Incremental Stiffness Research Articles

Reduced-basis stiffness inversion of a structure-foundation system via component-mode synthesis

The purpose of this paper is to propose a new formulation for incremental reduced-basis stiffness inversion via component-mode synthesis. An elastically supported shear building model is taken as an example of an assemblage of substructures. The superstructure, the swaying spring and the rocking spring are regarded as ­the constituent substructures. The story stiffnesses of the superstructure are expressed in terms of linear combinations of several stiffness basis distributions. Eigenmodes of a fixed-base shear beam model having a uniform cross-section combined with rigid modes due to swaying and rocking displacements of the foundation ­are adopted as component modes for an overall model. It is shown that inverse use of the component-mode synthesis technique in an incremental inverse problem enables the development of an efficient computational procedure for finding stiffness parameters in the design problem. An approximate expression for the seismic response to stationary random excitations is also presented for the reduced model. The validity of the proposed method is demonstrated through several elastically supported shear building models. Copyright © 2000 John Wiley & Sons, Ltd.

This Month in ANESTHESIOLOGY

This Month in ANESTHESIOLOGY

ウェーブレットによる非線形構造システムの同定

This paper addresses the use of wavelets for the identification of nonlinear structural dynamic systems. The proposed method accurately determines hysteresis curves and damping coefficients of the structural system without pre-assumptions of nonlinear characteristics of the system, whereby it enables clarification of the complex phenomena. In this method, incremental stiffness of the structural system is expressed as a series expansion of wavelets. Coefficients of the series are estimated by the least-squares method. This method has been applied to a variety of problems, which show the effectiveness of the method.

Secondary buckling analysis of spherical caps

The aim of this paper is to investigate the secondary buckling behaviour and mode-coupling of spherical caps under uniformly external pressure. The analysis makes use of a rotational finite shell element on the basis of strain-displacement relations according to Koiter's shell theory (Small Finite Deflections). The post-buckling behaviours after a bifurcation point are analyzed precisely by considering multi-mode coupling between several higher order harmonic wave numbers: and on the way of post-buckling path the positive definiteness of incremental stiffness matrix of uncoupled modes is examined step by step. The secondary buckling point that has zero eigen-value of incremental stiffness matrix and the corresponding secondary mode are obtained, moreover, the secondary post-buckling path is traced.

Second-Order Elastic Plane-Frame Analysis Using Finite-Element Method

Members of a plane-frame structure may be subjected to a combination of lateral loads, end moments, and axial forces. Analysis of such a structure may be classified as elementary, first-order, or second-order, depending on the load conditions, magnitude of displacements, and method of solution. A numerical procedure is presented for the solution of geometrically nonlinear problems considering second-order effects. After a brief literature review, the paper presents the assumptions pertinent to the research; the relationship between the engineering strain and Green strain tensor considering small strain and moderate to large rotations; finite-element consideration in the derivation of the incremental stiffness matrices using increments of total potential energy and the concepts of a variational method; a procedure for the solution of the nonlinear problems and its algorithm; numerical examples and comparison of the numerical results with published literature; and conclusions and results. The algorithm that is presented is useful in identifying the differences between the first- and second-order analysis of structural systems.

Large deflection dynamic analysis of space frames

A robust numerical procedure based on the incremental secant stiffness concept for a large deflection static and dynamic analysis of space frames is described. The proposed method is capable of handling the nonlinear static post-buckling and dynamic analysis of space frames undergoing arbitrarily large rotations in three dimensional space efficiently and accurately. The technique can be easily implemented in an analysis program for framed structures to upgrade its capacity in predicting dynamic and nonlinear behaviour of frames. Numerical examples are given to illustrate the versatility of the computer method in handling the large deflection and dynamic behaviour of clamped beam, curved cantilever and space frame.

Three-dimensional nonlinear analysis of elastic arches

This paper presents a curved finite element model for the three-dimensional nonlinear analysis of elastic arches. The model includes higher-order curvatures which make the order of the bending strains consistent with that of the membrane strains, and the same low-order polynomials are used for all the displacements. No approximations are made for the twist rotations, and the nonlinear strains and the incremental stiffness matrix include the exact coupling due to the twist rotations. All significant terms of strains are retained. Without these terms, superimposed rigid body motions may lead to overstiff solutions. The model also includes warping of the cross-section and the Wagner effect. Application of the model to several numerical examples and comparison with the existing experimental and analytical results demonstrate that the model is much more effective and efficient than other methods in terms of accuracy, the number of elements needed for convergence, and the ability to pass maximum load points. The model is also used to investigate the elastic flexural-torsional buckling and the elastic postbuckling behaviour of arches.

部分球形殼の2次座屈とモード・カップリングに関する解析的研究

The aim of this paper is to investigate the secondary buckling behaviour and mode-coupling of spherical caps under uniformly distributed pressure by using a rotational finite shell element. The post-buckling behaviours after bifurcation point are analyzed exactly by considering multi-mode coupling (with several higer order harmonic wave numbers), and on the way of post-buckling path the positive definiteness of incremental stiffness matrix of uncoupled modes is examined step by step. The secondary buckling point that has zero eigen-value of incremental stiffness matrix and secondary mode are obtained, moreover, the secondary post-buckling path is traced. Beside, with the obtained displacement fields, the total potential energy, strain energy including membrane and bending energy components and Astatic buckling loads are also calculated. From the above energy components, the complicated post-buckling behaviours are examined.

An improved two-node finite element for stability and natural frequencies of axial-loaded Timoshenko beams

The linear flexural stiffness, incremental stiffness, mass, and consistent force matrices for a simple two-node Timoshenko beam element are developed based upon Hamilton's principle, where interdependent cubic and quadratic polynomials are used for the transverse and rotational displacements, respectively. The resulting linear flexural stiffness matrix is in agreement with the exact 2-node Timoshenko beam stiffness matrix. Numerical results are presented to show that the current element can accurately predict the buckling load and natural frequencies of axially-loaded isotropic and composite beams for a wide variety of beam-lengths and boundary conditions. The current element consistently outperforms the existing finite element approaches in studies involving the buckling or vibration behavior of axially-loaded short beams.

Large deflection analysis of structures by an improved combined finite element-transfer matrix method

An improved combined finite element-transfer matrix (IFE-TM) method is applied to the static analysis of structures with large displacement. In the present method, the transference of the state vectors from the left to the right in the combined finite element-transfer matrix method is changed into the transference of incremental stiffness equations of every section from the left to the right, in order to overcome the fault that the FE-TM method is generally applied to structures with a regular boundary. Furthermore, in the present method, the propagation of round-off errors occurring in recursive multiplications of the transfer and point matrices is avoided. A load incremental method is combined with modified Newton-Raphson method for the non-linear equations. A program ITNONDL-J based on this method is developed on the microcomputer IBM-PC/AT. Finally, numerical examples are presented to demonstrate the accuracy as well as the capability of the proposed method for geometrically non-linear static analysis of structures with a random boundary.

Geometrically non-linear dynamic analysis of plates by an improved finite element-transfer matrix method on a microcomputer

An improved finite element-transfer matrix method is applied to the transient analysis of plates with large displacement under various excitations. In the present method, the transfer of state vectors from left to right in a combined finite element-transfer matrix method is changed into the transfer of generally incremental stiffness equations of every section from left to right. Furthermore, in this method, the propagation of round-off errors occurring in recursive multiplications of transfer and point matrices is avoided. The Newmark-<TEX>${\beta}$</TEX> method is employed for time integration and the modified Newton-Raphson method for equilibrium iteration in each time step. An ITNONDL-W program based on this method using the IBM-PC/AT microcomputer is developed. Finally numerical examples are presented to demonstrate the accuracy as well as the potential of the proposed method for dynamic large deflection analysis of plates with random boundaries under various excitations.

Least squares methods for the mechanics of nonhomogeneous granular assemblies

Using least squares methods consistent averages for stress and strain of a nonhomogeneous granular assembly are derived. The expression for the average of the strain increment serves as a constraint in a functional approach to solving the displacements and spins of particles of a granular assembly in a statistical manner. The method shows directly what microscopic angular distributions are needed to describe the internal state of an assembly. They are the distribution of contacts, the distribution of both the averages and the variations in the interactive properties of the grains. The method is then applied to an assembly with normal-interacting circular grains in two dimensions. The incremental stiffness and the average displacement of the contacts as a function of the angle are obtained for the case of non-rotating strains. The results show clearly that for this case the sliding mode of motion is the most relevant one for deviatoric loading.

Free vibration steady state response of geometrically nonlinear shallow arches

A study of the large amplitude, free vibration of geometrically non-linear shallow arches is presented. To obtain the non-linear steady state response a combination of finite elements and finite difference methods is employed. Finite elements idealization is used to express the geometrically non-linear and incremental stiffness matrices and the rest of the terms appearing in the equation of motion are expressed by finite difference equations relating displacement, velocity and acceleration. The resulting recurrence formula is then solved by assuming that for the first time interval, the transverse displacement away from the centre of the arch, is in the form of a sinusoidal equation. The complex relationships between frequency and large amplitude for shallow arches having various ratios of radius to thickness are thus obtained and the results are discussed in relation to the elastic geometrically non-linear load-deflection curves which are generally multi valued exhibiting local maximum and minimum and resulting into snap-through between various equilibrium states. These elastic geometrically non-linear curves are based on an analytical solution in which the shape of the arch is expressed in terms of an infinite Fourier series and the thrust in the shallow arch is assumed to be constant.

Nonlinear stretch reflex interaction during cocontraction.

1. We investigated the role of stretch reflexes in controlling two antagonist muscles acting at the interphalangeal joint in the normal human thumb. Reflex action was compared when either muscle contracted alone and during cocontraction. 2. The total torque of the flexor pollicis longus (FPL) and extensor pollicis longus (EPL) muscles was measured in response to an externally imposed extension of the interphalangeal joint. The initial joint angle and the amplitude of the extension were constant in all experiments, and the preload of the active muscle(s) was varied. Joint torque was measured at the peak of short-latency stretch reflex action during contraction of the FPL alone, contraction of the EPL alone, and during cocontraction. Incremental joint stiffness was calculated as the change in torque divided by the change in angle. 3. Incremental stiffness increased in proportion to the preload torque during single muscle contractions of either the FPL (lengthening disturbances) or the EPL (shortening disturbances). Thus stiffness was not regulated to a constant value in the face of varying loads for either single muscle stretch or release. 4. Incremental stiffness varied across the range of cocontraction levels while the net torque was maintained at approximately 0. Thus net torque alone did not determine the stiffness during cocontraction. 5. The contributions of each muscle to the net intrinsic torque during cocontraction were estimated by scaling the individual muscles' responses so that their sum gave the best fit (in a least-squares sense) to the cocontraction torque before reflex action. The solution is unique because the individual torques have opposite signs, but the stiffnesses add. This gave estimates of the initial torques of both muscles during cocontraction. 6. The contributions of the two muscles during cocontraction were used to estimate the active joint stiffness that would be expected if the two muscles were activated independently to the same levels as in the cocontraction trials. The stiffness measured at the peak of stretch reflex action during cocontraction trials differed from the sum of the stiffnesses of the two muscles when they were contracting alone. At low cocontraction levels, the measured stiffness was less than expected on the basis of summation of the action of the two muscles, whereas at high cocontraction levels, the measured stiffness was greater than expected. This demonstrates that there is nonlinear stretch reflex interaction. That is, reflex action for a pair of antagonists is not simply the linear sum of the reflex actions of the two muscles acting independently.(ABSTRACT TRUNCATED AT 400 WORDS)

Accuracy of semi-analytical sensitivities and its improvement by the “natural method”

The semi-analytical method is a very convenient tool for the computation of sensitivities in shape design. It was, however, shown in the past that this method may have serious accuracy problems, e.g. in the shape design of beams. In this paper the natural method is employed to locate the decisive defect in the incremental stiffness and to develop an alternative procedure that provides nondefective incremental stiffnesses. This approach is then extended to the more common Cartesian element description. Examples demonstrate that the error of the sensitivities in critical cases can be almost totally removed, if we employ the nondefective incremental stiffness formulation in the sensitivity computation.

Deformability of intact rock

Abstract A constitutive model is presented which uses elasticity, plasticity and continuum damage mechanics to describe the incremental stiffness of rock under general loading paths. Rock strength and elastic deformability are described in terms of a transformed stress tensor, using a continuum damage variable to account for the effect of microcracks on the capacity of rock to support distortional loading. The growth of damage is modelled using an incremental evolution law, which describes the effects of confinement and distortional load intensity on the potential of rock to undergo cracking. The dilation ratio predicted by a plasticity-based material model is determined exclusively by the flow rule; such that an appropriate plastic potential can be obtained by requiring the flow rule to satisfy the measured dilation response. It is argued that functions which satisfy this requirement of geologic materials cannot also predict their strength correctly, which implies that the flow rule has to be non-associative. This approach is applied to describe the inelastic deformability of rock. The material properties required in the model are tensile strength, elastic stiffness, shear resistance and dilation ratio. The last three vary with deformation and confinement. The evaluation of these properties is illustrated using cylindrical triaxial test data.

An Inverse Problem Associated With Modification of Incomplete Dynamic Systems

Given an incomplete set of natural frequencies and modes of a linear structure from a vibration test, the problem of computing the necessary changes in the structure to reach a desired spectra is considered. A criterion of optimality which gives the best approximation from a certain subspace is formulated. Then formulae for evaluating the family of all mathematical solutions are given, resulting in a class of mathematical incremental mass and stiffness matrices. Finally, we describe the problem of the realization of the mathematical solutions by constructing a model of a given type.

Non-linear analysis of framed structures with axial constraints

In tracing the lateral load-displacement characteristics of tall framed structures loaded up to their ultimate load one has to consider the non-linear effects resulting from material properties and geometry. Applying the incremental load approach, the problem is locally linearized at the ‘cost’ of recalculating the stiffness for each load increment. As a result, accumulation of numerical errors can degrade the accuracy of results. Matrix condensation, transformation of coordinates, and constraints on the axial deformations are combined to reduce the number of unknowns. Moreover, by introducing axial constraints, the numerical problem of ill-conditioned matrices due to combining large axial stiffnesses with relatively small flexural and rotational stiffnesses can be avoided. The inelastic moment-curvature and axial force-deformation behavior of structural members are modeled by Ramberg-Osgood curves. Partial geometric non-linearity is included through P- Δ effect, which considers the influence of the deformed structure on the incremental stiffness. Previous work has shown the basic concepts of the axial constraints and is extended to the non-linear analysis. The theoretical development of the analysis procedure has been incorporated into a computer program. Several examples are given to demonstrate the influence of the constraint level on the accuracy and efficiency of the analysis.

Incremental finite element solution for nonlinear problems of space trusses

Based on an updated total Lagrangian formulation, an accurate incremental stiffness matrix and corresponding finite element equation for nonlinear problems of space trusses are derived in this paper. The modified arc-length iteration technique is presented to solve the ultimate strength and the load-deflection curves of space trusses. The stiffness matrix has a symmetric and concise form in which the large defletion effects on structures can be included accurately and the material nonlinearity can be considered easily. Numerical examples show that the incremental finite element method presented in this paper can be used to analyse the elastoplastic large deflection problems of space trusses satisfactorily and effectively.

A Numerical Analysis of In-Plane Elastic-Plastic Bending of Curved Tubes. 1st Report. A Case of Pure Bending.

In-plane bending of curved tubes is analyzed numerically in the elastic-plastic range on the assumption of pure bending, uniform deformation along its longitudinal axis and the Kirchhoff-Love hypothesis. The equilibrium equations of pure bending of tubes, derived by E.L. Axelrad, are transformed into incremental forms and solved by the updated Lagrange method. The material of tubes is assumed to obey a power-hardening law and the Prandtl-Reuss flow theory. The curved tube is discretized by a rigid body-spring model, which is composed of rigid plate elements connected with nonlinear springs that are distributed over the ends. The in-plane stiffness and flexural rigidity of the tube wall are obtained by numerically integrating generalized incremental stiffness over the wall thickness. Numerical analyses have been carried out for various combinations of initial in-plane curvature, diameter and thickness of the curved tube. The present analyses simulate well the relation between bending moment and change in curvature, and flattening of cross sections which are obtained by bending experiments in aluminum tubes by Murota and Endo.