Membrane Stress State Research Articles

An enriched virtual element method for 2D‐3C generalized membrane shell model on surface

AbstractDealing with complex shell surfaces using the finite element method, we are often limited to simple geometric meshes such as triangles and quadrangles and have to refine the meshes to meet the calculation accuracy requirements, significantly increasing the calculation cost. The virtual element method (VEM), a new numerical method with high mesh flexibility, has been widely applied to many physical and mechanical problems. To the best of our knowledge, this method has yet to be studied for membrane shell models so far. In this paper, for the first time, we provide an enriched conforming VEM discrete scheme for the two‐dimensional three‐component (2D‐3C) generalized membrane shell (GMS) model proposed by Ciarlet et al. Furthermore, we prove the VEM discrete solution's existence and uniqueness for the GMS. Numerical experiments demonstrate that this discrete scheme is convergent and stable and can accurately represent the membrane stress state of conical, cylindrical, hyperbolical, and spherical shells clamped along a portion of their lateral faces. At the same time, we show the diversity of the grid subdivision. Thus, we develop the VEM for the GMS model successfully. In the future, we will continue to study the VEM for other shell models.

Manufacturing feasibility of a bend free ellipsoidal composite pressure vessel using automated fibre placement

Composite pressure vessels are attracting high research interest for their low weight and remarkable structural performance. Suppressing bending in pressure vessels offers increased pressure capacity by enabling the efficient membrane stress state. Variable angle tow (VAT) laminates have been shown theoretically to achieve this stress state in composite ellipsoids. This paper presents steps towards manufacturing such vessels using automated fibre placement (AFP). It was found that the bend free design is suited to tanks with diameters greater than ≈1400 mm when considering the VAT limitations of typical AFP equipment. When compared to a perfect ellipsoid, the addition of a polar boss reduced the portion of the tank satisfying the bend free criteria by 29.6 %, but also reduced deformation by 28.9 %. The layup considering overlaps was optimised, and manufacturing simulation and coupon manufacturing trials were used to verify thickness and steering radius predictions and visualise defect formation.

Momentless design of variable stiffness composite cylindrical vessels with elliptical heads

Membrane stress state is favorable for pressure vessels to resist internal pressure since the stress through shell thickness is constant. However, bending stresses resulted from bending moments of pressure vessels will destroy the favorable membrane stress state leading to nonuniform stress through shell thickness. Thus, suppressing bending moments in pressure vessels will be an efficient way to achieve improvement of load carrying capacity. A momentless design of variable stiffness composite cylindrical vessels with elliptical heads which are widely used in engineering has been presented in this study. The momentless governing equations concerning both the elliptical head and junction region of the pressure vessels under uniform pressure are derived. Numerical analyses are carried out on the pressure vessel models with different configurations. The variable angles of fiber orientations over these models are explicitly provided. The stress resultants and deformation behaviors together with the curvature changes of these models have been checked and compared. Results show that based on the variable fiber angles over the structure obtained from the presented mathematical design methodology, the bending moments are successfully suppressed in the variable stiffness composite pressure vessels. The momentless design method will ensure the pressure vessels to service in an efficient membrane state.

Microlattice Metamaterials with Simultaneous Superior Acoustic and Mechanical Energy Absorption.

The advent of 3D printing brought about the possibilities of microlattice metamaterials as advanced materials with the potentials to surpass the functionalities of traditional materials. Sound absorbing materials which are also tough and lightweight are of particular importance as practical engineering materials. There are however a lack of attempts on the study of metamaterials multifunctional for both purposes. Herein, wepresent four types of face-centered cubic based plate and truss microlattices as novel metamaterials with simultaneous excellent sound and mechanical energy absorption performance. High sound absorption coefficients nearing 1 and high specific energy absorption of 50.3 J g-1 have been measured. Sound absorption mechanisms of microlattices are proposed to be based on a "cascading resonant cells theory", an extension of the Helmholtz resonance principle that wehave conceptualized herein. Characteristics of absorption coefficients are found to be essentially geometry limited by the pore and cavity morphologies. The excellent mechanical properties in turn derive from both the approximate membrane stress state of the plate architecture and the excellent ductility and strength of the base material. Overall, this work presents a new concept on the specific structural design and materials selection for architectured metamaterials with dual sound and mechanical energy absorption capabilities.

Application of flexural and membrane stress analysis to distinguish tensile and compressive moduli of biologic materials

Three-point bending is often used during the mechanical determination of tissue material properties. When taken to failure, the test samples often experience high deformations. The objective of this study was to present beam and plate theories as analytical tools for determining tensile and compressive elastic moduli during the transition from flexure to membrane stress states. Samples of cartilage, a highly flexible connective tissue having differing tensile and compressive moduli, were tested. Three-point bending tests were conducted on auricular (ear) and costal (rib) cartilage harvested from pigs. The influence of span length variation and Poisson's ratio assumptions were statistically assessed. Tensile elastic moduli of the ear (3.886 MPa) and rib (6.131 MPa) were derived from high-deformation bending tests. The functional assessment described here can be applied as a design input approach for tissue reconstruction and tissue engineering, considering both hard and soft tissue applications.

Compressive membrane action in RC one-way slabs

The static boundary conditions of reinforced concrete members often prevent longitudinal expansion by way of friction, restraints, etc. This restraint to the longitudinal expansion of reinforced concrete members induces membrane stresses which can cause compressive membrane action (CMA), resulting in an increase in the structural capacities. In the literature CMA is often referred to as arching action. In this paper, the results of NLFE analyses of one-way reinforced concrete slabs with respect to arching action and composite action in bending due to the membrane stress state are investigated, interpreted, and compared with the thrust line. Using equilibrium conditions, algebraic relationships for the structural behaviour can be derived and compared with the analytical results. Furthermore, the influence of CMA on various parameters, in particular steel stress σsx, mid-slab deflection w, and slenderness h/l, is illustrated.

ELASTIC STABILITY OF THE SHELL CUT OUT FROM A CIRCULAR CYLINDER UNDER CANTILEVER-SUPPORTED CONDITION

In early days as in the 1940s, a partial cylindrical concrete shell (often referred to as long shell) was frequently used as a barrel roof structure. A simple guide was proposed as to the buckling of a long shell in the ASCE Manuals of Engineering Practice (1952), by making use of the theoretical buckling load of a circular cylinder subjected to the action of uniform axial pressure. It is to be noted that a strict membrane stress field is difficult to be maintained in a barrel-roof-type shell, because the number of stress unknowns is unfit for the number of available boundary conditions. Accordingly, for a long shell, use has been made of approximate membrane stress states in a “shallow” partial cylinder under external lateral pressure in not only a linear stability analysis but also a non-linear large deformation analysis. In a previous paper, the present author has shown that a cantilever-supported shell segment cut out from a circular cylinder with particular shapes and boundaries is able to satisfy the condition of a membrane stress field, under the action of a variety of loads; by assuming strain-displacement relations stay linear until buckling takes place, the stability of the shell segment can be analyzed as an eigenvalue problem. To be examined in this paper are two examples that are regarded as being typical and comprehensive. Each loading of self-weight and earthquake excitation is considered for varying thicknesses and shapes. Initiated with nonlinear equilibrium equations which are known as Love's Formulae and referred to as “general case” by Timoshenko, simplification is made by estimating non-linear terms with reference to the order of magnitude. Based on the assumption that the linearity of load-displacement relations remains valid until buckling takes place, above equations are led to equilibrium relations valid for an early post-buckling stage. These are made discrete through FEM formulation in Galerkin type weak form, being led to a system of equations in conformity with a linear eigenvalue problem. Further study is made to see how each parameter influences on partial matrices appearing in the overall FEM formulation. (In appendices at the end of the paper, the results of an analysis on an example of eigenvalue problem are compared with the results from the classical theory of elastic buckling, leading to perfect coincidence.) Linear buckling analyses are then conducted for each of the two load cases as eigenvalue problems by varying thickness and shapes as parameters. Results are shown for the load factors of buckling as contour lines in a plane of parameters; radial displacements of free edge lines are drown overlapped, representing modes of buckling for each combination of the parameters. It is intended to show general buckling characteristics of the shell in the form of graphics over the parameters varying widely. Finally, a large deformation step-by-step analysis is carried out, by converting the non-linear weak form led from the calculus of variation into a matrix form by making use of the predetermined values in the preceding step. An analysis is repeated for each load level, modifying the fore-going matrix step-by-step, until the displacement converges. Consequently, it is shown diagrammatically that buckling occurs at the minimum load level that the displacement does not converge but diverges, and that a large displacement appears suddenly with the same mode as that of linear buckling. Material behavior is assumed to be linearly elastic, throughout the paper.

Creep assessment in hyperelastic material by a 3D Neural Network Reconstructor using bulge testing

A new methodology based on a Neural Network approach is developed for three-dimensional reconstruction of pseudo-hemispherical geometry of a hyperelastic specimen during a creep bulge test. On the basis of this procedure, the inflated membrane stress and strain states are established to determine the material mechanical properties. A new and economic experimental apparatus, used to conduct the bulge test, is provided with a sliding crossbar for the dome image acquisition in order to detect its strain state. A pressure regulator is used to ensure constant pressure, essential for creep. The artificial neural network was trained by using the position (x, y, z) of the membrane points at different values of the pressure measured in the bulge test chamber. The main characteristics of the system, which consist of the experimental apparatus and the neural post-processing, are accurate characterization of the materials and the ability to analyze materials with moderate anisotropy. This method has been verified on carbon black-reinforced SBR. The hyperelastic parameters that were obtained are in agreement with the values reported in scientific literature.

Evolution and verification of a kinematic hypothesis for splitting of the strain energy

Splitting of the strain energy into its “non-membrane” and membrane percentage provides insight into the load-carrying mechanism of structures, subjected to proportional loading. It may be useful, for example, for sensitivity analysis of the initial postbuckling behavior of beams, arches, plates, and shells, and assemblies of such structures. The task of this work is to determine this percentage without computing insignificant numbers such as the values of the strain energy and its membrane part. It is hypothesized that this percentage is proportional to the acceleration of a fictitious particle, moving along a curve on the unit sphere. The curve is described by the vertex of the normalized “fundamental eigenvector” of the so-called “consistently linearized eigenvalue problem”. The proportionality factor is obtained from the initial condition for the “non-membrane” percentage of the strain energy, hypothesized as twice the initial velocity of the particle. The lower bound of this factor signals the constancy of this percentage with increasing load, whereas the upper bound indicates a monotonic increase or decrease up to its ab initio predictable value at a stability limit or to an unphysical asymptotic limiting value. The proof of the universal validity of the two hypotheses begins with their verification for the special cases of a membrane stress state and pure bending. The assertion that this is a sufficient condition for the universal validity of these hypotheses is subsequently verified for an example with a monotonically increasing “non-membrane” percentage of the strain energy. It is finally confirmed by an indirect proof of their validity for a non-monotonic course of this percentage. A by-product of this work are conditions for extreme values of the stiffness of structures, subjected to proportional loading.

The buckling sphere: A symbiosis of mechanics and geometry

The basis of this work is a novel symbiosis of mechanics of solids and spherical geometry to quantify and illustrate the variation of the “non-membrane” percentage of the strain energy in the prebuckling region of linear elastic beams, arches, plates and shells, and structures assembled of such one-dimensional and two-dimensional members. The zenith angle of an arbitrary point of a specific curve on an octant of the unit sphere, called buckling sphere, is related to this energy percentage. For the limiting case of buckling from a membrane stress state this curve degenerates to a point, characterized by zero values of both spherical coordinates. For all other stress states the azimuth angle increases with the proportionally increasing load. Its magnitude at the stability limit correlates with a global quantity that depends on both the “non-membrane” deformations and the stiffness of the structure at incipient buckling. The azimuth angle is computed with the help of the so-called consistently linearized eigenproblem, which is solved by means of the Finite Element Method. This eigenvalue problem is the basis for a hypothesis for the “non-membrane” percentage of the strain energy. In the theoretical part of the paper, the concept of the buckling sphere is presented. The subsequent numerical investigation consists of four examples, referring to buckling from a membrane stress state, a pure bending stress state, and a general stress state. The practical motive for this research is the intention to investigate the influence of “non-membrane” action just before buckling on the initial postbuckling behavior of elastic structures.

一対の内湾曲線で薄肉・厚肉に領域分割された部分円筒シェルの構造的特徴

In early days as in the 1940s, a partial cylindrical concrete shell (long shell) was frequently used as a barrel roof structure. Many detailed investigations have been carried out, since the advent of ASCE Manuals of Engineering Practice in 1952. As a result, it has been found difficult to realize a strict membrane stress field in the barrel-roof-type shell, because the number of stress unknowns is unfit for the number of available boundary conditions. In a previous paper, the present author showed that a shell segment cut out from a circular cylinder with particular shapes and boundaries was able to satisfy the condition of membrane stress field. In this paper, a segment of partial circular cylinder is investigated, which is partitioned into three parts, by a pair of concave curves into the central thick shell zone with both fixed boundaries parallel to the generator, and two outer thin shell zones. The left half of the circular shell segment under consideration is taken to be the mirror image of the right half, or vice versa. Each of the left and right thin shell zones has shapes and boundaries that satisfy the requirement of a membrane stress state, in a way similar to the case in the previous paper. The central thick shell zone, on the other hand, behaves like an arch in which breadth and depth vary in the circular direction, keeping a constant cross-sectional area. It is noted that similarity exists with regard to the differential equations which relate displacement (denoted as V) in the circular direction with the displacements in the normal direction (denoted as W), between each circular cylinder in membrane stress states and the circular arch with a varied cross-section. This fact assures continuity of displacements along the concave partition lines. Thus, given a method of distributing suitable thickness to each shell zone, the outer thin shell zones in membrane stress states are cantilever-supported by the inner thick shell zone treated as an arch with varying cross-section along the circular direction. After solving the outer thin shell zones by a membrane theory presented in the previous paper, and after introducing pertinent parameters to describe the adequate shell-thicknesses with related shell shape, the thick shell zone is analyzed by solving a set of differential equations on the basis of the arch theory led by eliminating the longitudinal distribution in the general shell theory on a circular cylinder (the Flügge's Theory), including these parameters. The results of these parametric analyses are herein presented, and it is confirmed that the partial circular cylinder partitioned into a thick shell and two thin shells by a pair of concave curves can be solved as the Membrane Shell-Arch structure theoretically. For comparison, a typical example is selected with consideration of the results of above parametric analyses, and is analyzed by the Finite Element Method of Weak Form, equivalent to the Galerkin Method, by minimizing directly the total potential energy of a general circular cylindrical shell in which finite elements are able to express not only membrane deformations, but also bending deformations. This comparison with the FEM results indicates the adequacy of the proposed Membrane Shell-Arch Theory. From another point of view, the method introduced here serves as a means of determining an economical thickness distribution, in the design of a shell roof to cover a large space.

High-Frequency Dynamic Analysis of Plates in Thermal Environments Based on Energy Finite Element Method

The energy density governing equation to analyze the high-frequency dynamic behavior of plates in thermal environments is derived in this paper, in which the thermal effects are considered to change the membrane stress state and temperature dependent material properties of plates. Then the thermal effects on the energy reflection and transmission coefficients are dealt with hereof. Based on the above, an EFEM (energy finite element method) based approximate approach for the energy analysis of coupled plates under nonuniform thermal environments is proposed. The approach could be conducted by three steps: (1) thermal analysis, (2) thermal stress analysis, and (3) forming element matrixes, joint matrixes, and the whole EFEM formulation for the energy analysis. The same mesh model is used for all the three steps. The comparison between EFEM results and classical modal superposition method results of simply supported plates in various uniform thermal environments and coupled plates in nonuniform thermal environments demonstrated that the derived energy governing equation and the proposed approach described well the smooth time- and locally space-averaged energy density. It is found that the distributions and levels of energy density are affected by thermal effects, and the variation trends are related to exciting frequency.

The use of a particle method for the modelling of isotropic membrane stress for the form finding of shell structures

The best known isotropic membrane stress state is a soap film. However, if we allow the value of the isotropic stress to vary from point to point then the surface can carry gravity loads, either as a hanging form in tension, or as a masonry shell in compression. The paper describes the theory of isotropic membrane stress under gravity load and introduces a particle method for its numerical simulation for the form finding of shell structures.

円筒から切り出されたシェルの片持ち支持条件下における膜応力的特徴

A cantilever-supported shell that is cut out from a circular cylinder is investigated. It is found that the shell with the specified boundary shape has a membrane stress field, which is obtained by solving analytically the differential equations. In order to confirm the results, typical examples are solved numerically using FEM based on Galerkin type weak form. Then, the effect of the plate bending neglected by membrane theory is investigated. Furthermore, it is verified that the influence caused by difference between the idealized boundary condition of the membrane theory and the exact fixed boundary condition of FEM is very small. Therefore, this shell is said mostly in membrane stress state, and has shell-like advantage on shape, space and ecomomy.

Buckling Analysis of Short Cylinders of Uniform Thickness under Uniform External Pressure

The failure of elastic perfect cylinders under uniform external pressure is caused by circumferential buckling. The effects of the cylinder length, of the restraining conditions at boundaries and of the length of the pressurized zone all have great influences on the critical buckling load. In the European Standard for Shells EN 1993-1-6 (2007), the critical circumferential buckling stress is obtained by introducing an external buckling factor which takes the influence of different boundary conditions into account. However, the restraint of the meridional rotation at boundaries was not considered separately for the factor. For short cylinders, any restraint of the boundaries affects the buckling mode and buckling strength greatly. In this paper, the effect of meridional end rotations on the buckling strength of short cylinders was studied. The pre-buckling membrane stress state given by membrane theory was assumed for all the analyses.

Bifurcation buckling from a membrane stress state

SUMMARY A membrane stress state may be considered as a general stress state in which mechanical constraints have been imposed. Therefore, it would appear that there are mathematical subsidiary conditions for bifurcation buckling from such a stress state. In this work, such conditions are derived in the frame of the Finite Element Method. The basic condition follows from disintegration of the second derivative of the mathematical formulation of the so-called consistently linearized eigenproblem with respect to a dimensionless load parameter. It is used for deriving another condition, characterized by the vanishing of a particular bilinear form. Linear stability analysis and bifurcation buckling from linear prebuckling paths are two special cases for which this condition is satisfied. Sensitivity analysis of bifurcation buckling of a two-hinged arch, subjected to a uniformly distributed static load, by varying the geometric form of its axis serves the purpose of non-trivial verification of the derived condition for the special case of a thrust-line arch. Copyright © 2011 John Wiley & Sons, Ltd.

Membrane theory treatment of eccentric flows in concentric hoppers

This paper presents an initial study of the effects of an accidental eccentric flow channel that sometimes develops in the conical hopper of a metal storage silo. A simple assumed pressure regime is adopted, based on studies of eccentric discharge in cylindrical silos, and the structural actions are analysed using shell membrane theory. The results are verified against a finite element analysis. A set of equations is derived which give the complete membrane stress state in the hopper under such an unsymmetrical pressure regime. Under symmetrical loading, silo hoppers are subject to biaxial tension, with failure normally governed by material rupture due to tensile meridional membrane stresses. It is found that the eccentric flow channel leads to dramatically increased meridional membrane stresses at the transition. Compressive membrane stresses are also found near the hopper outlet and at the sides of the flow channel near the transition. A reasonably close correlation is found between the derived membrane theory equations in a right circular conical hopper and linear elastic finite element predictions.

Buckling of Thin Cylindrical Shells Under Locally Elevated Compressive Stresses

Thin cylindrical shells used in engineering applications are often susceptible to failure by elastic buckling. Most experimental and theoretical research on shell buckling relates only to simple and relatively uniform stress states, but many practical load cases involve stresses that vary significantly throughout the structure. The buckling strength of an imperfect shell under relatively uniform compressive stresses is often much lower than that under locally high stresses, so the lack of information and the need for conservatism have led standards and guides to indicate that the designer should use the buckling stress for a uniform stress state even when the peak stress is rather local. However, this concept leads to the use of much thicker walls than is necessary to resist buckling, so many knowledgeable designers use very simple ideas to produce safe but unverified designs. Unfortunately, very few scientific studies of shell buckling under locally elevated compressive stresses have ever been undertaken. The most critical case is that of the cylinder in which locally high axial compressive stresses develop extending over an area that may be comparable with the characteristic size of a buckle. This paper explores the buckling strength of an elastic cylinder in which a locally high axial membrane stress state is produced far from the boundaries (which can elevate the buckling strength further) and adjacent to a serious geometric imperfection. Care is taken to ensure that the stress state is as simple as possible, with local bending and the effects of internal pressurization eliminated. The study includes explorations of different geometries, different localizations of the loading, and different imperfection amplitudes. The results show an interesting distinction between narrower and wider zones of elevated stresses. The study is a necessary precursor to the development of a complete design rule for shell buckling strength under conditions of locally varying axial compressive stress.

On the Forces Driving Plate Tectonics: Inferences from Absolute Plate Velocities and Intraplate Stress

Summary Absolute plate motions and intraplate stress both serve as tests of models for the forces acting on plate boundaries. Plate velocities relative to a presumedly fixed underlying mantle are calculated from the hypothesis that no net torque is exerted on the lithosphere. Intraplate stress is calculated by solving the equilibrium equations for thin elastic shells in the membrane state of stress. The absolute velocity fields predicted from a wide assortment of physical and geological models are all very similar. While the global pattern of absolute velocity is probably close to those predicted by such models, the absolute motions do not therefore provide a strong test of the driving mechanism. Comparison of predicted intraplate stress with the long-wavelength features of the global stress field, however, as determined by in situ measurements, earthquake mechanisms, and stressinduced geological structures, does prove to be a powerful test of possible driving forces. All absolute velocity models have several interesting properties. Lithosphere in the equatorial half of the Earth is moving significantly faster than lithosphere in the polar half. Some connection to the Earth's rotation is implied since this statement is demonstrably untrue for co-ordinate poles much different from the geographic pole. Subducted slabs are characterized by a slow horizontal translation perpendicular to strike that is independent of the plate convergence rate, confounding attempts to explain the dip angles of Benioff zones in terms of a uniform vertical sinking and a variable absolute velocity for the overthrust plate. Ridges must migrate at a wide range of velocities relative to their underlying source of new lithosphere; such rapid migration may be a necessary but is not a sufficient condition for ridge jumps. Force models considered in velocity and stress calculations include driving forces at spreading centres and subduction zones and various parameterizations of drag at the base of the lithosphere. From the rms absolute velocities of individual plates, there is a weak indication that pull by subducted lithosphere at trenches is an important driving force and that drag may be greater beneath continental than oceanic lithosphere. The predicted intraplate deviatoric stress cannot match the well-determined stress fields in North America and Europe unless the driving force exerted at ridges is at least comparable in magnitude to other forces in the system. The mid-plate stresses are very sensitive to the nature of drag at the base of the lithosphere and thus measured stresses may ultimately provide a sensitive test of absolute plate velocities.

Effect of temperature on capacitive RF MEMS switch performance—a coupled-field analysis

Three-dimensional multiphysics finite element analysis (FEA) was performed to investigate the reliability of RF MEMS switches at various operational temperatures. The investigated MEMS capacitive switch consists of a freestanding metal membrane actuated by a bottom electrode coated by a dielectric film. Coupled-field simulations between thermal, structural and electrostatic domains were performed. The simulations show that temperature significantly changes both the membrane stress state and out-of-plane geometry. In particular, the membrane buckles when temperature increase, from room temperature, takes place. The buckling temperature, i.e. the upper bound to the operational temperature, is a function of manufacturing residual stress state, membrane initial out-of-plane profile and a mismatch in materials coefficient of thermal expansion. The analysis also shows that temperature reduction, from room temperature to −40 °C, causes an increase in pull-in voltage to values that could compromise the switch reliability as a result of charge build-up in the dielectric layer. Our analyses illustrate that by proper designing of the membrane out-of-plane profile, it is possible to keep the pull-in voltage, at all operational temperatures, within allowable values. This design feature of RF MEMS switches offers an effective way to achieve reliable pull-in voltages in applications where large temperature variations are expected such as in satellites and airplane condition monitoring based on wireless communication.