Exact Shape Functions Research Articles

An efficient finite element computation using subparametric transformation up to cubic-order for curved triangular elements

PurposeA finite element computational methodology on a curved boundary using an efficient subparametric point transformation is presented. The proposed collocation method uses one-side curved and two-side straight triangular elements to derive exact subparametric shape functions.Design/methodology/approachOur proposed method builds upon the domain discretization into linear, quadratic and cubic-order elements using subparametric spaces and such a discretization greatly reduces the computational complexity. A unique subparametric transformation for each triangle is derived from the unique parabolic arcs via a one-of-a-kind relationship between the nodal points.FindingsThe novel transformation derived in this paper is shown to increase the accuracy of the finite element approximation of the boundary value problem (BVP). Our overall strategy is shown to perform well for the BVP considered in this work. The accuracy of the finite element approximate solution increases with higher-order parabolic arcs.Originality/valueThe proposed collocation method uses one-side curved and two-side straight triangular elements to derive exact subparametric shape functions.

Nonlinear finite element Analysis of laterally loaded piles in Layered Soils

This work is based on Winkler’s theory to analyze laterally loaded single piles using the finite element method. Soil nonlinearity is taken into account via nonlinear springs “p-y” curves. Exact element displacement shape functions and stiffness matrix are used for the element in the case of linear Winkler’s modulus of subgrade reaction. In the nonlinear stage, an averaging technique for the element secant Winkler modulus is used to calculate the shape functions and stiffness matrix. An iterative technique is used to take into account the non-linearity of the soil. Few elements are required to simulate the pile efficiently. Unlike other analytical methods, the current method can be used to analyze piles with any load-transfer curves with arbitrary variation with depth.

Geometrically nonlinear transient analysis of Timoshenko tapered beam-column element using a finite element formulation based on the exact shape functions

In this paper, a numerical formulation was developed to generate the consistent mass matrix for Timoshenko tapered beam-column element for geometrically nonlinear transient analysis. The basis of the suggested solution is the exact shape functions and their derivatives, which describe the non-uniformity of the element properties. To illustrate the variations in section properties along the tapered element length, the section properties were presented as exponential functions with tapering indices. The model can simulate elements with various solid and hollow cross-sections. The suggested formulation is integrated into a Visual Basic program to perform the analysis, along with numerous examples to verify its precision and effectiveness. As a practical application of the proposed formulation, a structural system was proposed for offshore wind turbines with high elevations, and a geometrical nonlinear dynamic analysis was conducted. The results demonstrated the proposed model’s ability to perform this type of complex calculation with minimal computational effort.

Inversion of dynamic displacement response of cable in the whole field based on single vibration measurement

Aiming at the problem that the dynamic displacement at critical positions of the cable cannot be measured due to economic or difficult installation, this paper proposes a method to invert the dynamic displacement of the cable in the whole field by using the dynamic displacement of a single measuring point. The method can realize the inversion of the lateral dynamic displacement response (L-DDR) and angular dynamic displacement response (A-DDR). Firstly, the exact mode shape functions of the L-DDR and A-DDR, considering bending stiffness, sag, inclination angle, and boundary conditions, are derived from the cable vibration control differential equation. The exact mode shape functions are a closed-form solution; thus, it can accurately calculate the cable mode shape at any frequency in real time. Then, an inversion method of the dynamic displacement of the cable in the whole field is proposed based on the amplitude ratio of the exact mode shape. This method can be used to invert the dynamic displacement of the cable in the frequency domain and time domain. Finally, through a numerical simulation and an actual cable test, it is verified that the accuracy of the proposed method is satisfactory. The cable dynamic displacement components at modal and external load frequencies are the main contributors to the dynamic displacement of the cable.

Relativistic Wormholes in Extended Teleparallel Gravity with Minimal Matter Coupling

AbstractThis study explores the wormhole solutions utilizing the notion of conformal symmetry in the background of the gravity. The exact shape functions are calculated using the Gaussian and Lorentzian distributions as the noncommutative source of the energy density. All the energy conditions are checked for the two different shape functions. The presence of exotic matter is confirmed due to the violation of the null energy conditions. The physical aspects of the epicyclic frequencies of test particles are discussed in the spacetime geometry of the obtained wormholes in the current analysis and it is shown that how they depend on the spacetime parameters. The red‐blue shifts for the photons emitted by the orbiting test particles are also investigated and their dependence on the spacetime parameters is discussed. Further, the stability of the developed structure of the traversable wormholes is investigated by using the Tolman‐Oppenheimer‐Volkoff equation.

On the Forced Vibration of Bending-Torsional-Warping Coupled Thin-Walled Beams Carrying Arbitrary Number of 3-DoF Spring-Damper-Mass Subsystems

This paper presents an analytical transfer matrix modeling framework for the forced vibration of a bending-torsional-warping coupling Euler-Bernoulli thin-walled beam carrying an arbitrary number of three degree-of-freedom (DOF) spring-damper-mass (SDM) subsystems. The thin-walled beam is divided into a series of distinct sub-beams whose ends are connected to the SDM subsystems. The transfer matrix for each sub-beam is developed based on the exact shape functions of the bending-torsional-warping coupling Euler-Bernoulli theory. Each SDM system is modelled by a set of effective springs based on the dynamic condensation method. The governing matrix equation is formulated based on the compatibility conditions of the placement and the force at the common interfaces of two adjacent sub-beams. Then, a closed-form expression for the frequency response function of the thin-walled beam system is proposed. The results computed by the proposed method achieve good agreement with those obtained by the conventional finite-element method, which shows the accuracy and reliability of the proposed method. The effects of the system parameters on the vibration transmission and vibration isolation properties of the thin-walled beam system are studied. The presented method can simultaneously consider arbitrary number of SDM subsystems and boundary conditions. Furthermore, none of the associated matrices are larger than 12 × 12, which provides a significant computational advantage.

Finite element formulation of Timoshenko tapered beam-column element for large displacement analysis based on the exact shape functions

ABSTRACT A numerical formulation was carried out in this paper to produce the tangent stiffness matrix for two-nodal tapered Timoshenko beam-column elements for geometrically nonlinear analysis. The proposed solution is based on the exact shape functions and their derivatives describing the non-uniformity of the element properties. The section properties were presented as exponential functions with tapering indices to illustrate the variations in section properties along the tapered element length. The model is applicable for elements with different solid and hollow cross-sections. The proposed formulation is embedded into a Visual Basic code to carry out the analysis accompanied by many examples for validating its accuracy and efficiency. The model results are compared with those of commercial software and cited references that showed high accurate results with a small number of elements.

Application of variation method in three dimensional stability analysis of rectangular plate using various exact shape functions

In this paper, a polynomial and trigonometric shape function are developed for the three-dimensional (3-D) stability analysis of a thick rectangular plate. This study has evaluated the effect of aspect ratio of the critical buckling load of a plate that is clamped on the first edge, free at the third edge, with the second and fourth edges simply supported respectively (CSFS) using a variational method. An expression of potential energy of thick plate was formulated using 3-D elastic principles thereafter, a compatibility equation of 3-D plate was derived through energy equation transformation to get the relations between the rotations and deflection. The solution of compatibility equations yields the exact plates shape function which is derived in terms of trigonometric and polynomial displacement and rotations. Similarly, by minimizing the energy equation with respect to the deflection, the direct governing equation was formulated. The solution of governing equation yields the deflection coefficient of the plate. By minimizing the potential energy equation with respect to deflection coefficient after the action deflection and rotations equation were substituted into it, a more realistic formula for calculation of the critical buckling load is established. The proposed method unlike the refined plate theory (RPT), considered all the six stress elements in the analysis. The result showed that the critical buckling loads from the present study using polynomial are slightly higher than those obtained using trigonometric theories signifying the more exactness of the latter. The result of the present study using the established 3-D model for both functions is satisfactory and closer to exact solution compared to the two- dimensional (2-D) RPT. The overall average percentage differences between the two functions recorded are 6.4%. This shows that at about 94% both approaches are the same and can be applied with confidence in the stability analysis of any type of plate with the boundary condition.

Exact finite beam element for open thin walled doubly symmetric members under torsional and warping moments

Starting with total potential energy variational principle, the governing equilibrium coupled equations for the torsional-warping static analysis of open thin-walled beams under various torsional and warping moments are derived. The formulation captures shear deformation effects due to warping. The exact closed form solutions for torsional rotation and warping deformation functions are then developed for the coupled system of two equations. The exact solutions are subsequently used to develop a family of shape functions which exactly satisfy the homogeneous form of the governing coupled equations. A super-convergent finite beam element is then formulated based on the exact shape functions. Key features of the beam element developed include its ability to (a) eliminate spatial discretization arising in commonly used finite elements, and (e) eliminate the need for time discretization. The results based on the present finite element solution are found to be in excellent agreement with those based on exact solution and ABAQUS finite beam element solution at a small fraction of the computational and modelling cost involved.

Casimir wormholes in 2+1 dimensions with applications to the graphene

In this paper we show that wormholes in (2+1) dimensions (3-D) cannot be sourced solely by both Casimir energy density and tension, differently from what happens in a 4-D scenario, in which case it has been shown recently, by the direct computation of the exact shape and redshift functions of a wormhole solution, that this is possible. We show that in a 3-D spacetime the same is not true since the arising of at least an event horizon is inevitable. We do the analysis for massive and massless fermions, as well as for scalar fields, considering quasi-periodic boundary conditions and find that a possibility to circumvent such a restriction is to introduce, besides the 3-D Casimir energy density and tension, a cosmological constant, embedding the surface in a 4-D manifold and applying a perpendicular weak magnetic field. This causes an additional tension on it, which contributes to the formation of the wormhole. Finally, we discuss the possibility of producing the condensed matter analogous of this wormhole in a graphene sheet and analyze the electronic transport through it.

Vibration analysis of nonlinear tapered functionally graded beams using point collocation method

Free transverse vibration of variable cross-section cantilever FG beams with nonlinear profiles is investigated in this paper. Four thickness functions, namely, linear, parabolic, sinusoidal, and exponential functions are assumed for variation of the cross-section of the beam. Linear and exponential grading rules for axially varying material properties are covered in this paper. The governing differential equation is solved using the weighted residual collocation method with the exact solution shape functions for the uniform beam as the trial functions. This choice for the trial functions showed an increase in the convergence rate. The Gauss–Legendre points are used as the collocation points to reduce the fluctuations in the convergence curve as usually encountered in the point collocation method. The effects of the taper parameter for all kinds of thickness functions on the natural frequencies are studied. The effects of various parameters, including taper parameter, profile, and grading function on the natural frequencies are investigated. Also, a series of finite element simulations are performed for comparison purposes. It is observed that the obtained results are in good agreement with the numerical simulations and available published data with vastly reduced computational cost as a result of using the collocation method.

Timoshenko beam formulation for in-plane behaviour of tapered monosymmetric I-beams: Analytical solution and exact stiffness matrix

The presence of curved centroidal axis in tapered monosymmetric I-beams results in coupling between the axial and flexural behaviours. In this paper, an extended Timoshenko beam formulation that incorporates the axial–flexure coupling in the governing differential equations is presented. The proposed formulation is applicable to generic 3D tapered I-beam, with monosymmetric/ bisymmetric cross-section, under in-plane loading. The governing equations are solved analytically and the results are validated with 3D finite element model for a range of tapered I-beams under distributed axial and transverse loadings. Further, the analytical solutions are used to develop an exact 1D tapered beam element (TBE) that can be used within the finite element framework. The main advantage of the proposed approach is that it allows tapered I-beam with monosymmetric section to be modelled as a straight-line segment along its web-centreline. The analytical solutions are also used to develop the exact shape functions for interpolating nodal variables and stress resultants within the element. The proposed beam element is validated by considering a statically indeterminate beam subjected to distributed axial and transverse loads. The displacements, rotations, stress resultants, normal and shear stresses obtained from the proposed beam element show good agreement with 3D FE model.

Influence of conicity on the free wave propagation in symmetric tapered periodic beam

In this study, the attenuation characteristics of the unit cell of an infinitely long double symmetric tapered beam (DSTB) is examined. The exact flexural shape functions of the DSTB is derived using the Bessel functions. The band-structures for rectangular and circular solid section with varying conicity is evaluated implementing the transfer matrix method. It is found that the tapered beam produces wider band gaps for an ideal selection of conicity for both rectangular and circular cross-section. It is also evidenced in this paper that the circular cross-section produces wider band gaps with higher level of attenuation in comparison to the rectangular section. Specifically, the first normalized attenuation bandwidth of 165% can be achieved in case of circular section and 97% in case of rectangular section. A representative cantilever finite beam structure comprised of the periodic DSTB also elucidates the reduction in flexural vibration for the same frequency ranges obtained from the dispersion curve of the infinite structure’s unit cell.

A new three-node element for bending, free vibration and buckling analysis of composite laminated beams based on FSDT theory

In this paper, a new three-node element with three degrees of freedom per node is proposed for bending, free vibration and buckling analysis of laminated beams. The element’s formulation is based on the first-order shear deformation theory (FSDT). For this aim, transverse displacement and rotation field of the element are selected from fifth and fourth order, respectively. Moreover, the shear strain is varied as quadratic function throughout the element. It is worth noting that the quadratic function can be used for axial displacement field. By employing equilibrium equation, curvature and shear strain relations of laminated beam with FSDT theory, the exact and explicit shape functions of the displacement fields are obtained. By utilizing the obtained shape functions, the explicit form of the stiffness matrix is calculated for the element. On the other hand, by using the governing equation of the free vibration and buckling of the beam, the explicit form of the translation and rotary mass matrices, and geometric stiffness matrix of the element are obtained. It should be mentioned, the proposed element is free of shear locking. Finally, several numerical tests fulfill to assess the robustness of the developed element. For this purpose, bending, free vibration and buckling analysis of laminated beams with different boundary conditions and aspect ratios are performed. The results of the numerical tests demonstrate high accuracy and efficiency of the proposed element for free vibration and buckling analysis of laminated beams.

Wave propagation analysis of laminated composite shell panels using a frequency domain spectral finite element model

This paper presents a frequency domain spectral finite element model (SFEM) for studying wave propagation in laminated composite shell panels. The SFEM formulation is derived using two different theories: (i) the first order shear deformation theory (FSDT) that takes shear deformation into consideration, and (ii) the classical shell theory (CST) that neglects shear deformation. The spectral approach here is based on the use of fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT). The spectral element uses exact shape functions and a frequency dependent exact dynamic stiffness matrix is obtained relating the nodal displacements to the nodal forces of an element. The formulation is validated by comparing the present results for wavenumber dispersion and natural frequencies with the published results. The present SFEM is then used to perform wavenumber dispersion, phase and group velocities, and wave propagation analyses of laminated composite shell panels. The numerical studies show that the wavenumbers obtained from the SFEM based on the CST are significantly different from those of the FSDT at high frequencies. The CST results for the group velocity and wave propagation response to tone burst excitation show significant deviation from the FSDT results for out-of-plane A0 flexural mode even at relatively low frequencies. The orthotropy ratio and lamination arrangements in a composite shell panel have significant effect on the group velocity as well as the flexural mode wave propagation response.

Response of a micro-capillary system exposed to a moving mass in magnetic field using nonlocal strain gradient theory

In this study, the dynamic behaviour of the micro dimensional Timoshenko beam (micro capillary system) exposed to a moving mass in a magnetic field is modelled using the non-local strain gradient theory. The motion equations of the Timoshenko micro-beam, which interacts with the moving mass, are obtained by considering the effects of strain gradient elasticity that provides stiffening and non-local elasticity that provide softening. The transverse Lorentz force acting on the beam in the magnetic field is obtained by Maxwell's equations and combined with the motion equations of the beam. These equations are converted into a weak form finite element equation by applying the exact shape functions of the two-node Timoshenko beam element in the Galerkin's method. Along with the effects of the magnetic field, the effects of the amount of different non-local parameters and the mass ratio and speed of the moving mass on the dynamics of the micro beam are presented. Experimental and theoretical studies have shown that the dynamic behaviour of the capillary structural systems is negatively affected by the effect of moving mass except for the non-local properties. Depending on the velocity and quantity of the mass and the nonlocal elasticity, the natural frequencies of the system drop excessively. When the mass ratio (moving mass/mass of the beam) is around 0.5, the frequencies have been observed to drop by as much as 60%, and the non-local elasticity and velocity of the mass are also effective in this. It has been shown that these disadvantages can be eliminated by the effect of the force obtained using the directed magnetic field. In fact, a desired system stiffness can be obtained by adjusting the magnitude of the magnetic field at the level required by the application.

Finite beam element with exact shape functions for torsional analysis in thin-walled single- or multi-cell box girders

In this paper, a new formulation of beam-type finite element for the non-uniform torsion of simple or continuous thin-walled single- or multi-cell box girder with open and closed cross-sections is developed, considering both torsional warping and shear deformation effects (primary shear deformation due to Saint-Venant torsion and secondary shear deformation due to restrained torsion). The warping displacement of the cross-section is defined as the product of warping function representing the warping moment along the girder axis and a series of segmental shape functions describing the warping shape of each wall of the cross-section. Two different warping functions haven been defined and incorporated in Model A and Model B. The girder is subjected to arbitrary distributed or concentrated twisting moment and restrained by the general torsional-bending boundary conditions. The governing differential equations pertaining to torsion and torsional warping are obtained through the principle of minimum potential energy and the proposed finite beam element is refined by selecting closed-form homogeneous solutions of the differential equations as interpolation functions. Numerical examples are presented regarding rectangular hollow section (RHS) girders and single- or multi-cell box girders with open and closed cross-sections and the results obtained are compared with those provided by pioneering work or examined by using commercial finite-element models to validate the proposed beam-type element and to demonstrate the wide range of applicability and the convenience of using it.

Material-Oriented Shape Functions for FGM Plate Finite Element Formulation.

A four-noded finite element of a moderately thick plate made of functionally graded material (FGM) is presented. The base element is rectangular and can be extended to any shape using a transformation based on NURBS functions. The proposed 2D shape functions are consistent with the physical interpretation and describe the states of element displacement caused by unit displacements of nodes. These functions depend on the FGM’s material parameters and are called material-oriented. The shape function matrix is based on a superposition displacement field of two plate strips with 1D exact shape functions. A characteristic feature of the proposed formulation is full coupling of the membrane and bending states in the plate. The analytical form of the stiffness matrix and the nodal load vector was obtained, which leads to the numerical efficiency of the formulation. The element has been incorporated into Abaqus software with the use of Maple program. The finite element shows good convergence properties for different FGM models in the transverse direction to the middle plane of the plate. During derivation of the 2D plate element the formally exact 1D finite element for transverse nonhomogeneous FGM plate strip was developed.

Application of the Finite Element Method for the Solution of Stability Problems of the Timoshenko Beam with Exact Shape Functions

In this article, the problems of the stability of a Timoshenko beam are solved by the finite element method, with application of exact shape functions. These shape functions are derived from the solution of homogeneous equations of equilibrium. The article presents expressions for elements of stiffness matrixes and the geometric rigidity of a Timoshenko beam. The approach that is used makes it possible to exclude the shear (locking) effect and obtain a high degree of correspondence between the numerical and analytical solutions with a small number of finite elements.

Development of size-dependent consistent couple stress theory of Timoshenko beams

In this paper, a linear size-dependent Timoshenko beam model based on the consistent couple stress theory is developed to capture the size effects. The extended Hamilton's principle is utilized to obtain the governing differential equations and boundary conditions. The general form of boundary conditions and the concentrated loading are employed to determine the exact static/dynamic solution of the beam. Utilizing this solution for the beam's deformation and rotation, the exact shape functions of the consistent couple stress theory (C-CST) is extracted, which leads to the stiffness and mass matrices of a two-node C-CST finite element beam. Due to the complexity and high computational cost of using the exact solution's shape functions, in addition to the Ritz approximate solution, a two primary variable finite element model of C-CST is proposed, and the corresponding general deformation and rotation fields, shape functions, mass and stiffness matrices are calculated. The C-CST is validated by comparing the prediction of different beam models for a benchmark problem. For the fully and partially clamped cantilever, and free-free beams, the size dependency of the formulations is investigated. The static solutions of the classical and consistent couple stress Timoshenko beam models are compared, and a criterion for selecting the proper model is proposed. For a wide range of material properties, the relation between the beam length and length scale parameter is derived. It is shown that the validity domain of the consistent couple stress Timoshenko model barely depends on the beam's constituent material.