Initial Stiffness Matrix Research Articles

Geometric stiffness effects on data recovery of an idealized mast/blanket model

The photovoltaic arrays for the international space station consist of a pre-tensioned blanket of solar collectors and a déployable mast. NASA uses the MSC/NASTRAN finite element program for modeling the dynamic response of the structure due to various loading conditions, such as plume impingement during shuttle docking. This finite element program uses the updated stiffness matrix (elastic plus geometric, or initial stress stiffness matrix) in determining the natural frequencies and mode shapes, as well as the dynamic response, of a pre-loaded structure. However, during the data recovery phase, when the moment and shear at the supports, and internal stresses are determined, geometric stiffness effects are neglected, and only the elastic stiffness is used in the calculation. The purpose of this study was to determine whether using the actual displacements, calculated based upon both elastic and geometric stiffness effects, would produce acceptable results in predicting shear and moment if the geometric stiffness effects were later omitted during data recovery. In this study, the PV array was idealized as a cantilever beam with an attached pre-tensioned cable. Static and dynamic analysis were performed, both using and neglecting the geometric stiffness matrix during data recovery. When considering the idealized mast/blanket model, neglecting the geometric stiffness contribution during data recovery led to a 32.2% difference in the vertical (shear) load at the fixed support, and an 8.8% difference in shear at the free end of the beam (compared to the inclusion of geometric stiffness effects in the analysis). The static analysis provided similar results and supported the “reasonableness” of the dynamic analysis. Due to the large discrepancies in the predicted stresses which can ocur when geometric stiffness contributions are neglected during data recovery, it should be imperative that when a structure contains pre-loaded elements, the geometric stiffness effects must be fully considered both in the determination of the natural frequencies and mode shapes, as well as in the subsequent calculation of nodal reactions and stresses.

An effective numerical procedure for inelastic analysis of structures under thermo-mechanical loading

An effective, stable and accurate thermo-elastic-plastic and creep analysis procedure is presented. First, based on initial elastic stiffness matrix method and accelerating convergence, an iterative scheme is proposed for inelastic analysis. Second, a high accuracy scheme for integrating the constitutive equations is developed. The stability, accuracy and applicability of the procedure are demonstrated by numerical examples.

A C0 three‐node shell element for non‐linear structural analysis

AbstractA C0 three‐node shell finite element well suited to non‐linear calculations is proposed. The element is based on Mindlin kinematics and the degenerated solid approach. Linear Lagrange functions are used for geometry and displacement interpolations. The formulation is made in the natural material frame. A strain interpolation avoids shear locking and an intermediate material frame related to the element sides is introduced in order to fix nodal transverse shear strain components. The modifications of strain interpolations concern both the non‐linear and linear parts of strain and are taken into account in ail calculations, among others in the expression of the initial stress stiffness matrix. A single set of integration points on the normal at the centre of gravity is sufficient, which is very interesting for numerical efficiency especially in the case of non‐linear analyses.

On nonlocal damage models for interface problems

Consistent with thermodynamic restrictions, nonlocal formulations of damage models are investigated with regard to their analytical and numerical capabilities for predicting the evolution of microcracking. To trace the post-limit structural response with an efficient numerical algorithm, an incremental-iterative solution strategy is constructed through the use of an initial elasticity stiffness matrix and an evolving-localization constraint. The constraint is related to a suitable measure of microcracking at the most severely damaged element that changes with the evolution of a localization zone. As a simple phenomenological approach, a nonlocal feature is incorporated into the constitutive model through the gradient of a scalar measure of damage, and a symmetry boundary condition is invoked for the nonlocal governing differential equation. In numerical calculations, the gradient is evaluated through a simple difference scheme. For applications to the interface problems with geologic media, the proposed procedure is verified with analytical solutions for one-dimensional cases, and illustrated with two-dimensional cases for which experimental observations are available. Some important theoretical and computational issues associated with nonlocal models are then discussed based on the results obtained.

A semi‐analytical solution procedure for predicting damage evolution at interfaces

AbstractA semi‐analytical solution procedure is developed to predict damage evolution at the interface between two dissimilar geologic materials. The procedure consists of an analytical field solution within each finite element, and a numerical scheme for simulating structural responses. For static problems, an incrementaliterative solution strategy is constructed through the use of an initial elasticity stiffness matrix, and an evolving‐localization constraint in terms of a suitable measure of damage at the most severely damaged element. For simplicity, one‐dimensional problems are considered to illustrate the features of the proposed procedure, and future research is discussed based on the preliminary results obtained.

A comparison of techniques for computing the buckling loads of stiffened shells

Three different methods are employed to estimate the buckling loads of several ring stiffened and orthotropic cylindrical shells using finite elements. The methods used are a nonlinear bifurcation analysis and two linearized buckling analyses, one that ignores the initial displacement stiffness matrix, and one that includes it. Large differences are observed between the predictions made by the two linearized buckling analyses for a range of shell geometries. Detailed studies of a shell with six stiffeners demonstrate that these differences are caused by different versions of the linearized eigenvalue problem, rather than by the use of different numerical formulations. These discrepancies are also observed for orthotropic shells when L R is small (⩽-1) and the degree of orthotropy ( E x E y is high (⩾-10). Investigations of the prebuckling behavior of some of the cylinders show that the problem is caused by significant nonlinear prebuckling deformations. This means that a nonlinear bifurcation analysis must sometimes be used to accurately estimate the buckling load of stiffened shells.

Nonlinear analysis of cable-stayed bridge by finite strip method

In this paper the finite strip method is employed to model the girder of a cable-stayed bridge. In order to simplify analysis, the entire girder is treated as a substructure with only a few degrees of freedom, corresponding to the points of cable attachments. After the stiffness matrix and equivalent load vector of the girder substructure are obtained, this matrix is assembled with linear elastic stiffness matrices of cables and pylons so as to give the initial stiffness matrix of the entire structure. With the nonlinearities due to sag and angle change of cables being taken into account, the final solution of a cable-stayed bridge can be obtained by using iterative methods with this stiffness matrix as initial values and iterating until the resistance of the structure to its deformation is in equilibrium with the external loads. The example of a double-plane cable-stayed bridge under an eccentric point load demonstrates the effectiveness of this method.

Vibration of curved plate assemblies subjected to membrane stresses

In a previous paper [1] the finite strip method was applied to the prediction of the natural frequencies of vibration of longitudinally invariant, rigidly connected assemblies of circularly curved and flat strips having diaphragm end supports. This work is extended here to include the presence of an initial membrane stress field. An individual curved strip may be subjected to a biaxial direct stress field comprising a uniform stress acting in the circumferential direction and a non-uniform stress acting in the longitudinal direction. The presence of the membrane stress field is accommodated in the analysis by the inclusion of an initial stress or geometric stiffness matrix. A further extension included here is a facility to delete in-surface inertia terms. Results are presented for the application of the strip method in predicting the frequencies of vibration of a circular cylinder subjected to a complicated membrane stress system.

Analysis of Partially Uplifted Base Mats

This is a systematic method of analyzing a partially uplifted base mat on an elastic foundation, without removing tensile soil springs. The initial system stiffness matrix is used throughout the operation. The method becomes beneficial and important when the size of the system stiffness matrix of a finite element model is relatively large and many loading conditions cause the mat to be uplifted partially in different ways. The generalized equilibrium equations can be incorporated with general structural analysis computer programs, such as STRUDL, SAP, etc.

Analysis of Nonlinear Behavior of Bar and Plate with Temperature Change by Finite Element Method

Structural members and machinery parts are often subjected to various kinds of heat treatment, such as welding, gas-cutting, quenching, etc. during the process of production. The heat treatment produces thermal stresses which may result in residual deformation and cracks in the members and parts. Similarly, those parts which are used at high temperature or steel structures at fires pose the problem on their critical strength.The thermal stress and deformation are produced by the change in the temperature distribution, and high temperature influences the mechanical properties of the material to a great extent. In order to investigate the mechanism of thermal stress and deformation, a thermal elastic-plastic analysis is required with consideration of the mechanical properties dependent upon temperature and the thermal history of the material. Furthermore, the deformation of a slender column or thin plate is usually large, so that the analysis should include nonlinearities in both geometrical and material relations.Concerning thermal stress analysis, the analytical solutions are limited especially when the problem involves the plastic strains, even though the fundamental relations have been established.To deal with general problems, the finite element method is a powerful tool. The authors have developed a basic theory for thermal elastic-plastic analysis based on the finite element method using the incremental procedure and have shown the usefulness of the method on problems of welding.In this paper, the theory is extended to the analysis of combined nonlinear problems considering the influence of temperature upon the properties of the material with the aid of the principle of virtual work. The basic equilibrium equations are obtained for a plate from which those for beams or columns are readily derived. In the equations, there are many additional terms appeared due to the nonlinearity of problem, such as load correction vector, equivalent nodal forces due to temperature changes, temperature stiffness matrix, initial stress stiffness matrix, initial deflection stiffness matrix, plastic stiffness matrix, etc. The functions of these matrices are clarified in examples of one dimensional members.

A finite element formulation for problems of large strain and large displacement

An incremental and piecewise linear finite element theory is developed for the large displacement, large strain regime with particular reference to elastic-plastic behavior in metals. The resulting equations, though more complex, are in a similar form to those previously developed for large displacement, small strain problems, the only additional term being an initial load stiffness matrix which is dependent on current loads. This similarity in form means that existing nonlinear general purpose programs may easily be extended to include finite strains. A large displacement, small strain formulation (as applicable to problems of structural stability) is obtained from this theory by assuming that changes in length of line elements and relative rotation of orthogonal line elements are negligible compared to unity. The simplified equations are in essential agreement with previous formulations in the literature. The only difference which is observed is the persistence of the initial load stiffness matrix, which may be significant in some cases.