Dynamic response of curved pipes

The problem of the linear elastic response of a thin curved pipe subjected to a local impulsive loading is considered. A series solution based on the Sanders shell theory is developed in the toroidal coordinate system. This solution covers the case of a pipe with arbitrary circular curvature of the torus center line. The loading is represented as a double series in the geometric variables, and direct time integration is carried out using the Newmark method. Sample results are presented for loadings concentrated at the intrados and extrados for both short and long pipes.     

Shell analysis of thin-walled pipes. Part II – Finite element formulation

A finite element formulation is developed for the analysis of thin-walled pipes based on thin shell theory. The formulation starts with a Fourier series solution of the equilibrium equations developed in a companion paper and develops a family of exact shape functions for each mode. The shape functions developed are used in conjunction with the principle of stationary potential energy and yield a finite element that is exact within the assumptions of the underlying shell formulation. The stiffness matrix contribution for each mode n is observed to be fully uncoupled from those based on other modes m ≠ n. The resulting finite element is shown to be free from discretization errors normally occurring in conventional finite elements. The applicability of the solution is illustrated through examples with various loading cases and boundary conditions. A comparison with other finite element and closed form solutions demonstrates the validity and accuracy of the current finite element.     

Stability of a toroidal fluid-containing shell

This paper presents a solution to the stability problem of a sectorial toroidal shell subject to fluid loading. The solution is based on the Budiansky shell stability theory specialized for toroidal coordinates, and is developed using the two-dimensional differential quadrature method. Numerical results are presented and these results are compared with results from the Donnell shell theory, and with ones from a finite element method solution. The study establishes the suitability of the differential quadrature method for problems of shell stability.     

The modal analysis of a pipe elbow with realistic boundary conditions

A vibration analysis for the determination of the natural frequencies and the associated eigenmodes of a pipe elbow with end-flanges or tangent terminations was performed. A numerical investigation of this problem was achieved with a semi-analytic definition finite ring element and a commercial finite element code. To assess the accuracy of the numerical solution for the elbow vibration, an experimental modal analysis was performed on a curved and on a straight pipe. The responses were processed by a data acquisition system which performs a fast Fourier transform on the time histories to convert them from a time to frequency domain, these leading to the extraction of natural frequencies and mode shapes associated with the test-specimen. The results were compared with the corresponding ones from the numerical approach and discussion about the results completes the paper.     

Toroidal elastic stress fields for pressurized elbows and pipe bends

The zero order and first order stress fields are determined by toroidal elasticity methods for an elbow or pipe bend subjected to internal pressure. The methods of toroidal elasticity, first introduced in 1980 at London, England are here made explicit. These methods formed the basis for the numerical results presented in the earlier paper.

The problem considered in this paper is one of the more difficult of the ten unit problems for a pressurized elbow or curved pipe acted upon by end loads and seismic accelerations. All of these ten unit problems have now been solved. The present paper complements the earlier paper. The solution presented here was initially completed in 1978.     

A computer program for the elastostatics of a toroidal shell using the differential quadrature method

This paper presents a general analysis of the elastostatic problem of a sectorial toroidal shell. The solution to the problem is based on the Mushtari–Vlasov–Donnell shell theory and is developed using the two-dimensional quadrature method. A computer program incorporating the theory is attached, and an example of its application is given for the case of a fluid loading. Numerical results are determined and these are validated by a comparison with results obtained from a finite element solution.     

Stress fields produced by a vertical seismic force acting on a 90° elbow or pipe bend

A vertical component of seismic force acts upon a 90° elbow or curved pipe bend lying in an X-Y plane. The stress field, due to the force of magnitude Z, is established and consists of an initial stress field S(0) and a corrective stress field. The solution satisfies the equilibrium and compatibility equations of toroidal elasticity (Ref. 1).     

Stress and stability analysis of toroidal shells

A finite element formulation applicable to the general shell of revolution is presented for the stress and stability analysis of toroidal pressure vessels under hydrostatic pressure. Considering the follower force effect of the external pressure, linear bifurcation buckling loads and corresponding mode shapes have been obtained in both axially and equatorially symmetric as well as antisymmetric buckling modes. Calculated critical values are compared with results of other investigations.     

The elastic analysis of a dent on pressurised pipe

The elastic analysis of a dent on a pressurised cylinder is investigated. The predetermined shapes of the dent are grouped into three categories according to the physical geometry of the dent width, namely the local dent, the short dent, and the long dent. The induced bending stresses associated with the dent are evaluated using Sanders's non-linear shell theory. The geometry of the dent is treated as an initial geometric imperfection and incorporated in the expressions of strain. The non-linear cylindrical shell equations thus obtained are linearised and then solved by a Fourier series expansion technique. Comparison of theoretical results with the finite element approach is made for a pipe of assumed dimensions and pressure. Good agreement of the results is observed. The analytical approach of this paper is suitable for pipe or cylinder with a dent depth of up to five times the wall thickness.     

The thermal and mechanical behaviour of structural steel piping systems

The temperature, the deformation and the stress field in thermo-mechanical problems play a very important role in engineering applications. This paper presents a finite element algorithm developed to perform the thermal and mechanical analysis of structural steel piping systems subjected to elevated temperatures. The new pipe element with 22 degrees of freedom has a displacement field that results from the superposition of a beam displacement, with the displacement field associated with the section distortion. Having determined the temperature field, the consequent thermal displacement produced in the piping systems due to the thermal variation can be calculated. The temperature rise produces thermal expansion and a consequent increase of pipe length in the structural elements. For small values of the ratio of the pipe thickness to mean radius, the thermal behaviour can be calculated with adequate precision using a one-dimensional mesh approach, with thermal boundary conditions of an axisymmetric type across the pipe section. With this condition, several case studies of piping systems subjected to elevated temperatures and mechanical loads are presented and compared with corresponding results from commercial finite element codes. The main advantage of this formulation is associated with reduced time for mesh generation with a low number of elements and nodes. Considerable computational effort may be saved with the use of this finite pipe element.     

An elasto-plastic finite element for steel pipelines

An efficient finite element for the modeling of inelastic behaviour of three-dimensional pipe systems is presented. The formulation is based on a two-node pipe element with 12 degrees of freedom. The element consists of an elastic portion and two potentially plastic 3D hinges of zero-length lumped at both nodes. The behaviour of the plastic hinges is characterized using recently derived and experimentally validated plastic interaction relations for pipe sections. The normality condition of plasticity is applied to the analytically derived yield hyper-surface at the stress resultant level in order to approximately simulate material elasto-plastic behaviour. The element models shear deformation effects both in the elastic and plastic ranges. Thus, it is suitable for predicting pipe behaviour subjected to high shearing forces. The model captures the essential features of pipe behaviour using a remarkably small number of degrees of freedom and is particularly suited for the analysis of long pipeline systems. Solutions for simple problems are provided and compared to several other well-established elements in the ABAQUS library in order to assess the validity of the results and demonstrate their scope of applicability.     

Buckling analysis of an orthotropic thin shell of revolution using differential quadrature

A method is developed to predict the buckling characteristics of an orthotropic shell of revolution of arbitrary meridian subjected to a normal pressure. The solution is given within the context of the linearized Sanders–Budiansky shell buckling theory and makes use of the differential quadrature method. Numerical results for buckling pressures and mode shapes are given for complete toroidal shells. Both completely free shells and shells with circumferential line restraints are covered. The loadings considered consist either of uniform pressure or circumferential bands of constant pressure. It is demonstrated that the differential quadrature method is numerically stable and converges. For isotropic toroidal shells, good agreement is observed with previously published analytical and finite element results. New results for buckling pressures and mode numbers are given for orthotropic shells and for band loaded shells.     

Experiment and numerical simulation of hydro-forming toroidal shells with different initial structure

In this paper, the effect of the initial structure on formation of toroidal shells is investigated for avoiding wrinkling in the hydro-forming process for manufacturing large elbows. The experimental results show that no wrinkling occurs on a toroidal shell with an octagonal cross-section. However, there are some tiny wrinkles on a toroidal shell with a hexagonal cross-section. The deformation process was simulated using an explicit dynamic finite element code LS-DYNA. The simulation results are basically in agreement with the experimental results. The reason for wrinkling is discussed. Research results show that wrinkling can be prevented by selecting an appropriate initial structure.     

Curvature effect on stress concentrations around circular hole in opened shallow cylindrical shell under external pressure

Circular holes are necessary either for access or for the passage of piping, ventilation ducts, uptakes, and electrical systems. A hole can be found in a missile skin, aircraft fuselage, ship hatch, boilers and submersible pressure hulls, amongst other numerous situations. Stress concentration around holes in shells is of relevant concern in the design stage. In particular, precisely analyzing the stress distributed around holes of the shell is critical for the shell's structural design. This work attempts to obtain stresses around a circular hole cut in an opened shallow cylindircal shell under external pressure. The finite element method is applied to examine how curvature and thickness influence stress concentrations around the circular hole. According to those results, the shell with a larger curved angle θ can resist the external pressure loading more effectively. In addition, if the curved angle θ is fixed, a gradual increase of the shell's thickness implies a gradual decrease of the stress. In addition, design data sheets regarding the stress distribution around the hole for different curved angles of the shell are developed, providing a valuable reference for the designer of a shell structure.     

The integrally hydro-forming process of pipe elbows

In this paper, a hydro-bulging technology for manufacturing pipe elbows is proposed to overcome difficulties in manufacturing large diameter elbows using traditional technology. The section profile variation of the toroidal shell during the forming process is discussed. It is demonstrated experimentally that manufacture of a pipe elbow using integrally hydro-bulging technology is feasible.     

Nonlinear thermoelastic frequency analysis of functionally graded CNT-reinforced single/doubly curved shallow shell panels by FEM

In this article, the nonlinear vibration frequencies of functionally graded carbon nanotube-reinforced composite doubly curved shell panels under elevated thermal environment are numerically investigated using finite element method. The doubly curved carbon nanotube-reinforced shell panel has been modeled mathematically using higher-order kinematics theory and Green–Lagrange geometrical nonlinear strains. The properties of the individual constituents of the graded composite are assumed to be temperature dependent. In addition, the properties of the media are obtained based on the modified rule of mixture. The carbon nanotubes are dispersed nonuniformly through the thickness direction. The large deformation kinematic effects on the structural responses are counted by including all the nonlinear higher-order terms in the formulation. The desired nonlinear responses are computed numerically using our in-house computer code in conjunction with the direct iterative scheme. The convergence and the accuracy of the present numerical model have been checked by solving various numerical examples. Nonlinear mechanical responses were affected by several other design parameters and explored numerically for the thickness ratios, volume fractions, temperature loading, type of geometries, and type of grading under the uniform thermal environment.     

In-plane bending of a curved pipe or toroidal tube acted on by end couples

Stresses, strains and displacements are determined for a toroidal tube or pipe bend acted upon by end bending moments such that the deformation is in the plane of the tube. The methods of toroidal elasticity are used so that the solution satisfies compatibility equations. The analysis in this paper is limited to the first-order state resulting from application of the method of successive approximation.     

Ram bending of a cylindrical pipe in the elastic range

The problem of ram bending of a straight cylindrical pipe is considered. Separate shell theory and finite element method (FEM) solutions are presented. The loading is idealized as a set of pads of uniform radial pressure, and results are given for the elastic range. Particular attention is paid to the FEM solution characteristics and the pipe springback behavior. The present study is a necessary preliminary step to the full elastic-plastic solution of the problem.     

Exact yield hyper-surface for thin pipes

A yield hyper-surface for pipe sections subjected to combinations of normal forces, internal and external pressure, twisting moments, biaxial bending moments and biaxial shearing forces is developed. The formulation is based on the fully plastic capacity of the pipe as determined by the maximum distorsional energy density yield criterion. The solution is obtained by maximizing a lower bound analysis and yields a yield hyper-surface that is exact within the limitations of the formulation. The developments are expressed as universal non-dimensional relationships suitable for limit states design of elevated pipes, submerged pipes, offshore platforms and structural tubular steel members. Previously established interaction relations for bending moments, axial forces and internal pressure are recovered as a special case of the general solution. The merits of using the yield hyper-surface to characterize generalized plastic hinge behavior in elasto-plastic pipe stress analysis are presented.