Shells had concerned with the problem of

Shells are thin structures that have high load carrying capacity;
this advantage can be used to transmit heavy super structural loads to weak
soil. Shallow spherical dome with variable thickness resting on Pasternak
foundation is studied considering external vertical pressure. The governing
differential equation is derived neglecting the strain energy of second strain
invariant of middle surface (Berger’s approach). Analytical solution of the
governing differential equation is presented. Parametric study is executed
considering thickness nonlinearity constant and Pasternak foundation constants.

Introduction:

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With the rapid
development in heavy super structures, shallow spherical dome is introduced as
a new foundation to carry heavy loads on weak soil. Domes have found wider
applications in engineering; much attention is currently focused on the
nonlinear bending and stability problems of those shells. Berger had assumed
that the strain energy due to second strain invariant of the middle surface of
plates can be neglected when driving governing differential equation 1. Eric
Ressiner had concerned with the problem of symmetrical bending of thin elastic
shells of revolution for shallow shells , general system of stress strain
relations were assumed to reduce the problem to two simultaneous second order
non-linear differential equations; The effect of uniform surface pressure is
considered assuming different elastic behavior in meridional and
circumferential direction 2. Plastic buckling of thin shells under external
pressure was investigated, using the shallow shell approach, characteristic
equations were obtained for simply supported type spherical plates and plastic
buckling coefficients were determined for various values of a shell geometry
parameter by Lakshmikantham and George Gerard 3.

The general solution
of Marguerre’s equation was obtained for shallow spherical shell for simply
connected region under various types of boundary conditions, numerical results
have been obtained for displacements, stress resultants and moments, also comparison
with similar results for circular plate under the same loading was shown by K.
V. Mital and P. S. Ttripathi 4. The influence of change in geometry on the
yield point load of shallow spherical shell was examined, the shell material was
taken as rigid, also the governing sets of equations and an improved
approximate solution were introduced by H. M. Haydl and A. N. Sherbourne 5. Shallow
caps under an axisymmetric external pressure distribution, which is inward in a
neighborhood of the pole and outward away from the pole, were treated by the
method of inner-outer

(asymptotic)
expansions, also the analysis was carried out for the special case of a
quadratically varying pressure distribution 6. Static and dynamic analysis of
fully clamped shallow spherical shells subjected to uniform pressure on concave
side and continuously supported on Pasternak foundation on the convex was
carried out, expression for deflection was obtained at different Lame constants
by D. N. Paliwal, S. N. Sinha and B. K. choudhary 7.

Ye Zhiming had
concerned with the nonlinear bending, stability and optimal design of
revolution shallow shells with variable thickness, solutions for nonlinear
bending and stability problems of revolution shallow shells with variable
thickness were presented 8.  Study on
the geotechnical behavior of shell footing using a nonlinear finite element
analysis with a finite element code “PLAXIS” was introduced by Bujang B.K. Huat
and Thamer A. Mohammed 9. New procedure for the buckling resistance assessment
of steel spherical dome shells subjected to uniform external pressure was
presented by Pawe? B?a?ejewski et al. 10, also comparison of the proposed
procedure with experimental results was carried out.