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.     

新的复合材料格栅加筋板的平铺等效刚度法

针对薄壁复合材料格栅加筋结构的受力特点,在改进原有力学假设的基础之上,推导了一种新的平铺等效刚度计算方法,它充分考虑了筋条和面板之间的相互作用.通过格栅单元结构布局形式的参数化表示,建立了通用的力学分析模型;该模型可用于分析各种结构布局形式和面板铺层方式下的结构总体屈曲问题,故对于航空航天结构设计非常有用.结合Rayleigh-Ritz方法,推导出了求解格栅加筋板屈曲载荷的通用线性特征方程;最后,分析了多种类型格栅结构的算例,并与现有的各种方法进行了比较,结果更为精确,因而对格栅加筋结构的优化设计具有很好的应用价值.     

The impact torsional buckling of elastic cylindrical shells with arbitrary form imperfection

A perturbation analysis for the impact torsional buckling of imperfective elastic cylindrical shells subjected to a step torque is given..The imperfection is supposed to be small and has arbitrary form.It is shown that only the imperfection which has the shape of static torsional buckling mode could influence the critical step torque.Finally a formula is presented for the critical step torque.     

薄壁加筋肋圆柱壳稳定性分析的参数化研究

针对在轴向载荷作用下的正置、正交网格形式的薄壁加筋肋圆柱壳结构，利用有限元程序，对薄壁加筋肋圆柱壳稳定性分析进行了参数化研究，得到了进行结构优化设计的准则，对于给定的设计载荷，当结构参数位于某一个局部失稳与整体失稳的临界区域时，结构的重量最轻。提出了基于有限元分析进行结构优化设计的策略，利用优化策略，获得了一薄壁加筋肋圆柱壳结构的优化设计结果，同时给出了粘合刚度简化模型与有限元计算结果的比较。     

Sensitivity of Buckling Load and Vibration Frequency with Respect to Shape of Stiffened and Unstiffened Plates

ABSTRACT

The method of calculating the sensitivities of buckling loads and vibration frequencies of nonlinear plates with respect to shape functions is presented. These shape functions describe the external shape of an unstiffened plate or the shape of a stiffener within the domain of a plate with fixed external boundaries. An adjoint variable method is used to determine the sensitivities. Thus, only one additional solution for an adjoint plate is required.     

Imperfection sensitivity of curved panels under combined compression and shear

The initial buckling load of curved panels under compressive loads is substantially reduced by the existence of imperfections, in particular geometric imperfections. It is therefore essential that these imperfections are considered in analysing components which incorporate such panels in order to accurately predict their buckling behaviour. Finite element analysis allows fully non-linear analysis of shells containing geometric imperfections, however, to obtain accurate results information is required on the exact size and shape of the imperfection to be modelled. In most cases this data is not available. It is therefore generally recommended that the imperfections are modelled on the first eigenmode and have an amplitude selected according to the manufacturing procedure. This paper presents the effects of varying imperfection shape and amplitude on the buckling and postbuckling behaviour of one specific case, a curved panel under combined shear and compression, to test the accuracy of such recommendations.     

加筋路径驱动的板壳自适应等几何屈曲分析

加筋薄壁结构常被用于航空航天结构的轻量化设计.随着结构尺寸和几何特征的增加,需要更加精细的网格来满足分析精度的要求.传统的等几何方法采用NURBS张量积形式的拓扑结构,使得在分析过程中难以实现局部细化,而全局细化则会增加不必要的自由度.为了提升加筋板壳结构的数值分析精度和效率,提出一种基于RPHT (rational polynomial splines over hierarchical T-meshes)样条的加筋板壳自适应等几何屈曲分析方法.样条网格可以沿着加筋路径进行自适应的局部细化,有效提升低自由度下加筋板壳结构等几何屈曲分析的精度.首先,蒙皮和筋条分别采用RPHT样条曲面和NURBS样条曲线进行建模,几何建模与数值仿真采用统一的几何语言,实现建模与分析的一体化.其次,采用几何投影算法和样条插值算法实现筋条与蒙皮之间的高效高精度强耦合,并建立基于加筋路径驱动自适应网格细化方法.最后,曲线加筋板和网格加筋壳两个算例验证本方法的高效性和鲁棒性,通过与基于NURBS的等几何分析进行对比,本方法能够明显降低分析模型的总自由度.     

Experimental buckling of internal integral ring-stiffened cylinders

Twenty aluminum cylinders with internal, integral tee-stiffener rings were tested under combinations of axisymmetrical axial load and external lateral pressure to determine buckling characteristics. Seven geometric types were tested; the primary variables were the ratios of cylinder radius to shell thickness, stiffener spacing to shell thickness, and stiffener spacing to stiffener depth. An eighth type, which had variable stiffener spacing and depth, was tested under a lateral pressure varying linearly in the axial direction. Strain-gage data were obtained to aid in evaluation of results. The test results agree well with the theoretical work used for the design.Paper was presented at 1966 SESA Annual Meeting held in Pittsburgh, Pa., on November 6–9.     

The initial postbuckling behavior of ring stiffened shells under hydrostatic pressure

In this article the initial postbuckling behavior of ring stiffened cylindrical shells under hydrostatic pressure is analyzed by Koiter's theory. The nonlinear bending equations consistent with boundary conditions have been used in prebuckling state. The eigenvalue problem is solved by Galerkin's method. The obtained buckling loads are compared with the results which are based on classical stability theory. As calculating examples, three typical outside-stiffened cylinders with different ring stiffener parameters are chosen. The results show that the strength of stiffener not only influences buckling load obviously, but also changes the imperfection-sensitivity of cylindrical shells.     

An experimental study on the static and impact torsional buckling of cylindrical shells

In this paper, experimental studies are carried out on the buckling of circular cylindrical thin shells under impact torque. Experiments of impact buckling are made on a Hopkinson torsional bar. The torsional bar gives a step torque on the shells. Through an analysis of the strain-time curve obtained in experiment, the dynamic buckling critical torqueM _er and buckling waves numbern of the shell with different geometric data and some qualitative results are obtained. The buckling behavior of circular cylindrical thin shells under static and impact torque is compared.     

Twist buckling and the foldable cylinder: an exercise in origami

Basic mechanisms for the buckling of a thin cylindrical shell under torsional loading are reviewed from a post-buckling perspective. Deflections are considered so far into the large-deflection range that the shell is allowed to fold to a flat two-dimensional form, in a mechanism reminiscent of a deployable structure. Critical and initial post-buckling effects are explored through concepts of energy minimization and hidden symmetries. For comparisons with the final large-deflection folded shape, a truss element program is employed. It is shown that, as buckling develops, the mode shape must change to accommodate both the symmetry-breaking aspects of the predominately inwards deflection, and the rotation of peak and valley lines of the buckle pattern necessary to accommodate the geometry of the final folded shape.     

Dynamic torsional buckling of a double-walled carbon nanotube embedded in an elastic medium

An elastic double-shell model based on continuum mechanics is presented to study the dynamic torsional buckling of an embedded double-walled carbon nanotube. Based on the presented model, a condition is derived to predict the buckling load of the embedded double-walled nanotube, and the effect of the van der Waals forces to the buckling load is discussed when an inner nanotube is inserted into an embedded outer one. In particular, the paper shows that the buckling load of the embedded double-walled nanotube is always between that of the isolated inner nanotube and that of the embedded outer nanotube for both dynamic and static torsional buckling, due to the effect of the van der Waals forces. This result is different from that obtained by the existing analysis neglecting the difference of the radii for the embedded double-walled nanotube, which indicates that disregarding the difference of the radii of multi-walled nanotubes cannot properly describe the effect of the van der Waals forces between interlayer spacing. In particular, for static torsional buckling of a double-walled nanotube, it is shown that the critical buckling load cannot only be enhanced, but also be reduced when inserting an inner nanotube into an isolated single-walled one. Additionally, it is shown that the elastic medium always increases the critical buckling load of double-walled nanotubes. The critical buckling load of embedded double-walled nanotubes for dynamic torsional buckling is proved to be no less than that for static torsional buckling.     

Predicting buckling loads from vibration data

There are analytical methods for predicting the buckling loads of columns with the boundaries ideally fixed, i.e., simply supported or built-in, or partially fixed. Vibration-test results may furnish a practical method of measuring the fixity. In this investigation a beam, that may or may not be loaded as a column, is assumed to have a torsional spring at each end such that a zero torsional stiffness corresponds to a simply supported end and an infinite torsional stiffness corresponds to a built-in end. From a Rayleigh-Ritz analysis, the buckling load and the fundamental frequency of the beam are each computed as a function of the torsional stiffness. This procedure leads to a one-to-one nondimensional relationship between the buckling load and the natural frequency. From these calculations, it is seen that regardless of the degree of clamping of one end relative to the other end, all that is needed to predict the buckling load within a 15-percent range is a knowledge of the theoretical buckling load of the simply supported column; the theoretical fundamental frequency of the simply supported beam; and the experimental fundamental frequency. Experimental results are presented to support the theory.     

Compressive strength of edge-loaded corrugated board panels

Postbuckling strength of simply supported corrugated board panels subjected to edge compressive loading has been studied experimentally using a specially developed test fixture. Although the load versus out-of-plane displacement response was highly sensitive to the presence of initial imperfections in the panels, the collapse loads did not vary much, which is attributed to the stable postbuckling behavior of the plates. Thin plates collapsed at nearly twice the buckling load, while thick panels collapsed at loads below the elastic critical buckling load. Local buckling of the facing on the concave side of the buckled plate was observed at load levels close to the collapse load. The plate collapse was triggered by compressive failure of the facings that initiated at the unloaded edges. A simplified design analysis was derived based on approximate postbuckling analysis and compared with an existing design formula for corrugated board panels and boxes.     

Torsional buckling of a double-walled carbon nanotube embedded in an elastic medium

The torsional buckling of a double-walled carbon nanotube embedded in an elastic medium is studied in this paper. The effects of surrounding elastic medium and van der Waals forces between the inner and outer nanotubes are taken into account. Using continuum mechanics, an elastic double-shell model is presented for the torsional buckling of a double-walled carbon nanotube. Based on the model, a condition is derived in terms of the buckling modes of the shell and the parameters describing the effect of van der Waals interaction and surrounding elastic medium. A simplified analysis is also carried out estimate the critical torque for torsional buckling of the double-walled carbon nanotube.     

Creep buckling of cross-ply symmetric laminated cylindrical panels

A creep buckling analysis of cross-ply symmetric laminated cylindrical panels is given in this paper.By means of theoretical analysis,a method to determine the critical load of creep buckling of the panels with simply supported boundary conditons is obtained.     

Combined torsional buckling of multi-walled carbon nanotubes coupling with axial loading and radial pressures

This paper reports the results of an investigation on combined torsional buckling of multi-walled carbon nanotubes (MWNTs) under combined torque, axial loading and radial pressures based on the continuum mechanics model, which takes into account the effect of the van der Waals interaction between adjacent tubes. A buckling condition is derived for determining the critical buckling torque and associated buckling mode. In particular, for combined torsional buckling of double-walled carbon nanotubes, an explicit expression is obtained and some detailed results are demonstrated. According to the innermost radius-to-thickness ratio, MWNTs are classified into three types: thin, thick, and (almost) solid. Numerical results are worked out for the critical buckling torque and associated buckling mode for all the three types of MWNTs subjected to various axial stresses (axial tensile stresses or axial compressive stresses), internal pressures, and external pressures. It is shown that, the axial tensile stress or the internal pressure will make the MWNTs resist higher critical buckling torque, while the axial compressive stress or external pressure will lead to a lower critical buckling torque. The effect of axial stress (axial tensile stress or axial compressive stress) on the critical buckling torque of MWNTs is very small for all the three types of MWNTs, while the effect of the internal pressure or external pressure is related to the types of MWNTs, which is strong for the thin MWNTs, moderate for the thick MWNTs, and small for the solid MWNTs. Numerical results also indicate that, the associated buckling mode is unique and dependent on the structure of MWNTs. Especially, for combined torsional buckling of MWNTs with very small axial stress and radial pressures, the buckling mode is just the one for the corresponding pure torsional buckling.     

Vibration localization in disordered periodically stiffened double-leaf panels

Vibration localization in periodically stiffened double-leaf multi-span panels is studied by employing the transfer matrix method. The localization factors of the ordered and disordered systems are calculated based on the Lyapunov exponent. The numerical results show that the propagation of vibration in rib-stiffened periodic double-leaf panels exhibits passbands and stopbands. The vibration localization phenomenon occurs and is enhanced with the increasing disorder of span-length. The torsional rigidities of the stiffeners have a significant effect on the pass bands and the localization factor. With the torsional rigidity of the stiffeners increasing, the vibration localization factor first decreases, then increases and finally tends to be the situation of the rib-stiffened single-leaf panels. It is also noted that for the double-leaf panels a passband appears among the lower dimensionless frequencies for some particular values of torsional rigidity of the stiffeners while a stopband always exists for the single-leaf panels.     

Three-dimensional theory of stability of a carbon nanotube in a matrix

The three-dimensional theory of stability of a carbon nanotube (CNT) in a polymer matrix is presented. The results are obtained on the basis of the three-dimensional linearized theory of stability of deformable bodies. Flexural and helical (torsional) buckling modes are considered. It is proved that the helical (torsional) buckling modes occur in a single CNT (the interaction of neighboring CNTs is neglected) and do not occur in nanocomposites (the interaction of neighboring CNTs is taken into account) __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 23–37, January 2006.     

The impact torsional buckling for the rigid plastic cylindrical shell

By using the energy criterion in[3],the impact torsional buckling for the rigid plastic cylindrical shell is studied.The linear dynamic torsional buckling equations for the rigid plastic shell is drived,and the critical impact velocity is given.