The method of calculations of a thick plate on the two-parameter layered foundation by the finiteelement method is presented. The numerical model allows to add a few (number of) foundationlayers. The expressions for the element stiffness matrices of the foundation are based on 18-nodezero-thickness interface elements. For modelling of thick plates the 9-node Mindlin element of theLagrange family is used. The formulation of the problem takes into account the shear deformation ofthe plate and unilateral contact conditions between plate and foundation. The tensionless characterof the foundation is achieved by removing from the global stiffness matrix the appropriate partof foundation stiffness attached to the node being in the separation stage. The advantages of theproposed algorithm are illustrated by numerical examples.
The present article investigates the dynamic behavior of a fully assembled turbogenerator system influenced by misalignment. In the past, most of the researchers have neglected the foundation flexibility in the turbogenerator systems in their study, to overcome this modelling error a more realistic model of a turbogenerator system has been attempted by considering flexible shafts, flexible coupling, flexible bearings and flexible foundation. Equations of motion for fully assembled turbogenerator system including flexible foundations have been derived by using finite element method. The methodology developed based on least squares technique requires forced response information to quantify the bearing–coupling–foundation dynamic parameters of the system associated with different faults along with residual unbalances. The proposed methodology is tested for the various level of measurement noise and modelling error in the system parameters, i.e., 5% deviation in E (modulus of elasticity) and ρ (density), respectively, for robustness of the algorithm. In a practical sense, the condition analyzed in the present article relates to the identification of misalignment and other dynamic parameters viz. bearing and residual unbalance in a rotor integrated with flexible foundation.
The two-variable refined plate theory is used in this paper for the analysis of thick plates resting on elastic foundation. This theory contains only two unknown parameters and predicts parabolic variation of transverse shear stresses. It satisfies the zero traction on the plate surfaces without using shear correction factor. Using the principle of minimum potential energy, the governing equations for simply supported rectangular plates resting on Winkler elastic foundation are obtained. The Navier method is adopted for solution of obtained coupled governing equations, and several benchmark problems under various loading conditions are solved by present theory. The comparison of obtained results with other common theories shows the excellent efficiency of this theory in modeling thick plates resting on elastic foundation. Also, the effect of foundation modulus, plate thickness and type of loading are studied and the results show that the deflections are decreased by increasing the foundation modulus and plate thickness.
Complex structures used in various engineering applications are made up of simple structural members like beams, plates and shells. The fundamental frequency is absolutely essential in determining the response of these structural elements subjected to the dynamic loads. However, for short beams, one has to consider the effect of shear deformation and rotary inertia in order to evaluate their fundamental linear frequencies. In this paper, the authors developed a Coupled Displacement Field method where the number of undetermined coefficients 2n existing in the classical Rayleigh-Ritz method are reduced to n, which significantly simplifies the procedure to obtain the analytical solution. This is accomplished by using a coupling equation derived from the static equilibrium of the shear flexible structural element. In this paper, the free vibration behaviour in terms of slenderness ratio and foundation parameters have been derived for the most practically used shear flexible uniform Timoshenko Hinged-Hinged, Clamped-Clamped beams resting on Pasternak foundation. The findings obtained by the present Coupled Displacement Field Method are compared with the existing literature wherever possible and the agreement is good.
The authors developed a simple and efficient method, called the Coupled Displacement method, to study the linear free vibration behavior of the moderately thick rectangular plates in which a single-term trigonometric/algebraic admissible displacement, such as total rotations, are assumed for one of the variables (in both X,Y directions), and the other displacement field, such as transverse displacement, is derived by making use of the coupling equations. The coupled displacement method makes the energy formulation to contain half the number of unknown independent coefficients in the case of a moderately thick plate, contrary to the conventional Rayleigh-Ritz method. The smaller number of undetermined coefficients significantly simplifies the vibration problem. The closed form expression in the form of fundamental frequency parameter is derived for all edges of simply supported moderately thick rectangular plate resting on Pasternak foundation. The results obtained by the present coupled displacement method are compared with existing open literature values wherever possible for various plate boundary conditions such as all edges simply supported, clamped and two opposite edges simply supported and clamped and the agreement found is good.