Imperfection Sensitivity Analysis Research Articles

High-fidelity numerical simulation and experimental validation of a 1600-mm-diameter axial loaded grid stiffened cylindrical shell

Stiffened cylindrical shells are commonly used in the launch vehicle, which are prone to buckling under axial compression load and extremely sensitive to imperfections. Establishing a high-fidelity finite element model is a prerequisite for correctly analyzing buckling load and imperfection sensitivity of stiffened cylindrical shells. This paper proposes a homogenization-based model updating approach for the grid stiffened cylindrical shell with many chamfers, which can establish a high-fidelity shell finite element model that maintains calculation efficiency and accuracy advantages. It uses the equivalent elastic constants of the unit cell as the link to modify the size parameters of the finite element model for the grid stiffened cylindrical shell. Moreover, the axially compressed buckling test of a 1.6-m-diameter orthogrid stiffened cylindrical shell is performed to verify the effectiveness of the proposed method. Through comparison, it could be found that the updated finite element model has a higher analysis accuracy. Specifically, the analysis error of the initial finite element model is −16.85% compared with the test result, while the error of the updated model is only −0.67%. The updated model can be used with full confidence for simulating mechanical performance such as the buckling process and the imperfection sensitivity analysis. Besides, it also indicates the necessity of considering the structural characteristics such as the chamfer in the optimization design process of the grid stiffened cylindrical shell structure.

Numerical and Experimental Buckling and Post-Buckling Analyses of Sphere-Segmented Toroidal Shell Subject to External Pressure

This study determined the buckling characteristics of sphere-segmented toroidal shells subjected to external pressure. The proposed toroidal vessel comprises six spheres and six rings. Two laboratory models with the same nominal dimensions were manufactured, measured, tested, and evaluated. To investigate whether sphere-segmented toroidal shells are imperfection-sensitive structures with closely spaced eigenvalues, the subspace algorithm was applied to evaluate the first 50 eigenmodes, and the modified Riks algorithm was used to obtain post-buckling characteristics. The results indicated that the deviation between the results of the experimental and numerical analyses was within a reasonable range. The proposed sphere-segmented toroidal shells were highly imperfection-sensitive structures with closely spaced eigenvalues. Subsequently, imperfection sensitivity analysis confirmed this conclusion. In numerical analyses, the first eigenmode could be considered as the worst eigenmode of sphere-segmented toroidal shells. The trend of the equilibrium path of sphere-segmented toroidal shells was consistent with spherical shells, revealing instability. In addition, ellipticity and completeness exerted a negligible effect on the buckling load of sphere-segmented toroidal shells.

Buckling of Composite Structures with Delaminations—Laminates and Functionally Graded Materials

In the present paper, buckling problems of constructions with single delamination are examined. Structures were made of unidirectional laminates and functionally graded materials (FGM). Two types of delaminations (closed and opened) were both investigated in experiments both in rectangular plates and axi-symmetric shells. The first part of the work is devoted to the formulation of contact problems (embedded, closed delaminations) with the aid of various functional inequalities. Then, computational models are discussed. To study the influence of the variable material configuration of FGMs, the fourth-order plate/shell relations were adopted. Finally, three particular problems examined are the buckling of flat rectangular plates, spherical shells, and compressed rectangular plates with elliptical delaminations. The experiments were conducted using imperfection sensitivity analysis and post-buckling non-linear analysis. The results demonstrate that the unsymmetric configurations of FGM structures lead to the reduction of buckling loads for structures with delaminations. For FG structures, those effects are described by the simple coefficient. Linear fracture mechanics were employed to distinguish the form of unilateral boundary problems (closed or opened). In the first case, the stable variations of the strain energy release rate GI with the delamination length variations were observed, whereas in the second case the unstable variations were observed.

Imperfection sensitivity study of the thermal–mechanical buckling of laminated composite cylinders using a novel reduced-order modeling method

Laminated composite cylinders subjected to axial compression exhibit the “snap-back” response, the buckling load of which is very sensitive to initial geometric imperfections. An operating environment with high temperatures influences both the buckling behavior and imperfection sensitivity of thin-walled cylinders. This paper proposes a novel reduced-order modeling method for nonlinear buckling and imperfection sensitivity analyses, considering the effects of the initial geometric imperfections and initial temperature field. The Koiter asymptotic theory is reformulated to construct a thermal–mechanical reduced-order model with initial geometric imperfections. The initial temperature field is converted to an additional degree of freedom during the construction of the reduced-order model. An a posteriori account of geometric imperfections is achieved for the thermomechanical reduced system of the perfect structure. A predictor–corrector process is developed to trace the thermoelastic geometrically nonlinear response of an axially compressed cylinder with geometric imperfections. Various numerical examples are designed to validate the high efficiency and satisfactory accuracy of the proposed method for imperfection sensitivity analyses. The optimization process based on the proposed method is applied to determine the most imperfection-sensitive laminate and the “worst” imperfection shape.

Shape optimisation of stainless steel corrugated cylindrical shells for additive manufacturing

Axially compressed circular cylindrical shells with large diameter-to-thickness ratios are highly susceptible to local buckling, and their load-carrying capacities are known to be very sensitive to initial geometric imperfections. Hence, severe knock-down factors on their theoretical buckling loads are typically prescribed in design specifications, which greatly impair their structural efficiency. With the aim of enhancing load-bearing resistance and reducing sensitivity to imperfections, the shape optimisation and assessment of compressed free-form wavy cylindrical shells, the realisation of which is now viable through additive manufacturing, are the subject of the present study. The adopted optimisation framework employs the Particle Swarm Optimisation (PSO) algorithm, integrating computer-aided geometric design, nonlinear numerical simulations and imperfection sensitivity analyses. The structural performance of the optimised free-form wavy shells is analysed and compared to that of reference circular shells, as well as other types of non-circular shell profiles, including sinusoidally corrugated shells, Aster shells and stringer-stiffened shells. The optimised free-form wavy shell profiles are shown to exhibit increases in ultimate stress of up to 136% compared with the reference circular shell profiles; in general, greater benefits are achieved for more slender cross-sections. In future work, the proposed optimised shells will be manufactured in stainless steel by means of powder bed fusion (PBF), and their structural performance will be further verified through physical experiments.

A novel and highly efficient strategy to determine the ‘Worst’ imperfection shape for buckling of cylindrical shell panels

The cylindrical shell panels subjected to in-plane compression experience more complex snap-back behavior associated with sharp turning around the limit-point load. The initial geometric imperfections play a significant role in the evaluation of the limit load along the nonlinear buckling response. Due to the highly computational cost for imperfection sensitivity analyses, the determination of a ‘worst’ imperfect shape that leads to the minimum limit load becomes a challenge. In this work, the initial geometrical imperfection shape is assumed to be the combination of numbers of closely-spaced buckling modes with different participation coefficients. The reduced order model based on the improved Koiter theory is constructed using all the closely-spaced modes, only once for ‘perfect’ structure and then applicable for any imperfections by just updating the independent imperfection term. Independent imperfection terms with gradient and Hessian forms are designed to be suitable for imperfections with small and relatively large magnitudes, respectively. The optimization strategy is developed to find out the optimal participation coefficients of closely-spaced modes that give the ‘worst’ imperfect shape. Two imperfection sensitivity indices, the limit load under a certain imperfection magnitude and the reduction factor of limit load under a certain interval of imperfection magnitudes, are defined and applied as the objective functions, respectively. The genetic algorithm is selected to find the optimal solutions efficiently benefiting from the a posteriori accounts of imperfections in function calls. Cylindrical shell panels with various configurations are applied to validate the good performance of the proposed strategy.

Alternative approach for imperfection-tolerant design optimization of stiffened cylindrical shells via energy barrier method

Imperfection sensitivity analysis of stiffened shells generally involves high computational cost, thus imperfection-tolerant design is a challenging issue. The concept of using the energy barrier to predict the buckling load of imperfect cylindrical shells has exhibited potential for practical application. In this study, the energy barrier method is used for stiffened cylindrical shells to obtain the minimum post-buckling load, and the numerical-based smeared stiffener method (NSSM) is employed to accelerate the analysis process. Then, an imperfection-tolerant design optimization framework for stiffened cylindrical shells is established via energy barrier method. Results show that the design configuration of a stiffened cylindrical shell with higher imperfection tolerance capacity can be obtained by the proposed method in a more efficient manner.

Post-buckling optimization of bending-induced variable stiffness composite cylinders considering worst geometric imperfections

Variable stiffness (VS) composite cylinders have attracted much more attention in the aerospace industry, due to their significantly improved mechanical properties. Most of the existing studies on the optimization of VS composite cylinders are based on linear buckling analysis, without considering the imperfection sensitivity and ignoring the potential of the non-linear post-buckling bearing of cylindrical shells. Therefore, the imperfection sensitivity analysis of VS composite cylinders under bending load is performed in the non-linear post-buckling regime. It is illustrated that the optimized result based on the linear buckling analysis is not robust, if imperfections are taken into account. Moreover, an optimization framework considering imperfection sensitivity is established, wherein the Worst Multiple Perturbation Load Approach (WMPLA) is used to calculate the lower-bound limit moment of composite cylinders. The critical moment of the optimized result with the worst imperfections has an improvement of 13.81%, and the knockdown factor (KDF) is improved by 27.27%. It is demonstrated to be necessary to consider the imperfection sensitivity based on the non-linear post-buckling analysis during the optimization of VS composite cylinders, and the considerable performance with improved anti-imperfection ability can be realized by tailoring the fiber orientation of VS cylinders.

Imperfection sensitivity of nonlinear primary resonance behavior in bi-directional functionally graded porous material beam

This research aims to analyze the imperfection sensitivity in the nonlinear resonance behavior of bi-directional functionally graded (BDFG) porous beam. The trigonometric and hyperbolic functions are chosen to describe the global and localized geometrical imperfections. Considering the first-order shear deformation theory and von-Kármán’s geometric nonlinearity, theoretical model is formulated to depict the coupled motion including the stretching, bending and shear deformations of BDFG porous beam. Differential quadrature based Galerkin’s procedure is proposed to formulate the reduced model. The nonlinear responses are obtained by using the numerical bifurcation method. Numerical results demonstrate that resonance response of beam under goes cyclic-fold bifurcations. Resonance behavior of beam may exhibit hardening, softening or softening-hardening type nonlinearity. Imperfection sensitivity analysis shows that the variation of imperfection modes, material gradient and porosity change the characteristic of frequency-response and force-response or even the number of bifurcation points. .

Thermally induced nonlinear stability and imperfection sensitivity of temperature- and size-dependent FG porous micro-tubes

An attempt is made in the current research to analyse the nonlinear thermal stability and imperfection sensitivity of functionally graded (FG) porous micro-tubes. Temperature-dependent properties of the geometrically imperfect micro-tube are graded across the radius of cross-section. It is assumed that the micro-tube with different end conditions is in contact with a two-parameter elastic foundation. The nonlinear component of the elastic foundation can be of the hardening or softening type. The equilibrium equations are obtained within the framework of von Karman nonlinear assumptions and high-order shear deformation tube theory. The governing equations are reformulated for the case of imperfect micro-tubes based on the modified couple stress theory. The system of nonlinear differential equations is solved using the two-step perturbation technique and Galerkin procedure. The analytical solutions are obtained for three different types of immovable boundary conditions which are clamped-rolling, simply-supported and clamped–clamped. The closed-form expressions are given to obtain the large deflection in the micro-tube as a function of the elevated temperature. Novel parametric studies are given to explore the thermal stability and imperfection sensitivity analysis of the perfect and imperfect micro-tubes, respectively. The effects of boundary conditions, couple stress components, porosity coefficient, elastic foundation, FG pattern, temperature dependence and geometrical parameters are studied.

Unfavorable geometric imperfections in steel plate girders subjected to localized loads

Recent experimental studies have shown that the current EN1993-1-5 patch loading resistance model underestimates the ultimate strength of steel plate girders in certain cases, especially for longitudinally stiffened steel plate girders and longer patch load lengths. In addition to this resistance model, the design standard allows the use of finite element analysis for the ultimate limit state. The existing design recommendations for an adequate ultimate limit state design are found insufficient. In the current paper, the imperfection sensitivity analysis of steel plate girders subjected to patch loading is performed. The paper presents a nonlinear finite element parametric study designed to determine the influence of a variety of geometric imperfections on the patch load resistance of longitudinally unstiffened and stiffened steel plate girders. Experimentally measured, buckling mode-affine, and hand-defined sinusoidal geometric imperfections were varied in the study in combination with varying patch load lengths and relative stiffnesses of the longitudinal stiffener.The ultimate strength of longitudinally unstiffened and stiffened steel plate girders increases as the patch load length is increased for all the geometric imperfections considered. For relatively small patch load lengths, the relative stiffness of the longitudinal stiffener has no significant influence on the ultimate load regardless of the geometric imperfection. For longer patch load lengths, the ultimate strength is highly dependent on geometric imperfections. The influence of the web panel aspect ratio is shown to be negligible for smaller patch load lengths regardless of the geometric imperfection, while for longer applied loads, the impact of this ratio is dependent on the geometric imperfection.Unfavorable geometric imperfections in steel plate girders subjected to patch loading are governed by the patch load length and the relative stiffness of the longitudinal stiffener. Different unfavorable geometric imperfections were found for different patch load lengths and relative stiffnesses. Local geometric imperfections defined in the upper web sub-panel yield lower ultimate strengths than global geometric imperfections (defined over the whole web depth). The lowest patch load resistances were returned for geometric imperfections that resembled the deformation at collapse (collapse-affine geometric imperfections). Although the concept of collapse-affine geometric imperfections seems impractical for design purposes (a geometrically and materially nonlinear analysis must be conducted before collapse-affine geometric imperfection shapes are determined), the use of these imperfection shapes is recommended.

A reduced model for nonlinear analysis and design of thin-walled structures prone to multi-modal buckling

A reduced model for nonlinear analysis and design of thin-walled structures prone to multi-modal buckling

Collapse analysis of a wind turbine tower with initial-imperfection subjected to near-field ground motions

As a slender thin-walled structure, wind turbines may be susceptible to pulse-type near-field seismic records. In order to study the effects of near-field records on the imperfection sensitivity and collapse modes of a 1.5 MW land-based wind turbine, two types of finite element model (FEM) are established and investigated through ABAQUS, i.e. FEM with and without initial welding imperfection. The horizontal displacement, horizontal acceleration, shear force and bending moment are investigated and compared. Finally, the incremental dynamic analysis is carried out to study the imperfection sensitivity and collapse analysis of this tower. The results show that the seismic response of this wind turbine subjected to pulse-type near-field seismic records are all significant larger than those under the excitation without velocity pulses. The presence of imperfections increases the variability of the seismic response of this wind turbine, and different defect amplitudes will affect the location of the plastic hinge on the tower. The plastic hinge is easy to occur at approximately 1/2–3/4 height of the tower under the strong pulse-type near-field earthquake excitation. The welding imperfection can significantly reduce the intensity of ground motion required for the nonlinear response of the wind turbine tower, especially for that under pulse-type near-field seismic excitation.

Incomplete reduced stiffness method for imperfection sensitivity of cylindrical shells

Cylindrical shells are commonly used in the industry structures, and they are prone to buckling under axial compression load. However, the practical critical loads are often lower than the theoretical value of buckling load due to the presence of imperfections. In this paper, an incomplete reduced stiffness method (iRSM) is proposed to perform the imperfection sensitivity analysis of cylindrical shells. In the iRSM, it is assumed that membrane stiffness components of the shell would be eroded at the critical buckling state, and a reduction function with two parameters is proposed to describe the mapping between membrane stiffness components and the geometric imperfections of the shell. Then, the buckling load of imperfect cylindrical shells can be obtained by solving the linear buckling problem of the reduced-stiffness system. For typical cylindrical shells, the reduction function is determined according to the measured imperfection and experimental results. By comparison with the experimental results and other conventional methods, it is demonstrated that the iRSM is a high-accurate, slightly conservative and fast method to perform the imperfection sensitivity of cylindrical shells.

Influence of geometric imperfections on the axially loaded composite conical shells with and without cutout

The nonlinear stability analysis of a structure with initial geometric imperfections is very complex, difficult, and unsolved yet in designing. At present, there are no unified design guidelines for imperfection sensitive metal shells (spherical, cylindrical, conical, and so forth), and the study about composite shell structures is often for a specific ply parameter and therefore less adaptable. In this paper, the imperfection sensitivity analysis of orthotropic conical shells with and without cutouts under compression is investigated numerically. The eigenmode-shape imperfections and dimple-shape imperfections are introduced to conical shells with different cutout sizes. In order to find the worst imperfections that can reduce the buckling load as far as possible, an optimization framework based on the multiple perturbation load approach is developed by the present authors to search the possible positions of dimple-shape imperfections; the effectiveness of the algorithm has been proved by comparing with other conventional approaches.Mid-surface imperfections, a type of realistic imperfection, transferred from a cylinder are considered for a conical shell with a cutout at last. This work can provide a reference value for conical shell design for practical engineering applications.

Limit load analysis and imperfection sensitivity of porous FG micro-tubes surrounded by a nonlinear softening elastic medium

An imperfection sensitivity analysis for the nonlinear post-buckling behavior of functionally graded (FG) porous micro-tubes is performed in this research. The case of geometrically imperfect micro-tubes surrounded by a nonlinear elastic medium under axial compressive load is analyzed. Properties of the micro-tube with uniform distributed porosity are FG across the radius of the cross-section. Two types of boundary conditions as simply-supported and clamped are considered. The high-order shear deformation theory of tubes is utilized to approximate the displacement field. Differential equations governing the equilibrium position of the micro-tube are extracted using the virtual displacement principle. These nonlinear equations are analytically solved by means of the two-step perturbation technique and Galerkin procedure. It is shown that when the imperfect micro-tube is in contact with a sufficiently soft foundation, the post-buckling path of the structure is unstable, and therefore the structure is imperfection sensitive. Since the imperfection sensitivity of micro-tubes is not reported in literature, results of this study are compared with buckling responses of perfect FGM tubes. The effects of porosity coefficient, power law index, length scale parameter, and geometrical parameters upon the limit buckling load of imperfect micro-tubes are investigated.

Accelerated Koiter method for post-buckling analysis of thin-walled shells under axial compression

Due to the high specific strength and stiffness, thin-walled shells are widely used in aerospace engineering structures. However, along with the increase of the structure size or the structure hierarchy, the computational cost of the post-buckling analysis of thin-walled shells would increase sharply and then the imperfection sensitivity analysis would become time-consuming. In this study, an accelerated Koiter method is proposed to improve the efficiency of post-buckling analysis and imperfection sensitivity analysis of thin-walled shells under axial compression. In the framework of the accelerated Koiter method, a single mode Koiter method considering pre-buckling nonlinear behavior is used to reduce the computational time of repeated imperfection analysis, which can accurately predict the effect of the small amplitude imperfection. Furthermore, in order to obtain the expanded point of Koiter method, a Combined Approximation (CA)-based iterative eigenvalue algorithm is constructed to obtain the pre-buckling state in the neighborhood of the bifurcation point and reduce computational cost by the reasonable step length prediction. Finally, the efficiency and accuracy of the proposed method are demonstrated by means of four illustrative examples, including the composite cylindrical shell, the isotropic cylindrical shell, the isotropic conical shell and the hierarchical stiffened cylindrical shell under axial compression.

Imperfection sensitivity analysis and DSM design of web-stiffened lipped channel columns experiencing local-distortional interaction

Cold-formed thin-walled steel members are commonly subjected to initial imperfection, which will greatly affect their load-carrying capacity. In this study, the load-carrying capacity and imperfection sensitivity of fixed-ended cold-formed steel web-stiffened lipped channel columns subjected to local-distortional (L-D) interactive failure (including secondary local bifurcation interaction (SLI), true L-D interaction (TI) and secondary distortional bifurcation interaction (SDI)) are investigated. A new imperfection simulation method combining finite element analysis (FEA) with the constrained finite strip method (cFSM) was firstly proposed, and then validated using experimental data from literatures. Next, 224 geometries of columns experiencing L-D interactive failure were chosen by the cFSM and their load-carrying capacity under normal state P num (with unavoidable minor initial imperfection) and those under imperfect state P imperfect (with various imperfections) were respectively obtained using FEA. The most unfavorable imperfection patterns for L-D interactive failure were summarized by comparing the P imperfect and P num of the members. Furthermore, the quality of the DSM-based design approaches for L-D interactive failure were evaluated by comparing P num with the failure load predictions of the selected columns provided by these design approaches. • An imperfection simulation method that combined FEM and constrained FSM was proposed. • The effects of imperfections on the thin-walled columns affected by L-D interaction were studied. • The accuracy and applicability of design approaches based on DSM were evaluated.

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.

A smeared stiffener based reduced-order modelling method for buckling analysis of isogrid-stiffened cylinder

• A reduced-order modelling method is proposed for nonlinear stability of isogrid-stiffened cylinders. • The reduced order model is constructed based on the smeared stiffener model. • Mode selection criteria are developed in the proposed method for imperfect cylinders. • Computational costs are significantly reduced in buckling and imperfection sensitivity analyses. Thin-walled isogrid-stiffened structures have been widely used as highly efficient structural components in aerospace engineering. In this work, the stability behavior of isogrid-stiffened cylinder is analyzed using the smeared stiffener model based reduced-order modelling (SSM-ROM) method. The smeared stiffener method proposed based on mechanical characteristics of isogrid-stiffener cell is applied to facilitate the finite element(FE) modelling of structure. The reduced-order models constructed based on Koiter asymptotic expansion are extended to be applicable for the smeared stiffener model. Mode selection criteria for cylinders with measured geometric-shape imperfections are developed to lower the computational cost in construction of the reduced-order model. The proposed method is implemented into the finite element framework and achieved completely using the in-house codes. The isogrid-stiffened cylinders with two different configurations(strong or weak stiffeners) are considered to validate the performance of the proposed method. Linear and nonlinear buckling analyses are first applied to test the numerical accuracy of the smeared stiffener model. The sensitivity of the buckling load in terms of magnitude of initial imperfections is then carefully studied using three different geometric imperfection shapes. Numerical results demonstrate the good performance of the proposed method in both structural buckling analysis and imperfection sensitivity analyses.