Showing papers on "Tangent stiffness matrix published in 1988"

Geometric and material non‐linear analysis of beam‐columns and frames using the minimum residual displacement method

Abstract: A geometric and material non-linear analysis procedure for framed structures is presented, using a solution algorithm of minimizing the residual displacements. This new non-linear solution technique is believed to be the optimum in the Newton–Raphson scheme since it follows the shortest path to achieve convergence. The concept of the effective tangent stiffness matrix is introduced and is found to be efficient, simple and logical in handling the non-linear analysis of frames with braced members and in separating multiple bifurcation points.

A simple method for the calculation of postcritical branches

Abstract: The practical behaviour of problems exhibiting bifurcation with secondary branches cannot be studied in general by using standard path‐following methods such as arc‐length schemes. Special algorithms have to be employed for the detection of bifurcation and limit points and furthermore for branch‐switching. Simple methods for this purpose are given by inspection of the determinant of the tangent stiffness matrix or the calculation of the current stiffness parameter. Near stability points, the associated eigenvalue problem has to be solved in order to calculate the number of existing branches. The associated eigenvectors are used for a perturbation of the solution at bifurcation points. This perturbation is performed by adding the scaled eigenvector to the deformed configuration in an appropriate way. Several examples of beam and shell problems show the performance of the method.

Elasto-plastic large deformation analysis of space-frames: a plastic-hinge and stress-based explicit derivation of tangent stiffnesses

Abstract: SUMMARY This paper deals with elasto-plastic large deformation analysis of space-frames. It is based on a complementary energy approach. A methodology is presented wherein: (i) each member of the frame, modelled as an initially straight space-beam, is sought to be represented by a single finite element, (ii) each member can undergo arbitrarily large rigid rotations, but only moderately large relative rotations; (iii) a plastic-hinge method, with arbitrary locations of the hinges along the beam, is used to account for plasticity, (iv) the non-linear bending-stretching coupling is accounted for in each member, (v) the applied loading may be non-conservative and (vi) an explicit expression for the tangent stiffness matrix of each element is given under conditions (i) to (v). Several examples, with both quasi-static and dynamic loading, are given to illustrate the accuracy and efficiency of the approaches presented.

On a non‐linear formulation for curved Timoshenko beam elements considering large displacement/rotation increments

Abstract: SUMMARY An incremental Total Lagrangian Formulation for curved beam elements that includes the effect of large rotation increments is developed. A complete and symmetric tangent stiffness matrix is obtained and the numerical results show, in general, an improvement over the standard formulation where the assumption of infinitesimal rotation increments is made in the derivation of the tangent stiffness matrix.

Bifurcations in finite element models with a non-associated flow law

Abstract: A numerical approach to bifurcation problems in soil mechanics is described. After locating the bifurcation point by a combination of an incremental-iterative loading procedure and an eigenvalue analysis of the tangent stiffness matrix, the solution is continued on the localization path by a suitable combination of the fundamental solution and the eigenvector belonging to the lowest eigenvalue. The procedures are applied in a bifurcation analysis of a cohesionless soil in a biaxial testing device. The results suggest that a diffuse bifurcation mode with a short wavelength is encountered whereupon a shear band gradually develops. The inclination angle of the shear band compares well with analytical formulae and with empirical data.

p-Almost tangent structures

Abstract: In this paper, we introduce a new type of geometric structures (calledp-almost tangent structures). They are a natural generalization of almost tangent structures. Moreover, the tangent budle ofp 1 -velocitiesT M of any manifoldM carries a canonicalp-almost tangent structure (hence the name). We interpret ap-almost tangent structure as a type ofG-structure and stablish its integrability in terms of the vanishing of some tensor fields associated with it. Finally, the existence of an adapted symmetric connection is proved.

Consistent linearization in elasto‐plastic shell analysis

Abstract: The present paper is directed towards elasto‐plastic large deformation analysis of thin shells based on the concept of degenerated solids. The main aspect of the paper is the derivation of an efficient computational strategy placing emphasis on consistent elasto‐plastic tangent moduli and stress integration with the radial return method under the restriction of ‘zero normal stress condition’ in thickness direction. The advantageous performance of the standard Newton iteration using a consistent tangent stiffness matrix is compared to the classical scheme with an iteration matrix based on the infinitesimal elasto‐plastic constitutive tensor. Several numerical examples also demonstrate the effectiveness of the standard Newton iteration with respect to modified and quasi‐Newton methods like BFGS and others.

Dynamic Analysis of Finitely Stretched and Rotated 3-D Space-Curved Beams

Abstract: A novel theory and its computational implementation are presented for the analysis of strongly nonlinear dynamic response of highly-flexible space-beams that undergo large overall motions as well as elastic motions with arbitrarily large rotations and stretches. The case of conservative force loading, which may also lead to configuration-dependent moments on the beam, is treated. A symmetric tangent stiffness matrix is derived at all times even if the distributed external moments exist. An example of transient dynamic response of the beam is presented to illustrate the validity of the theoretical methodology developed herein.

Large Deflection Analysis of Inelastic Plane Frames

Abstract: A general method for the large deflection analysis of inelastic plane frames is presented. The large deflection analysis is based on the Eulerian formulation in which a member tangent stiffness matrix is constructed with reference to the current deformed configuration. The exact differential equation and the moment‐thrust‐curvature relations of a member subjected to large relative deflection are used for deriving the member tangent stiffness matrix. A member tangent stiffness matrix for W‐sections of elasto‐plastic material is constructed as an example and its application to the large deflection analysis of inelastic frames is demonstrated by several examples. A simple form of the member tangent stiffness matrix is deduced from the above exact formulation by utilizing the assumption of small member relative deflections. It is shown that the simple one can yield excellent results compared to those obtained by the exact formulation. Hence it is reasonable to adopt the assumption of small member relative def...

An approach to probabilistic finite element analysis using a mixed-iterative formulation

Abstract: An efficient algorithm for computing the response sensitivity of finite element problems based on a mixed-iterative formulation is proposed. This method does not involve explicit differentiation of the tangent stiffness array and can be used with formulations for which a consistent tangent stiffness is not readily available. The method has been successfully applied to probabilistic finite element analysis of problems using the proposed mixed formulation, and this exercise has provided valuable insights regarding the extension of the method to a more general class of problems to include material and geometric nonlinearities.

Stability and Spurious Kinematic Modes in Strain-Softening Concrete

Abstract: Consequences of the use of softening or brittle crack models for the stability of numerical simulations of the mechanical behavior of concrete structures are investigated. It is shown that brittle as well as softening crack models may trigger the formation of spurious kinematic modes.

The Matrix Stiffness Method-Part 2

Abstract: In a final presentation of the matrix stiffness method of structural analysis, a general technique applicable to all classes of structure is outlined. The technique uses coordinate transformation and it is first necessary to discuss the axes of reference used to define the structure and its actions.

A Practical Iteration Strategy for Finite Deformation Post-Buckling Analysis

Abstract: A conventional linearized stiffness matrix without the coupling terms of the bending and the axial stiffnesses is used in deriving the governing nonlinear equations. Description of the nodal locations by their initial coordinates as well as their displacements, and transformations of the nodal quantities between space-fixed global coordinate system and element-fixed local one lead to nonlinear incremental force-displacement relations. Application of an iterative-corrective solution strategy to the nonlinear equation provides an expression of tangent stiffness matrix and residual forces. Several examples are given on the trace of load-deflection paths of framed structures.

Large Deformation, Post-Buckling Analyses of Large Space Frame Structures, Using Explicitly Derived Tangent Stiffness Matrices

Abstract: Large deformation and post-buckling analyses of structures have been studied by many researchers as an important subject in structural mechanics in the past decade or so. In all their studies, an incremental approach, either of the total Lagragean type or the updated Lagragean type, is employed. As the incremental approach is often based on the so-called tangent stiffness matrix, which reflects all the non-linear geometrical and mechanical effects, the majority of non-linear analyses of typical engineering structures, and especially truss- and frame-type large space structures, will be vastly simplified if an explicit expression (i.e., without involving assumed basis functions for displacements/stresses, and without involving element-wise numerical integrations) for the tangent stiffness matrix of an element can be derived.