1 Introduction The problem of the free vibrations of nonprismatic thin-walled beam systems is interesting for two reasons. The first reason is the need to describe more precisely and solve this mathematically difficult problem that, except for special cases, has no closed analytical solutions. The second reason is practical and stems from the necessity to rationally shape and economically design contemporary civil engineering structures built from thin-walled beams with variable geometrical and material parameters. The problems relating to thin-walled beam

### Józef Szybiński and Piotr Ruta

### Czesław Machelski

forces M r and N r , stemming from the unit load applied in the considered displacement point and in the radial direction. According to the reduction theorem used in structural mechanics, in this case, an equivalent model in the form of a planar arc with the geometry of the shell circumferential section, i.e. a 2D model, is used. The model is described further in this paper. Obviously, the algorithm for calculating displacement r from Eq. (1) in accordance with the procedure described earlier yields the same results as the direct calculation of r in the

### Tadeusz Majcherczyk, Zbigniew Niedbalski and Łukasz Bednarek

and reached the value of only 1.5 mm at the maximum ( Fig. 6 ). It may be observed that practically all measurement points in the roof strata dislocate with regular intensiveness. The only exception in this respect is the anchor installed at the depth of 2.48 m, which shows that in this very place, the rocks are subject to slight separation ( Fig. 7 ). It may be, therefore, argued that in this case, such a behaviour stems from the occurrence of a fairly homogeneous and non-stratified roof made of conglomerate and sandstone with varied granulation. Figure 6

### M. Bartlewska-Urban and T. Strzelecki

]. N is equivalent to shear modulus μ , and A are is equivalent to Lame coefficient λ ; R is the modulus of volume elasticity of the liquid; Q is a coupling coefficient stemming from the mutual interaction between the solid phase and the liquid phase, σ ij is a tensor of stress in the skeleton, related to the total RVE surface, defined as a fuzzy stress tensor, σ is the stress in the liquid, related to (similarly as the stress in the skeleton) the total surface area of the RVE cross section, also defined as fuzzy stress, ε ij is a skeleton deformation