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.
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 Figure 1 is considered, in which , , and represent the width, thickness, and inner radius of the beam. The beam is bounded along its edges by the boundaries and . The bottom surface of the beam is selected as the reference surface with the three orthogonal curvilinear coordinates , , and ; see Figure 1. In this paper, the beams are assumed to be isotropic and homogeneous and to vibrate freely in the plane. , , and denote the three displacement components in the axial, lateral, and normal directions, respectively.
(a) Coordinate system
(b) Sampling surface distribution
2.2. Plane Stress Assumption
As mentioned previously, the beam under consideration vibrates freely in the 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]where , , , and stand for the normal strains, and , , and 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:where , , and stand for the normal stresses; , , and are the shear component. is the material stiffness matrix.in which means Young’s module of the material andν represents Poisson’s ratio.
Furthermore, (2) can be rearranged in the form of matrix as
Therefore, the final stress-strain relations for the beam under plane stress hypothesis can be obtained aswhere .
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, 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. 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:Therefore, the basic variables in the axial and normal directions of an arbitrary sampling surface can be given bywhere and stand for the axial and transverse displacement components, respectively. stand for the time variable, is the circular frequency. As a consequence, displacement field of the beam under vibration can be calculated byand is Lagrange’s interpolation of degree :According to (1), strains on the th sampling surface can be found aswhere are determined by
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).
The energy functional for the curved beam under the circumstance of free vibration iswhere and denote the strain and kinetic energy function defined as follows:where stands for the material density. Substituting (5) and (12) into (14), the two energy functions can be further written as
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 arewhere and are the expansion coefficients; . 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]:where and denote the boundary values. and represent the penalty parameters. and are the parameters which define different restraint conditions. The boundary potential is introduced by means of Lagrange multiplier technique while the boundary potential 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 , 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 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
Finally, let the variation of the with respect to each coefficient ( and ) equal zero; the governing equations can be derived in a matrix form as where K and M stand for the final stiffness and mass matrices of order . 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 . 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, represents a circular beam with clamped and free restraints at the ends and , respectively. To unify the discussion, the dimensionless variable of frequency is introduced in the calculation (where /2). The beams are supposed to be made of steel ( GPa, , and kg/m3).
3.1. Validation
Table 2 gives the first five nondimensional frequency parameters 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 m, . For completeness, two thickness-to-radius ratios (i.e., = 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 , , and boundary conditions obtained by the current approach with those based on commercial FEM code. The results are calculated with the beam parameters = 1 m, = 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.
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 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
Figures 2–4 show the relative deviations of the 1st, 3rd, and 5th frequency parameters Ω for circular beams with different ratios of thickness-to-span length . The beam is supposed to be of unit span length; that is, . The ratio of is varied between 0.01 and 0.2. , , and boundaries are considered in the study. The results obtained by the current method of “” 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 = 1 m. “” 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 Ω of a circular beam based on the CBT and FSDT theory models, respectively. The thickness-to-span length ratio, , is varied from 0.01 to 0.2, corresponding to thin to thick beam configurations. Two boundary conditions, that is, and , are considered in the studies. The beam is supposed to be of unit span length and unit radius, that is, , 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 and 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 , where the classical boundary conditions are included. “” 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 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 Table 5 with a variety of classical restraints and span angles. The thickness ratio () of the beams is assumed to be constant as = 0.1, 20. Meanwhile, the span angle is taken as , 2π/3, andπ, respectively. “” displacement field is adopted in this study. First of all, it is seen that the circular beams with - 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 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 () are supposed to be clamped at one end () and elastically supported at the other end with stiffness rigidity (where , ). “” 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 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 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: , = 0.2. The beam is clamped at the end of and elastically restrained at the other end. The following three types of elastic supports are considered in the study: axially elastic restraint , transversely elastic restraint (), and elastic restraint in both directions (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 are the same as Figure 10.
Finally, Table 6 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 m, . “” 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 m, h/. 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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.