Mono-dimensional formulation of axial-symmetric spherical shells and characterization of the linear static behaviour

Abstract

In spherical shells, due to the axial-symmetry of the system, all the quantities involved in the elastic problem only depend on the curvilinear abscissa along the meridian lines. Hence, the mathematical domain of the model becomes mono-dimensional. However, in the classical model of axial-symmetric spherical shells, the kinematic problem is not the adjoint of the static problem. With the aim to clarify this aspect, a mono-dimensional linear model of axial-symmetric spherical shells has been proposed. The adopted mono-dimensional approach furnishes a model where the kinematic problem is the adjoint of the static problem. This model, able to describe the linear static behaviour of the shell, can be considered as a curved beam posed on an elastic variable soil. Specifically, the axis of the generic curved beam coincides with a meridian line and the elastic soil is related to the characteristics of the ring beams along the parallel lines. Several internal constraints and simplifying assumptions have been introduced. The comparison among approximate analytical models and a Finite Element one confirm the effectiveness of the proposed mono-dimensional approach. The most approximate model admits a closed-form solution. It appears to be perfectly capable of describing the classical oscillatory-damped behavior of beams on elastic soil. For this reason it has been used to introduce a criterion to classify spherical shells as long or short shells. Graphical abaci, that permit a handy classification, are then proposed, depending only on the geometrical characteristics of the shell.

Appendix: Bi-dimensional models of spherical shells

By referring to Fig. 1a the kinematic equations read:

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where the subscript m denotes the quantities related to the curved beam along the generic meridian line, while the subscript p refers to the strains of the ring beam along the generic parallel line.

The equilibrium equations read:

$$\begin{aligned} \begin{array}{l} \left\{ {\begin{array}{l} -(N_m\, r_0)'-\frac{V_m\,r_0}{R}-N_p\, \text {Sin}(\theta )=p_t\,r_0\\ \frac{N_m\, r_0}{R}-(V_m\,r_0)'+N_p\, \text {Cos}(\theta )=p_n\,r_0\\ -(M_m\,r_0)'-V_m\,r_0+M_p\, \text {Sin}(\theta )=0 \\ \end{array}} \right. \\ \end{array} \end{aligned}$$

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The constitutive equations read:

$$\begin{aligned} \left\{ {\begin{array}{c} {{N_m}}\\ {{N_p}}\\ {{V_m}}\\ {{M_m}}\\ {{M_p}} \end{array}} \right\} = \left[ {\begin{array}{ccccc} {\frac{{E h}}{{1 - {\nu ^2}}}}&{\frac{{\nu E h}}{{1 - {\nu ^2}}}}&0&0&0\\ {\frac{{\nu E h}}{{1 - {\nu ^2}}}}&{\frac{{E h}}{{1 - {\nu ^2}}}}&0&0&0\\ 0&0&{\frac{5}{6} G h}&0&0\\ 0&0&0&{\frac{{E {h^3}}}{{12(1 - {\nu ^2})}}}&{\frac{{\nu E {h^3}}}{{12(1 - {\nu ^2})}}}\\ 0&0&0&{\frac{{\nu E {h^3}}}{{12(1 - {\nu ^2})}}}&{\frac{{E {h^3}}}{{12(1 - {\nu ^2})}}} \end{array}} \right] \left\{ {\begin{array}{c} {{\varepsilon _m}}\\ {{\varepsilon _p}}\\ {{\gamma _m}}\\ {{\kappa _m}}\\ {{\kappa _p}} \end{array}} \right\} \end{aligned}$$

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