Elastic-plastic Buckling Research Articles

Elastic–Plastic Buckling of Scaled Model of Radial Ring-Stiffened Shallow Spherical Shells Based on Energy Method

Stiffened shallow spherical shell has the advantages including high strength and light weight, which allows it to be widely applied in areas including aerospace, submarine and civil engineering. For the design of scaled model for elastic–plastic buckling of a large radially–circumferentially stiffened shallow spherical shell under uniformly distributed pressure and for the purpose of similarity prediction, the similarity relation of the elastic–plastic buckling coefficient of the shell is derived; then by using the dense stiffening theory and considering the dual nonlinearity of geometry and materials, the energy increment formula for the structure under uniformly distributed external pressure is established; next, based on the similarity transformation of the energy increment of the structural system, the generalized similarity condition and similarity relation of elastic–plastic buckling of the structure under uniformly distributed external pressure are identified. Based on the scaling and equivalence of tensile–compressive stiffness and bending stiffness, the distortion scaled model for the prototype is designed. Through numerical simulation, the distortion similarity prediction for elastic–plastic buckling under uniformly distributed pressure of the radially–circumferentially stiffened shallow spherical shell that has square-shaped dimple defects has been conducted; based on the derived similarity relation of elastic–plastic buckling of stiffened plate, the similarity prediction for the elastic–plastic buckling of orthogonally stiffened plate in the plane under unidirectional uniformly distributed pressure has been conducted. The results indicate that the load–displacement curve, buckling load, and buckling mode of the distortion similarity scaled model, combined with the established radial displacement scaling factor expression, the scaling relation of tension–compression stiffness, and the similarity relation of buckling mode, can effectively predict the elastic–plastic buckling characteristics of its prototype structure. The research results can provide theoretical reference for the design and test of the scaled model with distortion of elastic–plastic buckling of similar stiffened large shell structures.

Elastic-plastic buckling of gold thin films into straight-sided blisters

We report on straight-sided buckles experimentally observed by optical and atomic force microscopies on gold ductile thin films deposited by physical vapor deposition on silicon wafers. It is shown that, whatever the buckle dimensions, their maximum deflections are higher than those expected by the elastic theory. Finite elements simulations taking into account plastic deformations in the film have been carried out. The numerical results are presented, discussed and compared to the experimental observations.

Energy-Dissipation Support Technology for Large Deformation Tunnels Based on the Post-Peak Behavior of Steel Plate Buckling: A Case Study

Large deformations can easily occur when tunneling through weak surrounding rock with high underground stresses. Under high-stress environments, the surrounding rock stores a large amount of strain energy, and the strain energy stored in the soft and weak surrounding rock enclosure is dissipated through the large traveling plastic deformation. Therefore, the effective use of the surrounding rock’s bearing capacity and energy dissipation is a feasible solution to the large deformation problem. Based on the energy principle, this study designs a high compressibility, large stroke energy-dissipation element, which transforms the traditional support into energy-dissipation support and solves the problem of large deformation by reasonably utilizing the “displacement” of the surrounding rock. The experimental results showed that the steel plate assembled structure had an initial peak value of about 15 MPa, a constant resistance of about 2 MPa with a large stroke, and a compression rate of more than 75%, with good post-compression flexural peak characteristics. Elastic–plastic buckling analyses of defective steel plate composite structures were carried out using finite element analysis software, and the deformation and force characteristics of a variety of steel plate composite structures after buckling under actual field conditions were investigated. Based on the 3D laser scanning system, the thickness of the upper and lower steel plates of the steel plate composite structure was set to 1 cm, the height of the vertical plate was set to 28 cm, and the thickness was set to 8 mm. The original rigid supporting method was converted into one of compression support by circumferentially embedding the steel plate composite structure into the initial support structure. The resulting support configuration offered a high safety factor because the pressure of the surrounding rock was released by the deformation of the steel plate composite structure.

Research on Elastic and Elastic-Plastic Buckling Load of Cylindrical Shell with an Inclined through Crack under Axial Compressive Load.

By experimental methods, 26 specimens were designed to conduct elastic and elastic-plastic buckling tests on cylindrical shells containing cracks. This study discusses the influence of factors such as the length-diameter ratio, the diameter-thickness ratio, the crack length, the inclination of the crack, etc., on the buckling load. Additionally, finite element models were established to compare with experimental results. For the PMMA cylindrical shell, the results showed that as the length-diameter ratio of the cylindrical shell increased, the buckling load first decreased and then increased. For the 6063 aluminum alloy cylindrical shell, with increasing length-diameter ratio, diameter-thickness ratio, and crack length of the cylindrical shell, the buckling load decreased accordingly. However, concerning the crack inclination, as the crack inclination increased, the buckling load increased accordingly. This indicates that the larger the crack inclination, the higher the load capacity of the cylindrical shell containing cracks. Through finite element simulations of cylindrical shells with cracks, it was found that through compressive mechanical properties, both elastic and elastic-plastic buckling loads yielded results that are closer to the experimental results. Additionally, the inclusion of contact effects in numerical simulations further improved the agreement with the experimental results, and the variation trend of the buckling load in the finite element simulation was consistent with the experimental results. The research findings provide valuable references for the assessment of load capacity in structures containing cracks.

Elastic–plastic bifurcation analysis of cylindrical shells with axisymmetric variable thickness

This study focuses on the elastic–plastic bifurcation analysis of axial compressed thin-walled cylindrical shells with variable thickness. The general asymptotic formula of the elastic–plastic critical buckling load, which is a function of the Coefficient of the Thickness Variation (CTV) and the buckling load of shells with constant thickness, is derived based on the hybrid perturbation-weighted residual method. The general formula can be reduced to the classical elastic buckling solution in Koiter (1994) when considering the elastic case with the first-order term. Furthermore, a finite element bifurcation analysis procedure is also presented by adopting the incremental J2 deformation theory of plasticity through a custom user-defined subroutine appended to the nonlinear finite element software ABAQUS. The robustness of this numerical scheme and the theoretical formula is confirmed by the decent agreement with the available experimental results in the literature. In the end, the influence of the thickness variation parameter and D/t on the elastic–plastic buckling load of shells with varying thickness is also discussed.

8-MW wind turbine tower computational shell buckling benchmark. Part 2: Detailed reference solution

An assessment of the elastic–plastic buckling limit state for multi-strake wind turbine support towers poses a particular challenge for the modern finite element analyst, who must competently navigate numerous modelling choices related to the tug-of-war between meshing and computational cost, the use of solvers that are robust to highly nonlinear behaviour, the potential for multiple near-simultaneously critical failure locations, the complex issue of imperfection sensitivity and finally the interpretation of the data into a safe and economic design.This paper presents a detailed reference solution to the computational buckling analysis of a standardised benchmark problem of an 8-MW multi-strake wind turbine support tower. Both linear and nonlinear analyses are performed, including advanced GMNIA with several different models of geometric imperfections. The crucial issue of interpreting the imperfection amplitude in a way that is compliant with the new prEN 1993-1-6 is discussed in detail. The solution presented herein is intended for use by analysts in both industry and academia for training, verification and calibration of finite element models and is intended to initiate a public repository of such computational solutions for metal civil engineering shell structures.This paper is the second of a pair. The first paper [37] presents a synthesis of 29 submissions to an international round-robin exercise performed on the same benchmark problem.

Experimental, numerical and analytical study to develop a design method for bending and shear resistances of 3D printed beetle elytron inspired sandwich plate (beetle elytron plate)

Beetle elytron plate (BEP) is a new type of sandwich structure inspired by the internal architecture of beetle elytra and characterized by trabeculae in the core to achieve better mechanical resistance compared to conventional honeycomb sandwich plate under lateral loads. To take advantage of this improved performance in design, this paper presents, for the first time, a series of static four-point bending and shear tests for buckling and post-buckling behaviour of small-scale 3D printed BEPs until failure. Tensile coupon tests were also carried out on the 3D printed material. The experimental results are then used to validate ABAQUS numerical modelling that previously could only be partially carried out due to a lack of experimental data. The test results confirm large increases in BEP resistances compared to conventional honeycomb sandwich plates of the same dimensions. The observed BEP failure mode under bending is post-buckling of the compressive skin. Assuming that the post-buckling stress of BEP can be calculated in the same way as conventional sandwich plate, and making use of the elastic–plastic buckling stress solution for BEP that had previously been proposed by the authors, this paper proposes a simple method to calculate the post-buckling stress of BEP for design purpose. For shear resistance, the shear buckling stress of the trabecular core of BEP can be calculated in the same way as for conventional honeycomb sandwich plate except for using the web width between the near edges of the two adjacent trabeculae. The proposed calculation methods give results in good agreement with experimental results, with the maximum difference being 8%. The difference would be even lower if the smooth 3D printed surface is under compression.

Closed-form solutions for the elastic–plastic buckling design of shell structures under external pressure

This paper deals with the elastoplastic buckling analysis of shell geometries traditionally encountered in pressure vessels subjected to external pressure loading. The objective is to propose in a unified way original analytical closed-form solutions giving rise to reliable results, with a good accuracy and a wide range of applications in a straightforward and low time-consuming way, for efficient dimensioning purposes. The present study is based on the plastic bifurcation theory, classically used in the context of conservative systems, where additional terms are incorporated here due to the follower external pressure acting normally to the deformed surface of the shell still after buckling. Cylindrical shells are first considered, which certainly represent the most fundamental components of pressure vessels in practice. Owing to the external pressure also acting on the closure ends of a cylinder, the influence of the induced axial compression is investigated thereafter. It appears to be non-negligible for short cylinders in elasticity and it becomes very significant in the case of plastic buckling. Then, closed-form solutions for the critical external pressure of a complete sphere are also obtained, both in elasticity and plasticity. Finally, all these new closed-form solutions are validated against reference numerical results obtained through finite element computations, and compared to current design rules.

Semi-analytical study of buckling response for grid-stiffened panels during creep age forming

The elastic–plastic buckling response of panels with grid stiffeners during creep age forming (CAF) has been studied using a semi-analytical method in this research. A simplified model for panels with multiple grid stiffeners under bending has been established and the effective width has been applied to consider the flexible skin effect and multiple stiffener effect. In the semi-analytical method, the equilibrium equation for the simplified model has been solved via differential quadrature method to acquire the buckling stress of grid-stiffened panels, using deformation theory (DT) and incremental theory (IT) of plasticity. The results with IT show good agreements with both the published experimental and non-linear FE results, demonstrating its effectiveness. Based on the method, the effects of geometric dimensions of grid-stiffened panels, multiple stiffeners and temperature on buckling response in the elastic and plastic regions have been studied. It is found that the transverse stiffener provides a 13.9% higher buckling stress of grid-stiffened panels than blade-stiffened panels, and buckling may occur in the heating stage during CAF. The proposed semi-analytical method based on IT provides accurate predictions of buckling stress during either CAF or cold forming, which can guide the parameter optimisation of grid-stiffened panels in design.

Experimental Study on Bearing Capacity of Compression Members of Space Grid Structures Reinforced by RPC

Insufficient bearing capacity of compression bars in space grid structures can significantly reduce the collapse resistance of structures and cause immeasurable losses. Reactive powder concrete (RPC) with high strength, micro-expansion, and good ductility is used to reinforce the compression members of grid structures by infilling steel tubes. Axial compression tests on five types of high-frequency welded pipes with different section sizes and initial stresses were carried out. The results showed that compared to steel tubes alone, the bearing capacity of steel tubes reinforced by RPC could be increased by 75.77 to 218.34%. With a decrease of either the initial stress or confinement coefficient, the contribution of the material reinforcement increased. The failure mode of the specimen after RPC grouting was the same as that of the non-reinforced steel pipe, which was dominated by elastic-plastic buckling. The grouting hole after reinforcement did not fail prior to the instability of the member, which suggests that necessary measures to repair the hole should be taken. Based on the design method of axial compression bearing capacity of concrete-filled steel tubular members in six codes in China and abroad, the test results of this paper and 242 RPC-filled steel tubes were compared and analyzed. Finally, an equation of nominal bearing capacity, which estimates the ultimate bearing capacity of compressive members strengthened by filled RPC in grid structures, was proposed.

Evaluating the ultimate bearing capacity of glued laminated bamboo hollow columns under eccentric compression

To improve the utilization rate of bamboo, glued laminated bamboo hollow column (GLBHC) was examined under eccentric compression to study the effects of different slenderness ratios, eccentric distances, and wall thicknesses on its ultimate bearing capacity. The nonlinear deformation of the material and the P-delta effect of the column occur under eccentric compression, which can affect its ultimate bearing capacity, so it is very important to analyze the elastic-plastic buckling of the column. Moreover, the discontinuity of the hollow column section aggravates the nonlinear deformation. To accurately estimate the ultimate bearing capacity of GLBHC, considering the difference in tensile and compression elastic modulus, two eccentric compression models were established. A formula for ultimate bearing capacity of the GLBHC under eccentric compression was proposed, and the calculated results were in good agreement with the test results.

Elastic–plastic buckling behaviour of beetle elytron plate with simple, fixed and flexible core supports

Beetle elytron plate (BEP) is a new type of sandwich structure inspired by the internal architecture of beetle elytra and characterised by trabeculae in the core to achieve higher load bearing capacity compared to the basic conventional honeycomb sandwich plate. This paper investigates elastic–plastic buckling behaviour of the compressive skin of BEP with flexible core. The development is based on adapting existing solutions for relevant structures through the following three transformation steps: (1) obtaining the elastic–plastic buckling stresses of BEP with simple and fixed supports by extending respective solutions for conventional sandwich plate; (2) modifying the elastic buckling stress solution for conventional sandwich plate with flexible core to BEP with flexible core through replacement of deformation shape functions; (3) assuming that the interpolation function between simple and fixed supports for BEP with flexible core for elastic buckling stresses is applicable to elastic–plastic buckling stresses. Despite complexity of the problem and simplification in the above mentioned transformation steps, the proposed analytical method is shown to be quite accurate by comparison against numerical simulation results using a validated ABAQUS model for a variety of BEP geometries (such as trabecular diameter, slenderness values of the compressive skin and the core layer) and different material properties as represented by Ramberg–Osgood model covering the entire spectrum of elastic buckling to plastic behaviour, with the maximum difference between the proposed analytical model and numerical simulation results being 7%.

Collapse Mechanism of Transmission Tower Subjected to Strong Wind Load and Dynamic Response of Tower-Line System

Transmission towers are prone to collapse under strong wind load, resulting in significant economic losses. In order to investigate the collapse mechanism and failure modes of the transmission tower under strong wind load and whether the wind vibration factor can greatly reflect the increasing effect of the fluctuating wind, the finite element method (FEM) is utilized to analyze the ultimate bearing capacity of a typical 220 kV transmission tower. The results show that the collapse of the tower under strong wind loads is usually due to the buckling of the leg members. When the reference wind speed is equal to 27 m/s, a small part of the main leg members reaches their yield strength, while the diagonal members are still in the elastic range, and the deformation of the transmission tower is unapparent at this wind speed. When reference wind speed is equal or greater than 30 m/s, the growing variety of main legs is totally into the plastic yield stage, and the overall deformation of this tower is visible. Therefore, the transmission tower is collapsed due to the large deformation caused by the elastic-plastic buckling of leg members. Based on the aforementioned study, a finite element model involving three transmission towers and four span transmission lines is established to analyze the dynamic response of the tower-line system below fluctuating wind. Results show that the wind-induced coefficients designed by current code not only notably satisfy the stress response of tower components subjected to fluctuating wind loads in the elastic phase but also accurately assess the collapse displacement of the transmission tower. The increasing effect of displacement on the top tower under fluctuating wind, unfortunately, could not considerably reply with the investigated factor, and the load-carrying capacity of the transmission tower in the plastic phase can be overestimated by static calculation results.

Seismic resilience of extra-large LNG tank built on liquefiable soil deposit capturing soil-pile-structure interaction

Assessment of seismic resilience of critical infrastructure such as liquefied natural gas (LNG) storage tanks, is essential to ensure availability and security of services during and after occurrence of large earthquakes. In many projects, it is preferred to build energy storage facilities in coastal areas for the ease of sea transportation, where weak soils such as soft clay and loose sand with liquefaction potential may be present. In this study, three-dimensional finite element model is implemented to examine the seismic response of a 160,000 m3 full containment LNG tank supported by 289 reinforced concrete piles constructed on liquefiable soil overlaying the soft clay deposit. The seismic soil-structure interaction analysis was conducted through direct method in the time domain subjected to the 1999 Chi-Chi and the 1968 Hachinohe earthquakes, scaled to Safe Shutdown Earthquake hazard level for design of LNG tanks. The analyses considered different thicknesses of the liquified soil deposit varying from zero (no liquefaction) to 15 m measured from the ground surface. The key design parameters inspected for the LNG tank include the acceleration profile for both inner and outer tanks, the axial, hoop and shear forces as well as the von Mises stresses in the inner tank wall containing the LNG, in addition to the pile response in terms of lateral displacements, shear forces and bending moments. The results show that the seismic forces generated in the superstructure decreased with increasing the liquefied soil depth. In particular, the von Mises stresses in the inner steel tank exceeded the yield stress for non-liquefied soil deposit, and the elastic–plastic buckling was initiated in the upper section of the tank where plastic deformations were detected as a result of excessive von Mises stresses. However, when soil liquefaction occurred, although von Mises stresses in the inner tank shell remained below the yield limit, localised stress concentrations were observed in the lower section of the tank near the base, increasing the risk of the elephant foot buckling. The lateral displacements, shear forces and bending moments in the piles increased with increasing depth of the liquefied soil. Indeed, increasing the pile lateral displacement amplified the bending moment at the pile head, thus resulting in increases in the pile bending moments especially when the liquefied soil depth exceeded one third of the entire soil deposit. In particular, the bending moment at the pile head exceeded the yield moment capacity of the pile and subsequent plastic hinges were formed. Moreover, when the thickness of the liquefied soil was more than half of the entire soil depth, the mobilised bending moments in the piles exceed the ultimate moment capacity of the pile and thus total failure of the piles were observed. In addition, in the absence of liquefied soil layer, the inertial interaction had a dominant impact on the pile response in this study. However, with increasing the thickness of the liquefied layer, further loads were developed in the piles due to amplified kinematic interaction, while the inertial interaction-induced loads decreased.

Structural Performance of Thin-Walled Twisted Box-Section Structure

The light weight and high strength-to-mass ratio of thin-walled boxed sections have incited interest in their widespread use in the construction of domes. However, the installation of these sections in forming the dome geometry has induced initial twists and curving features, to which their mechanical response has rarely been explored. Therefore, the structural performance of a structure with thin-walled twisted box sections is numerically studied in this paper, employing ANSYS, the verification of which is carried out through a comparison with experimental results. Additional components examined include the longitudinal stiffening rib, diaphragm, and web. The effects of variations in the thicknesses of these member plates on the mechanical behaviors are investigated. In general, the ultimate capacity of the structure is improved by increasing the thickness of the longitudinal stiffening rib, diaphragm, and web, but the strengthening effect of the stiffener is limited by a certain thickness enhancement. The common failure mode of the initial model is found to be an overall elastic-plastic buckling. A reduction in the thickness of the stiffener or web creates a curving deformation zone in the lower arch at the ultimate capacity, whereas the diaphragm thickness has little effect on the failure mode of the model.

In-plane buckling strength of high strength steel I-section arches

This paper investigates the in-plane inelastic buckling of high strength steel (HSS) circular arches with pinned or fixed end supports. A finite element (FE) model is developed to analyse in plane behaviour and buckling loads of HSS circular arches under general loading conditions which considers geometric non-linearity, material inelasticity, initial imperfection and residual stresses of welded HSS I-sections. The FE model is validated by comparison to analytical elastic buckling loads for arches in uniform compression, and experimental results reported in the literature on the out-of-plane elastic-plastic buckling of steel arches. It is found that current design codes for normal strength steel (NSS) structures require modification before application to HSS arches. Based on the results of the FE analysis, design equations are proposed which are in-line with existing provisions for NSS structures for both fixed and pin-ended arches under uniform compression and combined compressive and bending actions. The presented design rules give good lower-bound predictions for in-plane buckling strength of fixed and pinned HSS arches. Additionally, the influence of residual stresses and size of initial geometric imperfections on in-plane buckling strength are explored.

Elastic-plastic buckling of pipes with asymmetric dual corrosion defects subject to external pressure

In the deep-sea environment, pipelines are prone to buckling instability under the coupling effect of hydrostatic pressure and corrosion defects. This instability phenomenon is affected by the pipe dimensions, material parameters and corrosion defects. Based on the shell buckling theory, a semi-analytical method is proposed to analyze the elastoplastic buckling response of the pipe with dual corrosion defects. The buckling pressure of the pipe can be solved by the characteristic equation numerically. Then, the results of semi-analytical approach are in good agreement with those of experimental tests and FE analysis, which verifies the accuracy of the approach. The effects of geometric dimensions, material parameters, corrosion defect depth and angle on the buckling pressure of the pipe are analyzed in details. In addition, based on the FE model, the collapse mode and the post-buckling response of the pipe with asymmetric dual corrosion defects are considered. The application scope of the semi-analytical approach for pipes with different defect lengths and initial ovality is analyzed. Finally, the semi-analytical approach is extended for pipes with multiple corrosion defects.

Elastic-plastic buckling analysis of stiffened panel subjected to global bending in forming process

A new method has been proposed in this study for the elastic-plastic buckling analysis of stiffened panels under global bending. In this method, a simplified model of the stiffened panels has been built with the application of two theories of plasticity, the incremental theory (IT) and the deformation theory (DT). The effect of transverse shear deformation through the stiffener thickness has been considered using the Mindlin-Reissner plate theory. The governing differential equations have been solved by the differential quadrature (DQ) method and an iteration process has been adopted due to the non-linearity of material properties in the elastic-plastic buckling analysis. Non-linear finite element (FE) modelling of elastic-plastic buckling analysis has been carried out, and the FE results are between those based on DT and IT in general. When the reciprocal of strain hardening exponent mt increases to 20, the FE results are in a good agreement with DT results. Based on the proposed method and FE simulations, the effect of geometric parameters of stiffened panels (stiffener thickness to height ratio, stiffened panel length to height ratio, width to height ratio, and skin thickness to stiffener thickness ratio) on buckling behaviour in the elastic-plastic region has been investigated and discussed. The proposed method provides an efficient way for parameter optimisation in the structure design of stiffened panels for the aerospace applications.

An improved analytical solution for elastic and elastic-plastic buckling of pipes under hydrostatic pressures

An analytical solution for evaluating the critical pressure of pipes under hydrostatic pressure was proposed. The stability equations and the correct boundary conditions for 3D problem in cylindrical coordinates were established by using energy method. The stability equations were solved together with the derived correct boundary conditions and the solution of the critical pressure for elastic buckling of the pipes was expressed in closed form. By adopting the three-layers model for elastic-plastic buckling analysis, the solution was available for the critical pressure prediction of the pipes in elastic-plastic collapse. The effect of the initial ovality was incorporated in formulation. It shows that the increase in the magnitude of the axial compressive stress results in the further reduction in the critical pressure for fully elastic buckling while the pressures for elastic-plastic collapse are always higher than those of the 2D solutions based on the plane strain deformation. The influence of the initial ovality on the pressure capacity of the elastic-plastic buckling of the pipes was discussed and the results from the present approach were compared with the other solutions and experiments.

A semi-analytical approach to elastic–plastic buckling analysis of pipes with asymmetric local wall thinning

The buckling behavior of the corroded submarine pipeline subjected to external pressure is analyzed. A semi-analytical approach for the elastic–plastic buckling pressure of a pipe with asymmetric local wall thinning is developed. Based on this method, the buckling pressure is calculated by solving the characteristic equations of different buckling modes. When the depth of internal and external local wall thinning is equal, the semi-analytical approach can be used to determine the buckling pressure of pipe with symmetrical corrosion defect, and it also can be used for the buckling analysis of the pipe containing only an internal or external corrosion defect by making the external or internal defect depth being zero. Then, the accuracy and necessity of the proposed approach are verified by comparing with the finite element modeling, experimental tests and previous equations, respectively. The effects of various parameters, viz., radius-to-thickness ratio, defect angle and depth, corrosion defect position, and yield stress, on the buckling pressure are investigated in details. The applicability of the semi-analytical approach is discussed under the condition of both internal and external pressure and different defect lengths. Finally, the post-buckling behavior of corroded pipe is investigated in details.