Symmetric Imperfections Research Articles

Influence of stiffening rings on imperfection sensitivity of hyperboloid steel cooling towers

Stiffening rings are necessary strengthening measures for steel cooling towers. Research on the imperfection sensitivity of steel cooling towers considering different layout schemes of stiffening rings is lacking. In this study, finite element models of 160-m-high hyperboloid single-layer reticulated-shell steel cooling towers were established. A nonlinear buckling analysis of the cooling towers was performed considering different imperfection patterns and magnitudes. Simultaneously, a combined imperfection pattern that combines the lowest buckling mode shapes in all areas of the tower shell was proposed. The results showed that the imperfection sensitivity of the steel cooling tower is low and decreases with the increasing number of stiffening rings. The combined imperfection pattern is more unfavourable than the imperfection pattern of a specific buckling mode shape and is suitable for different layout schemes of stiffening rings. And the antisymmetric imperfection pattern is more unfavourable than the symmetric imperfection pattern. Additionally, the buckling resistance of a steel cooling tower decreases with the increasing imperfection magnitude, and the imperfection magnitude of Hout/300 can reasonably describe the adverse effects of imperfections on the structure. It is recommended that the combined antisymmetric imperfection pattern and the imperfection magnitude of Hout/300 are adopted for hyperboloid steel cooling towers.

Sensitivity analysis of FDA´s benchmark nozzle regarding in vitro imperfections - Do we need asymmetric CFD benchmarks?

Abstract Modern technologies and methods such as computer simulation, so-called in silico methods, foster the development of medical devices. For accelerating the uptake of computer simulations and to increase credibility and reliability the U.S. Food and Drug Administration organized an inter-laboratory round robin study of a generic nozzle geometry. In preparation of own bench testing experiment using Particle Image Velocimetry, a custom made silicone nozzle was manufactured. By using in silico computational fluid dynamics method the influence of in vitro imperfections, such as inflow variations and geometrical deviations, on the flow field were evaluated. Based on literature the throat Reynolds number was varied Rethroat = 500 ± 50. It could be shown that the flow field errors resulted from variations of inlet conditions can be largely eliminated by normalizing if the Reynolds number is known. Furthermore, a symmetric imperfection of the silicone model within manufacturing tolerance does not affect the flow as much as an asymmetric failure such as an unintended curvature of the nozzle. In brief, we can conclude that geometrical imperfection of the reference experiment should be considered accordingly to in silico modelling. The question arises, if an asymmetric benchmark for biofluid analysis needs to be established. An eccentric nozzle benchmark could be a suitable case and will be further investigated.

Effects of symmetric imperfections on the behavior of bistable struts

The behavior of a bistable strut for variable geometry structures was investigated in this paper. A three-hinged arch subjected to a central concentrated load was used to study the effect of symmetric imperfections on the behavior of the bistable strut. Based on a nonlinear strain–displacement relationship, the virtual work principle was adopted to establish both the pre-buckling and buckling nonlinear equilibrium equations for the symmetric snap-through buckling mode. Then the critical load for symmetric snap-through buckling was obtained. The results show that the axial force is in compression before the arch is buckled, but it becomes in tension after buckling. Thus, the previous formulas cannot be used for the analysis of post-buckling behavior of three-hinged shallow arches. Then, the principle of virtual work was also used to establish the post-buckling equilibrium equations of the arch in the horizontal and vertical directions as well as the static boundary conditions, which are very important for bistable struts.

Stability of prestressed stayed steel columns with a three branch crossarm system

Analytical and numerical investigations of the stability of prestressed stayed columns with three crossarms branching laterally and symmetrically are presented. It is shown that modal symmetry is broken automatically if the critical mode is antisymmetric, which distinguishes it from the stayed columns studied hitherto. The governing imperfection distribution that should be adopted within finite element analysis to capture the actual load-carrying capacity is also obtained. The findings suggest that when the critical buckling mode is symmetric, the governing imperfection distribution should also be symmetric. Conversely, if the critical buckling mode is antisymmetric, a symmetric imperfection distribution or an asymmetric imperfection distribution may be adopted as the most severe imperfection distribution depending on the system characteristics. This would enable designers to determine an accurate prediction of the actual strength through nonlinear finite element analysis.

Collapse of Corroded Pipelines under Combined Tension and External Pressure.

In this work, collapse of corroded pipeline under combined external pressure and tension is investigated through numerical method. Axially uniform corrosion with symmetric imperfections is firstly considered. After verifying with existing experimental results, the finite element model is used to study the effect of tension on collapse pressure. An extensive parametric study is carried out using Python script and FORTRAN subroutine to investigate the influence of geometric parameters on the collapse behavior under combined loads. The results are used to develop an empirical equation for estimating the collapse pressure under tension. In addition, the effects of loading path, initial imperfection length, yielding anisotropy and corrosion defect length on the collapse behavior are also investigated. It is found that tension has a significant influence on collapse pressure of corroded pipelines. Loading path and anisotropic yielding are also important factors affecting the collapse behavior. For pipelines with relatively long corrosion defect, axially uniform corrosion models could be used to estimate the collapse pressure.

Stacking sequence influence on imperfection sensitivity of cylindrical composite shells under axial compression

Space launcher vehicle structures are designed as thin walled cylindrical and conical structures which are prone to buckling and are sensitive towards geometrical imperfections. Small deviations in dimensions, which still are within manufacturing tolerances, may lead to a tremendous decrease in load carrying capacity. Thus, imperfections have to be considered during the design phase and this is commonly done using empirical knock down factors. Besides this approach, imperfections can be considered by applying numerical or analytical structural models. Composite materials are used to exploit the light weight potential of unstiffened thin walled structures. For this type of shell structure, the buckling load of the geometrically perfect shell and the imperfection sensitivity are significantly influenced by the laminate stacking sequence. In this paper, the influence of the laminate stacking sequence of composite shells with rotational symmetric imperfections on the buckling behavior is studied and laminate stacking sequences leading to the highest buckling loads of an imperfect shell structure are identified. These stacking sequences are evaluated further by applying non-rotational symmetric imperfections and localized imperfections and the stacking sequences leading to optimum designs of the geometrical perfect shell structure are considered as reference structures.

Elastic Buckling and Post-Buckling of Von Mises Planar Trusses

The article deals with the computational model of an elastic von Mises planar truss. The description of a mathematical concept, which is intended for the creation of computational programmes based on a finite number of segments, is presented. The mathematical solution is suitable for the analysis of load-deflection curves. Structural deformation is evaluated by seeking the minimal potential energy. The article examines the effects of change in the vertical displacement of the top joint on strut axes. The step by step incremental method is used in combination with the Newton-Raphson method. The presented study is aimed at the evaluation of the force in the bifurcation point, which determines the moment when loading of the model causes passing from the pre-critical effect (attainment of maximum vertical load action) to the post-critical effect. Symmetric and asymmetric initial axis imperfections are considered and relevant symmetric and asymmetric shapes of buckling are identified. The stability problem of the von Mises truss is discussed in connection with the random effects of imperfections.

Ultimate Strength of Transverse Plate Assemblies Under Uniaxial Loads

This paper addresses the influence of adjacent plates on the strength of a single plate by studying the strength of panels composed of three and five plates transversely connected and loaded along the longitudinal direction,. The influence of adjacent plates is studied in two limiting cases of the shapes of initial imperfections in those plates. First, a comparison between the strength of panels having asymmetric imperfections in adjacent plate elements and of single plates is made, which allows evaluating the ability of the single plate to properly simulate the panel strength. Following this, the results obtained for a panel model having symmetric imperfections in adjacent plates are confronted with the results of the asymmetric imperfections. The asymmetrical and symmetrical models of the panel give, respectively, the lowest and highest strengths possible (imperfections in adjacent plates with different amplitudes have intermediate strengths), allowing to establish limits to the variation of the strength. The results show that using a single plate for assessing the strength of a panel made of those plates can be conservative or nonconservative, depending on the plate slenderness and shape of initial deflections. It is concluded that for design purposes, it will in general be enough to consider assemblies of three plates, as the results for five plate assemblies are not much different.

Influence of initial geometric imperfections on the stability of thick cylindrical shells under internal pressure

AbstractThis paper investigates numerically and experimentally the influence of initial geometric imperfections on the critical loads of initially stretched thick hyperelastic cylindrical shells under increasing uniform internal pressure. Imperfections in shells can have a global or local character. First, two types of local imperfections are considered: (1) a local axially symmetric imperfection in the form of a ring and (2) a small rectangular imperfection. The influence of the imperfection thickness, position and size are analysed in detail. Results show that the critical load decreases as the imperfections increase in size or thickness and as they move from the boundaries to the centre of the shell. The influence of multiple local imperfections is also studied in the present paper. Finally, the influence of global imperfections is considered with the imperfections described as a variation of the shell curvature in the axial direction. The results show that thick hyperelastic shells may be sensitive to local and global imperfections. In all cases the experimental results are in good agreement with the numerical ones, corroborating the conclusions. Copyright © 2006 John Wiley & Sons, Ltd.

Dynamic buckling of a 2-DOF imperfect system with symmetric imperfections

Non-linear dynamic buckling of a two-degree-of freedom (2-DOF) imperfect planar system with symmetric imperfections under a step load of infinite duration (autonomous system) is thoroughly discussed using energy and geometric considerations. This system under the same load applied statically exhibits (prior to limit point) an unstable symmetric bifurcation lying on the non-linear primary equilibrium path. With the aid of the total energy-balance equation of the system and the particular geometry (due to symmetric imperfections) of the plane curve corresponding to the zero level total potential energy “surface” exact dynamic buckling loads are obtained without solving the non-linear initial-value problem. The efficiency and the reliability of the technique proposed herein is demonstrated with the aid of various dynamic buckling analyses which are compared with numerical simulation using the Verner–Runge–Kutta scheme, the accuracy of which is checked via the energy-balance equation.

Asymmetric Secondary Buckling in Monosymmetric Sandwich Struts

An interactive buckling model for sandwich struts accounting for buckle pattern localization is extended to cover such struts with differing face plate thicknesses. Although this does not affect the critical buckling characteristics of the structure, there is a significant change in the postbuckling behavior; formerly symmetric secondary buckling and imperfection sensitivity characteristics lose this quality as both become asymmetric.

Nonlinear buckling of imperfect systems with symmetric imperfections

The nonlinear stability of a 2-DOF imperfect elastic system with symmetric imperfections under a concentrated load is thoroughly discussed using an analytical approach. The perfect system loses its stability via an unstable symmetric bifurcation, which upon inclusion of imperfections is degenerated to a limit point. In this paper, it is shown that when the imperfections are symmetric, the system loses its stability through an unstable symmetric bifurcation point lying on the nonlinear prebuckling path. The corresponding critical (bifurcational) load is much lower than the limit point load related to symmetric buckling. The effect of initial imperfections on the critical bifurcational load (imperfection sensitivity) is also analytically discussed in detail. The paper is supplemented by various numerical results.

Local buckling of the elastic and viscoelastic coating around the penny-shaped interface crack

In the framework of the piecewise-homogeneous body model with the use of the three-dimensional geometrically non-linear exact equations of the theory of elasticity and viscoelasticity the local buckling (delamination) problem of the elastic and viscoelastic coating around a penny-shaped interface microcrack is studied. The method of solution of the considered problem is developed with the use of boundary form perturbation method, FEM and Laplace transform. All investigations are made on the sandwich circular plate and it is assumed that interface crack edges have an insignificant initial rotationally symmetric imperfection and as a buckling criterion the case where this imperfection starts to increase and grows indefinitely is taken. Numerical results illustrating the influence of the mechanical and geometrical parameters to the critical force for elastic coating and to the critical time for the viscoelastic coating are presented.

Structural optimization for specified nonlinear buckling load factor

The author’s method of design sensitivity analysis of nonlinear coincident buckling load factors and corresponding optimization method of finite dimensional elastic structures are shown to be applicable to a structure with moderately large number of degrees of freedom. The sensitivity formulations are written in general form for coincident buckling load factors including limit points and bifurcation points. Optimum designs are found for a spherical latticed dome subjected to a concentrated load and distributed loads, respectively. The results are compared with those by linear eigenvalue formulation. An approximate optimal solution against nonlinear buckling can be obtained by only scaling the solution under linear eigenvalue formulation even for the case where the effect of prebuckling deformation is very large. It is shown that the proposed method is practically applicable to optimum design with coincident critical points especially for the case of hill-top branching where bifurcation points are located at the limit point. In this case, imperfection sensitivity is not enhanced as a result of optimization. The reduction of maximum load factor due to a minor imperfection is shown to be in the same order as that to a major imperfection. Therefore symmetric imperfection as well as asymmetric imperfection should be considered in estimating the maximum loads.

On the buckling of the elastic and viscoelastic composite circular thick plate with a penny-shaped crack

Buckling problem of the elastic and viscoelastic rotationally symmetric thick circular plate with a penny-shaped crack is investigated. It is supposed that the crack edges have a small initial rotationally symmetric imperfection. The lateral boundary of the plate is clamped and the clamp compresses this plate circumferentially and inwards by a fixed radial displacement. The investigations are carried out in the framework of the exact geometrically non-linear equations of the theory of viscoelasticity and as a buckling criterion the case for which the initial imperfection of the crack edges start to increase indefinitely is taken. Numerical results are obtained using the Laplace transform and finite element method and are compared with the known ones for elastic composites.

Numerical models for nonlinear analysis of elastic shells with eigenmode-affine imperfections

Nonlinear finite-element analysis provides a powerful tool for assessing the buckling strength of shells. Since shells are generally sensitive to initial geometric imperfections, a reliable prediction of their buckling strength is possible only if the effect of geometric imperfections is accurately accounted for. A commonly adopted approach is to assume that the imperfection is in the form of the bifurcation buckling mode (eigenmode-affine imperfection) of a suitable magnitude. For shells of revolution under axisymmetric loads, this approach leads to the analysis of a shell with periodically symmetric imperfections. Consequently, sector models spanning over one or half the circumferential wave of the imperfection may be considered adequate. This paper presents a study which shows that a simple nonlinear analysis of the imperfect shell may not deliver the correct buckling load, due to the tendency of the shell to develop mode changes in the deformation process before reaching the limit point. This inadequacy exists not only with short sector models (half-wave or whole-wave models) but also with more complete models (half-structure or whole-structure models) for different reasons. The paper concludes with recommendations on the proper use of the four different kinds of models mentioned above in determining shell buckling strengths.

Nonlinear Stability and Dynamic Buckling of Autonomous Dissipative Systems

AbstractAn analytical approach for the stability and the dynamic buckling response of multiple‐parameter dissipative systems described by autonomous differential equations is presented. Attention is focused on general imperfect structural systems with or without symmetric imperfections including statically stable systems which display also an unstable complementary path. The significance of the unstable (physcial or complementary) path, of the basin of attraction of stable equilibria, and of the inset and outset manifolds of a saddle on the dynamic buckling mechanism is comprehensively examined with the aid of basic theorems of local dynamic analysis. A contradiction between local and global dynamic analysis as well as cases of dynamic buckling with sensitivity to initial conditions and damping, discontinuity and metastability phenomena are also explored. Three simple dissipative systems of structural engineering importance are used as illustrative models.

Nonlinear dynamic buckling and stability of autonomous structural systems

A general approach for the nonlinear dynamic buckling and stability of dissipative or nondissipative structural systems governed by autonomous ordinary differential equations (ODEs) is presented. Geometrically imperfect systems with or without symmetric imperfections, as well as statically stable systems, which, in addition to a monotonically rising (stable) equilibrium path exhibit an unstable complementary path, are considered. The role of the dynamic buckling mechanism of the basin of attraction of a stable equilibrium point (on the prebuckling path), as well as the inset (stable) and outset (unstable) manifolds of a saddle (on a physical or complementary unstable path), are comprehensively explained. A static energy criterion for determining dynamic buckling loads for vanishing (but nonzero) damping and lower bound estimates of exact dynamic buckling loads are presented. Metastability, loading discontinuity and chaos-like phenomena are also revealed. The analysis is supplemented by two illustrative models of practical engineering importance.

Limit Loads of Buried Pipelines With Asymmetric Initial Imperfections

The effect of asymmetric imperfection on the limit-load response of pipelines buried in shallow trenches is investigated. The pipeline is modeled as a long beam resting on a rigid foundation and a small strain, large displacement formulation is used. Three different asymmetric imperfection shapes for the beam are considered and the corresponding limit loads are compared with that for a symmetric imperfection. It is found that the shape of initial imperfection plays an important role. The difference between limit loads based on a symmetric imperfection and a nonsymmetric imperfection can be quite significant.

Rigidity of spherical bands

A method of numerical calculation of an imperfect spherical band (truncated spherical shell) subjected to external pressure is proposed and tested. It is shown that the calculated and experimental values of the critical loads agree with introduction into the calculation of local axially symmetric initial imperfections.