The Bending And Free Vibrations Of Moderately Thick Shallow Spherical Shells On The Winkler Elastic Foundations

This paper is concerned with the theoretical investigation of the bending and frees vibrations problems of moderately thick shallow spherical shells on the Winkler elastic foundations. First of all, From the fundamental equations of the moderate thick shallow spherical shells, through introducing Stress function F(r,θ) and displacement function x(r,θ) ,and looked Stress function F(r,θ), displacement function x(r,θ) and deflections function w(r,θ) as basic unknowns. Then, using the mixing method for the equations of moderately thick shallow spherical shells on the elastic foundation, then, the governing equations of quiet strength or free vibration can be determined. The analytic solutions of governing equations are expressed by Bessel's functions and harmonic functions. So, the displacements , internal forces and moments can be determined too.There are ten undetermined constants in final expressions. Under the condition of the certain boundary conditions, the Constants can be determined. The axial symmetry is out of shape, there are three undetermined Constants. Quantity of undetermined Constants is just equal to quantity of the certain boundary conditions. So, these constants can be determined. It is not the axial symmetry is out of shape, if n=1, there are four undetermined constants. If n>1, there are five undetermined constants. Quantity of undetermined constants is equal to quantity of the certain boundary conditions too. these constants can be determined. Just like this, the displacements, internal forces and moments can be determined.Trough using procedure of Matlab, this paper calculates the deflections and the curve of the deflections of moderately thick shallow spherical shells with the clamped edge on the Winkler foundation. The axial symmetry is out of shape (n=0), the deflections and the curve of the deflections can determined. The curve of the deflections is parabola. Curve of the deflections suit for the physical phenomenon. It is not the axialsymmetry is out of shape (?=1), unsymmetrical load on shells. Just for it, an unsymmetrical curve of the deflections determined. So curve of the deflections suit for the physical phenomenon too. This paper calculates the natural frequency and inspiring curve. Inspiring curve suited for the physical phenomenon.At last, this paper calculates the different deflections and the natural frequency with different thickness( h/R = 1/5 ^ h/R = 1/8 % h/R = 1/10). The curves of the deflection and inspiring curves accord with the physical phenomenon. For testing the exactness of theory that determined by this paper, the deflections and natural frequency under the theory of moderate thick shells compared with the deflections and natural frequency under the theory of thin shells .So, results can verify the correctness of the theory.