Element Stiffness Matrix Research Articles

Forced vibration assessment of moderately thick carbon nanotube-reinforced composite plates

Carbon nanotubes, distributed in composite sheets with various arrangements, can lead to the development of materials with functionally graded properties. In the present study, the method of spectral components was used to investigate the free vibration frequencies and dynamic responses of single-layer sheets reinforced with carbon nanotubes. The spectral component method offers significant advantages for high-frequency dynamic problems, as it requires fewer fine elements to solve boundary conditions. Based on the first-order shear theory, the governing equations of sheet vibration are derived. The discrete Fourier transform is applied to convert the differential equations from the time domain to the frequency domain. Using dynamic shape functions obtained from the exact solutions, the stiffness matrix of the spectral element is constructed. Dynamic frequency responses are then derived, and time-domain responses are obtained using the inverse Fourier transform. The study extracted the exact natural frequencies of functionally graded sheets and verified them with results from the literature, showing high accuracy with a minimal number of elements. The spectral finite element method of fundamental natural frequencies for different ratios of modulus of elasticity and width-to-thickness ratios obtained were investigated and compared with the results of research. The forced vibration was addressed, demonstrating that the method efficiently captures dynamic responses under different carbon nanotube distributions, with spectral displacements verified through numerical comparison. These findings demonstrate the capability and efficiency of the spectral finite element method for analyzing carbon nanotube-reinforced composite plates.

Exact Second-Order Stiffness Matrix of Axial-Loaded Beam–Columns on Winkler Foundation

This paper establishes the exact second-order stiffness matrix of axial-loaded beam–columns on Winkler foundation. This Winkler foundation model has been widely used in engineering applications. The equilibrium analysis of an axial-loaded beam–column on Winkler foundation is conducted, and the element flexural deformations and forces are solved exactly from the governing differential equation. Two dimensionless factors for the axial force and foundation spring stiffness are defined, respectively. The exact second-order element stiffness matrix of the axial-loaded beam–column on Winkler foundation is then formulated, showing the relationship between the element-end deformations (translation and rotation angle) and the corresponding forces (shear force and bending moment). Such an exact second-order stiffness matrix may be used for the buckling and second-order solutions with allowance for the use of one element per member for the exact solution. The classical buckling problem of axial-loaded beam–columns on Winkler foundation with typical element-end boundary conditions is then analyzed by the developed matrix stiffness method. By comparing with classical buckling criterion expressions of axial-loaded beam–columns on Winkler foundation with various boundary conditions, it is demonstrated that the derived exact second-order stiffness matrix can be used to obtain the exact buckling solutions systematically.

Plane straight element with semi-rigid connections: formulation from an energetic approach

Discrete connectors, such as screws, nails, and others, are commonly used to fasten various structural elements. The established hypothesis regarding the behavior of joints conceived with discrete connectors directly impacts the distribution of forces. For example, in the design of truss structures, the resisting moments that arise due to the arrangement of the connectors are often disregarded, as considering the semi-rigid behavior of the joints introduces greater complexities in the modeling required to determine displacements and internal forces. This work presents the stiffness matrix of a plane straight element with semi-rigid connections and its support reactions for three loading cases from the application of the Principle of Virtual Work (PVW). Semi-rigid connections are represented by translational and rotational linear elastic springs at both ends of the element, giving it six degrees of freedom. One example shows the application of this formulation using the Classical Displacement Method and the results are also obtained numerically via the Finite Element Method (FEM)This work presents the stiffness matrix of a plane straight element with semi-rigid connections and its support reactions for three loading cases from the application of the Principle of Virtual Work (PVW). Semi-rigid connections are represented by translational and rotational linear elastic springs at both ends of the element, giving it six degrees of freedom. One example shows the application of this formulation using the Classical Displacement Method and the results are also obtained numerically via the Finite Element Method (FEM).

Bracing systems for three-ribbed arch bridges against out-of-plane buckling

This paper is concerned with the design of bracing systems for three-ribbed parabolic arch bridges against out-of-plane buckling. Typical bracing systems for the three-ribbed arch bridges include K-bracing system and X-bracing system as well as transverse bracing system. The buckling criterion of the braced funicular three-ribbed arch bridge is derived from an exact matrix stiffness method (MSM) with a 14 × 14 s-order element stiffness matrix of three-dimensional beam-column elements that allows for torsional and warping deformations. The lateral torsional buckling load and mode are given by the lowest eigenvalue and eigenvector associated with the assembled structural stability stiffness matrix. A comparison study between different bracing system configurations suggests that the three-ribbed arches with the integral K- or X-bracing system (a K/X-bracing across the three arch ribs) have lower lateral torsional buckling loads than their arch counterparts with the independent K- or X-bracing system (the adjacent arch ribs are connected by K/X-bracing respectively), because the latter ones could provide more restraint on the middle arch rib to avoid early lateral torsional instability. Further, a bracing utilization efficiency index (defined as the normalized buckling capacity over material usage for bracing system) is proposed to quantify the effect of bracing systems in improving the lateral torsional buckling capacity of arch bridges. Highly efficient bracing systems that maximize the lateral torsional buckling load of three-ribbed arch structures with the lowest bracing material usage are then recommended.

Dynamic Modeling and Experimental Verification of an RPR Type Compliant Parallel Mechanism with Low Orders

Efficiency of calculating a dynamic response is an important point of the compliant mechanism for posture adjustment. Dynamic modeling with low orders of a 2R1T compliant parallel mechanism is studied in the paper. The mechanism with two out-of-plane rotational and one lifting degrees of freedom (DoFs) plays an important role in posture adjustment. Based on elastic beam theory, the stiffness matrix and mass matrix of the beam element are established where the moment of inertia is considered. To improve solving efficiency, a dynamic model with low orders of the mechanism is established based on a modified modal synthesis method. Firstly, each branch of the RPR type mechanism is divided into a substructure. Subsequently, a set of hypothetical modes of each substructure is obtained based on the C-B method. Finally, dynamic equation of the whole mechanism is established by the substructure assembly. A dynamic experiment is conducted to verify the dynamic characteristics of the compliant mechanism.

Susceptibility versus flexural stiffness in the stability of hybrid laminate columns with a rectangular cross section for a 1D model

Hybrid columns of FGM/FML type with inhomogeneous transverse structure perpendicular to their length are discussed. After that, laminated columns with different layups of laminate plies are discussed, with focus put on the effect exerted by many coupling of the stiffness submatrix on the lowest eigenvalue load. All columns were simply supported The study uses three methods for determining the lowest eigenvalues. The first two are analytical beam methods (i.e. 1D modelling approaches) based on the use of the Euler-Bernoulli theory. The proposed 1D methods are a generalisation of the well-known approach of determining the lowest eigenloads of isotropic columns. The first approach is to solve the eigenproblem directly from the differential equation of the deflection line. The second approach assumes that the curvature of the beam depends on the susceptibility of the beam. This is a fundamental assumption in the bending theory of beams and thin plates. The third method is the FEM (3D shell approach) which is used for the verification of the previous calculations. A comparison of the obtained results showed that the proposed 1D (one-dimensional) approach based on the full susceptibility matrix, which is the inverse stiffness matrix was as accurate as the 3D (three-dimensional) FEM shell model. The relative difference does not exceed 3%. Second 1D approach based on the stiffness matrix yields many times unacceptable differences relative to the reference results obtained by the FEM method. This comparison clearly demonstrates that an approach based on the assumption that curvature is directly proportional to susceptibility in flexural composite structures produces accurate results for all types of coupling.The existing literature extensively explores the impacts of stiffness matrix elements on both the linear and non-linear stability of laminate columns. However, there is a notable absence of studies focusing on the influence of susceptibility. This crucial aspect has been overlooked in prior studies. The present paper endeavors to fill this gap by shedding light on the influence of susceptibility, thereby emphasizing its significance in the analysis of laminate column stability.

Stability and dynamic analyses of hybrid laminates using refined higher-order zigzag theory

This article presents a comparative study on the stability and dynamic response of hybrid laminated composite plates over monolithically produced laminated composite plates. A Refined Higher-order Zigzag Theory (RHZT) is formulated and implemented using finite element method for the analysis of hybrid laminates. This formulation considers both the interlaminar shear stress continuity and shear-free boundary conditions at surfaces of the plates. The numerical implementation employs C 0 continuous eight-noded isoparametric plate elements considering geometric nonlinearity. The element stiffness matrix is formulated to consider both the linear and nonlinear terms of the strain-displacement equations. Derivation of the consistent mass matrix for a plate element is presented following the approximation of total kinetic energy and utilising Hamilton’s principle. Numerical examples are provided to demonstrate buckling and free vibration analyses (as eigenvalue problems). The results are numerically verified and validated using data from published literature. Finally, parametric studies are presented to demonstrate the effectiveness of fibre hybridisation approach on the buckling and free vibration behaviour of laminated composites.

Nonlinear inelastic behaviors of concrete-filled steel tubular beam-column structures

A new advanced method combining stability function and distributed plasticity model has been developed using Fortran programming language to predict the nonlinear inelastic behavior of concrete-filled steel tubalar (CFST) under static loading. The advantage of this method is the ability to accurately study the nonlinear behavior using only one or two beam-column elements per member instead of using solid and shell elements as traditional methods, thereby improving the model analysis time. The Generalized Displacement Control (GDC) algorithm, capable of analyzing beyond the limit point, will be used to solve the nonlinear equilibrium equations instead of the traditional Newton-Raphson algorithm. The element stiffness matrix is integrated through the Gauss-Lobatto numerical integration framework, while the nonlinear geometric effects P-Δ and P-δ are considered using stability functions and a corresponding geometric matrix. The reliability and accuracy of the proposed method are verified by comparing the analysis results with experimental data. The obtained results have demonstrated that using beam-column elements for simulation, the proposed method still provides accurate results while significantly reducing computational resources. Therefore, this new method holds promise as a useful tool for practical design and analysis of statically loaded CFST structures.

Homogenized Stiffness Matrix of Two-Node Elements through Experimental Flexibility tests

Homogenization is a numerical technique to obtain the equivalent mediated response of a complex material. However, there are some scenarios in which the complexity of the physical phenomena present in the real structure makes it difficult to generate a faithful Representative Volume Element (RVE). In these cases, it may turn easier and more reliable to perform a homogenization directly by experimental test results. For this reason, we present a procedure to build-up the whole stiffness matrix starting from experiments. This approach is here discussed to get a homogenised two-node beam element. The procedure requires some flexibility measurements, thus allowing fewer measurements if compared to stiffness-based approach. The method is verified with some experiments carried on a frame structure. The comparisons with a Finite Element model build up with stiffness matrix assembly demonstrates the validity and robustness of the proposed procedure.

Influence of Prestress on Vibration Frequency of Beam String Structures Based on Exact Matrix Stiffness Method

Beam string structures (BSS) are frequently employed in large-span spatial structures due to their high load-carrying efficiency and elegant appearance. Owing to the neglect of the influence of prestress, the natural frequency of the BSS is often overestimated which could be crucial to the dynamic behavior of BSS. This paper presents an exact matrix stiffness analysis method (MSM) for investigating the relationship between prestress and the natural frequency of the planar BSS. A novel dynamic stiffness matrix of beam–columns considering the natural frequency and axial force is used to develop the MSM. The dynamic analysis of the structural vibration stability of BSS is performed by using the global structural stiffness matrix which is assembled from the element stiffness matrices in the global coordinate system. The proposed MSM with the exact dynamic element stiffness matrix for the dynamic analysis of the BSS is verified by comparing with previous results based on the finite element method. The illustrative examples demonstrate that the prestress in the cable has a negative effect on the natural frequency of the BSS and should be considered in the dynamic analysis. As the axial force increases from zero to the buckling load [Formula: see text], the natural frequency of the BSS decreases from the maximum vibration frequency to zero. The influence of prestress on the vibration frequencies of BSS is particularly significant when the prestress to balance the loads is large, especially in the case of large dead and live loads (e.g. floor slabs).

Multi-criteria parametric optimization of the displacement and weight of a shell of minimal surface on a circular contour consisting of two inclined ellipses under thermal and power loading with consideration of geometric nonlinearity

Thin shells are well suited to optimal design problems, and the finite element method and gradient descent method make it possible to solve inverse problems in structural and applied mechanics. The calculation process takes into account geometric nonlinearity. In modern numerical studies, a formulation taking into account geometric nonlinearity in the finite element method is used, namely, a stepwise loading procedure. It becomes necessary to take into account the relations between displacement vectors and their derivatives and strain increments. These relations help to determine the stiffness matrix of the finite element at each loading step, which leads to a qualitative study of real displacements in the structure. The geometrically nonlinear calculation of a shell of minimum surface on a circular contour consisting of two inclined ellipses in a multi-criteria parametric optimization is performed by the finite element method. The finite element method is a universal variational method that is focused on solving the most complex problems of elasticity theory and applied and structural mechanics using separate calculation complexes. Stiffness matrix - for the whole body is formed on the basis of the finite element stiffness matrix. The system of solving equations of the finite element method is formed using the Lagrange's variational principle, according to which the total potential energy P of a finite element model of a body in a state of stability and equilibrium has a minimum value. As part of the sensitivity analysis, gradients of the design variables of the structure, displacements in the form of partial derivatives of these characteristics along the design variables, and shell thickness are calculated. The sensitivity information serves as the basis for building an optimal design algorithm using the gradient descent method of the objective function. Standard multi-criteria parametric optimization allows for an average of 10% steel savings, with geometric nonlinearity increasing to 20%. At this study site, 24.4% of sheet steel was saved, which is a significant relative saving. This methodology of the authors shows high results for innovative design and production of steel thin shells in Ukraine and around the world, and also makes it possible to apply several types of optimization to one research object.

Оценка наибольшего собственного числа глобальной матрицы жесткости регулярной сетки

The paper deals with condition of global stiffness matrices of regular mesh of finite elements. The estimation from above the greatest eigenvalue of such matrix offeres. The estimation is under construction on a local stiffness matrix of any finite element, hence, depends only from the size and the form of such element and does not depend on quantity of the final elements making a regular grid. At estimation construction the Gerschgorin’s theorem and that fact are used that local matrixes of rigidity of final elements of regular grids differ from each other only shift of blocks. On a numerical example it is shown that the constructed estimation possesses split-hair accuracy and at a considerable quantity of the elements entering into a grid, it is possible to consider it almost coinciding with the greatest own number. The behaviour of an estimation is shown at change of quality of the form of final elements.

Analytical Model for Non-linear M-θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\user2{M - }\\,\\,{\\varvec{\ heta}}$$\\end{document} Relationships of Dowel-Type Timber Connections Exposed to Fire

Recent experimental evidence has shown that wood–steel–wood dowel-type connections exhibit a semi-rigid behaviour even after 90 min of fire exposure. Because a semi-rigid behaviour influences the bending moment distribution among structural members, considering the moment–rotation relationships within frame analyses in which structural members are modelled as beam elements can enable a realistic fire response analysis that is significantly simpler than the three-dimensional finite element method. This study proposes an analytical methodology that accounts for the thermo-mechanical behaviour of timber and dowels, enabling the simulation of the non-linear moment–rotation relationships under fire conditions. The proposed analytical model divides dowels into a series of elements on an elastoplastic foundation and performs a direct stiffness method in a time-incremental procedure using an element stiffness matrix derived from beam-on-elastic-foundation theory. This study also presents the results of load-carrying tests on timber frames with dowel-type connections performed under ambient and fire conditions. The analytical results were consistent with the fire test results. Additionally, the analyses were also performed under three conditions in which the dowels were rigid, linear elastic, and elastoplastic bodies. These three results converged to the same value after 65 min of heating, which suggests that the ultimate states of beams with dowel-type connections exposed to fire can be modelled by assuming that dowels are rigid bodies.

A Constitutive Model for Stud Connection in Composite Structures

The complexity of finite element analysis for composite structures can be significantly reduced by representing the connector and adjacent concrete as a macroscopic element. Nevertheless, the prevailing macroscopic models for shear connections predominantly employ nonlinear elastic theory. This approach introduces inaccuracies in estimating structural stiffness and load-bearing capabilities, primarily due to its inability to precisely capture the cumulative effects of plastic damage. In response, this study introduces a novel macroscopic elastoplastic model grounded in plasticity theory, aimed at accurately characterizing the nonlinear behavior of stud connections subjected to concurrent shear and tensile forces. This paper meticulously delineates the implementation of the elastoplastic constitutive model using the backward Euler method for numerical integration. It further articulates the derivation of the consistent tangent stiffness, which aligns with the convergence efficiency of the Newton–Raphson iterative approach. The computation of the element stiffness matrix for a two-node element is executed via the governing equation inherent to the finite element method. An exemplar macroelement test conducted in ABAQUS affirms the implicit backward Euler scheme’s stability and consistency across varying tolerances. Validation of the elastoplastic model against empirical test outcomes corroborates its efficacy, demonstrating the model’s precision in predicting the load–displacement behavior of stud connections under the influence of shear and tensile forces.

Motor magnetic field analysis using the edge-based smooth finite element method (ES-FEM)

This paper proposes a smooth finite element method (S-FEM) for efficient and accurate analysis of motor magnetic fields. The edge-based smooth finite element (ES-FEM) formulations are derived for two-dimensional triangular element meshes suitable for multi-material motor structures, and then a magnetic flux density calculation method appropriate for S-FEM to calculate nonlinear electromagnetic fields is introduced. In the calculation of motor magnetic field, ES-FEM has a proper softening effect on the stiffness matrix of the finite elements, which makes the calculation more accurate and more efficient than FEM. The magnetic flux intensity calculation method introduced in this paper is not only appropriate for ES-FEM to calculate the nonlinear electromagnetic field of motor, but also can improve the calculation accuracy of magnetic flux density. In the calculation of nonlinear magnetic field of motor, ES-FEM can also improve the calculation accuracy of magnetic field analysis compared with FEM. Several numerical examples demonstrate the improving effect of the ES-FEM discussed above on the motor's magnetic field calculation.

Nonlinear dynamic collapse analysis of space semi-rigid frames using finite particle method

Based on the finite particle method, this work presents a general numerical method to perform the nonlinear dynamic collapse analysis of space semi-rigid structures. A space connection element composed of two particles and six zero-length springs is proposed to simulate the space semi-rigid connection. Geometrical nonlinearity is addressed through fictitious motion, while material and connection behavior nonlinearities are captured by updating the element stiffness matrix and the connection stiffness matrix, respectively. Member and connection fracture models are convenient for simulating fracture behaviors during collapse processes. To verify the effectiveness of the proposed method, the numerical results are compared with those from previous studies, and energy conversion analyses are adopted. The analysis results indicate that only member fractures occur at beam ends in the rigid and linear semi-rigid frames, while only connection fractures occur at beam ends in the nonlinear semi-rigid frame.

Thermal analysis of a perforated laminated plate with multiple random elliptical holes by specially formulated finite elements

The high-efficient and accurate analysis of anisotropic composite media weakened by a large amount of holes or cuts is still a great challenge in practical composite engineering. In this work, specially-proposed polygonal finite element enclosing an arbitrarily oriented elliptical hole in general anisotropic composite media is formulated for thermal analysis in the framework of hybrid finite element theory. Two independent temperature field approximations are defined for the present special element: Intra-element temperature field and element frame temperature field. The former defined inside the element is written in the form of truncated summation of the weighted special fundamental solutions, which are derived using the Stroh complex theory and conformal mapping whilst the latter is implemented by the conventional shape function interpolation. Then the hybrid variational functional is employed to combine them together to generate an element stiffness equation with respect to nodal temperature and an optional relationship between the interpolation coefficients. The element stiffness matrix is symmetric and involves element boundary integrals only. Compared to the traditional finite element, the distinctive element-boundary-integral and enclosed-elliptical-hole features make the present element significantly improve the meshing effort around an elliptical hole by reducing the number of elements and decreasing the meshing difficulty, and more importantly, effectively capture the critical variation of physical fields around the elliptical hole without locally-refined meshes. Moreover, the present special element can be incorporated directly into the conventional finite element system without any difficulty for extensive analysis. Finally, several typical examples are simulated by the present special elements to demonstrate their effectiveness.

A fast polynomial-FE method for the vibration of the composite laminate quadrilateral plates and shells based on the segmentation strategy

The finite element method is widely applicable to various kinds of models. Nonetheless, as the size and frequency domain of the solving model increase, it encounters challenges related to high memory consumption and slow solution speed. To enhance the computational efficiency while preserving its existing advantages, this study incorporates the Jacobi polynomial with a segmentation strategy into the finite element method for addressing the vibration analysis of composite laminate quadrilateral flat and curved plates. In this method, the dynamic stiffness matrix of a finite element is multiplied on both sides by a matrix composed of Jacobi polynomial basis functions, thereby transforming the expression form of the dynamic stiffness matrix from FE form to meshless form and reducing its size. Additionally, the segmentation strategy is introduced to divide the model into multiple blocks, enhancing convergence and calculation efficiency by altering the sparsity pattern of the dynamic stiffness matrix. Convergence research, validation studies, limitations studies, and parametric studies are conducted. The method demonstrates evident advantages in computational efficiency for random surface quadrilateral models requiring dense mesh description while maintaining a certain level of accuracy.

Numerical Analysis on the Static Performance of Gas Journal Bearing by Using Finite Element Method

In this paper, finite element method is used to calculate the static performance of gas journal bearing, in which rotation speed term is introduced into the stiffness matrix of linear triangular element to realize the performance calculation of the bearing with rotation speed. The results indicate that the average gas film thicknesses corresponding to the maximum load capacity and stiffness, and the minimum attitude angle increase with the growth of orifice diameter. Load capacity and stiffness significantly improved with the increase of rotation speed, eccentricity ratio and supply pressure when the bearing has thin average gas film thickness. Attitude angle increases with the growth of rotation speed, while the growth rate slows down or even decreases at high speed. The most effective way of reducing attitude angle is to increase supply pressure. It can be found that rotation speed affects attitude angle through changing gas pressure difference between two orifices, while other parameters have the same effect by changing gas pressure at orifice outlet.

Improved dynamic analysis of shear deformable shells using the hybrid displacement boundary element method

In this paper, the dynamic analysis of shear deformable shells was conducted using the hybrid displacement boundary element method (HD-BEM). The approach involved utilizing the fundamental solution as a trial function to derive the 8-node element stiffness matrix, while approximating the boundary variables with quadratic shape functions. To ensure efficient computations and avoid singular integrals, the source points were strategically positioned outside the domain. For accurate calculations of the proposed 8-node element mass matrix, only four Gauss points were relied upon. It is important to note that both the element mass and stiffness matrices demonstrate symmetry. Throughout the study, free, forced, and harmonic analyses were considered. Encouragingly, the results obtained from the study exhibited exceptional accuracy levels, even when employing a coarse discretization approach.