Buckling Of Rings Research Articles

On the dynamic propagation and arrest of buckles in pipe-in-pipe systems

Experiments and large scale numerical simulations are used to study the dynamics and the arrest of propagating buckles in pipe-in-pipe systems. In the first set of experiments the velocity of buckles initiated in a constant pressure environment is measured as a function of pressure using first water and then air as pressurizing media. For the outer and inner pipe parameters used, the buckle velocities correlated well with values measured in single pipes. The dynamic propagation experiments in air were then simulated numerically. Calculated buckle velocities followed the same trend as the measured values although they were somewhat higher. In the second set of experiments the effectiveness of internal ring buckle arrestors designed with previously developed quasi-static design procedures is re-evaluated under dynamic buckle propagation. In all cases examined the dynamic arresting efficiency was found to be higher than the quasi-static one. The same behavior was also observed in corresponding numerical simulations. This indicates that design of such devices using quasi-static design criteria should be conservative.

Optimum Design of a Belt-Conveyor Bridge Constructed as a Welded Ring-Stiffened Cylindrical Shell

In the structural optimization of a ring-stiffened cylindrical shell the unknown variables are the shell thickness as well as the thickness and the number of flat rings. The shell diameter enables to realize a belt-conveyor structure inside of the shell. The uniformly distributed vertical load consists of dead and live load. The design constraints relate to the local shell buckling strength, to the panel ring buckling and to the deflection of the simply supported bridge. The cost function includes the material and fabrication costs. The fabrication cost function is formulated according to the fabrication sequence and includes also the cost of forming of shell elements into the cylindrical shape as well as the cost of cutting of the flat plate ring-stiffeners. Since the shell thickness does not depend on the number of ring-stiff-eners (n), the nopt is calculated for a selected region of n.

Internal ring buckle arrestors for pipe-in-pipe systems

A new buckle arrestor concept for pipe-in-pipe systems is introduced and the results of a systematic study of its performance is presented. The concept involves either one single ring or a number of closely packed narrow rings placed in the annulus between the two pipes. Its effectiveness has been studied through a combination of experiments and analyses. The experiments involved two inch carrier tubes of three different D/ t values and internal rings of various dimensions. A number of experiments were first conducted using pipe-in-pipe systems. It was found that the inner tube had only a small effect on the crossover pressure of this arrestor and, as a result, in many of the following experiments inner tubes were not included. The crossover pressure of the ring arrestors was studied by varying their length, wall thickness and yield stress. Other parameters varied were the dimensions and properties of the two tubes and the gap between the arrestor and the carrier tube. The process resulted in an empirical design formula for the arresting efficiency expressed as a function of the key nondimensional variables of the problem. Large-scale finite element models which simulate the buckle crossover process have also been developed. They have been shown capable of reproducing experiments accurately. Such models can be used to prove an arrestor design developed through the empirical process described in the report.

BUCKLING OF THIN LAMINATED ORTHOTROPIC COMPOSITE RINGs/LONG CYLINDERS UNDER EXTERNAL PRESSURE

The buckling of isotropic rings under external pressure has attracted the interest of researchers since late 1950s. The formula for critical fluid buckling pressure of thin rings is very well known. This formula was directly extended to account for homogeneous orthotropic rings as well. The buckling of orthotropic cylindrical shells was also a subject of interest since the 1960s. However, the formulations developed, to date, require numerical solutions to obtain the critical pressure. In this work, a generalized closed form analytical formula for the buckling of thin orthotropic multi-angle laminated rings/long cylinders is developed. Standard energy based formulation is used to express the kinematics and equilibrium equations. Classical lamination theory is implemented to introduce the constitutive equations of thin shells. These equations are statically condensed, in terms of the ring's boundary conditions, to produce effective axial, coupling and flexural rigidities for the cases of rings and long cylinders. The critical buckling pressure may be calculated by hand using the derived equation in terms of these effective elastic rigidities. Comparisons are made with some existing results. Parametric studies are conducted to compare the present results with those of the buckling equations implemented by design standards. Various fiber orientations and stacking sequences are considered.

Non-linear inplane deformation and buckling of rings and high arches

A non-linear theory is presented for stretching and inplane-bending of isotropic beams which have constant initial curvature and lie in their plane of symmetry. For the kinematics, the geometrically exact one-dimensional (1-D) measures of deformation are specialized for small strain. The 1-D constitutive law is developed in terms of these measures via an asymptotically correct dimensional reduction of the geometrically non-linear 3-D elasticity under the assumptions of comparable magnitudes of initial radius of curvature and wavelength of deformation, small strain, and small ratio of cross-sectional diameter to initial radius of curvature ( h/R). The 1-D constitutive law contains an asymptotically correct refinement of O(h/R) beyond the usual stretching and bending strain energies which, for doubly symmetric cross sections, reduces to a stretch–bending elastic coupling term that depends on the initial radius of curvature and Poisson’s ratio. As illustrations, the theory is applied to inplane deformation and buckling of rings and high arches. In spite of a very simple final expression for the second variation of the total potential, it is shown that the only restriction on the validity of the buckling analysis is that the prebuckling strain remains small. Although the term added in the refined theory does not affect the buckling loads, it is shown that non-trivial prebuckling displacements, curvature, and bending moment of high arches are impossible to calculate accurately without this term.

Response of thick composite rings under uniform external pressure

The investigation reported herein is concerned with examining, via an experiment, the response of thick hoop wound AS4/PPS and S-2 Glass/ Epoxy rings subjected to external hydrostatic pressure. A custom pressurization testing chamber was manufactured for this purpose. Two different methods of applying the pressure was examined. In the first of these methods, constrained ring buckling (mode three) was observed in the initial stages of the test as the specimen was constrained from motion by the rubber membrane used to seal the hydraulic pressure. In the second type of loading method, unconstrained ring buckling (mode two) was observed. Microscale failure initiators such as fiber kinking or delamination between plies was visible in the rings that were unloaded upon buckling. A one dimensional shear deformation theory and a two dimensional elasticity theory were employed to examine the buckling and initial postbuckling behavior of the tested rings. Good agreement between experiment and theory is noted. The details of the analytical work pertaining to the ring buckling analysis is reported elsewhere.

Out‐of‐Plane Buckling of Restrained Thin Rings of General Open Section

This paper describes a closed-form solution based on thin-walled member theory for the out-of-plane buckling of thin rings of an open cross section. The theory considers the practical features of restraint and loading conditions of rings situated at axisymmetric shell intersections, and it is applicable to any thin-walled open-section form. Numerical comparisons between the closed-form solution and a finite-element shell buckling analysis are then presented to demonstrate the accuracy of the closed-form solution, and to study the effects on the buckling behavior of angle section rings of various parameters, including the ring width-to-thickness ratio, the load position, the restraint position, and the flange size. The effect of secondary warping on the buckling behavior of angle section rings is also discussed in some detail. Finally, a simple expression for use in design is established for the buckling strength of an angle section ring loaded and supported at its inner edge.

Conformation of 6-morpholino-3-azabicyclo[3.1.0]hexane derivatives

1 H NMR spectroscopic studies established the presence of a chair conformation for endo-3-methyl-6-morpholino-3-azabicyclo[3.1.0]hexane derivatives 8a and b. This was additionally confirmed by an X-ray structural analysis of 8b. The diastereomers 10a,b and the N-demethyl-endo-morpholino compounds 9a,b, however, were found to prefer a boat conformation. The correlation between dihedral angles and ring buckle of 3-azabicyclo[3.1.0]hexane 4 was determined by MNDOC semiempirical calculations. Further structural information about this bicyclic system was obtained by HF/6–31 G* calculations for the 3-azabicyclo[3.1.0]hexane parent compounds 4 and 5.

An Investigation of Dynamic Pulse Buckling of Thick Rings

The occurrence of dynamic buckling of thick rings responding to an impulse load is investigated by analytical and numerical solutions to the equation of motion and by nonlinear finite element analyses. An extension to the linearized analytical solution is made using a finite difference scheme which incorporates a nonlinear moment-curvature relationship to model the effects of elastoplastic behavior and strain-rate reversal on the buckle formation. The finite element solution to the problem is formulated with the nonlinear code, ADINA. A comparison of the results shows that the numerical solutions (and, in particular, the ADINA solution) predict a significant reduction in the amplitude of buckling response and an increase in the predominant wavelength of response with time, in comparison to the linear analytical solution. A limited comparison to published experimental results of dynamic pulse buckling of thick rings is also given.

The Design of Lightweight Pliable Hooped Petticoats. Part II: The Relation between Skirt Weight, Fullness, and Hoop Stiffness in Foundations for Crinoline-style Dresses

The classical theory of the buckling of rings is used to establish a relation between skirt weight, fullness, and hoop stiffness for critical strength, both for a single hoop and for a set of hoops in a crinoline foundation. In the latter case, the profile of the foundation is such that all hoops are equally loaded with respect to the buckling criterion. It is shown that the ratios of the flexural rigidity of the hoop to skirt weight and to fabric weight per unit area can be used as basic parameters, which are, respectively, inversely proportional to the cube and the fourth power of the circumference of the largest hoop. Experimentally, higher values of the parameter are acceptable before deformation occurs because the theory for a single ring does not take into account the effects of coupling between rings.

Plastic buckling of rings at steel silo transition junctions

Abstract Elevated steel bins and tanks typically consist of a cylindrical vessel, a conical hopper and a skirt. The intersection of these three shell segments is termed the transition junction and is subject to a large circumferential compression. A ring is often required at this junction. The high circumferential compression at the intersection may lead to a buckling failure of the ring. Elastic buckling of the ring at the transition junction has been studied recently, but very little is currently known about the plastic buckling of such a ring. This paper presents a theoretical investigation into the plastic buckling of the transition ring using a finite element analysis. Simple annular plate rings are studied first, and the results are compared with the few known predictions from an earlier study. The plastic buckling of T-section rings is then explored.

Growth of Buckling Instabilities During Radial Collapse of a Ring

Abstract Conditions leading to the growth of initial imperfections for rings or long cylindrical shells subjected to initial uniform inward impulsive velocity loading are investigated. The work is motivated by a need to prevent buckling of rings during the contracting ring test, which is used to determine intermediate strain rate compressive stress-strain data. A previous analysis by Abrahamson is extended to include deceleration of the ring during inward motion; and the results of this deceleration are found to greatly influence the growth of imperfections (buckling). Qualitative comparisons with experimental data are presented.

PLANAR BUCKLING AND POST-BUCKLING BEHAVIORS OF RINGS AND ARCHES SUBJECT TO DISPLACEMENT DEPENDENT LOADS

The tangent stiffness equation for a planar straight beam to reflect the displacement dependency of loading is derived. The load stiffness matrices for water pressure and the center directed uniform distributive loading for circular members are obtained explicitly. Using the stiffness equation derived, planar buckling and post-buckling of elastic rings and arches subject to a variety of loading patterns have been examined. It is found from the computations that the displacement dependency of loadings significantly affects not only the buckling strength, but also, even more, the post buckling finite displacement behavior of the structures.

Buckling of rings in column-supported bins and tanks

Theories for the out-of-plane buckling of rings under uniform circumferential compression are well established. However these theories are not applicable to rings in column-supported bins where the circumferential stress in the ring varies significantly over the cross-section and around the circumference. This paper deals with the out-of-plane buckling of annular plate rings in column-supported bins and tanks. The stress distributions in such rings are first examined using a finite element shell analysis. A closed-form solution for the buckling of rings under non-uniform circumferential stresses is then derived. Numerical results from the closed-form solution are compared with those from a finite element shell buckling analysis, and close agreement is found. The significant effect of stress non-uniformity on the buckling predictions is demonstrated. Finally, simplified equations are given which are suitable for structural design purposes, and which closely model the predictions of the more rigorous solution.

Unrestrained out-of-plane buckling of monosymmetric rings

Rings which are loaded by a uniformly distributed radial inward load may buckle out of the plane of the ring. This paper is concerned with out-of-plane buckling of rings of monosymmetric section with the axis of symmetry oriented radially. It is assumed that there are no restraints against buckling displacements. A new solution to the problem is derived and compared extensively with existing analytical solutions and finite element analysis. All the existing solutions are shown to be either limited in application or in eror. The new solution compares well with finite element analyses, provided there is no distortion of the cross section.

Instability Analysis of Pressure-Loaded Thin Arches of Arbitrary Shape

The governing differential equations and the virtual work expressions for the large displacement analysis of thin arches of arbitrary shape, subjected to pressure loads, are derived. The virtual work expressions are employed as the basis for formulation of finite element stiffness equations. Classical solutions are obtained, from the differential equations, for the buckling of circular rings under uniform “follower” (hydrostatic) and “dead” (constant direction) pressure loadings. Finite element solutions are calculated for elliptical rings for a wide range of axis ratios.

Deflection and buckling of rings with straight and curved finite elements

A plane circular ring is modeled by various elements, some straight and others curved. All formulations have two translations and one rotation as nodal dof. We consider static deflection, under point loading, and bifurcation buckling, under constant-direction loading and fluid pressure loading. Matrices that account for fluid pressure loading are derived. Numerical examples show that excellent results are given by a standard straight beam element whose length is that of the arc it subtends.

Closure to “Buckling of Rings Subjected to Interacting Loads”

This technical note deals with the interaction effects of multiple external and internal pressure loads, e.g., normal gas pressure, centrally directed pressure, etc., acting on a circular elastic ring buckling in its own plane. Several simple formulas for the critical load are derived and a number of unusual phenomena are discovered. Moreover, a misconception concerning the buckling of a ring subjected to the constant-directional pressure load is clarified.

The conformations of bicyclo[3.1.0]hexane derivatives by 1H and 13C NMR

AbstractThe 1H and 13C spectra of bicyclo[3.1.0]hexanes and thujanes have been recorded and assigned. Application of the Karplus equation has yielded dihedral angles, and a computer calculation of the angle of ring buckle as a function of the main dihedral angles has been carried out. The calculated angles of ring buckle agree well with known values in the bicyclo[3.1.0]hexanes, but for 1‐methylbicyclo[3.1.0]hexanes and thujanes the results are not self consistent. It is suggested that the bridgehead substituent causes the boat to twist, although the twist can be reduced by an axial methyl substituent on C‐4.

Dynamic plastic buckling of rings and cylindrical shells

An experimental investigation was undertaken into the dynamic plastic buckling of circular rings subjected to uniformly distributed external impulsive velocities. The experimental values of the average final radial deflections, critical mode numbers and dimensions of the permanent wrinkles in the mild steel and aluminum specimens are compared with various theoretical predictions and other experimental results.