Weak Form Quadrature Element Formulation Research Articles

Weak form quadrature element analysis of spatial geometrically exact composite beams with torsional warping

Spatial thin-walled laminated composite beams under large deformations with torsional warping effects are studied. A novel two-dimensional cross-section model for composite beams with shear and torsion-warping deformations is proposed wherein the three-dimensional warping and coupling effects caused by inhomogeneous anisotropic materials are considered. A weak form quadrature element formulation is established for nonlinear analysis of geometrically exact composite beams with torsional warping and shear deformations. By combining the two-dimensional cross-section model and the nonlinear one-dimensional beam model, the three-dimensional stress and strain fields can be obtained. Numerical examples are provided to investigate the torsional warping effects and verify the effectiveness of the present formulation. Results of the present beam formulation which are achieved at a significantly lower computational cost are in excellent agreement with those of shell and solid models of commercial codes.

Elastic buckling analysis of side-cracked thin plates by weak form quadrature elements

A weak form quadrature element formulation is established for elastic buckling analysis of thin plates with side cracks. The entire problem domain is divided into a square subdomain around the crack tip and several quadrilateral subdomains elsewhere. The stress distribution of a cracked plate is determined in pre-buckling analysis. In either pre-buckling analysis or buckling analysis of a cracked plate, the potential energy of the inner square subdomain is computed using asymptotic displacement expansion around the crack tip, while the potential energy elsewhere is computed using quadrilateral weak form quadrature elements. A generalized eigenvalue problem is dealt with to determine the critical loads and buckling modes. The first ten non-dimensional buckling loads and critical buckling modes for several typical problems are given to validate the present formulation.

Torsional warping analysis of arbitrary beam cross-sections by weak form quadrature elements

Torsional warping is essential for modern beam theories. The commonly used finite element method generally suffers from low convergence rates. A weak form quadrature element formulation for torsional warping analysis of beams with arbitrary cross-section is established. Two numerical examples are given to verify the effectiveness of the present formulation. Results demonstrate that the present formulation is of high accuracy and high efficiency.

A weak form quadrature element formulation of geometrically exact strain gradient shells

A geometrically exact strain gradient shell model and its weak form quadrature element formulation are proposed. The governing equations, boundary conditions and corresponding variational form of the shell model are presented. The generalized differential quadrature analogue is employed for derivative approximations of displacements and rotations with C1 continuity conditions brought about by the second-order strain gradient terms. A novel scheme based on auxiliary vectors is developed to represent and update the derivatives of constrained rotations on element edges for the numerical approach. The present quadrature shell element is feasible to incorporate strain gradient terms and free from shear and membrane locking problems due to its adjustable high-order approximation property in nonlinear shell analysis. To demonstrate the potential of this formulation for nonlinear analysis of shells with size effects, five benchmark numerical examples are presented. The results verify the feasibility of this formulation and illustrate the strain gradient effects on elastic shells undergoing large displacements and rotations.

A weak form quadrature element formulation of geometrically exact beams with strain gradient elasticity

In this article, a numerical scheme for geometrically nonlinear analysis of elastic strain gradient beams is presented based on the weak form quadrature element formulation in combination with the geometrically exact beam model. Compared to standard geometrically exact beam elements, the present formulation incorporates the derivatives of displacements and rotations as additional degrees of freedom at element boundary nodes due to the C1 continuity requirements brought about by the introduction of strain gradients. A generalized differential quadrature scheme based on the Hermite interpolation functions is employed to approximate the derivatives of displacements and rotations. The quaternion-based scheme for spatial rotations and rotation derivatives is introduced to realize a total Lagrange formulation without singularities. This formulation is shown to be feasible to analyze strain gradient beam structures undergoing large displacements and rotations with the advantages of retaining strain objectivity and circumventing shear locking problems. Six typical examples are given to demonstrate the validity of the formulation.

Analysis of geometrically exact beams with torsion-warping deformation by weak form quadrature elements

Geometrically exact beams with torsion-warping deformation are studied. With three-dimensional finite strains, the Wagner effects as well as initial curvature and twist are taken into account in a weak form quadrature element formulation. The cross-section warping function of arbitrary cross-sections is obtained by a two-dimensional weak form quadrature element analysis, enhancing the overall accuracy of the beam formulation. Typical examples with different cross-sections and loading cases are presented to illustrate the effectiveness and robustness of the formulation.

A quadrature element formulation of geometrically nonlinear laminated composite shells incorporating thickness stretch and drilling rotation

In this paper, a weak form quadrature element formulation of a geometrically nonlinear shell model is proposed and applied for analysis of laminated composite shell structures. Thickness stretch parameters of the shell are incorporated for introducing 3D constitutive relations in the formulation. A drilling rotation constraint on the basis of polar decomposition of a modified deformation gradient is enforced by the Lagrange multiplier method and employed for implementing spatial finite rotations. The present formulation is shown to be feasible to model complex structures and circumvent locking problems naturally. A series of numerical benchmark examples are presented to demonstrate the validity of the formulation.

Geometrically nonlinear dynamic analysis of FG-CNTRC plates subjected to blast loads using the weak form quadrature element method

Geometrically nonlinear dynamic analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) rectangular plates subjected to blast loads is conducted based on Reddy’s higher-order shear deformation theory using the weak form quadrature element method. The von-Kármán strain terms are introduced to consider the geometrically nonlinear effects. The polymer composite plate is reinforced by single-walled carbon nanotubes (SWCNTs) with the uniform and functionally graded distribution in the plate thickness direction. The effective material properties of the FG-CNTRC plates are estimated with the extended rule of mixture. The Newmark-β time integration scheme and the Newton–Raphson iteration technique are adopted to solve the nonlinear incremental dynamic equilibrium equation in temporal domain. Comparative and convergence studies are carried out to validate the accuracy, efficiency and numerical stability of the presented weak form quadrature element formulation. The effects of the carbon nanotube distribution and volume fraction, plate width-to-thickness ratio, plate aspect ratio, load type and boundary condition on the dynamic response of the FG-CNTRC plates under blast loads are systematically investigated through the parametric studies.

A weak form quadrature element formulation of geometrically exact shells incorporating drilling degrees of freedom

Geometrically nonlinear analysis of shell structures is conducted using weak form quadrature elements. A new geometrically exact shell formulation incorporating drilling degrees of freedom is established wherein rotation quaternions in combination with a total Lagrange updating scheme are employed for rotation description. An extended kinematic condition to serve as the drilling rotation constraint, derived from polar decomposition of modified mid-surface deformation gradient, is exactly satisfied in the formulation. Several benchmark examples are presented to illustrate the versatility and robustness of the present formulation.

Nonlinear dynamic response analysis of cylindrical composite stiffened laminates based on the weak form quadrature element method

This paper presents a weak form quadrature element formulation for the analysis of nonlinear dynamic response of cylindrical composite stiffened laminates subjected to impact loads. A total Lagrangian updating scheme in combination with Newmark’s numerical integration is adopted to solve the nonlinear governing equations. The formulation of present shell and beam model stems from the basic concept of Ahmad (1970). Several numerical examples are firstly presented to verify the effectiveness and accuracy of the weak form quadrature element formulation. Parametric studies on the effects of the impact as well as geometric parameters, boundary conditions and lamination sequences of the laminates are performed. It shows that lamination sequences, geometric parameters, impact load type and duration have remarkable influence on the dynamic response of structures.

A weak form quadrature element formulation for geometrically exact thin shell analysis

The present paper addresses a weak form quadrature element formulation for the geometrically exact thin shell model in which the Kirchhoff-Love hypothesis is adopted. The displacement derivative continuity conditions are enforced by the reconstruction of rotation variables at the edges of elements. By the utilization of rotation quaternions, a total Lagrange updating scheme is implemented for edge constraint director rotations. Several numerical examples are presented to illustrate the effectiveness of the proposed formulation and the significant reduction in the number of degrees of freedom in geometrically nonlinear thin shell analysis with large displacements and rotations.

Weak form quadrature element analysis on nonlinear bifurcation and post-buckling of cylindrical composite laminates

This paper presents a weak form quadrature element formulation in the analysis of nonlinear bifurcation and post-buckling of cylindrical composite laminates subjected to transverse loads. A total Lagrangian updating scheme is used in combination with arc-length method and the branch switching method is adopted to identify the whole post-buckling procedure of the laminates. The formulation of the shell model is based on the basic concept of Ahmad. The coincidence of discrete nodes and integration points in quadrature element endows it with compactness and conciseness in the nonlinear buckling analysis of cylindrical laminates. Shear and membrane locking problems are avoided in the quadrature element formulation for its high-order approximation property. Several numerical examples are firstly presented to verify the effectiveness and accuracy of the present formulation. Parametric studies of the effects including rise, thickness, boundary conditions and lamination scheme of the cylindrical composite laminates on the bifurcation and post-buckling behaviors are performed.

Finite deformation elasto-plastic consolidation analysis of soft clay by the weak form quadrature element method

A weak form quadrature element formulation is established for finite deformation consolidation problems of an elasto-plastic saturated soft clay. The total Lagrangian (TL) description scheme and the weak form description of Biot’s theory are adopted in the derivation of the formulation. The constitutive model of the soil skeleton is based on a multiplicative decomposition of the deformation gradient into elastic and plastic parts. The exponential flow relation between the velocity of pore fluid and hydraulic gradient is used to describe the continuity condition in Biot’s theory. Results of numerical examples are compared with those of ABAQUS and previous studies, and very good agreement is reached, demonstrating the reliability and efficiency of the present formulation. The effect of non-Darcian flow on consolidation in the finite strain range is discussed and it is shown that, with the increase of the non-Darcian model parameters, the rate of consolidation and the differential settlement decrease.

A weak-form quadrature element formulation for 3D beam elements used in nonlinear and postbuckling analysis of space frames

Abstract A weak-form quadrature element formulation is presented for the three-dimensional beam element for use in the geometrically nonlinear and postbuckling analysis of space frames. Starting from the virtual work equation of a beam in the linearized, incremental sense, the quadrature element method (QEM) is employed to derive the elastic stiffness, geometric stiffness, and induced moment matrices of the beam with due account taken of the large rotations in three-dimensional space. All the stiffness matrices are adopted in the incremental-iterative analysis using the generalized displacement control (GDC) method, with specific considerations for the predictor and corrector phases. By comparing the results obtained for all the benchmark problems studied with existing ones, it is demonstrated that the present formulation is capable of predicting large displacements and rotations, as well as the postbuckling paths of space frames. The present formulation is featured by the fact that it is simple, straightforward, and reliable.

A weak form quadrature element formulation for coupled analysis of unsaturated soils

A weak form quadrature element formulation is proposed for coupled analysis of unsaturated soils. In the formulation, integration points are coincident with nodes where all variables are obtained so that numerical implementation is made easy. Accuracy can be improved by the increase of the approximation order. An elasto-plastic constitutive model is used for description of the mechanical behavior, and hydraulic hysteresis of the soil water characteristic curve is considered. Results are compared with available analytical solutions, experimental data and numerical solutions in the literature, and good agreement is reached. Rapid convergence is reached as compared with the conventional finite element method. The effect of the ground water level is discussed.

A quadrature element formulation of an energy–momentum conserving algorithm for dynamic analysis of geometrically exact beams

A weak form quadrature element formulation of geometrically exact beams is established and employed for dynamic analysis along with the energy and momentum conserving algorithm. In addition to its high computational efficiency and circumvention of shear locking problems, the formulation also ideally preserves strain objectivity and conservation of energy, which are violated in some finite element formulations for rotation interpolations. Several benchmark examples are presented to demonstrate the effectiveness of the formulation.

Three dimensional analysis of unconfined seepage in earth dams by the weak form quadrature element method

It remains challenging to determine the unknown free surface in three dimensional unconfined seepage in earth dams. A number of iterations are frequently required which make the problem computationally expensive. In the present research, a weak form quadrature element formulation is presented for three dimensional analysis of unconfined seepage which is an extension of the recently established method for two dimensional seepage problems. “Free points” are introduced by the interpolation of which the free surfaces are smoothly approximated. Grid lines are constructed in the element and the “free points” are confined to the lines when updated. An interpolatory scheme for locating the exit points is proposed. Formulations and procedures of the method are given in detail. Results of numerical examples are compared with available analytical solutions and numerical solutions in the literature and agreement is reached demonstrating the efficiency and reliability of the present formulation.

Weak form quadrature element analysis of geometrically exact shells

This paper presents a weak form quadrature element formulation of the stress resultant geometrically exact shell model proposed by Simo. A total Lagrangian updating scheme of rotation is adopted by the use of rotational quaternion. In addition to its conciseness for the coincidence of discrete nodes and integration points, the weak form quadrature element formulation exhibits computational feasibility as well as avoidance of shear and membrane locking problems for its high-order approximation property in nonlinear shell analysis. Several numerical examples are presented to illustrate the effectiveness of the proposed formulation.

Consolidation analysis of non-homogeneous soil by the weak form quadrature element method

A weak form quadrature element formulation for consolidation analysis of non-homogeneous saturated soil is established based on Biot’s theory. Numerical examples are given, and the results are compared with the analytical solutions available or those from the commercial finite element software ABAQUS, demonstrating accuracy and rapid convergence of quadrature element solutions. The disparity between various treatments of non-homogenous soil is discussed, and the effectiveness and advantages of the quadrature element formulation in consolidation analysis of non-homogeneous soil are demonstrated.

A quaternion-based weak form quadrature element formulation for spatial geometrically exact beams

This paper deals with spatial beams undergoing large displacements and rotations. A weak form quadrature element formulation is put forward based on the geometrically exact beam model. Spatial rotations are represented by quaternion algebras to achieve a total Lagrangian formulation without singularities. Besides high computational efficiency, the present formulation retains strain objectivity and avoids locking problems that may occur in low-order finite element formulations. Several benchmark examples are studied and results are compared with those available in the literature, demonstrating the feasibility, accuracy and efficiency of the weak form quadrature element formulation.