Estimation of reciprocal residual flexibility from experimental modal data

Of the two main finite element approaches, the displacement method has progressed more rapidly than the force method in the area of element representation. Many authors have contributed to the displacement method with both static and dynamic stiffness matrices for beam elements, plane stress elements and plate bending elements. The plane stress and bending elements being rectangular, triangular or quadrilateral in form. A dynamic stiffness matrix consists of a static stiffness matrix and a mass matrix. In the force method, .static flexibility matrices have been developed for beam elements and plane stress elements. However, static flexibility matrices for plate bending elements have only recently been published (Kaufman and Hall, Morley, Robinson, Przemieniecki). For the past few years, the author has been investigating the rank force method for structural vibration analysis. In ref. 6 a dynamic flexibility matrix is presented for a beam element. This matrix consists of a static flexibility matrix and an inverse mass matrix. Ref. 5 contains the derivation of a dynamic flexibility matrix for a rectangular plate element in bending, twisting and shear. The author, in collaboration with Petyt, demonstrated how a dynamic stiffness and flexibility matrix can be extended to include material damping.

The paper focuses on damage detection of civil engineering structures and especially on concrete bridges. A method for structural health monitoring based on vibrational measurements is presented and discussed. Experimentally identified modal parameters (eigenfrequencies, mode shapes and modal masses) of bridge structures are used to calculate the inverse stiffness matrix, the so-called flexibility matrix. By monitoring of the stiffness matrix, damage can easily be detected, quantified and localized by tracking changes of its individual elements. However, based on dynamic field measurements, the acquisition of the flexibility matrix instead of the stiffness matrix is often the only choice and hence more relevant for practice. But the flexibility-based quantification and localisation of damage are often possible but more difficult, as it depends on the type of support and the location of the damage. These issues are discussed and synthetized, that is an originality of this paper and is believed useful for engineers in the damage detection of different bridge structures. First the theoretical background is briefly repeated prior to the illustration of the differences between stiffness and flexibility matrix on analytical and numerical examples. Then the flexibility-based detection is demonstrated on two true bridges with real-time measurement data and the results are promising.

The flexibility matrix that relates external loads and structural responses has been utilized in various damage identification algorithms that find structural changes expressed as stiffness degradation. Contrary to the stiffness matrix, the flexibility matrix can be accurately calculated using only several lower natural modes, which makes it feasible to experimentally estimate the flexibility matrix. To date, acceleration is the most popularly used physical quantity in the flexibility-based damage detection methods because of low noise, installation convenience and cost-effectiveness of acceleration transducers. As the acceleration responses correspond to translational degree-of-freedoms (DOF) in the numerical model, moment-resisting structures both with translational and rotational DOFs such as beams and plates are not appropriately represented by the acceleration-only approaches. This study presents an estimation method of the flexibility matrix of beam-type structures considering translational and rotational DOFs that can be measured by accelerometers and gyroscopes, respectively. The efficacy of the proposed method is numerically validated using a simply supported beam model.

Timoshenko beam elements have been the subject of numerous publications. The difficulty was that of arriving at a superconvergent element with four degrees of freedom, as is the case for the Bernuli‐Euler classical beam element. Two different approaches are presented here for the derivation of the shape functions. The first is based on the flexibility matrix, where utilizing the unit load method, including the term that accounts for the shear deformations in the virtual work expression, the stiffness matrix is derived. Then, a second method is presented to derive the exact shape functions, directly from the differential equations of the Timoshenko beam theory. The resulting shape functions are the same in both methods.

The wave-based substructuring approach for the efficient description of interface dynamics in substructuring

<div class="htmlview paragraph">With highly complex structures, it is sometimes desirable to derive a dynamic model of the system from experimental vibration data. This paper presents algorithms for synthesizing the mass and stiffness matrices from experimentally derived modal data in a way which preserves the physical significance of the individual mass and stiffness elements. The synthesizing procedures allow for the incorporation of other mass and stiffness data, whether empirical or based on the analyst's insight. The mass and stiffness matrices are derived for a cantilever beam example and are compared with those obtained using earlier techniques.</div>

Summary A problem of elastic structures stress-strain field determination is considered in this article. A theoretical background and computation algorithms of equilibrium finite element method are presented there. The different external effects are estimated, namely: load, prestressing, inicial strains and support settlements. The dual relationships (equilibrium and geometrical equations, stiffness and flexibility equations) of equilibrium element, also the expressions of stiffness and flexibility matrices are given. These relationships describe the stress-strain field of the finite element and allow to transforme the flexibility matrix to the stiffness matrix and on the contrary. There are presented direct and variational formulations of the problem. The algorithms of the forces method and displacements method are made for their solution. Easier is the algorithm of displacements method, because making equations of the forces method needs to solve the system of equilibrium equations. But the formulation and solving of the displacements equations can be changed by direct formulation of global flexibility matrix of the structure. In these cases, when the degree of static indeterminacy of the discrete model of structure is quite less than the number of freedom degrees, suth the change can be effective. The computational results illustrate the high accuracy of equilibrium elements not only with respect to forces, but also to displacements. The direct stiffness method in equilibrium elements makes it possible not only to simplify significantly implementation of the equilibrium element method, but also to include equilibrium elements in libraries of geometrically compatible elements.

A flexibility‐based component mode synthesis (CMS) is proposed for reduced‐order modelling of dynamic behaviour of large structures. The approach employs partitioning via the localized Lagrange multiplier method. The use of the localized Lagrange multipliers leads to, unlike the classical Lagrange multipliers, a linearly independent set of interface forces without any redundancies at multiply connected interface nodes. The flexibility‐based CMS method has shown significant advantages over the classical Craig–Bampton method. A key feature of the method is its substructural mode selection criterion that is independent of loading conditions. Unlike the majority of available CMS approaches, where one retains the full dimension of partition boundary degrees of freedom (DOFs), the flexibility‐based method allows to reduce significantly the interface DOFs. The reduction of interface DOFs represents the major contribution of the present communication. The efficiency of the proposed approach is demonstrated on an analysis of a simple plate partitioned and of a more complex 3D structure, both partitioned into several substructures. Copyright © 2006 John Wiley & Sons, Ltd.

Flexibility matrix identification using the moving vehicle induced responses for beam type bridge

A sub-domain scheme combined with a reduced system is proposed for non-matched interface problems. A twolevel condensation scheme (TLCS) is employed for the selection of primary degrees of freedom of each sub-domain. The degrees of freedom of each domain are divided into primary, secondary and interface degrees of freedom. After the reduced system is constructed in subdomain-level, the penalty frame method, which utilizes a conventional FEM solver, is employed for assembling non-matched subdomains. The present method can construct the reduced system of each domain without coupling with adjacent subdomains. Thus it is remarkably efficient in computation time and does not require a full memory of global system. Numerical examples demonstrate that the proposed method saves computational cost effectively and provides a reduced system which predicts the accurate eigenvalues of global system.

The coupled equations of motion of a system composed of a launch vehicle and multiple payloads are derived. In the process, it is shown how superfluous interface degrees of freedom on the booster side can be accommodated in the formulation. The discrete system equations are directly integrated, avoiding the solution of an expensive system eigenvalue problem. A modified Newmark-Chan-Beta numerical integration scheme is used to obtain the response. The unique form of the equations of motion allow for a quick and accurate solution both for the response and the internal loads. Although completely general, the presented technique is most effective when the number of interface degrees of freedom is relatively small compared to the overall number of degrees of freedom of the system. The technique is applied to the case of the S.T.S.-S.T.-OMS Kit system. Finally, an evaluation of the presented method is included.

A method is described for deriving constrained modes and frequencies from a reduced model based on a subset of the free-free modes plus the residual effects of neglected modes. The method involves a simple modification of the MacNeal and Rubin component mode representation to allow development of a verified constrained (fixed-base) structural model. Results for two spaceflight structures having translational boundary degrees of freedom show quick convergence of constrained modes using a measureable number of free-free modes plus the boundary partition of the residual flexibility matrix. This paper presents the free-free residual flexibility approach as an alternative test/analysis method when fixed-base testing proves impractical.

This paper proposes a straightforward criterion to warrant the displacement functions being used in the finite element approximation of circular arches. The criterion was established by studying the natural shape function, i.e. the exact solution of the deformed shape, of the circular arch element. The exact stiffness matrix [K]exact is derived from the natural shape and is confirmed to be the inverse of the well‐known flexibility matrix [F]exact in the curved beam theory. The present paper compares the inverse [K]−1 of the stiffness matrix derived from the assumed displacement function with the [F]exact. It is shown that the procedure also guarantees the implicit inclusion of rigid‐body modes in the pertinent stiffness matrix [K]. Case studies on typical approximate displacement functions assure the appropriateness as well as the ease of application of the proposed method.

Chapter Five - FLEXIBILITY, STIFFNESS AND MASS MATRICES

Formulation of a matrix‐valued force–displacement relationship which can take radiation damping into account is of major importance when modelling unbounded domains. This can be done by means of fundamental solutions in space and time in connection with convolution integrals or by means of a frequency dependent boundary element representation, but for discrete frequencies Ω only. In this paper a method for interpolating discrete values of dynamic stiffness matrices by a continuous matrix valued rational function is proposed. The coupling between interface degrees of freedom is fully preserved. Another crucial point in soil–structure interaction analysis is how to implement an approximation in the spectral domain into a time‐domain analysis. Well‐known approaches for the scalar case are based on the partial‐fraction expansion of a scalar rational function. Here, a more general procedure, applicable to MDOF‐systems, for the transformation of spectral rational approximations into the time‐domain is introduced. Evaluation of the partial‐fraction expansion is avoided by using the so‐called mixed variables. Thus, unknowns in the time‐domain are displacements as well as forces. Copyright © 2001 John Wiley & Sons, Ltd.