Large Amplitude Forced Vibration Analysis of Stiffened Plates under Harmonic Excitation

Large Amplitude Forced Vibration Analysis of Stiffened Plates under Harmonic Excitation

Anirban Mitra (Jadavpur University, India), Prasanta Sahoo (Jadavpur University, India) and Kashinath Saha (Jadavpur University, India)
DOI: 10.4018/978-1-4666-1867-1.ch002
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Large amplitude forced vibration behaviour of stiffened plates under harmonic excitation is studied numerically incorporating the effect of geometric non-linearity. The forced vibration analysis is carried out in an indirect way in which the dynamic system is assumed to satisfy the force equilibrium condition at peak excitation amplitude. Large amplitude free vibration analysis of the same system is carried out separately to determine the backbone curves. The mathematical formulation is based on energy principles and the set of governing equations for both forced and free vibration problems derived using Hamilton’s principle. Appropriate sets of coordinate functions are formed by following the two dimensional Gram-Schmidt orthogonalization procedure to satisfy the corresponding boundary conditions of the plate. The problem is solved by employing an iterative direct substitution method with an appropriate relaxation technique and when the system becomes computationally stiff, Broyden’s method is used. The results are furnished as frequency response curves along with the backbone curve in the dimensionless amplitude-frequency plane. Three dimensional operational deflection shape (ODS) plots and contour plots are provided in a few cases.
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Stiffened plates are important structural elements due to their enhanced stiffness and stability characteristics with the advantage of light weight. So, it is no surprise that they are extensively used in many branches of modern civil, mechanical, and structural engineering. They have wide application in constructing marine structures like floors of bridges, bridge decks, ship hulls etc and aircraft structures. In these applications stiffened plates are regularly subjected to static and time varying loads. Hence, analysis of stiffened plates under different loading conditions has always been an area of immense interest to researches. Also when a system exhibits large amplitude vibration, the analysis no longer remains linear and therefore, a nonlinear analysis is required to investigate its dynamic behaviour.

Research work on nonlinear behaviour of stiffened plates has gone through different phases. Different researchers have employed different techniques and methodologies in carrying out nonlinear analysis of stiffened plates. In one of the earlier works Rossow and Ibrahimkhail (1978) applied the constraint method to finite element static analysis of concentrically and eccentrically stiffened plates. Koko and Olson (1991) developed a new numerical technique for large deflection elastoplastic analysis of stiffened plates using super finite elements. They derived the governing differential equations using the virtual work principle and solution was achieved by Newton-Raphson iteration technique. D. V. Rao, Sheikh and Mukhopadhyay (1993) presented a finite element analysis of large deflection behaviour of stiffened plates using the isoparametric quadratic bending element. Mukhopadhyay (1994) analyzed stiffened plates in bending through a semi-analytic finite difference method. Bedair (1997) put forward a methodology for the analysis of multi-stiffened plates under lateral loading based on energy formulation and sequential quadratic programming (SQP) technique. Sapountzakis and Katsikadelis (2000) investigated elastic deformation of ribbed plates subjected to static, transverse and in-plane loading using the analog equation method to solve nonlinearly coupled equations. Geometric non-linear analysis of stiffened plates was carried out by Sheikh and Mukhopadhyay (2000) utilizing spline finite strip method and von Karman non-linear plate theory. Ojeda, Prusty, Lawrence and Thomas (2007) introduced a new approach based on finite element analysis for large deflection analysis of isotropic and composite stiffened plates with arbitrarily oriented stiffeners. Bruback and Hellesland (2008) using semi-analytical large deflection analysis studied the strength criterion, both in local and global bending, of stiffened plates under in-plane loading. Wutzow and Paiva (2008) employed integral equations and boundary element method (BEM) to perform a linear analysis of stiffened plates.

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