Trigonometric Shape Functions Research Articles

Buckling of functionally graded plates under thermal, axial, and shear in-plane loading using complex finite strip formulation

ABSTRACTA complex finite strip method was used to study the buckling of functionally graded plates (FGPs) under thermal and mechanical (longitudinal, transverse, and shear in-plane) loading. The mechanical characteristics of FGPs were assumed to vary through the thickness, according to power law distribution. The nonlinear temperature distribution in the direction of the plate thickness was assumed according to thermal conduction steady state conditions. In complex finite strip method, the polynomial Hermitian functions were assumed in the transverse direction and the complex exponential functions were used in the longitudinal direction to evaluate the standard and geometric stiffness matrices that have the ability of calculating the critical shear stress in contrast to trigonometric shape functions. The solution was obtained by the minimization of the total potential energy and solving the corresponding eigenvalue problem. In addition, numerical results for FGPs with different boundary conditions were presented and compared with those available in the literature and the interaction curves of mechanical and thermal buckling capacity of FGPs were obtained.

A Dynamic Finite Element for Coupled Extensional-Torsional Vibration of Uniform Composite Thin-Walled Beams

A Dynamic Finite Element (DFE) formulation for the free vibration analysis of extension-torsion coupled uniform composite thin-walled beams is presented. Employing the exact solutions of the differential equations governing the uncoupled vibrations of a uniform beam element, the analytical expressions for extensional and torsional dynamic trigonometric shape functions are derived. By exploiting the principle of virtual work and the frequency-dependent shape functions, the element dynamic stiffness matrix is developed. The application of the theory is demonstrated by a Circumferentially Uniform Stiffness (CUS) composite circular tube for which the influence of ply fibre-angle on the natural frequencies is studied. A variety of CUS configurations are studied and the correctness of the theory and the superiority of the proposed DFE over the conventional FEM methods are confirmed by numerical checks and the published results.

Buckling analysis in hybrid cross-ply composite laminates on elastic foundation using the two variable refined plate theory

This paper presents the effect of hybridization material on variation of critical buckling load with different cross-ply laminates plate resting on elastic foundations of Winkler and Pasternak types subjected to combine uniaxial and biaxial loading by using two variable refined plate theories. Governing equations are derived from the principle of virtual displacement; the formulation is based on a new trigonometric shape function of displacement taking into account transverse shear deformation effects vary parabolically across the thickness satisfying shear stress free surface conditions. These equations are solved analytically using the Navier solution of a simply supported. The influence of the various parameters geometric and material, the thickness ratio, and the number of layers symmetric and antisymmetric hybrid laminates material has been investigated to find the critical buckling loads. The numerical results obtained through the present study with several examples are presented to verify and compared with other models with the ones available in the literature.

Semi-analytical buckling analysis of omega stiffened panels under multi-axial loads

This paper presents a semi-analytical method for the buckling analysis of composite panels reinforced with omega stiffeners and subjected to combined loading conditions of biaxial loads and shear. The approach is based on the representation of the panel as an assembly of plate elements, allowing to capture local buckling modes involving skin and stiffener deflections. The panel model includes also the possibility of accounting for the stiffener foot. Trigonometric shape functions are introduced to describe the buckling patterns, while the buckling equations are derived through the application of the minimum potential energy principle. The comparison with Abaqus finite element analyses is presented, demonstrating, for a wide variety of test cases, percent differences below 9% and a good accuracy of the computed buckling modes. The computational speed up is of the order of 100, suggesting the use of the formulation in the context of preliminary design loops, sensitivity analyses or design optimizations.

Semi-analytical skin buckling of curved orthotropic grid-stiffened shells

With the introduction of composite materials to industrial aerospace applications, the research for innovative panel stiffening methods has gained significant interest. Possible candidates are grid-stiffened structures comprising of parallel and intersecting stiffeners forming regular polygonal patterns of skin fields. Since these thin-walled structures are critical to buckling, the structural stability is one of the driving criteria for minimum weight design. The present study investigates the local skin buckling of grid-stiffened structures known as orthogrid, isogrid, diamond grid and kagome grid with a semi-analytical Ritz energy method based on sets of trigonometric shape functions. The influence of the aspect ratio (stiffener angle), curvature and material orthotropy is shown for uni- and biaxial in-plane compression and shear. A self-stiffening effect of the grid-stiffened structures due to interaction with adjacent skin fields is identified, significantly increasing the buckling resistance of such structures. The presented results and trends support preliminary design tasks and the verification of detailed finite element analyses.

Novel basis functions for the partition of unity boundary element method for Helmholtz problems

SUMMARYThe BEM is a popular technique for wave scattering problems given its inherent ability to deal with infinite domains. In the last decade, the partition of unity BEM, in which the approximation space is enriched with a linear combination of plane waves, has been developed; this significantly reduces the number of DOFs required per wavelength. It has been shown that the element ends are more susceptible to errors in the approximation than the mid‐element regions. In this paper, the authors propose that this is due to the use of a collocation approach in combination with a reduced order of continuity in the Lagrangian shape function component of the basis functions. It is demonstrated, using numerical examples, that choosing trigonometric shape functions, rather than classical polynomial shape functions (quadratic in this case), provides accuracy benefits. Collocation schemes are investigated; it is found that the somewhat arbitrary choice of collocating at equally spaced points about the surface of a scatterer is better than schemes based on the roots of polynomials or consideration of the Fock domain. Copyright © 2012 John Wiley & Sons, Ltd.

Buckling Analysis and Optimization of Stiffened Composite Flat and Curved Panels

the skin and of the stiffener are represented by trigonometric shape functions, and the governing equations are derived applying the principle of the minimum potential energy and the method of Ritz. The formulation is used to obtain optimal configurations in terms of skin and stiffener lay-ups, stiffener cross sections, and geometry. Constraints can be imposed on the structural response in terms of buckling load and prebuckling stiffness as well as on technological requirements. The optimal design is presented for the buckling load maximization of a flat hatstiffenedpanelloadedincompressionandshearandfortheminimumweightunderbucklingconstraintsofacurved panel with open-section stringers loaded in compression. The results obtained from the semi-analytical formulation are then compared with finite element analyses.

POSTBUCKLING OF COMPRESSIVELY LOADED IMPERFECT COMPOSITE PLATES: CLOSED-FORM APPROXIMATE SOLUTIONS

In this paper, closed-form approximate solutions for the geometrically nonlinear behaviour of rectangular laminated plates with flexural orthotropy under longitudinal compression are presented. Based on the governing Marguerre-type differential equations postulated for imperfect plates, two plate configurations are discussed in detail, representing important application cases in practical engineering work. The first configuration is a laminated plate that is simply supported at all four edges (the so-called SSSS plate), while for the second configuration clamped unloaded longitudinal edges are considered (denoted as the SSCC plate). For both plate configurations, rather simple closed-form approximations in the form of trigonometric shape functions are employed for the description of the out-of-plane postbuckling plate deflections. Based on the chosen shape functions, the compatibility condition with respect to the in-plane strains is fulfilled exactly, while the out-of-plane equilibrium condition for a deflected plate element is not, but is solved using a Galerkin-type formulation instead. Eventually, very simple closed-form solutions for all postbuckling state variables (deflections, in-plane edge displacements, and effective widths) are derived that can be used very conveniently in engineering practice. The high accuracy of the presented analysis methods is established by comparison with the results of other authors.

Vibration analysis of multi-span plates using finite strip-elements

This paper discusses the use of the finite strip-element method for the vibration analysis of multi-span, thick plate structures. The method uses a combination of polynomial and trigonometric shape functions that enables all boundary conditions to be correctly treated. The global equations are derived using standard finite element procedures, and accurate solutions are obtained with the generation and solution of few equations.

3D ANALYSIS OF RIGHT ANGLE TUNNELS UNDER WAVE PROPAGATION EFFECT WITH BEM

Tunnel is integral part of the infrastructure of modern society and is used for a wide range of applications, including highway, sewage and water transport. This structure built in areas subject to earthquake activity and must withstand safely seismic waves. Seismic waves are characterized in terms of the deformations and strains imposed on the structure by the surrounding ground, often due to the interaction between two. Dynamic analysis of lined tunnel with bending in longitudinal axis is not a common case and is not exist in literature as rich as long lined tunnel with straight axis. In this paper the case was analyzed with BEM in full space time domain. Tunnel with bending in longitudinal axis and large backfilling is dynamically analyzed in three dimensional contexts by assuming linear or viscoelastic material behavior, full contact between soil and structure at their interface without modeling free surface of ground. An analytical boundary integral equation is developed for the efficient time domain dynamic response analysis of a bending cavity with arbitrary cross-section geometry. The numerical solution of boundary integral equation is obtained after discretizing both space and time variation. First curvilinear coordinates, which Z axis is along main longitudinal line, is introduced. Geometry of bend, displacement and traction is expressed in this coordinates. Complex shape of the cross section is mapping into a circular shape with a unit radius by trigonometric shape functions. This mapping enables us to discretize the boundary only along the longitudinal axis of tunnel. Therefore boundary is divided into N circular element this is because of advantages of circular element. Harmonic substantial behavior of deformation and traction, which expressed in cylindrical coordinates, results their variation with central angle expressed in Fourier series. This helps us to decrease the number of equation very much. Observation time is divided into M time step. Appropriate time step is chose because it is very important for numerical stability of transient dynamic problems. Weak and strong singularity occurs during integration process. Direct approach for the rigorous treatment and numerical evaluation is used. For this purpose circular element divided into subdivision elements and integration is done in each element. Singular points exist in one subdivision. Polar coordinates and CPV sense is used for evaluating strong singularity and three coordinates transformation is used for eliminating weak singularity. After all, equation should be solved for the total displacement and tractions at inner and outer surface of tunnel. This is accomplished by especial out of core block equation solver of gauss type for non-symmetric matrices. Numerical results are presented for the case of lined curve tunnel of circular cross section with some values in bending radius, tunnel radius and thickness of tunnel. Plain strain theory results are compared with the straight portion of the model.

Free Vibration Analysis of Sandwich Beams: A Dynamic Finite Element

The free vibration analysis of three-layered symmetric sandwich beams is studied. The governing differential equations of motion in free vibration are coupled both in axial and bending deformations. The weighted residual method is used and the resulting weak integral form of the governing equations is discretized using the Finite Element Method (FEM) formulation. The dynamic, frequency-dependent, trigonometric shape functions, derived from the solution of the uncoupled equations, are then used to develop the element matrices. The assembly of resulting Dynamic Finite Element (DFE) matrices leads to a nonlinear eigenvalue problem. A determinant search method is then used to compute the coupled natural frequencies and modes of a symmetric, thin face-layer sandwich beam. The DFE numerical results for the first four bending-axial coupled modes, obtained from a 20-element mesh, show good agreement with FEM, exact Dynamic Stiffness Matrix (DSM) method, and other results available in the literature. All the four modes are found to be dominated by bending deformation. The discussion of results is followed by some concluding remarks.

Numerical solution of curved pipes submitted to in-plane loading conditions

An alternative formulation to current meshes dealing with finite shell elements is presented to solve the problem of stress analysis of curved pipes subjected to in-plane bending forces. The solution is based on finite curved elements, where displacements are defined from a total set of trigonometric functions or a fifth-order polynomial, combined with Fourier series. Global shell displacements are achieved through the one associated with curved arch bending and the other referred to the toroidal thin-walled shell distortion. Beam-type displacement and in-plane rotation are uncoupled and separately formulated, using trigonometric shape functions, as in Timoshenko or Mindlin beam theory. To build up the solution, a simple deformation model was adopted, based on the semi-membrane concept of the doubly curved shells behaviour. Several studies are presented and compared with experimental and numerical analyses reported by other authors.

TRANSIENT HEAT CONDUCTION IN FUNCTIONALLY GRADED MATERIALS

The h-p version of the finite element method (FEM) is considered to determine the transient temperature distribution in functionally graded materials (FGM). The h-p version may be regarded as the marriage of conventional h-version and p-version. The graded Fourier p-element is used to set up the two-dimensional heat conduction equations. The temperature is formulated in terms of linear shape functions used generally in FEM plus a variable number of trigonometric shape functions representing the internal degrees of freedom (DOF). In the graded Fourier p-element, the function of the thermal conductivity is computed exactly within the conductance matrix and thus overcomes the computational errors caused by the space discretization introduced by the FEM. Explicit and easily programmed trigonometric enriched capacitance, conductance matrices and heat load vectors are derived for plates and cylinders by using symbolic computation. The convergence properties of the h-p version proposed and the results of the numbers of test problems are in good agreement with the analytical solutions. Also, the effect of the non-homogeneity of the FGM on the temperature distribution is considered.

An efficient BEM formulation for three-dimensional steady-state heat conduction analysis of composites

In this work, an efficient boundary element formulation has been presented for three-dimensional steady-state heat conduction analysis of fiber reinforced composites. The cylindrical shaped fibers in the three-dimensional composite matrix are represented by a system of curvilinear line elements with a prescribed diameter which facilitates efficient analysis and modeling together with the reduction in dimensionality of the problem. The variations in the temperature and flux fields in the circumferential direction of the fiber are represented in terms of a trigonometric shape function together with a linear or quadratic variation in the longitudinal direction. The resulting integrals are then treated semi-analytically which reduces the computational task significantly. The computational effort is further minimized by analytically substituting the fiber equations into the boundary integral equation of the material matrix with hole, resulting in a modified boundary integral equation of the composite matrix. An efficient assembly process of the resulting system equations is demonstrated together with several numerical examples to validate the proposed formulation. An example of application is also included.

In-plane vibration of plates with curvilinear plan-forms by a trigonometrically enriched curved triangular p-element

A trigonometrically enriched curved triangular p-element is developed and applied to free in-plane vibration analysis of plates with curvilinear plan-forms. New sets of trigonometric shape functions are presented. The main feature of these functions is their ability to overcome ill-conditioning which may occur using orthogonal polynomials of high order. In this method, meshes are constructed from a few large elements and curved segments are represented accurately by using the blending function method. The accuracy of the solution is improved simply by increasing the trigonometric order while keeping the mesh constant. Frequency values are first found for a closed sectorial plate and comparisons are made with exact solutions and with solutions from conventional linear and isoparametric quadratic triangular finite elements. Frequency values are also found for an open sectorial plate and comparisons are made with exact solutions. Furthermore, highly accurate frequency values are given for a square plate with a central circular opening. It is found that the method converges very fast downwards to the exact solution as the trigonometric order is increased and produces a much higher accuracy than linear and isoparametric quadratic triangular finite element solutions with far fewer d.o.f.s.

Steady state thermoelastic analysis of 3D solids with fiber inclusions by boundary element method

In this work, a boundary element based analysis methodology for the steady state uncoupled thermoelastic analysis of fiber reinforced three-dimensional solids is presented. For efficient analysis and modeling, the cylindrical shaped fibers are idealized by a system of curvilinear line elements with a prescribed diameter. The variations in the displacement, traction, temperature and flux fields in the circumferential direction are represented in terms of a trigonometric shape function together with a linear or quadratic variation in the longitudinal direction. This facilitates semi-analytical integration around the fiber which reduces the computational task significantly. The numerical accuracy of the proposed method of analysis is verified and some examples of thermoelastic analysis of fiber-reinforced solids are presented.

Transient heat conduction analysis of 3D solids with fiber inclusions using the boundary element method

AbstractA boundary element formulation is presented in this work for transient heat conduction analysis of three‐dimensional (3D) fiber‐reinforced materials. The cylindrical‐shaped fibers in a 3D matrix are idealized by a system of curvilinear line elements with a prescribed diameter. The variations in the temperature and flux fields in the circumferential direction are represented in terms of a trigonometric shape function together with a linear or quadratic variation in the longitudinal direction. This approach significantly reduces the modeling effort and the computing cost. The storage requirement for the convolution integrals is eliminated by adopting an accurate integration‐based convolution algorithm for the surface of the hole and the fibers as well as a fast convolution algorithm for the outer boundary recently developed by the present authors.Numerical examples are presented to demonstrate the accuracy and applicability of the proposed method of analysis of fiber‐reinforced materials. Copyright © 2007 John Wiley & Sons, Ltd.

Free vibration analysis of circular and annular sectorial thin plates using curve strip Fourier p-element

A curve strip Fourier p-element for free vibration analysis of circular and annular sectorial thin plates is presented. The element transverse displacement is described by a fixed number of polynomial shape functions plus a variable number of trigonometric shape functions. The polynomial shape functions are used to describe the element's nodal displacements and the trigonometric shape functions are used to provide additional freedom to the edges and the interior of the element. With the additional Fourier degrees of freedom (dof) and reduce dimensions, the accuracy of the computed natural frequencies is greatly increased. Results are obtained for a number of circular and annular sectorial thin plates and comparisons are made with exact, the curve strip Fourier p-element, the proposed Fourier p-element and the finite strip element. The results clearly show that the curve strip Fourier p-element produces a much higher accuracy than the proposed Fourier p-element, the finite strip element.

A Finite Circular Arch Element Based on Trigonometric Shape Functions

The curved-beam finite element formulation by trigonometric function for curvature is presented. Instead of displacement function, trigonometric function is introduced for curvature to avoid the shear and membrane locking phenomena. Element formulation is carried out in polar coordinates. The element with three nodal parameters is chosen on curvature. Then, curvature field in the element is interpolated as the conventional trigonometric functions. Shape functions are obtained as usual by matrix operations. To consider the boundary conditions, a transformation matrix between nodal curvature and nodal displacement vectors is introduced. The equilibrium equation is written by minimizing the total potential energy in terms of the displacement components. In such equilibrium equation, the locking phenomenon is eliminated. The interesting point in this method is that for most problems, it is sufficient to use only one element to obtain the solution. Four examples are presented in order to verify the element formulation and to show the accuracy and efficiency of the method. The results are compared with those of other concepts.

FREE VIBRATION ANALYSIS OF ROTATING FLEXIBLE BEAMS BY USING THE FOURIER p-VERSION OF THE FINITE ELEMENT METHOD

A p-version of the finite element method is applied to free vibration analysis of rotating beams in conjunction with the modeling dynamic method using the arc-length stretch deformation. In this study the flexible and the rigid body degrees of freedom (d.o.f.) are supposedly uncoupled, the linear equations of motion are derived for flapwise and chordwise bending with the integration of the gyroscopic effect. The hybrid displacements are expressed as the combination of the in-plane and out-of-plane shape functions. These are formulated in terms of linear and cubic polynomial functions used generally in FEM in addition to a variable number of trigonometric shape functions which represent the internal d.o.f. for the rotating flexible beams. The convergence properties of the rotating beam Fourier p-element and the influence of angular speed, boundary conditions and slenderness ratio on the dynamic response are studied. It is shown that using this element the order of the resulting matrices in the FEM is considerably reduced leading to a significant decrease in computational effort.