Sensitivity of frames to variations of hinges in dynamic and stability problems

Abstract

A b s t r a c tProblems of forced vibrations, eigenvibrations and stability of linear elastic frames and beams are considered in the paper. Using the adjoint variable method sensitivity derivatives with respect to small variations of the position and stiffness of elastic hinges are derived. The stractural response is measured by an integral functional on displacements or by the eigenvalues. Illustrative examples are presented. I n t r o d u c t i o nIn the majority of papers on structural optimization and sensitivity analysis attention has been focused on dimensional or form optimization. MrSz (1975) first formulated the problems of optimal design of supports and joints. This idea was further developed in a series of papers by Szel~g and Mr6z (1978) and Garstecki and Mr6z (1987). It has been shown that the generalized formulation of optimization problems taking into account not only dimensional or configuration variables, but also variable supporting conditions, can lead to better results since the structural response to external loads and to distortions is sensitive to support variations. Extensions of the formulation to problems of minimum stiffness of optimally located supports for dynamic and stability response has been presented by/~kesson and Olhoff (1988) and Olhoff and .~kesson (1991).Considering a support as a joint between a structure and a foundation, the above problems can be generalized for joints between substructures. Surprisingly, the latter problem has not been discussed in literature from the point of view of structural sylithesis. The effect of joint flexibility in frames subjected to static loads has been demonstrated by Chen and Lui (1987). Structural sensitivity for variable location and stiffness of elastic hinges in structures subjected to loads and initial distortions was studied by Garstecki (1988). The problem of the optimal location of hinges was discussed in the literature in the context of the optimal stiffness of beams (Masur 1970) and columns (Olhoff 1986). However, hinges appeared there as singular points of vanishing cross-section and resulted from simplifying assumptions, namely from the negligence of shear force or stress constraints. In engineering applications these hinges were avoided by the introduction of geometrical constraints to the formulation of the optimization problem.The present paper supplements the previous research by considering dynamic and stability problems. The sensitivity analysis is carried out with respect to variations of the position and stiffness of elastic hinges. For a better understanding only simple types of hinges are considered, where the discontinuity of the slope of deflection line occurs, however, the results can easily be generalized for hinges in which discontinuities of other kinematic fields occur, e.g. the discontinuity of the torsion angle. For the sake of completeness we will allow for the distributed variation of the beam's crosssection and a step-wise change of cross-section at the points where hinges occur. The formulation is limi...

Sensitivity analysis of sandwich beams and plates accounting for variable support conditions

“…Primes denote differentiation with respect to x. Note that setting φ = −w ′ in (7) we obtain (6), where influence of V z is neglected. In the case of a three layered sandwich beam the bending and shear stiffness coefficients B s and S are expressed as follows:…”

“…It can be achieved in sandwich structures because the properties of the component materials can be used reasonably. Therefore the optimal design of sandwich structures is a topic frequently taken up in literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14].…”

“…The influence of the initial distortions on the optimal design of support conditions in beams and frames was studied in [5]. In [6,9] the sensitivity analysis in the case of dynamically loaded structures allowing for variable joint parameters has been presented.…”

Identification of damage in beam and plate structures using parameter‐dependent frequency changes

“…Fritzen et al (1998), Friswell and Mottershead (1995) and Khot and Berke (1994). A general sensitivity study of natural frequencies and modes with respect to crack and holes was presented by Gudmundson (1982), Burczyn Âski andPolch (1994), and also by Garstecki and Thermann (1992) and Dems and Mro Âz (1989) for geometrically non-linear vibration and stability problems.…”

Structural damage identification using frequency and modal changes

“…The papers by Pandey et al [3], Hearn and Testa [4] provide good examples of these approaches. A general sensitivity study of natural frequencies and modes with respect to cracks and holes was presented, for instance, by Gudmundson [5], Garstecki and Thermann [6] as well as Dems and Mróz [7] for geometrically non-linear vibration and stability problems. The damage identification using some additional control parameters in order to increase the sensitivity of structural response with increasing damage and to maximize the assumed distance norm between the measured and structural model response was also presented in [2].…”