Post-buckling Stiffness Research Articles

Postbuckling Strengths of Composite Laminates with a Central Circular/Elliptical Cutout Under In-Plane Combined Loads

This paper deals with the effect of cutouts (i.e., circular and elliptical) aspect ratio on the buckling and postbuckling strengths, and failure characteristics of a simply-supported quasi-isotropic composite laminate subjected to in-plane shear and uni-axial compression combined with in-plane shear loads. The present study is carried out using self developed finite element program. The finite element formulation is based on the first order shear deformation theory and von Karmans assumptions to incorporate geometric nonlinearity. The resulting nonlinear algebraic equations are solved by Newton-Raphson method. The postbuckling strength of the laminate is defined in terms of the first-ply failure load of the lamina predicted by tensor polynomial form of the 3-D Tsai-Hill criterion. It is observed that for maximum buckling and first-ply failure strengths of the laminate with a cutout, it is preferred to have elliptical shape of the cutout as compared to the circular shape of the same area. It is also found that irrespective of load conditions and cutout shapes and sizes, shear load direction has significant effects on buckling and postbuckling stiffness of the laminate; although buckling stiffness is greater under negative shear, but the laminate shows stiffer postbuckling response under positive shear load.

The thermal-mechanical buckling and postbuckling design of composite laminated plate using a ROM-driven optimization method

The high temperature environment makes the buckling of composite laminated plate happen earlier. Further weight savings can be achieved once the plate has a favorable postbuckling stiffness. This study aims to address this issue by developing a ROM-driven lamination optimization to maximize the buckling and postbuckling performance of compressed plates under initial temperature field. The reduced-order model(ROM) constructed using the improved Koiter theory is reformulated to be suitable for thermal-mechanical buckling problems. A linear subspace of the force space is represented to be the span of the mechanical load, the thermal load and a set of predefined perturbation loads. The thermal load related to the initial temperature field is constant during the increase of in-plane compressive load. An independently additional degree of freedom corresponding to the thermal load is implemented into the construction of the reduced-order model. The buckling and postbuckling performance can be extracted from the solution of reduced-order model, and then subsequently selected as inputs to achieve the optimal laminate. The main novelty of this work is to achieve the thermal-mechanical buckling and postbuckling optimization, benefitting from the highly efficient single run of thermoelastic geometrically nonlinear analysis. Various numerical examples have been selected to validate the performance of the proposed method, considering the influences of length-to-width ratio, α 1/α 2, ply number and nonuniform temperature field.

Lightweight Design of Variable-Stiffness Cylinders with Reduced Imperfection Sensitivity Enabled by Continuous Tow Shearing and Machine Learning.

The present study investigates how to apply continuous tow shearing (CTS) in a manufacturable design parameterization to obtain reduced imperfection sensitivity in lightweight, cylindrical shell designs. The asymptotic nonlinear method developed by Koiter is applied to predict the post-buckled stiffness, whose index is constrained to be positive in the optimal design, together with a minimum design load. The performance of three machine learning methods, namely, Support Vector Machine, Kriging, and Random Forest, are compared as drivers to the optimization towards lightweight designs. The new methodology consists of contributions in the areas of problem modeling, the selection of machine learning strategies, and an optimization formulation that results in optimal designs around the compromise frontier between mass and stiffness. The proposed ML-based framework proved to be able to solve the inverse problem for which a target design load is given as input, returning as output lightweight designs with reduced imperfection sensitivity. The results obtained are compatible with the existing literature where hoop-oriented reinforcements were added to obtain reduced imperfection sensitivity in composite cylinders.

An efficient method for postbuckling optimization of composite laminated plates with holes under compression and shear loading

Laminated composite plates with hole are commonly used in aeronautic and aerospace structure design, subjecting to the in-plane compression and shear loading. The existing of hole reduces both the in-plane and out-of-plane stiffness and influences the postbuckling behavior of the plate. The structural lightweight design can be further explored when the external load is allowed entering into the postbuckling regime. However, the lamination optimizations to maximize the postbuckling stiffness have high computational burden when the standard nonlinear finite element method is applied in each function call for postbuckling analysis. Thus, we propose a reduced-order modeling method based on the improved Koiter perturbation theory for buckling and postbuckling analysis. The low-scale reduced order model, instead of the full-scale FE model, is constructed and solved to obtain the initial postbuckling response. The postbuckling stiffness is defined to be the slope of the load-end shortening relation curve at the bifurcation point. A multi-start searching strategy based on the gradient-based algorithm is developed to maximize the postbuckling stiffness, efficiently and accurately. Laminated composite plates with different hole sizes, loading conditions and ply numbers, are considered to validate the excellent performance of proposed method.

Nonlinear Finite Element Analysis of Thin-Walled Beam on Pre- and Post-Buckling Stiffness and Gauge Sensitivity Index

Lightweight design has a significant impact on reducing fuel consumption and harmful emission of conventional vehicles and improving driving range of electric vehicles. Reducing the thickness of components in vehicle bodies and closures is an efficient approach for weight reduction. Thickness reduction, however, will reduce structural stiffness, especially in the presence of lateral displacements of buckling when critical stress is reached. In this paper, nonlinear FEA models of a thin-walled beam with variable thickness are developed for calculating the changes of beam stiffness as to thickness reduction in the pre- and post-buckling stages. Next, these stiffness values are used to calculate gauge sensitivity of the beam, which changes with respect to beam thickness changes. It is concluded that the presence of buckling will reduce the beam stiffness, worsen the stress uniformity, and increase the gauge sensitivity value of the beam.

Buckling and postbuckling performance of composite fuselage panels with cutouts using continuous streamline fibres

In this paper the buckling and postbuckling behaviour of variable-stiffness composite fuselage panels with window openings based on streamline fibres is addressed. Streamlines of a fluid flow around a given body that represents the cutout are adopted to define the fibre trajectories that curve around the cutout in a continuous fashion. Plates and shells with circular and elliptical cutouts subjected to compressive and shear loads are investigated. Various design scenarios using different laminate layups are considered. As a first step, an optimum stacking sequence is obtained for maximum buckling capacity. Then, a postbuckling analysis is performed to evaluate the initial postbuckling stiffness that represents the strength of the laminates beyond buckling. The buckling and postbuckling performance of panels with streamline fibres is compared with that obtained for traditional panels with straight fibres. It is shown that significant improvements in the buckling capacity can be achieved using streamline fibres instead of straight ones. Moreover, laminates with streamline fibres optimized for maximum buckling loads exhibit a higher postbuckling strength than do the laminates with straight fibres.

An enhanced reduced-order model based optimization method for buckling and postbuckling performances of variable stiffness composite plates

The great potential of variable stiffness laminates in maximizing buckling load is obvious over straight fiber laminates, however the optimum design of such VAT laminates which has much more design variables and larger computational efforts, is still a challenge even for the modern computational method and capability. In addition, the postbuckling performance and the correlations with buckling resistance still need to be further studied for VAT laminates. This work proposes an enhanced reduced-order model based on the improved Koiter perturbation theory for buckling and postbuckling analysis of variable stiffness composite plates. The VAT laminas composed by “two-segment” fibers and “four-segment” fibers are considered, respectively, in construction of the reduced-order finite element model. The nonlinear prediction for initial postbuckling response of VAT laminate can be achieved, by solving the small-scaled reduced-order model. The nonlinear buckling load, postbuckling stiffness and postbuckling stiffness residual factor, are accurately and efficiently extracted from the initial postbuckling predictor, and then subsequently to be selected as inputs to the GA based optimization process for the design of VAT laminates. Extensive numerical investigations validate that the proposed method can find a combination of several varied angle fiber layers with constant thicknesses such that the combination results in an optimal stiffness distribution and provides a more favorable structural performance in both buckling and postbuckling regime for different demands of structure design.

Buckling behaviour of laminated viscoelastic composites under axial loads

This study investigates the buckling response of viscoelastic composite laminates under compression with the objective of better understanding the ply wrinkling behaviour at the early stage of thermoset composite manufacturing. The composite laminate is modelled as a 3-D viscoelastic solid using two different approaches in Abaqus©. As the application of the built-in viscoelastic model of Abaqus© is limited to isotropic materials, a more versatile orthotropic viscoelastic constitutive model based on differential form (DF) of viscoelasticity implemented as a user material subroutine (UMAT) is employed to elucidate the effect of ply anisotropy on the buckling response of uncured/partially cured unidirectional laminates. Numerical results are compared with the experimental data available in the literature to evaluate the accuracy of the proposed approach. Using the numerical model, the influence of various parameters including loading rates, the instantaneous elastic modulus, and in particular the effects of the ply anisotropy on the buckling behaviour of composite laminates are analysed. It has been found that unlike cured composites, the compressive stiffness of uncured prepregs is mainly dominated by the resin modulus rather than the fibre modulus. Additionally, it is shown that the waviness of fibres (fibre-bed effect) which stiffens the prepreg's transverse and shear properties increases its compressive stiffness slightly while the waviness effect on the post buckling stiffness is relatively significant. Hence, conducting compressive tests on uncured and partially cured prepregs as well as partially cured resin films with different thicknesses is suggested to validate the proposed modelling approach more rigorously.

Lateral-torsional buckling behaviour of concrete-filled high-strength steel tubular flange beams under mid-span load

Lateral-torsional buckling (LTB) behaviour of concrete-filled high-strength steel tubular flange beams (HS-CFTFBs) is experimentally and numerically investigated. Six specimens are tested in simple support with a lateral unrestrained concentrated load in mid-span. Among them, two are failure controlled through flexural yielding (FY), whereas the remaining specimens are controlled through LTB. Finite-element (FE) models are then constructed and the comparison with the experimental results indicates that they could reliably and accurately evaluate failure modes, load–displacement relationship and ultimate capacities of HS-CFTFBs. These verified FE models are consequently used to investigate the effect of flange depth, concrete and steel with various span lengths. Results suggest that the ultimate moment capacity evidently decreases as the span length increased, particularly for the high-strength steel beams. The increasing flange depth remarkably improved the LTB strength but merely slightly increased the amount of steel. The infilled concrete could improve the resistance to flange distortion and thus increase the ultimate capacity. However, the strength of concrete exhibits a slight influence on ultimate capacity. Increasing fy could improve the LTB capacity when the span length is relatively small, but the improvements decrease with increasing span length. When the failure mode is elastic LTB, increased fy could substantially improve post-buckling stiffness and ductility

Hard auxetic metamaterials

Auxetics are materials that contract laterally when compressed, rather than expand, in contrast to common experience. Here we show that common metals and plastics can be rendered auxetic through the introduction of a regular array of holes. Under compression, these hard holey materials bypass localized failure modes, such as shear banding, and instead deform via a global pattern transformation previously reported in elastomeric structures. Despite significant variations in internal structure, the pattern transformation responsible for auxetic behaviour in both metals and plastics is governed by the buckling of the slender struts that comprise the microarchitecture. Furthermore, in contrast to elastomeric structures, holey sheets made from hard materials exhibit significant negative post-buckling stiffness. This suggests that, beyond the geometrical nonlinearities associated with topological modifications, material nonlinearities which arise during plastic deformation offer further potential for altering the material properties of the constituent.

Exact analytical solutions to the problem of relative post-buckling stiffness of thin nonlocal graphene sheets

Abstract The main purpose of this research is to obtain analytical formulations for exact calculation of relative post-buckling stiffness of nonlocal graphene sheets. In addition to calculating the post-buckling stiffness reduction, the buckling and initial post-buckling responses of these structures, when they are subjected to end-shortening strain, have also been studied. To investigate these phenomena, a new technique called semi-Galerkin technique is used in which the out-of-plane deflection function is firstly postulated as the only displacement field and then, exact nonlocal stress function is calculated through a complete solution of the von-Karman compatibility equation. Finally, Galerkin's method is used to solve the unknown parameter considered in the proposed technique. The nano-sheets are modeled as an orthotropic layer with Kirchhoff assumptions and nonlocal differential elasticity theory is employed to achieve the buckling loads and exact relative stiffness values. For in-plane movements of the longitudinal edges of the nano-sheets, two essential and natural boundary conditions are adopted to be “straightly movable” or “freely movable”. The effects of aspect ratio and nonlocal parameter have been studied for each type of boundary conditions and for graphene sheets with different materials. The stress distribution along the length and the width of the nano-sheets is also investigated and discussed by the two local and nonlocal theories and for various values of nonlocal parameter.

Planar Mechanical Metamaterials with Embedded Permanent Magnets.

The design space of mechanical metamaterials can be drastically enriched by the employment of non-mechanical interactions between unit cells. Here, the mechanical behavior of planar metamaterials consisting of rotating squares is controlled through the periodic embedment of modified elementary cells with attractive and repulsive configurations of the magnets. The proposed design of mechanical metamaterials produced by three-dimensional printing enables the efficient and quick reprogramming of their mechanical properties through the insertion of the magnets into various locations within the metamaterial. Experimental and numerical studies reveal that under equibiaxial compression various mechanical characteristics, such as buckling strain and post-buckling stiffness, can be finely tuned through the rational placement of the magnets. Moreover, this strategy is shown to be efficient in introducing bistability into the metamaterial with an initially single equilibrium state.

Post-Buckling Analysis of a Rod Confined in a Cylindrical Tube

Understanding the buckling and post-buckling behavior of rods confined in a finite space is of both scientific and engineering significance. Under uniaxial compression, an initially straight and slender rod confined in a tube may buckle into a sinusoidal shape and subsequently evolve into a complicated, three-dimensional (3D) helical shape. In this paper, we combine theoretical and numerical methods to investigate the post-buckling behavior of confined rods. Two theoretical models, which are based on the inextensible and extensible rod theories, respectively, are proposed to derive the analytical expressions for the axial compressive stiffness in the sinusoidal post-buckling stage. The former is concise in formulation and can be easily applied in engineering, while the latter works well in a broader scope of post-buckling analysis. Both methods can give a good approximation to the sinusoidal post-buckling stiffness and the former model is proved to be a zeroth-order approximation of the latter. The flexible multibody dynamics method based on the Timoshenko's geometrically exact beam theory is used to examine the accuracy of the two models. The methods presented in this work can be used in, for example, drilling engineering in oil and gas industries.

Seismic upgrading of reinforced concrete frames with steel plate shear walls

The objective of this paper is to report on a study of the use of unstiffened thin steel plate shear walls (SPSWs) for the seismic performance improvement of reinforced concrete frames with deficient lateral rigidity. The behaviour of reinforced concrete frames during seismic activities was rehabilitated with an alternative and occupant-friendly retrofitting scheme. The study involved tests of eight 1/3 scale, one bay, two storey test specimens under cyclic quasi-static lateral loadings. The first specimen, tested in previous test program, was a reference specimen, and in seven other specimens, steel infill plates were used to replace the conventional infill brick or the concrete panels. The identification of the load-deformation characteristics, the determination of the level of improvement in the overall strength, and the elastic post-buckling stiffness were the main issues investigated during the quasi-static test program. With the introduction of the SPSWs, it was observed that the strength, stiffness and energy absorption capacities were significantly improved. It was also observed that the experimental hysteresis curves were stable, and the composite systems showed excellent energy dissipation capacities due to the formation of a diagonal tension field action along with a diagonal compression buckling of the infill plates.

Optimization of postbuckling behaviour of variable thickness composite panels with variable angle tows: Towards “Buckle-Free” design concept

Variable angle tow (VAT) laminates that generally exhibit variable stiffness properties not only provide extended design freedom, but also offer beneficial stress distributions. In this paper, the prospect of VAT composite panels with significantly reduced loss of in-plane compressive stiffness in the postbuckled state in comparison with conventional structures, is studied. Specifically, we identify that both thickness and local fiber angle variation are required to effectively define “Buckle-Free” panels under compression loading. In this work, the postbuckling behaviour of variable thickness VAT composite panels is analysed using an efficient and robust semi-analytical approach. Most previous works on the postbuckling of VAT panels assume constant thickness. The additional benefits of tailoring thickness variation in the design of VAT composite panels are seldom studied. However, in the process of manufacturing VAT laminates, either by using the conventional Advanced fiber Placement (AFP) machine (tow overlap) or the newly developed Continuous Tow Shearing (CTS) process (tow shrink) thickness build-up is inevitable. The postbuckling optimization for the design of VAT layups is conducted by a two-level framework using lamination parameters as intermediate design variables. The objective is to determine optimal lamination parameters and thickness distributions for maximizing the axial compressive stiffness of VAT laminates that are loaded in the postbuckling regime. The thickness variation due to both manufacturing of VAT laminates and for where it is independent of manufacturing process are considered. In accordance with the first-level optimal postbuckling solutions in terms of lamination parameters, we investigate a practical “Buckle-Free” VAT panel using a blended layup configuration. This blended VAT panel consists of a piecewise combination of segmental CTS layers and constant-thickness VAT layers. The prospect of taking advantage of a benign combination of stiffness and thickness to improve the overall compressive strength of VAT panels is studied. Finally, the optimal results are analysed to provide insight into the manufacturing of VAT laminates using either the AFP or the CTS process for improved postbuckling stiffness under compression loading.

Imperfection sensitivity of the size-dependent postbuckling response of pressurized FGM nanoshells in thermal environments

The purpose of the current study is to address the nonlinear buckling and postbuckling response of nanoscaled cylindrical shells made of functionally graded material (FGM) under hydrostatic pressure aiming to investigate the sensitivity to the initial geometric imperfection in the presence of surface effects and thermal environments. According to a power law distribution, the material properties of the FGM nanoshell are considered change through the shell thickness. Also, the change in the position of physical neutral plane corresponding to different volume fractions is taken into account to eliminate the stretching-bending coupling terms. In order to acquire the size effect qualitatively, the well-known Gurtin-Murdoch elasticity theory is incorporated within the framework of the classical shell theory. Using the variational approach, the non-classical governing equations are displayed and deduced to boundary layer type ones. Afterwards, explicit expressions for the size-dependent radial postbuckling equilibrium paths of imperfect FGM nanoshells are proposed with the aid of a perturbation-based solution methodology. It is displayed that by moving from the ceramic phase to the metal one, the critical buckling pressure decreases, but the postbuckling stiffness increases, because in contrast to the ceramic phase, the surface modulus and residual surface stress associated with the metal phase have the same sign.

Experimental Investigation of GFRP Laminate Thin-Walled Column Subjected to Uniform Compression

This paper presents the results of experimental investigation of thin-walled GFRP laminate columns with square cross-section subjected to uniform compression. Column under consideration are made of eight ply laminate with different quasi-isotropic layer arrangement. The following three cases of layers arrangement are taken into consideration: CS1 [45/-45/45/-45]S, CS2 [45/-45/0/0]Sand CS3 [45/-45/45/0]S. The analysed columns have the length of 250 mm and following cross-section dimensions (width × height × thickness of the wall): 82 × 82 × 2.08 mm.The main aim of experimental investigation has been to compare the way of buckling load and postbuckling behaviour determinations. The layer arrangement influence on buckling load and postbuckling stiffness have been also checked. Two different methods (P-w2method and "inflection point" method) of determining buckling forcePcrand two different measurement technique (i.e. strain-gauges technique and stereographic transformations technique) have been employed.

An equivalent stiffness (ES) method for initial design of tube-based energy absorbers under lateral quasi-static compression

A tube is often applied as an energy absorber in structures subject to impact loads. However, when several tubes are combined to absorb energy, how to arrange these tubes and match their dimensions has not been investigated. We provide a novel equivalent stiffness (ES) method for the initial design of tube-based energy absorbers, in which each tube is expressed by an equivalent spring with average post-buckling stiffness, and these springs are connected in series or parallel or hybrid. The plastic deformation and strain energy are distributed among them according to their relative ES value. The relationship between plastic energy and post-buckling stiffness is derived analytically and verified numerically. Finally, an application example is provided and the optimal tube arrangement and dimensions are obtained by combining the proposed ES method with an optimization code. The results demonstrate that this method can provide an initial design of tube-based energy absorbers efficiently.

Post-buckling finite strip analysis of thick functionally graded plates

In this paper, a novel semi-energy finite strip method (FSM) is developed based on the concept of first order shear deformation theory (FSDT) in order to attempt the post-buckling solution for thin and relatively thick functionally graded (FG) plates under uniform end-shortening. In order to study the effects of through-the-thickness shear stresses on the post-buckling behavior of FG plates, two previously developed finite strip methods, i.e., semi-energy FSM based on the concept of classical laminated plate theory (CLPT) and a CLPT full-energy FSM, are also implemented. Moreover, the effects of aspect ratio on initial post-buckling stiffness of FG rectangular plates are studied. It has been shown that the variation of the ratio of initial post-buckling stiffness to pre-buckling stiffness (S * /S) with respect to aspects ratios is quite independent of volume fractions of constituents in thin FG plates. It has also been seen that the universal curve representing the variation of (S * /S) with aspect ratio of a FG plate demonstrate a saw shape curve. Moreover, it is revealed that for the thin FG plates in contrast to relatively thick plates, the variations of non- dimensional load versus end-shortening is independent of ceramic-metal volume fraction index. This means that the post-buckling behavior of thin FG plates and the thin pure isotropic plates is similar. The results are discussed in detail and compared with those obtained from finite element method (FEM) of analysis. The study of the results may have a great influence in design of FG plates encountering post-buckling behavior.

An exact finite strip for the calculation of initial post-buckling stiffness of shear-deformable composite laminated plates

The first order shear deformation theory (FSDT) is used to present the buckling and initial post-buckling characteristics of symmetrically cross-ply laminated plates. In the buckling phase the Von-Karman’s equilibrium equation is solved exactly for the FSDT to obtain out-of-plane mode shapes and critical loads. The current post-buckling study is effectively a single-mode analysis, which is attempted by utilizing the so-called semi-energy method. The Von-Karman’s compatibility equation is solved exactly in the post-buckling phase with the assumption that the deflected form immediately after buckling is the same as that obtained for buckling. The Principle of Minimum Potential Energy is invoked to solve for the unknown coefficients in the assumed out-of-plane deflections and rotation functions.