Consistent Mass Matrices Research Articles

Free-Vibration and Buckling Analyses of Beams using Kriging-Based Timoshenko Beam Elements with the Discrete Shear Gap Technique

A family of locking-free Kriging-based Timoshenko beam elements with a new implementation of the discrete shear gap technique was recently developed (referred to as K-beam-DSG1 elements). Performance of the K-beam-DSG1 elements has been shown to be very satisfactory in the linear static analysis of beams for a wide variety of thicknesses. This paper presents further development of the K-beam-DSG1 elements to free vibration and bifurcation buckling analyses of prismatic and non-prismatic beams. Consistent Kriging-based mass matrices are used for free vibration analysis and similarly, consistent Kriging-based geometric stiffness matrices are used for buckling analysis. The results show that for most of the cases, the K-beam-DGS1 elements yield remarkably accurate natural frequencies and critical compressive loads using a reasonable number of elements to discretize the beam. For an axially functionally graded fixed-fixed supported beam, however, the elements fail to predict the critical load accurately.

Analytical and numerical investigation of beam-spring systems with varying stiffness: a comparison of consistent and lumped mass matrices considerations

This study examined the vibration behavior of a beam with linear spring attachments using finite element analysis. It aims to determine the natural frequency with both consistent/coupled mass and lumped mass matrices. The natural frequencies and corresponding mode shapes were correctly determined which formed the basis of any further noise vibration and severity calculations and impact or crash analysis. In order to obtain eigenfrequencies subject to the attached spring, the characteristic equation was obtained by eigenfunctions expansion whose roots were extracted using the root-finding technique. The finite element method by coupled and lumped mass matrices was then used to determine complete mode shapes against various eigenfrequencies. The mode shapes were then analyzed subject to supports with varying stiffness thereby comparing the analytical and numerical results in case of consistent and lumped masses matrices so as to demonstrate how the present analysis could prove more valuable in mathematical and engineering contexts. Utilizing a consistent mass matrix significantly enhanced accuracy compared to a lumped mass matrix, thereby validating the preference for the former, even with a limited number of beam elements. The results indicated that substantial deflection occurred at the beam's endpoints, supporting the dynamic behavior of the spring-beam system.

An explicit predictor/multicorrector time marching with automatic adaptivity for finite-strain elastodynamics

We propose a time-adaptive predictor/multi-corrector method to solve hyperbolic partial differential equations, based on the generalized-α scheme that provides user-control on the numerical dissipation and second-order accuracy in time. Our time adaptivity uses an error estimation that exploits the recursive structure of the variable updates. The predictor/multicorrector method explicitly updates the equation system but computes the residual of the system implicitly. We analyze the method's stability and describe how to determine the parameters that ensure high-frequency dissipation and accurate low-frequency approximation. Subsequently, we solve a linear wave equation, followed by non-linear finite strain deformation problems with different boundary conditions. Thus, our method is a straightforward, stable and computationally efficient approach to simulate real-world engineering problems. Finally, to show the performance of our method, we provide several numerical examples in two and three dimensions. These challenging tests demonstrate that our predictor/multicorrector scheme dynamically adapts to sudden energy releases in the system, capturing impacts and boundary shocks. The method efficiently and stably solves dynamic equations with consistent and under-integrated mass matrices conserving the linear and angular momenta as well as the system's energy for long-integration times.

An Alternative Approach to Obtaining Stiffness and Mass Matrices for Three-Dimensional Bernoulli-Euler and Timoshenko Beam-Columns

In the static analysis of beam-column systems using matrix methods, polynomials are using as the shape functions. The transverse deflections along the beam axis, including the axial- flexural effects in the beam-column element, are not adequately described by polynomials. As an alternative method, the element stiffness matrix is modeling using stability parameters. The shape functions which are obtaining using the stability parameters are more compatible with the system’s behavior. A mass matrix used in the dynamic analysis is evaluated using the same shape functions as those used for derivations of the stiffness coefficients and is called a consistent mass matrix. In this study, the stiffness and consistent mass matrices for prismatic three-dimensional Bernoulli-Euler and Timoshenko beam-columns are proposed with consideration for the axial-flexural interactions and shear deformations associated with transverse deflections along the beam axis. The second-order effects, critical buckling loads, and eigenvalues are determined. According to the author’s knowledge, this study is the first report of the derivations of consistent mass matrices of Bernoulli-Euler and Timoshenko beam-columns under the effect of axially compressive or tensile force.

Application of consistent geometric decomposition theorem to dynamic finite element of 3D composite beam based on experimental and numerical analyses

Analyzing static and dynamic problems including composite structures has been of high significance in research efforts and industrial applications. In this article, equivalent single layer approach is utilized for dynamic finite element procedures of 3D composite beam as the building block of numerous composite structures. In this model, both displacement and strain fields are decomposed into cross-sectional and longitudinal components, called consistent geometric decomposition theorem. Then, the model is discretized using finite element procedures. Two local coordinate systems and a global one are defined to decouple mechanical degrees of freedom. Furthermore, from the viewpoint of consistent geometric decomposition theorem, the transformation and element mass matrices for those systems are introduced here for the first time. The same decomposition idea can be used for developing element stiffness matrix. Finally, comprehensive validations are conducted for the theory against experimental and numerical results in two case studies and for various conditions.

Geometric nonlinear dynamic analysis of tapered steel members

In this study, a theoretical analysis is presented for estimating the in-plane geometric nonlinear elastic stability behavior of steel members with tapered elements under dynamic loads. Beam-column approach is adopted for modeling the structural members as beam-column elements. The formulation is based on the Eulerian description taking into consideration the influence of axial force on bending stiffness. The changes in member chord length due to axial deformation and flexural bowing are also considered. In the dynamic analysis, the system mass properties have been represented using both lumped and consistent mass matrices. The consistent mass matrix is derived in three components: translational, rotational, and axial inertia. The formulation of the mass matrices in local and global coordinate systems of tapered members, which incorporates geometric nonlinearity, has been presented. A parametric study is conducted to examine the effects of number of tapered elements, tapering the prismatic members, time step size, and tapering ratio. The dynamic responses of tapered members have been shown to be significantly affected by these parameters.

Locally bound-preserving enriched Galerkin methods for the linear advection equation

In this work, we introduce algebraic flux correction schemes for enriched (P1⊕P0 and Q1⊕P0) Galerkin discretizations of the linear advection equation. The piecewise-constant component stabilizes the continuous Galerkin approximation without introducing free parameters. However, violations of discrete maximum principles are possible in the neighborhood of discontinuities and steep fronts. To keep the cell averages and the degrees of freedom of the continuous P1/Q1 component in the admissible range, we limit the fluxes and element contributions, the complete removal of which would correspond to first-order upwinding. The first limiting procedure that we consider in this paper is based on the flux-corrected transport (FCT) paradigm. It belongs to the family of predictor-corrector algorithms and requires the use of small time steps. The second limiting strategy is monolithic and produces nonlinear problems with well-defined residuals. This kind of limiting is well suited for stationary and time-dependent problems alike. The need for inverting consistent mass matrices in explicit strong stability preserving Runge–Kutta time integrators is avoided by reconstructing nodal time derivatives from cell averages. Numerical studies are performed for standard 2D test problems.

An improved implicit method with dissipation control capability: The simple generalized composite time integration algorithm

In this paper, a simple and accurate implicit composite time integration scheme is newly developed for more efficient transient analyses of structural problems. The new scheme does not require the acceleration vector of the previous time step, and the initial acceleration vector is not needed, either. As a result, the factorization of the mass matrix is unnecessary, while factorizations of the effective coefficient matrix are still needed. Besides, the new method is designed to possess controllable algorithmic dissipation. Despite the absence of the acceleration vector of the previous time step, the numerical performance of the new method is equivalent to those of the existing methods. The excitation of an elastic bar problem and the simple nonlinear pendulum problem are numerically analyzed for the test of the new scheme. In the numerical experiments, the effects of consistent and lumped mass matrices are also discussed.

An analysis of implicit time integration schemes for wave propagations

The objective of this paper is to investigate the optimal use of some time integration schemes for the solution of transient wave propagation problems. We study the accuracy characteristics of the trapezoidal rule and the ρ∞-Bathe scheme considering various parameter sets (ρ∞,γ,CFL) with both consistent and lumped mass matrices. The ρ∞-Bathe scheme includes also the standard-, β1/β2-Bathe methods, the Newmark method and the trapezoidal rule. The study of the numerical dispersion shows that in the case of the consistent mass matrix, the ρ∞-Bathe scheme with a proper setting of (ρ∞,γ) and standard Bathe scheme provide similar dispersion errors and outperform the trapezoidal rule. The optimal CFL number of the ρ∞-Bathe scheme is about 25% larger than for the standard Bathe scheme. In addition, we show that using a lumped mass matrix and proper values of ρ∞<0, γ and CFL in the ρ∞-Bathe scheme, more accurate solutions can be obtained in some analyses.

Vibration analysis of femur bone by using consistent mass matrices and fast fourier transform analyzer

Vibration analysis is important for the evaluation of Natural frequencies and their effect on the bone for supportive equipment used while walking, running conditions. A femur bone is longest and taking the maximum load while standing, walking and running conditions. In present experimental research two types of conditions are used to evaluate natural frequencies of dead bone. While carried out experiment, FFT analyzer is used which gives desirable results of Natural frequencies. For analytical analysis, consistent mass matrices are used and natural frequencies are compared between analytical method and FFT analyzer. Average difference error between the two readings by consistent mass matrices and FFT analyzer is found to be less than one in first mode.

Dynamic brittle crack propagation modeling using singular edge-based smoothed finite element method with local mesh rezoning

This paper presents an effective numerical approach for dynamic brittle crack growth problems implemented on singular edge-based smoothed finite element method (sES-FEM). Both the consistent and lumped mass matrices are developed for five-node crack-tip elements for dynamic cases and their effects are compared on the numerical results. Further, to minimize the energy introduced or dissipated during continuous mesh rezoning, a balance recovery method is utilized in the computation. The interaction integral method is used to evaluate mixed dynamic stress intensity factors. Several numerical examples are presented to demonstrate the accuracy and applicability of the present approach in modeling dynamic crack propagation. The numerical results are examined in detail by comparisons with analytical solutions or experimental results, which shows the effectiveness of present method.

Bi-directional functionally graded thin-walled non-prismatic Euler beams of generic open/closed cross section Part II: Static, stability and free vibration studies

In the current study, deflections, buckling loads, natural frequencies and the corresponding mode shapes and forms for BDFG thin-walled tapered beams of generic open/closed/open-closed sections are examined using the finite element method. A tapered thin-walled beam finite element with the seven degrees of freedom at each node is investigated. A rigorous expression for strains based on the membrane theory of shells is considered. The flexural stiffness matrix, geometric stiffness matrix and the consistent mass matrices are developed in a companion paper. Using the present theory, one is able to investigate various torsional, flexural static and dynamic systems. Examples are presented and comparisons are made with the existing available literature. With the modifications of axial, coupled and bending rigidities one will be able to investigate composite beams with fibre reinforced composite laminates.

Free vibration analysis of spatial Bernoulli–Euler and Rayleigh curved beams using isogeometric approach

This paper deals with the linear free vibration analysis of Bernoulli–Euler and Rayleigh curved beams using isogeometric approach. The geometry of the beam as well as the displacement field are defined using the NURBS basis functions which present the basic concept of the isogeometric analysis. A novel approach based on the fundamental relations of the differential geometry and Cauchy continuum beam model is presented and applied to derive the stiffness and consistent mass matrices of the corresponding spatial curved beam element. In the Bernoulli–Euler beam element only translational and torsional inertia are taken into account, while the Rayleigh beam element takes all inertial terms into consideration. Due to their formulation, isogeometric beam elements can be used for the dynamic analysis of spatial curved beams. Several illustrative examples have been chosen in order to check the convergence and accuracy of the proposed method. The results have been compared with the available data from the literature as well as with the finite element solutions.

Vibration and static analysis of cracked and non-cracked non-prismatic frames by force formulation

Consistent mass and stiffness matrices are formulated for a new frame element. It is assumed that the frame member contains stable open edge cracks. All axial, bending, and shear effects are included in this study. Moreover, the proposed element has a non-prismatic property. Flexibility matrix of this structure consists of two parts. The first part is the flexibility of the non-cracked element, which is formulated in a basic coordinate system without rigid body motions. The additional flexibility due to cracks is the second part, which is calculated by using the fracture mechanics principles. The proposed element is applicable for both frame vibration and static analyses. It is a widely accepted assumption that cracking does not affect the distribution of the inertia components. In other words, mass matrices of the cracked and non-cracked elements are the same. Accuracy of this assumption is also reinvestigated in this study by using the developed consistent mass matrix. Various numerical examples are used to evaluate accuracy and capabilities of the novel element. Due to force method, the present formulation is locking free.

On the performance of GFEM with trigonometric enrichment in bidimensional dynamic elastoplastic modelling

In this work, an investigation is carried out concerning the performance of proposed enriched quadrilateral C0 element, developed by Generalized Finite Element Method, in dynamic elastoplastic models. The enriched quadrilateral C0 element is formulated by considering conventional shape function as global PU basis and trigonometric functions as the local basis. The enriched mesh distortion sensitivity is investigated in an elastodynamic problem. Other numerical aspects are also investigated in several benchmark analyses, involving stress concentration effect. The condition numbers of the stiffness and the consistent mass matrices are calculated for each enrichment level in different situations, and the error in norm L2 is calculated for displacement and stress. The results are compared with other numerical formulations encountered in the literature. Several numerical aspects of enriched quadrilateral C0 element formulated by GFEM are discussed and observed in the applications, and it performs competitiveness in comparison to other numerical formulations.

Numerical manifold method for vibration analysis of Kirchhoff's plates of arbitrary geometry

• Rectangular plate elements are proven to be able to analyze in the framework of NMM plates of any shape. • A rigorous and unified mass lumping scheme is proposed for vibration analysis of Kirchhoff's plates. • The proposed lumped mass matrices are shown to be able to supersede consistent mass matrices in any cases. Many rectangular plate elements developed in the history of finite element method (FEM) have displayed excellent numerical properties, yet their applications have been limited due to inability to conform to the arbitrary geometry of plates and shells. Numerical manifold method (NMM), considered to be a generalization of FEM, can easily solve this issue by viewing a mesh made up of rectangular elements as mathematical cover. In this study, ACM element (Adini and Clough element from A. Adini, R.W. Clough, Analysis of plate bending by the finite element method, University of California, 1960), a typical rectangular plate element is first integrated in the framework of NMM. Then, vibration analysis of arbitrary shaped thin plates is conducted employing the tailored NMM. Using the definition of integral of scalar functions on manifolds, we developed a mathematically rigorous mass lumping scheme for creating a symmetric and positive definite lumped mass matrix that is easy to inverse. A series of numerical experiments have been studied and analyzed, including free and forced vibration of thin plates with various shapes, validating the proposed mass lumping scheme can supersede the consistent mass formulation in those cases.

A Meshfree Higher Order Mass Matrix Formulation for Structural Vibration Analysis

An accurate meshfree formulation with higher order mass matrix is proposed for the structural vibration analysis with particular reference to the 1D rod and 2D membrane problems. Unlike the finite element analysis with an explicit mass matrix, the mass matrix of Galerkin meshfree formulation usually does not have an explicit expression due to the rational nature of meshfree shape functions. In order to develop a meshfree higher order mass matrix, a frequency error measure is derived by using the entries of general symmetric stiffness and mass matrices. The frequency error is then expressed as a series expansion of the nodal distance, in which the coefficients of each term are related to the meshfree stiffness and mass matrices. It is theoretically proved that the constant coefficient in the frequency error vanishes identically provided with the linear completeness condition, which does not rely on any specific form of the shape functions. Furthermore, a meshfree higher order mass matrix is developed through a linear combination of the consistent and lumped mass matrices, in which the optimal mass combination coefficient is attained via eliminating the lower order error terms. In particular, the proposed higher order mass matrix with Galerkin meshfree formulation achieves a fourth-order accuracy when the moving least squares or reproducing kernel (RK) meshfree approximation with linear basis function is employed; nonetheless, the conventional meshfree method only gives a second-order accuracy for the frequency computation. In the multidimensional formulation, the optimal mass combination coefficient is a function of the wave propagation angle so that the proposed accurate meshfree method is applicable to the computation of frequencies associated with any wave propagation direction. The superconvergence of the proposed meshfree higher order mass matrix formulation is validated via numerical examples.

Modal and Elastic Time-History Analysis of Frames with Tapered Sections by Non-Prismatic Elements

Low-cost robotic welding and wide availability of high strength steel plates of grades over 500[Formula: see text]MPa make the use of tapered members an economical alternative to conventional prismatic members for modern steel structures, as experienced by the authors in some practical projects in Hong Kong and Macau. This paper proposes a new and efficient numerical method for modal and elastic time-history analysis of the frames with tapered sections. A series of non-prismatic elements is derived on the basis of analytical expressions, and the exact consistent mass and tangent stiffness matrices are formulated. Five common types of tapered sections for practical applications, namely the circular solid, circular hollow, rectangular solid, rectangular hollow and doubly symmetric-I sections, are studied. Contrary to the conventional method using the approximate assumptions for the section properties along the member length, this research analytically expresses the flexural rigidity and cross-sectional area for the stiffness and mass matrices of an element. Further, the techniques for obtaining the dynamic performances, such as natural vibrations and time-history responses, of non-prismatic members are investigated. Finally, three examples are conducted for validating and verifying the accuracy of the proposed formulations. The present work can be used in the dynamic response analysis of frame structures with tapered sections in seismic zones.

Finite-element implementation for nonlinear static and dynamic frame analysis of tapered members

Non-prismatic members are popular for civil engineering structures. This paper derives a set of unified beam-column formulations for nonlinear static and dynamic analyses of the structures made of members with tapered sections, addressing the problems in engineering design practices. The element shape-functions are established upon the local-axes by extracting the rigid-body movements for simplifying mathematical expressions. To represent the variations in the stiffness gradients of tapered sections, the tapered-variability indexes are proposed. The generalized tangent stiffness and consistent mass matrices are developed based on the indexes. When analyzing non-prismatic members, the conventional method for handling member loads is inapplicable because it is derived for prismatic sections. Therefore, a new approach for converting the member loads acting on tapered members into the equivalent nodal forces is proposed based on the energy conservation principle. To consider the offsets in section axes, the eccentricity matrices are employed. For allowing large deflections, the incremental secant-stiffness method (ISM) based on Updated-Lagrangian (UL) description is proposed. Finally, extensive examples are provided for validating the accuracy and efficiency of the proposed element formulations in solving both the static and dynamic nonlinear problems.

Inverse mass matrix via the method of localized lagrange multipliers

SummaryAn efficient method for generating the mass matrix inverse of structural dynamic problems is presented, which can be tailored to improve the accuracy of target frequency ranges and/or wave contents. The present method bypasses the use of biorthogonal construction of a kernel inverse mass matrix that requires special procedures for boundary conditions and free edges or surfaces and constructs the free‐free inverse mass matrix using the standard FEM procedure. The various boundary conditions are realized by the the method of localized Lagrange multipliers. In particular, the present paper constructs the kernel inverse matrix by using the standard FEM elemental mass matrices. It is shown that the accuracy of the present inverse mass matrix is almost identical to that of a conventional consistent mass matrix or a combination of lumped and consistent mass matrices. Numerical experiments with the proposed inverse mass matrix are conducted to validate its effectiveness when applied to vibration analysis of bars, beams, and plain stress problems.