The paper considers the problem of the mechanics of a deformable solid about a cylindrical tank made of an alloy with shape memory (SMA) under a pressure during direct thermoelastic martensitic phase transformation under constant pressure. Both the momentless shell and the influence of the edge effect with rigid and articulated fastening are considered. The problem of relaxation in a similar shell during direct phase transformation has also been solved. In the second problem, internal pressure is applied to the shell in an austenitic phase state. Next, the shell material is cooled through the temperature range of the direct thermoelastic martensitic transformation. It is required to determine the necessary decrease in the process of such a transition of the uniformly distributed load so that the deflection of the shell remains unchanged. To describe the behavior of the shell material, we used the model of linear SMA deformation during phase transformations. The solution was obtained in the framework of the theory of thin isotropic shells and the assumption that the phase composition parameter at each moment of the process under consideration is uniformly distributed over the shell material, which corresponds to an unrelated statement of the problem for the case of uniform distribution of temperature over the material. The possibility of structural transformation in the shell material is not taken into account. It neglects the variability of the elastic moduli during the phase transition and the property of the SMA diversity resistance. To obtain an analytical solution to all the equations of the boundary value problem, the Laplace transform method using the volume fraction of the martensitic phase was used. After the transformation in the space of images, an equivalent elastic problem is obtained, solving which the Laplace images of the sought quantities are obtained in the form of analytical expressions, including operators that are Laplace images of elastic constants. These expressions are fractional rational functions of the Laplace image of the phase composition parameter. Returning to the space of originals by analytically decomposing the expressions for the sought quantities in the space of images into simple factors, we obtain the desired analytical solutions.