Small Initial Imperfections Research Articles

Lateral Buckling of an Elastic Pipe on a Frictional Seabed

Abstract Pipelines tend to buckle laterally under thermal expansion. In existing analytical solutions by Kerr and Hobbs, it is assumed that the seabed resistance q0 to lateral pipe movements is constant in magnitude and opposite in direction to the total displacement. Here, it is opposite to the velocity instead, i.e., the seabed is taken to be frictional rather than nonlinear elastic with a V-shaped potential function. A three-lobe (“mode 3f”) analytical solution is provided for the frictional case, using the same approximate end-of-buckle condition v = v′ = v″ = 0 used by Hobbs in his “mode 3” solution for the nonlinear elastic case. For both modes 3 and 3f solutions, the shape of the buckle does not change as it grows with increasing thermal expansion, though the scaling factors in the axial and lateral directions are different, i.e., the solutions are self-similar. A single finite element solution for the frictional case with an initial imperfection imposed by a bumper can be scaled to cover all such cases. It shows that the shape of the buckle depends on the amplitude of the initial triggering imperfection and is close to the mode 3f solution for very small initial imperfections. The difference between modes 3 and 3f is significant in regard to buckle shape and the relative size of the buckle lobes, but small in regard to the maximum bending moment for a given amount of thermal expansion accommodated by the buckle.

Buckling behaviour of FRP strengthened cylindrical metallic shells with cut-outs

This paper presents a detailed buckling study on FRP strengthening of metallic cylindrical shells with different geometric imperfections and centrally located unstiffened and stiffened cut-outs. Initially, bare and FRP strengthened aluminium cylindrical shells have been manufactured. Geometric imperfections have been measured and implemented in the numerical model. Interfacial bond quality has been assessed by two different non-destructive techniques. The buckling response of bare and FRP strengthened shells have been numerically simulated and validated with respective experiments. Subsequently, the difference in buckling response between FRP strengthened and bare aluminium shells with variations in cut-out’s size, orientation, shape, and corner radius has been analysed for unstiffened cut-outs. The effect of increase in stiffener’s thickness and overlap width for photo-frame type stiffening of rectangular cut-outs with and without closure has been investigated. The study has revealed interesting observations. (1) Effect of cut-out size on buckling capacity of the shell is a function of its imperfection magnitude. For shells with large initial imperfections, introducing a cut-out as well as with increase in its size, relatively smaller drop in buckling capacity is found; while shells with small initial imperfections resulted in a steep and large drop in buckling load even with a small cut-out size. (2) FRP strengthening is more effective for shells with cut-outs compared to shells without cut-outs, as it increases the flexural stiffness of shell against asymmetric buckles’ initiation and resists outward lip opening along longitudinal edge of cut-out. (3) For shells with large initial imperfections, elliptical cut-outs offer higher buckling resistance than rectangular cut-outs with varying corner radius but with FRP wrapping, rectangular cut-outs with rounded edges exhibit the best buckling performance. (4) In the case of shells with stiffened cut-out, both with or without closure, FRP strengthening is effective only for shells with smaller initial imperfections. Further findings and observations are discussed and concluded.

Asymmetric serviceability limit states of symmetrically loaded segmental tunnel rings: Hybrid analysis of real-scale tests

Serviceability limit states (SLS) of segmental circular tunnel rings, subjected to symmetric loading, may experience asymmetric deformations. This is investigated herein, based on real-scale experiments and structural analysis. Single segmental rings, consisting of six tubbings and six joints, were subjected to 24 point loads, simulating ground pressure until convergence-related SLS were reached. Structural analysis is based on transfer relations, representing analytical solutions of the linear theory of thin circular arches. One symmetric mode and two antisymmetric modes of rigid-body displacements of the investigated segmental rings are described analytically. The vertical-to-horizontal convergence ratio of the symmetric mode is compared with convergence ratios, measured close to the SLS obtained in the tests. This allows for categorization of the tests as symmetric and asymmetric problems. The structural behavior of two asymmetric tests is analyzed by means of a generalized hybrid method. The term “hybrid” applies insofar as the external loading and the data from monitoring of the displacements during the tests are used as input. It is “generalized” in order to allow for consideration of the asymmetric structural behavior, thus abandoning the restriction to symmetry. The relative rotations of the joints are determined such that they refer to symmetric and antisymmetric rigid-body displacements and that the simulated convergences reproduce the measured values. Applying this concept to the two tests, it is shown that the asymmetry develops in a progressive rather than in a stepwise fashion. It refers to different forms of structural behavior of joints that nominally transmit the same normal forces and bending moments. It is concluded that the asymmetry results from small initial imperfections rather than from asymmetric external loading.

Closed-form solution for nonlinear buckling analysis of FG-CNTRC cylindrical shells with initial geometric imperfections

The circular cylindrical shells have been widely used in modern engineering structures, especially in the aerospace industry such as the oil pipeline, the missile, spacecraft hull, storage tanks. In recent years, functionally graded carbon nanotube composites (FG-CNTRCs) have emerged, as a promising type of composites. Due to the increasing demands for high structures performance, this research paper proposes a closed-form solution to investigate the nonlinear buckling behavior of the FG-CNTRC cylindrical shells subjected to compressive load. The small initial imperfections of the FG-CNTRC cylindrical shells are also considered through analytical modeling. Effective properties of materials of the shells reinforced by single-walled carbon nanotubes (SWCNTs) are estimated through a micro-mechanical model based on the extended rule of mixtures. The Donnell shell theory and von-Karman nonlinear kinematics are used for nonlinear equilibrium equations. The novelty of this work is to exploit an exact solution via Galerkin procedure and term of the Airy stress function in order to reveal the impacts of the imperfection parameter, different types of CNTs distribution, the volume fraction of CNTs on nonlinear behavior and compressive equilibrium paths of FG-CNTRC cylindrical shells.

Effect of initial imperfections of struts on the mechanical behavior of tensegrity structures

Tensegrity has been widely accepted as a conceptual model for cell mechanics. Tensegrity models developed to study cell mechanical behavior have never considered the initial imperfection of the struts. However, the curved shapes of microtubules in living cells and the nonlinear cell mechanical behavior lend support to the necessity of introducing initial imperfections to cell tensegrity. Here we use a simple three-member tensegrity structure to investigate the mechanical behavior of tensegrity structures with initial imperfections. Our analytical results demonstrated that the stiffness of the tensegrity structure will be reduced significantly with even small initial imperfections. Further, a strong nonlinearity arises with increased initial imperfection both in terms of the maximum deformation of the strut as well as the stiffness behavior of the tensegrity. Finite element simulations of tensegrity structures with 2 struts and 3 struts show great consistency with the analytical solution. The initial imperfection method provides an intuitive way to introduce nonlinearity to cellular mechanical behavior. It has major implications in understanding load bearing capacity and force distribution in cytoskeleton.

Mechanical behavior of tensegrity structures with High-mode imperfections

The potential applications of tensegrity structures are widely accepted in fields, such as civil engineering and soft robotics. However, the curved shapes of struts in tensegrity inspired applications and the corresponding nonlinear mechanical behavior make it necessary to introduce initial imperfections, especially high-mode imperfections. This paper works out a method to analyze the initial imperfection analytically with the basic assumption that the strut length would not change with the deformation. Then a simple three-member tensegrity structure is harnessed here to investigate the mechanical behavior of tensegrity structures with the first three modes of initial imperfections. Our analytical results demonstrated that the stiffness of the tensegrity structure will be reduced significantly with even small initial imperfections. The analytical result of load-displacement curve shows that the deformation shows an upward trend with the rise of initial imperfection mode. Further, a strong nonlinearity arises with increased initial imperfection both in terms of the maximum deformation of the strut as well as the stiffness behavior of the tensegrity. Finite element simulations of tensegrity structures with 6 struts show great consistency with the analytical computation. The initial imperfection method provides an intuitive way to introduce nonlinearity to mechanical behavior. It has major implications in understanding load bearing capacity and force distribution in applications of tensegrity structures.

Tailored Buckling Constrained by Adjacent Members

This paper exploits the accuracy and versatility of additive manufacturing to display interesting buckling behavior in slender elastic columns. A set of parallel columns were printed to relatively high precision, and then subjected to axial loading. The load-deflection behavior is influenced by the post-buckled mutual contact between adjacent columns. Given the capability of incorporating prescribed (but small) initial geometric imperfections using additive manufacturing it is feasible to seed post-bucking behavior, effectively tailoring stiffness.

Eversion of bistable shells under magnetic actuation: a model of nonlinear shapes

We model in closed form a proven bistable shell made from a magnetic rubber composite material. In particular, we incorporate a non-axisymmetrical displacement field, and we capture the nonlinear coupling between the actuated shape and the magnetic flux distribution around the shell. We are able to verify the bistable nature of the shell and we explore its eversion during magnetic actuation. We show that axisymmetrical eversion is natural for a perfect shell but that non-axisymmetrical eversion rapidly emerges under very small initial imperfections, as observed in experiments and in a computational analysis. We confirm the non-uniform shapes of shell and we study the stability of eversion by considering how the landscape of total potential and magnetic energies of the system changes during actuation.

Liner wrinkling and collapse of girth-welded bi-material pipe under bending

Pipelines and flowlines that carry corrosive hydrocarbons are often protected by lining them internally with a thin layer of a corrosion resistant material. In the most economic method, the liner is brought in contact with a carbon steel carrier pipe by mechanical expansion. In applications involving severe plastic bending, such as winding onto a large diameter drum as is done in the reeling installation method, such a liner can wrinkle and collapse while the carrier pipe remains intact. Collapse has been shown to be sensitive to small initial geometric imperfections in the liner. A numerical framework for establishing the extent to which lined pipe can be bent before liner collapse was presented in [14–16]. This framework, suitably extended is used here to examine the effect of girth welds on liner collapse. The modeling starts by simulating the expansion process that plastically deforms the two tubes bringing them into contact. Bending plastically the composite structure leads to differential ovalization of the two tubes and detachment of the liner. The girth weld locally prevents this detachment creating a periodic boundary disturbance in the liner. With increasing bending the periodic disturbance grows and eventually yields to buckling into a shell-type diamond-shaped mode that causes the liner to collapse inside the intact outer pipe. The problem is investigated using a 12-inch carrier pipe base case. Comparing the collapse curvature of a girth-welded liner with imperfect liners free of welds that have the same collapse curvature, it is concluded that girth welds constitute a “weak” spot on the line. Results from a parametric study of factors that influence the collapse of a girth-welded are presented followed by recommendations.

Catastrophic failure of polymer melts during extension

Numerical flow modeling has been applied to study the break of monodisperse polymer melts during extension. These continuum mechanical based computations are within the ideas of the microstructural ’interchain pressure’ theory. Calculated breaks, a result of small initial sample imperfections, agree with experimental observations.

Slenderness ratio distribution and load-shortening behaviors of stiffened panels

This paper evaluates the distributions of three slenderness ratios of the plates, the stiffeners, and the stiffened panels, and presents a comparison of the load-shortening behaviors of the stiffened panels. The slenderness ratios, which represent the geometry and material properties of the stiffened panels, are obtained from deck and bottom platings of the midship area of 163 vessels, including 59 tankers, 46 bulkers, 28 product carriers, 15 container carriers, and 12 miscellaneous ships. Under the assumption that each slenderness ratio closely follows a normal distribution, average and upper/lower bound slenderness ratios are derived based on mean and mean plus/minus two times standard deviations. In order to compare the load-shortening capacities of the stiffened panels from simplified formulas of common structural rule (CSR) with those derived from a nonlinear FEA, a new parameter of relative average strain energy is introduced. It is concluded that the CSR formulas may be developed based on the very small initial imperfections of weld-induced initial deflection and residual stress.

Spontaneous Breakup of Extended Monodisperse Polymer Melts

We apply continuum mechanical based, numerical modeling to study the dynamics of extended monodisperse polymer melts during the relaxation. The computations are within the ideas of the microstructural "interchain pressure" theory. The computations show a delayed necking resulting in a rupture, as a result of small initial sample imperfections. These ruptures agree with experimental observations.

From wrinkles to creases in elastomers: the instability and imperfection-sensitivity of wrinkling

The stability of the wrinkling mode experienced by a compressed half-space of neo-Hookean material is investigated using analytical and numerical methods to study the post-bifurcation behaviour of periodic solutions. It is shown that wrinkling is highly unstable owing to the nonlinear interaction among the multiple modes associated with the critical compressive state. Concomitantly, wrinkling is sensitive to exceedingly small initial imperfections that significantly reduce the compressive strain at which the instability occurs. The study provides insight into the connection between wrinkling and an alternative surface mode, the finite amplitude crease or sulcus. The shape of the critical combination of wrinkling modes has the form of an incipient crease, and a tiny initial imperfection can trigger a wrinkling instability that collapses into a crease.

Buckling and postbuckling behavior of unstiffened slender curved plates under uniform shear

Buckling and postbuckling behavior of curved plates under in-plane shear are investigated. After revisiting classic elastic buckling results, the elastoplastic postbuckling behavior and the effects of curvature parameter and aspect ratio are simulated via geometrical and material nonlinear analyses. Imperfection sensitivity is studied for various imperfection shapes and magnitudes. An increase in curvature parameter raises the elastic buckling load, produces unstable buckling and reduces postbuckling reserves. The buckling load and shear capacity are higher in shorter plates. Small initial imperfections are found to have severe effects on the initial buckling load of plates with large curvature parameter, but little effect on ultimate postbuckling capacity.

Stability of cylindrical shells with initial imperfections under the action of external pressure

We use the equations of nonlinear theory of shallow shells to solve the problem of stability of thin elastic isotropic cylindrical shells, with small initial shape imperfections, that are under the action of external uniform pressure. The problem solution is constructed by the Rayleigh-Ritz method with the approximation of the shell midsurface displacement by double functional sums in trigonometric and beam functions. The system of nonlinear algebraic equations is solved by using the methods of continuation with respect to a close-to-best parameter. For the initial imperfections of the shells, we use their normalized deflections from the limit points of overcritical branches of the loading trajectories. We consider various cases of the shell fixation and support under loading by lateral and hydrostatic uniform pressure. We also construct the range of values of the critical pressure, which, with the maximal deviation of the shell shape from the cylindrical shape up to 30%, covers practically all known experimental data.

Influence of initial deflections on the stability of composite cylindrical shells interacting with a fluid flow

Composite cylindrical shells interacting with an internal fluid flow are analyzed for stability. It is assumed that the shells have small initial geometrical imperfections. The effect of axisymmetric and nonaxisymmetric initial deflections on the critical speeds of the fluid, which cause static (divergent) or dynamic (flutter) loss of stability, is studied

Wrinkling of Tubes by Axial Cycling

Circular tubes compressed into the plastic range first buckle into axisymmetric wrinkling. Initially, the wrinkle amplitude grows with increasing load, but induces a gradual reduction in axial rigidity that eventually leads to a limit load instability and collapse. For lower D/t tubes, the two instabilities can be separated by strain levels of a few percent. Persistent stress-controlled cycling can cause accumulation of deformation by ratcheting. Here, the interaction of ratcheting and wrinkling is investigated. In particular, it is asked if compressive ratcheting can first initiate wrinkling and then grow it to amplitudes associated with collapse. Experiments on SAF2507 super-duplex steel tubes with D/t of 28.5 have shown that a geometrically intact tube cycled under stress control initially deforms uniformly due to material ratcheting. However, in the neighborhood of the critical wrinkling strain under monotonic loading, small amplitude axisymmetric wrinkles develop. This happens despite the fact that the maximum stress of the cycles can be smaller than the critical stress under monotonic loading. In other words, wrinkling appears to be strain rather than stress driven, as is conventionally understood. Once the wrinkles are formed, their amplitude grows with continued cycling, and as a critical value of amplitude is approached, wrinkling localizes, the rate of ratcheting grows exponentially, and the tube collapses. Interestingly, collapse was also found to occur when the accumulated average strain reaches the value at which the tube localizes under monotonic compression. A custom shell model with small initial axisymmetric imperfections, coupled to a cyclic plasticity model, is used to simulate these cyclic phenomena successfully.

Influence of Axisymmetric Initial Imperfections of a Spherical Shell on its Critical Load

The work examines pre- and postbuckling behavior of a thin, elastic and shallow dome, which is clamped at own boundary and is under uniform external pressure. It is supposed, that before pressure application the dome has small axisymmetric initial imperfections. Solving of this boundary problem is based on Marguerre's equations and obtained by means Rayleigh-Ritz method and arch-length incremental method. Search of different forms of initial imperfections of a dome from spherical at the maximal deviation up to 30% of its thickness enables receiving of a range of its critical loadings which completely comprised all experimental values.

Some Effects of Imperfection Geometry on the Cyclic Distortion of Thermally Grown Oxides

Morphological imperfections on the surface of alloys that form thermally grown oxides (TGO) distort or rumple upon thermal cycling. Models have been developed that predict the dependence of this distortion on oxide growth phenomena and material properties. In the present study, the model is tested in the domain of small (micron-sized) initial imperfections. For this purpose, cyclic experiments have been performed on FeCrAlY substrates using protocols for incorporating small (linear and axi-symmetric) surface imperfections. The experiments revealed a peak rumpling-rate, occurring at characteristic imperfection amplitude of order 1 µm. For smaller imperfections, the growth-rates become immeasurably low. The model accurately reproduces these characteristics. Beyond this additional assessment of the model fidelity, the study has also demonstrated that asymmetric imperfections close during thermal cycling, correcting a misinterpretation that such features are cracks that filled with TGO.

Asymptotic-Numerical Method to Analyze the Postbuckling Behavior, Imperfection-Sensitivity, and Mode Interaction in Frames

The paper presents the formulation and illustrates the application of an asymptotic-numerical (semianalytical) method to analyze the geometrically nonlinear behavior of plane frames. The method adopts an “internally constrained” beam model and involves two distinct procedures: (1) an asymptotic analysis, which employs a perturbation technique to establish a sequence of systems of equilibrium differential equations and boundary conditions, and (2) the successive numerical solution of such systems, by means of the finite element method. This method can be applied to investigate the behavior of frames with arbitrarily complex configurations (member number and orientation) and leads to the determination of analytical expressions which provide: (1) the initial postbuckling behavior of perfect frames and (2) the nonlinear equilibrium paths of frames containing small initial imperfections or acted by primary bending moments, including the influence of eventual buckling mode interaction phenomena. In order to val...