PSI - Issue 47

Yaroslav Dubyk et al. / Procedia Structural Integrity 47 (2023) 863–872

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Yaroslav Dubyk et al./ Structural Integrity Procedia 00 (2023) 000 – 000

the earthquake. Saatcioglu et al. (2001) reported that the main factor for the collapse tanks under the earthquake was the inadequate and sometimes nonexistent design and construction guidelines. Conical shells vibrations study starts not so long time ago, due to the complexity of their solution Leissa (1973). More often, it were solutions using the Rayleigh-Ritz procedure, where there is a need to choose test function that should minimize functional in known functions superposition form that satisfy the boundary conditions Salmanizadeh et al (2022), Vescovini and Fantuzzi (2023). No less popular is Galerkin's method, in which differential equations are also solved approximately Yang et al. (2021) Liu et al. (2022). A new step in development of conical shells vibration study was given with composite materials wide application. In this case, the solution becomes somewhat more complicated since cones must now have interconnected layers with different thicknesses and elastic modulus. For example, in works Safarpour et al. (2019), Zarei (2020) solution was found using a combined numerical scheme. Many articles discuss the dynamic behavior of conical shells, where the solution is made by the finite element method Maji et al. (2020), Chadga et al. (2021). However, if analytical solutions is present, it is possible to make an alternative to numerical calculations and supplement them, for example, to estimate frequency diapason. Therefore, the objective of this work was formed: modernize exact solution for cylindrical shells Dubyk et al. (2020) in such way as to take into account the angle of inclination of the side surface and obtain a solution for a conical shell. Also, an effort was made to account for initial stresses like axial force and pressure. Comparisons with experimental data and othe r researcher’s results are given in this work. Using this solution, it is possible to obtain quick solution of dynamic problems for various boundary conditions Dubyk et al. (2018).

Nomenclature 

cone angle

mean radius, shell thickness and length

R, h, L , ,   E

Young's modulus, Poisson ratio and density of shell material

frequency



axial and circumferential normal forces

, x N N



shear force

 x N

axial and circumferential bending forces axial, circumferential and radial displacements wave number in circumferential and axial direction bending strains in axial and circumferential direction

, x Q Q  , , u v w

, m n

, xx   

bending strain of torque

x  

pressure, axial force and torque moment forces in shell from initial stresses

, , P N M

, ,   x x N N N

2. Conical shell system of equations 2.1. Mathematical formulation To derive the equations for the conical shell, the governing balance equations are presented according to the Donnell-Mushtari thin shell theory:   1 1 0 sin x x x N N N N hu x x x               (1)