Showing papers on "Tangent stiffness matrix published in 2014"

A mixed finite element procedure of gradient Cosserat continuum for second-order computational homogenisation of granular materials

Abstract: A mixed finite element (FE) procedure of the gradient Cosserat continuum for the second-order computational homogenisation of granular materials is presented. The proposed mixed FE is developed based on the Hu---Washizu variational principle. Translational displacements, microrotations, and displacement gradients with Lagrange multipliers are taken as the independent nodal variables. The tangent stiffness matrix of the mixed FE is formulated. The advantage of the gradient Cosserat continuum model in capturing the meso-structural size effect is numerically demonstrated. Patch tests are specially designed and performed to validate the mixed FE formulations. A numerical example is presented to demonstrate the performance of the mixed FE procedure in the simulation of strain softening and localisation phenomena, while without the need to specify the macroscopic phenomenological constitutive relationship and material failure model. The meso-structural mechanisms of the macroscopic failure of granular materials are detected, i.e. significant development of dissipative sliding and rolling frictions among particles in contacts, resulting in the loss of contacts.

Static Damage Identification of 3D and 2D Frames

Abstract: A new algorithm for static damage detection of three- and two-dimensional frames is presented in this paper. This approach is based on the minimization of difference between the measured and analytical static displacements of frames. The damage detection problem is solved as a nonlinear constrained structural optimization. In this strategy, the global structural stiffness matrix is parameterized. To achieve the goal, a new technique based on the eigen decomposition of the local elemental stiffness matrix is suggested. Structural damage is modeled as a reduction in cross-sectional properties of the elements. It is assumed that the stiffness matrix of the structure is perturbed due to damage. Hence, the damaged structural stiffness matrix is presumed to be the sum of the stiffness matrix of the undamaged structure and the perturbation matrix. Consequently, the sum of these matrices should be inverted in each iteration. Instead of the common ways of inversion, Sherman–Morrison–Woodbury formula is employed. P...

Development of the distinct lattice spring model for large deformation analyses

Abstract: SUMMARY This study develops the distinct lattice spring model (DLSM) for geometrically nonlinear large deformation problems The formulation of a spring bond deformation under a large deformation is derived under the Lagrange framework using polar decomposition The results reveal that the DLSM's stiffness matrix under small deformations is the tangent stiffness matrix of the DLSM under large deformations The formulation of the spring bond internal force under a given configuration is also presented and can be used to calculate the unbalanced force Using these formulations, three nonlinear solving methods (the Euler method, modified Euler method, and Newton method) are developed for the DLSM with which to tackle large deformation problems To investigate the performance of the developed model, three numerical examples involving large deformations are presented, the results of which are also in good agreement with the analytical and finite element method solutions Copyright © 2013 John Wiley & Sons, Ltd

CH of masonry materials via meshless meso-modeling

Abstract: In the present study a multi-scale computational strategy for the analysis of masonry structures is presented. The structural macroscopic behaviour is obtained making use of the Computational Homogenization (CH) technique based on the solution of the boundary value problem (BVP) of a detailed Unit Cell (UC) chosen at the meso-scale and representative of the heterogeneous material. The smallest UC is composed by a brick and half of its surrounding joints, the former assumed to behave elastically while the latter considered with an elastoplastic softening response. The governing equations at the macroscopic level are formulated in the framework of finite element method while the Meshless Method (MM) is adopted to solve the BVP at the mesoscopic level. The work focuses on the BVP solution. The consistent tangent stiffness matrix at a macroscopic quadrature point is evaluated on the base of BVP results for the UC together with a localisation procedure. Validation of the MM procedure at the meso-scale level is demonstrated by numerical examples that show the results of the BVP for the simple cases of normal and shear loading of the UC.

Numerical calculation of the tangent stiffness matrix in materials modeling

Abstract: The Finite Element Method in the field of materials modeling is closely connected to the tangent stiffness matrix of the constitutive law This so called Jacobian matrix is required at each time increment and describes the local material behavior It assigns a stress increment to a strain increment and is of fundamental importance for the numerical determination of the equilibrium state For increasingly sophisticated material models the tangent stiffness matrix can be derived analytically only with great effort, if at all Numerical methods are therefore widely used for its calculation We present our method to calculate the tangent stiffness matrix for the logarithmic strain measure The approach is compared with other commonly used procedures A significant increase in accuracy can be achieved with the proposed method (© 2014 Wiley-VCH Verlag GmbH & Co KGaA, Weinheim)

Solution of Post-Buckling & Limit Load Problems,Without Inverting the Tangent Stiffness Matrix & WithoutUsing Arc-Length Methods

Abstract: In this study, the Scalar Homotopy Methods are applied to the solution of post-buckling and limit load problems of solids and structures, as exemplified by simple plane elastic frames, considering only geometrical nonlinearities. Explicit- ly derived tangent stiffness matrices and nodal forces of large-deformation planar beam elements, with two translational and one rotational degrees of freedom at each node, are adopted following the work of (Kondoh and Atluri (1986)). By us- ing the Scalar Homotopy Methods, the displacements of the equilibrium state are iteratively solved for, without inverting the Jacobian (tangent stiffness) matrix. It is well-known that, the simple Newton's method (and the Newton-Raphson iteration method that is widely used in nonlinear structural mechanics), which necessitates the inversion of the Jacobian matrix, fails to pass the limit load as the Jacobian matrix becomes singular. Although the so called arc-length method can resolve this problem by limiting both the incremental displacements and forces, it is quite complex for implementation. Moreover, inverting the Jacobian matrix generally consumes the majority of the computational burden especially for large-scale prob- lems. On the contrary, by using the presently developed Scalar Homotopy Meth- ods, convergence near limit loads, and in the post-buckling region, can be easily achieved, without inverting the tangent stiffness matrix and without using complex arc-length methods. The present paper thus opens a promising path for conduct- ing post-buckling and limit-load analyses of nonlinear structures. While the simple Williams' toggle is considered as an illustrative example in this paper, extension

Inelastic Buckling Analysis of Frames with Semi-Rigid Joints

Abstract: An improved method for evaluating effective buckling length of semi-rigid frame with inelastic behavior is newly proposed. Also, generalized exact tangential stiffness matrix with rotationally semi-rigid connections is adopted in previous studies. Therefore, the system buckling load of structure with inelastic behaviors can be exactly obtained by only one element per one straight member for inelastic problems. And the linearized elastic stiffness matrix and the geometric stiffness matrix of semi-rigid frame are utilized by taking into account 4th terms of taylor series from the exact tangent stiffness matrix. On the other hands, two inelastic analysis programs(M1, M2) are newly formulated. Where, M1 based on exact tangent stiffness matrix is programmed by iterative determinant search method and M2 is using linear algorithm with elastic and geometric matrices. Finally, in order to verify this present theory, various numerical examples are introduced and the effective buckling length of semi-rigid frames with inelastic materials are investigated.

Tangent moduli of hot-rolled I-shaped axial members considering various residual stress distributions

Abstract: This paper presents an equation for the effective tangent moduli for steel axial members of hot-rolled I-shaped section subjected to various residual stress distributions. Because of the existence of residual stresses, the cross section yields gradually even when the member is subjected to uniform axial stresses. In the elasto-plastic stage, the structural response can be easily traced using rational tangent modulus of the member. In this study, the equations for rational tangent moduli for hot-rolled I-shaped steel members in the elasto-plastic stage were derived based on the general principle of force-equilibrium. For practical purpose, the equations for the tangent modulus were presented for conventional patterns of the residual stress distribution of hot-rolled I-shaped steel members. Through a series of material nonlinear analyses for steel axial members modeled by shell elements, the derived equations were numerically verified, and the presented equations were compared with the CRC tangent modulus equation, the most frequently used equation so far. The comparative study shows that the presented equations are extremely effective for accurately analyzing elasto-plastic behavior of the axially loaded members in a simple manner without using complex shell element models.

Inelastic and non-linear materials

Abstract: In Chapter 2 we presented a framework for solving general problems in solid mechanics. In this chapter we consider several classical models for describing the behaviour of engineering materials. Each model we describe is given in a strain-driven form in which a strain or strain increment obtained from each finite element solution step is used to compute the stress needed to evaluate the internal force, σ B T σ dΩ, as well as a tangent modulus matrix, or its approximation, for use in constructing the tangent stiffness matrix. Quite generally in the study of small deformation and inelastic materials (and indeed in some forms applied to large deformation) the strain (or strain rate) or the stress is assumed to split into an additive sum of parts. We can write this as e=e e +e i e = e e + e i (4.1) or σ=σ e +σ i σ = σ e + σ i (4.2) in which we shall generally assume that the elastic part is given by the linear model e e = D -1 σ e e = D - 1 σ (4.3) in which D is the matrix of elastic moduli.

Tangent Stiffness Matrix of Spatial Beam Element Considering Bend and Torsion Coupling Effect

Abstract: Based on the spatial beam-column differential equations, the slope deflection equations considering second-order and bend-torsion coupling effect are established. The additional moment caused by torsion and bend deflection are taken into account. Then the finite element pattern of spatial beam-column considering the couple effect of torsion and bend is given. The tangent stiffness matrix and relevant program for nonlinear analysis are further obtained. By the nonlinear calculation and stability analysis of single component which has high accuracy or precise solution, the precision of the FEM model given in this paper is verified by comparing the results with that given in references.

Dynamic Non - Linear Behaviour of Cable Stayed Bridges Under Seismic Loadings

Abstract: The cable stayed bridges represent key points in transport networks and their seismic behaviour need to be fully understood. This type of bridge, however is light and flexiable and has a low level of inherent damping. Consequenly, thery are susceptible to ambient excitation from seismic loads. Since the geometric and dynamic properities of the bridges as well as the characteristics of the excitations are complex, it is necessary to fully understand the mechanism of the interaction among the structural componenets with reasonable bridge shapes. This paper discuss the dynamic response of a cable stayed bridge under seismic loadings. All possible sources of nonlinearity, such cable sag, axial-force-bending moment interaction in bridge towers and girders and change of geometery of the whole bridge due to large displacement are based on the utilization of the tangent stiffness matrix of the bridge at the dead-load deformed state which is obtained from the geometry of the bridge under gravity load conditions ,iterative procedure is utilized to capture the non-linear seismic response and different step by step integration schemes are used for the integration of motion equations. In this study, three spans cable-stayed bridge with different cable systems has been analyzed by three dimensional nonlinearity finite element method. The three dimensional bridge model is prepared on SAP 2000 ver.14 software and time history analyses were performed to assess the conditions of the bridge structure under a postulated design earthquake of 0.5g. The results are demonstrated to fully understand the mechanism of the deck-stay interaction with the appropriate shapes of a cable stayed bridges.

PIA for Uniform Cubic B-spline Curve Interpolation with Prescribed Tangent Vector

Abstract: An uniform cubic B-spline curve is generated to interpolate the data points with prescribed tangent vectors based on the progressive iterative approximation. It starts with an initial Bspline curve which takes the given data points as the control points with even indexes and takes the end points of the tangent vectors as the control points with odd indexes. Then by adjusting its control points gradually with iterative formulas,a sequence of curves is obtained. The limit curve of the sequence will interpolate the data points with prescribed tangent vectors. The curve fits a given ordered point set and corresponding tangent vectors without solving a linear system.

Element stiffness matrix for Timoshenko beam with variable cross-section

Abstract: The variable cross-section members have been widely used in engineering practice for many years,thus it is necessary to investigate their element stiffness matrixes.In this paper,based on the principle of potential energy,the element stiffness matrix with approximation to second order are obtained, where the change rates of both the flexural and shear stiffness are treated as infinitesimal quantities(or Infinitesimal).It is noted that the effects of geometric nonlinearity due to axial force as well as shear deformation is considered in the matrix.In addition,based on the differential equilibrium equations of the members,the flexural and shear displacements modes with approximation to second order,expressed as cubic and quintic polynomial respectively,are also obtained.Moreover,the singularity of the element stiffness matrix and the expression of axial stiffness are discussed in detail.By comparing the obtained matrix results with some exact solutions,it is indicated that the accuracy of the obtained element stiffness matrix can be guaranteed.Finally,the convergence of this method is discussed by comparing with other methods in a case study.

Nonlinear analysis of plane frames using a corotational fomulation and plasticity by layers in a timoshenko beam el-ement

Abstract: The purpose of this work is to perform a nonlinear analysis of plane frame struc- ture using a corotational formulation and a layered plastic modeling. The plane frame is dis- cretized with a 2D Timoshenko beam element. Plasticity is introduced by rate-independent Von-Mises model with isotropic hardening. Numerical integration over the cross-section is performed for obtain the internal force vector and tangent stiffness matrix of these elements. At each integration point, the backward-Euler algorithm is used for integration in the consti- tutive equations. Some examples are used in order to check the performances in the elements and the path-following procedures.