One‐dimensional analysis of thin‐walled closed beams having general cross‐sections

This paper presents a new one‐dimensional theory for static and dynamic analysis of thin‐walled closed beams with general cross‐sections. Existing one‐dimensional approaches are useful only for beams with special cross‐sections. Coupled deformations of torsion, warping and distortion are considered in the present work and a new approach to determine sectional warping and distortion shapes is proposed. One‐dimensional C⁰ beam elements based on the present theory are employed for numerical analysis. The effectiveness of the present theory is demonstrated in the analysis of thin‐walled beams having pillar sections of automobiles and excavators. Copyright © 2000 John Wiley & Sons, Ltd.     

A finite element displacement formulation for gradient elastoplasticity

We present a second gradient elastoplastic model for strain‐softening materials based entirely on a finite element displacement formulation. The stress increment is related to both the strain increment and its Laplacian. The displacement field is the only field needed to be discretized using a C¹ continuity element. The required higher‐order boundary conditions arise naturally from the displacement field. The model is developed to regularize the ill‐posedness caused by strain‐softening material behaviour. The gradient terms in the constitutive equations introduce an extra material parameter with dimensions of length allowing robust modelling of the post‐peak material behaviour leading to localization of deformation. Mesh insensitivity is demonstrated by modelling localization of deformation in biaxial tests. It is shown that both the thickness and inclination of the shear‐band zone are insensitive to the mesh directionality and refinement and agree with the expected theoretical and experimental values. Copyright © 2001 John Wiley & Sons, Ltd.     

Modeling of functionally graded sandwich shells accounting for variation in transverse displacement

In this study, a simple C₀ isoparametric finite element formulation based on higher-order shear deformation theory is presented for static analysis of functionally graded material sandwich shells (FGMSS). To characterize the membrane-flexure behavior observed in a functionally graded shell, a displacement field involving higher-order terms in in-plane and transverse fields is considered. The proposed kinematics field incorporates for transverse normal deformation, transverse shear deformation, and nonlinear variation of the in-plane displacement field through the thickness to predict the overall response of the shell in an accurate sense. To develop the efficient C₀ formulation, the derivatives of transverse displacement are treated as independent field variables (nodal unknowns). Voigt's rule of mixture is employed to ascertain the mechanical properties of each layer's constituents along the thickness direction. A wide range of numerical problems are solved assuming various parameters: side-thickness ratio, curvature-side ratio, and gradation parameter, and their interactions with regard to static analysis of FGMSS are discussed in brief. Deflection and stresses incorporating different thickness schemes of sandwich shells are presented in the form of figures. To validate the results, a functionally graded shell without sandwich arrangement is considered. Since no results are available on static analysis of FGMSS, the present 2D model based on the finite element method might be helpful in assessing the applicability of other analytical and numerical models in this area in the future.     

A C1 finite element including transverse shear and torsion warping for rectangular sandwich beams

A new three‐noded C¹ beam finite element is derived for the analysis of sandwich beams. The formulation includes transverse shear and warping due to torsion. It also accounts for the interlaminar continuity conditions at the interfaces between the layers, and the boundary conditions at the upper and lower surfaces of the beam. The transverse shear deformation is represented by a cosine function of a higher order. This allows us to avoid using shear correction factors. A warping function obtained from a three‐dimensional elasticity solution is used in the present model. Since the field consistency approach is accounted for interpolating the transverse strain and torsional strain, an exact integration scheme is employed in evaluating the strain energy terms. Performance of the element is tested by comparing the present results with exact three‐dimensional solu‐tions available for laminates under bending, and the elasticity three‐dimensional solution deduced from the de Saint‐Venant solution including both torsion with warping and bending. In addition, three‐dimensional solid finite elements using 27 noded‐brick elements have been used to bring out a reference solution not available for sandwich structures having high shear modular ratio between skins and core. A detailed parametric study is carried out to show the effects of various parameters such as length‐to‐thickness ratio, shear modular ratio, boundary conditions, free (de Saint‐Venant) and constrained torsion. Copyright © 1999 John Wiley & Sons, Ltd.     

A 3‐node C0 triangular element for the modified couple stress theory based on the smoothed finite element method

In this paper, a 3‐node C⁰ triangular element for the modified couple stress theory is proposed. Unlike the classical continuum theory, the second‐order derivative of displacement is included in the weak form of the equilibrium equations. Thus, the first‐order derivative of displacement, such as the rotation, should be approximated by a continuous function. In the proposed element, the derivative of the displacement is defined at a node using the node‐based smoothed finite element method. The derivative fields, continuous between elements and linear in an element, are approximated with the shape functions in element. Both the displacement field and the derivative field of displacement are expressed in terms of the displacement degree of freedom only. The element stiffness matrix is calculated using the newly defined derivative field. The performance of the proposed element is evaluated through various numerical examples.     

A simple explicit–implicit finite element tearing and interconnecting transient analysis algorithm

A simple explicit–implicit finite element tearing and interconnecting (FETI) algorithm (AFETI‐EI algorithm) is presented for partitioned transient analysis of linear structural systems. The present algorithm employs two decompositions. First, the total system is partitioned via spatial or domain decomposition to obtain the governing equations of motions for each partitioned domain. Second, for each partitioned subsystem, the governing equations are modally decomposed into the rigid‐body and deformational equations. The resulting rigid‐body equations are integrated by an explicit integrator, for its stability is not affected by step‐size restriction on account of zero‐frequency contents (ω = 0). The modally decomposed partitioned deformation equations of motion are integrated by an unconditionally stable implicit integration algorithm. It is shown that the present AFETI‐EI algorithm exhibits unconditional stability and that the resulting interface problem possesses the same solution matrix profile as the basic FETI static problems. The present simple dynamic algorithm, as expected, falls short of the performance of the FETI‐DP but offers a similar performance of implicit two‐level FETI‐D algorithm with a much cheaper coarse solver; hence, its simplicity may offer relatively easy means for conducting parallel analysis of both static and dynamic problems by employing the same basic scalable FETI solver, especially for research‐mode numerical experiments. Copyright © 2011 John Wiley & Sons, Ltd.     

A phase field model of dynamic fracture: Robust field updates for the analysis of complex crack patterns

The numerical modeling of dynamic failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies and demands the formulation of additional branching criteria. This drawback can be overcome by a diffusive crack modeling, which is based on the introduction of a crack phase field. Following our recent works on quasi‐static modeling of phase‐field‐type brittle fracture, we propose in this paper a computational framework for diffusive fracture for dynamic problems that allows the simulation of complex evolving crack topologies. It is based on the introduction of a local history field that contains a maximum reference energy obtained in the deformation history, which may be considered as a measure of the maximum tensile strain in the history. This local variable drives the evolution of the crack phase field. Its introduction provides a very transparent representation of the balance equation that governs the diffusive crack topology. In particular, it allows for the construction of a very robust algorithmic treatment for elastodynamic problems of diffusive fracture. Here, we extend the recently proposed operator split scheme from quasi‐static to dynamic problems. In a typical time step, it successively updates the history field, the crack phase field, and finally the displacement field. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples, which show the evolution of complex crack patterns under dynamic loading. Copyright © 2012 John Wiley & Sons, Ltd.     

薄壁箱梁纵向剪滞翘曲函数精度选择的研究

根据箱形结构纵向翘曲位移函数设置的基本原理,选择一系列符合箱梁基本翘曲模式的翘曲位移函数,然后以最小势能原理为基础,综合考虑剪力滞后效应、剪切变形和转动惯量的影响,推导出相应翘曲位移函数箱形截面梁的振动控制微分方程和边界条件,据此得出箱形结构的固有频率方程。依据固有频率方程求出特定边界条件下相应翘曲位移函数箱形结构的自振频率,然后借助自振频率的大小对所设置翘曲位移函数的精确度做出评判。静力分析算例结果证明了翘曲位移函数精度选择的必要性,该文将解析解与数值解结果进行了比较,证明了方法的有效性。     

Higher‐order beam analysis of box beams connected at angled joints subject to out‐of‐plane bending and torsion

The difficulty in the analysis of thin‐walled beams by a beam theory comes from slowly decaying end effects associated with warping and distortion. However, a beam theory without considering such effects yields inaccurate solutions especially near beam ends. Numerical analysis using a higher‐order beam theory capable of representing such effects is now available, but the analysis of a series of box beams connected by angled joints still remains an unsolved problem because of the lack of a matching condition at the joint. The objectives of this investigation are to develop a field‐variable‐matching technique at an angled joint through a higher‐order beam theory and to implement it in the finite element formulation. Thin‐walled box beams in consideration are assumed to be subject to out‐of‐plane bending and torsion. Thus, the minimization of three‐dimensional displacement mismatch is used to relate the field variables at a joint intersection. The minimization condition turns out to represent coupling effects of different deformation kinematics such as torsion, bending, distortion and warping. Point‐wise displacement matching is not possible with a higher‐order beam theory. The validity of the proposed technique was verified by a finite element analysis using two‐node higher‐order beam elements applied to some benchmark problems. Copyright © 2008 John Wiley & Sons, Ltd.     

On consistent discretization of inertia terms for C0 structural elements with modified stiffness matrices

In the existing finite element calculations of dynamic problems using C⁰ structural elements, the inertia terms are evaluated without any reference to the modifications such as reduced integration, projections etc., typically needed in the discretization of the stiffness terms. A different discretization of inertia is discussed here. It is based on the following two observations. First, as shown in this work (at least for the beam problems), the modified stiffness matrix for a given C⁰ element can be obtained by standard, unmodified approach, in which degrees of freedom remain unchanged, but the shape functions are different. Those modified functions are of higher order and define the translational field within the element in terms of both translational and rotational parameters. Second, if standard consistent approach to the formulation of dynamic problems is to be followed, approximation of the displacement field used in the unmodified evaluation of the stiffness terms should also be used in discretization of the inertia terms. This implies that the modified higher-order functions should be employed when evaluating the element mass matrix for the C⁰ elements with modified stiffness matrices. As a consequence of this approach, consistency between formulation of the inertia and stiffness terms is restored. This leads to inertial coupling between rotational and translational degrees of freedom, which is absent in standard evaluation of inertia. It is demonstrated that this approach tends to improve accuracy of dynamic computations. © 1997 by John Wiley & Sons, Ltd.     

Singular‐ended spline interpolation for two‐dimensional boundary element analysis

A method of interpolation of the boundary variables that uses spline functions associated with singular elements is presented. This method can be used in boundary element method analysis of 2‐D problems that have points where the boundary variables present singular behaviour. Singular‐ended splines based on cubic splines and Overhauser splines are developed. The former provides C²‐continuity and the latter C¹‐continuity across element edges. The potentialities of the methodology are demonstrated analysing the dynamic response of a 2‐D rigid footing interacting with a half‐space. It is shown that, for a given number of elements at the soil–foundation interface, the singular‐ended spline interpolation increases substantially the displacement convergence rate and delivers smoother traction distributions. Copyright © 2000 John Wiley & Sons, Ltd.     

A two‐node curved axisymmetric shell element based on coupled displacement field

An efficient two‐node curved axisymmetric shell element is proposed. The element with three degrees of freedom per node accounts for the transverse shear flexibility and rotary inertia. The strain components are defined in a curvilinear co‐ordinate frame. The variation of normal displacement (w) along the meridian is represented by a cubic polynomial. The relevant constitutive relations and the differential equations of equilibrium in the meridional plane of the shell are used to derive the polynomial field for the tangential displacement (u) and section rotation (θ). This results in interdependent polynomials for the field variables w, u and θ, whose coefficients are coupled by generalized degrees of freedom and geometric and material properties of the element. These coupled polynomials lead to consistently vanishing coefficients for the membrane and transverse shear strain fields even in the limit of extreme thinness, without producing any spurious constraints. Thus the element is devoid of membrane and shear locking in thin limit of inextensible and shearless bending, respectively. Full Gaussian integration rules are employed for evaluating stiffness marix, consistent load vector and consistent mass matrix. Numerical results are presented for axisymmetric deep/shallow shells having curved/straight meridional geometries for static and free vibration analyses. The accuracy and convergence characteristics of this C⁰ element are superior to other elements of the same class. The performance of the element demonstrates its applicability over a wide range of axisymmetric shell configurations. Copyright © 1999 John Wiley & Sons, Ltd.     

Exact static element stiffness matrices of non‐symmetric thin‐walled curved beams

An evaluation procedure of exact static stiffness matrices for curved beams with non‐symmetric thin‐walled cross section are rigorously presented for the static analysis. Higher‐order differential equations for a uniform curved beam element are first transformed into a set of the first‐order simultaneous ordinary differential equations by introducing 14 displacement parameters where displacement modes corresponding to zero eigenvalues are suitably taken into account. This numerical technique is then accomplished via a generalized linear eigenvalue problem with non‐symmetric matrices. Next, the displacement functions of displacement parameters are exactly calculated by determining general solutions of simultaneous non‐homogeneous differential equations. Finally an exact stiffness matrix is evaluated using force–deformation relationships. In order to demonstrate the validity and effectiveness of this method, displacements and normal stresses of cantilever thin‐walled curved beams subjected to tip loads are evaluated and compared with those by thin‐walled curved beam elements as well as shell elements. Copyright © 2004 John Wiley & Sons, Ltd.     

Exact stiffness matrix of mono-symmetric composite I-beams with arbitrary lamination

The exact stiffness matrix, based on the simultaneous solution of the ordinary differential equations, for the static analysis of mono-symmetric arbitrarily laminated composite I-beams is presented herein. For this, a general thin-walled composite beam theory with arbitrary lamination including torsional warping is developed by introducing Vlasov’s assumption. The equilibrium equations and force–deformation relations are derived from energy principles. The explicit expressions for displacement parameters are then derived using the displacement state vector consisting of 14 displacement parameters, and the exact stiffness matrix is determined using the force–deformation relations. In addition, the analytical solutions for symmetrically laminated composite beams with various boundary conditions are derived as a special case. Finally, a finite element procedure based on Hermitian interpolation polynomial is developed. To demonstrate the validity and the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element results using the Hermitian beam elements and ABAQUS’s shell element.     

A systematic construction of B‐bar functions for linear and non‐linear mixed‐enhanced finite elements for plane elasticity problems

In a previous paper a modified Hu–Washizu variational formulation has been used to derive an accurate four node plane strain/stress finite element denoted QE2. For the mixed element QE2 two enhanced strain terms are used and the assumed stresses satisfy the equilibrium equations a priori for the linear elastic case. In this paper an alternative approach is discussed. The new formulation leads to the same accuracy for linear elastic problems as the QE2 element; however it turns out to be more efficient in numerical simulations, especially for large deformation problems. Using orthogonal stress and strain functions we derive B̄ functions which avoid numerical inversion of matrices. The B̄ ‐strain matrix is sparse and has the same structure as the strain matrix B obtained from a compatible displacement field. The implementation of the derived mixed element is basically the same as the one for a compatible displacement element. The only difference is that we have to compute a B̄ ‐strain matrix instead of the standard B ‐matrix. Accordingly, existing subroutines for a compatible displacement element can be easily changed to obtain the mixed‐enhanced finite element which yields a higher accuracy than the Q4 and QM6 elements. Copyright © 1999 John Wiley & Sons, Ltd.     

Four‐node semi‐EAS element in six‐field nonlinear theory of shells

We propose a new four‐node C⁰ finite element for shell structures undergoing unlimited translations and rotations. The considerations concern the general six‐field theory of shells with asymmetric strain measures in geometrically nonlinear static problems. The shell kinematics is of the two‐dimensional Cosserat continuum type and is described by two independent fields: the vector field for translations and the proper orthogonal tensor field for rotations. All three rotational parameters are treated here as independent. Hence, as a consequence of the shell theory, the proposed element has naturally six engineering degrees of freedom at each node, with the so‐called drilling rotation. This property makes the element suitable for analysis of shell structures containing folds, branches or intersections. To avoid locking phenomena we use the enhanced assumed strain (EAS) concept. We derive and linearize the modified Hu–Washizu principle for six‐field theory of shells. What makes the present approach original is the combination of EAS method with asymmetric membrane strain measures. Based on literature, we propose new enhancing field and specify the transformation matrix that accounts for the lack of symmetry. To gain knowledge about the suitability of this field for asymmetric strain measures and to assess the performance of the element, we solve typical benchmark examples with smooth geometry and examples involving orthogonal intersections of shell branches. Copyright © 2006 John Wiley & Sons, Ltd.     

Active control of FGM plates subjected to a temperature gradient: Modelling via finite element method based on FSDT

An efficient finite element formulation based on a first‐order shear deformation theory (FSDT) is presented for the active control of functionally gradient material (FGM) plates with integrated piezoelectric sensor/actuator layers subjected to a thermal gradient; this is accomplished using both static and dynamic piezothermoelastic analyses. The formulation based on FSDT can be applied to a range of relatively thin‐to‐moderately thick plates. A constant displacement‐cum‐velocity feedback control algorithm coupling the direct and inverse piezoelectric effects is applied to provide active feedback control of the integrated FGM plate in a self‐monitoring and self‐controlling system. Numerical results for the control of bending and torsional deflections and/or vibrations are presented for a FGM plate comprising zirconia and aluminium. The effects of constituent volume fraction and the influence of feedback control gain on the static and dynamic responses of the FGM plates are examined in detail. Copyright © 2001 John Wiley & Sons, Ltd.     

Strain gradient differential quadrature finite element for moderately thick micro-plates

In this study, we integrate the advantages of differential quadrature method (DQM) and finite element method (FEM) to construct a C¹-type four-node quadrilateral element with 48 degrees of freedom (DOF) for strain gradient Mindlin micro-plates. This element is free of shape functions and shear locking. The C¹-continuity requirements of deflection and rotation functions are accomplished by a fourth-order differential quadrature (DQ)-based geometric mapping scheme, which facilitates the conversion of the displacement parameters at Gauss-Lobatto quadrature (GLQ) points into those at element nodes. The appropriate application of DQ rule to non-rectangular domains is proceeded by the natural-to-Cartesian geometric mapping technique. Using GLQ and DQ rules, we discretize the total potential energy functional of a generic micro-plate element into a function of nodal displacement parameters. Then, we adopt the principle of minimum potential energy to determine element stiffness matrix, mass matrix, and load vector. The efficacy of the present element is validated through several examples associated with the static bending and free vibration problems of rectangular, annular sectorial, and elliptical micro-plates. Finally, the developed element is applied to study the behavior of freely vibrating moderately thick micro-plates with irregular shapes. It is shown that our element has better convergence and adaptability than that of Bogner-Fox-Schmit (BFS) one, and strain gradient effects can cause a significant increase in vibration frequencies and a certain change in vibration mode shapes.     

A two‐noded locking–free shear flexible curved beam element

A new two‐noded shear flexible curved beam element which is impervious to membrane and shear locking is proposed herein. The element with three degrees of freedom at each node is based on curvilinear deep shell theory. Starting with a cubic polynomial representation for radial displacement (w), the displacement field for tangential displacement (u) and section rotation (θ) are determined by employing force‐moment and moment‐shear equilibrium equations. This results in polynomial displacement field whose coefficients are coupled by generalized degrees of freedom and material and geometric properties of the element. The procedure facilitates quartic polynomial representation for both u and θ for curved element configurations, which reduces to linear and quadratic polynomials for u and θ, respectively, for straight element configuration. These coupled polynomial coefficients do not give rise to any spurious constraints even in the extreme thin regimes, in which case, the present element exhibits excellent convergence to the classical thin beam solutions. This simple C⁰ element is validated for beam having straight/curved geometries over a wide range of slenderness ratios. The results indicates that performance of the element is much superior to other elements of the same class. Copyright © 1999 John Wiley & Sons, Ltd.     

A three‐dimensional C1 finite element for gradient elasticity

In gradient elasticity strain gradient terms appear in the expression of virtual work, leading to the need for C¹ continuous interpolation in finite element discretizations of the displacement field only. Employing such interpolation is generally avoided in favour of the alternative methods that interpolate other quantities as well as displacement, due to the scarcity of C¹ finite elements and their perceived computational cost. In this context, the lack of three‐dimensional C¹ elements is of particular concern. In this paper we present a new C¹ hexahedral element which, to the best of our knowledge, is the first three‐dimensional C¹ element ever constructed. It is shown to pass the single element and patch tests, and to give excellent rates of convergence in benchmark boundary value problems of gradient elasticity. It is further shown that C¹ elements are not necessarily more computationally expensive than alternative approaches, and it is argued that they may be more efficient in providing good‐quality solutions. Copyright © 2008 John Wiley & Sons, Ltd.