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

1. IntroductionBeams 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 Figure 1 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 [theta] = [[theta].sub.0] and [theta] = [[theta].sub.1]. The bottom surface of the beam is selected as the reference surface with the three orthogonal curvilinear coordinates [theta], y, and z; see Figure 1. In this paper, the beams are assumed to be isotropic and homogeneous and to vibrate freely in the [theta]-z plane. u, v, and w denote the three displacement components in the axial, lateral, and normal directions, respectively.

2.2. Plane Stress Assumption. As mentioned previously, the beam under consideration vibrates freely in the [theta]-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]

[mathematical expression not reproducible], (1)

where [R.sub.z] = R + z, [[epsilon].sub.[theta]], [[epsilon].sub.y], and [[epsilon].sub.z] stand for the normal strains, and [[gamma].sub.yz], [[gamma].sub.[theta]z], and [[gamma].sub.[theta]y] 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:

[mathematical expression not reproducible], (2)

where [[sigma].sub.[theta]], [[sigma].sub.y], and [[sigma].sub.z] stand for the normal stresses; [[tau].sub.yz], [[tau].sub.[theta]z], and [[tau].sub.[theta]y] are the shear component. C' is the material stiffness matrix.

[C'.sub.11] = [C'.sub.2] = [C'.sub.23] = E(1 - v)/(1 + v)(1 - 2v),

[C'.sub.12] = [C'.sub.13] = [C'.sub.23] = v[C'.sub.11]/(1 - v),

[C'.sub.44] = [C'.sub.55] = [C'.sub.66] = E/2(1 + v), (3)

in which E means Young's module of the material and v represents Poisson's ratio.

Furthermore, (2) can be rearranged in the form of matrix as

[mathematical expression not reproducible], (4)

Therefore, the final stress-strain relations for the beam under plane stress hypothesis can be obtained as

[[sigma].sub.i] = C[[epsilon].sub.i], (5)

where C = [C'.sub.ii] - [C'.sub.io]([[C'.sub.00].sup.-1])[C'.sub.io].

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 Figure 1, [S.sub.1], ..., [S.sub.j], ..., [S.sub.J] 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:

[z.sub.1] = 0,

[z.sub.J] = h,

[z.sub.j] = [z.sub.1] + [z.sub.J]/2 - h/2 cps ([pi]2j - 3/2J - 4);

2 [less than or equal to] j [less than or equal to] J - 1. (6)

Therefore, the basic variables in the axial and normal directions of an arbitrary sampling surface can be given by

u ([theta] [z.sub.j], t) = [u.sub.j] ([theta]) [e.sup.i[omega]t],

w ([theta], [z.sub.j], t) = [w.sub.j] ([theta]) [e.sup.i[omega]t];

1 [less than or equal to] j [less than or equal to] J, (7)

where [u.sub.j]([theta]) and [w.sub.j]([theta]) stand for the axial and transverse displacement components, respectively. t stand for the time variable, [omega] is the circular frequency. As a consequence, displacement field of the beam under vibration can be calculated by

[mathematical expression not reproducible], (8)

and [L.sub.j](z) is Lagrange's interpolation of degree J -1:

[mathematical expression not reproducible]. (9)

According to (1), strains on the jth sampling surface can be found as

[mathematical expression not reproducible], (10)

where [M.sup.r.sub.j] are determined by

[mathematical expression not reproducible], (11)

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).

[mathematical expression not reproducible], (12)

The energy functional for the curved beam under the circumstance of free vibration is

[[PI].sub.S] = [T.sub.S] - [U.sub.s], (13)

where [U.sub.s] and [T.sub.s] denote the strain and kinetic energy function defined as follows:

[mathematical expression not reproducible], (14)

where [rho] stands for the material density. Substituting (5) and (12) into (14), the two energy functions can be further written as

[mathematical expression not reproducible]. (15)

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

[mathematical expression not reproducible], (16)

where [u.sup.n.sub.j] and [w.sup.n.sub.j] (n = 0, 1, ..., N) are the expansion coefficients; [DELTA][theta] = [[theta].sub.1] - [[theta].sub.0] . 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]:

[mathematical expression not reproducible], (17)

where [[bar.u].sub.l] and [[bar.w].sup.l] denote the boundary values. [k.sup.l.sub.u] and [k.sup.l.sub.w] represent the penalty parameters. [[eta].sup.l.sub.u] and [[eta].sup.l.sub.w] are the parameters which define different restraint conditions. The boundary potential [[PI].sub.b1] is introduced by means of Lagrange multiplier technique while the boundary potential [[PI].sub.b2] 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 [theta] = [[theta].sub.0], 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 [[PI].sub.b1] in (17) should be neglected and the penalty parameters will be determined at proper values [49].

Therefore, the final variational functional for the curved beam with general boundaries is defined as

[[PI].sub.total] ([u.sup.n.sub.j],[w.sup.n.sub.j]) = [[PI].sub.S] + [[PI].sub.b]; 0 [less than or equal to] n [less than or equal to] N, 1 [less than or equal to] j [less than or equal to] J. (18)

Finally, let the variation of the [[PI].sub.total] with respect to each coefficient ([u.sup.n.sub.j] and [w.sup.n.sub.j]) equal zero; the governing equations can be derived in a matrix form as

{K - [[omega].sup.2]M} G = 0, (19)

where K and M 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 [omega]. 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 [theta] = [[theta].sub.0] and [theta] = [[theta].sub.1], respectively. To unify the discussion, the dimensionless variable of frequency is introduced in the calculation [OMEGA] = [omega][R.sup.2.sub.m] [square root of (12[rho]/E[h.sup.2])] (where [R.sub.m] = R + h/2). The beams are supposed to be made of steel (E = 210 GPa, v = 0.3, and [rho] = 7800 kg/[m.sup.3]).

3.1. Validation. Table 2 gives the first five nondimensional frequency parameters [OMEGA] 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 [R.sub.m] = 1 m, [DELTA][theta] = 2[pi]/3. For completeness, two thickness-to-radius ratios (i.e., h/[R.sub.m] = 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 x 3" and "17 x 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 [R.sub.m] = 1m, h/[R.sub.m] = 0.3 and with "17 x 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.

3.2. Transverse Deformation Effects. The effects of transverse deformation on the vibration characteristics of curved beams are investigated in this section. In Figures 2-9, relative deviations between frequency parameters [OMEGA] 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

Deviations (%) = ([[OMEGA].sub.CBT/FSDT] -[[OMEGA].sub.2-D])/[[OMEGA].sub.2-D] x 100%. (20)

Figures 2-4 show the relative deviations of the 1st, 3rd, and 5th frequency parameters [OMEGA] for circular beams with different ratios of thickness-to-span length (h/[L.sub.[theta]]). The beam is supposed to be of unit span length; that is, [L.sub.[theta]] = .R[DELTA][theta] = 1. The ratio of h/[L.sub.[theta]] 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 x J = 17 x 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.

Figure 5 shows a similar study for clamped circular beams with various thickness-to-radius ratios and span angles. Geometrical dimensions used in the study are [R.sub.m] = 1 m. "N x J = 17 x 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.

Figures 6 and 7 consider the fundamental and fifth mode frequency parameters [OMEGA] of a circular beam based on the CBT and FSDT theory models, respectively. The thickness-to-span length ratio, h/[L.sub.[theta]], 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, [L.sub.[theta]] = 1, [R.sub.m] = 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 Table 4 ([DELTA][theta] = [pi]/2), where the classical boundary conditions are included. "N x J = 17x 9" displacement field is adopted in this study. From the results of Table 4, it becomes clear that the frequency parameters [OMEGA] 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 [OMEGA] of the certain circular beams are presented in Table 5 with a variety of classical restraints and span angles. The thickness ratio (h/[R.sub.m]) of the beams is assumed to be constant as h/[R.sub.m] = 0.1, 20. Meanwhile, the span angle [DELTA][theta] is taken as [DELTA][theta] = [pi]/3, 2[pi]/3, and [pi], respectively. "N x J = 20 x 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.

Figure 10 shows the deviations between the first, third, and fifth frequency parameters [OMEGA] 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 ([L.sub.[theta]] = 1) are supposed to be clamped at one end ([theta] = [[theta].sub.0]) and elastically supported at the other end with stiffness rigidity [k.sup.l.sub.u] = [k.sup.l.sub.[]omega] = [eta] (where [eta]/D [member of] [[10.sup.-2], [10.sup.8]], D = E[h.sup.3]/12). "N x J = 17 x 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 [eta]/D is smaller than [10.sup.-1]. However, the error increases sharply when it is increased from [10.sup.-1] to [10.sup.5]. Then, the error decreases and remains the same when [eta]/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.

Figures 11-13 show similar study for circular beams with different type of elastic boundary restraints. The following geometric parameters are used: [R.sub.m] = 1, h/[R.sub.m] = 0.2. The beam is clamped at the end of 0 = 06 and elastically restrained at the other end. The following three types of elastic supports are considered in the study: axially elastic restraint ([k.sup.l.sub.u] = [eta], [k.sup.l.sub.w] = 0), transversely elastic restraint ([k.sup.l.sub.u] = 0, [k.sup.l.sub.w] = and elastic restraint in both directions ([k.sup.l.sub.u] = [k.sup.l.sub.w] = [eta]). The changes of the relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models and those of the current 2D formulation with respect to elastic rigidity [eta]/D are the same as Figure 10.

Finally, Table 6 displays the lowest three nondimensional frequencies [OMEGA] 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 [R.sub.m] = 1m, [[theta].sub.0] = [pi]/2. "N x J = 17 x 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 [R.sub.m] = 1 m, h/[R.sub.m] = 0.1. These results can be used to verify new 1D refined beam theories for further studies.

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.

https://doi.org/10.1155/2017/2104015

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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Xueren Wang, (1) Xuhong Miao, (1, 2) Di Jia, (1) and Shengyao Gao (1)

(1) Naval Academy of Armament, Beijing 100161, China

(2) College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to Xueren Wang; wangxuerennaa@126.com

Received 1 July 2017; Revised 1 September 2017; Accepted 11 September 2017; Published 18 October 2017

Academic Editor: Nerio Tullini

Caption: FIGURE 1: Geometry and reference system for a curved beam and the diagrammatic sketch of sampling surface distribution.

Caption: FIGURE 2: Relative deviations between the first, third, and fifth frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for a complete free circular beam with various thickness ratios.

Caption: FIGURE 3: Relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for circular beams with various thickness ratios (F-C boundary condition).

Caption: Figure 4: Relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for circular beams with various thickness ratios (C-C boundary condition).

Caption: Figure 5: Relative deviations between frequency parameters [OMEGA] 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).

Caption: FIGURE 6: Frequency parameters [OMEGA] of circular beams with various thickness ratios based on the CBT and FSDT theory models (F-F boundary condition).

Caption: FIGURE 7: Frequency parameters [OMEGA] of circular beams with various thickness ratios based on the CBT and FSDT theory models (C-C boundary condition).

Caption: FIGURE 8: Modal shapes relative to the C-C circular curved beams of Table 4.

Caption: FIGURE 9: Modal shapes relative to the F-C circular curved beams of Table 5.

Caption: FIGURE 10: Relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities ([theta] = [[theta].sub.0]: clamped; [theta] = [[theta].sub.1]: ku = kw = [eta]).

Caption: Figure 11: Relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities ([theta] = [[theta].sub.0] : clamped; [theta] = [[theta].sub.1]: [k.sup.l.sub.u] = [eta], [k.sup.l.sub.w] = 0).

Caption: FIGURE 12: Relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities ([theta] = [[theta].sub.0]: clamped; [theta] = [[theta].sub.1]: [k.sup.l.sub.u] = 0, [k.sup.l.sub.w] = [eta]]).

Caption: Figure 13: Relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities ([theta] = [[theta].sub.0]: clamped; [theta] = [[theta].sub.1]: [k.sup.l.sub.u] = [k.sup.l.sub.w] = [eta]).

TABLE 1: Values of [[eta].sup.l.sub.u], [[eta].sup.l.sub.[omega]], [k.sup.l.sub.u], and [k.sup.l.sub.w] for different classical boundary conditions. Boundary Boundary conditions coefficients [[eta].sup.0. [[eta]. sub. sup.0.sub.u] [omega]] F (free): [[sigma].sub. 0 0 [theta]] = 0, [[tai].sub. [theta]z] = 0 S1 (simply supported): 0 1 [[sigma].sub.[theta]] =0, w = 0 S2 (simply supported): 1 0 u = 0, [[tai].sub. [theta]z] = 0 C (clamped): u = 1 1 0, w = 0 Penalty Boundary conditions parameters [k.sup.0.sub.u] [k.sup.0.sub.w] F (free): [[sigma].sub. 0 0 [theta]] = 0, [[tai].sub. [theta]z] = 0 S1 (simply supported): 0 [10.sup.3]E [[sigma].sub.[theta]] =0, w = 0 S2 (simply supported): [10.sup.3]E 0 u = 0, [[tai].sub. [theta]z] = 0 C (clamped): u = [10.sup.3]E [10.sup.3]E 0, w = 0 TABLE 2: Convergence of the lowest five frequency parameters [OMEGA] for C-C supported curved beams ([R.sub.m] = 1 m, [DELTA][theta] = 2[pi]/3). N J h/[R.sub.m] = 0.1 1 2 3 4 5 11 3 12.067 21.555 35.782 40.788 63.430 5 12.001 21.432 35.717 40.411 63.114 7 11.998 21.428 35.716 40.401 63.104 9 11.996 21.426 35.715 40.396 63.097 12 3 12.043 21.551 35.771 40.679 63.110 5 11.984 21.421 35.706 40.300 62.772 7 11.981 21.418 35.704 40.293 62.761 9 11.980 21.416 35.703 40.288 62.754 13 3 12.042 21.531 35.759 40.668 63.069 5 11.979 21.407 35.694 40.275 62.318 7 11.976 21.405 35.693 40.268 62.308 9 11.975 21.403 35.693 40.264 62.302 14 3 12.029 21.529 35.757 40.628 63.032 5 11.969 21.401 35.692 40.250 62.260 7 11.967 21.398 35.691 40.245 62.249 9 11.966 21.397 35.690 40.241 62.243 15 3 12.027 21.518 35.752 40.625 62.966 5 11.966 21.394 35.688 40.239 62.218 7 11.964 21.392 35.687 40.233 62.210 9 11.963 21.390 35.687 40.230 62.206 16 3 12.020 21.516 35.751 40.606 62.964 5 11.960 21.390 35.686 40.226 62.208 7 11.959 21.387 35.685 40.222 62.199 9 11.958 21.386 35.685 40.219 62.195 17 3 12.019 21.510 35.749 40.603 62.943 5 11.958 21.385 35.684 40.220 62.194 7 11.957 21.384 35.683 40.214 62.187 9 11.956 21.382 35.683 40.212 62.183 FSDT [3] 11.391 20.392 34.001 -- -- 2D [32] 11.470 20.575 34.105 38.963 60.102 N J h/[R.sub.m] = 0.2 1 2 3 4 5 11 3 10.959 14.434 24.614 28.932 38.069 5 10.791 14.369 24.234 28.727 37.409 7 10.788 14.368 24.231 28.724 37.403 9 10.786 14.368 24.229 28.722 37.400 12 3 10.945 14.433 24.609 28.924 38.004 5 10.781 14.367 24.224 28.714 37.338 7 10.779 14.366 24.219 28.712 37.332 9 10.778 14.366 24.217 28.711 37.330 13 3 10.943 14.432 24.596 28.921 37.993 5 10.777 14.365 24.215 28.709 37.326 7 10.774 14.365 24.211 28.705 37.317 9 10.773 14.364 24.209 28.704 37.314 14 3 10.937 14.432 24.594 28.918 37.982 5 10.772 14.365 24.212 28.705 37.316 7 10.769 14.363 24.206 28.701 37.309 9 10.769 14.363 24.205 28.701 37.307 15 3 10.936 14.431 24.588 28.918 37.981 5 10.771 14.364 24.206 28.703 37.314 7 10.767 14.363 24.201 28.698 37.304 9 10.766 14.363 24.200 28.697 37.302 16 3 10.933 14.431 24.587 28.917 37.976 5 10.767 14.363 24.205 28.700 37.308 7 10.764 14.362 24.198 28.696 37.299 9 10.763 14.362 24.197 28.695 37.298 17 3 10.933 14.431 24.585 28.916 37.976 5 10.767 14.363 24.202 28.699 37.308 7 10.763 14.362 24.196 28.694 37.297 9 10.762 14.361 24.195 28.693 37.295 FSDT [3] 10.271 13.622 23.107 -- -- 2D [32] 10.507 13.773 23.691 27.680 36.752 TABLE 3: The lowest six frequencies for circular beams with different boundary conditions, geometrical properties, and modeling methods (Hz, [R.sub.m] =1 m, h/[R.sub.m] = 0.3). BC Mode [DELTA][theta] = [pi]/4 Pre. FEM Diff. (%) F-F 1 1861.2 1861.3 0.003 2 3304.8 3304.8 0.001 3 3653.6 3653.9 0.007 4 5514.5 5515.4 0.016 5 5873.7 5874.5 0.014 6 6166.1 6166.4 0.005 F-C 1 373.90 373.99 0.024 2 1420.4 1420.6 0.014 3 1870.6 1870.8 0.009 4 3475.3 3475.8 0.013 5 4773.6 4774.1 0.010 6 5133.8 5134.0 0.005 C-C 1 1683.9 1684.5 0.033 2 2923.8 2924.6 0.028 3 3461.9 3462.8 0.027 4 4991.2 4992.3 0.022 5 6282.3 6283.3 0.017 6 6414.4 6414.6 0.003 BC Mode [DELTA][theta] = [pi]/2 [DELTA][theta] = 3[pi]/4 Pre. FEM Diff. (%) Pre. FEM Diff. (%) F-F 1 545.85 545.85 0.001 239.19 239.19 0.000 2 1333.9 1333.9 0.002 638.87 638.87 0.001 3 1800.9 1800.9 0.001 1177.4 1177.4 0.001 4 2285.0 2285.1 0.003 1355.5 1355.5 0.003 5 3207.8 3207.9 0.004 1790.9 1790.9 0.002 6 3344.4 3344.5 0.002 2289.5 2289.5 0.000 F-C 1 104.22 104.23 0.008 50.018 50.017 0.001 2 431.29 431.32 0.008 182.16 182.15 0.003 3 974.88 974.91 0.003 535.24 535.24 0.001 4 1368.0 1368.1 0.005 898.42 898.43 0.001 5 2131.7 2131.9 0.007 1173.9 1173.9 0.002 6 2556.9 2556.9 0.001 1692.3 1692.3 0.002 C-C 1 850.21 850.24 0.004 512.94 512.93 0.002 2 1079.5 1079.7 0.015 665.65 665.66 0.001 3 1805.4 1805.5 0.007 1159.1 1159.1 0.002 4 2056.6 2056.9 0.012 1227.9 1227.9 0.002 5 2984.2 2984.5 0.009 1732.5 1732.5 0.002 6 3280.8 3281.0 0.005 2148.3 2148.3 0.001 TABLE 4: The lowest six frequency parameters [OMEGA] of classically restrained circular beams with various thickness ratios ([R.sub.m] = 1 m, [DELTA][theta] = [pi]/2). h/[R.sub.m] Mode Boundary condition F-F F-S1 F-S2 F-C 0.05 1 8.3653 5.2974 1.8360 1.4976 2 23.694 18.869 11.065 7.1854 3 46.884 40.247 28.619 22.477 4 77.238 68.895 53.693 45.339 5 114.23 104.30 85.785 74.297 6 154.82 145.91 97.943 94.711 0.1 1 8.2897 5.2643 1.8326 1.4942 2 23.048 18.453 10.919 7.0603 3 44.492 38.450 27.682 21.456 4 71.197 63.988 48.914 40.384 5 77.261 77.261 50.605 49.486 6 102.04 93.921 78.471 70.567 0.2 1 8.0147 5.1403 1.8195 1.4802 2 20.994 17.077 10.404 6.6156 3 37.663 33.367 24.305 17.818 4 38.678 38.337 24.870 24.700 5 57.082 52.145 42.532 36.793 6 70.432 70.314 54.084 53.013 h/[R.sub.m] Mode Boundary condition S1-S1 S1-S2 S1-C 0.05 1 2.6801 7.5652 4.3680 2 14.464 23.307 17.702 3 34.040 46.609 38.807 4 60.960 77.028 66.582 5 94.751 97.943 92.155 6 134.88 114.06 106.77 0.1 1 2.6707 7.4945 4.3126 2 14.213 22.672 17.053 3 32.728 44.232 35.608 4 57.039 48.917 47.535 5 77.259 71.006 63.829 6 86.002 101.899 92.113 0.2 1 2.6343 7.2362 4.1083 2 13.354 20.658 14.846 3 28.877 24.349 23.209 4 38.320 37.789 32.799 5 47.221 54.074 49.629 6 67.144 57.039 54.498 h/[R.sub.m] Mode Boundary condition S2-S2 S2-C C-C 0.05 1 2.6800 9.5587 22.270 2 14.464 26.891 39.349 3 34.039 50.676 66.753 4 60.959 66.953 74.701 5 69.297 86.221 114.39 6 94.749 123.04 143.50 0.1 1 2.6706 9.2748 21.246 2 14.213 24.596 27.624 3 32.727 34.057 46.328 4 34.672 50.034 64.343 5 57.038 73.445 79.316 6 77.257 80.161 98.509 0.2 1 2.6343 8.2708 16.029 2 13.354 16.094 18.192 3 17.384 24.260 35.875 4 28.876 36.821 36.990 5 38.319 42.107 53.898 6 47.220 59.788 68.390 TABLE 5: The lowest six frequency parameters [OMEGA] of classically restrained circular beams with various span angles ([R.sub.m] = 1 m, h/[R.sub.m] = 0.1). [DELTA][theta] Mode Boundary condition F-F F-S1 F-S2 F-C [pi]/3 1 19.088 12.761 4.4804 3.2579 2 50.820 41.432 25.315 17.105 3 94.059 82.289 60.572 46.643 4 109.23 109.21 62.328 62.105 5 144.99 131.64 106.10 92.330 6 200.96 186.60 158.38 140.84 2[pi]/3 1 4.4805 2.6710 0.9739 0.8749 2 12.761 10.031 5.7637 3.6195 3 25.315 21.674 15.360 11.728 4 41.432 36.989 28.906 23.886 5 60.573 55.456 43.262 37.328 6 62.330 62.315 45.861 44.506 [pi] 1 1.8327 0.9507 0.4372 0.4349 2 5.2643 3.9432 2.1308 1.3692 3 10.919 9.1645 6.3020 4.6433 4 18.453 16.288 12.501 10.245 5 27.683 25.134 20.505 17.608 6 38.450 35.548 30.155 26.470 [DELTA][theta] Mode Boundary condition S1-S1 S1-S2 S1-C [pi]/3 1 7.4946 1.0378 11.416 2 32.728 18.234 38.524 3 71.007 50.481 60.019 4 109.21 62.312 81.544 5 118.59 93.844 128.08 6 172.44 144.88 158.76 2[pi]/3 1 1.0378 0.2623 1.9658 2 7.4946 3.6888 9.0668 3 18.234 12.365 20.381 4 32.728 25.042 34.259 5 50.482 41.228 42.524 6 62.313 43.263 55.402 [pi] 1 2.6707 1.0378 3.3476 2 7.4946 4.8346 8.4943 3 14.213 10.627 15.497 4 22.673 18.234 24.055 5 32.728 27.510 33.194 6 44.232 38.308 38.551 [DELTA][theta] Mode Boundary condition S2-S2 S2-C C-C [pi]/3 1 7.4943 20.441 34.046 2 32.727 34.047 47.270 3 34.672 59.680 90.478 4 71.005 101.17 106.39 5 109.20 110.73 139.31 6 118.58 154.51 189.04 2[pi]/3 1 1.0378 4.7592 11.419 2 7.4945 14.121 20.445 3 18.234 26.580 34.049 4 32.727 34.065 38.533 5 34.672 45.344 59.694 6 50.481 59.944 60.023 [pi] 1 2.6706 1.5851 4.3142 2 7.4945 5.6604 9.2781 3 14.213 11.769 17.059 4 22.672 19.579 24.603 5 32.728 28.529 34.059 6 34.673 34.072 35.618 TABLE 6: Frequency parameters [OMEGA] for elastically supported circular beams with different thickness ratios and restraint rigidities ([R.sub.m] = 1 m, [[theta].sub.0] = [pi]/2). ku = [eta], kw = 0 h/[R.sub.m] [eta]/D 1 2 3 0.01 [10.sup.0] 1.5013 7.2263 22.817 [10.sup.1] 1.5249 7.2266 22.817 [10.sup.2] 1.7442 7.2296 22.818 [10.sup.3] 3.1816 7.2654 22.825 [10.sup.4] 6.8030 9.5884 22.907 [10.sup.5] 7.0247 21.852 30.250 0.05 [10.sup.0] 1.5109 7.1856 22.477 [10.sup.1] 1.6253 7.1873 22.477 [10.sup.2] 2.4911 7.2050 22.483 [10.sup.3] 6.0299 7.7159 22.538 [10.sup.4] 7.0667 18.929 23.904 [10.sup.5] 7.7839 22.923 44.333 0.10 [10.sup.0] 1.5207 7.0608 21.456 [10.sup.1] 1.7408 7.0650 21.458 [10.sup.2] 3.1733 7.1131 21.481 [10.sup.3] 6.6891 9.3939 21.720 [10.sup.4] 7.4853 19.902 25.792 [10.sup.5] 8.7622 23.296 32.768 0.15 [10.sup.0] 1.5282 6.8657 19.865 [10.sup.1] 1.8473 6.8741 19.872 [10.sup.2] 3.7026 6.9755 19.942 [10.sup.3] 6.7367 10.615 20.652 [10.sup.4] 7.8630 17.816 24.479 [10.sup.5] 8.6753 19.710 26.673 ku = 0, kw = [eta] h/[R.sub.m] [eta]/D 1 2 3 0.01 [10.sup.0] 1.5040 7.2283 22.818 [10.sup.1] 1.5515 7.2464 22.823 [10.sup.2] 1.9406 7.4289 22.877 [10.sup.3] 3.3673 9.2160 23.442 [10.sup.4] 4.2514 15.578 29.613 [10.sup.5] 4.3720 17.693 38.640 0.05 [10.sup.0] 1.5243 7.1955 22.480 [10.sup.1] 1.7397 7.2863 22.506 [10.sup.2] 2.8562 8.2070 22.780 [10.sup.3] 4.1096 13.588 25.951 [10.sup.4] 4.3410 17.262 36.620 [10.sup.5] 4.3653 17.659 38.610 0.10 [10.sup.0] 1.5468 7.0804 21.461 [10.sup.1] 1.9337 7.2624 21.514 [10.sup.2] 3.3372 9.0383 22.066 [10.sup.3] 4.1854 15.042 27.958 [10.sup.4] 4.2996 16.861 34.871 [10.sup.5] 4.3113 17.034 35.536 0.15 [10.sup.0] 1.5660 6.8947 19.872 [10.sup.1] 2.0917 7.1678 19.946 [10.sup.2] 3.5429 9.6553 20.751 [10.sup.3] 4.1440 14.969 26.889 [10.sup.4] 4.2159 15.961 28.964 [10.sup.5] 4.2233 16.055 29.097 ku = kw = [eta] h/[R.sub.m] [eta]/D 1 2 3 0.01 [10.sup.0] 1.5066 7.2283 22.818 [10.sup.1] 1.5770 7.2467 22.823 [10.sup.2] 2.1392 7.4312 22.878 [10.sup.3] 4.4370 9.2166 23.451 [10.sup.4] 9.5523 15.677 29.817 [10.sup.5] 17.584 25.942 40.813 0.05 [10.sup.0] 1.5374 7.1957 22.480 [10.sup.1] 1.8517 7.2878 22.507 [10.sup.2] 3.5131 8.2091 22.786 [10.sup.3] 7.4008 13.649 26.053 [10.sup.4] 16.954 18.929 37.704 [10.sup.5] 18.203 33.998 52.083 0.10 [10.sup.0] 1.5725 7.0808 21.462 [10.sup.1] 2.1335 7.2658 21.516 [10.sup.2] 4.4148 9.0427 22.094 [10.sup.3] 9.3746 15.112 28.313 [10.sup.4] 17.240 20.798 37.990 [10.sup.5] 19.364 26.704 43.372 0.15 [10.sup.0] 1.6040 6.8956 19.873 [10.sup.1] 2.3677 7.1741 19.954 [10.sup.2] 4.9852 9.6663 20.838 [10.sup.3] 10.523 15.059 28.171 [10.sup.4] 17.085 18.064 36.130 [10.sup.5] 19.121 19.856 39.345 TABLE 7: Frequency parameters [OMEGA] for elastically supported circular beams with different span angles and restraint rigidities ([R.sub.m] =1 m, h/[R.sub.m] = 0.1). [[theta].sub.0] [eta]/D ku = [eta], kw = 0 1 2 3 [pi]/3 [10.sup.0] 3.2671 17.105 46.643 [10.sup.1] 3.3484 17.110 46.647 [10.sup.2] 4.0665 17.154 46.688 [10.sup.3] 8.0451 17.639 47.081 [10.sup.4] 15.321 23.449 49.869 [10.sup.5] 18.965 31.838 55.957 2[pi]/3 [10.sup.0] 0.9258 3.6197 11.729 [10.sup.1] 1.2960 3.6210 11.731 [10.sup.2] 3.1142 3.6536 11.753 [10.sup.3] 3.6599 9.1322 12.165 [10.sup.4] 3.9841 12.050 23.150 [10.sup.5] 4.5641 13.485 25.680 [pi] [10.sup.0] 0.5264 1.3719 4.6438 [10.sup.1] 1.0030 1.4177 4.6480 [10.sup.2] 1.3161 3.0900 4.7182 [10.sup.3] 1.3396 4.5807 9.8024 [10.sup.4] 1.4308 4.9059 10.622 [10.sup.5] 1.5524 5.4708 11.429 [[theta].sub.0] [eta]/D ku = 0, kw = [eta] 1 2 3 [pi]/3 [10.sup.0] 3.3052 17.116 46.646 [10.sup.1] 3.6978 17.217 46.674 [10.sup.2] 6.0927 18.272 46.962 [10.sup.3] 10.265 27.077 50.196 [10.sup.4] 11.295 37.199 59.270 [10.sup.5] 11.403 38.393 59.956 2[pi]/3 [10.sup.0] 0.9197 3.6507 11.737 [10.sup.1] 1.2006 3.9285 11.815 [10.sup.2] 1.7696 5.9362 12.680 [10.sup.3] 1.9434 8.6156 18.070 [10.sup.4] 1.9635 9.0227 20.175 [10.sup.5] 1.9656 9.0622 20.360 [pi] [10.sup.0] 0.4487 1.4211 4.6595 [10.sup.1] 0.5063 1.8014 4.8097 [10.sup.2] 0.5555 2.9075 6.2241 [10.sup.3] 0.5642 3.3007 8.2147 [10.sup.4] 0.5651 3.3429 8.4675 [10.sup.5] 0.5652 3.3471 8.4916 [[theta].sub.0] [eta]/D ku = kw = [eta] 1 2 3 [pi]/3 [10.sup.0] 3.3143 17.117 46.646 [10.sup.1] 3.7784 17.222 46.679 [10.sup.2] 6.5964 18.303 47.008 [10.sup.3] 13.084 27.089 50.748 [10.sup.4] 23.265 37.875 68.723 [10.sup.5] 31.839 42.697 83.774 2[pi]/3 [10.sup.0] 0.9683 3.6508 11.737 [10.sup.1] 1.5359 3.9290 11.818 [10.sup.2] 3.3494 5.9983 12.716 [10.sup.3] 7.8507 9.3882 18.576 [10.sup.4] 9.2396 17.386 26.000 [10.sup.5] 10.492 19.461 32.929 [pi] [10.sup.0] 0.5367 1.4241 4.6599 [10.sup.1] 1.0033 1.8520 4.8159 [10.sup.2] 2.3074 3.3600 6.3712 [10.sup.3] 3.1616 6.9331 9.9866 [10.sup.4] 3.4291 8.1148 15.259 [10.sup.5] 4.0046 8.7902 16.261

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Title Annotation: | Research Article |
---|---|

Author: | Wang, Xueren; Miao, Xuhong; Jia, Di; Gao, Shengyao |

Publication: | Shock and Vibration |

Article Type: | Report |

Date: | Jan 1, 2017 |

Words: | 10219 |

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