Influence of imperfection distributions for cylindrical stiffened shells with weld lands

Thin-Walled Structures 93 (2015) 177–187

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Influence of imperfection distributions for cylindrical stiffened shells with weld lands Peng Hao a, Bo Wang a,n, Kuo Tian a, Kaifan Du a, Xi Zhang b a State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116023, China b Beijing Institute of Astronautical Systems Engineering, Beijing 100076, China

art ic l e i nf o

a b s t r a c t

Article history: Received 3 March 2015 Received in revised form 16 March 2015 Accepted 17 March 2015

The influence of imperfection distributions considering manufacturing characters on the buckling response of stiffened shells has not been satisfactorily understood. Stiffened shells with three types of weld lands were established, including axial weld lands, circumferential and sequential axial weld lands, as well as staggered axial weld lands. As a concept of equivalent imperfection, dimple-shape imperfections produced by perturbation loads were adopted to substitute the measured imperfections, in order to reduce experimental and computational costs. Firstly, the influence of imperfection positions on the collapse load was examined for single perturbation load. Then, the influence of imperfection distributions was investigated for multiple perturbation load based on Monte Carlo Simulation. Finally, detailed comparison of three types of weld lands was made from the point-of-view of load-carrying capacity and imperfection sensitivity. Results can provide general instructions about imperfectioncritical areas for axially compressed stiffened shells, which are particularly crucial for the manufacturing process. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Stiffened shell Buckling Imperfection distribution Weld land

1. Introduction Axially compressed stiffened shells are the major structural component of launch vehicles to resist buckling and collapse. However, there exists large discrepancies between experimental and theoretical predictions of collapse loads for thin-walled structures, and it has been generally recognized that initial imperfections (i.e., small deviations from perfect structures) are the main attribution, which can dominate the buckling and post-buckling behavior of realworld thin-walled structures. Thus, a great deal of research was put forward to investigate the influence of various forms of initial imperfections [1,2]. So far, the most frequently used guideline to handle the effects of imperfections is NASA SP-8007 [3], where knockdown factors (KDFs) for structural design were determined by the lower bound curve based on a large collection of experimental results in 1960s. With the rapid advances of material system and manufacturing technology, these KDFs were proven to be over conservative [4]. In this case, a project named as Shell Buckling Knockdown Factor (SBKF) was funded by NASA Engineering and Safety Center for buckling-critical launch vehicle structures [4–6], aiming to develop and validate new shell buckling design methods

n

Corresponding author. E-mail addresses: [email protected] (P. Hao), [email protected] (B. Wang). http://dx.doi.org/10.1016/j.tws.2015.03.017 0263-8231/& 2015 Elsevier Ltd. All rights reserved.

accounting for imperfection sensitivity. Later, a Framework Plan called New Robust Design Guideline for Imperfection Sensitive Composite Launcher Structures (DESICOS) was funded by European Commission [7–9], whose ultimate objective is to establish a new design approach for imperfection sensitive composite launcher structures. Recently, a National Basic Research Program of China called Lightweight Design Theory and Method of Stiffened Shells including Imperfection Sensitivity was approved for future heavy-lift launch vehicles [10–14], and this study is also part of this research program. Among these research projects, it should be noted that, compared to the experiment-based approach, analysis-based approach is considered as a promising tool to investigate the safety margins attached to structural stability designs from a point-of-view of economy, since the experimental cost is usually hard to afford, especially for large-diameter structures. For thin-walled structures, geometric imperfections refer to deviations from the shape of perfect geometry, which are generally grouped into three broad categories: realistic, stimulating and worst imperfections [15]. Specifically, realistic imperfections can be determined by contact or non-contact optical measurement methods [4,16], and advanced imperfection measurement systems for unstiffened shells and stiffened panels have been developed by Degenhardt et al. [17]. Until now, a great deal of research has been carried out based on measured imperfections [18–21]. The essence of such a method is to associate characteristic imperfections with certain manufacturing processes, and then predict the nonlinear buckling phenomenon accurately by means of numerical analyses. In the

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established International Imperfection Data Bank, the measured initial geometric imperfections of unstiffened shells were decomposed into Fourier series for nominally identical specimens [22–25]. Unfortunately, the bottleneck of this method is that complete knowledge of the measured initial geometric imperfections is usually not available within the context of design process, let alone the accommodated mathematical model, moreover, measured imperfections of stiffened shells are desperately scarce in open literatures, due to the complex measurement method and extremely high expend. In addition, the measured imperfection sensitivity analysis may suffer from heavy computational efforts, especially for stiffened shells, because extremely refined meshes are required to describe and capture the nodal deviations from perfect geometry; stimulating imperfections can be considered as a type of artificial imperfection, leading to a characteristic physical buckling behavior. Typically, the incorporation of geometric imperfections using eigenmode shapes is a commonly applied technique [26,27]. Besides, Hühne et al. [28] developed the Single Perturbation Load Approach (SPLA) to create a local dimple-shape imperfection, which can produce a physically meaningful buckling behavior that is similar to the one observed in tests. Since the SPLA can be regarded as a type of equivalent imperfections, extremely refined meshes are not required, rather than measured imperfections, as discussed in Section 3.2. Then, the SPLA was extended to the design of composite conical structures under axial compression [29]. Furthermore, a combined methodology of the SPLA with a stochastic app roach was proposed by Degenhardt et al. [9]. As another significant supplement of the SPLA, the concept of Multiple Perturbation Load Approach (MPLA) was introduced by Arbelo et al. [8]; worst imperfections can be determined mathematically by means of optimization methods [30,31]. Based on a finite number of perturbation loads, an optimization framework to identify the worst realistic imperfection was presented by the authors for cylindrical unstiffened shells [11], aiming to provide references for the improved KDFs. More recently, Worst Multiple Perturbation Load Approach (WMPLA) was developed and extended to the design of stiffened shells with and without cutouts by the authors [13]. In addition, reliability-based method has been adopted to find more rational KDFs [32], where random geometric imperfections, material property and thickness variations, and even non-uniform axial loading were incorporated into the buckling prediction of cylindrical shells [33,34]. Also, a stochastic method using the first-order second-moment analysis was presented for unstiffened shells based on measured imperfections [19]. To the authors' knowledge, previous studies of imperfection sensitivity for thin-walled structures were mainly concentrated on the buckling response to various forms of imperfections. However, the influence of imperfection distributions considering manufacturing characters has not been studied intensively so far. Actually, establishing a correlation between imperfection distributions and structural performances is particularly significant for guiding the manufacture process and technology of stiffened shells. With regard to metallic stiffened shells, several segments are manufactured independently, and then assembled to complete barrels by welding, which adds the flexibility in the choice of stiffener configurations [35]. Rotter and Teng [36] examined the elastic buckling strengths of cylindrical shells under axial compression. Thornburgh [37] investigated the effects of the axial weld lands on the buckling response of cylindrical shells, and it was found that the buckling load is very sensitive to the specific location and geometry of stiffeners near the axial weld lands. Nevertheless, the influence of imperfection distributions on structural performances for cylindrical shells with weld lands is still far from being satisfactorily understood or solved, and such knowledge is particularly crucial for the fields of safety assessment and manufacturing. In this study, the influence of imperfection distributions for stiffened shells with three types of weld lands was thoroughly investigated. Nonlinear explicit dynamic analysis method was

utilized to predict the buckling and post-buckling behavior of stiffened shells under axial compression. As a concept of equivalent imperfection, dimple-shape imperfections produced by perturbation loads were adopted to substitute the measured imperfections, in order to reduce both the experimental and computational costs. Firstly, a 3-m-diameter orthogrid stiffened shell with axial weld lands was established according to Ref. [38]. Dimple-shape imperfections were compared and validated by the available experimental results and measured imperfections. The influences of imperfection positions and distributions on the collapse load were then examined based on single and multiple perturbation loads, respectively. Several critical understandings were gained by full exploration of the probability density function (PDF) of collapse loads based on Monte Carlo Simulation (MCS). Then, a 5-m-diamter orthogrid stiffened shell with circumferential and sequential axial weld lands was built, aiming to investigate the effects of imperfection distributions when circumferential weld lands are considered. Furthermore, a 5-m-diameter orthogrid stiffened shell with circumferential and staggered axial weld lands was developed to discuss the effects of distributions of axial weld lands. Finally, detailed comparison of three types of weld lands was made from the point-of-view of load-carrying capacity and imperfection sensitivity. Numerical results can provide general instructions about imperfection-critical areas for axially compressed stiffened shells with different types of weld lands, which require special attention in the manufacturing process.

2. Methodology 2.1. Nonlinear post-buckling analysis In this study, nonlinear post-buckling analysis of stiffened shells was performed by using the explicit dynamic method. For an explicit dynamic analysis, by utilization of the explicit time integration with central difference method, the equation of motion can be written as




 M C 2M M C ext int U U þ ¼ F  F þ  K U   t t t Δt 2 2Δt t þ Δt Δt 2 Δt 2 2Δt t  Δt ð1Þ

where, M is the mass matrix, C is the damping matrix, K is the stiffness matrix, a is the vector of nodal acceleration, V is the vector of nodal velocity, U is the vector of nodal displacement, t is is the vector of applied the time, Δt is the time increment, Fext t external force, Fint is the vector of internal force. t As is evident from Eq. (1), Ut þ Δt depends only upon the timedependent variables Ut and Ut  Δt , thus it can be concluded that no convergence checks are needed when solving these equations. As such, nonlinear explicit dynamic analysis enables the prediction of load-displacement path for stiffened shells, from pre-buckling to post-buckling field, until collapse occurs [39–42]. However, since a very small element size can result in a huge increase of computational effort, the selection of an appropriate element size is usually a trade-off between accuracy and efficiency. Moreover, various forms of geometric imperfections can be caused by manufacturing, transport, installation and even serving processes of stiffened shells, which increase the nonlinearity of buckling and collapse behavior under axial compression, and thus enhance the significance of performing the explicit dynamic analysis. In the implemented numerical analysis procedure, geometric imperfections can be taken into account by shifting the radial coordinate of each node according to nodal displacement vector. Geometry of an imperfect stiffened shell can be expressed

P. Hao et al. / Thin-Walled Structures 93 (2015) 177–187

as X ¼ Xp þ Ximp

ð2Þ

Ximp ¼ δN

ð3Þ

where, Xp is the nodal coordinate vector of perfect geometry, Ximp is the nodal displacement vector caused by imperfection, δ is the maximal amplitude of deviations from perfect geometry along radial direction, N is the nodal coordinate vector of the basic imperfection shape.

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require high experimental cost, in addition, extremely refined meshes are required to describe the nodal deviations from perfect geometry, which may also cause serious computational effort in the nonlinear explicit dynamic analysis. Dimple-shape imperfections produced by perturbation loads are thus considered as a promising concept of equivalent imperfection to substitute the measured imperfections.

3. Influence of imperfection distributions for cylindrical stiffened shells with axial weld lands

2.2. Perturbation load approach 3.1. Model description As already mentioned in the introduction, SPLA can produce a physically meaningful buckling behavior for thin-walled structures, and it has been demonstrated that the SPLA shows many advantages over the eigenmode-shape imperfections for the prediction of KDFs [43]. The numerical analysis procedure of the SPLA includes three steps: Firstly, the deformation field of thin-walled structures is calculated by means of the nonlinear static analysis, where a given perturbation load is applied at the middle of shell length; secondly, the deformation field is then introduced to the perfect geometry by shifting the nodal coordinates; finally, the nonlinear post-buckling analysis is performed based on the imperfect geometry, aiming to obtain the actual load-carrying capacity of stiffened shells. Moreover, an optimization framework of determining the worst realistic imperfection was proposed based on a finite number of perturbation loads, named as WMPLA, which has the potential to provide improved KDFs for thin-walled structures [11,13]. In its numerical procedure, the optimization formulation can be expressed as Find : X ¼ ½F P ; N 1 ; N 2 ; :::; Nn 

ð4Þ

Minimize : P co

ð5Þ

Subject to : X li r X i r X ui ; i ¼ 1; 2; :::; n þ1

ð6Þ

where FP is the amplitude of the perturbation load, Nn is the position number of the nth perturbation load, Pco is the collapse load, X li and X ui are the lower and upper bounds of the ith variable, respectively. For various types of thin-walled structures, the method to describe and determine the possible positions of perturbation loads may be different, e.g. for unstiffened shells: effective distance [11]; for stiffened shells: eigenmode shape [13]; for stiffened shells with cutout: nodal coordinates [13]. Due to the high computational efforts caused by the nonlinear explicit dynamic analysis, surrogate model is usually recommended to be utilized in the WMPLA, for the purpose of releasing computational burden involved in the optimization iterations. Since the measured imperfections of stiffened shells usually

An aluminum orthogrid stiffened shell with a diameter of 3070.6 mm and a length of 2387.6 mm was established in this section according to Ref. [38], which is representative of modern launch vehicles, as shown in Fig. 1. This shell is constructed by three integrally machined 25.4-mm-thick curved panels, and then welded together along adjacent axial stiffeners. The skin thickness ts ¼2.5 mm, the stiffener thickness tr ¼8.9 mm, the height of axial stiffeners ha ¼20.3 mm, the height of circumferential stiffeners hc ¼5.7 mm, the spaces of circumferential and axial stiffeners dc ¼114.3 mm and da ¼76.2 mm, respectively. Two steel rings with a width of 63.5 mm and a depth of 88.9 mm are attached to both ends of the stiffened shell, for the purpose of easy clamping in the experiment. Since the grades of the aluminum alloy and steel were not provided in Ref. [38], typical material properties were selected for this stiffened shell: Young's modulus E ¼68.0 and 199.0 GPa, Poisson's ratio υ ¼0.33 and 0.30, for the aluminum alloy and steel, respectively. The FE model of this shell was established in the commercial software ABAQUS. Specifically, circumferential stiffeners were discretized by beam elements, aiming to increase the minimum element size and thus reduce the computational cost. Other parts were discretized by shell elements with reduced integration. The shell was simply supported at both ends, and all the loading-end nodes were coupled with a central reference point by a rigid body link, which can enforce a uniform loading condition. Since no uniform loading conditions may be implemented in the case of imperfect structures, uniform displacement of the loading end was applied to the reference point instead, and increased gradually until global collapse occurred. Nonlinear explicit dynamic analysis was then performed, and the predicted collapse load of this stiffened shell is 4849 kN, which agrees well with the result in Ref. [38]. The load versus end-shortening curve for the perfect geometry is shown in Fig. 2, together with the deformed shapes at the collapse load. In the study of Arbocz and Williams [38], geometric imperfections of shell surface were obtained by the full-field contact measurement with an accuracy of 0.05 mm, which can be represented by means of a

Fig. 1. Schematic of the 3-m-diameter stiffened shell with axial weld lands. (a) numerical model; (b) test specimen [38].

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two-dimensional Fourier series wðx; yÞ ¼ tWðx; yÞ ¼ t

N X i¼0

þt

N X X k;l ¼ 0

wðx; yÞ ¼ tWðx; yÞ ¼ t

cos

Aio cos

iπ x L


 kπ x ly ly Akl cos þ Bkl sin L R R

N X X k;l ¼ 0

sin


 kπ x ly ly C kl cos þ Dkl sin L R R

ð7Þ

ð8Þ

There are 343 and 72 data points in the shell surface along axial and circumferential directions, respectively. As can be seen from Fig. 3, axial weld lands produce a very characteristic imperfection pattern consisting of one axial half-wave and nine circumferential full waves, with the maximum imperfection amplitude occurring near the weld lands. Differing from Ref. [38], the measured geometric imperfections were imported into the perfect FE model by shifting the coordinates of each node on the shell surface herein. Specifically, each node on the stiffeners was shifted adaptively according to the corresponding node on the shell surface. Since extremely refined meshes are required to describe and capture the nodal deviations from perfect geometry, the mesh convergence study indicated that a mesh size of 10 mm can provide a sufficient accuracy with an affordable computational cost. By performing the nonlinear explicit dynamic analysis, the

Fig. 2. Load versus end-shortening curves of 3-m-diameter stiffened shells with perfect geometry and measured imperfections.

predicted collapse load of the stiffened shell with measured imperfections is 3195 kN, which is within the range of the available experimental results provided in Ref. [38] (i.e. 3047– 3530 kN). The load versus end-shortening curve and deformed shapes at the collapse load for the stiffened shell with measured imperfections are also shown in Fig. 2. Due to the presence of geometric imperfections, the deformation pattern varies considerably from the one with perfect geometry. 3.2. Influence of imperfection positions for single perturbation load In order to examine the influence of imperfection positions on the collapse load, typical imperfection shapes should be selected firstly. Since the dimple shape produced by the SPLA is similar to the one that forms in a compressed shell at the onset of buckling as observed in tests, single perturbation load was firstly employed, with the perturbation load applied at the middle position of shell length. Since the SPLA can be regarded as a type of equivalent imperfections, extremely refined meshes are not required. Thus, a mesh size of 20 mm was utilized in the following investigation to release the computational burden. Because the circumferential symmetry of this stiffened shell is broken by introducing the axial weld lands, the circumferential position of the perturbation load should be examined deliberately. Four typical positions were selected along circumferential direction with a step of 151, as shown in Fig. 4. To be specific, position 1 is located at the weld land, and position 4 is located at the middle bay between two axial weld lands. Imperfection sensitivity curves of this shell for these four positions are shown in Fig. 5, which can generally be divided into two stages: one extremely steep stage and one relatively smooth stage. After a critical value of perturbation load, the collapse load gets nearly constant. This is due to the fact that the local snap-through imperfections produced by these perturbation loads have almost the same geometry prior to the global buckling, and this can be better understood based on the discussion in Castro et al. [43]. For relatively small perturbation loads (0oFP o2.0 kN), stiffened

Fig. 4. Typical imperfection positions of the stiffened shell with axial weld lands.

Fig. 3. Measured imperfections of the 3-m-diameter stiffened shell with axial weld lands.

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shell is very sensitive to imperfections that are far away from the weld land, since position 4 produces the lowest collapse load among four positions, while imperfections located at other areas cause similar knockdown effects. This is because of the fact that the stiffness of the region that is far away from the weld land is lower, and the imperfection amplitude will be larger when the perturbation load is located at the middle bay. With the increase of perturbation load (2.5oFP o6.0 kN), the imperfection located at position 1 produces the lowest collapse load. The lower bound of collapse loads is 3627 kN, as listed in Table 1. Because the axial weld land is the major loading path of stiffened shell under axial compression, when the

Fig. 5. Single perturbation load imperfection sensitivity curves of the stiffened shell with axial weld lands. (Position 1–Position 4).

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perturbation load is located near the weld land, the reduction of load-carrying capacity will be more significant. 3.3. Influence of imperfection distributions for multiple perturbation load To cover the effects of more general imperfections, WMPLA for stiffened shells was further developed [11,13], where the positions of multiple perturbation load were described in terms of node coordinates, including a circumferential angle coordinate θi and an axial coordinate Zi. As part of this study, before the search of the worst multiple perturbation load, an attempt has been made to find the equivalent multiple perturbation load pattern and amplitude for the measured imperfections, which enhances the physical understanding of the WMPLA. As is evident from Fig. 3, three distinct imperfection clusters can be observed near each weld land, along with a great deal of small imperfections scattered throughout the shell surface. To represent this type of measured imperfections, three perturbation loads were employed. The positions of three perturbation loads were selected at the middle bay of three weld lands along axial direction, which are (90.01, 1193.8 mm), (210.01, 1193.8 mm) and (330.01, 1193.8 mm), respectively, and the corresponding imperfection shape is provided in Fig. 6. By varying the imperfection amplitude, the imperfection sensitivity curve can be obtained, as shown in Fig. 7. The deformed shapes at collapse loads for several typical perturbation loads are also given in Fig. 7. A similar collapse pattern can be found when the perturbation load is greater than 2.5 kN. The collapse load is 3108 kN when the amplitude of three perturbation loads is 4.0 kN, which is equivalent to the measured imperfections. In this case, the maximal Mises stress and deformation of the stiffened shell reaches 296 MPa and 10.5 mm when the perturbation loads are applied. Although it is indeed critical to apply such a large perturbation load during an axial compression

Table 1 Comparison of collapse loads of stiffened shells with various types of weld lands.

Stiffened shell with axial weld lands Stiffened shell with circumferential and sequential axial weld lands Stiffened shell with circumferential and staggered axial weld lands

Pco [kN]n

Psp [kN]nn

Pmp [kN]nnn

Pme [kN]nnnn

4849 9553 8759

3627 7360 7383

2990 6349 6381

3493 7293 7143

n

Pco is the collapse load of stiffened shell with perfect geometry. Psp is the lower bound of collapse loads of stiffened shell with singe perturbation load. nnn Pmp is the lower bound of collapse loads of stiffened shell with multiple perturbation load. nnnn Pme is the mean value of collapse loads of stiffened shells obtained by MCS. nn

Fig. 6. The imperfection shape of the equivalent multiple perturbation load.

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experiment, the deformation shape produced by the perturbation loads can also be introduced to the stiffened shell prior to the experiment. However, the latter approach may cause a stress-free dimple [5], and the difference with the former approach still needs to be further investigated. The collapse load decreases to 3046 kN when the amplitude of three perturbation loads is 6.0 kN, which achieves close estimates of the lower bound of experimental results (i.e. 3047 kN). This also provides a reference for selecting an appropriate imperfection amplitude in the WMPLA. To investigate the influence of multiple perturbation load distributions, Monte Carlo Simulation (MCS) was carried out, assuming that imperfection positions are uniformly distributed on the shell surface. As can be observed from Fig. 7, the collapse load decreases very slowly when the perturbation load is greater than 2.5 kN. From a point-of-view of conservatism, a perturbation load of 6.0 kN was employed. In order to release the computational burden, the surrogate-based technique was utilized herein. In the design of experiment (DOE), a sample size of 300 was generated using the Optimal Latin Hypercube Sampling (OLHS) method [44]. Then, the Kriging model was built based on the sampling data, and MCS was performed with 106 samples. The PDF of the collapse load can be obtained, as shown in Fig. 8. The mean value of the collapse load is 3493 kN, and the coefficient of variation is 0.057. With the decrease of collapse loads along horizontal axis, the imperfection distribution becomes more critical to stiffened shells. For the lower tail of the PDF curve (the collapse load is lower than 3240 kN), there are two main types of imperfection distributions according to the statistical analysis, which account for almost 10% of the integral of PDF curve: (1) one

Fig. 7. Equivalent multiple perturbation load imperfection sensitivity curve of the stiffened shell with axial weld lands.

perturbation load is located near the axial weld land, and the other two are placed at two sides of the first one; (2) three perturbation loads are usually scattered along circumferential direction (none of them is near the axial weld land), and located at the middle bay of the stiffened shell along axial direction. Such types of imperfection distributions are very critical to stiffened shells with axial weld lands, which should be eliminated or avoided in the manufacturing process; (3) two perturbation loads are located at either the upper or lower ends of the stiffened shell, while the other one is located at the middle bay of the stiffened shell along axial direction, which is the most representative imperfection distribution of the major component of the PDF curve; (4) one perturbation load is placed near either the upper or lower ends of the stiffened shell, and the other two perturbation loads are still located at the middle bay; (5) two perturbation loads are placed at similar circumferential positions, and the other one is located relatively far away from these two perturbation loads. In general, it is clear that those imperfection distributions with all the perturbation loads located at the same panel would produce a smaller knockdown of load-carrying capacity, compared to those with the perturbation loads scattered at different panels. For the upper tail of the PDF curve (the collapse load is larger than 3760 kN), two typical imperfection distributions can be found: (6) two perturbation loads are placed at similar circumferential positions but different axial positions, and the other one is located near either the upper or lower ends of the stiffened shell; (7) all the positions of three perturbation loads are very close to either the upper or lower ends of the stiffened shell. With regard to practical manufacturing process, middle panels should be fabricated intensively to reduce the probability of introducing additional imperfections or defects. Also, the regions near axial weld lands should be paid great attention, even for those near the upper or lower ends. Subsequently, WMPLA was utilized to identify the worst distribution of multiple perturbation load. The optimization was carried out using Multi-Island Genetic Algorithm (MIGA) based on the previously built Kriging model, with the objective of minimizing the collapse load of imperfect stiffened shells. The double-loop strategy (including inner optimization and outer update) can guarantee the confidence of the optimum design, because the worst imperfection distribution has already been validated by nonlinear explicit dynamic analysis. The convergence of surrogate-based optimization is considered to be achieved, if the relative error between the results obtained from surrogate model and the ones from nonlinear explicit dynamic analysis is less than 0.1%. For more detailed information about the implemented surrogate-based optimization, please see Ref. [13]. The history of outer updates of the search process is shown in Fig. 9, together with the distribution of the obtained worst multiple perturbation load. The corresponding collapse load is 2990 kN, and the positions of three perturbation loads are (82.21, 1169.7 mm), (171.61, 1132.9 mm) and (359.91, 1200.3 mm), respectively. To be

Fig. 8. Probability density function of the stiffened shell with axial weld lands.

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specific, one perturbation load is located near the axial weld land, and the other two perturbation loads are placed at two sides of the first one, with a distance to the first perturbation load of almost 801. Such an imperfection distribution is proven to be severe for the preserving of load-carrying capacity, since it can facilitate the development of buckling deformations.

4. Influence of imperfection distributions for cylindrical stiffened shells with circumferential and axial weld lands 4.1. Cylindrical stiffened shells with circumferential and sequential axial weld lands The previous section discussed the influence of imperfection distributions on the load-carrying capacity of stiffened shells with axial weld lands. Actually, with the increase of length dimension, stiffened shells are usually divided into several segments for easy processing, and this has also brought some additional barriers for analyzing the influence of imperfection distributions. Thus, a 5-mdiameter orthogrid stiffened shell with circumferential and sequential axial weld lands was then established, which has four panels along circumferential direction and three segments along axial direction, as shown in Fig. 10. Each panel has a length of 1200 mm and a circumferential degree of 901. Adjacent panels were assembled together by welding, and the weld lands were assumed to be unstiffened thick plates [37]. It should be noted that this type of weld land is convenient for the assembly of two adjacent panels, and thus commonly used in modern fuel tanks of launch vehicles. The skin thickness ts ¼4.5 mm, the thickness of weld land tw ¼8.5 mm, the width of weld land b ¼150.0 mm, the height of weld land hw ¼ 120.0 mm, the stiffener thickness

Fig. 9. History of outer updates of the stiffened shell with axial weld lands.

183

tr ¼ 9.0 mm, the stiffeners height h¼15.0 mm, the spaces of circumferential and axial stiffeners dc ¼208.0 mm and da ¼259.1 mm, respectively. Typical properties of the aluminum alloy used were assumed as follows: Young's modulus E¼ 68.2 GPa, Poisson's ratio υ ¼ 0.33, yield stress σs ¼350 MPa, ultimate stress σb ¼435 MPa, elongation is 0.1. The boundary and load conditions are identical with those in Section 3.1. The mesh convergence study indicated that a mesh size of 25 mm can provide a sufficient accuracy with a sensible tradeoff on computational cost. The predicted collapse load of the stiffened shell with perfect geometry is 9553 kN. The influence of imperfection distributions was then investigated, and single perturbation load was firstly employed. According to the symmetry along circumferential direction, four typical positions were selected. Specifically, position 4 is located at the weld land, and position 1 is located at the middle bay of a panel. Four imperfection sensitivity curves are shown in Fig. 11. The tendency of these curves are typically similar to those in Fig. 5, while the scatter in Fig. 11 is more remarkable. For small-amplitude imperfections, the positions far away from the weld land yield lower collapse loads. For large-amplitude imperfections, the positions near the axial weld land are dominant. Over all, the lower bound of collapse loads is 7360 kN, as listed in Table 1. Due to the presence of circumferential weld lands, the influence of imperfection positions along axial direction needs to be further examined. Thus, another four positions were selected along axial direction for position 1 and position 4, respectively, as also shown in Fig. 10. The corresponding imperfection sensitivity curves are shown in Figs. 12 and 13, respectively. The lower bound of the collapse loads for position 1 is 4.4% and 4.2% smaller

Fig. 11. Single perturbation load imperfection sensitivity curves of the stiffened shell with circumferential and sequential axial weld lands (Position 1–Position 4).

Fig. 10. Schematic of the 5-m-diameter stiffened shell with circumferential and sequential axial weld lands.

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than those for positions 1–1 and 1–2, respectively. Also, similar results can be observed for position 4. This indicates that the circumferential weld lands can reduce the imperfection sensitivity to a certain extent. In addition, the symmetric positions about the middle bay produce extremely similar imperfection sensitivity curves. With the increase of perturbation load, the lower bounds of all the curves except for position 1 tend to a same collapse load. Consequently, it can be concluded that the imperfections located at the middle bay usually cause a larger reduction of load-carrying

Fig. 12. Single perturbation load imperfection sensitivity curves of the stiffened shell with circumferential and sequential axial weld lands (Position 1–1 – Position 1–5).

capacity, compared to other axial positions. Results indicate that the middle panels of stiffened shell along axial direction are relatively more sensitive to imperfections, which require special attention in the manufacturing process. Also, a sample size of 300 was generated using the OLHS method, and the Kriging model was constructed. MCS was then performed based on the Kriging model with 106 samples, and the PDF of the collapse load is shown in Fig. 14. The mean value of the collapse load is 7293 kN, and the coefficient of variation is 0.039. Similarly, the lower tail of the PDF curve (the collapse load is lower than 6930 kN) contains two main types of imperfection distributions, accounting for almost 10% of the integral of PDF curve. For both (1) and (2), two perturbation loads are located near the middle bay of axial weld lands, while the other one is generally placed at the center of a panel. The imperfection distributions within the major component of the PDF curve are slightly different from the ones of the previous stiffened shell in Sections 3. For (3)– (5), two perturbation loads are located at the middle bay of the stiffened shell along axial direction, while the rest one can be applied at arbitrary position of the upper or lower panels, because the circumferential weld lands have the ability to reduce imperfection sensitivity, which potentially increases the probability of falling in this region of the PDF curve. For the upper tail of the PDF curve (the collapse load is larger than 7740 kN), all the positions of three perturbation loads are placed at either the upper or lower panels of the stiffened shell. Such pattern of imperfection distributions accounts for about 5% of the integral of PDF curve. Then, WMPLA was employed to identify the worst imperfection distribution. MIGA was utilized to carry out the optimization based on the Kriging model. The history of outer updates of the search process is shown in Fig. 15, together with the distribution of the obtained worst multiple perturbation load. The imperfection positions are (88.91, 1769.1 mm), (134.71, 2121.6 mm) and (174.01, 1922.3 mm), respectively, and the corresponding collapse load is 6349 kN. To be specific, three perturbation loads are located at one middle panel along axial direction with an almost uniform distance, and two of them are very close to the middle bay of axial weld lands, which is typically similar to the worst imperfection distribution provided in Section 3.2. 4.2. Cylindrical stiffened shells with circumferential and staggered axial weld lands

Fig. 13. Single perturbation load imperfection sensitivity curves of the stiffened shell with circumferential and sequential axial weld lands (Position 4–1 – Position 4–5).

For the purpose of reducing stress concentrations near axial weld lands, staggered axial weld lands [45] are a promising choice, since the load path along axial weld lands is blocked and diffused. This also enriches the potential of the discussion on imperfection distributions. Based on the FE model in Section 4.1, another stiffened shell was established, and the main difference between two models is that the middle segment along axial direction is

Fig. 14. Probability density function of the stiffened shell with circumferential and sequential axial weld lands.

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Fig. 15. History of outer updates of the stiffened shell with circumferential and sequential axial weld lands.

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Fig. 17. Single perturbation load imperfection sensitivity curves of the stiffened shell with circumferential and staggered axial weld lands (Position 1–Position 4).

Fig. 16. Schematic of the 5-m-diameter stiffened shell with circumferential and staggered axial weld lands.

rotated 45 degrees in circumferential direction, named as staggered axial weld land pattern, as shown in Fig. 16. The boundary and load conditions together with mesh size are identical with those in Section 4.1. The predicted collapse load of the stiffened shell with perfect geometry is 8759 kN. Similarly, single perturbation load was employed to investigate the influence of imperfection distributions, and the obtained imperfection sensitivity curves are shown in Fig. 17. As can be seen, four curves exhibit a distinctly different tendency compared to the ones of the previous two models, showing a greater degree of scatter. With regard to position 1, the curve can approximately be divided into three straight lines. For the first line, the collapse load remains almost unchanged with the increase of perturbation loads. For the second line, the relationship between the collapse load and the perturbation load is almost linear or bi-linear. For the third line, the collapse load does not decrease with the further increase of perturbation load. This indicates that position 4 is extremely sensitive to imperfections for such a distribution of weld lands. The lower bound of collapse loads is 7383 kN, as listed in Table 1. In general, although the collapse load of the perfect stiffened shell with staggered axial weld lands is lower than the one with sequential axial weld lands, this shell is more robust to dimple-shape imperfections with various positions. Thus, it can be concluded that staggered axial weld lands are conducive to tolerate small-amplitude imperfections for stiffened shells, and this advantage is achieved due to the presence of the distributed imper fection-critical areas (axial weld lands). Similarly, another four positions were selected along axial direction for positions 1 and 4, respectively. The corresponding imperfection sensitivity curves are shown in Figs. 18 and 19. As is evident from Fig. 18, the imperfections located at the middle bay

Fig. 18. Single perturbation load imperfection sensitivity curves of the stiffened shell with circumferential and staggered axial weld lands (Position 1–1 – Position 1–5).

cause a larger reduction of load-carrying capacity, compared to other axial positions. Moreover, with regard to Fig. 19, the stiffened shell is very sensitive to the imperfections located at position 4, however, positions 4–2 and 4–4 only produce a very small knockdown effect for small-amplitude imperfections, and this may be attributed to the influence of circumferential weld lands. Then, a sample size of 300 was generated using the OLHS method. Also, MCS was performed based on the Kriging model with 106 samples, with the PDF of the collapse load shown in Fig. 20. The load-carrying capacities of stiffened shells with staggered axial weld lands are inferior to those with sequential weld lands in general, since the mean value of the collapse load is 7143 kN, and the coefficient of variation is 0.035. This is due to the fact that the collapse load of the stiffened shell with staggered axial weld lands is very low, although this axial weld land pattern is beneficial for imperfection tolerance. The typical imperfection distributions are also provided in Fig. 20. Compared to the stiffened shell with circumferential and sequential axial weld lands, the main difference lies in the lower and upper tails of the PDF curve. For the lower tail (the collapse load is lower than 6820 kN, accounting for about 10% of the integral of PDF curve), including (1) and (2), the locations of two perturbation loads are very close to the axial weld lands of the middle panel along axial

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direction, and the other one is placed at another middle panel. Although the middle segment along axial direction was rotated 45 degrees in circumferential direction, imperfections near the axial weld lands of the middle segment are still very critical. For the major component of the PDF curve, the imperfection distributions are very similar to the ones of the stiffened shell with sequential axial weld lands. In general, imperfections scattered at adjacent panels would produce a smaller knockdown effect on the collapse load, since such an imperfection pattern has a limited capacity to accelerate the development of buckling deformations. For the upper tail (the collapse load is larger than 7480 kN, accounting for about 10% of the integral of PDF curve), including (6) and (7), all the positions of three perturbation loads are located at either the upper or lower panels of the stiffened shell. WMPLA was subsequently employed to identify the worst imperfection distribution. The distribution of the obtained worst multiple perturbation load is shown in Fig. 21, with the imperfection positions of (88.61, 1733.6 mm), (250.11, 2028.4 mm) and (358.11, 1805.8 mm), respectively. The corresponding collapse load is 6381 kN, which is even higher than that of the stiffened shell with sequential axial weld lands. For the worst imperfection distribution, the positions of two perturbation loads are very close to the axial weld lands of the middle panel along axial direction, and the other one is placed at another middle panel along axial direction. The distribution of the worst multiple perturbation load is slightly different from that of stiffened shells with sequential axial weld lands.

Fig. 19. Single perturbation load imperfection sensitivity curves of the stiffened shell with circumferential and staggered axial weld lands (Position 4–1 – Position 4–5).

5. Concluding remarks In this study, stiffened shells with three types of weld lands were established, including axial weld lands, circumferential and sequential axial weld lands, as well as staggered axial weld lands. The influences of imperfection positions and distributions on the collapse load were discussed successively based on single and multiple perturbation loads, where the amplitude of the equivalent imperfection was determined by comparison with the available experimental results and measured imperfections. For each stiffened shell, MCS and WMPLA were then performed based on the Kriging model, and several critical understandings were gained by full exploration of the PDF of collapse loads. In general, the imperfections located near the middle bay or axial weld lands usually cause a larger reduction of load-carrying capacity, while the circumferential weld lands have the ability to reduce the imperfection sensitivity. In addition, those imperfection distributions with all the perturbation loads scattered at different panels would produce a larger knockdown of loadcarrying capacity, compared to those with the perturbation loads located at the same panel. Although stiffened shell with staggered axial weld lands is beneficial for imperfection tolerance, the collapse load of its perfect geometry is relatively lower, and thus the actual load-carrying capacity is still inferior to that with sequential weld lands. Results can provide general instructions about imperfection critical areas for axially compressed stiffened shells with different weld land patterns, which require special attention in the manufacturing process.

Fig. 21. History of outer updates of the stiffened shell with circumferential and staggered axial weld lands.

Fig. 20. Probability density function of the stiffened shell with circumferential and staggered axial weld lands.

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