Static Structural Response Research Articles

Optimal Linearization of Morison‐Type Wave Loading

The linearization of Morison-type wave loading is to replace \Iu\N|\N\Iu\N|\N by \Icu\N, where \Iu\N is a zero mean, stationary Gaussian random process and \Ic\N is a constant. The coefficient \Ic\N is traditionally determined by a use of the statistical least-square error linearization; that is minimizing \IE\N[(\Iu\N|\N\Iu\N|\N-\Icu\N)²]. In this study, the optimal linearization refers to a coefficient \Ic\N, chosen in such a way that the statistics of structural response calculated from the linearized force will be identical to that calculated from the originally nonlinear force. To accomplish this, the optimal linearization must not only consider the ocean wave properties, but incorporate the structural properties. While most offshore structural engineers believe that the least-square linearization may cause significantly nonconservative error only when the fundamental structural frequency is nearly three times the peak wave frequency, this study shows that worse errors may occur for other frequency ratios.

Damped composite structures

The use of damped composites in aircraft holds the promise of achieving vibrational-resistant structures at a reduction in structural weight. This accomplishment can only be reached by integrating damping methods into the design process. This paper reports the results of a parametric study that examines the effect of damping applications on the static and dynamic response of structures. In particular, it is found that in spite of the low modulus of commercially available damping materials and the dependence of properties on environmental conditions, damping designs have a broad range of potential application.

Correlation between Static and Dynamic Response of Model Masonry Structures

The paper presents correlations between response of the same structural system subjected to either dynamic shaking or static lateral forces. Two one-quarter scale test structures were constructed with identical designs and were tested in the laboratory using two different methods. The first structure was subjected to simulated earthquake motions on a shaking table while the second structure was forced to displace through the same history at static rates using computer controlled servohydraulic actuators. Characteristics of dynamic lateral force distributions are examined first and followed by a description of techniques used for the static loading experiment. Correlations are made between observed response of the statically and dynamically tested twin structures to suggest differences in strength, stiffness and energy dissipation that may arise with each test method.

Numerical modelling of the non-linear static response of clad cable net structures

An efficient numerical approach is presented for the analysis of cladding-network interaction in pretensioned cable roof structures under static loads. The approach is based on the sub-region mixed energy principle, which allows for the linear and non-linear aspects of the behaviour of the structure to be treated separately. Membrane action of the cladding is represented by a five-force model calculated by the finite element method. The governing equilibrium and compatibility equations are solved using the dynamic relaxation technique with kinetic damping. Computations are carried out for an experimental model of a hyperbolic paraboloid net with a discrete membrane surface and the numerical and experimental results are compared.

Some advances in reduction methods for nonlinear structural statics

A reduced basis technique is presented for predicting a static response of nonlinear structures. A new set of basis vectors and a new strategy for the solution of the reduced system of equations are proposed. Numerical results demonstrate the effectiveness of the proposed algorithms and suggest their potential for the analysis of large-scale problems.

The efficiency of numerical methods for the analysis of prestressed nets and pin-jointed frame structures

The non-linear static response of pretensioned cable net and pin-jointed frame structures is analysed using the minimum potential energy principle. The relative efficiencies of the stiffness matrix and dynamic relaxation methods in solving specific structural examples are examined. The performance of the dynamic relaxation algorithm with kinetic and viscous damping is discussed.

Further generalization of an equivalent plate representation for aircraft structural analysis

Recent developments from a continuing effort to provide an equivalent plate representation for aircraft structural analysis are described. Previous work provided an equivalent plate analysis formulation that is capable of modeling aircraft wing structures with a general planform such as cranked wing boxes. However, the modeling is restricted to representing wing boxes having symmetric cross sections. Further developments, which are described, allow modeling of wing cross sections having asymmetries that can arise from airfoil camber or from thicknesses being different in the upper and lower cover skins. An implementation of thermal loadings, which are described as temperature distributions over the planform of the cover skins, has been included. Spring supports have been added to provide for a more general set of boundary conditions. Numerical results are presented to assess the effect of wing camber on the static and dynamic response of an example wing structure under pressure and thermal loading. These results are compared with results from a finite element analysis program to indicate how well a cambered wing box can be represented with an equivalent plate formulation.

Structural design of cylindrical railguns

A systematic structural design approach to the development of cylindrical electromagnetic launchers has been formulated, using classical analytical methods in preliminary feasibility studies and finite-element methods to simulate static and dynamic structural response. Three-dimensional stress analysis results indicate that conventional assumptions associated with two-dimensional analysis for design are generally valid. Frequency-domain analysis of the subject structure and associated design loads can be used to estimate the significance of inertial effects under dynamic loading. Results of the study show that significant improvements in system performance are possible despite the limitations of nonconducting structural materials currently available for railgun fabrication.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Effect of parametric uncertainties on wind excited structural response

This paper addresses the influence of parametric uncertainties on the wind excited response of structures. Based on the available experimental data from both laboratory and field studies, the variability in the parameters related to the wind environment and meterological data, kinematics of wind flow field, wind-structure interaction and structural properties is assessed. The random variability in the parameter space is propagated to ascertain its influence on the structural response statistics utilizing a Monte Carlo simulation technique. By means of an example, the influence of parametric uncertainties on the dynamic response of a tall reinforced concrete chimney is presented. The analysis of simulated data suggests a need for further improvement in the modeling of wind-structure interaction, prediction of natural frequencies and damping, and a reduction in the variability of extreme wind estimates.

Nonlinear structural analysis via reduced basis technique

A reduced basis technique is presented for predicting a static response of nonlinear structures. The idea of taking a set of correction vectors of the classical Newton-Raphson iterative scheme coupled with the lowest eigenvectors of the updated tangent stiffness matrices, to be the basis vectors, is proposed and proved to be acceptable. The effectiveness of the proposed approach is demonstrated by means of numerical examples.

Fixed‐point iteration to nonlinear finite element analysis. Part I: Mathematical theory and background

AbstractThis paper sets the stage for the implementation of the Fixed‐Point Iteration (FPI) to nonlinear finite element analysis as an alternative to the existing Newton‐Raphson Method (NRM) or its derivatives. The superiority of the former method over the latter one is such that it enables one to obtain nonlinear structural static or dynamic responses without inverting the structural stiffness matrix.In the first part of the paper, a new convergence correction/acceleration factor has been developed for the FPI when applied to a single nonlinear algebraic equation. This new factor causes the iteration function of the equation under consideration to rotate about an axis that passes through one of its fixed points or roots. Using this observation, the slope of the iteration function can be adjusted in the neighbourhood of a specific root to ensure the convergence of the FPI. It is found that the optimum choice of the new factor corresponds to a zero slope, evaluated at the root, of the iteration function. The rate of convergence and the error estimate of this form of the FPI is developed and compared with the NRM. The equilibrium positions of a nonlinear loaded softening spring have been obtained by both methods as an illustrative numerical example to measure the effect of the new factor on the convergence rate.The second part of the paper extends the above concept to find the solution of a linear system of algebraic equations using the FPI. This leads to a better diagonal approximate inverse for the Jacobi iteration, or method of simultaneous displacements. If the elements of the solution vector of a specific system are all equal, the new Jacobi iteration becomes an exact method and the solution is obtained in one iteration. The concept is also extended to the Gauss‐Seidel iteration, or method of successive displacements. Systems involving symmetric as well as nonsymmetric coefficient matrices have been used as numerical examples and are presented. For future implementation to nonlinear finite element analysis, the active column or the skyline (or the non‐zero profile) FPI algorithms are developed for programming considerations.

Fixed‐point iteration to nonlinear finite element analysis. Part II: Formulation and implementation

AbstractThe finite element formulation and implementation of the Fixed‐Point Iteration (FPI) to linear/nonlinear structural static or dynamic analysis are developed. The direct and tangent formulations are presented and compared with the Newton–Raphson method (NRM). ‘Modified’ FPI algorithms have also been derived. A graphical interpretation of the method is introduced and suggested to call the FPI ‘the Saw method’. Mixing both the FPI and NRM is shown to be possible and may be useful in some applications. The overall strategies, iterative algorithms, and the appropriate norm convergence criteria necessary to implant the FPI into existing finite element programs are also included in the development.The superiority of the FPI over the NRM as seen from the development and the formulation lies in three major factors. First, the existing assembly process of element matrices is eliminated completely from the nonlinear finite element analysis. Secondly, the Gauss elimination or Crout's method is also eliminated. In the finite element terminology, this means that nonlinear structural static or dynamic responses can he obtained without recourse to the inverse of the structural stiffness matrix. Thirdly, the FPI can also be applied equally to linear structural analysis. Hence, the assembly process and the programming and storage associated with it can be removed from the existing finite element programs.While the FPI can solve problems that the NRM can, it will also be able to handle some engineering problems where the latter cannot. Buckling problems and problems where the force–displacement curve changes curvature are examples where the FPI is expected to be efficient.

Mechanical Nonlinear Shear Wall Model

The nonlinear performance of a shear wall building can often only be described by full-scale testing or nonlinear analysis. Full-scale tests are time consuming and expensive; nonlinear analysis often is mathematically complex so that limiting assumptions are necessary to simplify the problem. This study describes a mechanical model which can simulate the nonlinear static or dynamic response of a shear wall or diaphragm. An assembly of these models can then simulate entire buildings and is suitable for the experimental investigation of the linear and nonlinear, static or dynamic response of shear wall structures.

Extension of constrained incremental newton—raphson scheme to generalized loading fields

This paper develops numerical strategies which enable the constrained incremental Newton-Raphson scheme to handle the static response of structure to loading fields with completely generalized histories. This is made possible through the use of specially warped hyperelliptic constraint surfaces which control successive or clustered load steps in the vicinity of loading events with specific timing schedules. Such an approach enables improved convergence and stability characteristics. Due to the generality of the methodology, pre- and postbuckling behaviour caused by both kinematic and material nonlinearity can be handled. To demonstrate the scheme, the results of several bench-mark problems are also presented. These include situations involving nonlinear kinematics as well as highly history-dependent elastic-plastic and thermoelastic-plastic material behaviour.

Structural design sensitivity analysis with generalized global stiffness and mass matrices

First- and second-order derivatives of measures of structural response with respect to design variables are calculated using the generalized global stiffness and mass matrix formulation of structural equations. The adjoint variable method is extended for first-order design sensitivity calculation and a mixed direct differentiationadjoint variable method is employed to calculate second derivatives of static structural response measures. The variational or virtual work form of the structural equations is shown to be well suited for design sensitivity analysis with generalized global stiffness and mass matrices, avoiding the requirement for explicit reduction of the matrices. The method is illustrated using a simple structure with a multipoint constraint.

Note on the Design Differentiability of the Static Response of Elastic Structures

Abstract This paper provides a new proof of design sensitivity of the static response of some typical structures. These structures (beams, plates, and plane elastic solids) have been described previously. A proof of design sensitivity of the inverse state operator was provided there, and design sensitivity of static response was derived. The proof presented here is simpler and self-contained.

Random Response to Flow-Induced Forces

The response statistics of a linear structure subjected to excitation induced by a random flow with a non zero-mean velocity is examined. The technique of statistical linearization is applied on the nonlinear equation of motion of the structure. Approximate solutions for the structural response statistics are obtained. The Pierson-Moskowite is used to described the energy versus frequency distribution of the randomly flucuating component of the fluid motions. Results of several studies regarding parameters such as the fluid density and the structural stiffness are presented. The accuracy and efficiency of the analytical approach is assessed by comparison with data obtained by Monte Carlo simulations of the random structural response.

Cranes in storm winds

In the first part of this paper the theoretical effect of wind gusts on crane stability is examined. It is assumed that the crane is a system with a single degree of freedom and that the wind applies a constant (overturning) moment supplemented by a short duration (gust) moment. It concludes that the approach to crane design adopted in the majority of Codes and Standards which takes a constant wind velocity and a static structural response is acceptable for most cranes. The second part of the paper develops a method, based upon that used in CP3: Ch. V: Pt. 2, for selecting appropriate out-of-service design wind speeds for ‘nomad’ cranes, such as builders' tower cranes, which are used in a variety of locations during their useful life.

Nonlinear analysis via global–local mixed finite element approach

AbstractA computational algorithm, based on the combined use of mixed finite elements and classical Rayleigh–Ritz approximation, is presented for predicting the nonlinear static response of structures; The fundamental unknowns consist of nodal displacements and forces (or stresses) and the governing nonlinear finite element equations consist of both the constitutive relations and equilibrium equations of the discretized structure. The vector of nodal displacements and forces (or stresses) is expressed as a linear combination of a small number of global approximation functions (or basis vectors), and a Rayleigh–Ritz technique is used to approximate the finite element equations by a reduced system of nonlinear equations. The global approximation functions (or basis vectors) are chosen to be those commonly used in static perturbation technique; namely a nonlinear solution and a number of its path derivatives. These global functions are generated by using the finite element equations of the discretized structure.The potential of the global–local mixed approach and its advantages over global–local displacement finite element methods are discussed. Also, the high accuracy and effectiveness of the proposed approach are demonstrated by means of numerical examples.

Reduced Basis Technique for Nonlinear Analysis of Structures

A reduced basis technique and a computational' algorithm are presented for predicting the nonlinear static response of structures. A total Lagrangian formulation is used and the structure is discretized by using displacement finite element models. The nodal displacement vector is expressed as a linear combination of a small number of basis vectors and a Rayleigh-Ritz technique is used to approximate the finite element equations by a reduced system of nonlinear equations. The Rayleigh-Ritz approximation functions (basis vectors) are chosen to be those commonly used in the static perturbation technique namely, a nonlinear solution and a number of its path derivatives. A procedure is outlined for automatically selecting the load (or displacement) step size and monitoring the solution accuracy. The high accuracy and effectiveness of the proposed approach is demonstrated by means of numerical examples.