Buckling Model Research Articles

A bi-Helmholtz type of two-phase nonlocal integral model for buckling of Bernoulli-Euler beams under non-uniform temperature

It is well-acknowledged by the scientific community that Eringen’s nonlocal integral theory is not applicable to nanostructures of engineering interest due to conflict between equilibrium and constitutive requirements. In this paper, a well-posed two-phase nonlocal integral elasticity with the bi-Helmholtz kernel is developed to study the size-dependent buckling response of Bernoulli-Euler beams under non-uniform temperatures. The governing equation is derived by invoking the variational principle of virtual work, and the temperature effect is equivalent to the thermal load along the axial direction, which is determined by nonlocal heat conduction. The two-phase nonlocal integral constitutive equation is transformed into a differential one equipped with four constitutive boundary conditions, and then exact solutions for the buckling loads of the beam with various boundary edges are obtained. Numerical results are validated by comparing them with those from local elasticity. Moreover, the effects of parameters related to the two-phase nonlocal elastic model and the nonlocal heat conductive model are investigated.

Buckling Behavior of Thin-Walled Stainless-Steel Lining Wrapped in Water-Supply Pipe under Negative Pressure

This paper presents a study about the buckling behavior of thin stainless-steel lining (SSL) for trenchless repair of urban water supply networks under negative pressure. The critical buckling pressure and displacement (p–δ) curves, temperature changing curves, hoop and axial strain of the lining monitoring section and the strain changes with system pressure (p–ε) of the lining under the action of different diameters, different lining wall thickness and different ventilation modes were obtained through five groups of full-scale tests. The variation principles of the post-buckling pressure and the reduction regularity of the flowing section of the lining were further investigated. By comparing different pipeline buckling models and introducing thin-shell theory, the buckling model of liner supported by existing pipe was established. The comparison between the test results and thin-shell theory indicates that one of the significances of the enhancement coefficient k value is to change the constraint condition of the aspect ratio, l/R, thus increasing the critical buckling pressure of the lining. Finally, an improved lining buckling prediction model (enhancement model) is presented. A previous test is used as a case study with the results showing that the enhanced model is able to predict critical buckling pressure and lobe-starting amount of the liner, which can provide guidance for trenchless restoration of the liner with thin-walled stainless steel.

Nonlocal vibration and buckling of two-dimensional layered quasicrystal nanoplates embedded in an elastic medium

A mathematical model for nonlocal vibration and buckling of embedded two-dimensional (2D) decagonal quasicrystal (QC) layered nanoplates is proposed. The Pasternak-type foundation is used to simulate the interaction between the nanoplates and the elastic medium. The exact solutions of the nonlocal vibration frequency and buckling critical load of the 2D decagonal QC layered nanoplates are obtained by solving the eigensystem and using the propagator matrix method. The present three-dimensional (3D) exact solution can predict correctly the nature frequencies and critical loads of the nanoplates as compared with previous thin-plate and medium-thick-plate theories. Numerical examples are provided to display the effects of the quasiperiodic direction, length-to-width ratio, thickness of the nanoplates, nonlocal parameter, stacking sequence, and medium elasticity on the vibration frequency and critical buckling load of the 2D decagonal QC nanoplates. The results show that the effects of the quasiperiodic direction on the vibration frequency and critical buckling load depend on the length-to-width ratio of the nanoplates. The thickness of the nanoplate and the elasticity of the surrounding medium can be adjusted for optimal frequency and critical buckling load of the nanoplate. This feature is useful since the frequency and critical buckling load of the 2D decagonal QCs as coating materials of plate structures can now be tuned as one desire.

Optimal design of trapezoid stiffeners of composite cylindrical shells subjected to hydrostatic pressure

Optimization of a composite cylindrical pressure hull subjected to hydrostatic pressure with trapezoidal stiffeners is considered in this paper. The composite cylindrical shell is fabricated with carbon fiber reinforced epoxy composite while the stiffener is made of aluminum alloy. First of all, the analytical buckling model for composite cylinders with stiffeners subjected to hydrostatic pressure is derived. Subsequently, the finite element method is used to verify the accuracy of the analytical solution through some examples. After verification, the analytical solution is coupled with the genetic algorithm and then used to optimize the cross-sectional shape of the stiffeners of the composite pressure hull to obtain the maximum buckling pressure. During the optimization process, the relationship between the buckling pressure and each geometrical parameter of the stiffener has been analyzed. The inertia moment of the stiffener section is found to have a good linear relationship with the buckling pressure. Therefore, the inertia moment is used as an objective function to optimize the cross-sectional shape of the stiffeners of the composite pressure hull, and the feasibility of this method has been proved by comparing the optimization results. With high efficiency, this method can be applied to the optimal design of the stiffener shape of a composite pressure shell.

A size-dependent thermal buckling model for micro-beams based on modified gradient elasticity

To investigate the size effect in thermal buckling behaviors of micro-beams, a new size-dependent thermal buckling model based on modified gradient elasticity is presented. The governing equation and boundary conditions of the model are derived by applying the variation principle. Numerical examples of different supported beams are solved. The critical buckling temperature rises and normalized deflection of different modes are presented. The results show that the smaller the beam size is, the bigger the critical buckling temperature rise is. The internal length scales significantly affect the critical buckling temperature rises of each mode, and this effect is mainly controlled by the strain gradient on thickness direction. The difference of critical buckling temperature rises caused by choosing different higher-order boundary conditions cannot be ignored. Compared with other size-dependent models, the presented model is featured by evidently physical meaning and simple form.

Non-linear finite element optimization for inelastic buckling modelling of smooth rebars

This paper presents an optimization methodology to simulate the monotonic and cyclic response of steel reinforcement smooth bars when subjected to inelastic buckling. A finite element (FE) model of steel rebars, based on non-linear fibre sections and an initial geometrical imperfection, is adopted. The multi-step optimization proposed herein to identify the main parameters of the material constitutive models is based on genetic algorithms (GA) and Bayesian model updating. The methodology consists of comparing available experimental tests from literature with the corresponding numerical results. New empirical relationships and probabilistic distributions of the optimized model parameters, such as post-yielding hardening ratio, isotropic hardening in compression and tension, plus initial curvature, are presented. Finally, utilizing both the GA-based and Bayesian-based calibration, an improvement of an existing analytical model for inelastic buckling of smooth steel rebars is proposed. Such analytical modelling can be efficient and reliable for future building codes and assessment guidelines for existing buildings.

A multi-scale model for global buckling and local wrinkling interaction with application to sandwich beams

In this work, we present a new 2D Reduced model capable of taking into account the coupling between local instabilities and global buckling which occur in sandwich structures. To establish this model, we use a multi-scale approach based on the technique of Fourier series with slowly varying coefficients which allows transforming a 2D sandwich model into 2D Reduced model, of which only the envelope of instability is numerically evaluated. It is like the Ginzburg–Landau envelope equation, but it provides the post buckling behavior and takes into account the coupling between global and local instabilities. The proposed model is validated by comparison with the 2D Complete model and the analytical solution.The non-linear system is solved by the Asymptotic Numerical Method (ANM), which is an efficient, robust and accurate continuation method for solving non-linear systems.

NONLOCAL CONTINUUM ADAPTIVE ELASTIC BONE-COLUMN BUCKLING MODEL

It is well argued that stability-initiated failure dominates, especially in older bone, because of the instability of single trabeculae which is prone to inelastic buckling at stresses far less than expected for strength-based failure. It is also well known that when several horizontal struts have disappeared, trabecula fails due to compression-buckling load. In this contribution, our main goal is to improve, from theoretical point of view, the mechanistic understanding of bone buckling failure which is known to be at the core of important clinical problems. For that and with respect to previous works, an attempt is made in order to establish a simplified adaptive-beam buckling model, formulated within the context of the nonlocal adaptive continuum mechanics, from which numerical computations were performed in order to get a better knowledge about bone-column buckling mechanism affected by both bone density and bone density gradient distributions restricted to Euler–Bernoulli beam theory. An attempt is made to compare the experimental data with the response of our simplified model. For that, controlled buckling tests of single trabeculae were carried out from three medial tibia end sections (knee joint).

On the elastic buckling of cross-ply composite closed cylindrical shell under hydrostatic pressure

This paper focuses on the elastic buckling of cross-ply composite closed cylindrical shell subjected to hydrostatic pressure. The analytical model for buckling of composite cylindrical shell is derived by using the First Order Shear Deformation Theory (FOSDT) and considering the pressure stiffness. Several types of shells including thin, thick, long, and short shells are investigated. Two composite materials i.e. Carbon/Epoxy and Glass/Epoxy and several layup configurations are employed. The results of the derived FOSDT-based model are compared with those obtained from a Shell Buckling Equation (SBE) used in earlier studies and the Classical Shell Theory (CST). For verification, the results of the derived FOSDT-based model are compared with the results found in literature and also with the results obtained from finite element analysis. The difference among the results of the SBE and FOSDT, and the CST and FOSDT increases with the increase of the thickness in the case of all layup configurations and materials. For thin shells, the difference between the results of the CST and FOSDT almost vanishes, while for thin and large shells, the difference between the results of the SBE and FOSDT is still significant.

Dynamic electromagnetic buckling analysis of pulsed magnets

The dynamic electromagnetic buckling behaviors of pulsed magnets are investigated by both theoretical analysis and experiments for the first time in this paper. The electromagnetic dynamic buckling models of pulsed magnets, which include the dynamic effects of inertia, stress wave propagation, etc., are developed to analyze the effect of the practical operating conditions. Representative coils have been developed and tested by the discharge experiments of high voltage and current, and the theoretical results are highly consistent with experimental results, which shows that the models proposed in this paper are accurate, effective and reliable. Moreover, the experimental results are the direct and solid evidence for the existence of buckling in pulsed magnets. It is found that the windings in different locations of pulsed magnets exhibit a variety of buckling behaviors from Euler buckling to asymmetric collapse, which reflect the complexity of the buckling analysis of pulsed magnets. Lastly, the influences of Lorentz force distribution, inertia effect, stress wave propagation, material properties and geometry to the buckling behaviors are also discussed based on the proposed models, and a laminated laying method of fiber reinforcement is proposed to improve the buckling strength of pulsed magnets.

Numerical models of two-dimensional buckling and bending mechanisms and implications for oroclines

The question of whether oroclines form by orogen-parallel buckling or orogen-perpendicular bending is a matter of debate. To address this problem, we conducted two-dimensional numerical models that investigate differences in the characteristics of oroclines caused by buckling, bending or a combination thereof. Our models consider a simple setup of a viscous layer embedded in a less viscous matrix and subjected to buckling and bending conditions. Models were run with varied layer thicknesses and viscosity contrasts. We applied a spectrum of deformation regimes, with pure buckling and pure bending as end members. Buckling models produced significant layer compression in inner fold hinges, whereas bending models produced widespread extension. Fold geometry also differed between buckling and bending models, as observed through relationships in fold amplitude, fold width, layer thickness, and interlimb angles. A comparison of the numerical results with three real-world examples (Kazakhstan, Gibraltar and Cantabrian oroclines) reveals significant similarities in geometrical features (particularly for the Kazakhstan and Gibraltar oroclines), and some discrepancies. The models show that geometric relationships, such as amplitude-width, thickness-width, thickening, and interlimb angles, provide limited information on the deformation. However, the distribution of stress across the fold hinge and limbs is a primary factor that might be used to understand whether oroclines formed by buckling or bending.

High performance model for buckling of functionally graded sandwich beams using a new semi-analytical method

In this paper, a new semi-analytical approach based on the scaled boundary finite element method (SBFEM) is proposed to solve buckling problem of functionally gradient material (FGM) sandwich beams. Material properties of each individual layer are assumed to be continuously graded along the thickness according to a power law function with respect to the volume fractions. Based on the layer-wise theory, the two-dimensional constitutive model is directly performed in the proposed formulations without any kinematic assumptions of plate theory. The buckling governing equations of FGM sandwich beams based on the SBFEM are derived using the weighted residual method and solved by the means of the eigenvalue analysis. The advantage of the present approach is that only the longitudinal dimension of the beam structure needs to be discretized using one-dimensional higher-order spectral element so that it can be dealt with as a one-dimensional mechanical system while maintaining the analytical characteristics in the thickness direction, which possesses a key feature to make structure modelling more effective and accuracy. To evaluate the validity of proposed formulations, a series of numerical examples involving the element convergence and parametric analysis are carried out and the results are compared with existing solutions available in open literature. The numerical studies confirm the accuracy and adaptability of the presented method for the buckling analysis of FGM sandwich plates.

The helical buckling and extended reach limit of coiled tubing with initial bending curvature in horizontal wellbores

The extended reach limit of coiled tubing (CT) in horizontal wellbores is an important factor restricting the downhole engineering operations of unconventional oil and gas wells. This paper establishes an extended reach limit calculation method of CT with initial bending curvature in a horizontal wellbore. First, this paper assumes that when the initial bending curvature is lower than a certain value, which is called the critical initial bending curvature, the CT will show a sinusoidal buckling configuration when tripping into a horizontal wellbore; otherwise, it will show a helical buckling configuration. Second, based on these two initial configurations, a buckling model and a contact force model are derived. A method of calculating the critical initial bending curvature is also given. Third, a calculation method of the extended reach limit of CT is given on the basis of the buckling and contact force models. The calculation results show that the results from the methodology presented in this paper are consistent with published experimental data. Finally, based on this method, the factors that may affect the extended reach limit of CT are analyzed. The results show that reducing the initial bending curvature or friction factor can effectively increase the extended reach limit of CT. Therefore, the methods presented in this paper can be used to not only predict the extended reach limit of CT but also optimize downhole engineering design parameters.

The Influence of Axial Compression Ratio on the Seismic Behavior of RC Frame Column

In order to analyze the quantitative influence of the axial compression ratio on the seismic performance of reinforced concrete (RC) frame column, this paper carried out the numerical analysis of the quasi-static simulation of a RC frame column and analyzed its hysteresis performance, bearing capacity, stiffness degradation, energy consumption capacity and ductility capacity under different axial compression ratios, based on OpenSees finite element software, considering the buckling and fatigue damage model. The results show that the smaller the axial compression ratio, the fuller the hysteresis curve and the slower the stiffness degradation of the column. The ultimate bearing capacity of the column increases by 24.6% when the axial compression ratio increases from 0.3 to 0.8, but bearing capacity decreasing and stiffness degradation is faster and faster. When the deformation does not exceed the ultimate displacement, the equivalent viscous damping at each displacement level of high axial compression ratio column is greater than that of the low axial compression ratio, but the total hysteretic energy decreases about 64.2% at maximum amplitude. The ultimate displacement gradually decreases with the increase in the axial compression ratio, and the ductility factor decreases about 55.9% at maximum amplitude. The results can be referenced by the seismic design and analysis of RC frame.

Modified strip model for indirect buckling restrained shear panel dampers

The indirect buckling restrained shear panel damper (IBRSPD) comprises stiffeners and energy dissipation units that are processed from a hot-rolled H-beam. Under shear loading, the web plates of the units induce a stable energy dissipation capacity to resist earthquakes, and the stiffeners can effectively limit the buckling of the web panels. Then, the partial tension field (PTF) is generated in the web plates owing to the indirect buckling restraint, which leads to the free boundary condition in the vertical edges. This mechanism has not been considered in previous models of the shear panel damper (SPD). In this study, a modified strip (MS) model is proposed to characterize the web plate behavior, and equations for the inclination angle of the PTF action are presented. According to the development degree of the PTF area, four stages are determined, and the components of the shear strength, namely, the shear buckling strength (SBS) and the PTF action, are calculated for each stage. Through monotonic analyses, a comparison between the finite element model and the MS model is conducted to verify the validation of the newly proposed model. In addition, the contributions of the SBS and PTF components are analyzed under cyclic loading while considering the loading history and the pinching effect. With respect to the energy dissipation capacity, the height of the IBRSPD is discussed in relation to the energy dissipation efficiency. Finally, the MS model is adopted in a seismic analysis to investigate the reduction of the seismic response.

Experimental study on seismic behavior of double-skin composite wall with L-shaped connectors

In this paper, the authors' research group propose a novel double-skin composite wall (DSCW), which is composed of boundary columns, steel faceplates, L-shaped connectors and infill concrete. To further study the seismic performance of DSCW specimens, six high shear-span ratio DSCWs were designed and manufactured for a quasi-static test. In the experiment, the welding spacing, welding forms, and the thickness of the steel faceplate are used to set forth parameters; moreover, the failure mode, deformation performance and energy dissipation capacity of the DSCWs are analyzed. Test results show that the failure mode of each specimen was flexural failure, as shown by the fractured steel plate at the bottom of the boundary columns, the crushed infill concrete at the tear, and the specimens' buckled steel faceplates in different degrees. The average value of the yield drift ratio was 1/177; the average value of ultimate drift ratio was 1/39, and the average value of displacement ductility coefficient was 4.46. A good energy dissipation capacity was shown by the specimens as it increased gradually during the testing process. The results show that the local buckling of the steel faceplate can be delayed by reducing the welding spacing and adopting the blossom-form or dense bottom welding arrangement; thus, improve the ductility and energy dissipation capacity of the specimens. When the thickness of the steel faceplate is increased, the local buckling of the steel faceplate is less likely to occur, and the bearing capacity is improved. Based on the theoretical analysis, a buckling stress calculation model is proposed to calculate the buckling stress of the steel faceplate at the bottom of the specimen.

A bio-inspired robotic fish utilizes the snap-through buckling of its spine to generate accelerations of more than 20g

Inspired by the fastest observed live fishes, we have designed, built and tested a robotic fish that emulates the fast-start maneuver of these fishes and generates acceleration and velocity magnitudes comparable to those of the live fishes within the same time scale. We have designed the robotic fish such that it uses the snap-through bucking of its spine to generate the fast-start response. We have used a dynamic snap-through buckling model and a series of experiments on a beam under snap-through buckling to describe the robotic fish’s motion. Our under-actuated robot relies on passive dynamics of a continuous beam to generate organic waveforms. In its transient fast-start maneuver, our robotic fish produces mode shapes very similar to those observed in live fishes, by going through a snap-through bifurcation. We have also used a nonlinear structural model subjected to a non-conservative eccentric compressive force, which is constrained to act tangential to the structure at all times, coupled with a simple fluid dynamic model to approximate the transient behavior of the robot. We relate the numerical results from our nonlinear model to the dynamics observed in the live system proposing an updated kinematic model to understand the mode shapes observed in the fast-start maneuver of the live fishes. We also report on deploying the robotic fish in a river.

A simple closed-form analytical model for the column buckling of omega-stinger-stiffened panels with periodic boundary conditions

During the preliminary design of stiffened panels, the stability behaviour is critical and both buckling modes, global and local, have to be considered in order to avoid panel configurations in which the minimum stiffness of the stringers is not achieved. In the present work, a new simple computational model is presented that computes the critical column buckling load of an omega-stringer-stiffened panel with periodic boundary conditions. This is achieved by obtaining the effective stiffness properties for an equivalent composite column. The model is able to predict column buckling conservatively as numerical studies show and can easily be used, e.g. as a constraint for optimization studies.

Mixed Finite-Element Formulation Based on Refined Sinusoidal Model for Buckling of Layered Beams

In the last 3 decades, sinusoidal theory has been increasingly utilized to research mechanical behaviors of layered composite and sandwich structures. Nevertheless, the existing sinusoidal model will encounter trouble in precisely yielding the buckling loads of layered structures composed of layers with different material properties. Thus, a refined sinusoidal model is offered for the buckling analysis of composite and sandwich structures that can simulate the zigzag effect of the in-plane displacement and meet the free conditions of transverse shear stresses on the surfaces. In the light of the elegant sinusoidal model, a three-node beam element has been constructed to work out the discrete eigenvalue equation coming from the stability behavior. Making use of a mixed variational theorem from the literature, the finite-element formulation can meet beforehand the continuous conditions of transverse stress at the interfaces. The three-dimensional finite element method (3D-FEM) results are utilized to evaluate the precision and efficiency of the proposed approach through a numerical example. The proposed finite-element formulation can produce satisfactory results with lower calculational cost, and some interesting conclusions are presented. However, the results for stability of layered structures made of plies with different material properties will be overestimated by utilizing the models violating the continuous prerequisite of interlaminar stresses.

Effects of strain gradients in the onset of global buckling in slender walls due to earthquake loading

Global buckling of slender walls, reported only in a few laboratory tests before 2010, became a critical issue in design of reinforced concrete buildings after it was observed following the 2010 Mw 8.8 Chile earthquake and the 2011 Mw 6.3 New Zealand earthquake. Researchers have proposed theoretical buckling models based on prismatic columns subjected to uniform tension/compression cycles, where the key parameters are slenderness ratio, number of curtains of reinforcement, and maximum tensile strain before buckling during load reversal. These models have shown sufficient accuracy in comparison with laboratory tests on columns under such loading conditions. However, buckling in walls is more complex because of variation of strains through the wall depth and variation of moment along the wall height. Nonlinear finite elements are used to evaluate the effects of these more complex loadings on buckling of wall boundary elements. Analyses showed that the maximum tensile strain (averaged over the wall out-of-plane unsupported height) required to buckle the wall during load reversal does not depend on the moment variation along the wall height. Moreover, for typical wall lengths, the wall boundary behaves like an isolated column subjected to axial force cycles, with minimal apparent bracing provided by the wall web. This allows to analyze a broad range of practical cases for buckling susceptibility using simplified approaches based on buckling models of axially loaded columns.