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Rail vehicles generate huge longitudinal impact loads in collisions. If unreasonable matching exists between the compressive strength of the intermediate coupler and the structural strength of the car body, the risk of car body structure damage and train derailment will increase. Herein, a four-stage rigid–flexible coupling finite element model of the coupler is established considering the coupler buckling load. The influence of the coupler buckling load on the train longitudinal–vertical–horizontal buckling behavior was studied, and the mechanism of the train horizontal buckling instability in train collisions was revealed. Analysis results show that an intermediate coupler should be designed to ensure that the actual buckling load is less than the compressive load when the car body structure begins to deform plastically. The actual buckling load of the coupler and the asymmetry of the structural strength of the car body in the lateral direction are two important influencing factors for the lateral buckling of a train collision. If the strength of the two sides of the car body structure in the lateral direction is asymmetrical, the deformation on the weaker side will be larger, and the end of the car body will begin to deflect under the action of the coupler force, which in turn causes the train to undergo sawtooth buckling. [ABSTRACT FROM AUTHOR]

Abstract Rail vehicles generate huge longitudinal impact loads in collisions. If unreasonable matching exists between the compressive strength of the intermediate coupler and the structural strength of the car body, the risk of car body structure damage and train derailment will increase. Herein, a four-stage rigid–flexible coupling finite element model of the coupler is established considering the coupler buckling load. The influence of the coupler buckling load on the train longitudinal–vertical–horizontal buckling behavior was studied, and the mechanism of the train horizontal buckling instability in train collisions was revealed. Analysis results show that an intermediate coupler should be designed to ensure that the actual buckling load is less than the compressive load when the car body structure begins to deform plastically. The actual buckling load of the coupler and the asymmetry of the structural strength of the car body in the lateral direction are two important influencing factors for the lateral buckling of a train collision. If the strength of the two sides of the car body structure in the lateral direction is asymmetrical, the deformation on the weaker side will be larger, and the end of the car body will begin to deflect under the action of the coupler force, which in turn causes the train to undergo sawtooth buckling.

The concept of local buckling of compressed isotropic cylindrical shells is developed in this paper. Buckling of axially compressed cylindrical shells under different types of local perturbations has been studied in many theoretical and experimental researches. Different methodologies, based on these studies, were suggested for design buckling load estimation. However, in most calculations and experiments only two classical boundary conditions were considered: fixed displacement of the shell edges in the axial direction or fixed and uniformly distributed compressive load. Both of them are hard or even impossible to realize in practical applications. In the present paper we studied the shell behaviour with 6 different types of boundary conditions, which included the above mentioned two types. We considered shell loading through rigid plates with different types of possible displacements (degrees of freedom) and through stiffening ribs at the edges of the shell. Two theoretical methods were applied: numerical (finite element method) and analytical (Pogorelov's geometrical method). Lateral force was applied as a perturbation and stability of the shell under load combination was studied. The results of calculations were compared with obtained experimental data for validation of numerical solutions. Then metastability and post-buckling behaviour of the structure was studied using theoretical and experimental methods. In particular the interval of existence and stability of post-buckling equilibrium states of the shells with one or several buckles was studied systematically. The post-buckling equilibrium paths and corresponding energy barriers for all mentioned types of boundary conditions were analyzed. Parametric analysis of boundary value problem allowed to establish the main (Batdorf) structure parameter. Formula for design buckling load was suggested and discussed. • The effect of boundary conditions on stability of compressed cylindrical shells under local perturbations is studied. • Results obtained by numerical method are validated by comparison with experimental data. • Post-buckling behaviour and metastability of the structure was studied using analytical, experimental and numerical methods. • Analysis of boundary value problem allowed to establish the main (Batdorf) structure parameter. • Formula for design buckling load is suggested and discussed. [ABSTRACT FROM AUTHOR]

Abstract The suggested before energy barrier criterion for design buckling pressure is extended to the case of clamped shallow spherical caps. It is developed for shells with geometrical imperfections and a new adjusted formula is obtained for design buckling load of imperfect structures. The method is applied to shells with elastic-perfectly plastic material. It is validated by experimental data and proved by numerical analysis based on finite element method. Calculation results were obtained and analysed for different values of shell geometrical parameter. A simple formula for design buckling pressure is suggested and validated. [ABSTRACT FROM AUTHOR]

The simple analytical model of spherical shell buckling suggested in the previous publication in the TWS journal is applied to studying the response of the structure to different perturbations given as random variables. Among them probing radial force, energy barrier and geometrical imperfections are considered as factors decreasing uniform buckling pressure. The model is expanded to the case of imperfect shells. The propagation of uncertainty of perturbations is studied by estimation the standard deviation of the buckling load. Monte Carlo method and the propagation of error formula based on Tailor expansion are used. The application of the methods to design buckling pressure estimation is discussed. • The used model allows comprehensive analysis of the shell behavior under different types of local perturbations. • Local perturbations are considered as random variables. • The statistics of response of the shell is studied. • The propagation of input uncertainty (error) is investigated. • This methodology can be used for estimation of design buckling load. [ABSTRACT FROM AUTHOR]

Energy perturbation is applied as a measure in the stability analysis of a spherical shell subjected to uniform external pressure. The energy barrier (buckling energy) is calculated using asymptotic method, which allows to estimate sensitivity of the structure to the perturbation. It is possible to define the level of external pressure separating areas of small and extensive sensitivity of the structure to the perturbations. This pressure level can be recommended as a design buckling load for spherical caps. The obtained result corrects NASA SP-8032 recommendations which are over-conservative. The asymptotic approach is validated using ANSYS software package. In addition, the influence of boundary conditions on buckling load is estimated using the software. [ABSTRACT FROM AUTHOR]

An important property of localization in buckling of spherical shells under external pressure is discussed. It is shown that the localization is possible for structures with nonlinear softening. An analytical model of local buckling of spherical shell is developed. Rayleigh–Ritz method is used at small and moderate deflections. At large deflections corresponding to relatively small pressure (less than 20% of classical bucking load) it is based on asymptotic method. The asymptotic model is then expanded to the practically important range of the load. The response of the structure to local perturbations of different types (including radial probing force, prescribed deflection at the shell pole, and energy barrier) is studied. Special attention is paid to the energy barrier which is required for structure transition from initial equilibrium state to the post-buckling dimple-like state. Energy barrier criterion is used as a measure of metastability of the structure and applied for estimation of load level separating high and low sensitivity of the shell to local perturbations. Based on this pressure value, formulae for design buckling load are proposed and deliberated. Similarities and differences of local buckling of spherical shells under external pressure and axially compressed cylindrical shells are discussed. • Localization phenomena in buckling of spherical shells under external pressure is discussed. • A simple analytical model of local buckling of the shells is suggested. • The model allows comprehensive analysis of the shell behavior under different types of local perturbations. • This analysis can be used for estimation of design buckling load. • The model is proved by comprehensive comparison with numerical data. [ABSTRACT FROM AUTHOR]

Buckling of unstiffened axially compressed cylindrical shells under different types of local perturbations has been studied in many theoretical and experimental researches in the last decade. Software packages based on the finite element method were used for analysis of structure behavior. Different approaches, based on these studies, were suggested for knockdown factor and design buckling load estimation. However, numerical analysis may be costly and complicated especially at design stage when optimal solution should be found under variety of practical restrictions. The analytical model was suggested before by author of this paper. It was based on Pogorelov's geometrical method which is valid at large deflections. Even though the model described all qualitative properties of the buckling phenomena well, it yielded a result with noteworthy error especially at the range of moderate deflections. In the present paper the model is significantly improved by using some results of numerical analysis. It is validated by comprehensive quantitative comparison with published numerical and experimental data. Different important properties of shell buckling and post-buckling behavior are studied to demonstrate that the model allows performing a comprehensive investigation of the shell local buckling. Based on simple model analysis the criterion for design buckling load estimation can be chosen. • A simple analytical model of local buckling of compressed cylindrical shells is suggested. • The model allows comprehensive analysis of the shell behavior under different types of local perturbations. • This analysis can be used for estimation of design buckling load. • The model is proved by comprehensive comparison with numerical and experimental data. [ABSTRACT FROM AUTHOR]

The results of unique experimental studies of buckling of dented axially compressed unstiffened cylindrical shells are provided in the suggested paper. The dimple-like geometrical imperfections of different shape, size, deflection amplitude, and number of dents were created on cylindrical shells before axial loading. The initial deflection amplitude varied in the wide range, up to several shell thicknesses. In all cases the plateaus were discovered on the buckling load initial deflection diagrams. Corresponding axial loads were close to local lower buckling loads of perfect shells. These loads are defined as minimum values of axial force when the post-buckling equilibrium states of the perfect structure with one or several dimples exist. It was shown that the lower local buckling loads slightly depend on geometrical imperfections and therefore can be used for estimation of design buckling load of the structure. Over 50 specimens were tested. Pre-buckling and post-buckling behavior of imperfect shells was studied. • Result of unique experimental study of buckling of dented compressed cylindrical shells are provided. • The dimple-like geometrical imperfections of different shape, size, deflection amplitude were considered. • Plateaus were discovered on the buckling load initial deflection diagrams. • Lower bound estimation of influence of local dents is suggested. [ABSTRACT FROM AUTHOR]

Energy barrier as a measure of metastability of axially compressed cylindrical shell is developed in the case of elasto-plastic material of the structure. The concept of local shell buckling is used for estimation of design buckling load. It does not require measuring initial geometrical shell imperfections which is challenging or sometimes impossible in the case of large scale real structures. Parametric analysis of the structure is performed and corresponding formula is derived for design buckling load which splits up the zone of high sensitivity of the shell to imperfections. The obtained formula is compared with published experimental results and Eurocode EN 1993-1-6 (2007) recommendations. • The concept of local buckling and energy method is applied to elasto-plastic axially compressed cylindrical shells. • Parametric analysis is performed using numerical simulation. • Formula for knockdown factor is suggested for design buckling load estimation. • Obtained result is validated by comparison with experimental data and Eurocode approach. [ABSTRACT FROM AUTHOR]

According to the concept of local buckling of compressed isotropic cylindrical shells, the buckling process begins with appearance of one or several dimple-like buckles. Then, depending on the load level, it leads to local or general shell buckling. The latter causes the loss of structure bearing capacity and catastrophic consequences. Therefore the stability of axially compressed cylindrical shells under different additional local perturbations was investigated intensively by many authors. Based on the performed studies, new formulae were suggested for design buckling load which improved the more conservative NASA SP-8007 (1968) recommendations. The present research is focused on analysis of post-buckling behaviour of the shell with one local dimple-like buckle. The conditions of existence and stability of such post-buckling equilibrium state were investigated. Special attention was paid to the influence of the shell material plasticity and geometrical parameter of the structure on its post-buckling behaviour. Based on this analysis the calculation methodology of design buckling load of the structure is suggested for the case of the shell with plastic material. • Effect of plasticity on stability of compressed cylindrical shells is studied in the scope of concept of local shell buckling. • Post-buckling behaviour of the shell with local buckle is investigated taking into account plasticity of the shell material. • Results obtained by numerical method are validated by comparison with experimental data. • The calculation methodology of design buckling load of plastic shell is suggested in the paper. [ABSTRACT FROM AUTHOR]

The asymptotic method for spherical shells at large deflections was developed in the case of composite shell subject to external pressure. Reissner's equations describing axially symmetric deformation of deep shells with arbitrary deflections and rotation angles were considered as initial ones. They were significantly simplified and asymptotic formulae for deformation energy, external pressure and stresses depending on deflection amplitude were obtained. The energy barrier criterion was applied for the composite shell for estimation of the shell metastability. Formula for design buckling pressure was derived for composite shells. It improves the significantly more conservative NASA SP-8032 recommendations. The full inversion of the shell was studied as well. Asymptotic formulae were obtained for evaluation of stresses and load deflection dependencies. They can be useful for design of diaphragms for positive expulsion propellant spherical tanks of launch vehicles and for other applications. [ABSTRACT FROM AUTHOR]