Showing papers on "Direct stiffness method published in 2018"

Elastic buckling analysis of cold-formed steel built-up sections with discrete fasteners using the compound strip method

Abstract: In this paper, the compound strip method is applied to the stability analysis of cold-formed steel built-up sections. A beam element with adjustable stiffness properties is adopted to represent the utilised fastener and the associated stiffnesses of the connection elements are incorporated in the global stiffness matrix of the built-up sections. The presented method allows for modelling arbitrarily-located discrete fasteners in the context of the semi-analytical finite strip method. The proposed numerical technique is verified against finite element solutions through various numerical examples and shown to be both accurate and versatile. Some typical and also complex built-up sections with various fastener configuration and end boundary conditions are analysed to evaluate the influence of fastener spacing. The extent of composite behaviour in built-up sections is determined by investigating the enhancement of buckling capacity and changes in the corresponding buckling modes. The simplicity of the proposed technique expedites extensive parametric studies of cold-formed built-up sections and facilitates the search for optimal placement of fasteners and choice of section geometry.

A method for stiffness analysis of overconstrained parallel robotic mechanisms with Scara motion

Abstract: This paper presents a general method for analyzing stiffness of overconstrained parallel robotic mechanisms with Scara motion. In the method, the stiffness model of a limb is derived by applying Castigliano's second theorem to strain energy of the limb with a structural decomposition strategy, and the stiffness matrix of a parallel mechanism is established based on stiffness models of limbs and the static equilibrium equation of the moving platform. Comparisons show that the stiffness model obtained from the proposed method is very close to the counterpart obtained from finite element analysis (FEA). In addition, a new index is proposed to evaluate stiffness performance of a parallel mechanism in a given configuration based on strain energy under external unit forces and moments. With this index, the dimensions of a parallel mechanism can be optimized and the path of a given task can be planned to obtain high stiffness.

Simulate intersecting 3D hydraulic cracks using a hybrid “FE-Meshfree” method

Abstract: In this work, a discontinuous hybrid “FE-Meshfree” method is developed to simulate three-dimensional (3D) cracking. Then, an algorithm of Local Adaptive Background Sub-element (LABS-element) is developed in the context of hybrid “FE-Meshfree” method to simulate intersecting cracks. Compared to the extended finite element method (XFEM), the present hybrid “FE-Meshfree” method is not required to increase the size of global stiffness matrix, introduce extra unknowns, and construct special enrichment functions. The present algorithms of LABS-element and “FE-Meshfree” method are validated by intensive numerical tests, which achieves the balance between accuracy and flexibility. Additionally, a hydraulic and mechanical (H-M) coupling model is generated, in which the deformation of rocks and the propagation of cracks are solved by the hybrid “FE-Meshfree” method, meanwhile the fluid flow in cracks is solved by a fluid simulator based on the principle of parallel-plate flow model. This H-M coupling model is then used to investigate the propagation of fluid-driven cracks in rock mass with multiple pre-existing cracks, and the observed fluid pressure feedback is potentially useful during shale oil/gas well simulation.

A method for stiffness modeling of 3R2T overconstrained parallel robotic mechanisms based on screw theory and strain energy

Abstract: Stiffness (or compliance) performances of three-rotation and two-translation (3R2T) overconstrained parallel mechanisms have an important influence on their applications in bio-inspired robots with precision operations. However, it is difficult to build stiffness models of the class of mechanisms due to fairly complicated structures, and an effective and efficient modeling method has not been reported. This paper presents an approach to stiffness modeling of 3R2T overconstrained parallel robotic mechanisms. First, expressions of applied wrenches exerted on limbs and joints of a mechanism under an external load are solved based on screw theory. Then, the stiffness matrix of a limb is derived by applying Castigliano’s second theorem to strain energy of the limb, which is solved by introducing a strategy of structural decomposition. Third, the stiffness model of a mechanism is built based on stiffness matrices of all limbs and the principle of virtual work on the moving platform. Finally, the effectiveness of the proposed method is verified based on the fact that the computational compliance matrices are very close to those from finite element analysis (FEA) models. The proposed method can be used to build conveniently stiffness models with high accuracy for 3R2T overconstrained parallel mechanisms in precision positioning applications.

Structural Analysis : Principles, Methods and Modelling

Abstract: Introduction Structural analysis and design Structural idealisation Structural members and elements Structural systems Types of loads Supports for structures Statics of structures: Equilibrium and support reactions Introduction Coordinate systems Force Moment of a force Resultant force and moment Reactions Free-body diagram Equilibrium equations for planar structures External statical determinacy and stability Internally stable structures Determination of reactions Equilibrium and reactions in three-dimensional structures Problems Internal actions of beams and frames Introduction Internal actions at a cross-section Sign convention of internal actions Determination of internal actions and statical determinacy Axial force, shear force and bending moment diagrams Problems Statically determinate trusses Introduction Assumptions for truss analysis Sign convention and notation An introduction to the method of joints Method of joints in matrix form Method of sections Statical indeterminacy and stability of trusses Deformation of trusses Trusses with loaded members Space trusses Problems Euler-Bernoulli beam model Introduction Equilibrium of a small length of beam Kinematic (or strain-displacement) equations Constitutive equations Method of double integration Governing differential equations (as a function of displacements) Relationship between bending moment, shear force and member loading Problems Slope-deflection methods Introduction Method of double integration with step functions Moment-area method Conjugate beam method The slope-deflection equations Problems Work-energy methods Strain energy The work theorem Virtual work Virtual work applied to trusses Virtual work applied to beams and frames Castigliano's theorem Problems The force method Introduction The force method applied to trusses The force method applied to beams and frames Problems Moment distribution Introduction Basic concepts Continuous beams Frames without sidesway Frames with sidesway Problems Truss analysis using the stiffness method Overview of the stiffness method Sign convention, notation, coordinate systems and degrees of freedom Derivation of the stiffness matrix in local coordinates Transformation between local and global coordinate systems Truss element in global coordinates Assembling Solution procedure Calculation of internal actions Nodal coordinates Space truss Problems Beam analysis using the stiffness method The beam element Derivation of the stiffness matrix Beam element in global coordinates Assembling of the stiffness elements Member loads Solution procedure and post-processing Problems Frame analysis using the stiffness method The frame element Derivation of the element stiffness matrix Transformation between local and global coordinate systems Frame element in global coordinates Member loads Assembling, solution and post-processing Problems Introduction to the finite element method Introduction Euler-Bernoulli beam model Timoshenko beam model Problems Introduction to the structural stability of columns Introduction Assumptions Critical load from equilibrium Critical load from potential energy Buckling of an elastic column Effective buckling length Buckling stresses Imperfections in columns Problems Introduction to nonlinear analysis Introduction Nonlinear material properties Illustrative examples Nonlinear analysis using the Newton-Raphson method Finite element analysis using the Newton-Raphson method Problems Appendices Index

A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations

Abstract: Due to the nonlocal property of the fractional derivative, the finite element analysis of fractional diffusion equation often leads to a dense and non-symmetric stiffness matrix, in contrast to the conventional finite element formulation with a particularly desirable symmetric and banded stiffness matrix structure for the typical diffusion equation. This work first proposes a finite element formulation that preserves the symmetry and banded stiffness matrix characteristics for the fractional diffusion equation. The key point of the proposed formulation is the symmetric weak form construction through introducing a fractional weight function. It turns out that the stiffness part of the present formulation is identical to its counterpart of the finite element method for the conventional diffusion equation and thus the stiffness matrix formulation becomes trivial. Meanwhile, the fractional derivative effect in the discrete formulation is completely transferred to the force vector, which is obviously much easier and efficient to compute than the dense fractional derivative stiffness matrix. Subsequently, it is further shown that for the general fractional advection---diffusion---reaction equation, the symmetric and banded structure can also be maintained for the diffusion stiffness matrix, although the total stiffness matrix is not symmetric in this case. More importantly, it is demonstrated that under certain conditions this symmetric diffusion stiffness matrix formulation is capable of producing very favorable numerical solutions in comparison with the conventional non-symmetric diffusion stiffness matrix finite element formulation. The effectiveness of the proposed methodology is illustrated through a series of numerical examples.

Nonlinear analysis of reinforced concrete plane frames exposed to fire using direct stiffness method

Abstract: This article presents a framework based on the direct stiffness method for nonlinear thermo-mechanical analysis of reinforced concrete plane frames subjected to fire. It accounts for geometric nonl...

Flutter Instability of Cracked Rotating Non-Uniform Beams Subjected to Distributed Follower Force

Abstract: In this paper, the effect of an open edge crack on the instability of rotating non-uniform beams subjected to uniform distributed tangential compressive load is studied. The local stiffness due to the presence of crack is considered in the global stiffness matrix of the structure using the finite element method. The cracked beam element is modeled as two equal sub-beam elements connected by a massless rotational spring. Based on the fracture mechanics, the strain energy release rate and the stress intensity factors are employed to investigate the stiffness of the rotational spring. Then, the modified shape functions are developed to reflect the crack stiffness in the finite element analysis. To validate the accuracy of the finite element model and results obtained, comparisons have been made between the results obtained and those available in the literature. The effects of several parameters, including the linear and nonlinear thickness variations, angular velocity, crack location and size, on the instability of cracked rotating non-uniform cantilevers are also examined. The results show that the location of crack significantly influences the critical magnitude of the follower force that destabilizes the cantilevers. In addition, geometric non-uniformity reduces the stability of the cracked cantilevers. For the same amount of cantilever mass, different patterns of mass distribution result in different stability diagrams.

Buckling analysis of thin-walled members with open-branched cross section via semi-analytical finite strip transfer matrix method

Abstract: Thin-walled members with open-branched cross section have been used in many modern engineering structures, and their buckling performance has been widely studied. In this paper, two transfer methods are developed to tackle the transfer problems at the junction of the open-branched cross-section of thin-walled members by using semi-analytical finite strip transfer matrix method (FS-TMM) which is a combined use of semi-analytical finite strip method (SA-FSM) and transfer matrix method (TMM) for the buckling analysis. Compared to traditional SA-FSM, this method has a smaller matrix and higher computational efficiency due to no global stiffness matrix generated. An asymmetric E-section member, a symmetric I-section member and a X-section member with loaded edges simply supported are analyzed by the derived formulation. All the results are compared with SA-FSM's or finite element method's results to prove the reliability and efficiency of this method.

On stability and reflection-transmission analysis of the bipenalty method in contact-impact problems: a one-dimensional, homogeneous case study

Abstract: Summary The stability and reflection-transmission properties of the bipenalty method are studied in application to explicit finite element analysis of one-dimensional contact-impact problems. It is known that the standard penalty method, where an additional stiffness term corresponding to contact boundary conditions is applied, attacks the stability limit of finite element model. Generally, the critical time step size rapidly decreases with increasing penalty stiffness. Recent comprehensive studies have shown that the so-called bipenalty technique, using mass penalty together with standard stiffness penalty, preserves the critical time step size associated to contact-free bodies. In this paper, the influence of the penalty ratio (ratio of stiffness and mass penalty parameters) on stability and reflection-transmission properties in one-dimensional contact-impact problems using the same material and mesh size for both domains is studied. The paper closes with numerical examples which demonstrate the stability and reflection-transmission behaviour of the bipenalty method in one-dimensional contact-impact and wave propagation problems of homogeneous materials. This article is protected by copyright. All rights reserved.

Dynamic modal prediction and experimental study of thin-walled workpiece removal based on perturbation method

Abstract: Because of the removal of the material, the material quality and stiffness matrix of the thin-walled parts will be directly changed and the dynamic characteristics will be significantly changed in the aviation thin-walled parts of the NC machine. When the structural parameters caused by the removal process frequently change slightly, it is necessary to solve the generalized feature problem, by using the classical method to obtain the dynamic characteristics repeatedly or the measured mode. However, it is clear that these methods are time consuming, labor intensive, and are not suitable for efficient prediction of dynamic modes. The method of matrix perturbation is proposed in order to solve the natural modal of the modified thin-walled structure. The finite element method is used to analyze the free vibration of the thin-walled structured blades. Based on the frequency, stiffness matrix, mass array, and vibration data which were extracted from the analysis results, the natural frequency of the modified structural parameters is calculated by the matrix perturbation method; then, the calculated results are compared with the finite element calculation results as well as the measured results. These results show that the process of applying the matrix perturbation method to modify the natural mode is effective, reliable, and time saving in the dynamic mode prediction of thin-walled NC machine, which lays the foundation for the establishment of three-dimensional stability domain. Finally, it can also provide theoretical basis for the resonance suppression of milling process by analyzing the evolutional rule of thin-walled part milling.

Assessment of Effects of Reductions of Lateral Stiffness along Height on Buildings Modeled as Elastic Cantilever Shear Beams

Abstract: A simplified model useful for assessing economic losses due to moderate seismicity events in urban areas has been developed by studying the behavior of buildings before yielding their structural system, allowing for nonuniform stiffness along their height. In particular, buildings are modeled as cantilever shear beams with uniform mass and parabolic reduction of lateral stiffness. This particular stiffness distribution is relevant, as it could be expected to occur in buildings where earthquake action is a critical structural design criterion. The equation of motion governing the dynamic behavior of the proposed model is solved analytically, finding mode shapes in terms of first and second zero-order Legendre functions. The solution is verified by comparing it with results obtained from fine mesh finite element models. The effect of reducing the lateral stiffness is then studied in the first five modes of vibration. Results include modal periods, mode shapes, modal participation factors, and derivatives of...

Hysteretic beam element with degrading smooth models

Abstract: This work presents the development of a hysteretic beam element in the context of the finite element method, that is suitable for the inelastic dynamic analysis of framed structures. The formulation proposed is able to capture the main characteristics of hysteresis in structural systems and mainly accounts for stiffness degradation, strength deterioration and pinching phenomena, as well as for non-symmetrical yielding that often characterizes their behavior. The proposed formulation is based on the decoupling of deformations into elastic and hysteretic parts by considering additional hysteretic degrees of freedom, i.e., the hysteretic curvatures and hysteretic axial deformations. The direct stiffness method is employed to establish global matrices and determine the mass and viscous damping, as well as the elastic stiffness and the hysteretic matrix of the structure that corresponds to the newly added hysteretic degrees of freedom. All the governing equations of the structure, namely the linear global equations of motion and the nonlinear evolution equations at elemental level that account for degradations and pinching, are solved simultaneously. This is accomplished by converting the system of equations into state space form and implementing a variable-order solver based on numerical differentiation formulas (NDFs) to determine the solution. Furthermore, hysteretic loops and degradation phenomena are easily controlled by modifying the model parameters at the element level enabling simulations of a more realistic response. Numerical results are presented and compared against experimental results and other finite element codes to validate the proposed formulation and verify its ability to simulate complex hysteretic behavior exhibiting cyclic degradations.

Direct Stiffness Method

Abstract: At the core of the finite element modelling process are a diverse possible range of solution approaches for any particular problem. Each of these approaches are adapted for the type of problem that one is interested in, for example structural, fluid, thermal or acoustic problems. The commonest type of problems that FEM addresses are the structural and solid mechanics problems and the direct stiffness method is the heart of the solution strategy. This chapter describes the principles of the direct stiffness method. Simple truss elements are introduced as the crudest finite elements for demonstrating the direct stiffness method, although other more advanced discretization finite elements can also be used. The mechanics of the direct stiffness method will be explained. In particular, the discussion highlights the use of nodal properties for the truss elements to determine displacements, velocities, internal and external forces, etc. for a given truss system. The chapter concludes with practical example problems.

Element Stiffness Matrix

Abstract: First, the element stiffness matrix [k] for a 2-node bar is generated using three approaches: direct, variational, and weighted residuals. For the weighted residuals method, emphasis is placed on the use of the Galerkin's method. We conclude from this exercise that the element stiffness matrix can be directly obtained from integration of the element shape functions. Second, we calculate numerical entries of element stiffness matrices by using the Gauss numerical integration scheme for the simplest and most commonly used element types. Also introduced are the full-integration, selective-reduced-integration, and reduced-integration schemes. When the reduced-integration scheme is used, concerns regarding the potential hourglass mode are discussed. Third, for a new element type, which can be decomposed into two or more basic element types, we discuss the principle of superposition for creating the stiffness matrix of this new element type. Finally, the rotational matrix is presented to transfer vectors and stiffness matrices from locally derived stiffness matrices, which are based on the natural coordinate system, to the global coordinate system.

QTT-isogeometric solver in two dimensions

Abstract: The goal of this paper is to develop a numerical algorithm that solves a two-dimensional elliptic partial differential equation in a polygonal domain using tensor methods and ideas from isogeometric analysis The proposed algorithm is based on the Finite Element (FE) approximation with Quantized Tensor Train decomposition (QTT) used for matrix representation and solution approximation In this paper we propose a special discretisation scheme that allows to construct the global stiffness matrix in the QTT-format The algorithm has $O(\log n)$ complexity, where $n=2^d$ is the number of nodes per quadrangle side A new operation called z-kron is introduced for QTT-format It makes it possible to build a matrix in z-order if the matrix can be expressed in terms of Kronecker products and sums An algorithm for building a QTT coefficient matrix for FEM in z-order "on the fly", as opposed to the transformation of a calculated matrix into QTT, is presented This algorithm has $O(\log n)$ complexity for $n$ as above