Determination of the orthotropic plate parameters of stiffened plates and grillages in free vibration

Summary It is well known that the analysis of vibration of orthogonally stiffened rectangular plates and grillages may be simplified by replacing the actual structure by an orthotropic plate. This needs a suitable determination of the four elastic rigidity constants D _x, D _y, D _xy, D ₁ and the mass of the orthotropic plate. A method is developed here for determining these parameters in terms of the sectional properties of the original plate-stiffener combination or the system of interconnected beams. Results of experimental work conducted on aluminium plates agree well with the results of the theory developed here.Notation a, b span and width of stiffened plates, grillages and orthotropic plates - c, spacing of beams in the y and x directions - D rigidity of an unstiffened isotropic plate - D _x, D _y rigidities in the x and y directions of an orthotropic plate - D _xy torsional rigidity of an orthotropic plate - D ₁ a parameter associated with a poisson type ratio - e depth of the common neutral axis of the beam-plate combination below the middle surface of the plate - E, G modulus of elasticity and modulus of shear - H =D ₁+2D _xy - i, j integers - I, , I _i, I _j moment of inertia of beams - J, J _i, J _j polar moment of inertia of beams - m, n integers referring to mode numbers in the x and y directions - p _{m n} natural frequency of an orthotropic plate - r, s number of beams in the transverse and longitudinal direction - T _g, T _o, T _s kinetic energy of grillages, orthotropic plates and stiffened plates - u frequency parameter of grillages - V _g, V _o, V _s potential energy of grillages, orthotropic plates and stiffened plates - W=W(x, y, t) deflection of an orthotropic plate - w=w(x, y) amplitude of vibration of orthotropic plates, stiffened plates and grillages - x, y, x _i, y _i cartesian coordinates - X _m, Y _n beam eigen functions - X _mj, Y _ni X _mand Y _nat x=x _jand y=y _i - X _m, Y _n; X_m, Y_n first and second derivatives of X _mand Y _nrespectively - Y _mn plate eigen functions - _n, _n parameters occurring in the expression for the beam functions - , , _i, _j mass per unit length of beams - _mn = frequency parameter of beam and slab bridges - = - Poisson's ratio - , mass per unit area of unstiffened plates and orthotropic plates - _mn natural frequency of stiffened plates and grillages