Closed-Form Solution of Axisymmetric Slender Elastic Toroidal Shells

Toroidal shells are widely used in structural engineering. The governing equations of toroidal shells are very complicated because of its variable coefficients with singularity. To find their analytical solution, traditionally, the complex form governing equations were proposed and some useful solutions were obtained. Unfortunately, no any closed-form solution has even been obtained for either general or slender toroidal shells. This paper focus on a special case of toroidal shells, i.e., slender symmetrical toroidal shells. For the first time, the closed-form solution of this kind of shell has been successfully obtained from displacement form governing equations. The closed-form solution is demonstrated for the example of thermal compensation devices. The correction of well-known Dahl formula for slender toroidal shell has been proposed based on the solution obtained in this paper.     

Equivalent Mohr–Coulomb and generalized Hoek–Brown strength parameters for supported axisymmetric tunnels in plastic or brittle rock

The evaluation of equivalent Mohr–Coulomb (M–C) strength parameters to the prototype Hoek–Brown (H–B) ones for tunnels has been tackled in different ways for many years. The extension of the H–B criterion to the generalized one has made the challenge even greater. Most of the latest methods did not account for the effect of the support pressure and none gave formulae for equivalent parameters of supported or brittle rock. Here, an almost exact explicit solution for the evaluation of the critical pressure, of a tunnel in a rock mass satisfying the generalized H–B criterion, is initially investigated. Then, formulae are derived for the evaluation of equivalent parameters, of either elastoplastic or elastic–brittle plastic rock. They are based either on a best fitting procedure of the two envelopes or on the equation of selected responses of the models. Supported tunnels in equivalent M–C rock masses are then validated against those excavated in the prototype H–B rock masses.     

An energy approach for predicting springback of metal sheets after double-curvature forming, Part I: axisymmetric stamping

In our previous study (Xue P, Yu TX, Shu E. International Journal of Materials Processing Technology 1999; 89-90:65-71.), based on the membrane theory of shells of revolution and an energy method a mechanics model and corresponding analytical procedure have been proposed to predict springback of circular and square metal sheets after a double-curvature forming operation. The strain hardening of the material is incorporated into the mechanics model. In the present paper, the method is extended to the cases, in which bending effect, as well as bending-and-unbending effect are taken into account. It is shown that the procedure developed is capable of quantitatively predicting the strain distribution and springback of metal sheets after axisymmetric stamping with a relatively minor effort of calculation and a good accuracy. The effect of stretching force applied at the edge of the plate on springback is also considered. Excellent agreement is found between the theoretical prediction of springback and experiment results.     

Axisymmetric Deformation of a Pressurized Thin Elastic Membrane with Nonuniform Thickness

Many circular elastic membranes used in practice involve nonuniform thickness, which is a main source of error that affects the functionality of the membrane. An analytical framework is developed in this paper to solve the deflection profile of a pressurized circular membrane with nonuniform thickness. The nonlinear solution is established within the deformed configuration. The results are compared with finite-element simulation with good agreement.     

Axisymmetric Vibrations of Reinforced Orthotropic Shallow Spherical Caps

Axisymmetric vibrations of reinforced shallow. spherical caps manufactured from orthotropic materials are considered. The closed form solution is obtained for the natural frequency of the cap with a clamped and immovable circular edge by assuming that the motion component parallel to the cap boundary plane (in plane) is negligible. Parametric studies are performed to assess the effect of various geometric and structural parameters on the natural frequency of the cap and, most importantly, to identify the most influencing parameters of the problem. From the generated data, it is concluded that the national frequency increases with increasing extensional stiffness and eccentricity of reinforcements and to a lesser extent with increasing bending stiffness of reinforcements. Other important parameters include the base circle radius and the initial rise of the cap.     

Axisymmetric pressure boundary loading for finite deformation analysis using p-FEM

Follower loads, i.e. loads which depend on the boundary displacements by definition, frequently occur in finite deformation boundary-value problems. Restricting to axisymmetrical applications, we provide analytical and numerical solutions for a set of problems in compressible Neo-Hookean materials so to serve as benchmark problems for verifying the accuracy and efficiency of various FE methods for follower load applications. Thereafter, the weak formulation for the follower-load in 3-D domain is reduced to an axisymmetrical setting, and, subsequently, consistently linearized in the framework of p-FEMs, exploiting the blending function mapping techniques. The set of axisymmetric benchmark solutions is compared to numerical experiments, in which the results obtained by a p-FEM code are compared to these obtained by a state-of-the-art commercial h-FEM code and to the “exact” results. These demonstrate the efficiency and accuracy of p-FEMs when applied to problems in finite deformations with follower loads.     

General Axisymmetric Active Earth Pressure by Method of Characteristics—Theory and Numerical Formulation

Active pressure for axisymmetric problem under general conditions is formulated and solved by an iterative finite difference solution of the characteristic equations in the present paper. A general lateral stress coefficient is used instead of the Haar-von Karman hypothesis and numerical modeling has also confirmed that Harr-von Karman hypothesis is a reasonable assumption for the present problem if sufficient wall movement is allowed. The shape of the failure zone and variation of active pressure with depth and wall friction are investigated. Some interesting results on the active pressures arising from the arch action are found. Principle of superposition of the effect of soil weight, surcharge, and cohesive strength is discussed. Finally, the results in the present study are applied to bore pile construction and it is demonstrated that no casing for construction is required under some conditions.     

Dual boundary element method for axisymmetric crack analysis

In this paper a dual boundary element formulation is developed and applied to the evaluation of stress intensity factors in, and propagation of, axisymmetric cracks. The displacement and stress boundary integral equations are reviewed and the asymptotic behaviour of their singular and hypersingular kernels is discussed. The modified crack closure integral method is employed to evaluate the stress intensity factors. The combination of the dual formulation with this method requires the adoption of an interpolating function for stresses after the crack tip. Different functions are tested under a conservative criterion for the evaluation of the stress intensity factors. A crack propagation procedure is implemented using the maximum principal stress direction rule. The robustness of the technique is assessed through several examples where results are compared either to analytical ones or to BEM and FEM formulations.     

Plane Strain and Axially Symmetric Consolidation in Unsaturated Soils

A method is presented for the analysis of coupled consolidation in unsaturated soils due to loading under conditions of plane strain as well as axial symmetry. The method is based on the transformation of the governing differential equations by the Fourier transform, when the soil system is deformed under plane strain conditions, or Hankel transform for problems exhibiting axial symmetry. The effect of such transformations is to simplify considerably the solution from a computational point of view. In addition, using these transformations the same differential equations can be used to analyze consolidation under both the above conditions. Results are presented to point out some aspects of the consolidation in unsaturated soils generated by the application of strip as well as circular loads.     

Thermal Stresses in Spherical Shells

This paper presents an iterative finite-difference technique for the analysis of axisymmetric spherical shells with variable wall thickness. The formulation is based on thin elastic shell theory. One-dimensional finite-difference points are used to discretize the shell into strips along the meridian, and an iterative technique is employed to determine the normal and meridional displacements. The stress resultants and bending stresses are then evaluated. Unlike existing analytical and finite-difference techniques, the proposed method is applicable with ease to any variation in rigidity along the meridian, to general loading conditions, and to steep and shallow shells. Results are presented and compared with those of the finite-element method.