Stiffness Formulation Research Articles

Structural Damage Detection using Best Approximated Dead Load Redistribution

Presented in this article is a novel damage detection method based on best approximation of dead load stress redistribution due to damage. It is a simple, economical, fast, and passive method that requires no artificial means of loading. The measurement is made of strain readings due to dead load at different stages of the structure during its life cycle. The approach is by finding a ‘null plane’, which is a unique identity of each structural member, such that the measurement is closest to its orthogonal projection on this null plane. The detection procedure resembles the process of finding the shortest distance from a point to a plane in elementary geometry. Both theoretical proof with a flexibility formulation and practical procedure with a stiffness formulation are given. The feasibility of the proposed method, together with the false positive test, are studied in a numerical example of a synthetic frame structure. Measurement noise influence is also conducted using perturbed strain measurement.

A new composite structure impact performance assessment program

While previous researchers have conducted their study on the relative impact performance of composite structures from a force or an energy standpoint only, this proposed Composite Structure Impact Performance Assessment Program (CSIPAP) suggests a multi-parameter methodology to gain further insight in the impact behavior of composite structures. These are peak and critical force; critical and dissipated energy; contact duration and coefficient of restitution (COR), which is direct indication of effective structural stiffness; and residual stiffness (normalized contact duration) which yields a plot that bears a striking resemblance with the normalized Compression After Impact (CAI) strength. Using a determinate impactor/target system as baseline configuration, the program is applied toward the understanding of the role played in an impact event by fundamental impactor and target parameters. The equations previously derived for the prediction of the force–energy and residual stiffness curves are shown to apply to the configurations tested, thus confirming their general validity. A modification to the existing effective structural stiffness formulation, which does not account for impactor characteristics, is proposed, and it comprises the impactor material, size and mass characteristics.

TECHNICAL NOTE A FRAMEWORK FOR MULTI-SITE DISTRIBUTED SIMULATION AND APPLICATION TO COMPLEX STRUCTURAL SYSTEMS

In this technical note, the development of a framework for multi-site distributed simulations is presented. The algorithm is suitable for any combination of physical (laboratory) and analytical (computer) distributed simulations of structures, their foundations and the underlying sub-strata subjected to static and dynamic loading. Two examples of multi-site testing and multi-platform simulation are given. The main contribution in this note is the separation between time-step integration and stiffness formulation, which enables the use of static analysis and testing as modules of the main control module referred to as the simulation coordinator. The approach proposed is intuitive, simple and efficient. It is therefore recommended for use in distributed analysis using different programs, distributed testing facilities (e.g. the NEES equipment sites) or a combination of analysis and testing.

Enhanced stiffness modeling, identification and characterization for robot manipulators

This paper presents the enhanced stiffness modeling and analysis of robot manipulators, and a methodology for their stiffness identification and characterization. Assuming that the manipulator links are infinitely stiff, the enhanced stiffness model contains: 1) the passive and active stiffness of the joints and 2) the active stiffness created by the change in the manipulator configuration, and by external force vector acting upon the manipulator end point. The stiffness formulation not accounting for the latter is known as conventional stiffness formulation, which is obviously not complete and is valid only when: 1) the manipulator is in an unloaded quasistatic configuration and 2) the manipulator Jacobian matrix is constant throughout the workspace. The experimental system considered in this study is a Motoman SK 120 robot manipulator with a closed-chain mechanism. While the deflection of the manipulator end point under a range of external forces is provided by a high precision laser measurement system, a wrist force/torque sensor measures the external forces. Based on the experimental data and the enhanced stiffness model, the joint stiffness values are first identified. These stiffness values are then used to prove that conventional stiffness modeling is incomplete. Finally, they are employed to characterize stiffness properties of the robot manipulator. It has been found that although the component of the stiffness matrix differentiating the enhanced stiffness model from the conventional one is not always positive definite, the resulting stiffness matrix can still be positive definite. This follows that stability of the stiffness matrix is not influenced by this stiffness component. This study contributes to the previously reported work from the point of view of using the enhanced stiffness model for stiffness identification, verification and characterization, and of new experimental results proving that the conventional stiffness matrix is not complete and is valid under certain assumptions.

The dynamic element method for analysis of frame and cable type structures

With the dynamic element method (DEM), results more accurate than with the conventional finite element method (FEM) are obtained with the same number of degrees of freedom. This is due to the introduction of shape functions of polynomial type, introducing frequency dependence into the mass matrix expressions. It is demonstrated how this affects free vibration analysis, including the solution of nonlinear eigenvalue problems. Various numerical techniques for solving these polynomial problems are discussed. The polynomial matrix formulations for stiffness and mass matrices are given for a beam, a bar and a cable element. Numerical examples demonstrate how the DEM can be implemented for modelling of frame type structures and its efficiency is compared to that of the conventional FEM.

On the Dynamics of Elastohydrodynamic Mixed Lubricated Ball Bearings. Part I: Formulation of Stiffness and Damping Coefficients

The problems of stiffness and damping characteristics of isothermal elastohydrodynamic mixed lubricated point contact are evaluated numerically considering surface roughness effect including asperity contact load. A set of equations under steady-state and dynamic conditions is derived from the classical Reynolds equation, using linear perturbation method. The elasticity equation and steady-state Reynolds equation are solved simultaneously for finding the steady-state pressure distribution, using finite difference method. Then, the set of perturbed equations is solved for the dynamic pressure distribution in the contact. A Gaussian surface roughness is adopted to model both surface roughness and mixed lubrication. Total load capacity of the contact is calculated from the lubricant film pressure and contact pressure distribution. Results are compared with those of smooth isothermal cases. The stiffness and damping coefficients of the contact are determined using the dynamic pressures. The asperity contact stiffness is calculated separately. Effect of various design parameters on stiffness and damping characteristics of a ball bearing is investigated.

A framework for multi-site distributed simulation and application to complex structural systems

In this technical note, the development of a framework for multi-site distributed simulations is presented. The algorithm is suitable for any combination of physical (laboratory) and analytical (computer) distributed simulations of structures, their foundations and the underlying sub-strata subjected to static and dynamic loading. Two examples of multi-site testing and multi-platform simulation are given. The main contribution in this note is the separation between time-step integration and stiffness formulation, which enables the use of static analysis and testing as modules of the main control module referred to as the simulation coordinator. The approach proposed is intuitive, simple and efficient. It is therefore recommended for use in distributed analysis using different programs, distributed testing facilities (e.g. the NEES equipment sites) or a combination of analysis and testing.

Expression of Load Transfer Paths in Structures

A concept of a parameter U* has been introduced by the authors to express load transfer paths in a structure. In this paper, matrix formulation of internal stiffness shows that the value of U* expresses a degree of connection between a loading point and an internal arbitrary point. Stiffness fields, stiffness lines, and stiffness decay vectors are defined using newly introduced U* potential lines. A concept of a load path can be expressed as a stiffness line that has a minimum stiffness decay vector. A simple model structure is calculated using FEM for an application of U* analysis. The distribution of U* values shows that a diagonal member between a loading point and a support point Plays an important rold for the load transfer.

Load Path Analysis of Structures and Generalized Support Conditions

A concept of a parameter U* has been introduced by the authors to express load transfer paths in a structure. In this paper, a new extended expression of U* is introduced, and it is shcown that the new parameter can be applied to structures with complicated multiple support conditions. The present expression that is derived by matrix formulation of internal stiffness and initial deformation includes the definition of U* in the previous report. Examples of U* calculation for structures with multiple support conditions are demonstrated. Calculation results for a simple structure and an actual truck cab body show the effectiveness of the present method of U* calculation.

A solution including skin effect for stiffness and stress field of sandwich honeycomb core

Honeycomb sandwich panels have been increasingly used in every possible field, and their efficient load carrying capacity attributes have attracted considerable attention. All previous studies have been focused mainly on stiffness, neglecting for the most part skin effects. This paper represents an important further contribution by developing an analytical model that permits the computation of stiffnesses as well as interfacial stresses considering the skin-effect for hexagonal honeycomb sandwich, subjected to in-plane and out-of-plane forces. An explicit analytical model is derived based on equilibrium equations, where boundary conditions imposed by the skin effect are appropriately considered. The accuracy of the solution is verified through close correlations with existing stiffness formulations and finite element results. The skin effect on both stiffness and interfacial stress distribution is analytically defined. The present model is then used to carry out a parametric study on interfacial stresses, and to detect the critical sections in the structure where further consideration should be given for design purposes. The method provided in this study can be used for accurate analysis and design of sandwich structures.

Dynamic stiffness formulation and free vibration analysis of a three-layered sandwich beam

A dynamic stiffness theory of a three-layered sandwich beam is developed and subsequently used to investigate its free vibration characteristics. This is based on an imposed displacement field so that the top and bottom layers behave like Rayleigh beams, whilst the central layer behaves like a Timoshenko beam. Using Hamilton’s principle the governing differential equations of motion of the sandwich beam are derived for the general case when the properties of each layer are dissimilar. For harmonic oscillation the solutions of these equations are found in exact analytical form, taking full advantage of the application of symbolic computation, which has also been used to obtain the amplitudes of axial force, shear force and bending moment in explicit analytical forms. The boundary conditions for responses and loads at both ends of the freely vibrating sandwich beam are then imposed to formulate the dynamic stiffness matrix, which relates harmonically varying loads to harmonically varying responses at the ends. Using the Wittrick–Williams algorithm the natural frequencies and mode shapes of some representative problems are obtained and discussed. The important degenerate case of a symmetric sandwich beam is also investigated.

Elastoplastic Modeling of Metal Matrix Composites Containing Randomly Located and Oriented Spheroidal Particles

Micromechanics-based effective elastic and plastic formulations of metal matrix composites (MMCs) containing randomly located and randomly oriented particles are developed. The averaging process over all orientations upon three elastic governing equations for aligned particle-reinforced MMCs is performed to obtain the explicit formulation of effective elastic stiffness of MMCs with randomly oriented particles. The effects of volume fraction of particles and particle shape on the overall elastic constants are studied. Comparisons with the Hashin-Shtrikman bounds and Ponte Castaneda-Willis bounds show that the present effective elastic formulation does not violate the variational bounds. Good agreement with experimental elastic stiffness data is also illustrated. Furthermore, the orientational averaging procedure is employed to derive the overall elastoplastic yield function for the MMCs. Elastoplastic constitutive relations for the composites are constructed on the basis of the derived composite yield function. The stress-strain responses of MMCs under the axisymmetric loading are also investigated in detail. Finally, elastoplastic comparisons with the experimental data for SiCp/Al composites are performed to illustrate the capability of the proposed formulation.

A POSTERIORI ERROR ESTIMATION AND H-ADAPTIVE REFINEMENT TECHNIQUES FOR TRANSIENT DYNAMIC ANALYSIS OF STIFFENED PLATE/SHELL PANELS

This paper presents a posteriori error estimation and h-adaptive refinement techniques for transient dynamic analysis of stiffened plates/shells using the finite element method (FEM). We furnish the formulation of stiffness and mass matrices for finite element (FE) models, QL9S2 and QUAD4S2 for dynamic analysis of plates/shells with arbitrarily-located concentric/eccentric stiffeners. Procedures for computing a posteriori errors for spatial and temporal errors have been presented for transient dynamic problems. An h-adaptive refinement strategy for stiffened plate/shell panels by employing QL9S2 and QUAD4S2 FE models has been discussed. An adaptive time stepping scheme, which is to be used with the time errors for quality control of the time steps, has also been presented. Numerical studies have been conducted to evaluate the efficacy of the error estimators and the adaptive mesh refinement and time stepping algorithm. The spatial error estimator for transient dynamic analysis is found to exhibit monotonic convergence at all time steps. The temporal error estimator for transient dynamic analysis in association with the adaptive time stepping is able to compute more accurate and reliable time steps to keep the time errors within the specified tolerance limits.

Elastic In-Plane Stiffness for a Circular Cut Reduced Beam Section (RBS)

One newer moment connection that has been introduced and validated (prequalified) for use in high seismic areas is the so-called "dogbone"ť, or reduced beam section (RBS). The purpose of this paper is to briefly review the basic in-plane stiffness formulation for a prismatic beam, develop more rigorously the full non-prismatic, first-order elastic stiffness changes caused by the circular RBS, provide its simplified representation, and finally conclude with the general RBS stiffness effect suggestions for design. Other RBS seismic design and performance issues of this efficient and popular detail will not be addressed, as they are already well covered in other available sources.

Extended Hyperbolic Model for Sand-to-Concrete Interfaces

A relatively simple, four-parameter extended hyperbolic model for interfaces was developed for use in soil-structure interaction analyses. The model accommodates arbitrary stress path directions and includes three important elements: (1) development of a yield surface during interface shear; (2) a formulation for yield-inducing shear stiffness that is applicable to any stress path orientation; and (3) a formulation for unloading-reloading shear stiffness. The model was evaluated against the results of shear tests performed at the interface between three different types of sand and a concrete surface under a variety of stress paths. Comparisons between measured and calculated interface response indicate that the model provides accurate estimates of the response of sand-to-concrete interfaces.

Dynamic Stiffness Formulation and Its Application for a Combined Beam and a Two Degree-of-Freedom System

This paper is concerned with the dynamic stiffness formulation and its application for a Bernoulli-Euler beam carrying a two degree-of-freedom spring-mass system. The effect of a two degree-of-freedom system kinematically connected to the beam is represented exactly by replacing it with equivalent stiffness coefficients, which are added to the appropriate stiffness coefficients of the bare beam. Numerical examples whose results are obtained by applying the Wittrick-Williams algorithm to the total dynamic stiffness matrix are given and compared with published results. Applications of the theory include the free vibration analysis of frameworks carrying two degree-of-freedom spring-mass systems.

Nonlinear analysis of cable-stayed spatial latticed structures

The combination of spatial latticed structures (hereafter SLS) and flexible cables, the cable-stayed spatial latticed structures (hereafter CSLS) can cross longer span. According to variation principle, a novel geometric nonlinear formulation for 3-D bar elements considering large displacement and infinitesimal rotation increments with second-order precision is developed. The cable nonlinearity is investigated and it is taken that the secant modulus method can be considered as an exact method for a cable member. The tower column with which the cables link is regarded as a special kind of beam element, and, a new simplified stiffness formulation is presented. The computational strategies for the nonlinear dynamic response of structures are given, and the ultimate load carrying capacities and seismic responses are analyzed numerically. It is noted that, compared with corresponding spatial latticed shells, the cable-stayed spatial latticed shells have more strength and more stiffness, and that the verical seismic responses of both CSLS and CLS are remarkably greater than the horizontal ones. In addition, the computation shows that the stiffness of tower column influences the performance of CSLS to a certain extent and the improvement of structural strength and stiffness of CSLS is relevant not only to cables but also to tower columns.

A co-rotational 8-node assumed strain shell element for postbuckling analysis of laminated composite plates and shells

The formulation of a nonlinear composite shell element is presented for the solution of stability problems of composite plates and shells. The formulation of the geometrical stiffness presented here is exactly defined on the midsurface and is efficient for analyzing stability problems of thin and thick laminated plates and shells by incorporating bending moment and transverse shear resultant forces. The composite element is free of both membrane and shear locking behaviour by using the assumed natural strain method such that the element performs very well as thin shells. The transverse shear stiffness is defined by an equilibrium approach instead of using the shear correction factor. The proposed formulation is computationally efficient and the test results showed good agreement. In addition the effect of the viscoelastic material is investigated on the postbuckling behaviour of laminated composite shells.

Rigid body mode and spurious mode in the dual boundary element formulation for the Laplace problems

In this paper, the general formulation for the static stiffness is analytically derived using the dual integral formulations. It is found that the same stiffness matrix is derived by using the integral equation no matter what the rigid body mode and the complementary solutions are superimposed in the fundamental solution. For the Laplace problem with a circular domain, the circulant was employed to derive the stiffness analytically in the discrete system. In deriving the static stiffness, the degenerate scale problem occurs when the singular influence matrix can not be inverted. The Fredholm alternative theorem and the SVD updating technique are employed to study the degenerate scale problem mathematically and numerically. The direct treatment in the matrix level is achieved to deal with the degenerate scale problems instead of using a modified fundamental solution. The addition of a rigid body term in the fundamental solution is found to shift the zero singular value for the singular matrix without disturbing the stiffness.

A co-rotational 8-node assumed strain element for large displacement elasto-plastic analysis of plates and shells

The formulation of a non-linear shear deformable shell element is presented for the solution of stability problems of stiffened plates and shells. The formulation of the geometrical stiffness presented here is exactly defined on the midsurface and is efficient for analyzing stability problems of thick plates and shells by incorporating bending moment and transverse shear resultant force. As a result of the explicit integration of the tangent stiffness matrix, this formulation is computationally very efficient in incremental nonlinear analysis. The element is free of both membrane and shear locking behaviour by using the assumed strain method such that the element performs very well in the thin shells. By using six degrees of freedom per node, the present element can model stiffened plate and shell structures. The formulation includes large displacement effects and elasto-plastic material behaviour. The material is assumed to be isotropic and elasto-plastic obeying Von Mises's yield condition and its associated flow rules. The results showed good agreement with references and computational efficiency.