Abstract
Variable kinematic beam theories are used in this paper to carry out vibration analysis of isotropic thin-walled structures subjected to non-structural localized inertia. Arbitrarily enriched displacement fields for beams are hierarchically obtained by using the Carrera Unified Formulation (CUF). According to CUF, kinematic fields can be formulated either as truncated Taylor-like expansion series of the generalized unknowns or by using only pure translational variables by locally discretizing the beam cross-section through Lagrange polynomials. The resulting theories were, respectively, referred to as TE (Taylor Expansion) and LE (Lagrange Expansion) in recent works. If the finite element method is used, as in the case of the present work, stiffness and mass elemental matrices for both TE and LE beam models can be written in terms of the same fundamental nuclei. The fundamental nucleus of the mass matrix is opportunely modified in this paper in order to account for non-structural localized masses. Several beams are analysed and the results are compared to those from classical beam theories, 2D plate/shell, and 3D solid models from a commercial FEM code. The analyses demonstrate the ineffectiveness of classical theories in dealing with torsional, coupling, and local effects that may occur when localized inertia is considered. Thus the adoption of higher-order beam models is mandatory. The results highlight the efficiency of the proposed models and, in particular, the enhanced capabilities of LE modelling approach, which is able to reproduce solid-like analysis with very low computational costs.
1. Introduction to Refined Beam Theories
In engineering practice, problems involving non-structural masses are of special interest [1]. An important example is that of aerospace engineering. In aerospace design, in fact, non-structural masses are commonly used in finite element (FE) models to incorporate the weight of the engines, fuel, and payload, see, for example, [2–5]. In this paper, the effects due to localized inertia on free vibration of thin-walled beams are investigated through one-dimensional (1D) higher-order models. A brief overview about the evolution of refined beam theories is given below.
A number of refined beam theories have been proposed over the years to overcome the limitation of classical beam models such as those by Euler [6] (hereinafter referred to as EBBM) and Timoshenko [7, 8] (hereinafter referred to as TBM). If the rectangular cartesian coordinate system shown in Figure 1 is adopted and we consider bending on the -plane, the kinematic field of EBBM can be written as follows: where and are the displacement components of a point belonging to the beam domain along and coordinates, respectively. and are the displacements of the beam axis, whereas is the rotation of the cross-section about the -axis (i.e., ) as shown in Figure 2(a). According to EBBM, the deformed cross-section remains plane and orthogonal to the beam axis.
(a)
(b)
EBBM neglects cross-sectional shear deformation phenomena. Generally, the shear stresses play an important role in several problems (e.g., short beams and composite structures) and neglecting these terms can lead to incorrect results. One may want to generalize (1) and overcome the EBBM assumption of the orthogonality of the cross-section. The improved displacement field results in the TBM are as follows: TBM constitutes an improvement over EBBM since the cross-section does not necessarily remain perpendicular to the beam axis after deformation and one degree of freedom (i.e., the unknown rotation ) is added to the original displacement field (see Figure 2(b)).
Classical beam models yield reasonably good results when slender, solid section, and homogeneous structures are subjected to bending. Conversely, the analysis of deep, thin-walled, and open section beams may require more sophisticated theories to achieve sufficiently accurate results, see [9]. One of the main problems of TBM is that the homogeneous conditions of the transverse stress components at the top/bottom surfaces of the beam are not fulfilled, as shown in Figure 3. One can impose, for instance, (2) in order to have null transverse strain component () at . This leads to the third-order displacement field known as the Reddy-Valsov beam theory [10] as follows: where and are cubic functions of the coordinate. It should be noted that although the model of (3) has the same number of degrees of freedom (DOFs) of TBM, it overcomes classical beam theory limitations by foreseeing a quadratic distribution of transverse stresses on the cross-section of the beam.
The above theories are not able to include any kinematics resulting from the application of torsional moments. The simplest way to include torsion consists of considering a rigid rotation of the cross-section around the -axis (i.e., ), see Figure 4. The resulting displacement model is where is the displacement component along the -axis. According to (4), a linear distribution of transverse displacement components is needed to detect the rigid rotation of the cross-section about the beam axis.
Beam models that include all the capabilities discussed so far can be obtained by summing all these contributions. By considering the deformations also in the -plane, one has where , , , and are cubic functions. In the case of rectangular cross-section, the cubic functions from Vlasov’s theory are , , , and , where and are the dimensions of the cross-section along the - and -axis, respectively. The beam models discussed so far are not able to account for many higher-order effects, such as the second-order in-plane deformations of the cross-section.
Over the last century, many refined beam theories have been proposed to overcome the limitations of classical beam modelling. A commendable and comprehensive review on beam theories can be found in [11, 12]. Different approaches have been used to improve the beam models, which include the use of warping functions based on de Saint-Venant's solution [13–16], the variational asymptotic solution (VAM) [17–22], and the generalized beam theory (GBT) [23–28]. A displacement field which is able to take into account the cross-section deformation by means of warping functions is where and are the warping functions, whereas and are the transverse shear strains measured on the beam axis. As a general guideline, one can state that the richer the kinematic field is, the more accurate the 1D model becomes [29]. The main disadvantages of a richer displacement field are the increase of equations to be solved and the choice of the terms to be added since this choice is generally problem dependent.
CUF represents a tool to tackle the problem of the choice of the expansion terms. Let be the transposed displacement vector. According to CUF, the generic displacement field can be expressed in a compact manner as an -order expansion in terms of generic functions, , as follows: where are the functions of the coordinates and on the cross-section. is the vector of the generalized displacements, stands for the number of terms used in the expansion. In line with (7), it is clear that (1) to (5) consist of MacLaurin expansions that uses 2D polynomials as base functions, where and are positive integers. This class of models is referred to as TE (Taylor-Expansion). It should be noted that (1), (2), and (4) are particular cases of the linear () TE model, which can be expressed as where the parameters on the right-hand sides (, , , , etc.) are the displacements and the rotations of the beam reference axis. Higher-order terms can be taken into account according to (7). For instance, the displacement fields of (3) and (5) can be seen as particular cases of the third-order () TE model as follows: The possibility of dealing with any-order expansion makes the TE CUF able to handle arbitrary geometries, thin-walled structures, and local effects as it has been shown for static [30, 31], linearized stability analyses [32, 33], and free-vibrations of both metallic [34–36] and composite structures [37–39].
Recently, a new class of CUF models has been developed by the first author and his coworkers [40, 41]. In this class of models, Lagrange-like polynomials are used to discretize the displacement field on the cross-section. These models are referred to as LE (Lagrange Expansion) and they have been used to develop a component-wise (CW) modelling approach in some recent works. Static analyses on isotropic [40] and composite structures [42, 43] have revealed the strength of LE models in dealing with open cross-sections, localized boundary conditions, and layer-wise descriptions of composite structures. Moreover, static and dynamic analyses of reinforced-shell wing structures [44, 45] as well as civil engineering framed constructions [46, 47] by 1D CW models have been carried out and the results have shown the enhanced capabilities of LE in obtaining 3D accuracy with very low computational costs. Furthermore, the LE models have shown their enhanced capabilities in dealing with load factors and localized inertia in static analysis of thin-walled beams in a companion paper [48].
In the present paper, TE and LE models are extended to include localized inertia in the free vibration analysis of homogeneous isotropic thin-walled beams. In the next section, LE models are formulated by using CUF. 1D refined elements including localized inertia are then obtained by classical finite element method (FEM). Numerical results are subsequently discussed. Finally, the main conclusions are outlined.
2. Higher-Order Models Based on Lagrange Polynomial Expansions
The degrees of freedom of the TE models (displacements and -order derivatives of displacements) described above are defined along the axis of the beam. The unknown variables are only pure displacements if Lagrange polynomials are adopted as expansion functions () in (7). This class of CUF models, which is referred to as LE, was recently introduced in [40], where triangular three-(L3) and six-node (L6) elements as well as quadrilateral four-(L4), nine-(L9), and 16-node (L16) elements were used to discretize the displacement unknowns on the cross-section of beam structures. In this paper, L9 elements are used. In the case of an L9 element, the cross-sectional interpolation functions are given by where and vary from to , whereas and are the coordinates of the nine points whose locations in the natural coordinate frame are shown in Figure 5. The displacement field given by an L9 element is therefore where are the displacement variables of the problem and represent the translational displacement components of each of the nine points of the L9 element. The choice of using Lagrange-type polynomials to discretize the cross-section displacement field leads to the following advantages.(i)Each of the problem unknowns (e.g., ) has a precise physical meaning; that is, they are pure translational displacements. This is not true in the case of TE models.(ii)The beam model can be refined in a smart way; that is, unknown variables can be arbitrarily placed in subdomains over the cross-section area (e.g., close to loadings). This is realized by discretizing the beam cross-section with a number of L-elements as shown in Figure 6.
3. Finite Element Formulation
3.1. Preliminaries
Referring to the coordinate frame shown in Figure 1, let the cross-section of the structure be denoted by and let the beam boundaries over be . The stress, , and strain, , components are grouped as follows: In the case of small displacements with respect to a characteristic dimension of , linear strain-displacement relations can be used as follows: where is the following linear differential operator: Constitutive laws were exploited to obtain stress components as follows: In the case of isotropic material, the matrix is where and are the Lamé’s parameters. If Poisson’s ratio and Young modulus are used, one has and .
3.2. Fundamental Nuclei
The FE approach was adopted to discretize the structure along the -axis. This process is conducted via a classical finite element technique, where the displacement vector is given by stands for the shape functions and for the nodal displacement vector, For the sake of brevity, the shape functions are not reported here. They can be found in many books on finite elements, for instance, in [49]. Elements with four nodes (B4) were adopted in this work; that is, a cubic approximation along the axis was assumed. The choice of the cross-section discretization for the LE class (i.e., the choice of the type, the number, and the distribution of cross-section elements) or the theory order, , for the TE class is completely independent of the choice of the beam finite element to be used along the axis of the beam.
The stiffness and mass matrices of the elements were obtained via the principle of virtual displacements as follows: where stands for the strain energy, is the work of the inertial loadings, and stands for the virtual variation. The virtual variation of the strain energy was rewritten using (13), (15), and (17) as follows: where is the stiffness matrix in the form of the fundamental nucleus. Its components are provided below and they are referred to as , where is the row number () and is the column number (). The fundamental nucleus has to be expanded according to the summation indexes and in order to obtain the elemental stiffness matrix.
The virtual variation of the work of the inertial loadings is where stands for the density if the material, and is the acceleration vector. Equation (22) is rewritten using (17) as follows: where is the fundamental nucleus of the mass matrix. Its components are provided below and they are referred to as , where is the row number () and denotes column number (). Consider the following: It should be noted that no assumptions on the approximation order or on the choice of functions (TE or LE) have been made in formulating and . It is therefore possible to obtain refined beam models without changing the formal expression of the nuclei components. This is the key-points of CUF which allows, with only nine coding statements, the implementation of any-order of multiple class theories.
In the present paper, the effect due to non-structural masses is also investigated. Localized inertia can in principle be arbitrarily placed into the 3D domain of the beam structure. In the framework of the CUF, this is easily realized by adding the following term to the fundamental nucleus of the mass matrix: where is the identity matrix and is the value of the non-structural mass, which is applied at point .
The undamped dynamic problem can be derived by substituting (20) and (23) into (19). After global mass and stiffness FE matrices are assembled, one has Introducing harmonic solutions, it is possible to compute the natural frequencies by solving an eigenvalues problem as follows: where is the th eigenvector. The eigenvalue problem of (27) is solved by using ARPACK libraries [50], which are based on an algorithmic variant of the Arnoldi process [51] called Implicitly Restarted Arnoldi Method (IRAM) [52].
4. Numerical Results
This section investigates the efficiency of the present approach applied to modal analysis of beam structures with localized inertia. The attention is focused on the capability of the approach to deal with refined solutions with low computational costs. The results were compared with classical beam theories and FE models from the commercial code MSC . Common C- and I-shaped beams as well as hollow-rectangular boxes were considered. The adopted material was an aluminium alloy and it had the following characteristics: Young modulus equal to GPa, Poisson’s ratio equal to , and density Kg/m3. For all the problems addressed, ten four-node (B4) cubic 1D Lagrangian finite elements were used along the beam axis for both TE and LE models.
4.1. C-Section Beam
The analysis of a cantilever C-shaped beam was carried out as the first assessment. The cross-section of the structure is shown in Figure 7. The geometrical data were as follows: m, , and m. The length of the beam, , was equal to m.
Table 1 shows the first ten natural frequencies for each implemented model. Columns 2 and 3 refer to the classical models EBBM and TBM. The natural frequencies by fourth-order (), sixth-order (), and eighth-order () refined TE beam model are given in columns 4 to 6. Two different distributions of L9 elements above the cross-section were considered (L9 and L9) in the case of LE modelling approach, as shown in Figure 8, and the results are given in the 7th and 8th columns of Table 1. In columns 9 and 10, MSC shell and solid models are given for comparison purposes, and they are,respectively, referred to as FEM-2D and FEM-3D in the table. FEM-2D model was obtained by using 4-node QUAD elements, whereas FEM-3D model was obtained by using 8-node CHEXA elements. For each model, the number of DOFs is given in the last row of Table 1. The following comments can be made:(1)As expected, classical models are not able to provide torsional and coupled modes.(2)A higher than fourth-order TE model is necessary to correctly detect torsional modes.(3)As indicated in Table 1, where superscripts are used to denote the kind of each vibrational mode, bending, torsional, and coupled bending/torsional modes are detected by LE models in accordance with MSC 2D and 3D models. An appropriate distribution of the L9 elements above the cross-section is effective in improving the accuracy of the solution.(4)The 10 L9 LE model provides a solid-like solution with a high reduction of the number of the DOFs.
(a) 6 L9
(b) 10 L9
A second beam configuration was considered and a non-structural mass was added on the C-shaped beam at the coordinates (0, , ), as Figure 9 shows. The weight of the non-structural mass was equal to Kg. For this load case, Table 2 reports the main vibrational frequencies by different models for clamped-free boundary conditions. The number of the DOFs for each model implemented is also given in the last row of the table. Figures 10 and 11 show the 8th mode shape by both LE and FEM-3D models, and the effects due to the localized non-structural mass are highlighted. Table 3 shows the natural frequencies of the C-shaped beam undergoing clamped-clamped boundary conditions. For this case, the same localized inertia as above was placed at as shown in Figure 9. The following comments arise.(1)As expected, the application of the non-structural mass is responsible for bending/torsional mode couplings.(2)Classical models and lower-order TE models are not effective for the problem under consideration.(3)The proposed LE model is able to detect the MSC shell and solid results.(4)The number of the DOFs is very low in the case of CUF models if compared both to the shell and solid models.
| (a) L9 LE model (29.861 Hz) |
(b) FEM-3D model (29.756 Hz)
| (a) L9 LE model (27.169 Hz) |
(b) FEM-3D model (26.716 Hz)
4.2. I-Shaped Beam
A cantilever I-shaped beam was considered as the second example. The cross-section geometry is shown in Figure 12. The dimensions and were, respectively, equal to m and m, whereas the thickness was m. The length of the structure, , was m.
The first ten natural frequencies, calculated according to different models, are given in Table 4. Classical and refined TE as well as LE models are considered. Two different LE models were implemented and they differed in the cross-section L9 elements discretization, as shown in Figure 13. In columns 8 to 10, MSC beam, shell, and solid models are given for comparison purposes. MSC 1D model (hereinafter referred to as FEM-1D) was obtained with two-node CBAR elements, whereas shell and solid elements were constructed using the same elements as in the previous analysis case, and they are referred to as FEM-2D and FEM-3D, respectively.
| (a) L9 |
| (b) L9 |
The consistent correspondence between the 12 L9 LE model and the solid model was further investigated by means of the Modal Assurance Criterion (MAC), whose graphic representation is shown in Figure 15(a). The MAC is defined as a scalar representing the degree of consistency between two distinct modal vectors (see [53]) as follows: where is the th eigenvector of model A, whereas is the th eigenvector of model B. The modal assurance criterion takes on values from zero (representing no consistent correspondence) to one (representing a consistent correspondence). The following remarks can be made:(1)Unlike classical beam theories, higher-order TE and LE models are able to foresee torsional and coupled modes.(2)Unlikely CUF 1D models, FEM-1D model is not able to produce the solid-like solution.(3)The cross-section discretization refinement for LE models is an effective method that leads to the 3D solution.(4)Bending, torsional, and coupled modes are detected by LE models in accordance with FEM-2D and FEM-3D models with a high reduction of the number of the DOFs.(5)The MAC analysis underlines the perfect correspondence between the LE model and the solid one for the considered load case.
In the second load case, a non-structural mass was added at the tip cross-section of the I-shaped beam, as shown in Figure 14. The weight of the localized inertia was equal to Kg and it was applied at (, , ). Table 5 gives the first 10 vibrational frequencies for classical, TE, LE, and MSC models. The application of the non-structural mass produces an increase in the number of the coupled modes, as expected. The MAC matrix between the 12 L9 model and the FEM-3D model for the load case under consideration is also calculated and it is represented in Figure 15(b). It should be underlined that(1)Classical models are ineffective in detecting the effects due to non-structural masses that are not placed in correspondence of the shear center;(2)The increase in the order for TE models provides greater accuracy on the evaluation of the coupled frequencies;(3)Even though local inertia is considered, LE models can produce solid-like solution, as shown by the MAC matrix. In fact, a very good agreement was found for the FEM-3D and LE models. LE analyses are therefore clearly able to detect 3D-elasticity solutions. However, some mixings in modes 3 and 4 are evident.
(a) Without localized inertia
(b) With localized inertia
4.3. Hollow-Rectangular Beam
The cantilever hollow-square cross-section beam shown in Figure 16 was investigated as the last example. The geometrical data were as follows: m, m, and m. The length of the beam, , was equal to 3.2 m. Table 6 reports the first 10 natural frequencies, together with the number of DOFs for each model implemented. Columns 2 and 3 show the results from the classical beam theories. The TE models from the fourth-order to the eighth-order Taylor-like expansion are considered in columns 4 to 6. The LE models are reported in columns 7 and 8. The LE models were obtained by adopting two different discretizations above the cross-section of the beam, as Figure 17 shows. In Figure 17, a different notation is adopted with respect to Figures 8 and 13. The nodes of the L-elements are, in fact, not depicted in Figure 17, for the sake of clearness. In columns 9 and 10 of Table 6, MSC shell and solid models are given for comparison purposes and they are referred to as FEM-2D and FEM-3D, respectively. For this analysis case, the mesh discretizations for FEM-2D and FEM-3D models are shown in Figure 18. Figure 19 shows a comparison between some selected modal shapes by the L9 LE and FEM-3D models.
| (a) L9 |
| (b) L9 |
(a) FEM-2D
(b) FEM-3D
| (a) Mode 2, L9 LE model (57.023 Hz) |
(b) Mode 2, FEM-3D model (54.551 Hz)
| (c) Mode 10, L9 LE model (109.87 Hz) |
(d) Mode 10, FEM-3D model (101.58 Hz)
The vibrations of the box beam were further analyzed and a non-structural mass with a weight of Kg was added at the coordinates (, , −/2), as shown in Figure 20(a) (Case A). A comparison between LE and MSC models is given in Table 7. The correspondence between LE L9 and FEM-3D models is underlined through MAC analyses, which are shown in Figure 21. MAC matrices for both the cases without and with localized inertia (Case A) are provided in Figure 21. Some modal shapes by LE and FEM-3D models for the load case considered are shown in Figure 22.
(a) Case A
(b) Case B
(a) Without localized inertia
(b) With localized inertia (Case A)
| (a) Mode 1, L9 LE model (6.04 Hz) |
(b) Mode 1, FEM-3D model (5.98 Hz)
| (c) Mode 4, L9 LE model (43.20 Hz) |
(d) Mode 4, FEM-3D model (52.90 Hz)
In the last load case, two non-structural masses were added at (, , −/2) and (0, , /2), as shown in Figure 20(b) (Case B). For this case, the weight of each non-structural mass was equal to Kg. The results from LE and MSC analyses are given in Table 8 for Case B. The following statements hold:(1)As it is also clear from previous analyses, CUF 1D models are able to foresee cross-sectional deformations, and thus local shell-like modes of the box beam are detected.(2)At least an eight-order TE model () is necessary to correctly detect shell-like natural frequencies.(3)The LE models accurately detect the solid solution. Both global and local modes are correctly found with a significant reduction of computational costs.(4)For the case with no localized inertia, MAC matrix shows a perfect correspondence between LE and FEM-3D models. However, modes 3 and 4 as well as modes 8 and 9 are exchanged between the two models.(5)Good agreement is evident from the MAC matrix even though non-structural masses are applied. In this case, no mode exchanges appear but some mixings are evident since coupled phenomena occured.
5. Conclusions
The effects due to non-structural localized inertia on the vibration of isotropic thin-walled beams have been investigated in this work. The ineffectiveness of classical beam theories and related FEs in dealing with non-structural masses which are not placed in the shear center of the cross-section has been demonstrated. In fact, localized inertia has been shown to be responsible for torsion, bending/torsion couplings, and local phenomena such as cross-sectional deformations. For these reasons, the use of higher-order beam theories is mandatory.
In this paper, higher-order beam theories have been developed by using the Carrera Unified Formulation (CUF), which is a hierarchical approach allowing for the automatic formulation of FE arrays deriving from arbitrarily enriched kinematic fields. By using an appropriate indexing notation, stiffness and mass matrices are in fact written in terms of fundamental nuclei which depend neither on the approximation order nor the class of the beam theory implemented. Accordingly, in the present work, non-structural localized masses have been introduced and arbitrarily placed over the 3D beam domain by exploiting the higher-order capabilities of CUF. Two CUF beam classes have been considered for the proposed analyses, TE and LE. In TE class, beam theories are formulated by using Taylor-like polynomials to discretize the cross-sectional generalized displacements. On the other hand, LE models have only pure displacement unknowns and Lagrange polynomials are used to locally refine the cross-section displacement field.
A number of homogeneous thin-walled beams subjected to localized inertia have been addressed in the proposed study and the results compared to those from 2D plate/shell and 3D solid FE models by the commercial code MSC . The analyses highlight the enhanced capabilities of CUF theories with respect to classical ones when dealing with localized inertia effects. In particular, it is demonstrated that LE models yield 3D results with very low computational efforts.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.