Flexibility Matrices Research Articles

Dynamic and Seismic Analysis of Foundations based on Free Field B-Spline Characteristic Response Histories

In this research, a direct time domain formulation for analysis of unbounded media and foundations is developed that treats dynamic excitations and ground motion in a uniform manner. The boundary element method with higher order B-Spline fundamental solutions is employed to compute the characteristic responses of the surface of the elastodynamic domain. Subsequently, time histories of the system response to general excitations are computed by a superposition scheme that accommodates in a uniform manner arbitrary time histories of external loads and/or ground motion. The characteristic responses are computed in the form of time dependent flexibility matrices of the media that are sparse due to the finite duration of the B-Spline excitation signal and characteristics of the wave propagation. The duration of the B-Spline impulse response is limited to a few time steps. Significant savings in computing time and storage requirements are achieved. The proposed approach also greatly reduces the size of the problems under consideration and still fully considers the effects of the free field. The significance of nonrelaxed boundary conditions and correct representation of the free field is established. The method is demonstrated and validated through applications pertaining to the analysis of foundations and inclusions subjected to transient loads and seismic excitations.

A METHOD FOR RECOGNITION AND LOCATION OF STRUCTURAL DAMAGE1*

ABSTRACT A systematic introduction is made of two types of method for recognition and location of structural damage, one of which is based on strain-type parameters (such as strain, modal strain, curvature modal shape, and so on), and the other is based on displacement-type parameters (such as displacement, modal displacement, flexibility matrices, and so on). The following phenomenon is discovered: that the methods based on displacement-type parameters harbor the possibility of giving a wrong damage location. In consideration of the local feature of damage, some new damage location formula based on local parameter changes is suggested. Damage propagation is the result of interaction between loads and structural details rather than an inner property of the structure, so it is necessary to consider the common effect of loads and structural details when locating damage in engineering structures. *Communicated by I. Elishakoff.

The construction of free–free flexibility matrices for multilevel structural analysis

This article considers a generalization of the classical structural flexibility matrix. It expands on previous papers by taking a deeper look at computational considerations at the substructure level. Direct or indirect computation of flexibilities as “influence coefficients” has traditionally required pre-removal of rigid body modes by imposing appropriate support conditions, mimicking experimental arrangements. With the method presented here the flexibility of an individual element or substructure is directly obtained as a particular generalized inverse of the free–free stiffness matrix. This generalized inverse preserves the stiffness spectrum. The definition is element independent and only involves access to the stiffness generated by a standard finite element program and the separate construction of an orthonormal rigid-body mode basis. The free–free flexibility has proven useful in special application areas of finite element structural analysis, notably massively parallel processing, model reduction and damage localization. It can be computed by solving sets of linear equations and does not require processing an eigenproblem or performing a singular value decomposition. If substructures contain thousands of d.o.f., exploitation of the stiffness sparseness is important. For that case this paper presents a computation procedure based on an exact penalty method, and a projected rank-regularized inverse stiffness with diagonal entries inserted by the sparse factorization process. These entries can be physically interpreted as penalty springs. This procedure takes advantage of the stiffness sparseness while forming the full free–free flexibility, or a boundary subset, and is backed by an in-depth null space analysis for robustness.

Damage assessment in reinforced concrete beams using eigenfrequencies and mode shape derivatives

The use of changes in dynamic system characteristics to detect damage has received considerable attention during the last years. This paper presents experimental results obtained within the framework of the development of a health monitoring system for civil engineering structures, based on the changes of dynamic characteristics. As a part of this research, reinforced concrete beams of 6 meters length are subjected to progressing cracking introduced in different steps. The damaged sections are located in symmetrical or asymmetrical positions according to the beam tested. The damage assessment consists in relating the changes observed in the dynamic characteristics and the level of the crack damage introduced in the beams. It appears from this analysis that eigenfrequencies are affected by accumulation of cracks in the beams and that their evolutions are not influenced by the crack damage locations; they decrease with the crack damage accumulation. The MAC factors are less sensitive to crack damage compared to eigenfrequencies, but give an indication of the symmetrical or asymmetrical nature of the induced crack damage. Next to this, the COMAC factors, the strain energy evolution and the changes in flexibility matrices are also examined as to their capability for detection and location of damage in the RC beams, the strain energy method appears to be more precise than the others.

A UNIFIED METHOD FOR CONSTRUCTING THE DYNAMIC BOUNDARY STIFFNESS AND BOUNDARY FLEXIBILITY FOR ROD, BEAM AND CIRCULAR MEMBRANE STRUCTURES

In this paper, the dynamic boundary stiffness and boundary flexibility for rod, beam and circular membrane structures are analytically derived using the dual integral formulation. Two approaches, the real- and the imaginary-part kernels are employed to determine the dynamic boundary stiffness and boundary flexibility. The continuous system for a circular membrane can be transformed into a discrete system with a circulant matrix. Based on the properties of circulant, the analytical solution for dynamic boundary stiffness and dynamic boundary flexibility in the discrete system can be derived. The exact formulae for the dynamic boundary stiffness and boundary flexibility matrices of a rod, a beam and a circular membrane are obtained. Also, the calculation for the static flexibility using the pseudo-inverse technique is discussed.

Bandwidth optimization for rectangular matrices

In this paper a comparative study of different algorithms is provided for nodal and element ordering of finite element models. The matrix force method of structures is presented for in-plane force systems of three-node triangular and four-node rectangular plane stress elements. Ordering algorithms for rectangular matrices are incorporated into the turn-back approach and the bandwidth and sparsity of self stress and flexibility matrices are compared through many examples. This comparison is made for plane stress finite element models having various configurations.

Cycle bases of graphs for sparse flexibility matrices

An efficient algorithm is presented for the formation of cycle bases of graphs corresponding to sparse cycle-member incidence matrices, leading to the formation of highly sparse flexibility matrices. The algorithm presented employs a new expansion process and uses an efficient graph-theoretical method for controlling the independence of the selected cycles.

Structural Damage Detection Using Localized Flexibilities

The present paper offers structural damage detection methods based on the relative changes in localized flexibility properties. The localized flexibility matrices are obtained either by applying a decomposition procedure to an experimentally determined global flexibility matrix or by processing the output signals of a vibration test in a substructure-by-substructure manner. Three methods are evaluated in the present paper: a free-free substructural flexibility method, a deformation-based flexibility method, and a strain-basis flexibility method. These methods are applied to a model ten-story building problem, an experimental bridge damage data, and a hybrid experimental-simulated engine structure. The present methods are found to correctly identify damage locations.

Comparative study of damage identification algorithms applied to a bridge: I. Experiment

Over the past 30 years detecting damage in a structure from changes in global dynamic parameters has received considerable attention from the civil, aerospace and mechanical engineering communities. The basis for this approach to damage detection is that changes in the structure's physical properties (i.e., boundary conditions, stiffness, mass and/or damping) will, in turn, alter the dynamic characteristics (i.e., resonant frequencies, modal damping and mode shapes) of the structure. Changes in properties such as the flexibility or stiffness matrices derived from measured modal properties and changes in mode shape curvature have shown promise for locating structural damage. However, to date there has not been a study reported in the technical literature that directly compares these various methods. The experimental results reported in this paper and the results of a numerical study reported in an accompanying paper attempt to fill this void in the study of damage detection methods. Five methods for damage assessment that have been reported in the technical literature are summarized and compared using experimental modal data from an undamaged and damaged bridge. For the most severe damage case investigated, all methods can accurately locate the damage. The methods show varying levels of success when applied to less severe damage cases. This paper concludes by summarizing some areas of the damage identification process that require further study.

Comparing Model Update Error Residuals and Effects on Model Predictive Accuracy

The effect of error residual choice on the predictive capability of an updated structural model is examined. First, analytical expressions are developed that relate errors in dynamically measured static flexibility matrices, stiffness matrices, and modal matrices. The analysis shows that flexibility, stiffness, and modal error residuals correspond to unique weightings on the error in the measured modal data. Next, the flexibility weighting indicator and the stiffness weighting indicator are defined to provide a simple means of ranking individual modes in terms of their potential contribution to errors in a measured static flexibility or stiffness matrix. Finally, expressions are developed for bounding model prediction errors when static flexibility, stiffness, or modal error residuals are used. The results show that the amount of uncertainty in predicted static displacements, static loads, and mode shapes and frequencies is directly related to the choice of error residual and is different in each case. These results are also illustrated using numerical simulations for a modified version of Kabe's problem, which includes model form errors.

Experimental Determination of Local Structural Stiffness by Disassembly of Measured Flexibility Matrices

A new method is presented for identifying the local stiffness of a structure from vibration test data. The method is based on a projection of the experimentally measured flexibility matrix onto the strain energy distribution in local elements or regional superelements. Using both a presumed connectivity and a presumed strain energy distribution pattern, the method forms a well-determined linear least squares problem for elemental stiffness matrix eigenvalues. These eigenvalues are directly proportional to the stiffnesses of individual elements or superelements, including the cross-sectional bending stiffnesses of beams, plates, and shells, for example. An important part of the methodology is the formulation of nodal degrees of freedom as functions of the measured sensor degrees of freedom to account for the location offsets which are present in physical sensor measurements. Numerical and experimental results are presented which show the application of the approach to example problems.

The construction of free–free flexibility matrices as generalized stiffness inverses

We present generalizations of the classical structural flexibility matrix. Direct or indirect computation of flexibilities as ‘influence coefficients’ has traditionally required pre-removal of rigid body modes by imposing appropriate support conditions. Here the flexibility of an individual element or substructure is directly obtained as a particular generalized inverse of the free–free stiffness matrix. This entity is called a free–free flexibility matrix. It preserves exactly the rigid body modes. The definition is element independent. It only involves access to the stiffness generated by a standard finite element program as well as a separate geometric construction of the rigid body modes. With this information, the computation of the free–free flexibility can be done by solving linear equations and does not require the solution of an eigenvalue problem or performing a singular value decomposition. Flexibility expressions for symmetric and unsymmetric free–free stiffnesses are studied. For the unsymmetric case two flexibilities, one preserving the Penrose conditions and the other the spectral properties, are examined. The two versions coalesce for symmetric matrices.

Correlation studies on an RC frame shaking‐table specimen

This paper presents the correlation of the results of a new model for the dynamic analysis of reinforced concrete (RC) frames with the experimental time history of a two storey RC frame shaking-table specimen. The frame member model consists of separate subelements that describe the deformations due to flexure, shear and bond slip in RC structural elements. The subelements are combined by superposition of flexibility matrices to form the frame element. A non-linear solution method which accounts for the unbalance of internal forces between different subelements during a given load increment is used with the model. The ability of the proposed model to describe the dynamic response of frame structures under earthquake excitations is evaluated by comparing the analytical results with experimental evidence from a two-storey, one bay reinforced concrete frame tested on the shaking-table. The model parameters for the shaking-table specimen are derived from available experimental evidence and first principles of reinforced concrete. The effect of reinforcing bar slip on the local and global dynamic response of the test structure is assessed. © 1997 John Wiley & Sons, Ltd.

A direct flexibility method

We present a Direct Flexibility Method (DFM) for the solution of finite element equations. This method is based on a decomposition of the finite element model into substructures, which may reduce to individual elements. Substructures are preprocessed by the Direct Stiffness Method (DSM) to generate free-free flexibility matrices for floating substructures. The interface problem is solved for the interface forces and the solution recovered over substructure interiors. The DFM shares with the DSM the advantages of being automatic, maintaining locality and sparseness, efficiently handling continuum elements, and requiring only the availability of element stiffness libraries. The new method appears to be advantageous for specific applications. These include: massively parallel processing, inverse problems, treatment of rigid members and inclusions, and use of under-integrated elements without spurious-mode stabilization.

Correlation studies on an RC frame shaking-table specimen

SUMMARY This paper presents the correlation of the results of a new model for the dynamic analysis of reinforced concrete (RC) frames with the experimental time history of a two storey RC frame shaking-table specimen. The frame member model consists of separate subelements that describe the deformations due to flexure, shear and bond slip in RC structural elements. The subelements are combined by superposition of flexibility matrices to form the frame element. A non-linear solution method which accounts for the unbalance of internal forces between di⁄erent subelements during a given load increment is used with the model. The ability of the proposed model to describe the dynamic response of frame structures under earthquake excitations is evaluated by comparing the analytical results with experimental evidence from a twostorey, one bay reinforced concrete frame tested on the shaking-table. The model parameters for the shaking-table specimen are derived from available experimental evidence and first principles of reinforced concrete. The e⁄ect of reinforcing bar slip on the local and global dynamic response of the test structure is assessed. ( 1997 John Wiley & Sons, Ltd.

COMPUTING STATICALLY COMPLETE FLEXIBILITY FROM DYNAMICALLY MEASURED FLEXIBILITY

A method is presented for computing a statically complete structural flexibility matrix from a dynamically measured flexibility matrix. When computed from a limited set of measured modal data, dynamically measured flexibility cannot reproduce the correct static force–displacement relations of the structure, i.e., it is not “statically complete.” A previously developed algorithm is used to include the effects of residual flexibility in the dynamically measured flexibility matrix, so that certain entries in the measured flexibility matrix can be considered to be statically complete. The method presented in this paper computes the remainder of the entries in the statically complete flexibility matrix by first forming a static flexibility matrix using assumed stiffness parameters and elemental connectivity, then scaling is such that is approximates the corresponding statically complete entries in the measured flexibility matrix. The method requires the solution of linear systems of equations only. The method is derived and applied to both numerical and experimental measured flexibility matrices, and the improved accuracy of the static flexibility over the dynamically measured flexibility is demonstrated.

Method for Structural Model Update Using Dynamically Measured Static Flexibility Matrices

A new method is presented for parametric correction of full-order analytical stiffness matrices from reduced-order dynamically measured static flexibility matrices. The measured static flexibility matrix is formed using modal data in conjunction with a residual flexibility estimate. The algorithm corrects model parameters by minimizing a static flexibility matrix error residual constructed for measurement degrees of freedom only. By utilizing a flexibility matrix residual instead of one derived using stiffness matrices, numerical problems associated with inverting a rank-deficient measured flexibility matrix are avoided. Posing the problem in terms of flexibility matrices also avoids the problems of modal correspondence, mode selection, and modal truncation. In addition, incorporating static reduction relationships into the error residual makes prior eigenvector expansion or model reduction unnecessary. Two formulations for the error residual are presented: an explicit inverse formulation and one derived from a pseudoinverse relationship. The second leads to an exact linear update problem when all of the model degrees of freedom are measured. Numerical simulation results demonstrate that both formulations are capable of localizing and quantifying local stiffness errors in a full-order finite element model from reduced-order measurements. Experimental results for a cantilever beam structure are also presented.

Method for structural model update using dynamically measured static flexibility matrices

Method for structural model update using dynamically measured static flexibility matrices

Combinatorial Methods for the Formation of Sparse Flexibility Matrices

Two algorithms are designed for the formation of cycle bases of the graph models of rigid–jointed skeletal structures leading to sparse flexibility matrices. These combinatorial methods employ the Greedy Algorithm for the formation of suboptimal cycle bases with the least possible lengths.

Numerical study of parameters influencing the response of flexible retaining walls

A practical numerical model is described for analysis of flexible retaining walls. In terms of capabilities, the model fits between traditional limit equilibrium methods and full finite element approaches; it overcomes many of the limitations associated with the former but is not equipped with the versatility offered by the latter. Using an approach similar to that adopted in boundary-element based models, the wall stiffness is represented by a series of elastic beam elements whose stiffness is combined with that of the prestressed struts and the soil to form, the overall stiffness matrix. The stiffness matrix of the soil is obtained by inversion of flexibility matrices generated by interpolation and sealing of flexibility matrices calculated for a simplified soil model using finite element methods. The soil behaves linearly elastically, as long as the pressures correspond to stress levels lying between the limits. Where the lateral displacement of the wall corresponds to a pressure outside of these allowable limits, correction forces are added until the resulting pressures are within the active or passive pressures. Arching is permitted by considering the forces acting on the wall compared with the forces consistent with possible failure surfaces within the soil. Other features that can be accomodated by the model include struts, variations in water table, and the effects of surcharges. The proposed model has been shown to capture the displacement, anchor loads, and lateral stresses for several field problems. Based on these studies and other field applications of the model a number of points have been observed that are of practical interest; these points are separately listed. Key words: numerical analysis, retaining wall, anchor, arching, soil–structure interaction, deflection.