Single Perturbation Load Approach Research Articles

Analysis using finite element method of the buckling characteristics of stiffened cylindrical shells

ABSTRACT The impact of imperfection/flaws in axially compressed/externally pressurised stiffened cylindrical shells with various shapes was studied using finite element analysis (FE). The study examined imperfection methods such as the Single Perturbation Load Approach (SPLA) and Smooth Inward Dimple (SID) to validate them against existing experimental data. It also explored the influence of initial imperfections in terms of depth, magnitude, size, and location on the buckling strength of the shell. Both methods showed reasonable agreement with published experimental data, with differences ranging from 2% to 13%. The shells’ load-carrying capacity was affected by initial geometric imperfections, particularly in terms of depth, magnitude, size, and location. The study’s findings add to our understanding of how structural imperfections impact the initial phases of designing, fabricating, and analysing axially compressed or externally pressurised stiffened cylindrical shell structures in terms of their magnitude, size, and location.

Buckling Knockdown Factors of Composite Cylinders under Both Compression and Internal Pressure

The internal pressure of a thin-walled cylindrical structure under axial compression may improve the buckling stability by relieving loads and reducing initial imperfections. In this study, the effect of internal pressure on the buckling knockdown factor is investigated for axially compressed thin-walled composite cylinders with different shell thickness ratios and slenderness ratios. Various shell thickness ratios and slenderness ratios are considered when the buckling knockdown factor is derived for the thin-walled composite cylinders under both axial compression and internal pressure. Nonlinear post-buckling analyses are conducted using the nonlinear finite element analysis program, ABAQUS. The single perturbation load approach is used to represent the geometric initial imperfection of thin-walled composite cylinders. For cases with the axial compressive force only, the buckling knockdown factor decreases as the shell thickness ratio increases or as the slenderness ratio increases. When the internal pressure is considered simultaneously with the axial compressive force, the buckling knockdown factor decreases as the slenderness ratio increases but increases as the shell thickness ratio increases. The buckling knockdown factors considering the internal pressure and axial compressions are higher by 2.67% to 38.98% compared with the knockdown factors considering the axial compressive force only. The results show the significant effect of the internal pressure, particularly for thinner composite cylinders, and that the buckling knockdown factors may be enhanced for all the shell thickness ratios and slenderness ratios considered in this study when the internal pressure is applied to the cylinder.

Lower-bound axial buckling load prediction for isotropic cylindrical shells using probabilistic random perturbation load approach

Due to high sensitivity to various imperfections, buckling loads of thin-walled cylindrical shells subjected to axial load vary dramatically. In order to predict lower-bound buckling loads for axially loaded cylindrical shells rationally, a probabilistic analysis approach named Probabilistic Random Perturbation Load Approach (PRPLA) is developed in this study. Firstly, a Back-Propagation Neural Network (BPNN) based method is established to describe measured imperfection patterns. Next, Random Single Perturbation Load Approach (RSPLA) is loaded upon BPNN-based depicted traditional imperfection patterns to construct a stochastic dimple imperfection. The aforementioned scattering traditional imperfections, as well as a variety of scattering non-traditional imperfections, are then sampled using Monte-Carlo simulation to generate cylindrical shell models differentiating from a nominal one. The probabilistic distribution of lower-bound buckling loads is obtained by finite element analysis. A nominal shell's realistic lower-bound buckling load is determined by choosing a specified reliability level lastly. The results show that describing measured imperfection patterns via BPNN is very close to real ones, and PRPLA presented is an improved method to find lower-bound buckling loads efficiently compared with NASA SP-8007 and many commonly used numerical approaches.

Numerical Derivation of Buckling Knockdown Factors for Isogrid-Stiffened Cylinders with Various Shell Thickness Ratios

This study is aimed at providing a numerical derivation of the shell knockdown factors of isogrid-stiffened cylinders under axial compressive loads. The present work uses two different analysis models such as the detailed model with modeling of numerous stiffeners and the equivalent model without modeling of stiffeners for isogrid-stiffened cylinders. The single perturbation load approach is used to represent the geometrically initial imperfection of the cylinder. Postbuckling analyses using the displacement control method are conducted to calculate the global buckling loads of a cylinder. The shell knockdown factor is numerically derived using the obtained global buckling loads without and with the initial imperfection of the isogrid-stiffened cylinder. The equivalent model is more efficient than the detailed model in terms of modeling time and computation time. The present knockdown factor function in terms of the shell thickness ratio (radius to thickness) for the isogrid-stiffened cylinder is significantly higher than NASA’s knockdown factor function; therefore, it is believed that the present knockdown factor function can facilitate in developing lightweight launch vehicle structures using isogrid-stiffened cylinders.

Derivation of knockdown factors for grid-stiffened cylinders considering various shell thickness ratios

PurposeThe purpose of this paper is to derive knockdown factor functions in terms of a shell thickness ratio (i.e. the ratio of radius to thickness) for conventional orthogrid and hybrid-grid stiffened cylinders for the lightweight design of space launch vehicles. Design/methodology/approachThe shell knockdown factors of grid-stiffened cylinders under axial compressive loads are derived numerically considering various shell thickness ratios. Two grid systems using stiffeners – conventional orthogrid and hybrid-grid systems – are used for the grid-stiffened cylinders. The hybrid-grid stiffened cylinder uses major and minor stiffeners having two different cross-sectional areas. For modeling grid-stiffened cylinders with various thickness ratios, the effective thickness (teff) of the cylinders is kept constant, and the radius of the cylinder is varied. Thickness ratios of 100, 192 and 300 are considered for the orthogrid stiffened cylinder, and 100, 160, 200 and 300 for the hybrid-grid stiffened cylinder. Postbuckling analyses of grid-stiffened cylinders are conducted using a commercial nonlinear finite element analysis code, ABAQUS, to derive the shell knockdown factor. The single perturbation load approach is applied to represent the geometrical initial imperfection of a cylinder. Knockdown factors are derived for both the conventional orthogrid and hybrid-grid stiffened cylinders for different shell thickness ratios. Knockdown factor functions in terms of shell thickness ratio are obtained by curve fitting with the derived shell knockdown factors for the two grid-stiffened cylinders. FindingsFor the two grid-stiffened cylinders, the derived shell knockdown factors are all higher than the previous NASA’s shell knockdown factors for various shell thickness ratios, ranging from 100 to 400. Therefore, the shell knockdown factors derived in this study may facilitate in the development of lightweight structures of space launch vehicles from the aspect of buckling design. For different shell thickness ratios of up to 500, the knockdown factor of the hybrid-grid stiffened cylinder is higher than that of the conventional orthogrid stiffened cylinder. Therefore, it is concluded that the hybrid-grid stiffened cylinder is more efficient than the conventional orthogrid-stiffened cylinder from the perspective of buckling design. Practical implicationsThe obtained knockdown factor functions may provide the design criteria for lightweight cylindrical structures of space launch vehicles. Originality/valueDerivation of shell knockdown factors of hybrid-grid stiffened cylinders considering various shell thickness ratios is attempted for the first time in this study.

Imperfection sensitivity analysis for a composite bowed-out shell under axial compression

In this article, we present a systematic work to investigate the imperfection sensitivity of composite bowed-out shells with different layup patterns under axial compression. Two types of geometric imperfections, including eigenmode-shaped imperfections (produced by a first-order eigenmode imperfection approach and an N-order eigenmode imperfection approach) and dimple-shaped imperfections (produced by a single perturbation load approach and a multiple perturbation load approach), are introduced into the finite element model to predict their knock-down factors. For the eigenmode-shaped imperfections, we show that the knock-down factors predicted by the first-order eigenmode imperfection approach are riskier than the ones predicted by the N-order eigenmode imperfection approach. When adopting the single perturbation load approach, we reveal that the direction of a dimple on the shell makes a negligible effect on axial pressure bearing capacity, while the amplitude of a dimple on the shell plays a significant role in affecting the knock-down factors. Using the multiple perturbation load approach as an extension of the single perturbation load approach, we uncover that the knock-down factors predicted by the multiple perturbation load approach are more conservative than these achieved by the single perturbation load approach. In addition, we also find that the composite bowed-out shells are more sensitive to dimple-shaped imperfection than eigenmode-shaped imperfections. This work provides helpful findings for designing an airplane body and marine risers.

Experimental validation of cylindrical shells under axial compression for improved knockdown factors

For cylindrical shells under axial compression, the essence of initial geometric imperfections is the superposition of local out-of-plane deformations of various forms, which may facilitate the development of buckling deformations, thus leading to a significant knockdown of the load-carrying capacity. It is very challenging for existing methods to provide an accurate prediction of the lower bound on a load-carrying capacity before the structure is fabricated. Therefore, it is crucial to find a type of assumed imperfection that will allow us to approximate lower bounds for shells in the design stage. Five 1-m-diameter unstiffened shells, termed W1-W5, are designed, analysed and tested. The measured imperfection approach, single-perturbation load approach (SPLA), worst multiple-perturbation load approach (WMPLA), and a Combined Approach for measured imperfections and superimposed radial point load imperfections are compared with test results. The results show that the SPLA-based methods produce higher KDFs than the test results and are sensitive to the distribution of the measured imperfections. In contrast, the KDFs predicted by the WMPLA and the Combined Approach are similar to one another and very close to the test results. From the comparison results, it can preliminarily be concluded that the WMPLA is able to envelop the small- and large-amplitude measured imperfections, which has the potential to predict a rational lower bound on the buckling loads of unstiffened cylindrical shells. The WMPLA should be extended to the design of other types of thin-walled structures with caution, because the manufacturing signature may be distinctly changed for different processes, and the buckling tests of other types of structures would be carried out in future study.

Design of axially loaded isotropic cylindrical shells using multiple perturbation load approach – Simulation and validation

Thin-walled cylindrical shells subjected to axial load are prone to buckling and very sensitive to the geometric imperfections. In order to determine a rational design load for axially loaded cylindrical shells, different analytical and numerical design methods have been developed. As an extension of single perturbation load approach (SPLA), multiple perturbation load approach (MPLA) is a promising method but has not been thoroughly studied. In this paper, the buckling tests of three steel cylindrical shell specimens are conducted. Combined with experimental and numerical results, the advantages and limitations of several commonly used design methods are discussed in detail, including NASA SP-8007, the measured imperfection approach, SPLA and MPLA. The results validate that SPLA is indeed risky to be adopted as a design method for isotropic metallic cylindrical shells. Meanwhile, it indicates that MPLA can not only improve the knock-down factor but also give a more safe design load than SPLA. The comprehensive numerical investigations are performed to study the effects of each perturbation load parameter on MPLA, including the number of perturbation loads, their relative position and magnitude. It is found that the symmetry of perturbation loads plays an important role in the global buckling load of cylindrical shell. Moreover, some guidance for the future application of MPLA is provided and the four perturbation load approach (4PLA) is proposed as an improved method for the preliminary design of isotropic metallic cylindrical shells.

Postbuckling analyses and derivations of Knockdown factors for hybrid-grid stiffened cylinders

In this work, postbuckling behaviors of grid-stiffened cylinders under compressive loads are examined in an attempt to establish an analytical method to derive the shell Knockdown factor for space launch vehicle structures. Two grid systems using stiffeners – traditional orthogrid and hybrid-grid systems – are considered for the stiffened cylinders. The hybrid-grid system consists of major and minor stiffeners, which have two different cross-sectional sizes. Additionally, the baseline and minimum weight models are considered for each grid-stiffened cylinder model. The single perturbation load approach is used to represent the geometrical imperfection of cylinders. A commercial finite element analysis code, ABAQUS, is used for the postbuckling analyses. The present numerical simulations investigate the global and local instabilities for both stiffened cylinders using orthogrid and hybrid-grid systems. The global buckling load for the hybrid-grid stiffened cylinder is higher than that for the traditional orthogrid stiffened cylinder for both the baseline and minimum weight models. However, the postbuckling behaviors for the hybrid-grid stiffened cylinders are more complex, as compared to those for the orthogrid stiffened cylinders. Additionally, the shell Knockdown factors are derived using obtained numerical analysis results for the grid-stiffened cylinders. Since the shell Knockdown factor of the hybrid-grid stiffened cylinder is higher than that of the traditional orthogrid stiffened cylinders for both the baseline and minimum weight models, the hybrid-grid stiffened cylinder is more efficient in terms of the buckling design when compared to a traditional orthogrid stiffened cylinder. Furthermore, the derived Knockdown factors in the present work are higher than the values by NASA's buckling design criteria.

Derivations of Knockdown Factors for Cylindrical Structures Considering Different Initial Imperfection Models and Thickness Ratios

This work conducts postbuckling analyses of axial-loaded cylinders and derives shell knockdown factors for the buckling design of space launch vehicle structures. Simple isotropic cylinders without stiffeners are considered as analysis models, and a commercial nonlinear finite element analysis code, ABAQUS, is used for the present postbuckling analyses. Two different methods, single perturbation load approach (SPLA) and multiple perturbation load approach (MPLA), are used for modeling the geometrically initial imperfection of a cylinder. In addition, various shell-thickness ratios (ratio of radius to thickness) are considered to derive knockdown factors using SPLA. In the postbuckling analyses, local and global bucklings are investigated and, using the obtained analysis results, the shell knockdown factors are derived for the buckling design of launch vehicle structures. The MPLA, with more perturbation loads, provides lower knockdown factors. Furthermore, for the three different thickness ratios, the knockdown factors derived using the SPLA all are higher than the values using NASA’s buckling design criteria, and they are nearly constant with respect to shell-thickness ratios.

Design of sandwich-walled cylindrical shell structure with initial imperfections

Sandwich-walled cylindrical shell has been built with lattice-like stiffeners by adding another layer of skin in the inner surface of shell. Taking advantage of high mechanical property of lattice-like stiffeners, such as high specific stiffness, high specific strength etc., sandwich-walled cylindrical shell is studied for high load-carrying capacity in this paper. In this study, major parameters of sandwich-walled cylindrical shell is discussed for the effect of load-carrying capacity, and the imperfection sensitivity analysis of sandwich-walled cylindrical shells is carried out based on the eigenmode-shape imperfection approach and the single perturbation load approach. Comparing with the traditional stiffened cylindrical shell, the sandwich-walled cylindrical shell has a higher load-carrying efficiency. The reason of phenomenon is high bending stiffness of lattice-like stiffeners. And then, the optimization design of sandwich-walled cylindrical shells is carried out. It is results that optimum design of sandwich-walled cylindrical shell improved knockdown factor and load-carrying capacity. It can be concluded that sandwich-walled cylindrical shell is a potential structure to be more efficient structure in the future aerospace and aircraft designs.

Sizing strategy for stringer and orthogrid stiffened shells under axial compression

ABSTRACTThin-walled stiffened shell structures are used in primary structures of space launcher vehicles. These structures are prone to buckling and thus fail due to a loss of structural stability. Generally, it has to be distinguished between two major modes of instability for stiffened shell structures: a global buckling of the entire structure and a local buckling of skin fields, longitudinal and circumferential stiffeners. Due to the large number of variables when designing stiffened shell structures, their preliminary design is a demanding task. To allow an efficient preliminary design, sizing strategies can be developed. For this purpose, analytical methods, which allow to assess the local and global instability of stiffened shell structures, are employed. In this article, sizing strategies based on efficient analytical methods are introduced and applied to identify suitable designs of stringer stiffened and orthogrid stiffened shell structures. To study the imperfection sensitivity of stringer and orthogrid stiffened shell structures, numerical computations are performed using the single perturbation load approach and mode shape imperfections. Finally, weight strength curves are derived for stringer and orthogrid stiffened shell structures.

Numerically and experimentally predicted knockdown factors for stiffened shells under axial compression

Stiffened shells in launch vehicles are very sensitive to various forms of imperfections. In this study, the imperfection sensitivity of a 4.5mdiam isogrid stiffened shell under axial compression is investigated. The measured imperfection, NASA SP-8007 and several types of assumed imperfections, including eigenmode-shape imperfection and dimple-shape imperfections (produced by the single perturbation load approach (SPLA) and worst multiple perturbation load approach (WMPLA)), are introduced into FE model to predict the knockdown factors (KDFs), respectively. Then, the buckling test of this full-scale stiffened shell under axial compression is carried out to validate the above numerical approaches. It can be found that the KDF predicted by the WMPLA is very close to the test results, while the ones predicted by eigenmode-shape imperfection and NASA SP-8007 are extremely conservative. Besides, the measured imperfection and other assumed imperfections are proven to be risky, because these methods overestimate the actual load-carrying capacity. Finally, it can be concluded that the WMPLA is a potential and efficient approach to predict KDFs in the design stages for future launch vehicles.

Design of cylindrical shells using the Single Perturbation Load Approach – Potentials and application limits

The Single Perturbation Load Approach (SPLA) is a promising deterministic procedure on the basis of mechanical considerations to determine reasonable design loads for cylindrical shells in axial compression. In this paper, two main issues are identified that should be understood better in order to appreciate the potential and the limits of application of the SPLA. Firstly, the question whether a single perturbation load in general represents a “worst case” imperfection is addressed. Secondly, the influence of the stiffness properties of the shell on the quality of the SPLA predictions is studied. Finally, an indicator is suggested which identifies, based on the cylinder stiffness properties, whether the SPLA is conservative for a considered shell or not.

Probabilistic perturbation load approach for designing axially compressed cylindrical shells

The buckling load of cylindrical shells is heavily dependent on geometric imperfections and other non-traditional imperfections such as scattering wall thicknesses or loading imperfections. In this paper, the probabilistic perturbation load approach (PPLA) is proposed. This procedure is independent from costly measurements of geometric imperfections and is able to include various kinds of non-traditional imperfections by integrating the single perturbation load approach into a semi-analytical probabilistic framework. It is shown that the PPLA leads to design loads which are lower than the experimental buckling loads and much less conservative than the ones obtained from NASA SP-8007 in all investigated cases.

Buckling of axially compressed CFRP truncated cones with additional lateral load: Experimental and numerical investigation

Truncated thin-walled conical shells are often used as transition parts between cylinders of different diameters. Parts of space launcher transport systems are one example for the application of conical shells. Buckling of such thin-walled imperfection sensitive structures is a very important phenomenon to be considered during their design phase. Existing design guidelines, NASA SP-8007 for cylinders and NASA SP-8019 for cones, dated from the late 1960’s are currently used in the aerospace industry and employ conservative lower-bound knock-down factors. These empirically based lower-bound methods do not include important mechanical properties of laminated composite materials, such as the stacking sequence. New design approaches that allow taking full advantage of composite materials and take into account specific manufacturing methods are therefore required.The Single Perturbation Load Approach (SPLA) is an alternative proposal for a deterministic procedure for the design of thin-walled cylinders and cones under axial compression that accounts for geometric imperfections. The study deals with the buckling experiments on axially compressed, unstiffened carbon fiber-reinforced polymer (CFRP) truncated cones with an additional lateral load, performed by DLR for validation of the SPLA applied to this type of structure. Three geometrically identical cones with different layup were designed, manufactured and tested. During testing a digital image correlation system was employed and load-shortening data is extracted. The experimental results are compared with the FEA results.

Investigation on the Geometric Imperfections driven Local Buckling Onset in Composite Conical Shells

Buckling is a critical failure phenomenon for structures, and represents a threat for thin shells subjected to compressive forces. The global buckling load, for a conical structure, depends on the geometry and material properties of the shell, on the stacking sequence, on the type of applied load and on the initial geometric imperfections. Geometric imperfections, occurring inevitably during manufacturing and assembly of thin-walled composite structures, produce a reduction in the carrying load capability with respect to the design value. This is the reason why investigating these defects is of major concern in order to avoid over-conservative design structures. In this paper, the buckling behavior a conical structure with 45° semi-vertical angle is numerically investigated. The initial imperfections are taken into account by using different strategies. At first, the Single Perturbation Load Approach (SPLA), which accounts for defects in the form of a lateral load, normal to the surface, has been adopted. Then, the actual measured defects have been applied to the structure by using the Real Measured Mid-Surface Imperfections (MSI) approach. Investigations on cylindrical shells using the first strategy have already shown the occurrence of a particular phenomenon called “local snap-through”, which represents a preliminary loss of stiffness. In order to better understand this phenomenon for conical shells, both the aforementioned techniques have been used to provide an exhaustive overview of the imperfections sensitiveness in conical composite shells. This study is related to part of the work performed in the frame of the European Union (EU) project DESICOS.

Constant single-buckle imperfection principle to determine a lower bound for the buckling load of unstiffened composite cylinders under axial compression

Unstiffened cylindrical shells are very sensitive to geometric imperfections such as production-related deviations from the ideal geometry (conventional imperfections) as well as imperfect boundary conditions, material and wall thickness imperfections (non-conventional imperfections). The load carrying capability of unstiffened shells is reduced significantly by those imperfections. The NASA SP-8007 design guideline from 1968 is currently used for shell design. This guideline provides knock-down factors for the buckling loads and is based on experimental data of isotropic and orthotropic test specimen. Therefore the structural behavior of composite material is not considered adequately.Based on the single-perturbation load approach (SPLA), this paper introduces a new physical based approach, to determine a lower bound for the buckling load of unstiffened composite cylindrical shells with respect to conventional and non-conventional imperfections, the constant single-buckle imperfection (CSBI) principle. The CSBI principle is based on the theory of the metastability of the geometric imperfect cylindrical shell which is introduced within this paper. The results indicate that the CSBI principle has the potential to provide an improved shell design in order to reduce weight and cost of thin-walled shells.

Imperfection-insensitive design of stiffened conical shells based on equivalent multiple perturbation load approach

Stiffened conical shells in launch vehicles are very sensitive to various forms of imperfections. As a type of equivalent imperfections, several perturbation load approaches are used to investigate the influence of dimple-shape imperfections on the load-carrying capacity of stiffened conical shells. Firstly, the effect of axial location of dimple is examined by single perturbation load approach (SPLA), since the stiffness of stiffened conical shells varies along axial direction. Then, worst multiple perturbation load approach (WMPLA) is employed to find the lower bound of the collapse load of stiffened conical shells, and also provides the knowledge to determine the number of dimples in the multiple perturbation load approach (MPLA). After that, the optimization of stiffened conical shells for imperfection-insensitive design is carried out, where the equivalent MPLA is adopted during the optimization process to reduce the computational cost. Illustrative example indicates that stiffened conical shells exhibit more complicated imperfection sensitivity compared to cylindrical shells, and the proposed optimization framework can find an imperfection-insensitive design under structural weight constraint in an efficient manner.

Numerical approaches to stability analysis of cylindrical composite shells based on load imperfections

Purpose – Current procedures of buckling load estimation for thin-walled structures may provide very conservative estimates. Their refinement offers the potential to use structure and material properties more efficiently. Due to the large variety of design variables, for example laminate layup in composite structures, a prohibitively large number of tests would be required for experimental assessment, and thus reliable numerical techniques are of particular interest. The purpose of this paper is to analyze different methods of numerical buckling load estimation, formulate simulation procedures suitable for commercial software and give recommendations regarding their application. All investigations have been carried out for cylindrical composite shells; however similar approaches are feasible for other structures as well. Design/methodology/approach – The authors develop a concept to apply artificial load imperfections with the aim to estimate as good as possible lower bounds for the buckling loads of shells for which the actual physical imperfections are not known. Single and triple perturbation load approach, global and local dynamic perturbation approach and path following techniques are applied to the analysis of a cylindrical composite shell with known buckling characteristics. Results of simulations are compared with published experimental data. Findings – A single perturbation load approach is reproduced and modified. Buckling behavior for negative values of the perturbation load is examined and a pattern similar to a positive perturbation load is observed. Simulations with three perturbation forces show a decreased (i. e. more critical) value of the buckling load compared to the single perturbation load approach. Global and local dynamic perturbation approaches exhibit a behavior suitable for lower bound estimation for structures with arbitrary geometries. Originality/value – Various load imperfection approaches to buckling load estimation are validated and compared. All investigated methods do not require knowledge of the real geometrical imperfections of the structure. Simulations were performed using a commercial finite element code. Investigations of sensitivity with respect to a single perturbation load are extended to the negative range of the perturbation load amplitude. A specific pattern for a global perturbation approach was developed, and based on it a novel simulation procedure is proposed.