. Substituting (5) and (12) into (14), the two energy functions can be further written as(15)Us=12∬S∫ζ1ζJC11∑i∑jLiLjRiRj∂ui∂θ∂uj∂θ+2∑i∑jLiLjRiRj∂ui∂θwj+∑i∑jLiLjRiRjwiwj+2C12∑i,r∑jMriLrLjRjwi∂uj∂θ+2C12∑i,r∑jMriLrLjRjwiwj+C22∑i,s∑j,rMsiMrjLsLjwiwj+C33∑i,s∑j,rMsiMrjLsLjuiuj+C33∑i,s∑jMsiLsLjRjui∂wj∂θ-2C33∑i,s∑jMsiLsLjRjuiuj+C33∑i∑j,rMrjLiLrRi∂wi∂θuj+C33∑i∑jLiLjRiRj∂wi∂θ∂wj∂θ-2C33∑i∑jLiLjRiRj∂wi∂θuj+C33∑i∑jLiLjRiRjuiujRzdθdydz,Ts=12∬S∫ζ1ζJ-ρω2∑i∑jLiLjuiuj+∑i∑jLiLjwiwjRzdθdydz.


## 2.4. Modified Fourier Series Approximation

The modified Fourier series approximation is introduced to represent the possible deformations of the curved beams. Particularly, each of the basic beam variables is mathematically described as a set of modified Fourier series including a standard cosine Fourier series as well as certain auxiliary functions [36–41]. The auxiliary terms are introduced for the purpose of removing the entire possible discontinuities with the basic beam variables and their derivatives at the edges to form a mathematically complete set and then ensure the convergence and speed up the calculation [39, 42–45]. In addition, the governing equations of the beams are derived and numerically solved by a modified variational principle for the sake of making arbitrary boundary conditions applicable.As mentioned previously, the displacement variables at an arbitrary sampling surface in the modified form of Fourier series are(16)ujθ=∑n=0N-2ujncos⁡nπθΔθ+ujN-1θθΔθ-12+ujNθ2θ/Δθ-1Δθ,wjθ=∑n=0N-2wjncos⁡nπθΔθ+wjN-1θθΔθ-12+wjNθ2θ/Δθ-1Δθ,where ujn and wjn(n=0,1,…,N) are the expansion coefficients; Δθ=θ1-θ0.N represents the truncation number.The boundary conditions of the curved beams are supposed to be of essential type. The necessary boundary equations can be stated in functional form as follows by applying the penalty technique and Lagrange multipliers [6, 46–49]:(17)Πb=Πb1+Πb2;Πb1=∫∫z1zJ∑l=12-1lηulσθu-u-l+ηwlτθzw-w-lθ=θldydz,Πb2=12∫∫z1zJ∑l=12ηulkulu-u-l2+ηulkwlw-w-l2θ=θldydz,where u-l and w-l denote the boundary values. kul and kwl represent the penalty parameters. ηul and ηwl are the parameters which define different restraint conditions. The boundary potential Πb1 is introduced by means of Lagrange multiplier technique while the boundary potential Πb2 is introduced by the aid of the penalty technique to ensure a uniform formulation to tackle general boundaries [6] and to ensure a computational stability in computational process. Taking the end of θ=θ0, for example, the values of the penalty parameters and boundary coefficients for different classical restraint conditions are shown in Table 1. For elastic boundary conditions, the boundary potentials Πb1 in (17) should be neglected and the penalty parameters will be determined at proper values [49].Table 1
Values ofηul, ηwl, kul, and kwl for different classical boundary conditions.
Boundary conditions
Boundary coefficients
Penalty parameters
η
u
0
η
w
0
k
u
0
k
w
0
F (free):σθ=0, τθz=0
0
0
0
0
S1 (simply supported):σθ=0, w=0
0
1
0
103E
S2 (simply supported):u=0, τθz=0
1
0
103E
0
C (clamped):u=0, w=0
1
1
103E
103ETherefore, the final variational functional for the curved beam with general boundaries is defined as(18)Πtotalujn,wjn=Πs+Πb;0≤n≤N,1≤j≤J.Finally, let the variation of theΠtotal with respect to each coefficient (ujn and wjn) equal zero; the governing equations can be derived in a matrix form as (19)K-ω2MG=0,whereK andM stand for the final stiffness and mass matrices of order 2(N+1)∗J.G denotes the vector of the unknown generalized displacements. Thus, solutions can be obtained directly by the eigenvalue decomposition of (19) and the roots of the decomposition are the square of eigenfrequency ω. The mode shape of the curved beam corresponding to each eigenfrequency can be constructed by substituting the corresponding eigenvector back into the displacement variables given in (16) and then substituting it in the displacement distribution formula given in (8).


## 3. Numerical Results and Discussion

Several examples for thick curved beams with different geometrical dimensions and boundary restraints are presented to verify the flexibility of the method. The transverse deformation effects are systematically investigated as well. To unify the discussion, character string X-Y (X/Y = F, S, C) is used to represent the boundary conditions of the beams. For example,C-F represents a circular beam with clamped and free restraints at the ends θ=θ0 and θ=θ1, respectively. To unify the discussion, the dimensionless variable of frequency is introduced in the calculation Ω=ωRm212ρ/Eh2 (where Rm=R+h/2). The beams are supposed to be made of steel (E=210 GPa, ν=0.3, and ρ=7800 kg/m3).


### 3.1. Validation

Table2 gives the first five nondimensional frequency parameters Ω of C-C supported circular beams. The numbers of the sampling surfaces and serious truncation are increased from 11 to 17 and 3 to 9, respectively. The geometric parameters of the beam are Rm=1 m, Δθ=2π/3. For completeness, two thickness-to-radius ratios (i.e., h/Rm = 0.1 and 0.2) corresponding to the moderately thick and thick beam configurations are considered in the study. As observed from Table 2, with the increase of truncated number, the natural frequencies tend to be constant values quickly. The maximum differences between the results based on the “11 × 3” and “17 × 9” computational schemes are less than 0.2%, which confirms the high convergence of the present method. The DQM results based on the FSDT [3] and the Ritz solutions with 2D elasticity theory [32] are also listed in the table. It is observed that the present solutions match well with those predicted by Malekzadeh et al. [3] and Jin et al. [32]. The slight differences between the three groups of results show the satisfied accuracy of the proposed approach. Table 3 compares the first six natural frequencies (Hz) of a circular beam with F-F, F-C, and C-C boundary conditions obtained by the current approach with those based on commercial FEM code. The results are calculated with the beam parameters Rm = 1 m, h/Rm = 0.3 and with “17 × 9” truncation scheme. Calculations based on FEM commercial software ANSYS (PLANE82, 0.025 m) are used as the benchmark solutions. In Table 3, it is obvious that the present method produces good results comparing with FEM.Table 2
Convergence of the lowest five frequency parametersΩ for C-C supported curved beams (Rm=1 m, Δθ=2π/3).
N
J
h
/
R
m
=
0.1
h
/
R
m
=
0.2
1
2
3
4
5
1
2
3
4
5
11
3
12.067
21.555
35.782
40.788
63.430
10.959
14.434
24.614
28.932
38.069
5
12.001
21.432
35.717
40.411
63.114
10.791
14.369
24.234
28.727
37.409
7
11.998
21.428
35.716
40.401
63.104
10.788
14.368
24.231
28.724
37.403
9
11.996
21.426
35.715
40.396
63.097
10.786
14.368
24.229
28.722
37.400
12
3
12.043
21.551
35.771
40.679
63.110
10.945
14.433
24.609
28.924
38.004
5
11.984
21.421
35.706
40.300
62.772
10.781
14.367
24.224
28.714
37.338
7
11.981
21.418
35.704
40.293
62.761
10.779
14.366
24.219
28.712
37.332
9
11.980
21.416
35.703
40.288
62.754
10.778
14.366
24.217
28.711
37.330
13
3
12.042
21.531
35.759
40.668
63.069
10.943
14.432
24.596
28.921
37.993
5
11.979
21.407
35.694
40.275
62.318
10.777
14.365
24.215
28.709
37.326
7
11.976
21.405
35.693
40.268
62.308
10.774
14.365
24.211
28.705
37.317
9
11.975
21.403
35.693
40.264
62.302
10.773
14.364
24.209
28.704
37.314
14
3
12.029
21.529
35.757
40.628
63.032
10.937
14.432
24.594
28.918
37.982
5
11.969
21.401
35.692
40.250
62.260
10.772
14.365
24.212
28.705
37.316
7
11.967
21.398
35.691
40.245
62.249
10.769
14.363
24.206
28.701
37.309
9
11.966
21.397
35.690
40.241
62.243
10.769
14.363
24.205
28.701
37.307
15
3
12.027
21.518
35.752
40.625
62.966
10.936
14.431
24.588
28.918
37.981
5
11.966
21.394
35.688
40.239
62.218
10.771
14.364
24.206
28.703
37.314
7
11.964
21.392
35.687
40.233
62.210
10.767
14.363
24.201
28.698
37.304
9
11.963
21.390
35.687
40.230
62.206
10.766
14.363
24.200
28.697
37.302
16
3
12.020
21.516
35.751
40.606
62.964
10.933
14.431
24.587
28.917
37.976
5
11.960
21.390
35.686
40.226
62.208
10.767
14.363
24.205
28.700
37.308
7
11.959
21.387
35.685
40.222
62.199
10.764
14.362
24.198
28.696
37.299
9
11.958
21.386
35.685
40.219
62.195
10.763
14.362
24.197
28.695
37.298
17
3
12.019
21.510
35.749
40.603
62.943
10.933
14.431
24.585
28.916
37.976
5
11.958
21.385
35.684
40.220
62.194
10.767
14.363
24.202
28.699
37.308
7
11.957
21.384
35.683
40.214
62.187
10.763
14.362
24.196
28.694
37.297
9
11.956
21.382
35.683
40.212
62.183
10.762
14.361
24.195
28.693
37.295
FSDT [3]
11.391
20.392
34.001
—
—
10.271
13.622
23.107
—
—
2D [32]
11.470
20.575
34.105
38.963
60.102
10.507
13.773
23.691
27.680
36.752Table 3
The lowest six frequencies for circular beams with different boundary conditions, geometrical properties, and modeling methods (Hz,Rm=1 m, h/Rm=0.3).
BC
Mode
Δ
θ
=
π
/
4
Δ
θ
=
π
/
2
Δ
θ
=
3
π
/
4
Pre.
FEM
Diff. (%)
Pre.
FEM
Diff. (%)
Pre.
FEM
Diff. (%)
F
-
F
1
1861.2
1861.3
0.003
545.85
545.85
0.001
239.19
239.19
0.000
2
3304.8
3304.8
0.001
1333.9
1333.9
0.002
638.87
638.87
0.001
3
3653.6
3653.9
0.007
1800.9
1800.9
0.001
1177.4
1177.4
0.001
4
5514.5
5515.4
0.016
2285.0
2285.1
0.003
1355.5
1355.5
0.003
5
5873.7
5874.5
0.014
3207.8
3207.9
0.004
1790.9
1790.9
0.002
6
6166.1
6166.4
0.005
3344.4
3344.5
0.002
2289.5
2289.5
0.000
F
-
C
1
373.90
373.99
0.024
104.22
104.23
0.008
50.018
50.017
0.001
2
1420.4
1420.6
0.014
431.29
431.32
0.008
182.16
182.15
0.003
3
1870.6
1870.8
0.009
974.88
974.91
0.003
535.24
535.24
0.001
4
3475.3
3475.8
0.013
1368.0
1368.1
0.005
898.42
898.43
0.001
5
4773.6
4774.1
0.010
2131.7
2131.9
0.007
1173.9
1173.9
0.002
6
5133.8
5134.0
0.005
2556.9
2556.9
0.001
1692.3
1692.3
0.002
C
-
C
1
1683.9
1684.5
0.033
850.21
850.24
0.004
512.94
512.93
0.002
2
2923.8
2924.6
0.028
1079.5
1079.7
0.015
665.65
665.66
0.001
3
3461.9
3462.8
0.027
1805.4
1805.5
0.007
1159.1
1159.1
0.002
4
4991.2
4992.3
0.022
2056.6
2056.9
0.012
1227.9
1227.9
0.002
5
6282.3
6283.3
0.017
2984.2
2984.5
0.009
1732.5
1732.5
0.002
6
6414.4
6414.6
0.003
3280.8
3281.0
0.005
2148.3
2148.3
0.001


### 3.2. Transverse Deformation Effects

The effects of transverse deformation on the vibration characteristics of curved beams are investigated in this section. In Figures2–9, relative deviations between frequency parameters Ω calculated by the CBT/FSDT theory models [5] and the present 2D approach for circular beams with various different geometries and boundary conditions are considered. The “deviations (%)” between the results are defined as(20)Deviations%=ΩCBT/FSDT-Ω2-DΩ2-D×100%.Figure 2
Relative deviations between the first, third, and fifth frequency parametersΩ based on the CBT/FSDT theory models to those of 2D theory for a complete free circular beam with various thickness ratios.Figure 3
Relative deviations between frequency parametersΩ based on the CBT/FSDT theory models to those of 2D theory for circular beams with various thickness ratios (F-C boundary condition).Figure 4
Relative deviations between frequency parametersΩ based on the CBT/FSDT theory models to those of 2D theory for circular beams with various thickness ratios (C-C boundary condition).Figure 5
Relative deviations between frequency parametersΩ based on the CBT/FSDT theory models to those of 2D theory for circular beams with different thickness-to-radius ratios (C-C boundary condition).Figure 6
Frequency parametersΩ of circular beams with various thickness ratios based on the CBT and FSDT theory models (F-F boundary condition).Figure 7
Frequency parametersΩ of circular beams with various thickness ratios based on the CBT and FSDT theory models (C-C boundary condition).Figure 8
Modal shapes relative to the C-C circular curved beams of Table4.Table 4
The lowest six frequency parametersΩ of classically restrained circular beams with various thickness ratios (Rm=1 m, Δθ=π/2).
h
/
R
m
Mode
Boundary condition
F
-
F
F
-
S
1
F
-
S
2
F
-
C
S
1
-
S
1
S
1
-
S
2
S
1
-
C
S
2
-
S
2
S
2
-
C
C
-
C
0.05
1
8.3653
5.2974
1.8360
1.4976
2.6801
7.5652
4.3680
2.6800
9.5587
22.270
2
23.694
18.869
11.065
7.1854
14.464
23.307
17.702
14.464
26.891
39.349
3
46.884
40.247
28.619
22.477
34.040
46.609
38.807
34.039
50.676
66.753
4
77.238
68.895
53.693
45.339
60.960
77.028
66.582
60.959
66.953
74.701
5
114.23
104.30
85.785
74.297
94.751
97.943
92.155
69.297
86.221
114.39
6
154.82
145.91
97.943
94.711
134.88
114.06
106.77
94.749
123.04
143.50
0.1
1
8.2897
5.2643
1.8326
1.4942
2.6707
7.4945
4.3126
2.6706
9.2748
21.246
2
23.048
18.453
10.919
7.0603
14.213
22.672
17.053
14.213
24.596
27.624
3
44.492
38.450
27.682
21.456
32.728
44.232
35.608
32.727
34.057
46.328
4
71.197
63.988
48.914
40.384
57.039
48.917
47.535
34.672
50.034
64.343
5
77.261
77.261
50.605
49.486
77.259
71.006
63.829
57.038
73.445
79.316
6
102.04
93.921
78.471
70.567
86.002
101.899
92.113
77.257
80.161
98.509
0.2
1
8.0147
5.1403
1.8195
1.4802
2.6343
7.2362
4.1083
2.6343
8.2708
16.029
2
20.994
17.077
10.404
6.6156
13.354
20.658
14.846
13.354
16.094
18.192
3
37.663
33.367
24.305
17.818
28.877
24.349
23.209
17.384
24.260
35.875
4
38.678
38.337
24.870
24.700
38.320
37.789
32.799
28.876
36.821
36.990
5
57.082
52.145
42.532
36.793
47.221
54.074
49.629
38.319
42.107
53.898
6
70.432
70.314
54.084
53.013
67.144
57.039
54.498
47.220
59.788
68.390Figure 9
Modal shapes relative to the F-C circular curved beams of Table5.Table 5
The lowest six frequency parametersΩ of classically restrained circular beams with various span angles (Rm=1 m, h/Rm=0.1).
Δθ
Mode
Boundary condition
F
-
F
F
-
S
1
F
-
S
2
F
-
C
S
1
-
S
1
S
1
-
S
2
S
1
-
C
S
2
-
S
2
S
2
-
C
C
-
C
π/3
1
19.088
12.761
4.4804
3.2579
7.4946
1.0378
11.416
7.4943
20.441
34.046
2
50.820
41.432
25.315
17.105
32.728
18.234
38.524
32.727
34.047
47.270
3
94.059
82.289
60.572
46.643
71.007
50.481
60.019
34.672
59.680
90.478
4
109.23
109.21
62.328
62.105
109.21
62.312
81.544
71.005
101.17
106.39
5
144.99
131.64
106.10
92.330
118.59
93.844
128.08
109.20
110.73
139.31
6
200.96
186.60
158.38
140.84
172.44
144.88
158.76
118.58
154.51
189.04
2
π/3
1
4.4805
2.6710
0.9739
0.8749
1.0378
0.2623
1.9658
1.0378
4.7592
11.419
2
12.761
10.031
5.7637
3.6195
7.4946
3.6888
9.0668
7.4945
14.121
20.445
3
25.315
21.674
15.360
11.728
18.234
12.365
20.381
18.234
26.580
34.049
4
41.432
36.989
28.906
23.886
32.728
25.042
34.259
32.727
34.065
38.533
5
60.573
55.456
43.262
37.328
50.482
41.228
42.524
34.672
45.344
59.694
6
62.330
62.315
45.861
44.506
62.313
43.263
55.402
50.481
59.944
60.023
π
1
1.8327
0.9507
0.4372
0.4349
2.6707
1.0378
3.3476
2.6706
1.5851
4.3142
2
5.2643
3.9432
2.1308
1.3692
7.4946
4.8346
8.4943
7.4945
5.6604
9.2781
3
10.919
9.1645
6.3020
4.6433
14.213
10.627
15.497
14.213
11.769
17.059
4
18.453
16.288
12.501
10.245
22.673
18.234
24.055
22.672
19.579
24.603
5
27.683
25.134
20.505
17.608
32.728
27.510
33.194
32.728
28.529
34.059
6
38.450
35.548
30.155
26.470
44.232
38.308
38.551
34.673
34.072
35.618Figures2–4 show the relative deviations of the 1st, 3rd, and 5th frequency parameters Ω for circular beams with different ratios of thickness-to-span length (h/Lθ). The beam is supposed to be of unit span length; that is, Lθ=RΔθ=1. The ratio of h/Lθ is varied between 0.01 and 0.2. F-F, C-C, and F-C boundaries are considered in the study. The results obtained by the current method of “N×J=17×8” truncation scheme are selected as benchmark. From the figures, we can see that there is a clear increment of frequency parameter for the larger thickness-to-span length ratio and the increment becomes more prominent for higher modes. The maximum difference can be as much as 35%. Furthermore, results on the basis of CBT are generally higher than those based on FSDT model and the 2D elasticity theory because the effects of shear deformations are more significant in thick beams. It is due to the fact that hypotheses in the CBT will introduce additional stiffness in the modeling in fact. This investigation shows that the CBT can be grossly error for the modeling of moderately thick and thick curved beams. In addition, it is obvious that the results based on the FSDT are more accurate than those of CBT since the effects of traverse deformation are included.Figure5 shows a similar study for clamped circular beams with various thickness-to-radius ratios and span angles. Geometrical dimensions used in the study are Rm = 1 m. “N×J=17×9” displacement field is adopted for the 2D solutions in this study. As expected, the effects of transverse normal and shear deformations decrease as the span angle increases. The relative deviations between results based on the CBT model and the current 2D approach are also very big and the maximum difference can be as much as 50%. It can be observed that that the effects of the transverse normal and shear deformation varied with mode number and (span) length-to-radius ratio. Generally, lower (span) length-to-radius ratio values will lead to larger modeling deviation of vibration behavior since transverse effects are more significant for short beams.Figures6 and 7 consider the fundamental and fifth mode frequency parameters Ω of a circular beam based on the CBT and FSDT theory models, respectively. The thickness-to-span length ratio, h/Lθ, is varied from 0.01 to 0.2, corresponding to thin to thick beam configurations. Two boundary conditions, that is, F-F and C-C, are considered in the studies. The beam is supposed to be of unit span length and unit radius, that is, Lθ=1, Rm=1 m. From the figures, we can see that the effects of the shear deformation increase generally as the thickness-to-span length ratio increases. When the thickness-to-span length ratio is equal to 0.1, the difference between the CBT and FSDT results can be as many as 21.6% and 25.3% for the F-F and C-C boundary conditions, respectively. In addition, it is obvious that the transverse shear deformation has greater influence on the higher modes.Elasticity solutions for circular beams with different geometrical dimensions and several sets of classical boundary conditions are presented in the following presentation. The lowest six frequencies considering three different thickness-to-radius ratios are presented in Table4  (Δθ=π/2), where the classical boundary conditions are included. “N×J=17×9” displacement field is adopted in this study. From the results of Table 4, it becomes clear that the frequency parameters Ω decrease when the thickness of the beams is increased. However, it should be pointed out that the beam’s natural frequencies (Hz) are increasing actually because the stiffness of a beam increases generally when its thickness is increased. For the sake of completeness, the first three modal shapes for the beam with C-C boundary conditions are presented in Figure 8. The figure indicates that the modal shapes of thick beams are characterized by complex coupling between the extension, bending, and shearing modes.The lowest six frequency parameters Ω of the certain circular beams are presented in Table5 with a variety of classical restraints and span angles. The thickness ratio (h/Rm) of the beams is assumed to be constant as h/Rm = 0.1, 20. Meanwhile, the span angle Δθ is taken as Δθ=π/3, 2π/3, andπ, respectively. “N×J=20×9” displacement field is adopted in this study. First of all, it is seen that the circular beams with C-C boundary conditions have highest frequency parameters among all boundary cases. The frequency parameters of the beam decrease when the span angle increases because when the span angle increases, the flexibility of the beam increases synchronously. Figure 9 gives the three lowest mode shapes for the circular beam of Table 6 with FC restraint. The figure reveals that the change in the span angle can directly affect the mode shapes of the beam. The modes of the beam are noted to be determined by bending, shear, and normal deformation, which could not be determined by the CBT and FSDT theory models.Table 6
Frequency parametersΩ for elastically supported circular beams with different thickness ratios and restraint rigidities (Rm=1 m, θ0=π/2).
h
/
R
m
η
/
D
k
u
=
η
,
k
w
=
0
k
u
=
0
,
k
w
=
η
k
u
=
k
w
=
η
1
2
3
1
2
3
1
2
3
0.01
100
1.5013
7.2263
22.817
1.5040
7.2283
22.818
1.5066
7.2283
22.818
101
1.5249
7.2266
22.817
1.5515
7.2464
22.823
1.5770
7.2467
22.823
102
1.7442
7.2296
22.818
1.9406
7.4289
22.877
2.1392
7.4312
22.878
103
3.1816
7.2654
22.825
3.3673
9.2160
23.442
4.4370
9.2166
23.451
104
6.8030
9.5884
22.907
4.2514
15.578
29.613
9.5523
15.677
29.817
105
7.0247
21.852
30.250
4.3720
17.693
38.640
17.584
25.942
40.813
0.05
100
1.5109
7.1856
22.477
1.5243
7.1955
22.480
1.5374
7.1957
22.480
101
1.6253
7.1873
22.477
1.7397
7.2863
22.506
1.8517
7.2878
22.507
102
2.4911
7.2050
22.483
2.8562
8.2070
22.780
3.5131
8.2091
22.786
103
6.0299
7.7159
22.538
4.1096
13.588
25.951
7.4008
13.649
26.053
104
7.0667
18.929
23.904
4.3410
17.262
36.620
16.954
18.929
37.704
105
7.7839
22.923
44.333
4.3653
17.659
38.610
18.203
33.998
52.083
0.10
100
1.5207
7.0608
21.456
1.5468
7.0804
21.461
1.5725
7.0808
21.462
101
1.7408
7.0650
21.458
1.9337
7.2624
21.514
2.1335
7.2658
21.516
102
3.1733
7.1131
21.481
3.3372
9.0383
22.066
4.4148
9.0427
22.094
103
6.6891
9.3939
21.720
4.1854
15.042
27.958
9.3746
15.112
28.313
104
7.4853
19.902
25.792
4.2996
16.861
34.871
17.240
20.798
37.990
105
8.7622
23.296
32.768
4.3113
17.034
35.536
19.364
26.704
43.372
0.15
100
1.5282
6.8657
19.865
1.5660
6.8947
19.872
1.6040
6.8956
19.873
101
1.8473
6.8741
19.872
2.0917
7.1678
19.946
2.3677
7.1741
19.954
102
3.7026
6.9755
19.942
3.5429
9.6553
20.751
4.9852
9.6663
20.838
103
6.7367
10.615
20.652
4.1440
14.969
26.889
10.523
15.059
28.171
104
7.8630
17.816
24.479
4.2159
15.961
28.964
17.085
18.064
36.130
105
8.6753
19.710
26.673
4.2233
16.055
29.097
19.121
19.856
39.345Figure10 shows the deviations between the first, third, and fifth frequency parameters Ω based on the CBT/FSDT theory models and those of the current 2D formulation for circular beams for various values of thickness-to-radius ratios and restraint rigidities. The beams (Lθ=1) are supposed to be clamped at one end (θ=θ0) and elastically supported at the other end with stiffness rigidity ku1=kw1=η (where η/D∈10-2,108, D=Eh3/12). “N×J=17×9” displacement field is adopted in this study. From the figure, it is obvious that the effects of the transverse normal and shear deformation have a great influence on the frequencies of the beam when subjected to elastic boundary conditions. According to Figure 10, we can see that the errors of the CBT and FSDT are acceptable when the restraint rigidity η/D is smaller than 10−1. However, the error increases sharply when it is increased from 10−1 to 105. Then, the error decreases and remains the same when η/D tends to be infinity. The error of the CBT/FSDT can be as much as 180%, 120%, and 85% for the worst case in each study.Figure 10
Relative deviations between frequency parametersΩ based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities (θ=θ0: clamped; θ=θ1: ku=kw=η).Figures11–13 show similar study for circular beams with different type of elastic boundary restraints. The following geometric parameters are used: Rm=1, h/Rm = 0.2. The beam is clamped at the end of θ=θ0 and elastically restrained at the other end. The following three types of elastic supports are considered in the study: axially elastic restraint ku1=η,kw1=0, transversely elastic restraint (ku1=0,kw1=η), and elastic restraint in both directions (ku1=kw1=η). The changes of the relative deviations between frequency parameters Ω based on the CBT/FSDT theory models and those of the current 2D formulation with respect to elastic rigidity η/D are the same as Figure 10.Figure 11
Relative deviations between frequency parametersΩ based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities (θ=θ0: clamped; θ=θ1: ku1=η, kw1=0).Figure 12
Relative deviations between frequency parameters Ω based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities (θ=θ0: clamped; θ=θ1: ku1=0, kw1=η).Figure 13
Relative deviations between frequency parametersΩ based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities (θ=θ0: clamped; θ=θ1: ku1=kw1=η).Finally, Table6 displays the lowest three nondimensional frequencies Ω for circular beams with a variety of geometric constants and restrained rigidities. The elastic boundary conditions studied in the table are the same as those of Figures 11–13. The geometrical dimensions used in the calculation are Rm=1 m, θ0=π/2. “N×J=17×9” displacement field is adopted in this study. The table reveals that the frequencies of the beam will increase when the rigidity of the restraint increases. This is because when the restraint rigidity increases, the stiffness of the beam increases synchronously while the mass remains unchanged. Table 7 shows similar studies for the beams with different span angles. The geometrical parameters and material properties used in the calculation are Rm=1 m,h/Rm=0.1. These results can be used to verify new 1D refined beam theories for further studies.Table 7
Frequency parametersΩ for elastically supported circular beams with different span angles and restraint rigidities (Rm=1 m, h/Rm=0.1).
θ
0
η
/
D
k
u
=
η
,
k
w
=
0
k
u
=
0
,
k
w
=
η
k
u
=
k
w
=
η
1
2
3
1
2
3
1
2
3
π
/
3
100
3.2671
17.105
46.643
3.3052
17.116
46.646
3.3143
17.117
46.646
101
3.3484
17.110
46.647
3.6978
17.217
46.674
3.7784
17.222
46.679
102
4.0665
17.154
46.688
6.0927
18.272
46.962
6.5964
18.303
47.008
103
8.0451
17.639
47.081
10.265
27.077
50.196
13.084
27.089
50.748
104
15.321
23.449
49.869
11.295
37.199
59.270
23.265
37.875
68.723
105
18.965
31.838
55.957
11.403
38.393
59.956
31.839
42.697
83.774
2
π
/
3
100
0.9258
3.6197
11.729
0.9197
3.6507
11.737
0.9683
3.6508
11.737
101
1.2960
3.6210
11.731
1.2006
3.9285
11.815
1.5359
3.9290
11.818
102
3.1142
3.6536
11.753
1.7696
5.9362
12.680
3.3494
5.9983
12.716
103
3.6599
9.1322
12.165
1.9434
8.6156
18.070
7.8507
9.3882
18.576
104
3.9841
12.050
23.150
1.9635
9.0227
20.175
9.2396
17.386
26.000
105
4.5641
13.485
25.680
1.9656
9.0622
20.360
10.492
19.461
32.929
π
100
0.5264
1.3719
4.6438
0.4487
1.4211
4.6595
0.5367
1.4241
4.6599
101
1.0030
1.4177
4.6480
0.5063
1.8014
4.8097
1.0033
1.8520
4.8159
102
1.3161
3.0900
4.7182
0.5555
2.9075
6.2241
2.3074
3.3600
6.3712
103
1.3396
4.5807
9.8024
0.5642
3.3007
8.2147
3.1616
6.9331
9.9866
104
1.4308
4.9059
10.622
0.5651
3.3429
8.4675
3.4291
8.1148
15.259
105
1.5524
5.4708
11.429
0.5652
3.3471
8.4916
4.0046
8.7902
16.261


## 3.1. Validation

Table2 gives the first five nondimensional frequency parameters Ω of C-C supported circular beams. The numbers of the sampling surfaces and serious truncation are increased from 11 to 17 and 3 to 9, respectively. The geometric parameters of the beam are Rm=1 m, Δθ=2π/3. For completeness, two thickness-to-radius ratios (i.e., h/Rm = 0.1 and 0.2) corresponding to the moderately thick and thick beam configurations are considered in the study. As observed from Table 2, with the increase of truncated number, the natural frequencies tend to be constant values quickly. The maximum differences between the results based on the “11 × 3” and “17 × 9” computational schemes are less than 0.2%, which confirms the high convergence of the present method. The DQM results based on the FSDT [3] and the Ritz solutions with 2D elasticity theory [32] are also listed in the table. It is observed that the present solutions match well with those predicted by Malekzadeh et al. [3] and Jin et al. [32]. The slight differences between the three groups of results show the satisfied accuracy of the proposed approach. Table 3 compares the first six natural frequencies (Hz) of a circular beam with F-F, F-C, and C-C boundary conditions obtained by the current approach with those based on commercial FEM code. The results are calculated with the beam parameters Rm = 1 m, h/Rm = 0.3 and with “17 × 9” truncation scheme. Calculations based on FEM commercial software ANSYS (PLANE82, 0.025 m) are used as the benchmark solutions. In Table 3, it is obvious that the present method produces good results comparing with FEM.Table 2
Convergence of the lowest five frequency parametersΩ for C-C supported curved beams (Rm=1 m, Δθ=2π/3).
N
J
h
/
R
m
=
0.1
h
/
R
m
=
0.2
1
2
3
4
5
1
2
3
4
5
11
3
12.067
21.555
35.782
40.788
63.430
10.959
14.434
24.614
28.932
38.069
5
12.001
21.432
35.717
40.411
63.114
10.791
14.369
24.234
28.727
37.409
7
11.998
21.428
35.716
40.401
63.104
10.788
14.368
24.231
28.724
37.403
9
11.996
21.426
35.715
40.396
63.097
10.786
14.368
24.229
28.722
37.400
12
3
12.043
21.551
35.771
40.679
63.110
10.945
14.433
24.609
28.924
38.004
5
11.984
21.421
35.706
40.300
62.772
10.781
14.367
24.224
28.714
37.338
7
11.981
21.418
35.704
40.293
62.761
10.779
14.366
24.219
28.712
37.332
9
11.980
21.416
35.703
40.288
62.754
10.778
14.366
24.217
28.711
37.330
13
3
12.042
21.531
35.759
40.668
63.069
10.943
14.432
24.596
28.921
37.993
5
11.979
21.407
35.694
40.275
62.318
10.777
14.365
24.215
28.709
37.326
7
11.976
21.405
35.693
40.268
62.308
10.774
14.365
24.211
28.705
37.317
9
11.975
21.403
35.693
40.264
62.302
10.773
14.364
24.209
28.704
37.314
14
3
12.029
21.529
35.757
40.628
63.032
10.937
14.432
24.594
28.918
37.982
5
11.969
21.401
35.692
40.250
62.260
10.772
14.365
24.212
28.705
37.316
7
11.967
21.398
35.691
40.245
62.249
10.769
14.363
24.206
28.701
37.309
9
11.966
21.397
35.690
40.241
62.243
10.769
14.363
24.205
28.701
37.307
15
3
12.027
21.518
35.752
40.625
62.966
10.936
14.431
24.588
28.918
37.981
5
11.966
21.394
35.688
40.239
62.218
10.771
14.364
24.206
28.703
37.314
7
11.964
21.392
35.687
40.233
62.210
10.767
14.363
24.201
28.698
37.304
9
11.963
21.390
35.687
40.230
62.206
10.766
14.363
24.200
28.697
37.302
16
3
12.020
21.516
35.751
40.606
62.964
10.933
14.431
24.587
28.917
37.976
5
11.960
21.390
35.686
40.226
62.208
10.767
14.363
24.205
28.700
37.308
7
11.959
21.387
35.685
40.222
62.199
10.764
14.362
24.198
28.696
37.299
9
11.958
21.386
35.685
40.219
62.195
10.763
14.362
24.197
28.695
37.298
17
3
12.019
21.510
35.749
40.603
62.943
10.933
14.431
24.585
28.916
37.976
5
11.958
21.385
35.684
40.220
62.194
10.767
14.363
24.202
28.699
37.308
7
11.957
21.384
35.683
40.214
62.187
10.763
14.362
24.196
28.694
37.297
9
11.956
21.382
35.683
40.212
62.183
10.762
14.361
24.195
28.693
37.295
FSDT [3]
11.391
20.392
34.001
—
—
10.271
13.622
23.107
—
—
2D [32]
11.470
20.575
34.105
38.963
60.102
10.507
13.773
23.691
27.680
36.752Table 3
The lowest six frequencies for circular beams with different boundary conditions, geometrical properties, and modeling methods (Hz,Rm=1 m, h/Rm=0.3).
BC
Mode
Δ
θ
=
π
/
4
Δ
θ
=
π
/
2
Δ
θ
=
3
π
/
4
Pre.
FEM
Diff. (%)
Pre.
FEM
Diff. (%)
Pre.
FEM
Diff. (%)
F
-
F
1
1861.2
1861.3
0.003
545.85
545.85
0.001
239.19
239.19
0.000
2
3304.8
3304.8
0.001
1333.9
1333.9
0.002
638.87
638.87
0.001
3
3653.6
3653.9
0.007
1800.9
1800.9
0.001
1177.4
1177.4
0.001
4
5514.5
5515.4
0.016
2285.0
2285.1
0.003
1355.5
1355.5
0.003
5
5873.7
5874.5
0.014
3207.8
3207.9
0.004
1790.9
1790.9
0.002
6
6166.1
6166.4
0.005
3344.4
3344.5
0.002
2289.5
2289.5
0.000
F
-
C
1
373.90
373.99
0.024
104.22
104.23
0.008
50.018
50.017
0.001
2
1420.4
1420.6
0.014
431.29
431.32
0.008
182.16
182.15
0.003
3
1870.6
1870.8
0.009
974.88
974.91
0.003
535.24
535.24
0.001
4
3475.3
3475.8
0.013
1368.0
1368.1
0.005
898.42
898.43
0.001
5
4773.6
4774.1
0.010
2131.7
2131.9
0.007
1173.9
1173.9
0.002
6
5133.8
5134.0
0.005
2556.9
2556.9
0.001
1692.3
1692.3
0.002
C
-
C
1
1683.9
1684.5
0.033
850.21
850.24
0.004
512.94
512.93
0.002
2
2923.8
2924.6
0.028
1079.5
1079.7
0.015
665.65
665.66
0.001
3
3461.9
3462.8
0.027
1805.4
1805.5
0.007
1159.1
1159.1
0.002
4
4991.2
4992.3
0.022
2056.6
2056.9
0.012
1227.9
1227.9
0.002
5
6282.3
6283.3
0.017
2984.2
2984.5
0.009
1732.5
1732.5
0.002
6
6414.4
6414.6
0.003
3280.8
3281.0
0.005
2148.3
2148.3
0.001


## 3.2. Transverse Deformation Effects

The effects of transverse deformation on the vibration characteristics of curved beams are investigated in this section. In Figures2–9, relative deviations between frequency parameters Ω calculated by the CBT/FSDT theory models [5] and the present 2D approach for circular beams with various different geometries and boundary conditions are considered. The “deviations (%)” between the results are defined as(20)Deviations%=ΩCBT/FSDT-Ω2-DΩ2-D×100%.Figure 2
Relative deviations between the first, third, and fifth frequency parametersΩ based on the CBT/FSDT theory models to those of 2D theory for a complete free circular beam with various thickness ratios.Figure 3
Relative deviations between frequency parametersΩ based on the CBT/FSDT theory models to those of 2D theory for circular beams with various thickness ratios (F-C boundary condition).Figure 4
Relative deviations between frequency parametersΩ based on the CBT/FSDT theory models to those of 2D theory for circular beams with various thickness ratios (C-C boundary condition).Figure 5
Relative deviations between frequency parametersΩ based on the CBT/FSDT theory models to those of 2D theory for circular beams with different thickness-to-radius ratios (C-C boundary condition).Figure 6
Frequency parametersΩ of circular beams with various thickness ratios based on the CBT and FSDT theory models (F-F boundary condition).Figure 7
Frequency parametersΩ of circular beams with various thickness ratios based on the CBT and FSDT theory models (C-C boundary condition).Figure 8
Modal shapes relative to the C-C circular curved beams of Table4.Table 4
The lowest six frequency parametersΩ of classically restrained circular beams with various thickness ratios (Rm=1 m, Δθ=π/2).
h
/
R
m
Mode
Boundary condition
F
-
F
F
-
S
1
F
-
S
2
F
-
C
S
1
-
S
1
S
1
-
S
2
S
1
-
C
S
2
-
S
2
S
2
-
C
C
-
C
0.05
1
8.3653
5.2974
1.8360
1.4976
2.6801
7.5652
4.3680
2.6800
9.5587
22.270
2
23.694
18.869
11.065
7.1854
14.464
23.307
17.702
14.464
26.891
39.349
3
46.884
40.247
28.619
22.477
34.040
46.609
38.807
34.039
50.676
66.753
4
77.238
68.895
53.693
45.339
60.960
77.028
66.582
60.959
66.953
74.701
5
114.23
104.30
85.785
74.297
94.751
97.943
92.155
69.297
86.221
114.39
6
154.82
145.91
97.943
94.711
134.88
114.06
106.77
94.749
123.04
143.50
0.1
1
8.2897
5.2643
1.8326
1.4942
2.6707
7.4945
4.3126
2.6706
9.2748
21.246
2
23.048
18.453
10.919
7.0603
14.213
22.672
17.053
14.213
24.596
27.624
3
44.492
38.450
27.682
21.456
32.728
44.232
35.608
32.727
34.057
46.328
4
71.197
63.988
48.914
40.384
57.039
48.917
47.535
34.672
50.034
64.343
5
77.261
77.261
50.605
49.486
77.259
71.006
63.829
57.038
73.445
79.316
6
102.04
93.921
78.471
70.567
86.002
101.899
92.113
77.257
80.161
98.509
0.2
1
8.0147
5.1403
1.8195
1.4802
2.6343
7.2362
4.1083
2.6343
8.2708
16.029
2
20.994
17.077
10.404
6.6156
13.354
20.658
14.846
13.354
16.094
18.192
3
37.663
33.367
24.305
17.818
28.877
24.349
23.209
17.384
24.260
35.875
4
38.678
38.337
24.870
24.700
38.320
37.789
32.799
28.876
36.821
36.990
5
57.082
52.145
42.532
36.793
47.221
54.074
49.629
38.319
42.107
53.898
6
70.432
70.314
54.084
53.013
67.144
57.039
54.498
47.220
59.788
68.390Figure 9
Modal shapes relative to the F-C circular curved beams of Table5.Table 5
The lowest six frequency parametersΩ of classically restrained circular beams with various span angles (Rm=1 m, h/Rm=0.1).
Δθ
Mode
Boundary condition
F
-
F
F
-
S
1
F
-
S
2
F
-
C
S
1
-
S
1
S
1
-
S
2
S
1
-
C
S
2
-
S
2
S
2
-
C
C
-
C
π/3
1
19.088
12.761
4.4804
3.2579
7.4946
1.0378
11.416
7.4943
20.441
34.046
2
50.820
41.432
25.315
17.105
32.728
18.234
38.524
32.727
34.047
47.270
3
94.059
82.289
60.572
46.643
71.007
50.481
60.019
34.672
59.680
90.478
4
109.23
109.21
62.328
62.105
109.21
62.312
81.544
71.005
101.17
106.39
5
144.99
131.64
106.10
92.330
118.59
93.844
128.08
109.20
110.73
139.31
6
200.96
186.60
158.38
140.84
172.44
144.88
158.76
118.58
154.51
189.04
2
π/3
1
4.4805
2.6710
0.9739
0.8749
1.0378
0.2623
1.9658
1.0378
4.7592
11.419
2
12.761
10.031
5.7637
3.6195
7.4946
3.6888
9.0668
7.4945
14.121
20.445
3
25.315
21.674
15.360
11.728
18.234
12.365
20.381
18.234
26.580
34.049
4
41.432
36.989
28.906
23.886
32.728
25.042
34.259
32.727
34.065
38.533
5
60.573
55.456
43.262
37.328
50.482
41.228
42.524
34.672
45.344
59.694
6
62.330
62.315
45.861
44.506
62.313
43.263
55.402
50.481
59.944
60.023
π
1
1.8327
0.9507
0.4372
0.4349
2.6707
1.0378
3.3476
2.6706
1.5851
4.3142
2
5.2643
3.9432
2.1308
1.3692
7.4946
4.8346
8.4943
7.4945
5.6604
9.2781
3
10.919
9.1645
6.3020
4.6433
14.213
10.627
15.497
14.213
11.769
17.059
4
18.453
16.288
12.501
10.245
22.673
18.234
24.055
22.672
19.579
24.603
5
27.683
25.134
20.505
17.608
32.728
27.510
33.194
32.728
28.529
34.059
6
38.450
35.548
30.155
26.470
44.232
38.308
38.551
34.673
34.072
35.618Figures2–4 show the relative deviations of the 1st, 3rd, and 5th frequency parameters Ω for circular beams with different ratios of thickness-to-span length (h/Lθ). The beam is supposed to be of unit span length; that is, Lθ=RΔθ=1. The ratio of h/Lθ is varied between 0.01 and 0.2. F-F, C-C, and F-C boundaries are considered in the study. The results obtained by the current method of “N×J=17×8” truncation scheme are selected as benchmark. From the figures, we can see that there is a clear increment of frequency parameter for the larger thickness-to-span length ratio and the increment becomes more prominent for higher modes. The maximum difference can be as much as 35%. Furthermore, results on the basis of CBT are generally higher than those based on FSDT model and the 2D elasticity theory because the effects of shear deformations are more significant in thick beams. It is due to the fact that hypotheses in the CBT will introduce additional stiffness in the modeling in fact. This investigation shows that the CBT can be grossly error for the modeling of moderately thick and thick curved beams. In addition, it is obvious that the results based on the FSDT are more accurate than those of CBT since the effects of traverse deformation are included.Figure5 shows a similar study for clamped circular beams with various thickness-to-radius ratios and span angles. Geometrical dimensions used in the study are Rm = 1 m. “N×J=17×9” displacement field is adopted for the 2D solutions in this study. As expected, the effects of transverse normal and shear deformations decrease as the span angle increases. The relative deviations between results based on the CBT model and the current 2D approach are also very big and the maximum difference can be as much as 50%. It can be observed that that the effects of the transverse normal and shear deformation varied with mode number and (span) length-to-radius ratio. Generally, lower (span) length-to-radius ratio values will lead to larger modeling deviation of vibration behavior since transverse effects are more significant for short beams.Figures6 and 7 consider the fundamental and fifth mode frequency parameters Ω of a circular beam based on the CBT and FSDT theory models, respectively. The thickness-to-span length ratio, h/Lθ, is varied from 0.01 to 0.2, corresponding to thin to thick beam configurations. Two boundary conditions, that is, F-F and C-C, are considered in the studies. The beam is supposed to be of unit span length and unit radius, that is, Lθ=1, Rm=1 m. From the figures, we can see that the effects of the shear deformation increase generally as the thickness-to-span length ratio increases. When the thickness-to-span length ratio is equal to 0.1, the difference between the CBT and FSDT results can be as many as 21.6% and 25.3% for the F-F and C-C boundary conditions, respectively. In addition, it is obvious that the transverse shear deformation has greater influence on the higher modes.Elasticity solutions for circular beams with different geometrical dimensions and several sets of classical boundary conditions are presented in the following presentation. The lowest six frequencies considering three different thickness-to-radius ratios are presented in Table4  (Δθ=π/2), where the classical boundary conditions are included. “N×J=17×9” displacement field is adopted in this study. From the results of Table 4, it becomes clear that the frequency parameters Ω decrease when the thickness of the beams is increased. However, it should be pointed out that the beam’s natural frequencies (Hz) are increasing actually because the stiffness of a beam increases generally when its thickness is increased. For the sake of completeness, the first three modal shapes for the beam with C-C boundary conditions are presented in Figure 8. The figure indicates that the modal shapes of thick beams are characterized by complex coupling between the extension, bending, and shearing modes.The lowest six frequency parameters Ω of the certain circular beams are presented in Table5 with a variety of classical restraints and span angles. The thickness ratio (h/Rm) of the beams is assumed to be constant as h/Rm = 0.1, 20. Meanwhile, the span angle Δθ is taken as Δθ=π/3, 2π/3, andπ, respectively. “N×J=20×9” displacement field is adopted in this study. First of all, it is seen that the circular beams with C-C boundary conditions have highest frequency parameters among all boundary cases. The frequency parameters of the beam decrease when the span angle increases because when the span angle increases, the flexibility of the beam increases synchronously. Figure 9 gives the three lowest mode shapes for the circular beam of Table 6 with FC restraint. The figure reveals that the change in the span angle can directly affect the mode shapes of the beam. The modes of the beam are noted to be determined by bending, shear, and normal deformation, which could not be determined by the CBT and FSDT theory models.Table 6
Frequency parametersΩ for elastically supported circular beams with different thickness ratios and restraint rigidities (Rm=1 m, θ0=π/2).
h
/
R
m
η
/
D
k
u
=
η
,
k
w
=
0
k
u
=
0
,
k
w
=
η
k
u
=
k
w
=
η
1
2
3
1
2
3
1
2
3
0.01
100
1.5013
7.2263
22.817
1.5040
7.2283
22.818
1.5066
7.2283
22.818
101
1.5249
7.2266
22.817
1.5515
7.2464
22.823
1.5770
7.2467
22.823
102
1.7442
7.2296
22.818
1.9406
7.4289
22.877
2.1392
7.4312
22.878
103
3.1816
7.2654
22.825
3.3673
9.2160
23.442
4.4370
9.2166
23.451
104
6.8030
9.5884
22.907
4.2514
15.578
29.613
9.5523
15.677
29.817
105
7.0247
21.852
30.250
4.3720
17.693
38.640
17.584
25.942
40.813
0.05
100
1.5109
7.1856
22.477
1.5243
7.1955
22.480
1.5374
7.1957
22.480
101
1.6253
7.1873
22.477
1.7397
7.2863
22.506
1.8517
7.2878
22.507
102
2.4911
7.2050
22.483
2.8562
8.2070
22.780
3.5131
8.2091
22.786
103
6.0299
7.7159
22.538
4.1096
13.588
25.951
7.4008
13.649
26.053
104
7.0667
18.929
23.904
4.3410
17.262
36.620
16.954
18.929
37.704
105
7.7839
22.923
44.333
4.3653
17.659
38.610
18.203
33.998
52.083
0.10
100
1.5207
7.0608
21.456
1.5468
7.0804
21.461
1.5725
7.0808
21.462
101
1.7408
7.0650
21.458
1.9337
7.2624
21.514
2.1335
7.2658
21.516
102
3.1733
7.1131
21.481
3.3372
9.0383
22.066
4.4148
9.0427
22.094
103
6.6891
9.3939
21.720
4.1854
15.042
27.958
9.3746
15.112
28.313
104
7.4853
19.902
25.792
4.2996
16.861
34.871
17.240
20.798
37.990
105
8.7622
23.296
32.768
4.3113
17.034
35.536
19.364
26.704
43.372
0.15
100
1.5282
6.8657
19.865
1.5660
6.8947
19.872
1.6040
6.8956
19.873
101
1.8473
6.8741
19.872
2.0917
7.1678
19.946
2.3677
7.1741
19.954
102
3.7026
6.9755
19.942
3.5429
9.6553
20.751
4.9852
9.6663
20.838
103
6.7367
10.615
20.652
4.1440
14.969
26.889
10.523
15.059
28.171
104
7.8630
17.816
24.479
4.2159
15.961
28.964
17.085
18.064
36.130
105
8.6753
19.710
26.673
4.2233
16.055
29.097
19.121
19.856
39.345Figure10 shows the deviations between the first, third, and fifth frequency parameters Ω based on the CBT/FSDT theory models and those of the current 2D formulation for circular beams for various values of thickness-to-radius ratios and restraint rigidities. The beams (Lθ=1) are supposed to be clamped at one end (θ=θ0) and elastically supported at the other end with stiffness rigidity ku1=kw1=η (where η/D∈10-2,108, D=Eh3/12). “N×J=17×9” displacement field is adopted in this study. From the figure, it is obvious that the effects of the transverse normal and shear deformation have a great influence on the frequencies of the beam when subjected to elastic boundary conditions. According to Figure 10, we can see that the errors of the CBT and FSDT are acceptable when the restraint rigidity η/D is smaller than 10−1. However, the error increases sharply when it is increased from 10−1 to 105. Then, the error decreases and remains the same when η/D tends to be infinity. The error of the CBT/FSDT can be as much as 180%, 120%, and 85% for the worst case in each study.Figure 10
Relative deviations between frequency parametersΩ based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities (θ=θ0: clamped; θ=θ1: ku=kw=η).Figures11–13 show similar study for circular beams with different type of elastic boundary restraints. The following geometric parameters are used: Rm=1, h/Rm = 0.2. The beam is clamped at the end of θ=θ0 and elastically restrained at the other end. The following three types of elastic supports are considered in the study: axially elastic restraint ku1=η,kw1=0, transversely elastic restraint (ku1=0,kw1=η), and elastic restraint in both directions (ku1=kw1=η). The changes of the relative deviations between frequency parameters Ω based on the CBT/FSDT theory models and those of the current 2D formulation with respect to elastic rigidity η/D are the same as Figure 10.Figure 11
Relative deviations between frequency parametersΩ based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities (θ=θ0: clamped; θ=θ1: ku1=η, kw1=0).Figure 12
Relative deviations between frequency parameters Ω based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities (θ=θ0: clamped; θ=θ1: ku1=0, kw1=η).Figure 13
Relative deviations between frequency parametersΩ based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities (θ=θ0: clamped; θ=θ1: ku1=kw1=η).Finally, Table6 displays the lowest three nondimensional frequencies Ω for circular beams with a variety of geometric constants and restrained rigidities. The elastic boundary conditions studied in the table are the same as those of Figures 11–13. The geometrical dimensions used in the calculation are Rm=1 m, θ0=π/2. “N×J=17×9” displacement field is adopted in this study. The table reveals that the frequencies of the beam will increase when the rigidity of the restraint increases. This is because when the restraint rigidity increases, the stiffness of the beam increases synchronously while the mass remains unchanged. Table 7 shows similar studies for the beams with different span angles. The geometrical parameters and material properties used in the calculation are Rm=1 m,h/Rm=0.1. These results can be used to verify new 1D refined beam theories for further studies.Table 7
Frequency parametersΩ for elastically supported circular beams with different span angles and restraint rigidities (Rm=1 m, h/Rm=0.1).
θ
0
η
/
D
k
u
=
η
,
k
w
=
0
k
u
=
0
,
k
w
=
η
k
u
=
k
w
=
η
1
2
3
1
2
3
1
2
3
π
/
3
100
3.2671
17.105
46.643
3.3052
17.116
46.646
3.3143
17.117
46.646
101
3.3484
17.110
46.647
3.6978
17.217
46.674
3.7784
17.222
46.679
102
4.0665
17.154
46.688
6.0927
18.272
46.962
6.5964
18.303
47.008
103
8.0451
17.639
47.081
10.265
27.077
50.196
13.084
27.089
50.748
104
15.321
23.449
49.869
11.295
37.199
59.270
23.265
37.875
68.723
105
18.965
31.838
55.957
11.403
38.393
59.956
31.839
42.697
83.774
2
π
/
3
100
0.9258
3.6197
11.729
0.9197
3.6507
11.737
0.9683
3.6508
11.737
101
1.2960
3.6210
11.731
1.2006
3.9285
11.815
1.5359
3.9290
11.818
102
3.1142
3.6536
11.753
1.7696
5.9362
12.680
3.3494
5.9983
12.716
103
3.6599
9.1322
12.165
1.9434
8.6156
18.070
7.8507
9.3882
18.576
104
3.9841
12.050
23.150
1.9635
9.0227
20.175
9.2396
17.386
26.000
105
4.5641
13.485
25.680
1.9656
9.0622
20.360
10.492
19.461
32.929
π
100
0.5264
1.3719
4.6438
0.4487
1.4211
4.6595
0.5367
1.4241
4.6599
101
1.0030
1.4177
4.6480
0.5063
1.8014
4.8097
1.0033
1.8520
4.8159
102
1.3161
3.0900
4.7182
0.5555
2.9075
6.2241
2.3074
3.3600
6.3712
103
1.3396
4.5807
9.8024
0.5642
3.3007
8.2147
3.1616
6.9331
9.9866
104
1.4308
4.9059
10.622
0.5651
3.3429
8.4675
3.4291
8.1148
15.259
105
1.5524
5.4708
11.429
0.5652
3.3471
8.4916
4.0046
8.7902
16.261


## 4. Conclusions

This paper proposes an accurate modified Fourier series-based sampling surface approach for the analytical evaluation of the vibration characteristics of thick curved beams. The approach is valid for arbitrary thickness configuration and maintains its simplicity and uniform in any type of boundary conditions (i.e., classical boundary condition, elastic support, or their combination). The theoretical models of the beams are based on the 2D theory of elasticity including the effects of both transverse shear and normal deformations. Under the current framework, the transverse beam domain is discretized by a set of nonequally spaced sampling surfaces and the displacement components coinciding with these surfaces are mathematically described as an set of modified Fourier series including the certain auxiliary terms which are used to form a mathematically complete set and guarantee the results convergent to the exact solutions. The governing equations of the beams are derived and numerically solved using a modified variational principle by the use of the penalty technique as well as Lagrange multipliers. Elasticity solutions including transverse shear and normal effects are compared with the corresponding one-dimensional results in terms of the classical and first-order shear deformation theories. The influences of transverse normal and shear deformation on the vibration characteristics are systematically evaluated. The results show that the proposed method is applicable for thick circular beams with arbitrary boundary conditions.



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*Source: 2104015-2017-10-18.xml*