The Vibration Correlation Technique (VCT) is a non-destructive method to predict buckling loads for imperfection-sensitive structures. While successfully used to validate numerical models and predict experimental buckling loads, recommendations for defining the VCT experiment are scarce. Here, its sensitivity towards the number of load steps and the maximum load level measured is studied, and an uncertainty quantification of the measured frequency affecting the VCT prediction is performed First, a series of finite element (FE) models representing nominally identical cylinders, and validated by buckling experiments, are used to perform a sensitivity study. When no frequency deviations are introduced in the FE results, a positive correlation between the VCT predictions and the maximum load used for measurements is found, the number of load steps used being only relevant in reducing the errors. Introducing frequency deviations deterred the predictions correlation with the maximum load, while using more load steps reduced this influence. Second, a sensitivity study based on experimental data confirmed most of the trends previously observed using the FE results, the exception being a poor prediction sensitivity as a function of the maximum load, owing to several cylinders for which the VCT method gave predictions that progressively decreased with increasing the load.

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For most structural parts of real launcher structures buckling is the critical design criterion. Due to the high imperfection sensitivity of these structures and to the unknown geometric imperfections during the design phase, it is still today a challenge to predict a reliable design buckling load and to experimentally and non-destructively evaluate the load carrying capacity of real structures. The space industry is looking for new and alternative less-conservative design methods, and non-destructive experimental strategies. This paper presents a summary of different examples to numerically and experimentally predict the buckling load of imperfection sensitive structures. The numerical strategies herein covered are based on the fast Ritz-method, developed for conical and cylindrical structures applicable for linear and non-linear buckling, and static calculations. The experimental examples are all based on the non-destructive buckling estimation enabled by means of the Vibration Correlation Technique (VCT). An overview of experiments on different types of cylindrical shells (unstiffened, stringer-stiffened and grid) with different materials (composite and metallic) and for different load cases and their combination (axial compression, internal pressure and bending) is presented. The examples are on academic laboratory level and on qualification tests of real full-scale space structures.

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Considering the design of aerospace structures, an experimental campaign is essential for validating the sizing methodology and margins of safety. Particularly for buckling-critical cylindrical shells, the traditional buckling test could lead the specimen to permanent damage. Therefore, the validation of nondestructive experimental procedures for estimating the buckling load of imperfection-sensitive structures from the prebuckling stage is receiving more attention from the industry. In this context, this paper proposes an experimental verification of the robustness of a vibration correlation technique developed for imperfection-sensitive structures. The study comprises three nominally identical unstiffened composite laminated cylindrical shells. Each specimen is tested 10 times for buckling at DLR and, the reproducible results — within a small range of deviation between them — corroborate the equivalence of the cylinders. For the robustness assessment of the vibration correlation technique, two different buckling test facilities are considered. Furthermore, the material properties are recalculated through composite composition rules and the influence of enhanced theoretical buckling loads on the VCT predictions is verified. The experimental campaigns corroborate that the vibration correlation technique provides appropriate estimations representing the influence of the different test facilities; moreover, enhanced theoretical buckling loads can improve the predictions for some of the test cases.@en

Thin-walled structures are widely used in aerospace, offshore, civil, marine and other engineering industries. Buckling of such thin-walled imperfection sensitive structures is a very important phenomenon to be considered during their design phase. Existing design guidelines, being the most known the NASA SP-8007 for cylinders 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 are therefore required. This study deals with buckling experiments of axially compressed, unstiffened carbon fiber–reinforced polymer (CFRP) cylinders with and without an additional lateral load. Two geometrically identical cylinders with the same layup were designed, manufactured and tested. Before testing, the thickness of the cylinders was measured with ultrasonic inspection and the geometry was measured utilizing a 3D scanning system based on photogrammetry. During testing, a digital image correlation system was employed to monitor deformations, strain gage readings and load-shortening data was taken. Modelling of shape mid-surface and thickness imperfections as well as fiber volume fraction correction are included into the Finite Element Analysis (FEA) of the test structures, and the experimental results are compared against FEA results.

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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.@en

The design and manufacture of unstiffened composite conical structures is very challenging, as the variation of the fiber orientations, lay-up and the geometry of the ply pieces have a significant influence on the thickness imperfections and ply angle deviations imprinted to the final part. This paper deals with the manufacture of laminated composite cones through the prepeg/autoclave process. The cones are designed to undergo repetitive buckling tests without accumulating permanent damage. The aim is to define a process that allows the control of fiber angle deviations and the removal of thickness imperfections generated from gaps and overlaps between ply pieces. Ultrasonic scan measurements are used to proof the effectiveness of the proposed method.@en

Nondestructive methods, to calculate the buckling load of imperfection sensitive thin-walled structures, are one of the most important techniques for the validation of new structures and numerical models of large scale aerospace structures. The vibration correlation technique (VCT) allows determining the buckling load for several types of structures without reaching the instability point, but this technique is still under development for thin-walled plates and shells. This paper presents and discusses an experimental and numerical validation of a novel approach, using the vibration correlation technique, for the prediction of realistic buckling loads on unstiffened cylindrical shells loaded in compression. From the experimental point of view, a batch of three composite laminated cylindrical shells are fabricated and loaded in compression up to buckling. An unsymmetric laminate is adopted in order to increase the sensitivity of the test structure to initial geometric imperfections. In order to characterize a relationship with the applied load, the first natural frequency of vibration and mode shape is measured during testing using a 3D laser scanner. The proposed vibration correlation technique allows one to predict the experimental buckling load with a very good approximation, without actually reaching the instability point. Furthermore, a series of numerical models, including non-linear effects such as initial geometric and thickness imperfection, are carried-out in order to characterize the variation of the natural frequencies of vibration with the applied load and compare the results with the experiment findings. Additional experimental tests are currently under development to further validate the proposed approach for metallic and balanced composite structures.

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Since the development of the first theories to predict the buckling induced by axial compression in shells sensitive to imperfections, a significant discrepancy between theoretical and experimental results has been observed. Donnell and Koiter are among the first authors demonstrating, for these structures, the relevant influence of the geometrical imperfections on the reduction of the buckling load. Currently, the preliminary design of imperfections sensitive shell structures used in space applications is carried out according to the NASA SP-8007guideline. However, several studies have proven that this guideline leads to over-conservative design configurations when considering the geometrical and material imperfections existing in real cones. Since the pioneer work of Arbocz, alternative methods have been investigated to overcome this issue. Among the different approaches, in this paper, the Single Perturbation Load Approach (SPLA), originally developed byHühne as a deterministic way to calculate the knock-down factor of imperfection sensitive shells, is further studied. Indeed, a numerical investigation about the application of the SPLA to the simulation of the mechanical behavior of imperfection sensitive composite conical structures under axial compression is presented. This study is related to part of the work performed in the frame of the European Union (EU) project DESICOS.@en

A semi-analytical model to predict the non-linear behavior of unstiffened cylinders and cones considering initial geometric imperfections and various loads and boundary conditions is presented. The formulation is developed using the Classical Laminated Plate Theory (CLPT) and Donnell's equations, solving for the complete displacement field. The non-linear static problem is solved using a modified Newton-Raphson algorithm with line-search. A numerical integration scheme for the non-linear matrices is proposed and details regarding the implementation of the proposed method are given. Two methods to include measured imperfections into the analyses are presented and for one method the effect of using different approximation levels for the imperfection field on the non-linear response is investigated, and a minimum approximation accuracy that should be used is determined. The semi-analytical results are verified using finite elements and previous models from the literature. The implemented routines are distributed on-line and are based on a matrix notation simply applicable to other problems.

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Thin-walled cylindrical composite shell structures can be applied in space applications, looking for lighter and cheaper launcher transport system. These structures are prone to buckling under axial compression and may exhibit sensitivity to geometrical imperfections. Today the design of such structures is based on NASA guidelines from the 1960’s using a conservative lower bound curve generated from a database of experimental results. In this guideline the structural behavior of composite materials may not be appropriately considered since the imperfection sensitivity and the buckling load of shells made of such materials depend on the lay-up design. It is clear that with the evolution of the composite materials and fabrication processes this guideline must be updated and / or new design guidelines investigated. This need becomes even more relevant when cutouts are introduced to the structure, which are commonly necessary to account for access points and to provide clearance and attachment points for hydraulic and electric systems. Therefore, it is necessary to understand how a cutout with different dimensions affects the buckling load of a thin-walled cylindrical shell structure in combination with other initial geometric imperfections. In this context, this paper present some observations regarding the buckling load behavior vs. cutout size and radius over thickness ratio, of laminated composite curved panels and cylindrical shells, that could be applied in further recommendations, to allow identifying when the buckling of the structure is dominated by the presence of the cutout or by other initial imperfections.@en

A semi-analytical model for the non-linear analysis of simply supported, unstiffened laminated composite cylinders and cones using the Ritz method and the Classical Laminated Plate Theory is proposed. A matrix notation is used to formulate the problem using Donnell's and Sanders' non-linear equations. The approximation functions proposed are capable to simulate the elephant's foot effect, a common phenomenon and a common failure mode for cylindrical and conical structures under axial compression. Axial, torsion and pressure loads can be applied individually or combined, and solutions for linear static, linear buckling and non-linear buckling analyses are presented and verified using a commercial finite element software. The presented non-linear buckling analyses used perturbation loads to create the initial geometric imperfections, showing the capability of the method for arbitrary imperfection patterns. The linear stiffness matrices are integrated analytically and for the conical structures an approximation is proposed to overcome the non-integrable expressions.@en

Nondestructive methods, to calculate the buckling load of imperfection sensitive thin-walled structures, such as large-scale aerospace structures, are one of the most important techniques for the evaluation of new structures and validation of numerical models. The vibration correlation technique (VCT) allows determining the buckling load for several types of structures without reaching the instability point, but this technique is still under development for thin-walled plates and shells. This paper presents and discusses an experimental verification of a novel approach using vibration correlation technique for the prediction of realistic buckling loads of unstiffened cylindrical shells loaded under axial compression. Four different test structures were manufactured and loaded up to buckling: two composite laminated cylindrical shells and two stainless steel cylinders. In order to characterize a relationship with the applied load, the first natural frequency of vibration and mode shape is measured during testing using a 3D laser scanner. The proposed vibration correlation technique allows one to predict the experimental buckling load with a very good approximation without actually reaching the instability point. Additional experimental tests and numerical models are currently under development to further validate the proposed approach for composite and metallic conical structures.

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Nondestructive experimental methods to calculate the buckling load of imperfection sensitive thin-walled structures are one of the most important techniques for the validation of new structures and numerical models of large scale aerospace structures. Vibration correlation technique (VCT) allows determining equivalent boundary conditions and buckling load for several types of structures without reaching the instability point. VCT is already widely used for beam structures, but the technique is still under development for thin-walled plates and shells. This paper intends to explain the capabilities and current limitations of this technique applied to two types of structures under buckling conditions: flat plates and cylindrical shells prone to buckling. Experimental results for a flat plate and a cylindrical shell are presented together with reliable finite element models for both cases. Preliminary results showed that the VCT can be used to determine the realistic boundary conditions of a given test setup, providing valuable data for the estimation of the buckling load by finite element models. Also numerical results herein presented show that VCT can be used as a nondestructive tool to estimate the buckling load of unstiffened cylindrical shells. Experimental tests are currently under development to further validate the approach proposed herein.

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Imperfection sensitive structures such as unstiffened or skin-dominant shell structures are commonly used for aeronautic and aerospace applications. Cylindrical shells are dominating satellite launcher structures and a reliable methodology to calculate their behaviour in the early stages of design is fundamental to achieve optimum results. Launcher design requires fast and precise prediction of structural weight as well its weight distribution already in the early design phase, because in that phase different concepts of the whole launcher system have to be evaluated in order to identify the optimal one. The prediction has to be precise, because less reliable ones might lead to basic changes, later in the detailed design phase, which might also influence the design of the whole system. Such changes in later design phases are extremely costly in terms of time and money; they definitely have to be avoided. The dimensioning criterion with the design of launcher structures is buckling not before ultimate load, thus they do not have an exploitable post-buckling area. The most critical aspect for numerical buckling prediction is the structure's sensitivity to geometric and loading imperfections. Currently, imperfection sensitive shell structures prone to buckling are designed according to the NASA SP-8007 guideline [1], from 1968, using its conservative lower bound curve. In this guideline the structural behaviour of composite materials is not appropriately considered, since the imperfection sensitivity and the buckling load of shells made of such materials depend on the lay-up design. There is no specific design guideline for imperfection sensitive composite structures prone to buckling. NASA performed high investments for the last 5 years with one project called "Shell Buckling Knock-down Factor" (SBKF) in order to develop a new guideline to calculate the knock-down factor of cylindrical shells prone to buckling [2], and also the European project DESICOS [3] (New Robust DESign Guideline for Imperfection Sensitive Composite Launcher Structures) is working on new methodologies to estimate the ultimate load of such structures. An example of applicability of these new design guidelines could be the next generation of the European launchers family "Ariane" in order to maintain the actual position in the satellite launchers market [4].

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The importance of taking into account geometric imperfections for cylindrical and conical thin-walled structures prone to buckling had been already recognized by the first authors dealing with new formulations. Nowadays, the analysts still use empirically based lower-bound methods such as the NASA SP-8007 guideline to calculate the required knock-down factors (KDFs), which does include important mechanical properties of laminated composite materials, such as the stacking sequence. New design approaches that allow taking full advantage of composite materials are required. The single perturbation load approach (SPLA), a new deterministic approach first proposed by Hühne, will be investigated with unstiffened composite conical structures varying the geometry, lamina and layup. The SPLA's capability for predicting KDF is compared with the NASA approach. The SPLA was applied to the geometrically perfect structures and to the structure with geometric imperfections of two types, mid-surface imperfections and thickness imperfections. The study contributes to the European Union (EU) project DESICOS, whose aim is to develop less conservative design guidelines for imperfection sensitive thin-walled structures.

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The importance of taking into account geometric imperfections for cylindrical and conical thin-walled structures in buckling had been already recognized a long time ago. Nowadays, the designers still use empirically based lower-bound methods such as the NASA SP-8007 guideline to calculate the required knock-down factors (KDFs). New design approaches that allow taking full advantage of using composite materials are required. The single perturbation load approach (SPLA), a new deterministic approach developed by Hühne for cylindrical shells, will be investigated with unstiffened composite conical structures. The SPLA's capability for predicting KDF is compared with the NASA approach. The SPLA was applied to perfect conical shells with different semi-vertex angles, and three different positions of the perturbation load were considered. The study contributes to the European Union (EU) project DESICOS, whose aim is to develop less conservative design guidelines for imperfection sensitive thin-walled composite structures.@en

This paper presents the application of the Ritz method for the analysis of laminated composite cylinders and cones using the classical laminated plate theory (CLPT) and the first shear deformation theory (FSDT). The Donnell and Sander kinematic approximations are investigated for the CLPT. It is shown that the equations proposed by Sanders when applied with the CLPT increase the range of applicability of the CLPT in comparison with the Donnell kinematics. The developed models are suitable to obtain linear and non-linear responses of conical and cylindrical shells under displacement or load controlled axial compression and geometric imperfections. This paper brings the investigation of static and buckling responses and the commercial finite element solver Abaqus was used to validate all the linear and non-linear results.

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The Vibration Correlation Technique (VCT) is a nondestructive experimental method that can be used for the estimation of realistic boundary conditions and to improve the correlation of numerical models used to estimate the buckling load of shell structures. This paper presents initial experimental results of vibration frequencies and modes shapes of a cylindrical shell loaded in compression, up to buckling. The experimental results are used to point out the influence of the boundary conditions and material properties on the correlation with a finite element model. The results show that the application of calibrated boundary conditions and material properties are not enough to guarantee a good numerical-experimental correlation, requiring further studies suggested herein.@en

Semi-analytical models for the linear buckling analysis of unstiffened laminated composite cylinders and cones with flexible boundary conditions are presented. The Classical Laminated Plate Theory and the First-order Shear Deformation Theory are used in conjunction with the Donnell's non-linear equations to derive the buckling equations. Axial, torsion and pressure loads can be applied individually or combined in the proposed models. The stiffness matrices are integrated analytically and for the conical shells an approximation is proposed to overcome non-integrable expressions. Comparisons with the literature show that the classical base functions available for axial compression cannot capture the buckling modes for non-orthotropic laminates. For torsion loads these classical shape functions do not catch the buckling modes even when applying the assumption of pure orthotropy, and it is shown how the proposed models correlate well with experimental data from the literature and finite element results. The use of elastic constraints at the boundaries allows the simulation of different boundary conditions in a versatile way and it is shown how those constants can be adjusted in order to change from one type of boundary condition to another.@en

The important role of geometric imperfections on the decrease of the buckling load for thin-walled cylinders had been recognized already by the first authors investigating the theoretical approaches on this topic. However, there are currently no closed-form solutions to take imperfections into account already during the early design phases, forcing the analysts to use lower-bound methods to calculate the required knock-down factors (KDF). Lower-bound methods such as the empirical NASA SP-8007 guideline are commonly used in the aerospace and space industries, while the approaches based on the Reduced Stiffness Method (RSM) have been used mostly in the civil engineering field. Since 1970s a considerable number of experimental and numerical investigations have been conducted to develop new stochastic and deterministic methods for calculating less conservative KDFs. Among the deterministic approaches, the single perturbation load approach (SPLA), proposed by Hühne, will be further investigated for axially compressed fiber composite cylindrical shells and compared with four other methods commonly used to create geometric imperfections: linear buckling mode-shaped, geometric dimples, axisymmetric imperfections and measured geometric imperfections from test articles. The finite element method using static analysis with artificial damping is used to simulate the displacement controlled compression tests up to the post-buckled range of loading. The implementation of each method is explained in details and the different KDFs obtained are compared. The study is part of the European Union (EU) project DESICOS, whose aim is to combine stochastic and deterministic approaches to develop less conservative guidelines for the design of imperfection sensitive structures.

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