Membrane Theory Of Shells Research Articles

The Dynamic Behaviour of a Finite Length Straight Pipe Subject to Waterhammer : 1st Report, For Small Elastic Deformation of the Pipe Wall

A practical mathematical model capable of predicting the dynamic strains in a finite length straight pipe subject to waterhammer has been investigated. The membrane shell theory, neglecting the radial inertial force, is employed and calculated numerically by the "two step Lax-Wendroff method" after having transformed the displacement equation of motion into equations of conservation for strain and particle velocity. An experimental investigation was carried out for two oil-filled straight pipes whose boundary conditions were open-fixed/closed-fixed and open-fixed/closed-free, and pressure fluctuations and axial and circumferential dynamic strains were measured. The experimental and analytical results are discussed in detail, and the proposed mathematical model and method of calculation are found to be sufficiently appropriate to evaluate the dynamic response of the pipe wall.

The problem of the flow of a plastic material along a surface

The equations of the perfect, rigid-plastic membrane theory of shells whose thickness is variable and unknown, are studied. The equations were studied earlier in /1–5/ for the case of a smooth surface of flow or for regular modes; singular modes are discussed below. It is shown that in the case of such modes and equations in question split into two systems. The first system yields the forces and the shell thickness, and after this the velocities are found from the second system. The Tresca flow conditions and the maximum reduced stress are studied as examples.

The dynamic behaviour of a finite length straight pipe subject to waterhammer (1st report, For small elastic deformation of the pipe wall)

A practical mathematical model capable of predicting the dynamic strains in a finite length straight pipe subject to waterhammer has been investigated. The membrane shell theory neglecting the radial inertial force is employed and calculated numerically by the "two step Lax-Wendroff method" after having transformed its displacement equations of motion into equations of conservation form on strain and particle velocity. An experimental equations of motion into equations of conservation form on strain and particle velocity. An experimental investigation is carried out for the two oil-filled straight pipes whose boundary conditions are open-fixed / closed-fixed and open-fixed / closed-free, and pressure fluctuations and axial and circumferential dynamic strains are measured. The free, and pressure fluctuations and axial and circumferential dynamic strains are measured. The experimental and analytical results are discussed in detail, then the present mathematical model and calculative method are found to be sufficiently appropriate to evaluate the dynamic response of the pipe wall.

Numerical solution to pneumatic forming problems of axisymmetric superplastic thin shells

A numerical method is presented for analysis of the pneumatic forming process of axisymmetric superplastic thin shells. The governing non‐linear partial differential equations are derived from membrane shell theory, using total Lagrangian description and considering a hardening creep material model. The method of lines used in the computational algorithm, involves two steps: first the Cauchy problem in time is considered, then the boundary problem is solved by finite differences. The effectiveness of the proposed algorithm is illustrated by several examples.

EQUILIBRIUM OF A SINGULAR-STRESS SURFACE APPEARING IN A FIBRE-REINFORCED MATERIAL

General equilibrium conditions of a singular-stress surface appearing in a compressible material with internal kinematical constraints of inextensibility in the preferred direction are established for finite deformations as well as for incremental deformations by borrowing the equilibrium equations from the membrane theory of shells. Plane-deformation versions of those conditions are also obtained. To demonstrate some utility of the conditions, we applied them to a simple bifurcation problem, that is, an axisymmetric buckling of an axially fibre-reinforced circular hollow cylinder under axial compression.

Nonlinear torsion of thin-walled bars of variable, open cross-sections

An analysis of the nonlinear torsion of thin-walled bars of variable, open cross-section is presented. The geometrical description of the mid-surface of a bar and its strain expressions are based on membrane shell theory with internal constraints. The finite element method is used to formulate stiffness matrices for a thin-walled bar element. A comparison between the present method, other analytical solutions and experimental results is made by means of a numerical example.

On the general solution of stress equation for the membrane theory of shells of revolution

A new method is presented to solve the equation of Nemenyi-Truesdell, in the membrane theory of thin shells of revolution. It is an ordinary linear differential equation of the second order, for which resolution methods of Truesdell and Mitrinovitch are known. The proposed method is based on the transformation of the given equation into a linear integral equation of Volterra type. This allows to write the general solution of stress equation only in terms of equation of the meridian generating the shell.

The boundary integral equation method applied to shallow membrane shells of positive Gaussian curvature

The boundary integral equation method (BIEM) is developed for the analysis of shallow membrane shells with positive Gaussian curvatures. Shells with constant thickness and constant curvatures are considered. In the infinite domain, fundamental solutions are obtained which correspond to generalized concentrated tangential forces in the x and y coordinate directions. The Betti-Maxwell reciprocal theorem and Green's second identity are used to obtain the boundary integral equations of the solution presented. This approach, which is applied for the first time in membrane shell theory, seems to be a powerful alternative to domain type methods. Shells with various boundary conditions, loadings and arbitrary plan forms can be considered. It is also possible to add the effects of thermal fields and openings in the shells. The potential of the method is demonstrated by means of a worked example.

Elastic torsion of thin walled bars of variable cross sections

An application of the finite element method to the theory of thin walled bars of variable cross sections has been presented in this paper. A solution of this problem is based on the linear membrane shell theory with the application of Vlasov's assumptions. A bar is divided into elements along its longitudinal axis and then, a shell mid-surface of the element is approximated by arbitrary triangular Subelements. Nodal displacements of the element are assumed to be polynomials of the third order and the equivalent stiffness matrix is obtained. Calculated nodal displacements enable an analysis of normal and shearing stresses.

Effect of out-of-roundness on stresses in a cooling tower

The effect of a bulge in a hyperboloidal cooling tower on the stresses due to gravity and prescribed wind load is studied, by means of shell membrane theory. The geometry of a slightly out-of-round shell is derived, and the equilibrium equations are modified to include the effects of out-of-roundness. An iterative procedure is employed to solve the equilibrium equations numerically. Tables of stresses are given for a specific shell subjected to various moderate-sized bulges. The effect of a moderate-size bulge on the stresses in a cooling tower subjected to wind load and self weight is relatively small. The main effect of the out-of-roundness is to redistribute the stresses in the shell by a process of shedding load in the neighborhood of the bulge.

On work hardening and springback in plane strain draw forming

Based on a simple forcebalance method, an approximate analysis of the plane strain, deep drawing of sheet metal is carried out under the assumptions of Coulomb friction and the membrane theory of shells. Using this analysis with an estimate of the ratio of ultimate to yield load in plane strain enables us to predict the maximum work hardening of the part. This hardening is shown to depend upon the draw bead settings, the frictional conditions on the die and punch, the wall angle of the draw die, and certain uniaxial material properties, chiefly, the ratio of ultimate to yield stress, but also including the rate sensitivity and the anisotropy parameter. The limitations of this theory (mainly, the neglect of bending effects) are discussed. Comparison is made between predicted work hardening and measured yield strengths in a bumper facebar of dual phase steel. Springback of a gently curved part is calculated for preloads and postloads. It is shown that postloads are more effective, and that they can, in principle, reduce springback to low levels in any typical automotive sheet metal. In practice, it is difficult to transmit forces of sufficient magnitude to the region under the punch for certain materials.

A Numerical Analysis of the Hydraulic Bulging of Circular Disks Into Axisymmetric Dies

The hydraulic bulging of peripherally clamped, thin, circular disks into axisymmetric dies is studied by means of an incremental finite element method, based on membrane shell theory and formulated to account for finite strains and rotations. The material is treated as an isotropic, elastic-plastic solid obeying the von Mises yield criterion and plastic-potential flow law. The analysis was first performed for a flat-bottomed die, assuming Coulomb friction between the material and the die base. Experimental data were gathered from aluminium disks deformed into a die having a flat, thick glass base. The glass permitted a continuous assessment of the deformation profile and the contact boundary between the aluminium and the glass, using Moire´ topography. The agreement between the experimental observations and theoretical predictions is good.

Large strain analysis of an inflating membrane

This paper describes a numerical method for computing stresses and deformation, using the finite element, in a thin sheet under hydro-static pressure. Due to the complexity of the finite strain theory in formulation, the analysis is restricted to the membrane shell theory with axisymmetric deformation. The Lagrangian description of motion referred to a set of convected coordinates is used in the formulation. A computer program based upon the modified tangent stiffness method, which also verifies the equilibrium conditions has been written. We first applied it to a nonlinear elastic membrane deformed under pressure inflated to a stretch ratio greater than five. The computed results check with those in the literature. Further computation has been made for the stresses and deformation of metal sheets under hydro-static bulge pressure. For treating the inflation after the occurrence of instability, an alternate formulation to increase the volume enclosed by the shell instead of pressure is included in the present increment solution procedure.

Large elasto-plastic strain analysis of flanged hole forming

This paper describes an analysis of stresses and deformation using the finite element method in a flanged hole produced by punch stretching during a sheet metal forming process. Due to the complexity of the finite strain theory for inelastic materials, the analysis is restricted to the membrane shell theory with axisymmetric deformation. The Lagrangian description of motion referred to a set of convected coordinates is used in the formulation. A computer program based upon the modified tangent stiffness method with an equilibrium check has been written. For the purpose of verification of the program, we applied it to compute the finite inelastic deformation of a circular sheet caused by stretch from a hemispherical punch. The present computer solution is in excellent agreement with those in the literature. The program was applied to analyze flanged hole forming with four different punch shapes. The results reveal that the strain path during the forming process is not affected by the punch shape but the maximum punch load depends on the punch shape.

Elasto/Viscoplastic Analysis of Thin Circular Plates Under Large Strains and Large Deformations

In this paper the elasto/viscoplastic analysis of thin circular plates under large strains and large deformations is studied by the use of the finite-element method based on the membrane shell theory. As the constitutive relation for the materials Perzyna’s equation which in the plastic range takes into account the viscosity of the material is employed. The criterion for yielding used in this analysis is the von Mises yield theory. The geometric nonlinearity is treated with incremental method and the solutions at any stage are obtained by summation of the incremental values. The experiments are carried out for the thin circular aluminum plates, and the variations of the deformations and the stresses with loading rate are analyzed. The elasto/viscoplastic solutions from the prediction method agree fairly well with the values experimentally determined for the circular plates. The method can be used to generate plasticity solutions in a simple manner when stationary conditions are reached.

Longitudinal waves in liquid-filled tubes—I: Theory

Pulse propagation in axisymmetric elastic and viscoelastic tubes has been analyzed on the basis of extensions of existing one-dimensional bar-like and membrane-shell theories upon addition of viscous behavior of the components. A combined analytical-experimental procedure has also been developed for the determination of relevant mechanical properties. It has been found that, for the range of parameters considered, the fluid viscosity can be ignored, whereas that of the container plays a significant role; on the other hand, the radial inertia can be neglected. In a companion paper, the predictions of the theory are found to be in good correspondence with experimental results.

The Pumping Mechanism of the Nematode Esophagus

The radial orientation of the myofilaments in the nematode esophagus raises interesting questions as to how such a structure can function as a pump. A physical model of the esophagus of Ascaris lumbricoides was developed and the membrane theory of shells applied in order to relate the observed dimensional changes to myofilament force, pressure stresses, and membrane elastic constants. By stressing the excised esophagus passively with osmotic pressure, the esophagus was shown to be elastically anisotropic with the ratio of circumferential to longitudinal elastic constants, E(psi)/E(l) approximately 2.74. When this value was incorporated, the model predicted the ratio of the respective strains, epsilon(psi)/epsilon(l), to be 0.52 during an equilibrium contraction of the esophagus. This agreed with the experimental value, 0.46 +/- 0.10, measured during occasional, prolonged muscle contractions. When measured during normal pumping, on the other hand, the value of epsilon(psi)/epsilon(l) was 0 +/- 0.10. This indicated that a nonequilibrium condition normally occurs in which a greater myofilament force per unit area of lumen membrane is not balanced by internal pressure and therefore acceleration of the lumen contents and negative intraluminal pressure occurs.The pumping action of esophagi dissected from Ascaris was observed to be normally peristaltic and periodic. Contraction was initiated by a spontaneous depolarization that propagated at 4.0 +/- 0.20 cm/s along the esophageal membrane. A wave of localized increases in the internal pressure of the muscle and localized changes in external dimensions was observed. A subsequent spontaneous repolarization, which propagated at 5.8 +/- 0.23 cm/s, triggered relaxation of the muscle during which the localized pressure and dimensional changes returned to resting values. A mechanism was deduced in which fluid is drawn into and moved along the lumen by the wave of contraction. During the wave of relaxation, the lumen contents are pressurized and injected into the intestine by elastic restoring forces.

Free Vibration of Neutrally Buoyant Inflatable Cantilevers in the Ocean Environment

Free vibration analysis of the neutrally buoyant inflated cantilevers, made of plastic sandwiched films, is presented, accounting for the added inertia and nonlinear hydrodynamic drag. The significant feature of the analysis is the reduction of the shell equations (the membrane, Fliigge's, and Herrmann-Armenakas') into a single equation which is similar in form to that for a vibrating beam with rotary inertia effects. The natural frequencies obtained are compared with the experimental results and those predicted by the Rayleigh-Ritz method in conjunction with the Washizu and membrane shell theories. The analyses show, and the experimental program confirms, that Fltigge's shell equation in reduced form is capable of predicting free vibration behavior quite accurately. However, the reduction technique should be applied with care, since in several cases it leads to misleading results, e.g., in the case of the Herrmann-Armenakas theory, generally accepted to be one of the most elaborate.

Numerical Solution of a Deep-Drawing Problem

A rigid-plastic analysis of the axisymmetric deep-drawing problem is made using a special numerical technique. The effects of work-hardening, friction, and normal anisotropy have been included. Incremental strain theory is used to obtain results for both bending and membrane shell theories. These results are then compared. The authors feel that membrane theory gives very good results and should be used in the future to analyze any relatively thin cups.

Stress analysis of the aortic valve during diastole: Important parameters

Assuming that the aortic valve cusps have a very small thickness compared with its two other dimensions, the membrane shell theory is used to evaluate the displacements and stress resultants taking place in the cusps under diastolic pressure. Two geometries are examined; spherical and ellipsoid of revolution. The calculated values of these displacements and stresses are compared with recent data from the literature that have been obtained with various other theories. It is concluded that the results do not depend significantly on the theory used. Since the cusp material is anisotropic it is essential to introduce the directional anisotropy of the cusps as it affects greatly the displacements. The most striking finding is that very small changes in the values of the geometric parameters that only slightly alter the shape of the cusps introduce enormous changes on the stress resultants. An accurate three-dimensional record of the cusp surfaces during the diastolic cycle is the essential requirement for accurate determination of the valve stresses.