Creep analysis of cylindrical shell by a finite difference method

In this paper, the nodal line finite difference method (NLFDM) is extended to the analysis of a thin circular cylindrical shell, as shown in Fig. 1, undergoing free vibrations [1]. The material of the circular cylindrical shell is assumed to be linear elastic isotropic. The complexity of a theoretical analysis of a vibration problem depends largely on the number of degrees of freedom of the structural system in question. Since a thin circular cylindrical shell is a continuous system it has an infinite number of degrees of freedom. The NLFDM [10] transforms this continuous system into a system having a finite number of degrees of freedom. While carrying out the dynamic (or static) analysis by the NLFDM, the number of degrees of freedom depends on the number of the used nodal lines. The natural frequencies and modes are treated in more detail because they are basic to understanding the dynamic response under any kind of excitation. It will be shown that the number of natural frequencies and that of normal modes, for axial wave number different from zero, are each equal to three times the number of nodal lines used. The orthogonality relationships of the normal modes obtained by the nodal line finite difference method are also included. It should be remarked that these relationships are slightly different from those obtained by the finite element method; in fact the overall matrix of coefficients replaces the stiffness matrix, and any two different modes are orthogonal with resp ect to the unit matrix instead of the mass matrix.

In this paper, a new solution method is proposed for determining the natural frequency of a given mode for a finite-length circular cylindrical thin shell with a circumferential part-through crack. The governing equation of the cracked cylindrical shell is derived by integrating the line-spring model with the classical thin shell theory. The proposed method calculates the natural frequency from an initial trial to satisfy both the governing equations and appropriate boundary conditions through an optimization process. The initial trial is proposed to satisfy the governing equations by using the beam modal function to determine the modal wavenumbers and mode shapes of cylindrical shells in the axial direction, assuming the flexural mode shapes of cylindrical shells in the axial direction to be of the same form as that of a flexural vibration beam with the same boundary conditions. Four representative sets of boundary conditions are considered: simply supported (SS-SS), clamped-clamped (C-C), clamped-simply supported (C-SS), and clamped-free (C-F). Compared with the finite element (FE) method, the proposed solution method is verified to provide an accurate and efficient way to calculate the dynamic characteristics of both intact and cracked cylindrical shells.

This paper investigates the linear free vibration of axially moving simply supported thin circular cylindrical shells with constant and time-dependent velocity considering the effect of viscous structure damping. Classical shell theory is employed to express strain-displacement relation. Linear elasticity theory is used to write stress–strain relation considering Hook’s Law. Governing equations in cylindrical coordinates are derived using the Hamilton principle. Equilibrium equations are rewritten with the help of Donnell–Mushtari shell theory simplification assumptions. Motion equations for displacements in axial and circumferential directions are solved analytically concerning to displacement in the radial direction. As the displacement in the radial direction is the combination of driven and companion modes, the third motion equation is discretized using the Galerkin method. The set of ordinary differential equation obtained from the Galerkin method is solved using the steady-state method, which in practice leads to the prediction of the exact frequencies of vibration. By employing multiple scale method the critical speed values of a circular cylindrical shell and several types of instabilities are discussed. The numerical results show that by increasing the mean velocity, the system always loses stability by the divergence instability in different modes, and the critical speed values of lower modes are higher than those of higher modes. As well as the unstable regions for the resonances between velocity function fluctuation frequencies and the linear combination of natural frequencies is gained from the solvability condition of second order multiple scale method. The accuracy of the method is checked against the available data.

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

The two-dimensional problem of plane sound wave impulse scattering on the thin elastic circular cylindrical shell cracked along its infinite length. The shell is empty inside and surrounded by water from outside. The Kirchhoff - Love theory for description of elastic shell dynamic equilibrium is used. The shell defect influence on the echo-signal structure is investidated.

On the creep analysis of pressurized circular cylindrical shells

The extended Newton method combined with the method of finite-differences is shown to be a powerful means to the steady-state creep analysis of shells of revolution. The creep theory of von Mises type and the power creep law are assumed. As numerical examples, a simply-supported circular cylindrical shell with open ends and a clamped spherical shell, subjected to internal pressure respectively, are analysed. The results of calculation of the cylindrical shell, in particular, are compared with those of semi-infinite sandwich shell obtained by Rabotnov.

In this article the buckling of specially orthotropic laminated composite cylindrical shells is investigated. The nonlinear equilibrium equations based on the Sander's assumption of simplified nonlinear strain-displacement relations and the linearized stability equations for a circular cylindrical thin shell are considered. The equations include the rotations and transverse shear force. The resulting equations are improved compared to the Donnell stability equations. The mechanical and thermal buckling loads of a thin composite circular cylindrical shell based on Donnell and improved stability equations are obtained.

Buckling of cracked cylindrical thin shells under combined internal pressure and axial compression

This paper presents a numerical technique for elastic analysis of a thin circular cylindrical shell with end plates under uniform line load along a generator. This technique is based upon the discrete element method. An accurate set of the governing differential equations due to Vlasov is used. The derivatives with respect to the circumferential coordinate are replaced by finite difference relationships. The end plate is analyzed as a free body under unit concentrated edge loads and the resulting coefficients are used to satisfy continuity conditions at the shell-plate junction. A computer program to implement this technique is developed and results are compared with those published in the literature. A number of new results are presented.

Wind‐induced dynamic behavior of a silo‐like, thin circular cylindrical shell is studied to establish a rational analytical method. The ovalling oscillations are measured with elastic cylinders in a wind tunnel. It is shown that the ovalling characteristics in a smooth flow are quite different from those observed in a turbulent flow. Measurements of the time‐space correlations of the pressure fluctuations reveal that the pressure field on the cylinder surface has a convective nature in the smooth flow, while it is considerably random in the turbulent flow. On the basis of the expermental results, the dynamic response of elastic cylinders is analyzed by using a statistical method. As for the overall trend, the predicted results agree well with the experimental observations. Furthermore, the analytical results suggest that the dynamic response is strongly affected by the convection velocity of the pressure field; this is also consistent with the experimental results.

Dynamic response of a pre-stressed cylindrical shell to a moving load

Finite-difference (FD) schemes have been used widely for solving wave equations in seismic exploration. However, the conventional FD schemes hardly guarantee high accuracy at both small and large wavenumbers. In this paper, we propose an optimal time-space domain FD scheme for acoustic vertical transversely isotropic (VTI) wave modeling. The optimal FD coefficients for the second-order spatial derivatives are derived by approaching the time-space domain dispersion relation of acoustic VTI wave equations with the combination of the Taylor-series expansion and the sampling interpolation. We perform numerical dispersion analyses and acoustic VTI modeling using the optimal time-space domain FD scheme. The numerical dispersion analyses show that the optimal FD scheme has smaller dispersion than the conventional FD scheme at large wavenumbers, and also preserves high accuracy at small wavenumbers. The acoustic VTI modeling examples also demonstrate that the optimal time-space domain FD scheme has greater accuracy compared with the conventional time-space domain FD scheme for the same modeling parameters.

Reconstruction of a distributed force applied on a thin cylindrical shell by an inverse method and spatial filtering

A quantative comparison between the boundary integral equation (BIE) method and the finite difference (FD) method is presented in which each technique is applied to an elliptic boundary‐value problem (BVP) containing a boundary singularity. Two types of singularity have previously been analysed theoretically, namely those due to a discontinuous boundary potential, which we shall refer to as S1, and those due to a sudden change from the specification of boundary potential flux, an S2 singularity. In this paper the analysis is presented for a third type of boundary singularity, namely an S3 singularity: that arising from a discontinuous boundary flux. Such a condition is frequently encountered in the field of heat transfer where, for example, a system or pipe has a change of lagging material.In general, it is found that the BIE method is superior, with regards to the computational time required to achieve a certain degree of accuracy, over standard FD methods even when there is a boundary singularity. Further, the BIE method determines the solution near the singularity much more accurately than the FD method. The FD method does, however, have advantages for a very restrictive class of problems; for example, when the boundary conditions are of the Dirichlet type and the boundary geometry is rectangular. In this case an optimum relaxation parameter can easily be obtained. A soon as Neumann conditions are prescribed, the BIE is far more efficient than the FD, whatever the boundary geometry.It is concluded that, for fast, accurate solutions of general Laplacian boundary‐value problems, the BIE is appreciably superior to the FD and this is even more pronounced when there is a boundary singularity.