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• Improved variant of the reduced stiffness method is presented, the localized reduced stiffness method (LRSM). • The LRSM is a versatile and effective lower-bound method for the design of buckling critical thin-walled shells. • Physical explanation for the plateau or lower-bound buckling load of cylinders under axial compression is identified. • The LRSM is applied and validated with experimental results for different shells and different load cases. • The LRSM is easy to implement in commercial finite element codes and delivers 24% to 176% higher KDFs for the buckling load when compared with currently used empirical guidelines. Thin-walled shells like cylinders, cones and spheres are primary structures in launch-vehicle systems. When subjected to axial loading, bending or external pressure, these thin-walled shells are prone to buckling. The corresponding critical load heavily depends on deviations from the ideal shell shape. In general, these deviations are defined as geometric imperfections, and although imperfections exhibit comparatively low amplitudes, they can significantly reduce the critical load. Considering the influence of geometric imperfections adequately into the design process of thin-walled shells poses major challenges for structural design. The most common procedure to take into account the influence of imperfections is based on classical buckling loads obtained by a linear analysis which are then corrected by a knockdown factor. The knockdown factor represents a statistical lower-bound with respect to data obtained experimentally for different types of thin-walled shells. This article presents a versatile and simple numerical design approach for buckling of critical shell structures. The new design procedure is based on the reduced stiffness method and leads to significantly improved critical load estimations in comparison to lower-bounds obtained empirically. An analysis example is given which is based on the launch-vehicle stage adapter (LVSA) of NASAs Space Launch-system (SLS). [ABSTRACT FROM AUTHOR]

In this paper, a generalized discrete elastica including both bending and shear elastic interactions is developed and its possible link with nonlocal beam continua is revealed. This lattice system can be viewed as the generalization of the Hencky bar-chain model, which can be retrieved in the case of infinite shear stiffness. The shear contribution in the discrete elastica is introduced by following the approach of Engesser (normal and shear forces are aligned with and perpendicular to the link axis, respectively) and that of Haringx (shear force is parallel to end section of links), both supported by physical arguments. The nonlinear analysis of the shearable-bendable discrete elastica under axial load is accomplished. Buckling and post-buckling of the lattice systems are analyzed in a geometrically exact framework. The buckling loads of both the discrete Engesser and Haringx elastica are analytically calculated, and the post-buckling behavior is numerically studied for large displacement. Nonlocal Timoshenko-type beam models, including both bending and shear stiffness, are then built from the continualization of the discrete systems. Analytical solutions for the fundamental buckling loads of the nonlocal Engesser and Haringx elastica models are given, and their first post-buckling paths are numerically computed and compared to those of the discrete Engesser and Haringx elastica. It is shown that the nonlocal Timoshenko-type beam models efficiently capture the scale effects associated with the shearable-bendable discrete elastica. [ABSTRACT FROM AUTHOR]

Grid structures are extensively used in many engineering structures as an orthotropic layer with at most in-plane anisotropy. In this study, the laminated grids structures are considered. These structures are composed of various numbers of thin composite grid layers. This concept yields to numerous laminated grid configurations with disparate coupling effects, which can be used to tailor the stiffness, compare to conventional grids. In the present study, four known grid patterns with equal weight and thickness constitute the laminated grids. The genetic algorithm (GA) is employed to optimize the stacking sequence and pattern composition of the laminated grid. The buckling load is taken as a fitness function and the pattern and orientation of the grid layer are considered as the design variables. The first-order shear deformation and classical laminated plate theories are considered along with Ritz method to obtain the elastic buckling loads. The optimization results are presented for various boundary conditions and aspect ratios. The results show, using the laminated grid considerably improves the axial buckling load of the plates. Moreover, the plate boundary conditions and aspect ratios play an imperative role in the optimum pattern and stacking sequence of the laminated grid plates. [ABSTRACT FROM AUTHOR]

The effects of creep of the core material on the global and local geometrically nonlinear behavior of sandwich panels under axial and/or lateral loading conditions is presented, within the framework of the high-order sandwich theory (HSAPT). The theoretical model combines the concepts of the hereditary (convolution) integral of viscoelasticity based on the principle of superposition, with the HSAPT. From the expansion of the relaxation moduli into Prony series, an incremental exponential law that corresponds to the rheological generalized Maxwell model is obtained, and which allows for a step-by-step time analysis that accounts for the change in deformations and stresses with time. The results show that creep of the core leads to a significant reduction in the local and global buckling capacity of sandwich panels. It is shown that sandwich panels under a combined axial and lateral loading can undergo global buckling with time due to creep, under a sustained load that is less than 50% of their buckling load. Similar trend is also observed for the local buckling (wrinkling) of the compressed face sheet when the panel is subjected to a sustained uniformly distributed load. In both cases, buckling commenced at a relatively low stress levels but it is associated with an unlimited increase of the stresses with time that eventually lead to material failures and total failure of the sandwich panel. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an analytical solution for the linear elastic buckling analysis of simply supported battened columns subjected to axial compressed loading. The critical buckling load is derived by using the classical energy method. Unlike most of existing work, the present approach considers not only the shear effect but also the discrete effect of battens on the global buckling behaviour of the columns. The present analytical solution is validated using the data obtained from the finite element analysis. The results show that the number of battens has significant influence on the critical buckling load of battened columns, particularly when the relative rigidity of the batten to the main member is small. It is shown that the critical buckling load increases with the number of battens, the combined bending and shear rigidity of battens, but decreases with the increased membrane stiffness of the two main members, and the increased distance between the centroids of the two main members. [Copyright &y& Elsevier]

Abstract: This paper presents an analytical study on the influence of in-plane forces on the dynamic characteristics of a plate with grooves in different sizes. A sub-structuring approach, the reverse receptance method, is applied to solve the eigenvalues of an in-plane forces loaded plate with two parallel grooves and two orthogonal grooves, respectively. On considering coupling and interaction between two orthogonally connected grooves, the point receptance arrays are employed. Accuracy is evaluated by comparing the analytical solutions with numerical results obtained from the finite element model. Parametric study reveals, the deeper the groove, the lower the natural frequencies and buckling loads of the grooved plate will be. With in-plane tensions applied and increased linearly, the grooved plate is stiffened gradually and its natural frequencies increase accordingly. With equal bi-axial tensions, the natural frequencies of the grooved plates increase twice as much as those with uni-axial tension do. With in-plane compressive forces on the plate, the situations are reversed. The lowest natural frequency of a grooved plate with sufficient tensions can be higher than that of a non-grooved unloaded plate. When the groove size and the loading condition are the same, the orthogonal grooves decrease the natural frequencies and the buckling loads of a plate more than the parallel grooves do. The in-plane forces do not affect the mode shapes of the grooved plate; but the grooves do. The analytical evaluation demonstrates the accuracy and applicability of the proposed approach. [Copyright &y& Elsevier]

Abstract: Thermal post-buckling analysis of uniform, isotropic, slender and shear flexible columns is presented using a rigorous finite element formulation and a much simpler intuitive formulation. The ends of the columns are axially restrained to move and consequently any temperature rise above the stress free condition of the column produces an equivalent constant compressive mechanical load that causes the column to buckle at a critical temperature. Further increase in temperature beyond critical temperature results in the thermal post-buckling phenomenon. As a result of constraints imposed on the axial displacement at the ends of the column, the post-buckling phenomenon is governed by the von-Karman strain displacement relation applicable to one dimensional problems. Empirical formula for ratio of nonlinear axial load to critical load (equivalent constant mechanical load for a given temperature rise) as a function of the central deflection are obtained using both the rigorous finite element and intuitive formulations for various boundary conditions. The boundary conditions considered are the classical such as hinged–hinged, clamped–clamped and clamped–hinged conditions and nonclassical boundary conditions like the hinged–guided or the clamped–guided conditions. Post-buckling analysis results pertaining to nonclassical boundary conditions are meagre in the literature. It is observed that results obtained from both the formulations are in excellent agreement for all boundary conditions considered. Also the accuracy and simplicity of the intuitive formulation is aptly demonstrated to slender and shear flexible columns. [Copyright &y& Elsevier]

The effect of the prebuckling in-plane deformations on the elastic flexural-torsional buckling of laterally fixed circular arches is studied in this paper. The finite strains and the energy equation for the flexural-torsional buckling of arches have been derived based on an accurate orthogonal rotation matrix. A closed form solution for the elastic flexural-torsional buckling resistance of laterally fixed arches in uniform bending, including the effects of the prebuckling deformations, is obtained. It is found that the notion that the prebuckling deformations increase the flexural-torsional buckling moment of an arch or of a beam is not necessarily correct for a laterally fixed arch or beam in uniform bending, in deference to a laterally pinned arch. When a laterally fixed arch is subjected to positive uniform bending, the effects of the prebuckling deformations decrease the buckling moment, and the reduction of the buckling moment increases with an increase of the included angle and of the out-of-plane slenderness ratio of the arch. When a laterally fixed arch is subjected to negative uniform bending, the effects of the prebuckling deformations decrease the absolute value of its buckling moment when the included angle is very small, but increase the absolute value of the buckling moment when the included angle exceeds a certain value. The increase in the absolute value of the buckling moment increases with an increase of the included angle and of the out-of-plane slenderness ratio of the arch. When the ratio of the out-of-plane to the in-plane second moments of area of the cross-section is not small, both the reduction of the buckling moment of a laterally fixed arch in positive uniform bending and the increase of the buckling moment of a laterally fixed arch in negative uniform bending, are substantial. [Copyright &y& Elsevier]

Based on the mathematical formulations for predicting forming limits induced by buckling, wrinkling and bursting of free-expansion tube hydroforming, a theoretical “Process Window Diagram” (PWD) is proposed and established in this paper. The theory developed in the first part of the present work was formulated within the context of free-expansion tube hydroforming with both combined internal pressure and end feeding. The PWD is designed to provide a quick assessment of part producibility for tube hydroforming. The predicted PWD is validated against experimental results conducted for 6260-T4 60×2×320 (mm) aluminum tubes. An optimal loading path is also proposed in the PWD with an attempt to define the ideal forming process for aluminum tube hydroforming. Parametric studies show that the PWD has a strong dependency on tube geometry, material property and process parameters. To the authors’ knowledge, this is the first attempt that a PWD is being formulated theoretically. Such a concept can be advantageous in deriving design solutions and determining optimal process parameters for tube hydroforming processes. [Copyright &y& Elsevier]

Three possible failure modes have been identified in tube hydroforming: buckling, wrinkling and bursting. A general theoretical framework is proposed for analyzing these failure modes as an elastoplastic bifurcation problem. This framework enables advanced yield criteria and various strain-hardening laws to be readily incorporated into the analysis. The effect of plastic deformation on the geometric instability in tube hydroforming, such as global buckling, axisymmetric wrinkling and asymmetric wrinkling, is precisely treated by using the exact plane stress moduli tensor. A mathematical formulation for predicting the localized condition for bursting failure is established herein. Furthermore, the critical conditions governing the onset of buckling, axisymmetric wrinkling and asymmetric wrinkling are derived in closed-form expressions for the critical axial compressive stresses. Closed-form solutions for the critical stress are developed based on Neale–Hutchinson''s constitutive equation and an assumed deformation theory of plasticity. It is demonstrated that the onset of asymmetric wrinkling always requires a higher critical axial compressive stress than the axisymmetric one under the context of tube hydroforming with applied internal pressure and hence the asymmetric wrinkling mode can be excluded in the analysis of tube hydroforming. Parametric studies show that buckling and axisymmetric wrinkling are strongly dependent on geometric parameters such as t0/r0 and r0/ℓ0, and that axisymmetric wrinkling is the predominant mode for short tubes while global buckling occurs for long slender tubes. [Copyright &y& Elsevier]

The finite element method is applied to study the static and dynamic stability of an aerofoil cross-section rotating blade subjected to an axial periodic force. The effects of coupling due to the center of flexure (shear center) distance from the centroid, rotational speed, stagger angle and disk radius on the stability are considered. Critical load parameters are calculated and dynamic instability regions of the blade with different reference values are illustrated graphically. The numerical results indicate that the coupling effect is very important in the third and fourth modes depending on dY distance. The increase in stagger angle makes the blade less stable. However, the increase in rotational speed and disk radius make the blade more stable. [Copyright &y& Elsevier]