Stability of shallow anisotropic shells of revolution

Abstract

The paper outlines a method of analyzing layered anisotropic shells of revolution for stability using complex Fourier series. This simplifies the derivation of the basic equations compared with complete trigonometric Fourier series. Anisotropic shells in the form of a torus segment are analyzed for stability. This method allows optimizing the structure of the material and the geometry of the shell Keywords: stability, torus-like shells of revolution, Fourier series, complex numbers, anisotropy of material, external pressure, axial compression, torsionIntroduction. The stability of anisotropic cylindrical shells under loading of various types was analyzed in [8][9][10][11][12][13]. Stability problems were solved using the Donnell-Mushtari-Vlasov theory (DMV). The basic feature of the solutions is that the unknown functions are represented by series of a complete (in the range of the circumferential coordinate j p Î ( , ) 0 2 ) system of trigonometric functions. Such series are used to describe spiral buckling modes of anisotropic shells induced by coupled tensile, shear, bending, and torsional strains. In the case of isotropic and orthotropic shells, the bending and torsion problems are uncoupled. Therefore, anisotropy of shells complicates the analysis: the order of systems of ordinary differential equations is doubled after reducing the dimension of problems. This is also true of other problems for anisotropic shells [3].To approximate the deflection (W) and force (F) functions, we will use more compact notation of complex trigonometric series. We will analyze shallow anisotropic shells of revolution for stability. The equilibrium equations of such shells differ from those for cylindrical shells only by terms multiplied by 1 1 / R , where R 1 is the meridian radius [2, 5, 6]. Solving these equations, we obtain new mechanical results though the method is very similar to that outlined in [8][9][10][11]. However, the chief result of the present study is the conformation that complex series can be used to reduce the number of governing equations.1. Problem Formulation. The shallow shells of revolution under consideration are generated by revolving a rather shallow arc around the axis of revolution z [1], the chord OO1 of the meridian is parallel to the axis of revolution (Fig. 1). The coordinate system x, y on the mid-surface of the shell is such that the x-axis is aligned with the meridian of this surface and the y-axis with the parallel circle. For surfaces of revolution, meridians and parallels are the principal lines of curvature, and

Stability of anisotropic shells of revolution of positive or negative Gaussian curvature

“…Anisotropic shells generated by revolving a shallow circular arc around the axis parallel to its chord was analyzed for stability in [18] using the theory of shallow anisotropic shells and assuming membrane precritical state. This approach was earlier applied to isotropic shells of similar geometry [7,15,20].The present paper continues the publications [14,18,19]. We will outline a procedure for stability analysis of anisotropic shells of revolution with an arbitrary meridian and a nonlinear precritical state.…”

“…The present paper continues the publications [14,18,19]. We will outline a procedure for stability analysis of anisotropic shells of revolution with an arbitrary meridian and a nonlinear precritical state.…”

“…For this coordinate, we take the angle j that is reckoned from the initial meridian along the circumference so that the coordinate systems a a 1 2 , ,z is right-handed. Since buckling modes of shells made of materials with one symmetry plane are such that the axes of the dents do not coincide with the coordinate axes, the solution can be expanded into a Fourier series of a complete system of trigonometric functions or into complex Fourier series [18,21]. The latter series is preferable as more compact.…”

“…We will also use the numerical discrete-orthogonalization method [3] to solve inhomogeneous and homogeneous boundary-value problems. This method [5,6] was widely used in calculations [4,8].We will use the procedure for stability analysis of the same shells as in [18]. This will allow us to evaluate the applicability of the approximate method, which was also used in [7,15,20].…”

“…We will use the procedure for stability analysis of the same shells as in [18]. This will allow us to evaluate the applicability of the approximate method, which was also used in [7,15,20].…”