THE EQUILIBRIUM FINITE ELEMENTS IN COMPUTATION OF ELASTIC STRUCTURES

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.

Chapter Five - FLEXIBILITY, STIFFNESS AND MASS MATRICES

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.

A problem of ideal elastoplastic structures stress-strain field determination is considered The dual general and discrete mathematical models of analysis problem are made on the basis of the extremal energy principles and finite element method. In these models the possible discontinuity of displacements and the dissipation of energy in the place of those discontinuities, also the different external effects (load, initial strains and support settlements) are estimated. At first, on the basis of the mixed functional and mixed finite elements the discrete expressions of fundamental relationships (geometric equations, yield conditions) and the discrete mathematical model of mixed formulation of the problem are made. This mathematical model corresponds to the minimum total energy principle for a kinematically admissible displacements. The dual static formulation of the problem is obtained by Lagrange's multipliers method; this corresponds to the minimum complementary energy principle. The kinematic formulation of the problem is obtained in the case of linear yield conditions. These mathematical models permit to determine the lower values of the stress and displacements of structures. It has shown that the approximation of geometric equations and yield conditions by Bubnov-Galiorkin's collocation method gives the more accurate results.

The equilibrium dual discrete mathematical models of the problems of limit load analysis and optimization are investigated in the article. These models are presented in terms of static and kinematic formulation using equilibrium finite elements. In these mathematical models the possible discontinuities of displacement velocities are evaluated and the velocity of energy dissipation is estimated not only within the volume of finite elements, but at the plastic surfaces between elements. At first, on the basis of the energy principle of the maximum external power [1,2] the general mathematical models (3) and (7) of static formulation of limit load analysis and optimization problems are created. In these models the yield conditions are controlled not only within the volume, but also at the surfaces of finite elements. The equilibrium finite elements and interpolation functions of strains (9) are used for discretization of these models. The constancy of external power is taken as the optimum criterion. The discrete expressions of fundamental relationships—equilibrium and geometric equations, yield conditions (10)-(12) for finite element and (14)-(17) for the discrete model of a body are developed. The discrete expressions of yield conditions are given using the classic collocation methods: collocation at the point, collocation at the sphere (element) and Bubnov-Galiorkin's collocation method [11,12]. The equilibrium equations of discrete structure are developed on the basis of virtual displacement principle while geometrical equations are derived using virtual force principle. In contrast to the approach of other authors, yield conditions and geometrical equations are described not only within finite elements, but also at the surfaces between elements. That helps to design the dual discrete mathematical models of the problems (21)-(26), in which the discontinuities of displacement velocities and the velocity of energy dissipation in the place of those discontinuities are estimated. The mathematical models (22), (24), (26), (29) and (32) of kinematic problem formulation are developed from sensible static formulations by Lagrange's multiplier method. The modified mathematical models (27)-(29) are presented. In these models the equilibrium equations are eliminated or the geometrical equations are transformed into compatible equations of plastic stress velocities, in this way decreasing the number of equations and unknown values. The dependence of the numerical results (limit load) of the frame on the approximation degree of bending moments, as well as on the discretization method of yield conditions are illustrated. In table 1 the values of limit loading parameter F 0 and their error of calculation ΔF 0 (per cent, in comparison with the analytic solution F 0 = 0,4662M 0) are presented. They are given for the first and second order finite elements with linear and parabolic distribution of bending moments using different discrete yield conditions and a different number of finite elements. The numerical result shows, that the discretization of yield conditions by Bubnov-Galiorkin's method gives the best accuracy and stable solutions. By discretizing yield conditions using the point's collocation and collocation at the element, the accurancy of numerical results depends not only on the number of elements, but also on a more or less successful choice of finite elements net.

A simple and robust methodology is presented to identify damages in a structure using changes in vibration data. A comparison is made among damage indicators such as natural frequencies, mode shape data, curvature damage factors and flexibility matrices to study their efficacy in damage assessment. Continuous ant colony optimization (ACOR) technique is used to solve the inverse problem related to damage identification. The outcome of the simulated results demonstrates that the flexibility matrix as a damage indicator provides better damage identification.

Proportional flexibility-based damage detection for buildings in unknown mass scenarios: The case of severely truncated modal spaces

Preliminary Considerations Nodal Variables - Stiffness and Flexibility Matrices of a Structure Element Variables - Total Stiffness Matrix of an Element Direct Stiffness Method Supplementary Procedures to the Stiffness Method Basic Element Variables The Equations of Equilibrium and the Equations of Compatibility of a Structure Computation of the Displacements of Statically determinate Framed Structures The Flexibility Method The Displacement or Stiffness Method Appendix A. The International System of Units Appendix B. End Actions in Fixed-At-Both-Ends Elements Appendix C. Vector Quantities Appendix D. Computation of the Total Stiffness Matrix of an Element and the Equivalent Actions to be Placed at the Nodes of a Structure Using Shape Functions.

A technique is presented for estimating the residual flexibility between nonexcited structural degrees of freedom from experimental structural vibration data. Using this method, one can include the residual flexibility estimated from modal measurements in the computation of measured flexibility for experiments with incomplete reciprocity, i.e., when the response and excitation measurement sensors are not fully collocated. The method can also be used to estimate the unknown entries in the residual flexibility matrix for experimental component mode synthesis when excitations are not provided at all interface degrees of freedom. A general solution is presented that contains an unknown positive semidefinite contribution. The general solution satisfies modal orthogonality in the limit that all of the structural degrees of freedom are instrumented and when the positive semidefinite contribution lies in a nullspace defined by the stiffness matrix and the modal flexibility. With a limited number of measurements, modal orthogonality is shown to be satisfied to the extent that the measured modes are preserved by static condensation. A rank-deficient solution is presented that allows the residual to be used in the computation of the flexibility matrix without further modeling assumptions. Numerical and experimental results that demonstrate the application of the method to both flexibility matrix convergence and experimental component mode synthesis are presented.

The statical behaviour of a spatial bar of an elastic and isotropic material under arbitrary distributed loads having a non‐circular helicoidal axis and cross‐section supported elastically by single and/or continuous supports is studied by the stiffness matrix method based on the complementary functions approach. By considering the geometrical compatibility conditions together with the constitutive equations and equations of equilibrium, a set of 12 first‐order differential equations having variable coefficients is obtained for spatial elements of helicoidal axes. The stiffness matrix and the element load vector of a helicoidal bar with a non‐circular axis and arbitrary cross‐section are obtained taking into consideration both the presence of an elastic support and the effects of the axial and shear deformations. For helicoidal staircases, the significance of both axial and shear deformations and eccentricities existing in wide and shallow sections are also investigated. The developed model has been coded in Fortran‐77, which has been applied to various example problems available in the relevant literature, and the results have been compared.

A new method for the element stiffness matrix of arbitrary planar bars

<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>

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.

This paper introduces the direct global stiffness matrix (DGSM) method for analyzing the dynamics of cylindrical shells with axial symmetry. The dynamic stiffness matrix of a straight, circular cross-section cylindrical element is formulated in terms of ring forces and displacements at each end on a per circumferential mode basis. The stiffness matrix is based on an exact wave-type solution of the thin shell Donnell equations. More complicated shell systems are analyzed by assembling any number of cylindrical elements. For example, a cylinder with a gradual variation in radius may be modeled by a series of shell elements welded together at their ends. Total system dynamics are found at any given frequency by inverting the global system matrix formed from the individual stiffness element matrices and summing over circumferential mode number. Results for a test case are presented to demonstrate and verify the implementation. [Research sponsored by ONR.]

The development of the electronic digital computer and the application of matrix algebra has made it possible for the structural engineer to analyze complex or highly redundant structures which previously had been prohibitive to analyze by the hand method of calculation. Papers recently written have utilized either the “flexibility” or “stiffness” matrix approach. It is the purpose of this paper to point out the common origin of these methods in energy and to discuss procedures for calculating the total energy of a structure. The concept of the “total” structure will be considered. Applications include analysis of a truss with elastic supports and a continuous beam.