Static analysis of reinforced thin-walled plates and shells by means of finite element models

In this paper, variable kinematic one-dimensional (1D) structural models have been used to analyze thin-walled structures with longitudinal stiffeners and static loads. These theories have hierarchical features and are based on the Carrera Unified Formulation (CUF). CUF describes the displacement field of a slender structure as the product of two function expansions, one over the cross-sectional coordinates, Taylor (TE) or Lagrange (LE) expansions were used here, and one along the beam axis. The results obtained using the refined 1D models have been compared with those from classical finite element analyses that make use of plates/shells and solids elements. The performances of classical and refined structural models have been compared in terms of accuracy and computational costs. The results show that the use of the LE over the cross-section allows the strain/stress fields to be evaluated accurately for all the structural components. The comparisons with the results obtained using the classical models highlight how, the use of 1D refined models, allows the number of degrees of freedom (DOF) to be reduced, meanwhile, the accuracy of the results can be preserved.     

Diagnosis of the load-carrying capacity of pipes containing a longitudinal crack when loaded by an axial force and internal pressure

A method is developed for evaluating the load-carrying capacity and safe life of pipes with a through longitudinal crack when loaded by an axial force and internal pressure. The method is based on the use of previously developed nomograms of strain and limit load. The first nomogram is calculated on the basis of constitutive relations of the theory of plasticity, while the second is calculated in accordance with a fracture criterion for biaxial loading. The proposed method makes it possible to determine one of three parameters — critical crack length, limiting pressure, or limiting axial load — without preliminary calculations if the other two parameters are already known. As an example, nomograms are presented for a steel 15G2 pipe having a diameter of 1420 mm and wall thickness of 21.5 mm and containing a longitudinal crack. Translated from Problemy Prochnosti, No. 12, pp. 16–23, December, 1994.     

Symmetricability of pressure stiffness matrices for shells with loaded free edges

For the case of free edges which are loaded, follower forces remaining normal to the middle surface of a shell throughout the deformation history do not have a load potential. In finite element analysis, this results in an unsymmetric pressure stiffness matrix. Depending on the structure of the available computer program, implementation of an equation solver permitting solution of unsymmetric simultaneous systems of algebraic equations may be a tedious task. This explains the significance of the topic of symmetricability of pressure stiffness matrices, turning out to be of special importance in the case of static buckling under the assumption of a linear prebuckling path. At first, incremental equations for tracing the nonlinear load–displacement path are derived. Thereafter, the buckling condition is deduced. Then, it is demonstrated that symmetrization of the pressure stiffness matrix is admissible if the so-obtained ‘buckling pressure’ differs ‘very little’ from the ‘buckling pressure’ resulting from an alternative symmetric ‘buckling matrix’, as is shown to be the case for a simply supported cylindrical shell with a free upper edge, subjected to hydrostatic external pressure. The alternative symmetric ‘buckling matrix’ is a consequence of deleting the virtual work term, causing the unsymmetry of the pressure stiffness matrix, in the expression for the external virtual work. A mechanical interpretation of this virtual work term is given. It is shown to be equal to the difference of virtual work of the original pressure load and of a ‘substitute pressure-field’, of the form of a Fourier series of the former. This explains why, normally, the buckling coefficient resulting from the ‘substitute pressure-field’ represents a good approximation to the buckling coefficient stemming from the original pressure load.     

Ductile failure of cylindrical bodies with axial cracks loaded by internal pressure

The article considers problems of finding the limit pressure leading to ductile failure of a cylinder with an axial crack. Actual formulas are obtained for a thin-walled shell and for cylinders with considerable thickness. The article shows their satisfactory agreement with the results of full-scale tests.Translated from Problemy Prochnosti, No. 2, pp. 16–20, February, 1990.     

Flow forming of thin-walled precision shells

Flow forming is an innovative form of cold and chipless metal forming process, used for the production of high precision, thin-walled, net-shaped cylindrical components. During this process, the length of a thick walled tube, commonly known as a preform, is increased with a simultaneous decrease in the thickness of the preform without any change in the internal diameter. Forming of the preform is carried out with the help of one or more rollers over a rotating mandrel. By a pre-determined amount of thickness reduction in one or more number of forming passes, the work material is plastically deformed in the radial direction by compression and made to flow in an axial direction. The desired geometry of the workpiece is achieved when the outer diameter and the wall of the preform are decreased, and the available material volume is forced to flow longitudinally over the mandrel. Over the last three and a half decades the flow forming technology has undergone several remarkable advancements. The versatility of the process makes it possible to produce a wide variety of axi-symmetric, nearer to the net-shape tubular parts with a complex profile using minimum tooling changes. In this review article, process details of flow forming have been elaborated. The current state-of-the-art process has been described, and future developments regarding research and industrial applications are also reviewed.     

External surface cracks in shells under cyclic internal pressure

The stress analysis and fatigue crack growth behaviour of a part‐through‐cracked double‐curvature thin‐walled shell is examined. An external surface crack is assumed to lie in one of the principal curvature planes of the shell, and to present a semi‐elliptical shape. The stress intensity factors (SIFs) along the crack front for different elementary opening stresses acting on the crack faces are determined through a three‐dimensional finite element analysis. Then approximate values of SIF in the case of a cracked pressure vessel are computed by employing the above results together with the superposition principle and the power series expansion of the actual opening stress. Finally, a numerical simulation procedure is carried out to predict the crack growth under cyclic internal pressure. Some results are compared with those of other authors.     

Free vibration analysis of reinforced thin-walled plates and shells through various finite element models

Abstract

This article deals with free vibration analysis of thin-walled structures reinforced by longitudinal stiffeners using refined one-dimensional (1D) models.The 1D theory, which is used in the present article, has hierarchical features and it is based on the Carrera Unified Formulation (CUF). The displacement field over the cross section is obtained by means of Taylor (TE) or Lagrange (LE) expansions. Finite element (FE) method is applied along the beam axis to obtain weak form solutions of the related governing equations. The obtained results are compared with those from classical finite element formulations based on plate and shell (2D), beam (1D), and solid (3D) elements that are available in commercial software. When solid formulation is used to build the FE solutions, stringers and skin are modeled with only 3D elements while, in the 2D-1D FE models, shell and beam elements are used for skin and stringers, respectively. Three benchmark problems are analyzed: a flat plate, a curved panel, and a thin-walled cylinder. When TE models are used, different orders of expansion, N, are considered, where N is a free parameter of the formulation. As far as Lagrange expansions are concerned, four-node (LE 4) and nine-node (LE 9) elements are used to build different meshes on the cross section. The results show that the present 1D models are able to analyze the dynamic behavior of complex structures and can detect 3D effects as well as very complex shell-like modes typical of thin-walled structures. Moreover, the 1D-CUF elements yield accurate results with a low number of degrees of freedom.     

On the stability of thin-walled, corrugated, circular cylindrical shells under external pressure

Summary Thin-walled, corrugated, circular cylindrical shells under external pressure have repeatedly buckled at much lower values than has been predicted from simple orthotropic shell theory, where a flexural mode of buckling is assumed, in-plane to the cross-section of the tube. In the present paper it is shown that there exists always a combined, torsional–flexural, out-of-plane mode of buckling, frequently corresponding to an extremely low critical pressure. The need to design these tubes also with respect to torsional stiffness is emphasized, and a formula for the buckling pressure is proposed. Dedicated to Professor Franz Ziegler on the occasion of his 70th birthday     

Dynamic short-wave buckling of thin-walled cylindrical shells upon local action of an external pressure pulse

We present the results of experimental investigations of dynamic instability (buckling) of thin-walled cylindrical shells upon local action of an external pressure impulse. In order to create the local short-term loading, we used pulsed radiation from a CO ₂ laser. The test pieces were smooth and reinforced shells made from aluminum alloys. The nature of the buckled wave in the treated region and the magnitude of the critical impulse correspond to the short-wave type of buckling in the shell and are determined by the amplitude of the pressure pulse. Axial static compression of the shell leads to a decrease in the critical impulse and an increase in the number of waves in the meridional direction within the treated region.Translated from Problemy Prochnosti, No. 4, pp. 36–43, April, 1995.     

Response of impulsively loaded cylindrical shells

Experiments were performed to identify the structural response modes of thin cylindrical shells, with and without internal pressure, subjected to external radial impulsive loads. For unpressurized shells the response modes consisted of dynamic pulse buckling followed by large inward deflections of the loaded surface. In shells with high internal pressure, these response modes were followed by an outward motion driven by the internal pressure. Energy balance equations were formulated which modelled the important features of the observed response mechanisms. These equations were then used to determine the shell load-damage relationships and to predict failure.