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Effect of thickness variation of a curved beam on the critical buckling load parameter for various cross-sections. y ¼ 601, b t ¼ b R , t t ¼ t R (' C1), b t ¼ b R , t R 6 ¼t t (J C2 and C4), b t 6 ¼b R , t t 6 ¼t R (n C3 and B C5).
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In this study, in-plane stability analysis of non-uniform cross-sectioned thin curved beams under uniformly distributed dynamic loads is investigated by using the Finite Element Method. The first and second unstable regions are examined for dynamic stability. In-plane vibration and in-plane buckling are also studied. Two different finite element mo...
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... Fig. 5, it is seen that when the variation of cross-section diminishes and approaches the uniform crosssection, the curved beam becomes stiffer; as a result, the critical buckling load increases and takes the value of the uniform cross-sectioned curved beam. From this figure, it can be said that static stability values of curved beams having ...
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Mathematically, the buckling phenomenon can be defined as a bifurcation in the solution to the equations of static equilibrium that leads to failure. In practice, buckling is characterized by a sudden failure of a structural member under high compressive stresses; meanwhile, the actual stress at the point of failure is less than the ultimate compre...
Citations
... Based upon the formulated surface elastic-based curved beam model, At first, the validity of the developed curved beam formulation is examined. Accordingly, after removing the expressions associated with the Gurtin-Murdoch continuum elasticity, the non-dimensional buckling loads of a clamped isotropic curved beam under a uniform compressive load in radial direction is calculated and compared with those presented by Timoshenko and Gere [80] based upon an exact analytical solution, Luu et al. [81] via an isogeometric method and Ozturk et al. [82] using the finite element technique. In accordance with Table 1, a very good agreement is revealed which confirms the accuracy of the constructed numerical curved beam formulations. ...
In the present exploration, the effect of surface stress type of size dependency on the nonlinear in-plane stability characteristics of functionally graded (FG) laminated composite curved nanobeams subjected to uniform radial pressure together with a temperature rise is examined. To accomplish this motivation, the Gurtin-Murdoch continuum elasticity is applied within a higher-order shear flexible curved beam theory in the presence of geometric type of nonlinearity. The effective properties of the employed nanocomposites are approximated via the Halpin-Tsai homogenization scheme corresponding to different lamination patterns and are considered in the principle of virtual work to establish the surface elastic-based nonlinear differential equations of the stability problem. It is found that by taking the surface stress effect into account, the values of the radial load related to the first and second bifurcation points get larger. Also, it causes to increase the curved beam deflection associated with the first bifurcation point, while it leads to decrease the radial deflection at the second bifurcation point. Accordingly, the effect of surface stress type of size dependency resulted in a reduction in the slope of bifurcation path related to the nonlinear radial load-deflection equilibrium curve of a curved nanobeam. It can be also observed that the temperature rise causes to increase the radial loads associated with the upper limit and the first bifurcation points, while it leads to reduce the values of radial load related to the lower limit and the second bifurcation points. In this regard, there is a unique intersection point as all equilibrium curves traced for different temperature rises pass through that.
... Different types of arches (parabolic, sinusoidal and elliptical) are presented in [4]. The dynamic analysis of arches with variable cross-sections was considered in [8,9]. Arches of an arbitrary curve are very often analyzed by using the isogeometric approach. ...
The paper presents arch structures modeled by finite elements in which the nodes can be flexibly connected. Two-node curved elements with three degrees of freedom at each node were used. Exact shape functions were adopted to obtain stiffness and consistent mass matrices but they were modified by introducing rotational flexibility in the boundary nodes. Calculations of statics and dynamics of arches with different positions of flexible joints and different values of rotational stiffness of the joints were carried out.
... where kge is the element geometric stiffness matrix. Since the laminated composite arch (curved) frame structure is composed of straight frame beam elements, element mass, elastic stiffness, and geometric stiffness matrices are needed to be transformed in therms of reference coordinates as is shown in Figure 5. Hence, those matrices are transformed considering the 3-DOF element by using Equation (17). where T is the transformation matrix for 3DOF, which is, ...
In this study, the dynamic stability analysis of the laminated composite single-bay and two-bay curved frame structures has been investigated. For this purpose, a computer code is written in MATLAB to evaluate the natural frequency values, critical buckling load, and first unstable regions of the laminated composite curved frames. The effects of radii of curvature and stacking order on the static and dynamic stability of both single and two bay arch frame structures are investigated. Besides, the effects of the stacking order are investigated by considering five different stacking sequences. The results of the present study are compared with the results obtained numerically via ANSYS for validation. It is concluded that the radius of curvature has a small effect on the first five natural frequency values, buckling loads, first unstable regions of the structure, whereas the fiber orientation considerably has a considerable impact on such static and dynamic properties.
... Nonlinear responses of thin circular arches undergoing large displacements have been studied in many works. Some analytical solutions [1,[9][10][11]15] and finite element results are available [6,8,13,14]. Different boundary conditions of the two ends, i.e., pinned-pinned, fixed-fixed, and pinned-fixed ends, are considered. ...
Circular arches are often used as parts of engineering structures due to their effectiveness in resisting vertical loads. An understanding of nonlinear structural behavior of circular arches is crucial to obtaining their good designs. Points of interest are the effects of the rise-to-span ratio on their nonlinear structural behavior. In this paper, large displacement analysis of pinned-fixed circular arches with different rise-to-span ratios using an isogeometric approach is presented. In the analysis, the Euler-Bernoulli beam theory is used. The exact reference axis of each arch and its deformed axis are represented by non-uniform rational B-spline (NURBS) curves. The effects of the rise-to-span ratio on the nonlinear structural behavior of the arches are studied. Some well-established examples, exhibiting snap-through and snap-back phenomena, are considered, and the accuracy and efficiency of the proposed approach are discussed.
... The fundamental natural frequencies and corresponding mode shapes were extracted for different combinations of the opening angle (θ), the thickness and width tapering ratios, (tt/tr) and (bt/br) and the mean radius of the beam (R). The frequencies extracted are in good agreement with reported analytical solutions [2]. ...
... The objectives of the present investigations are to review the literature related to the analytical and numerical investigations on the dynamic analysis of nonuniform curved beam problems [2][3][4][5][6][7] and to carry out a parametric FEM study on the vibration behavior of nonuniform curved beams. The different parameters considered were the opening angle (θ), the thickness and width tapering ratios, (tt/tr) and ...
This paper presents the details an extensive parametric study carried out on non-uniform cross-sectioned thin curved beams using Finite Element Method (FEM). The parameters considered were the opening angle (θ), the thickness and width tapering ratios, (tt/tr) and (bt/br) and the mean radius of the beam (R). The FEM analysis was carried out using the front-end commercial software ANSYS (Ver: 10.0) [1] with its modal analysis capabilities. The Block Lanczos algorithm with subspace iteration technique was used in the extraction of natural frequencies and corresponding mode shapes. The fundamental natural frequencies and corresponding mode shapes were extracted for different combinations of the opening angle (θ), the thickness and width tapering ratios, (tt/tr) and (bt/br) and the mean radius of the beam (R). The frequencies extracted are in good agreement with reported analytical solutions [2].
... Meanwhile, a series of other high-order curved beam elements were developed and studied, such as Lagrangian curved beam element for the curved beams with varying curvature based on the general Hamilton principle [18], mixed stress finite beam element based on the Hellinger-Reissner functional [19] and hybrid curved beam element [20][21][22][23]. Some other numerical methods are also developed to study the free vibrations of curved beams, such as the differential quadrature method (DQM) [24].Besides, it is worth noting that although the exact solutions of the non-uniform curved beams are few, the existing numerical solutions are also usually obtained without considering the shear deformation and rotation inertia, such as Tong et al. [25], ÖztÜrk et al. [26] and Lee et al. [27]. ...
This paper investigates the vibration characteristics of a circular ring with an arbitrary number of concentrated elements based on the Hamilton principle. The shear and inertia effects are introduced to the variational functional of system kinetic and potential energy by adopting the generalized shell theory. The concentrated elements are treated as the concentrated masses with elastic boundary condition. The system vibration displacements are analytically expanded in the form of Fourier series and substituted into the variational functional to obtain the equation of motion. Computed results are compared with those solutions obtained from the finite element program ANSYS to validate the accuracy of the present method. Effects of the asymmetrical concentrated elements on the convergence of circumferential wavenumbers are discussed. Moreover, the coupling characteristics of different circumferential wavenumbers caused by the asymmetrical or symmetrical concentrated elements and their influences on the vibration response characteristics of the system under simple harmonic excitations are studied.
... The defined finite elements were used to solve many numerical examples (including arches with variable parameters) in which eigenfrequencies and eigenforms were determined. Oztürk et al. [15], using the assumption about arch axial nondeformability and assuming the shape functions in the form of a combination of trigonometric functions and classical polynomials, developed a finite element and employed it to solve the free vibration-and-dynamic stability problem for the arch. An element developed using exact (for the static problem) shape functions (Litewka and Rakowski [16]) was employed in Litewka and Rakowski [17]. ...
The problem of spatial vibrations, both aperiodically forced and free vibrations, of an arch with an arbitrary distribution of material and geometric parameters is considered. Approximation with Chebyshev series was used to solve a conjugated system of partial differential equations describing the problem. The system of differential equations was solved using an algorithm generating a recursive infinite system of equations, developed by S. Paszkowski in “Numerical applications of Chebyshev polynomials” (in Polish), Warsaw PWN, 1975. Since the coefficients of the obtained system of equations are defined by closed analytical formulas they can be directly used to solve any nonprismatic arch, without it being necessary to solve again the considered problem. The algorithm is highly accurate; i.e., already at a small approximation base it yields results agreeing with exact analytical solutions (obviously for problems in the case of which such solutions can be derived). In order to demonstrate this the eigenfrequencies and eigenvectors obtained for a circular prismatic arch were compared with their precise values determined from the exact analytical solutions. The results yielded by the proposed method were also compared with the results obtained by other methods and by other authors. As an illustration, the proposed method was used to solve a more complex problem, i.e., the problem of the free and aperiodically forced vibrations of a nonprismatic arch with its axis described by a catenary curve. In the example the effect of the lack of cross-sectional symmetry of the arch on the form of the system’s spatial free and forced vibrations was analysed.
... The complexity of curved beam models, on the other hand, has gradually grown over the years. Approaches have evolved from early models based on a number of simultaneous restricting assumptions such as isotropic materials [22][23][24][25][26], the absence of coupling between different DOF [27][28][29][30], prismatic beams and/or homogeneous cross-section properties [31][32][33], and constant curvature [34][35][36], among others, towards models where some of these restraints have been increasingly relaxed. One important innovation that greatly facilitates the modelling of curved beams is the use of the Frenet-Serret frame field, which describes the beam kinematics based on a local reference frame given by the tangent, normal and bi-normal vectors along the curved axis (see e.g. ...
... etc., can be calculated via the recursive algorithm given in [45], where sub-indices have been omitted for clarity: t into equations (20) and using equations (38) to calculate the required derivatives, the right-hand side of Eq. (36) can be expressed in terms of the coefficient vectors C C C C , , , t n b θ , which are the new variables to be determined from the force/momentum balance (36). The result can be written in the form of a matrix equation: ...
This paper presents a unified theory for modelling composite thin-walled beams (TWB) of arbitrary planar axial curvature, variable cross-section and general material layup, complemented by the development of an Isogeometric Analysis (IGA) formulation for the discretization and solution of the equilibrium equations. To this end, the standard formulation of composite TWB with rectilinear axes is combined with a general framework for describing the kinematics of arbitrary three-dimensional curves based on the Frenet-Serret frame field. The theory includes explicit terms accounting for curvature gradients within the IGA stiffness matrix, allowing for the treatment of cases with highly curved geometry. Also included is an advanced shear-modification adjustment previously derived for rectilinear TWB, here reformulated for the case of curved TWB, improving the description of the in-plane shear-strain coupling and thus increasing the accuracy for cases with axial-bending-torsional structural coupling. Results from three numerical test cases indicate that this unified formulation effectively transfers all the advantages associated with rectilinear TWB to curved TWB models, yielding an accuracy comparable to more complex models while maintaining a competitive computational economy.
... Recently, buckling analysis of linearly web tapered I-beams has been of interest to many researchers, who used a power series method (Leung, 2009), the Ayrton-Perry formulation (Marques et al., 2012), and the finite-element method (Lei and Shu, 2008;Beyer et al., 2015). Based on the finite-element method, stability analysis of linearly tapered beams along thickness and width using various methodologies has been presented (Karabalis and Beskos, 1983;Eisenberg and Reich, 1989;Ozturk et al., 2006). ...
Vibration and stability analysis of nonlinearly tapered cone beams coupled with a piezoelectric layer under compressive axial load is conducted using a new semi-theoretical model based on the Adomian decomposition method and a modified mathematical procedure. The method is applied to tapered structures with perfectly surface-bonded piezoelectric layers and general boundary conditions to analytically derive the natural frequencies and mode shape functions for flutter and buckling analysis. Furthermore, from the optimum structural design perspective, the effects of follower force, geometrical tapering ratio, boundary condition, and applied voltage on piezoelectric layers on the structural stability are thoroughly studied and presented. Simulation results show that the stability of the beam can be noticeably enhanced by the external voltage because of a pair of tensile loads locally induced by the piezoelectric effect. Moreover, for certain boundary conditions and applied voltages, the nonlinear tapered design has much greater buckling and flutter capacities than does the uniform beam. Consequently, the ideal structural design for effective stability enhancement can be reached efficiently.
... The critical buckling load obtained here is presented in Table 3 along with the available exact analytical results and numerical solutions shown in Refs. [62,63,65,66]. It is seen that the present results are in very good agreement. ...
