Element Stiffness Matrix Research Articles

Dynamic Fracture Modeling of Impact Test Specimens by the Polygon Scaled Boundary Finite Element Method

In the polygon scaled boundary finite element method, the modeling of crack can be simplified and the stress intensity factors are extracted from the semi-analytical solution directly. These salient features are applied to develop an efficient method to model the impact test specimens, one of which occurs without crack propagation, the other with crack propagation. The notched bend specimens are discretized by polygon elements with no local refinement around the crack tip. The mass and stiffness matrix of polygon elements are derived based on the SBFEM, and then the elastodynamic equations of the global system are established. The Newmark integration method is applied to obtain the dynamic responses of the specimens. For the case with crack propagation, an efficient local remeshing algorithm is used to track the crack tip and model the crack propagation. The dynamic stress intensity factors are computed directly from the instantaneous displacement field and crack velocity. The numerical results of the two specimens correspond well with the experimental data and other numerical results reported in the literature. The effects of the time step, mesh density and damping coefficient are also investigated. Moreover, the displacement and stress contours are extracted from the dynamic solution, which is helpful for the interpretation of the experimental observations.

ТРЕУГОЛЬНЫЙ КОНЕЧНЫЙ ЭЛЕМЕНТ С ШЕСТЬЮ СТЕПЕНЯМИ СВОБОДЫ В УЗЛЕ

The work is devoted to obtaining the stiffness matrix of a high-precision, flat finite element with 6 degrees of freedom in the node, for solving plane elasticity problems by the finite element method.
 The scientific literature describes higher order elements. However, the theoretical results of these studies are quite far from their practical application. This paper gives a very detailed derivation of the stiffness matrix of a high-precision finite element. Similarly, a stiffness matrix of a tetrahedral finite element with 12-degree-of-freedom node can be obtained. To test the obtained stiffness matrix, a program based on the FEM was written, with the help of which the cantilever beam is calculated. The error in calculating displacements is only 0.22%.
 Conclusion: the stiffness matrix presented in the paper can be used with great success in numerical methods for calculating the stress-strain state.

ЧИСЛЕННЫЙ ПРИМЕР ПОСТЕПЕННОГО ПРЕОБРАЗОВАНИЯ МАТРИЦ ЖЕСТКОСТИ ТРЕУГОЛЬНОГО КЭ БАЛКИ-СТЕНКИ И ЕГО ДКЭ.

The paper considers the numerical example of gradual transformation of the stiffness matrix of triangular finite element of deep beam with the help of stiffness matrix of its additional finite elements. Material FE (concrete) reveals from two (under compression) to four (under tension) physical nonlinear properties before collapse. Initial stiffness matrix changes due to the every one. Therefore two variants of its numerical transformation are considered. Both variants confirm the possibility of realization of the analysis of structures with several nonlinear properties at strength limit stae when the Additional Finite Element Method (AFEM) is used.

2D acoustic wavefield simulation by improved stiffness and mass matrices of bilinear square element

Abstract In the frequency domain, we optimize the entries of the element stiffness matrix computed on a square element with the purpose of approximating the 2D acoustic wave equation with the second spatial order. The optimized matrices are computed from the minimization of the normalized phase velocity as a function of propagation angle and the number of grid points per wavelength, leading to a remarkable reduction in the numerical error. The optimized number of 4.1 sample points in a wavelength, with a maximum error in velocities <0.3%, offering the simulation of large-scale realistic models. The superior performance of the optimized matrices to those from the lumped, consistent and eclectic matrices, is evident from the numerical examples presented, with consequent reduction not only in the numerical dispersion/anisotropy but also in an improved resolution. It is noteworthy that optimized matrices give modeling results with better accuracy than models from the high-order spectral element method. Our methodology is easily extendable to the accurate discretizing of the general 3D elastic wave equation that optimizes the bandwidth of the impedance matrix, availing computational resources for accurately modeling the compressional and shear wave velocities, density, as well as multi-parameter inversion.

Shell-to-Beam Numerical Homogenization of 3D Thin-Walled Perforated Beams.

Determining the geometric characteristics of even complex cross-sections of steel beams is not a major challenge nowadays. The problem arises when openings of various shapes and sizes appear at more or less regular intervals along the length of the beam. Such alternations cause the beam to have different stiffnesses along its length. It has different bending and shear stiffnesses at the opening point and in the full section. In this paper, we present a very convenient and easy-to-implement method of determining the equivalent stiffness of a beam with any cross-section (open or closed) and with any system of holes along its length. The presented method uses the principles of the finite element method (FEM), but does not require any formal analysis, i.e., solving the system of equations. All that is needed is a global stiffness matrix of the representative volumetric element (RVE) of the 3D representation of a beam modeled with shell finite elements. The proposed shell-to-beam homogenization procedure is based on the strain energy equivalence, and allows for precise and quick determination of all equivalent stiffnesses of a beam (flexural and shear). The results of the numerical homogenization procedure were compared with the existing analytical solution and experimental results of various sections. It has been shown that the results obtained are comparable with the reference results.

Improved Finite Element Method for Inflated Beams with Local Wrinkles

An improved finite element method is proposed for inflatable beams with a global finite deformation and a local wrinkling behavior, based on the corotational approach and tension field theory. The finite deformation is decomposed into a rigid-body displacement in the global coordinate system and a small strain measured in the local coordinate system, by using the corotational approach. The tangent stiffness matrix of a three-node triangular membrane element is presented. With the modified bimodulus constitutive law, a wrinkling model is constructed for determination of the status of element (tension, wrinkled, or slack). To improve the rate of convergence, a consistent tangent material stiffness matrix is embedded into the local coordinate system. Numerical results are in good agreement with the existing experimental data. In some case, the proposed method can obtain more accurate deflection of inflatable beams than the postbuckling analysis of thin shells. A quadratic rate of convergence is observed for the proposed method, and thus the computational efficiency is improved greatly. The limitation of the method is also discussed.

Flutter analysis of sandwich panel with the constrained viscoelastic layer considering the effects of the imperfection and the core thickness deformation

The supersonic flutter analysis of panels treated with the viscoelastic constrained layer damping (CLD) is performed with a special focus on the effects of the initial geometric imperfection and the core through-the-thickness deformation on the flutter boundaries. For kinematic modeling of the CLD core, a higher-order shear deformation theory is used where the through-the-thickness deformation is also taken into account. The core viscoelastic property is also described using the frequency-dependent complex-modulus approach and the first-order Piston theory is employed to determine the aerodynamic pressure. The discretized equations of motion are then derived using the finite element method (FEM). The integrations required in the FEM for obtaining each element stiffness matrix are also facilitated by approximating the imperfection function using the quintic Hermite interpolation function. Moreover, due to the large dimension of the final global matrices, the model is reduced using the oscillatory damped modes of the panel, which greatly reduces the computation time needed for flutter analysis. Numerical studies are performed for two different viscoelastic materials (VEMs), which shows that an efficient flutter suppression could not always be achieved with any type of VEMs. The thickness deformation effect is also found to be most prominent when the CLD length is about 80% of the panel length. Moreover, the mode crossing phenomenon that occurs by increasing the imperfection amplitude is found to significantly reduce the flutter boundaries.

Exact Solutions for Torsion and Warping of Axial-Loaded Beam-Columns Based on Matrix Stiffness Method.

The typically-used element torsional stiffness GJ/L (where G is the shear modulus, J the St. Venant torsion constant, and L the element length) may severely underestimate the torsional stiffness of thin-walled nanostructural members, due to neglecting element warping deformations. In order to investigate the exact element torsional stiffness considering warping deformations, this paper presents a matrix stiffness method for the torsion and warping analysis of beam-columns. The equilibrium analysis of an axial-loaded torsion member is conducted, and the torsion-warping problem is solved based on a general solution of the established governing differential equation for the angle of twist. A dimensionless factor is defined to consider the effect of axial force and St. Venant torsion. The exact element stiffness matrix governing the relationship between the element-end torsion/warping deformations (angle and rate of twist) and the corresponding stress resultants (torque and bimoment) is derived based on a matrix formulation. Based on the matrix stiffness method, the exact element torsional stiffness considering the interaction of torsion and warping is derived for three typical element-end warping conditions. Then, the exact element second-order stiffness matrix of three-dimensional beam-columns is further assembled. Some classical torsion-warping problems are analyzed to demonstrate the established matrix stiffness method.

Damage Identification for Shear-Type Structures Using the Change of Generalized Shear Energy

Structural damage identification has become an important topic in the field of civil engineering in recent years. The shear-type structure, such as shear frame structure, is a common type used in civil engineering. In this paper, a damage identification method based on the change of generalized shear energy is proposed for shear-type structures. The main steps of the proposed method are as follows. Firstly, the element stiffness matrix in the structural finite element model is decomposed to obtain the elementary shear force vector. Secondly, the elementary generalized shear energy is calculated by the dot product of the vibration mode shape vector and the elementary shear force vector. Thirdly, structural damage locations can be determined by the changes of elementary generalized shear energy. Finally, more accurate damage localization and quantification are achieved by solving the mode shape sensitivity equation. A 20-storey numerical example and a three-storey experimental model are used to demonstrate the proposed damage identification algorithm. From the numerical and experimental results, it was found that the proposed approach can accurately identify the location and extent of the damage in the shear structures even if the data contain noise. It has been shown that the presented algorithm may be useful in the damage identification of shear-type structures.

Impact Coefficient Analysis of Curved Box Girder Bridge Based on Vehicle-Bridge Coupling

In the current vehicle-bridge dynamics research studies, displacement impact coefficients are often used to replace the moment and shear force impact coefficients, and the vehicle model is also simplified as a moving-load model without considering the contribution of vehicle stiffness and damping to the system in some concerned research studies, which cannot really reflect the mechanical behavior of the structures under vehicle dynamic loads. This paper presents a vehicle-bridge coupling model for the prediction of dynamic responses and impact coefficient of the long-span curved bending beam bridge. The element stiffness matrix and mass matrix of a curved box girder bridge with 9 freedom degrees are directly deduced based on the principle of virtual work and dynamic finite element theory. The vibration equations of vehicle-bridge coupling are established by introducing vehicle mode with 7 freedom degrees. The Newmark-β method is adopted to solve vibration response of the system under vehicle dynamic loads, and the influences of flatness of bridge surface, vehicle speed, load weight, and primary beam stiffness on the impact coefficient are comprehensively discussed. The results indicate that the impact coefficient presents a nonlinear increment as the flatness of bridge surface changes from good to terrible. The vehicle-bridge coupling system resonates when the vehicle speeds reach 60 km/h and 100 km/h. The moment design value will maximally increase by 2.89%, and the shear force design value will maximally decrease by 34.9% when replacing moment and shear force impact coefficients with the displacement impact coefficient for the section internal force design. The load weight has a little influence on the impact coefficient; the displacement and moment impact coefficients are decreased with an increase in primary beam stiffness, while the shear force impact coefficient is increased with an increase in primary beam stiffness. The theoretical results presented in this paper agree well with the ANSYS results.

Nonlinear Numerical Simulation of Finite Elements Based on Fiber Beam Elements With Shear Effects for Structures

The classical fiber beam model based on the Euler-Bernoulli beam theory ignores the influence of shear deformation on the section. To get a more accurate beam element model, based on the fiber beam element with shear effects and the Timoshenko beam theory, the stiffness matrix of the fiber beam element was deduced, and the dual effects of geometric nonlinearity and material nonlinearity were considered at the same time, combined with the elastoplastic incremental theory. Then, the nonlinear finite element analysis theory for the structure under the complex stress state of compression, bending and shear was established. Finally, a program was coded on MATLAB to conduct finite element numerical simulation of the typical compression-bending-shear members of reinforced concrete and rectangular concrete-filled steel tube, and the nonlinear full-process load-displacement curves were obtained. The analysis of the numerical examples show that, the established nonlinear finite element analysis theory is universal, feasible and correct.

Stress state of hydrotechnical facilities resistant to natural disasters

The equations of the theory of the stress-strain state of an elastic body for practical engineering structures of the agro-industrial complex do not have analytical solutions. Therefore, the development of numerical methods for determining the strength parameters of agricultural facilities is an urgent task. Among the numerical methods of strength calculations, the finite element method is currently widely used. In a Cartesian coordinate system, a finite element of a hexahedral shape is used to determine the stress-strain state of an elastic body under bulk loading in two formulations: in the formulation of the method of displacements with nodal unknowns in the form of displacements and their derivatives, and in a mixed formulation with nodal unknowns in the form of displacements and stresses. Approximation of displacements through nodal unknowns in obtaining the finite element stiffness matrix was performed using a form function, the elements of which were Hermite polynomials of the third degree. When obtaining the deformation matrix, displacements and stresses of the internal points of the finite element were approximated in terms of nodal unknowns using bilinear functions. The calculation example shows a significant advantage of using a finite element in a mixed formulation.

Dynamic Response Analysis of Asynchronous Deicing of Quad Bundle Conductor Spacer System During DC Ice Melting

In order to study the melting and falling off of the iced conductor under DC ice melting process by considering a lgj-400 / 50 transmission conductor as a research object, Ansys LS-DYNA model is used for simulating the response of materials under short periods of high-intensity loads and a nonlinear structural dynamics model is used to establish the finite element model of split conductor system in the actual process. In the process of finite element analysis, solid45 element is used to model the conductor layer by layer. During the calculation process, the life and death element method is used to transform its element stiffness matrix to near zero icing element for accurately simulating the actual situation during ice melting. Currently, the research on the deicing of the split conductor mostly equates the split conductor to a single conductor for calculation. In the actual ice melting operation, the split conductor system has a phenomenon of asynchronous deicing, so it is necessary to analyze the deicing of different sub-conductors. The results show that the jump height of the deicing conductor under DC deicing is smaller than that of the normal deicing. The jump height and tension of conductor deicing decrease with an increase in the deicing current. When the sub-conductors of the four-bundle conductor spacer system are not deiced synchronously, the two sub-conductors on the diagonal and two sub-conductors below are deiced, the split conductor system is relatively stable. When the sub-conductors under other working conditions are not deiced synchronously, the four-bundle conductor spacer system has lateral swing, resulting in torsional risk, which causes conductor wear and strand breaking accidents.

NODAL REACTIONS AND COEFFICIENTS OF THE STIFFNESS MATRIX OF A FINITE ELEMENT BASED ON THE REPRESENTATION OF DISPLACEMENTS BY POLYNOMIALS

The study of prismatic bodies with constants along one of the coordinates of mechanical and geometric parameters is most appropriate to conduct on the basis of the semianalytical finite element method (NMSE). Its essence is a combination of finite element sampling and decomposition of displacements in the characteristic direction by a system of trigonometric coordinate functions.In [8, 15], a variant of the semivanalytic finite element method for the calculation of prismatic bodies when used as a system of coordinate functions of Fourier series was developed. The use of trigonometric series provides maximum efficiency of the semi-analytical finite element method, however, only the boundary conditions corresponding to the object's support on an absolutely rigid in its plane and flexible diaphragm can be satisfied at the ends of the body.As a result of the performed researches, the basis of the representation of displacements by polynomials is obtained, which allows to significantly expand the range of boundary conditions at the ends of the body. In this case, it isnot possible to reduce the solution of the initial spatial boundary value problem to a sequence of two-dimensional problems, so a reasonable choice of appropriate polynomials becomes especially important.Both the conditionality of the matrix of the system of separate equations and, consequently, the convergence of integration algorithms for its solution, and the universality of the approach to the possibility of satisfying different variants of boundary conditions at the ends of the body depend on their correct choice.In addition, the question of methods of integration in the calculation of the coefficients of the stiffness matrix of a finite element (CE), which is quite common, due to the significant complexity of this procedure.

Cable structures: An exact geometric analysis using catenary curve and considering the material nonlinearity and temperature effect

Structures formed by cables constitute structural systems of great practical application in engineering, such as suspension bridges, cable-stayed bridges, cable cars, transmission lines in which case they are used as supporting elements that conduct electricity, overcoming the spans between the towers or in stays for the so-called cable-stayed towers, among other applications. Cable structures have a highly non-linear behavior both from a geometric point of view, arising from their initial equilibrium configuration, and from a material point of view, since the constitutive laws of the cables present an elastoplastic behavior. The objective of this work is to present a geometrically exact formulation for static analysis of cable structures considering the catenary configuration. For this, both geometric and material nonlinearities are included in the development of the the catenary cable element formulation. The effect of temperature variation is also considered. A rigorous updated Lagrangian description combined with the corotational approach is employed to obtain the consistent tangent stiffness matrix of the exact cable element. The proposed methodology was implemented in the software program called Advanced Structural Analysis System (ASTRAS), which is an academic software based on the Finite Element Method (FEM). Finally, numerical examples are analyzed using the present procedure and the obtained results are compared with those found in the literature and given by commercial softwares in order to demonstrate the consistency and precision of the formulation.

ANSYS implementation of a coupled 3D peridynamic and finite element analysis for crack propagation under quasi-static loading

This study presents an approach to couple peridynamic (PD) and finite element method (FEM) for 3D deformation and crack propagation in ANSYS framework. The stiffness matrix of the PD element is derived by using the peridynamic least squares minimization (PD LSM) method based on the small deformation assumption. The PD governing equations are constructed by using MATRIX 27 element native to ANSYS. The coupling between MATRIX 27 elements and traditional finite elements is achieved through the coupled degrees of freedom (DOF) command available in ANSYS. The crack propagation is implemented by monitoring the number of broken bonds and updating the stiffness matrix of the PD element. The accuracy of the coupled PD-FE approach is verified by comparison against the 3D FE displacement predictions. The validity of the coupled PD-FE approach is established by simulating the wedge splitting test.

The Image-Based Finite Element Evaluation of the Deformed State

The article presents one of the possible approaches to modeling objects with anisotropic properties based on images of the study area. Data from such images are taken into account when building a numerical model. In this case, material inhomogeneity can be included by integrating the local stiffness matrix of each finite element with a certain weight function. The purpose of the presented work is to develop a finite element for the formation of a computational ensemble and simulation of mechanical behavior taking into account the data of two-dimensional medical images. To implement the proposed approach, we used the assumption that there is a correlation between the values in the image pixels and the elastic properties of the material. Meshing was based on a four-node plane finite element. This approach allows using the quantitative phase or scanning electronic images, as well as computed tomography data. A number of test problems for compression of elementary geometry samples were calculated. The distal part of the rat femur was considered as a model problem. A computed tomography scan of the sample was used to construct a numerical model taking into account the inhomogeneity of the material distribution inside the organ. The distribution field of the nodal displacements based on data obtained from the images of the study area is presented. Within the framework of a model problem, we considered how a computer tomograph resolution influences the quality of the obtained results. For this purpose, calculations were carried out based on compressed input medical images.

Development of deep learning-based joint elements for thin-walled beam structures

This study presents a new modeling technique to estimate the stiffness matrix of a thin-walled beam-joint structure using deep learning. When thin-walled beams meet at joints, significant sectional deformations occur, such as warping and distortion. These deformations should be considered in the one-dimensional beam analysis, but it is difficult to explicitly express the coupling relationships between the beams’ deformations connected at the joint. This study constructed a deep learning-based joint model to predict the stiffness matrix of a higher-order one-dimensional super element that presents the relationships. Our proposition trains the neural network using the eigenvalues and eigenvectors of the joint's reduced stiffness matrix to satisfy the correct number of zero-strain energy modes overcoming the randomly perturbed error of the deep learning. The deep learning-based joint model produced compliance errors mostly within 2% for a given structural system and the maximum error of 4% in the worst case. The newly proposed methodology is expected to be widely applicable to structural problems requiring the stiffness of a reduction model.

Modeling technique for the linear‐elastic material behavior of short fiber‐reinforced composites using random fields

AbstractThe microstructure of short fiber‐reinforced composites is probabilistic due to the manufacturing process. The induced spatial fluctuation of the material properties cannot be covered by a homogeneous modeling concept since this technique leads to a deterministic representation of the material. In this study, a numerical approach is presented that includes the fluctuation of the material properties on the component level. The modeling approach consists of two steps. First, the probabilistic characteristics of the microstructure are derived by numerical simulation on the mesoscale. The obtained effective material properties and the correlation structure of the stiffness matrix elements are used to discretize second‐order Gaussian random fields for the material representation on the component level in a second step. To validate all results nanoindentation and tensile tests are conducted. It is shown that the suggested approach leads to excellent results comprising the overall structural response as well as the spatial fluctuation of the material properties. Hence, the introduced technique is suitable for modeling the linear‐elastic behavior of short fiber‐reinforced composites.

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.