Birinapant cost In this paper we used

In this paper, we used a variation of the classical theory considering shells to be surfaces with material normals, constructed on the basis of Lagrangian mechanics [11–14]. Different applications of this theory have been described in Refs. [15–17]. The second part of the paper deals with static stability of the U-shaped bellows. We have performed a theoretical simulation of the ultimate internal pressure in the bellows at which adjacent states of equilibrium occur. The simulation results have been verified via numerical experiment using the Finite Element Method in the ANSYS software.
Thin shell theory equations A shell is regarded as a material surface whose particles have five degrees of freedom: three translations and two rotations. The movement is determined by the vector of small displacements u and the vector of small rotation θ in the tangent plane. We introduce a vector of shell normal variation (the overdot indicates a small increment under strain). A generalized force, corresponding the vector φ, is introduced in the Birinapant cost for the work done by the external distributed moment:
The surface strain is determined by the tensors
Here ∇ is the Hamiltonian operator on the surface, and is the second metric tensor. The quantities in brackets with the subscripts and superscripts denote, respectively, the components in the tangent plane (┴), the symmetric part (S) and the transposition (T). The rotation is associated with the displacement (according to the Kirchhoff kinematic hypothesis):
The principle of virtual work allows to derive the entire system of equations. The forces and moments in the shell are introduced as Lagrange multipliers: τ and µ are the symmetric force and moment tensors; Q is the shear forces vector. These tensors and vector lie in the tangent plane. The equations of balance of forces and moments follow from the variational setting: and so do the boundary conditions at the edge in the general form: where q and m× are the external distributed load and the moment on the surface; P0 and M0 are the external distributed load and the moment at the edge. A segment of the inner edge with the length dl and the normal ν is acted on (from ν) by the force and the moment
The elastic relations for a shell made of isotropic material have the form
Here a ≡ ∇r is the first metric tensor on the surface (the unit tensor in the tangent plane); the coefficients C1, C2, D1, D2 were taken the same as in the Kirchhoff plate; E is the elastic modulus of the shell material, h is its thickness; νis Poisson\'s ratio. There are no elastic relations for the shear forces vector Q in the classical theory, so instead, we have relation (2) for this vector.
System of equations for shells of revolution Let us consider a shell whose surface is formed by rotation of a meridian around the x axis [11]. The meridian is set by a dependence of the cylindrical coordinates on the arc coordinate:, , and its location on the surface is determined by the angle θ (Fig. 1). The radius vector of the surface points is given by where i, j, k are the Cartesian unit vectors along the x, y, and z axes, respectively; the unit vector tangent to the parallel:
We have, for the unit vectors tangent and normal to the meridian in its plane, here ω is the curvature of the meridian, and is the curvature of the parallel. The components of external loads are equal to zero in an axisymmetric setting, while the shell displacement vector has two components:
Using formulae (1) and (2), we can determine the revolution, the elongation strain and the bending strain
Using Eq. (5), we can write the relations for the forces and the moments:
The system of components is completed by the equations of balance (3):
Three scalar conditions imposed on the shell edge follow from boundary conditions (4). For the clamped case these are the conditions and . The tension force , the shear force and the bending moment are set for a free edge with the normal .