Membrane Stress Field Research Articles

Formation of post-buckling shear mechanisms in stiffened web panels of slender steel plate girders

A series of validated finite element models are used to investigate the formation of mechanisms in stiffened slender web panels of steel plate girders under pure shear. The prototype girder is based on the seminal tests by Basler and has a web slenderness of 267. Six model cases are analyzed with variations in web panel aspect ratio (1, 1.5, and 3), steel yield strength (Grades 36 and 50), initial imperfection magnitude, and flange thickness. When loaded in shear, the web panels exhibit a 3-stage response: (1) elastic behavior, (2) web mechanism formation, and (3) panel mechanism. In Stage 1, the web initially exhibits an elastic in-plane shear response with no distinct buckling bifurcation. During this stage, the web’s initial out-of-plane imperfections become engaged in second-order bending along its compression diagonal. Concurrently, tension develops in the opposite diagonal as a non-uniform membrane stress field, with highest intensities at locations where second-order bending is low. In Stage 2, a web mechanism is formed when connected bands of thru-thickness von Mises yielding develop across the tension diagonal. These bands emerge at locations where tension membrane stresses interact with locations of maximum second-order bending stress in the buckled shape. The shear load at the end of web mechanism formation is recommended as a target for plastic limit state design because it marks a significant decrease in shear stiffness. In Stage 3, von Mises yielding continues to saturate the web panel, and the bounding flanges and transverse stiffeners become increasingly engaged in load redistribution from the plastified web. Any hardening increase in shear resistance during the panel mechanism stage is modest in magnitude (up to ∼10% for some cases in this study) and is heavily dependent on the sizing of the flanges and stiffeners to carry redistributed forces.

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.

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.

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.

LOCAL AND INTERACTIVE POST-BUCKLING OF RHS THIN-WALLED MEMBERS — COMPARING A NEW SPECIAL BEAM FINITE ELEMENT WITH SHELL FE MODELS

This paper presents a special thin-walled plane beam finite element that accounts for the in-plane cross-section local deformation. The element is based on the superposition of a classical beam displacement field and of an additional field describing local effects, with an approximation on the local second-order membrane stress field. The theoretical formulation is summarized and an application of the resulting numerical tool to the post-buckling analysis of RHS thin-walled members with moderate local and global slenderness susceptible to both global and local buckling is then performed. Different types of analyses are presented (computation of critical bifurcation loads, geometrically non-linear analysis, geometrically and materially non-linear analysis). The results obtained with the proposed beam finite element are compared to values provided by shell FE models.

4절점 응축 셸 요소를 이용한 복합재 적층 구조물의 전단응력 예측

We propose an accurate and efficient estimation method of transverse shear stresses for analysis and design of laminated composite structures by 4-node quadrilateral degenerated shell elements. To get proper distributions of transverse shear stresses in each layer, we use 3-dimensional equilibrium equations instead of constitutive equations with shear correction factors which vary diversely according to the shapes of shell sections. Three dimensional equilibrium equations are integrated through the thickness direction with complete polynomial membrane stress fields, which are recovered by REP (Recovery by Equilibrium in Patches) recovery method. The 4-node quadrilateral degenerated shell element used in this paper has drilling degrees of freedom and shear stresses derived from assumed strain fields that are set up at natural coordinate systems. The numerical results demonstrate that the proposed estimation method attains reasonable accuracy and efficiency compared with other methods and FE analysis using 4-node degenerated shell elements.

Locking-free finite element methods for shells

We propose a new family of finite element methods for the Naghdi shell model, one method associated with each nonnegative integer k. The methods are based on a nonstandard mixed formulation, and the kth method employs triangular Lagrange finite elements of degree k+2 augmented by bubble functions of degree k + 3 for both the displacement and rotation variables, and discontinuous piecewise polynomials of degree k for the shear and membrane stresses. This method can be implemented in terms of the displacement and rotation variables alone; as the minimization of an altered energy functional over the space mentioned. The alteration consists of the introduction of a weighted local projection into part, but not all, of the shear and membrane energy terms of the usual Naghdi energy. The relative error in the method, measured in a norm which combines the H I norm of the displacement and ro tation fields and an appropriate norm of the shear and membrane stress fields, converges to zero with order k+1 uniformly with respect to the shell thickness for smooth solutions, at least under the assumption that certain geometrical coefficients in the Nagdhi model are replaced by piecewise constants.

Stability Analysis of a Spinning Elastic Disk Under a Stationary Concentrated Edge Load

The natural frequencies and stability of a spinning elastic disk subjected to a stationary concentrated edge load are investigated both numerically and analytically. It is found that a stationary, conservative, compressive edge load decreases the natural frequencies of the forward and backward traveling waves, but increases the natural frequency of the so-called reflected wave. It is also found that the significance of the effect of the conservative edge load on the natural frequencies and stability of the spinning disk is solely through the transverse component of the edge load on the boundary, and not through the membrane stress field it produces inside the disk. In addition, the compressive edge load induces a stationary-type instability before the critical speed, and induces a merged-type instability after the critical speed when a reflected wave meets a forward or a backward wave. The expression for the derivative of the eigenvalues of the spinning disk with respect to the edge load is derived to verify the numerical results.

The Effects of a Space-Fixed Friction Force on the In-Plane Stress and Stability of Transverse Vibrations of a Spinning Disk

The membrane stress field in a spinning disk induced by a stationary circumferential friction force is first derived in the form of an infinite series. It is then shown, both by analysis and numerical computation, that this membrane stress field has no effect on the stability of transverse vibration of the spinning disk.

Vibration of curved plate assemblies subjected to membrane stresses

In a previous paper [1] the finite strip method was applied to the prediction of the natural frequencies of vibration of longitudinally invariant, rigidly connected assemblies of circularly curved and flat strips having diaphragm end supports. This work is extended here to include the presence of an initial membrane stress field. An individual curved strip may be subjected to a biaxial direct stress field comprising a uniform stress acting in the circumferential direction and a non-uniform stress acting in the longitudinal direction. The presence of the membrane stress field is accommodated in the analysis by the inclusion of an initial stress or geometric stiffness matrix. A further extension included here is a facility to delete in-surface inertia terms. Results are presented for the application of the strip method in predicting the frequencies of vibration of a circular cylinder subjected to a complicated membrane stress system.

Buckling of rectangular mindlin plates

The elastic buckling of rectangular Mindlin plates is considered using two related methods of analysis. These methods are the Rayleight-Ritz method and one of its piece-wise forms, the finite strip method. Arbitrary combinations of the standard boundary conditions of clamped, simply-supported and free edges are accommodated by the use in the assumed displacement fields of the normal modes of vibration of Timoshenko beams. The applied membrane stress field leading to buckling can comprise biaxial direct stress plus shear stress. A range of numerical applications is described for isotropic and transversely isotropic plates of thin and moderately thick geometry. The results obtained using the two methods compare closely to one another and to other published results where these are available. A direct relationship between unidirectional buckling stress and frequency of vibration is demonstrated for a category of plates having one pair of opposite edges simply supported.

Increase of the first natural frequency and buckling load of plates by optimal fields of initial stresses

Departing from the common way of optimization of vibrating structures by an optimal mass distribution this paper deals with the problem of maximizing the fundamental frequency of structures by optimizing fields of initial stresses without varying the given appropriate shape of the structure. Thin elastic circular and rectangular plates are considered, which may be loaded by external inplane forces, and optimal initial membrane stress fields are calculated, which produce values of the first natural frequency of the free bending vibrations as high as possible. As a constraint, the strain energy caused by the field of initial stresses in the resting, externally unloaded plate is given. The computation leads to a max-min-problem solved numerically with the help of gradient methods. The optimal fields of initial stresses of the buckling plates are calculated as extreme cases in the same manner.

Investigation of pointed domes—I. Theory

A theoretical method of shell analysis has been utilized and, with the help of certain innovations, the study of pointed domes subjected to pointed loading is presented in this paper. To derive an analytical expression for the pointed dome, the general theory of shell of revolution is used to take into account the effect of membrane and bending stress field. Additionally the general classical approach of analytical multi-shell analysis is utilized to obtain the solution. This solution is applied to a pointed dome with a given dimension and the membrane stress field was calculated. The results thus obtained give a good approximation of the internal force field in the central portion of the dome. However this is not true near the lower boundary and near the kinked vertex of the dome. The values of internal meridional bending moment and meridional shear have also been calculated. These indicate that the bending field diminishes in magnitude as we move away from the kinked region.

Investigation of pointed domes—II. Experiment

The aim of the experiment was to study the behavior of pointed domes subjected to point loading at the vertex. A concrete dome reinforceed with 1 mm bars running both in the meridional and parallel circular direction was constructed. Electrical resistance strain gages were mounted at several locations along the meridional and parallel direction of the shell both inside and outside. The structure was loaded within the elastic limit. The experimental values of membrane stress field are calculated and compared with the theoretical values as is described in part I of this report. (see pp. 81–85). These results are in close agreement in regions some distance away from the lower boundary and the pointed apex. But the difference between the two results becomes noticeable near the lower boundary and the vertex of the dome. Further analysis of the experiment findings indicates that the bending strain field is slight at regions that are some distance away from the lower boundary and to the apex of the dome.

Pressure Stability of Orthotropic Prolate Spheroids

Through the use of general instability theory for doubly curved orthotropic shells, a mathematical relation was developed to predict external pressure buckling of stiffened prolate spheroids. The procedure was applied to two experiments which were found to be in fair agreement with theory. In general, the method is applicable to doubly curved shells with orthotropic material properties (composites) as well as geometric orthotropicity, and subject to arbitrary membrane stress fields. It is not limited to pressure alone. Furthermore, because of the closed form of the solutions to many problems, the procedure would be particularly useful for optimization purposes.