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In [1] the stiffness of bridges and cantilevers made of natural chrysotile asbestos nanotubes has been studied by means of scanning probe microscopy. The stiffness is defined as a ratio the value of local load (applied to the tube) to the value of the displacement. The nanotubes with different materials for fillings are analyzed. The experiments show that the stiffness of the tube depends on the materials for filling. The tubes with water are softer than the tubes without filling materials and the tubes filled with mercury are more rigid than tubes without filling materials. It was shown in [1] that the classical theory of beam bending can not explain the experimental results, but the experimental results well agree with the Timoshenko-Reissner theory (at least qualitatively), when interlaminar shear modulus of elasticity changes for different filling materials. When additional factors such as lamination of structure and cylindrical anisotropy are taken into account [2] the theory of Rodionova-Titaev-Chernykh (RTC) permits to obtain much more reliable results. In this work the authors also applied one more nonclassical shell theory, namely the shell theory of Paliy-Spiro (PS) developed for medium - thickness shells and considered radial pressure. The comparison of nonclassical shell theories (RTC and PS) with experimental data and FEM calculations are presented in the report.