Elastoplastic buckling and post-buckling analysis of sandwich columns

Sandwich structures are widely used in many industrial applications thanks to their interesting compromise between lightweight and high mechanical properties. This compromise is realized thanks to the presence of different parts in the composite material, namely the skins which are particularly thin and stiff relative to the homogeneous core material and possibly core reinforcements. Owing to these geometric and material features, sandwich structures are subject to global but also local buckling phenomena which are mainly responsible for their collapse. The buckling analysis of sandwich materials is therefore an important issue for their mechanical design. In this respect, this paper is devoted to the theoretical study of the local/global buckling and post-buckling behavior of sandwich columns under axial compression. Only symmetric sandwich materials are considered with homogeneous and isotropic core/skin layers. First, the buckling problem is analytically addressed, by solving the so-called bifurcation equation in a 3D framework. The bifurcation analysis is performed using an hybrid model (the two faces are represented by Euler–Bernoulli beams, whereas the core material is considered as a 2D continuous solid), considering both an elastic and elastoplastic core material. Closed-form expressions are derived for the critical loadings and the associated bifurcation modes. Then, the post-buckling response is numerically investigated using a 2D finite element bespoke program, including finite plasticity, arc-length methods and branch-switching procedures. The numerical computations enable us to validate the previous analytical solutions and describe several kinds of post-critical responses up to advanced states, depending on geometric and material parameters. In most cases, secondary bifurcations occur during the post-critical stage. These secondary modes are mainly due to the modal interaction phenomenon and give rise to unstable post-buckled solutions which lead to final collapse.     

Exact buckling solution for two-layer Timoshenko beams with interlayer slip

This paper deals with the buckling behavior of two-layer shear-deformable beams with partial interaction. The Timoshenko kinematic hypotheses are considered for both layers and the shear connection (no uplift is permitted) is represented by a continuous relationship between the interface shear flow and the corresponding slip. A set of differential equations is obtained from a general 3D bifurcation analysis, using the above assumptions. Original closed-form analytical solutions of the buckling load and mode of the composite beam under axial compression are derived for various boundary conditions. The new expressions of the critical loads are shown to be consistent with the ones corresponding to the Euler–Bernoulli beam theory, when transverse shear stiffnesses go to infinity. The proposed analytical formulae are validated using 2D finite element computations. Parametric analyses are performed, especially including the limiting cases of perfect bond and no bond. The effect of shear flexibility is particularly emphasized.     

轻质金属点阵夹层板热屈曲临界温度分析

本文针对均匀温度场下四边简支和四边固支金属点阵夹层板的临界热屈曲温度进行了求解和参数影响分析。将点阵夹芯等效为均匀连续体,并且将夹层板的剪切刚度近似为点阵夹芯的抗剪切刚度,忽略夹芯的抗弯刚度且认为夹层板主要由面板来提供抗弯刚度。对于无法获得解析解的四边固支条件,通过对未知变量进行双傅里叶展开的方法求解了Ressiner夹层板模型的临界屈曲温度,理论分析结果与有限元计算结果吻合良好。进一步分析了不同边界条件、点阵胞元构型、点阵材料相对密度、面板厚度等对临界屈曲温度的影响规律。     

Buckling analysis of sandwich plates with FGM face sheets resting on elastic foundation with various boundary conditions: an analytical approach

This paper presents an analytical investigation on the buckling analysis of symmetric sandwich plates with functionally graded material (FGM) face sheets resting on an elastic foundation based on the first-order shear deformation plate theory (FSDT) and subjected to mechanical, thermal and thermo-mechanical loads. The material properties of FGM face sheets are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic material. An analytical approach is used to reduce the governing equations of stability and then solved using an analytical solution which is named as power series Frobenius method for symmetric sandwich plates with six different boundary conditions. A detailed numerical study is carried out to examine the influence of the plate aspect ratio, side-to-thickness ratio, loading type, sandwich plate type, volume fraction index, elastic foundation coefficients and boundary conditions on the buckling response of FGM sandwich plates. This has not been done before and serves to fill the gap of knowledge in this area.     

Design optimization of truss-cored sandwiches with homogenization

Lightweight metallic sandwich plates comprising periodic truss cores and solid facesheets are optimally designed against minimum weights. Constitutive models of the truss core are developed using homogenization techniques which, together with effective single-layer sandwich approaches, form the basis of a two-dimensional (2D) single-layer sandwich model. The 2D model is employed to simulate the mechanical behaviors of truss-cored sandwich panels having a variety of core topologies. The types of loading considered include bending, transverse shear and in-plane compression. The validities of the 2D model predictions are checked against direct FE simulations on three-dimensional (3D) truss core sandwich structures. Optimizations using the 2D sandwich model are subsequently performed to determine the minimum weights of truss-cored sandwiches subjected to various failure constraints: overall and local buckling, yielding and facesheet wrinkling. The performances of the optimized truss core sandwiches with 4-rod unit cell and solid truss members and pyramidal unit cell with hollow truss members are compared with benchmark lightweight structures such as honeycomb-cored sandwiches, tetrahedral core sandwiches and hat-stiffened single layer plates.     

Structural response of pyramidal core sandwich columns

An experimental and analytical investigation is carried out to examine the in-plane compressive response of pyramidal truss core sandwich columns. The identified failure mechanisms include Euler buckling, shear buckling and face wrinkling. The operative mechanism is dependent on the properties of the bulk material and geometry of the sandwich columns and analytical formulae are derived for each of these modes. Failure maps are constructed for sandwich columns made from an elastic ideally-plastic material and AISI 304 stainless steel which has a strongly strain hardening response. Pyramidal core sandwich columns made from 304 stainless steel have been designed using these mechanism maps and the measured responses are compared with the analytical predictions. Finally, optimal single layer and multi-layer pyramidal sandwich column designs that minimize the weight for a given load carrying capacity are calculated using the developed analytical models for the failure of the sandwich columns. The results demonstrate that pyramidal core sandwich columns outperform the currently used hat-stiffened column design.     

Three-dimensional thermoelastic analysis of finite length laminated cylindrical panels with functionally graded layers

Based on the 3D thermoelasticity theory, the thermoelastic analysis of laminated cylindrical panels with finite length and functionally graded (FG) layers subjected to three-dimensional (3D) thermal loading are presented. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The variations of the field variables across the panel thickness are accurately modeled by using a layerwise differential quadrature (DQ) approach. After validating the approach, as an important application, two common types of FG sandwich cylindrical panels, namely, the sandwich panels with FG face sheets and homogeneous core and the sandwich panels with homogeneous face sheets and FG core are analyzed. The effect of micromechanical modeling of the material properties on the thermoelastic behavior of the panels is studied by comparing the results obtained using the rule of mixture and Mori–Tanaka scheme. The comparison studies reveal that the difference between the results of the two micromechanical models is very small and can be neglected. Then, the effects of different geometrical parameters, material graded index and also the temperature dependence of the material properties on the thermoelastic behavior of the FG sandwich cylindrical panels are carried out.     

Theoretical analysis of the deformation of SMP sandwich beam in flexure

Shape memory polymers (SMPs) can have a large frozen strain but with a very small recovery stiffness in comparison with shape memory metals or ceramics. To provide more deployable stresses for the application of actuators, sandwich beams consisting of a SMP core and two thin metallic skins were considered. The packaging behaviors of two types of SMP sandwich beams, aluminum/SMP/aluminum and steel/SMP/steel, were discussed. Due to the high compliance of SMP core on packaging condition that the testing temperature is above the activation temperature of the material, buckling and post-buckling are the essential deformation mechanisms of SMP sandwich beams under bending. Theoretical solutions were derived in studying such non-linear behaviors, including the initiation of critical buckling, post-buckling response, and final failure modes. Systematic parameter’s analyses, e.g., buckling half-wavelength, amplitude, location of the neutral-strain surface in different packaging curvatures, were also presented.     

Elastic structural response of prismatic metal sandwich tubes to internal moving pressure loading

The analytical and numerical modeling of the structural response of a prismatic metal sandwich tube subjected to internal moving pressure loading is investigated in this paper. The prismatic core is equivalent to homogeneous and cylindrical orthotropic solids via homogenization procedure. The sandwich tube with the “effective” homogenized core is modeled using multi-layer sandwich theory considering the effects of transverse shear deformation and compressibility of the core; moreover, the solutions are obtained by using the precise integration method. Several dynamic elastic finite element (FE) simulations are carried out to obtain the structural response of the tube to shock loading moving at different velocities. The comparison between analytic solutions and FE simulations demonstrates that the transient analytical model, based on the proposed sandwich model, is capable of predicting the critical velocity and the dynamic structural response of the sandwich tube with the “effective” homogenized core with a high degree of accuracy. In addition, the critical velocity predicted using FE simulations of the complete model is not in agreement with that of the effective model. However, the structural response and the maximum amplification factors obtained using FE simulations of the complete model are nearly similar to that of the effective model, when the shock loading moves at the critical velocity. The influences of the relative density on the structural response are studied, and the capabilities of load bearing for sandwich tubes with different cores are compared with each other and with the monolithic tube. The results indicate that Kagome and triangle-6 are preferred among five topologies.     

Thermoelastic buckling behavior of thick functionally graded rectangular plates

Thermoelastic buckling behavior of thick rectangular plate made of functionally graded materials is investigated in this article. The material properties of the plate are assumed to vary continuously through the thickness of the plate according to a power-law distribution. Three types of thermal loading as uniform temperature raise, nonlinear and linear temperature distribution through the thickness of plate are considered. The coupled governing stability equations are derived based on the Reddy’s higher-order shear deformation plate theory using the energy method. The resulted stability equations are decoupled and solved analytically for the functionally graded rectangular plates with two opposite edges simply supported subjected to different types of thermal loading. A comparison of the present results with those available in the literature is carried out to establish the accuracy of the presented analytical method. The influences of power of functionally graded material, plate thickness, aspect ratio, thermal loading conditions and boundary conditions on the critical buckling temperature of aluminum/alumina functionally graded rectangular plates are investigated and discussed in detail. The critical buckling temperatures of thick functionally graded rectangular plates with various boundary conditions are reported for the first time and can be served as benchmark results for researchers to validate their numerical and analytical methods in the future.     

Static buckling appraisal of an echinodome

The buckling behavior of an underwater shell of revolution structure of optimum form—an echinodome—is examined under axisymmetric and symmetric point loads both experimentally and theoretically. For the concentrated loadings, experimental predictions of critical buckling are based on the Southwell technique and a possible alternative method is suggested. Bifurcation and nonlinear collapse buckling analyses are described theoretically. Within the bifurcation investigation both linear and nonlinear approaches are included. The effects of base fixity on the instability of the structure are considered. Comparisons are made with earlier external-pressure loading effects and the relative significance of the two forms of loading are discussed.     

Comparisons of test and theory for nonsymmetric elastic-plastic buckling of shells of revolution

Experimental and analytical buckling pressures are presented for very carefully fabricated thin cylindrical shells with 45, 60 and 75° conical heads and for cylindrical shells with torispherical heads pierced by axisymmetric cylindrical nozzles of various thicknesses and diameters. Nonsymmetric buckling occurs at pressures for which some of the material is loading plastically in the neighborhoods of stress concentrations caused by meridional slope discontinuities. The buckling pressures for the cone-cylinder vessels are predicted within 2.6 per cent and for the pierced torispherical vessels within 4.4 per cent with use of BOSOR5, a computer program based on the finite difference energy method in which axisymmetric large deflections, nonlinear material properties and nonsymmetric bifurcation buckling are accounted for. The predicted buckling pressures of the pierced torispherical specimens are rather sensitive to details of the analytical model in the neighborhood of the juncture between the nozzle and the head. The buckling pressures of the cone-cylinder vessels can be accurately predicted by treatment of the wall material as elastic, enforcement of the full compatibility conditions at the juncture in the prebuckling analysis, and release of the rotation compatibility condition in the bifurcation (eigenvalue) analysis.     

Test method for measuring strength of a curved sandwich beam

A fixture for testing curved sandwich beams in flexure was designed and evaluated. The test specimen is a continuous sandwich beam consisting of a central circular 90° region connected by two straight legs. The fixture was designed according to the four-point flexure principle to produce a pure bending moment in the curved region. The validity of the test fixture in producing the desired loading was examined by fitting a curved aluminum bar of similar bending stiffness as the sandwich beams considered. Strain gage readings were successfully compared to predictions from curved homogeneous beam theory. In addition, the deflection of the beam at the loading points was analyzed using straight and curved beam theory for the various sections of the beam, and predictions were compared to measured load-displacement response. Good agreement was achieved between experimental and analytical results lending confidence to the test principle. Curved sandwich beams consisting of glass/polyester face sheets over a PVC foam core were tested to failure and the loading response of the beams and their failure behavior are discussed. It was found that the beams failed at the upper face/core interface due to radial tension stress.     

Wrinkling of thick orthotropic sandwich plates under general loading conditions

Summary  This contribution presents an efficient analytical model as well as a FE computation of the critical load, which leads to local stability failure (wrinkling) in sandwich structures. The analytical model assumes an orthotropic face layer and a thick transversely isotropic core. In the last section, a more general core material model is considered. Common core materials (foams and honeycombs) can be described with good accuracy within this model. The main advantage of the solution is the consideration of general loading conditions for the orthotropic face layer as well as in-plane deformations of the core. The results of the FE calculations and the analytical model are in good agreement with each other. Received 7 January 1999; accepted for publication 15 June 1999     

FREE VIBRATION ANALYSIS OF LATTICE SANDWICH BEAMS UNDER SEVERAL TYPICAL BOUNDARY CONDITIONS

Free vibration problems of lattice sandwich beams under several typical boundary conditions are investigated in the present paper. The lattice sandwich beam is transformed to an equivalent homogeneous three-layered sandwich beam. Unlike the traditional analytical model in which the rotation angles of the face sheets and the core are assumed the same, different rotation angles are considered in this paper to characterize the real response of sandwich beams. The analytical solutions of the natural frequencies for several typical boundary conditions are obtained. The effects of material properties and geometric parameters on the natural frequencies are also investigated.     

Supercritical and subcritical buckling bifurcations for a compressible hyperelastic slab subjected to compression

We study the buckling bifurcation of a compressible hyperelastic slab under compression with sliding–sliding end conditions. The combined series-asymptotic expansions method is used to derive the simplified model equations. Linear bifurcation analysis yields the critical stress value of buckling, which gives a non-linear correction to the classical Euler buckling formula. The correction is due to the geometrical non-linearities coupled with the material non-linearities. Then through non-linear bifurcation analysis, the approximate analytical solutions for the post-buckling deformations are obtained. The amplitude of buckling is expressed explicitly in terms of the aspect ratio, the incremental dimensionless engineering stress, the mode of buckling and the material constants. Most importantly, we find that both supercritical and subcritical buckling could occur for compressible materials. The bifurcation type depends on the material constants, the geometry of the slab and the mode numbers.     

Imperfection sensitivity of axially compressed stringer reinforced cylindrical sandwich panels

This paper presents some numerical results of the effects of several nondimensional parameters on the buckling and initial post buckling behaviors of shallow sandwich panels under axial compression. Results are presented that show these effects due to transverse shearing resistance of the core material, different face-sheet thicknesses, and different core thicknesses. Further effects on the buckling and initial postbuckling behaviors of sandwich panels are presented due to the torsional resistance of longitudinal edge stiffeners.The results show that the range of flatness parameter, δ/d, for which sandwich panels remain imperfection-insensitive increases with increases in transverse shearing resistance of the core material and with larger core thicknesses. These results also indicate that this range of δ/d is smallest when the face-sheet thicknesses are equal. Finally, as in the case of homogeneous panels, torsional resistance of the longitudinal edge stiffeners has the effect of making the sandwich panel less imperfection-sensitive.     

Buckling of a sandwich symmetrical circular plate with varying mechanical properties of the core

This paper is devoted to analytical and numerical studies of global buckling of a sandwich circular plate. The mechanical properties of the plate core vary along its thickness, remaining constant in the facings. The middle surface of the plate is its symmetrical plane. The mathematical model of the plate is presented. The field of displacements is formulated using the proposed nonlinear hypothesis that generalizes the classical hypotheses. The equations of equilibrium are formulated based on the principle of stationary total potential energy. The proposed mathematical model of the displacements considers the shear effect. The numerical model of the plate is also formulated with a view to verify the analytical one. Numerical calculations are carried out for the chosen family of plates. The values of the critical load obtained by the analytical and numerical methods are compared. The effects of the material properties of the core and the change of the plate radius on the critical load intensity are presented.     

夹层FGM圆柱壳在扭转载荷作用下的弹性稳定性

采用半解析方法研究了两端简支的功能梯度夹层圆柱壳在端部扭转载荷作用下的弹性稳定性.考虑圆柱壳的里外表层为均匀材料,中间层为材料性质沿厚度方向连续变化的功能梯度材料,并且在界面处的材料性质保持连续. 基于Flügge薄壳理论,建立了位移形式的结构静态屈曲控制方程.根据边界条件将位移表示为三角级数形式,获得包含柱壳端部扭转载荷参数的近似线性代数特征值问题,并通过数值方法求得了表征结构失稳特征的临界载荷. 数值结果表明,临界载荷随着半径与厚度比的增加而减小,随着功能梯度中间层的弹性模量的平均值的增加而增加.      

Analytical modeling and finite element simulation of the plastic collapse of sandwich beams with pin-reinforced foam cores

Analytical predictions are presented for the plastic collapse strength of lightweight sandwich beams having pin-reinforced foam cores that are loaded in 3-point bending. Both polymer and aluminum foam cores are considered, whilst the facesheet and the pins are made of either composite or metal. Four different failure modes are account for: metal facesheet yield or composite facesheet microbuckling, facesheet wrinkling, plastic shear of the core, and facesheet indentation beneath the loading rollers. A micromechanics-based model is developed and combined with the homogenization approach to calculate the effective properties of pin-reinforced foam cores. To calculate the elastic buckling strength of pin reinforcements, the pin-reinforced foam core is treated as assemblies of simply supported columns resting upon an elastic foundation. Minimum mass design of the sandwich is then obtained as a function of the prescribed structural load index, subjected to the constraint that none of the above failure modes occurs. Collapse mechanism maps are constructed and compared with the failure maps of foam-cored sandwich beams without pin reinforcements. Finite element simulations are carried out to verify the analytical model and to study the performance and failure mechanisms of the sandwich subject to loading types other than 3-point bending. The results demonstrate that the weaker the foam is, the more optimal the pin-reinforced foam core becomes, and that sandwich beams with pin-reinforced polymer foam cores are structurally more efficient than foam- or truss-cored sandwich beams.