Weak Form Quadrature Element Method Research Articles

Transverse free vibration analysis of thin sectorial plates by the weak form quadrature element method

The weak form quadrature element method is applied to free vibration analysis of thin sectorial plates with arbitrary vertex angles and boundary conditions. To tackle the strong stress singularity around the vertex, analytical displacement descriptions are introduced into the inner sectorial subdomain, while the outer annular subdomain is modeled by a single weak form quadrature thin plate element. The continuity on the interface between the two subdomains is enforced afterwards. Eventually, a generalized eigenvalue formulation is established after introducing Hamilton’s principle. The first six non-dimensional frequency parameters for various vertex angles and boundary conditions are obtained and compared with available results. Several typical free vibration modes are plotted. The accuracy, convergence rate, and computational cost of the present formulation are discussed at length.

Free vibration analysis of thin plates with side cracks by the weak form quadrature element method

The weak form quadrature element method is applied to vibration analysis of thin plates with side cracks. Eigenfunctions are introduced into the inner square domain around the crack tip, while the rest of the problem domain is analyzed by the weak form quadrature element method (QEM). The continuity of displacements and slopes on the interface between the inner and outer domains is enforced. A generalized eigenvalue formulation is established after introducing Hamilton’s principle. The first ten nondimensional frequency parameters for several typical problems are obtained and compared with those available in the literature. The corresponding modes of free vibration are plotted, and the influence of the crack on vibration frequencies and modes is discussed at length.

Nonlinear Bifurcation and Post-buckling Analysis of Cylindrical Composite Stiffened Laminates Based on Weak Form Quadrature Element Method

Nonlinear Bifurcation and Post-buckling Analysis of Cylindrical Composite Stiffened Laminates Based on Weak Form Quadrature Element Method

Determination of asymptotic field coefficients for multi-material antiplane V-notches by the weak form quadrature element method

Weak form quadrature element analysis (QEA) of multi-material antiplane V-notches is conducted. Angular eigenfunctions are introduced into the inner subdomains where stress singularity exists, while the outer domain is divided into several quadrature elements according to geometric boundary conditions and material properties. Algebraic equations are established and coefficients are solved for after introducing the minimum potential energy principle. The influence of the first eight terms on improvement of far field stress and displacement estimations is demonstrated. As compared to the computational costs of other methods, much fewer degrees of freedom are needed in the present formulation to achieve the same accuracy of results, exhibiting the merits of QEA of fracture problems.

Free vibration analysis of Timoshenko pipes conveying fluid with gravity and different boundary conditions

Abstract In this paper, the free vibration of Timoshenko pipes conveying fluid with gravity and different boundary conditions is analyzed by using the weak form quadrature element method. The dimensionless frequency and critical flow velocity have been calculated considering various parameters such as the mass ratio of pipe to fluid, slenderness ratio and gravity. The numerical results with different numbers of discrete points are tested, and the results show that the method has a fast convergence speed. Some numerical results are compared with the existing literature, which proves that the weak form quadrature element method has high accuracy and computational efficiency. This paper shows that the weak form quadrature element method is a preferred method for analyzing fluid-conveying pipes with various boundary conditions.

An improved weak-form quadrature element (IWQE) method for static and dynamic analysis of non-homogeneous plane trusses

Based on Chebyshev interpolation and Gauss-Lobatto quadrature together with the variational principle, an improved weak-form quadrature element (IWQE) method is developed and further customized for analyzing plane truss structures. Compared to existing WQE method which adopts Lagrange interpolation and Gauss quadrature, this developed IWQE method has demonstrated robustness for elements with a large number of quadrature points. Compared to finite element (FE) method whose stiffness matrix of an element is positive and semi-definite, the stiffness matrix of the IWQE method of an element is always positive and definite. Therefore, the displacements of an element's internal nodes can be uniquely expressed by using the values of its end nodes only. Accordingly, the sizes of the assembled matrices only depend on the number of end nodes and the orders of the assembled matrices for the whole structure can be greatly reduced. Such attributes can substantially increase the computational efficiency. To validate the developed IWQE method, the static and dynamic analysis of a plane truss structure with non-homogeneous properties are taken as an example for case study. Compared to the other numerical methods, such as differential quadrature element (DQE), WQE and FE, this IWQE method developed in present work demonstrates better convergence, accuracy and robustness, providing a more efficient and powerful numerical tool for analyzing structures with non-homogeneous attributes.

Determination of asymptotic field coefficients for V-notches by the weak form quadrature element method

Weak form quadrature element methods (QEM) are chosen for determination of asymptotic field coefficients for in-plane V-notch problems. The entire domain is divided into two parts the potential energies of which are computed separately. Stress and displacement descriptions are introduced into the sector domain around the notch tip. Coefficients are extracted from the algebraic equations directly after introducing the minimum potential energy principle. The effect of higher order terms on the local stress and displacement descriptions near V-notch tip is analyzed. High adaptivity and fast convergence of the QEM are demonstrated. Comparisons of several benchmark example results and those in the literature are made to validate the present formulation.

A novel formulation for the weak quadrature element method for solving vibration of strain gradient graded nonlinear nanobeams

A novel formulation of the weak form quadrature element method, referred to as the locally adaptive weak quadrature element method, is proposed to develop elements for nonlinear graded strain gradient Timoshenko and Euler–Bernoulli nanobeams. The equations of motion are obtained based on Hamilton principle while accounting for the position of the physical neutral axis. The proposed elements use Gauss quadrature points to ensure full integration of the variational statement. The proposed formulation develops matrices based on the differential quadrature method which employs Lagrange-based polynomials. These matrices can be modified to accommodate any number of extra derivative degrees of freedom including third-order beams and higher-order strain gradient beams without requiring an entirely new formulation. The performance of the proposed method is evaluated based on the free vibration response of the linear and nonlinear strain gradient Timoshenko and Euler–Bernoulli nanobeams. Both linear and nonlinear frequencies are evaluated for a large number of configurations and boundary conditions. It is shown that the proposed formulation results in good accuracy and an improved convergence speed as compared to the locally adaptive quadrature element method and other weak quadrature element methods available in the literature.

Weak-Form Quadrature Element Method: A Comparative Review of Different Formulations and Its Comprehensive Assessment

Weak-Form Quadrature Element Method: A Comparative Review of Different Formulations and Its Comprehensive Assessment

Weak form quadrature element analysis of crack-tip asymptotic field coefficients

Coefficients of crack-tip asymptotic fields are computed using the weak form quadrature element method (QEM). The entire domain is divided into several integrable subdomains and expressions of the crack tip asymptotic displacement field are introduced into those near the crack tip. Algebraic equations with coefficients of the crack-tip asymptotic field and nodal displacements are established using the minimum potential energy principle. Coefficients of arbitrary order can be extracted directly by solving the equations. The first ten order coefficients for several typical examples are given in this paper. As compared with the finite element method (FEM) and the boundary collocation method (BCM), fewer degrees of freedom are needed in the present approach to achieving the same accuracy. In addition, the merits of succinctness and directness of the weak form quadrature element method for fracture analysis are demonstrated.

Analysis of pillow effect for isosceles triangular membranes of mesh reflector antenna by the weak form quadrature element method

The pillow effect is a key weakening factor in the surface accuracy of mesh antennas. Previous studies, however, have significant limitations in such antennas because they have been largely based on specific assumptions, which have failed to investigate the effects of mesh membrane material characteristics on the pillow effect. Thus, in this study, static analysis of a triangular membrane was performed using the weak form quadrature element method to explore the effects of material parameters in various directions on the pillow effect in polar coordinates. Because of its lack of central symmetry, the strain–displacement relations from Föppl–von Kármán membrane theory in polar coordinates were improved for calculations in triangular membranes. In polar coordinates, the triangular membrane domain was mapped into one sector for the discretion of integrals using bilinear interpolation. The convergence of the method results was examined, and the accuracy was demonstrated by comparing the computed deflections to those from the finite element method. Based on the present method, the effects of material properties on the pillow effect in different sections of a triangular membrane were examined, as was the influence of the shape of an isosceles triangle.

A weak form temporal quadrature element formulation for linear structural dynamics

PurposeVarious time integration methods and time finite element methods have been developed to obtain the responses of structural dynamic problems, but the accuracy and computational efficiency of them are sometimes not satisfactory. The purpose of this paper is to present a more accurate and efficient formulation on the basis of the weak form quadrature element method to solve linear structural dynamic problems.Design/methodology/approachA variational principle for linear structural dynamics, which is inspired by Noble's work, is proposed to develop the weak form temporal quadrature element formulation. With Lobatto quadrature rule and the differential quadrature analog, a system of linear equations is obtained to solve the responses at sampling time points simultaneously. Computation for multi-elements can be carried out by a time-marching technique, using the end point results of the last element as the initial conditions for the next.FindingsThe weak form temporal quadrature element formulation is conditionally stable. The relation between the normalized length of element and the suggested number of integration points in one element is given by a simple formula. Results show that the present formulation is much more accurate than other time integration methods and its dissipative property is also illustrated.Originality/valueThe weak form temporal quadrature element formulation provides a choice with high accuracy and efficiency for solution of linear structural dynamic problems.

Nonlinear forced vibration analysis of the composite shaft-disk system combined the reduced-order model with the IHB method

In this paper, the internal resonance phenomena of a composite shaft-disk system with multi-degrees-of-freedom are analyzed. The force caused by the unbalanced mass of the disk is considered as an external excitation force. The shaft is simply supported. Shear deformation and gyroscopic effects are considered. The strain–displacement relationship of the shaft element is expressed using the Timoshenko beam theory. Each node has 5 degrees of freedom. SHBT (simplified homogenized beam theory) is applied to calculate the stiffness of the composite shaft. WQEM (weak form quadrature element method) is used to construct the element matrices, and the system matrices are established using the element matrix assembly rule of the FEM (finite element method). The reduced-order model is applied to reduce the calculation time. IHB (incremental harmonic balance) method is utilized to solve the nonlinear equations of motion of the composite shaft-disk system. The nonlinear vibration characteristics of the Jeffcott rotor are analyzed using the proposed method and compared with the results of previous researches, and the results are very similar. Based on these considerations, the nonlinear vibration phenomena of the composite shaft-disk system with multi-degrees-of-freedom are considered at the several resonance points.

Analysis of the nonlinear forced vibration and stability of composite beams using the reduced-order model

The purpose of this paper is to analyze the nonlinear vibration of composite beams and the stability of the solution by using the reduced-order model and the incremental harmonic balance (IHB) method. Timoshenko beam theory is used to indicate the displacement of the beam element. Each nodal point has three degrees of freedom. Simplified homogenized beam theory is used to calculate the equivalent moduli of each ply of the composite beam. Element matrices are created using the weak form quadrature element method, and the equation of motion at the element is created using Lagrange’s equation. A system matrix is created using the element matrix assemble rule of the finite element method. In order to reduce the calculation time, a reduced-order model is used. The nonlinear forced vibration equation is solved using the IHB method. The results calculated using the non-reduced-order model and the reduced-order model are compared, and the results are very close. Based on this, the reduced-order model is used to analyze the nonlinear vibrations of composite beams at the first resonance point, and stability tests are conducted for the calculated solutions using the multivariable Floquet theory.

Stability Analysis of Composite Shafts Considering Internal Damping and Coupling Effect

A mathematical model is proposed to analyze the stability of composite shafts, considering the internal damping, transverse shear deformation and Poisson’s coupling effect at the same time. The strain–displacement relations are described using the Timoshenko beam theory, and the strain energy and kinetic energy are expressed using the weak form quadrature element method (QEM), for which each nodal point has 4 degrees of freedom. Then, the motion equation of the system is established using the Lagrange equation. The instability thresholds of the composite shaft are determined using the proposed model. The results were compared with those calculated by the equivalent modulus beam theory (EMBT), equivalent single layer theory (ESLT) and simplified homogenized beam theory (SHBT). Good agreement has been achieved. Therefore, the proposed model can be effectively used for the dynamic analysis of composite shafts.

A high-order FEM formulation for free and forced vibration analysis of a nonlocal nonlinear graded Timoshenko nanobeam based on the weak form quadrature element method

The purpose of this paper is to provide a high-order finite element method (FEM) formulation of nonlocal nonlinear nonlocal graded Timoshenko based on the weak form quadrature element method (WQEM). This formulation offers the advantages and flexibility of the FEM without its limiting low-order accuracy. The nanobeam theory accounts for the von Kármán geometric nonlinearity in addition to Eringen’s nonlocal constitutive models. For the sake of generality, a nonlinear foundation is included in the formulation. The proposed formulation generates high-order derivative terms that cannot be accounted for using regular first- or second-order interpolation functions. Hamilton’s principle is used to derive the variational statement which is discretized using WQEM. The results of a WQEM free vibration study are assessed using data obtained from a similar problem solved by the differential quadrature method (DQM). The study shows that WQEM can offer the same accuracy as DQM with a reduced computational cost. Currently the literature describes a small number of high-order numerical forced vibration problems, the majority of which are limited to DQM. To obtain forced vibration solutions using WQEM, the authors propose two different methods to obtain frequency response curves. The obtained results indicate that the frequency response curves generated by either method closely match their DQM counterparts obtained from the literature, and this is despite the low mesh density used for the WQEM systems.

Low velocity impact analysis of high-order rectangular FG-CNTRC plates using the weak form QEM

The functionally graded carbon nanotubes reinforced composites (FG-CNTRC) plates have excellent properties and broad application prospects. In the analysis of low velocity impact response of functionally graded carbon nanotubes reinforced composite plates, innovative numerical formulations are urgently needed. As a new high-order numerical method, the weak form quadrature element method (QEM) has its unique advantages and has been widely used in the engineering structural field. In this paper, the nonlinear Hertzian contact law is employed to describe the impact process between a rigid impactor and the rectangular FG-CNTRC plate, that was modelled with the Reddy’s high-order shear deformation theory. By using the weak form QEM, the calculation model for the impact system is established, and the numerical solution is carried out with the help of Newmark-β step-by-step time domain integration method and Newton-Raphson iteration technology. The effectiveness and accuracy of the weak form QEM formulation were verified through the comparison with the existing literature and the relevant low velocity impact properties of FG-CNTRC plate are explored.

Free vibration of FGM annular sectorial plates with arbitrary boundary supports and large sector angles

An original work is presented for the free vibration analysis of thin functionally graded material (FGM) annular sectorial plates with arbitrary boundary supports using the weak form quadrature element method (QEM). Novel C 1 quadrature annular sectorial plate element is developed based on the physically neutral surface thin plate theory in cylindrical coordinate system. Three displacement functions are used to avoid the introduction of the mixed second order derivative degree of freedom at corner points. The derived explicit formulations with a varying node number can be applied for any type of node. A way to define the elastic spring constants is proposed to ensure the independent of the non-dimensional frequency parameter on the power-law exponent. The convergence and accuracy of the present model are validated by comparing the obtained results with those available in literatures. It is demonstrated that the proposed model is simple, straightforward and has good performance. A wide range of numerical results are reported to demonstrate the effect of the sector angle, ratio of inner/outer radii, and boundary conditions on the vibration behavior of the thin annular sectorial plate. Some tabulated results should be new which may serve as a reference for verifying numerical solutions obtained by other methods.

Geometrically nonlinear quadrature element analysis of spatial curved beams

A novel curved beam quadrature element is presented for geometrically nonlinear analysis of spatial curved beams. Starting from the incremental virtual work equation of the curved beam, the weak form quadrature element method (QEM) is employed to derive the elastic stiffness, geometric stiffness, and induced moment matrices of the curved 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 testing the constructed curved beam quadrature element with four benchmark problems, it is demonstrated that the element avoids the shear and membrane locking phenomena due to its convenient feature of high-order approximation. In addition, the element is capable of predicting large displacements and rotations, as well as postbuckling paths of spatial beams. Especially, the presented element is featured by unifying the analyses for both curved and straight beams.

Flexural and eigen-buckling analysis of steel-concrete partially composite plates using weak form quadrature element method

Flexural and eigen-buckling analyses for rectangular steel-concrete partially composite plates (PCPs) with interlayer slip under simply supported and clamped boundary conditions are conducted using the weak form quadrature element method (QEM). Both of the derivatives and integrals in the variational description of a problem to be solved are directly evaluated by the aid of identical numerical interpolation points in the weak form QEM. The effectiveness of the presented numerical model is validated by comparing numerical results of the weak form QEM with those from FEM or analytic solution. It can be observed that only one quadrature element is fully competent for flexural and eigen-buckling analysis of a rectangular partially composite plate with shear connection stiffness commonly used. The numerical integration order of quadrature element can be adjusted neatly to meet the convergence requirement. The quadrature element model presented here is an effective and promising tool for further analysis of steel-concrete PCPs under more general circumstances. Parametric studies on the shear connection stiffness and length-width ratio of the plate are also presented. It is shown that the flexural deflections and the critical buckling loads of PCPs are significantly affected by the shear connection stiffness when its value is within a certain range.