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Dynamic buckling of fiber composite shells under impulsive axial compression
Thin-Walled Structures 43 (2005) 499–514 www.elsevier.com/locate/tws
Dynamic buckling of fiber composite shells under impulsive axial compression Chiara Bisagni* Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La Masa 34, 20156 Milano, Italy Received 19 December 2003; accepted 27 July 2004 Available online 22 October 2004
Abstract The paper deals with dynamic buckling due to impulsive loading of thin-walled carbon fiber reinforced plastics (CFRP) shell structures under axial compression. The approach adopted is based on the equations of motion, which are numerically solved using a finite element code (ABAQUS/Explicit) and using numerical models validated by experimental static buckling tests. To study the influence of the load duration, the time history of impulsive loading is varied and the corresponding dynamic buckling loads are related to the quasi-static buckling loads. To analyse the sensitivity to geometric imperfections, the initial geometric imperfections, measured experimentally on the internal surface of real shells, are introduced in the numerical models. It is shown numerically that the initial geometric imperfections as well as the duration of the loading period have a great influence on the dynamic buckling of the shells. For short time duration, the dynamic buckling loads are larger than the static ones. By increasing the load duration, the dynamic buckling loads decrease quickly and get significantly smaller than the static loads. Since the common practice is to assume that dynamic bucking loads are higher than the static ones, which means that static design is safe, careful design is recommended. Indeed, taking the static buckling load as the design point for dynamic problems might be misleading. q 2004 Elsevier Ltd. All rights reserved. Keywords: Dynamic buckling; Finite element analysis; Fiber composite cylindrical shells; Impulsive axial compression; Geometric imperfections
* Tel.: C39 02 2399 8390; fax: C39 02 2399 8334. E-mail address: [email protected]. 0263-8231/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2004.07.012
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1. Introduction The stability of composite cylindrical shells is of fundamental interest in aircraft and missile design as thin-walled cylindrical shells have constituted primary structural parts for many years and because the composite materials are nowadays extremely attractive due to their considerable strength-to-weight ratio. They are subjected to various loading both static and dynamic. The investigations in the past two decades were primarily confined to the static stability. This effort has led to a reasonably good understanding of the response of the composite shells to static loading, including the effects of initial geometric imperfections and lamina stacking sequence. However, dynamic buckling is not yet clearly defined. It relates to the behaviour of the structures subjected to pulse loads, and represents the loss of stability or the deformation of a structure to unbounded amplitudes as the result of a transient response to an applied impact loads. Analytical work on dynamic buckling problems for axially loaded cylindrical shells was first carried out by Volmir [1], who obtained solutions of a two degrees of freedom system using Galerkin’s method. Then Coppa and Nash [2] and Roth and Klosner [3] used the potential energy method to study a limited degrees of freedom system, Karagiozova and Jones [4] studied a multi-degrees of freedom model for dynamic buckling of an elastic–plastic structure, while Tamura and Babcock [5] analysed the dynamic buckling of cylindrical shells with geometric imperfections, applying the Budiansky–Roth criterion [6]. More recently, Simitses [7] investigated the effect of static preloading on the dynamic critical loads; Lindberg [8] summarised and demonstrated the range of applicability of two dynamic buckling criteria choosing as example a cylindrical shell under dynamic pulse axial loads from nearly impulsive loads to step loads of infinite duration; Gilat et al. [9] analysed the axisymmetric response of nonlinearly elastic cylindrical shells subjected to dynamic axial loads using an incremental formulation. Finite element analyses were carried out by Mustafa et al. [10] for the prediction of the dynamic buckling response of thick tubes of various lengths, incorporating an idealised geometric imperfection in the radius, and by Pegg, who investigated the dynamic pulse buckling of infinite cylinders of various radius to thickness ratios [11] and of ring-stiffened cylinders [12], using the code ADINA. Very few experimental data can be found in literature, regarding dynamic buckling. Abramovich and Grunwald [13] performed an experimental test series on laminated composite plates, while Cui et al. [14] investigated the elastic–plastic dynamic buckling properties of rectangular mild steel plates under in-plane fluid–solid slamming. Zimcik and Tennyson [15] investigated experimentally the dynamic response of thin-walled circular cylindrical shells (with and without controlled initial shape imperfections) to transient dynamic square-pulse loading of varying time duration. Yaffe and Abramovich [16] investigated, both numerically and experimentally, the buckling of aluminium cylindrical stringer stiffened shells under axial dynamic applied loading. But, until now, no test results were obtained to form a sound experimental database for this phenomenon. There are then few works concerned with dynamic buckling of composite cylindrical shells and they do not cover all the fundamental aspects of the composite shells.
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Schokker et al. [17] investigated the dynamic instability of interior ring stiffened composite shells under hydrostatic pressure, using a potential energy function and then Sridharan and Kasagi [18] established imperfection sensitivity of typical ring-stiffened shells and the interaction of overall and local instabilities always under hydrostatic pressure. Also Tabiei and coworkers [19,20] studied the critical suddenly applied lateral pressure for cylindrical laminated shells with and without statically applied preloading. They compared explicit vs implicit finite element solutions and used the shell volume instead of displacement to assess buckling. Shaw et al. [21] investigated the dynamic stability of an imperfect fiber reinforced composite cylindrical shell under torsional impulsive load in the form of a step function. They used Donnell-von Ka´rma´n type nonlinear equations, and their attention is concentrated to compare two kinds of criteria to determine the critical dynamic loads. Regarding the composite cylindrical shell structure behaviour at axial loading, Eslami et al. [22] proposed a general layerwise theory that may be employed for dynamic buckling analysis of imperfect multilayered composite circular cylindrical shells, while Lam and Ng [23] studied the dynamic stability of thin laminated cylindrical shells under combined static and periodic forces using Love’s theory for thin shells. Argento and Scott presented a theoretical development [24] and numerical results [25] for the parametric resonance of layered anisotropic circular cylindrical shells. The shell’s ends are clamped and subjected to axial loading consisting of a static part and a harmonic part. Liaw and Yang [26] used a thin shell finite element to study the dynamic buckling behaviour of a class of thin shell problems, including axially compressed, laminated, asymmetrically imperfect, cylindrical shells. Huyan and Simitses [27] considered the problem of dynamic buckling of geometrically imperfect cylindrical shells under axial compression and bending moment and employed the finite element code ANSYS to generate dynamic responses and the equations of motion approach to determine dynamic critical loads. The objective of the present paper is to investigate the dynamic buckling due to impulsive loading of thin-walled CFRP shells structures, using a commercially available finite element code. Indeed, the numerical finite element approach allows investigation of dynamic buckling of many complex cases to which the simpler theories are not fully applicable. Cylindrical shells dynamically loaded by axial compression are investigated, because they exhibit almost all the features, which may influence buckling behaviour, and which can bring about trouble with computational prediction. Two four-ply CFRP laminated cylindrical shells, with the same material properties and geometric dimensions of real specimens already used for experimental static buckling tests [28–30] are investigated. The contribution of this paper to the state of the art is summarised by the following points: 1. Three different types of finite element analyses are compared to study the static buckling: eigenvalue analysis, non-linear Riks method and dynamic analysis. 2. The time history of impulsive loading with finite duration and constant in magnitude is varied, and the corresponding dynamic buckling loads are related to the static values.
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3. The effect of lamina stacking sequence on buckling load of cylindrical shells and the sensitivity to geometric imperfections are analysed for static and dynamic buckling loads. 4. Static and dynamic deformations are investigated for the considered laminated cylinders. Because the mode shapes are unknown, selected small portion of shells with appropriate boundary conditions cannot be used for the analyses and consequently the whole cylindrical shells are considered. In any case, the present work gives a contribution to study the problems related to the behaviour of laminated cylindrical shells subjected to dynamic loading, but a lot of experimental work still has to be performed to understand completely these phenomena.
2. Fiber composite cylindrical shells Two four-ply laminated cylindrical shells, with the same material properties and geometric dimensions of real specimens already used for experimental static buckling tests performed by the same author [28–30], are investigated. The cylindrical shells are made of carbon fiber reinforced plastics (CFRP) and are characterised by a mean radius of 350 mm, a length of 700 mm and a thickness of 1.32 mm, with a nominal radius-to-thickness ratio of 265. Two different stacking sequences are investigated: [08/458/K458/08] for the first shell and [458/K458]S for the second one, where 08 corresponds to the axial direction of the shells and where the stacking sequences is taken from outside to inside. Each ply is 0.33 mm thick and has the material properties reported in Table 1.
3. Methodology The buckling analysis of the fiber composite shells is carried out using two commercially available finite element codes: ABAQUS/Standard and ABAQUS/Explicit. ABAQUS [31] is a general purpose finite element program with linear static, dynamic and non-linear analysis capabilities. When solving dynamic problems, it uses two basic methods, the implicit and explicit integration operators, implemented in ABAQUS/ Standard and ABAQUS/Explicit respectively. The static buckling is investigated by eigenvalue analysis, non-linear static method, and dynamic analysis [32]. At first, the results of the three different types of finite element analysis are compared and the initial geometric imperfections are not considered, then Table 1 Mechanical properties of the CFRP ply Elastic modulus E11 (N/mm2)
Elastic modulus E22 (N/mm2)
Shear modulus G12 (N/mm2)
Poisson’s ratio n12
Density (kg/m3)
Thickness (mm)
52,000
52,000
2350
0.302
1320
0.33
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the geometric imperfections are accounted for in the finite element model to evaluate their effects on critical loads. Thereafter the dynamic buckling loads are estimated for impulsive loading using the equations of motion approach, numerically solved by the explicit integration operator implemented in ABAQUS/Explicit. The loads are suddenly applied with finite duration and constant in magnitude. Different values of load duration are considered and the corresponding dynamic buckling loads are related to the static ones.
4. Finite element model In the present study, the finite element model considers complete cylinders for the two four-ply laminated cylindrical shells. Indeed, neither a quarter of cylinder nor half cylinder are considered, so as not to influence the buckling load values and the buckling shapes, imposing symmetric conditions. The cylindrical specimens are modelled by 4-nodes shell elements. This is due primarily to the fact that ABAQUS/Explicit does not present 8-nodes elements and so 4-nodes elements are chosen even in ABAQUS/Standard to compare results obtained using equal elements in both codes. Consequently, S4R elements are used. They are 4-nodes reduced-integration shell elements with a large-strain formulation [31]. A convergence analysis was performed [29], and then a model containing 11,440 elements (220 elements in the circumferential direction and 52 elements in the axial direction) and 11,660 nodes, each having six degrees of freedom, has been chosen because it represents a reasonable choice between good accuracy of results and computational time. The boundary conditions are taken equal to those ones that are applied during the experimental buckling tests [30]: all three translations and three rotations are suppressed at the bottom side, while only the axial translation is free at the top side. The axial compression is transferred to uniform nodal forces along the circumference.
5. Static buckling The static buckling is investigated by eigenvalue analysis, non-linear static method, and dynamic analysis for the two fiber composite cylindrical shells. At first, the results of the three different types of finite element analyses are compared and the initial geometric imperfections are not considered, then the geometric imperfections are accounted for in the finite element model to evaluate their effects on critical loads. 5.1. Eigenvalue analysis performed using ABAQUS/Standard The linear buckling analysis predicts the theoretical buckling strength (the bifurcation point) of an ideal linear elastic structure. The buckling load computed using this method represents generally the upper limit of the buckling strength of the structure and is unconservative. The advantages of this method over the non-linear buckling analysis are
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that it is computationally much simpler and it serves as a guideline to carry out more accurate non-linear analyses. However, for structures with significant non-linearities, the actual buckling load and the buckling mode may be considerably different from those predicted by a linear analysis. In the finite element analysis, the problem of linear buckling is addressed by including the effect of the differential stiffness to the linear stiffness matrix. The differential stiffness results from including higher order terms of the strain–displacement relationships. From a physical standpoint, the differential stiffness matrix represents the linear approximation of softening the linear stiffness matrix in the case of compressive axial load, and stiffening the linear stiffness matrix in the case of tensile load. The differential stiffness is a function of element type, geometry and applied loads. Hence, it is also called the geometric stiffness matrix. Mathematically, linear buckling analysis of a structure requires the solution of the eigenvalue problem: ð½K C l½Kd ÞfFg Z f0g where [K] is the linear stiffness matrix, [Kd] the differential stiffness matrix, and l and {F} denote the associated eigenvalue and eigenvector, respectively. In terms of buckling analysis, the eigenvector {F} represents the buckling mode shape and the associated eigenvalue l indicates the multiple of [Kd] needed to make singular the equation, that is to cause buckling. In carrying out a linear buckling analysis the deflections have to be small, the element stresses have to remain elastic, the distribution of the internal element forces due to the applied loads have to remain constant and the follower force effect has not to be included in the differential stiffness. To perform eigenvalue analysis, the finite element code ABAQUS/Standard is used. The results of the eigenvalue analysis are reported in Table 2, where they are compared to the analytical solution, described in detail in [33]. The static buckling modes of the two shells, obtained by the eigenvalue analysis, are represented in Fig. 1. The buckling shapes for the CFRP shells depend on the stacking sequences. Indeed the cylinder with lay-up orientation [08/458/K458/08] exhibits a diamond shape with 14 circumferential waves and seven axial semi-waves, while the cylinder with lay-up orientation [458/K458]s exhibits an axisymmetric shape with 11 axial semi-waves.
Table 2 Comparison among buckling loads, obtained using different FE analyses
Analytical solution ABAQUS/Standard: eigenvalue analysis ABAQUS/Standard: RiKs analysis ABAQUS/Explicit: dynamic analysis
Buckling load (kN) cylinder [08/458/K458/08]
Buckling load (kN) cylinder [458/K458]S
240.00 248.60 251.16 244.28
118.58 120.56 119.13 119.03
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Fig. 1. Static buckling mode: cylinder [08/458/K458/08] and cylinder [458/K458]s.
5.2. Non-linear analysis performed using ABAQUS/Standard In linear mechanics of deformable bodies, displacements are proportional to loads. The essence of buckling, however, is a disproportionate increase in displacements resulting from a small increase in load. Consequently, buckling analysis is a subtopic of non-linear rather than linear mechanics. Even though the stress–strain relations remain linear, during buckling, the structure experiences large deformations and its changing geometric configuration causes the structure to respond non-linearly. Hence, though it is more time consuming and computationally more expensive, a geometric non-linear analysis is more accurate. The non-linear buckling analysis in ABAQUS/Standard is performed using the modified Riks procedure to solve the non-linear governing equations and activates automatic bisection of load increments [31,34,35]. In the analysis, fine loads increments are used as the loads approach the expected critical buckling load, because the basic approach in a non-linear buckling analysis is to constantly increment the applied loads until the solution begins to diverge. For this reason, a preliminary linear buckling analysis is useful as a predictor of the buckling load. The results of the non-linear analysis are presented in Table 2 with differences from the analytical solution and the eigenvalue analysis of about 5%. 5.3. Quasi-static analysis performed using ABAQUS/Explicit The finite element code ABAQUS/Explicit performs dynamic analysis, using a Lagrangian formulation and integrating the equations of motion in time explicitly by means of central differences. It can be also used for static analysis simulating the dynamics of a slow compression test and allowing for the investigation of the load vs. displacement curve [29]. The result of the quasi-static analysis is reported in Table 2, where it is compared with the previous results. The values obtained performing different types of finite element
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analyses are in good agreement. Indeed the difference with the values obtained by other methods is within 3%. Considering that the cylinder is sensitive to geometrical imperfections and hence also sensitive to the methodology, the former results are satisfactory. Therefore, a dynamic analysis using an explicit integration operator can be used very efficiently to study the buckling behaviour of cylindrical shells including laminated shells under axial compression loads. 5.4. Effect of initial geometric imperfections on critical loads Owing to the significant discrepancy between analytically predicted buckling loads and the ones observed in experiments, it is of great technical importance to clarify the effect of initial imperfections on the buckling of circular cylindrical shells, so as to compare later the results for the dynamic analyses. For this purpose, the midsurface of the cylinder under investigation is regarded to be the reference surface where initial imperfections are applied. A cylindrical coordinate system is employed, and the nodal coordinate in the radial direction is taken as rZRCw8, where R is the radius of the cylinder and w8 is the amplitude of the imperfection shape. The data of the geometric imperfections, measured before the tests on the internal surface of the real specimens using a laser scanning system [28–30], are so introduced into the finite element models. The measured initial geometric imperfections present a mean value of imperfection amplitude equal to 0.084 mm, that corresponds to a ratio between imperfection amplitude and shell thickness equal to 0.063. The numerical results are compared to the experimental values of buckling loads obtained during static axial compression buckling tests performed using a loading rig under position control [28–30]. The values, obtained using the non-linear Riks method (ABAQUS/Standard) and a slow dynamic analysis (ABAQUS/Explicit), are reported in Table 3 for a cylinder [08/458/K458/08]. These numerical buckling loads are compared to the experimentally measured one, and the differences result of about 15–20%. The ratio between the buckling loads obtained for the cylinders with imperfections and those of the cylinders without imperfections, known as knock-down factor, is an index of the sensitivity of the buckling load with respect to the considered geometric imperfections. It is possible to find out the high sensitivity of the cylindrical shells buckling load to the geometric imperfections: the buckling load is around 85% of that one of a shell without imperfections. The post-buckling deformation shows nine circumferential waves and two axial waves, like the one observed in the experimental tests (Fig. 2). Table 3 Numerical-experimental correlation for cylinder [08/458/K458/08] Buckling load (kN) Experimental test Riks analysis Dynamic analysis
163.46 187.11 198.17
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Fig. 2. Experimental and numerical post-buckling static deformation.
6. Dynamic buckling analysis Dynamic buckling due to axial load with finite duration is here investigated for the two fiber composite cylindrical shells using the code ABAQUS/Explicit. The effect of laminate lay-up on buckling load and the sensitivity to geometric imperfections are analysed. The time history of impulsive loading with finite duration and constant in magnitude is varied, and the corresponding dynamic buckling loads are related to the quasi-static buckling loads. 6.1. Methodology for dynamic critical conditions There are three methods used by the various investigators in estimating critical conditions for dynamically loaded elastic systems [1,7,24] Their concepts and details are briefly described as follows. 6.1.1. Total energy-phase plane approach Critical conditions are related to characteristics of the system phase plane, and the emphasis is on establishing sufficiency conditions for stability (lower bounds) and sufficiency conditions for instability (upper bounds). For a one-degree-of-freedom system, the energy balance equation is plotted in the phase plane for various values of the load parameter. When load parameters are small, the curve is a closed one and the motion is bounded and, therefore, it is stable. As the load parameter increases, a value is reached at which an escaping motion is possible and the system is in critical condition. 6.1.2. Total potential energy approach Critical conditions are related to characteristics of the system total potential. Through this approach, also, the lower and upper bounds of critical conditions are established. The energy balance equation is used and the total potential energy is plotted versus the system generalised coordinate for various values of the load parameter in this approach. When escaping motion is possible, a critical condition exists.
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6.1.3. Equation of motion approach—Budiansky–Roth criterion The equations of motion are solved for various values of the load parameter, thus obtaining system responses. The load parameter, at which there exists a large change in responses, is called critical. This approach is easily adapted to computational methods such as finite element codes and, therefore, it has become very popular in applications. In addition, it has met with success when applied to systems that exhibit violent buckling (limit point and bifurcation point with unstable post-buckling branch) under static loads. Cylindrical shells are typical violent buckling configurations, hence, this criterion is employed in the present work. In the dynamic analysis, the basic equations of motion of a cylindrical shell are _ C ½Kfug Z fFðtÞg ½Mfu€ g C ½Cfug where [M] is the mass matrix, [C] the damping matrix, [K] the stiffness matrix including _ the nodal velocity vector, fug € the non-linearity, {u} the nodal displacement vector, fug nodal acceleration vector and {F(t)} the load vector. The equations of motion are solved for various values of the loading and the value at which there is a significant jump in the response is assumed critical. When monitoring the system response through displacements of selected points for small values of the loading parameter, small oscillations are observed, the amplitudes of which gradually increases as the loading is increased. When the loading reaches its critical value, the maximum amplitude experiences a large jump. Therefore, implementation of this criterion requires to solve the equations of motion for different values of the loading parameter, then plot the displacement amplitude versus loading curve from which the critical loading value is determined.
6.2. Finite element analyses The critical conditions for dynamic buckling are so estimated by a sudden large change in dynamic responses, according to the Budiansky–Roth criterion [6]. Consequently, to get the dynamic critical load of a cylindrical shell subjected to suddenly applied axial compression with finite duration, it is necessary to solve the equations of motion for several load parameter values, so as to obtain the dynamic responses. The dynamic critical load is determined as the lowest load at which there is a large sudden change in transient response. At first some analyses were performed to compare the dynamic analyses performed with the implict method implemented in ABAQUS/Standard and with the explicit method implemented in ABAQUS/Explicit. The results show that both operators basically predict the same behaviour, but the explicit method ABAQUS/Explicit requires only a fraction of the CPU time used by the implicit scheme. Indeed, in the explicit dynamics procedure used by ABAQUS/Explicit, the use of small increments, dictated by the stability limit, is advantageous for dynamic buckling analysis, because it allows the solution to proceed without iterations and without requiring tangent stiffness matrices to be formed. Even if the analysis may take an extremely large
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number of increments, each increment is relatively inexpensive, resulting in an economical solution. Thus, all future investigations considering a dynamic analysis are performed using ABAQUS/Explicit. The axial compression is transferred to uniform nodal forces along the circumference and the load is suddenly applied with finite duration and constant in magnitude. Five different time durations are analysed: TZ1, 2.5, 5, 10, and 15 ms. The analyses are carried out with and without initial imperfections and with a mesh of the whole cylindrical shell of 220!52 4-nodes elements. To be able to get good results from the analyses, the points for which displacements are to be monitored are to be carefully chosen, otherwise the plots produced may be rather obscure and confusing. Instead of monitoring the radial displacements of several points from the midsection of the cylinder, where the largest displacement values were expected, the axial displacement is considered in this study. Fig. 3 presents the axial displacement of a point of the loaded side of the laminated cylinder [08/458/K458/08] without initial imperfection and subjected to suddenly applied axial compression of finite duration TZ10 ms with three load magnitudes. The curve corresponding to 132 kN is a regular response and the cylinder vibrates about its equilibrium position. From 132 to 134 kN, there is a great change between the two responses. This is the buckling condition and the load average equal to 133 kN is regarded as the dynamic buckling load of the shell. Using this criterion to estimate the dynamic buckling loads, dynamic analyses are performed for different time durations T.
Fig. 3. Axial displacement for load duration TZ10 ms—cylinder [08/458/K458/08].
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Table 4 Dynamic buckling loads versus load duration—Cylinder [08/458/K458/08] Load duration T (ms)
Dynamic buckling load (kN) (without imperfection)
Ratio of dynamic to static load (without imperfection)
Dynamic buckling load (kN) (with imperfection)
Ratio of dynamic to static load (with imperfection)
1 2.5 5 10 15
350 215 155 133 120
1.41 0.86 0.62 0.53 0.48
230 144 125 117 108
1.16 0.72 0.63 0.59 0.54
6.3. Results Dynamic buckling due to axial load with finite duration was investigated for the two fiber composite cylindrical shells, using the code ABAQUS/Explicit. At first, the cylinders are analysed without geometric imperfections, then the geometric imperfections measured on the real shells are introduced. The results of the analyses are reported for the cylinder [08/458/K458/08] and for the cylinder [458/K458]S in Tables 4 and 5, respectively, where they are also compared to the static buckling loads. The dynamic buckling loads versus load duration are then reported in Fig. 4 for the two lay-ups. It may be observed that the dynamic critical loads of a cylinder decrease quickly with increasing load duration and converge to a certain value. This value is defined as the critical load for the load case of infinite duration. The diagrams in Fig. 4 indicate also that shell stiffness increases for a short duration of loading. Indeed, for short time duration, the dynamic buckling loads are higher than the static buckling loads. As already shown by Zimcik and Tennyson [15] and by Eslami et al. [22], it is believed that these phenomena are due to the stress wave vibration between the impacted and fixed ends of the shell. This phenomenon is more evident in the case of the laminated cylinder [08/458/K458/08]. With increasing load duration, the dynamic buckling loads decrease quickly and get significantly smaller than the static ones. The present investigation gives smaller ratios of the dynamic buckling loads to the static ones in comparison with the results obtained by Huyan and Simitses [27], but the results are in good agreement to the ones obtained by Eslami et al. [22] and already performed by the author and Zimmermann on different composite cylindrical shells [36]. Table 5 Dynamic buckling loads versus load duration—Cylinder [458/K458]S Load duration T (ms)
Dynamic buckling load (kN) (without imperfection)
Ratio of dynamic to static load (without imperfection)
Dynamic buckling load (kN) (with imperfection)
Ratio of dynamic to static load (with imperfection)
1 2.5 5 10 15
135 118 102 85 81
1.12 0.98 0.85 0.71 0.67
120 107 93 70 66
1.03 0.92 0.79 0.60 0.57
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Fig. 4. Dynamic buckling loads versus duration: cylinders [08/458/K458/08] and cylinder [458/K458]s.
So in general the structures experience dynamic buckling loads larger than the static buckling loads, provided their duration is very short. If their duration is longer, the dynamic buckling load is smaller than the static one, which means that taking the static buckling load as the design point for dynamic problems might be misleading. The ratio between the dynamic buckling load and the static buckling load is of practical interest for the designer, since it provides a direct indication of the load carrying capacity of the structural elements exposed to rapidly applied loads relative to almost statically applied loads. Increasing the length of the loading period leads to a decrease in the ratio.
Fig. 5. Dynamic deformation—Cylinder [458/K458]s—Load duration TZ2.5 ms.
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Furthermore, the dynamic buckling loads, like the static ones, decrease with increasing imperfection amplitude. The sensitivity to initial geometric imperfection depends on the lay-up of the laminated shells. The imperfection amplitude has virtually the same effect on static critical load as on the dynamic critical load of infinite duration. Fig. 5 reports the dynamic deformation of laminated cylinder [458/K458]S for a load duration T equal to 2.5 ms.
7. Conclusion Dynamic buckling due to impulsive loading of thin-walled CFRP cylindrical shells has been investigated. The equations of motion approach, which associates the dynamic buckling of a structure with the state at which a small change in the magnitude of the loading leads to a large change in the structure response, has been successfully applied using the finite element code ABAQUS/Explicit. It has allowed to calculate the dynamic buckling loads for different values of load duration in a reliable and easy way. The dynamic buckling loads result strongly dependent on the load duration. Indeed for short time duration, the dynamic buckling loads are higher than the static ones, while by increasing the load duration, the dynamic buckling loads decrease quickly and get significantly smaller than the static loads. The initial geometric imperfections, measured on the internal surface of the real shells using a laser scanning system, have been introduced in the numerical models. The dynamic buckling loads, like the static ones, decrease because of the geometric imperfection and the laminate set-up of the shells influences this behaviour. The present study brings an important conclusion for design purposes. Since the common practice is to assume that dynamic bucking loads are higher than the static ones, which means that static design is safe, careful design is recommended. Indeed, taking the static buckling load as the design point for dynamic problems might be misleading. These conclusions are based on a numerical study, that will be confirmed by further computations and by experimental dynamic buckling tests with different values of load duration so to generate widely applicable design criteria for dynamic buckling of composite shells. For this aim work is currently under way on performing experimental dynamic buckling tests by means of a deceleration horizontal sled normally used for crash tests. The load intensity and the length of the loading period can be altered by changing the impact velocity and by increasing or decreasing the mass of the sled. The results of the tests will be reported in due time.
Acknowledgements The author wishes to thank Dr Rolf Zimmermann of DLR (German Aerospace Center), Institut fu¨r Strukturmechanik, Braunschweig, Germany, for having introduced to the interesting field of dynamic buckling of composite shells, through a research supported by the European Union under a Marie Curie Grant inside the programme Training and Mobility of Researcher (Contract No. ERBFMBI CT97 2465).
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References [1] Volmir AS. On the stability of dynamically loaded cylindrical shells. Dokladi Akademii Nauk 1958. SSSR, 123(1–6): 806–808 (in Russian); also translated in Soviet Physics Doklady, 1958. 3(6): 1287–1289. [2] Coppa AP, Nash WA. Dynamic buckling of shells structure subject to longitudinal impact. General Electric Co., Technical Documentary Rept. FDL-TDR-64-65, Philadelphia, PA, 1964. [3] Roth RS, Klosner JM. Nonlinear response of cylindrical shells subjected to dynamic axial loads. AIAA J 1964;2(10):1788–94. [4] Karagiozova D, Jones N. Multi-degrees of freedom model for dynamic buckling of an elastic–plastic structure. Int J Solids Struct 1996;33(23):3377–98. [5] Tamura YS, Babcock CD. Dynamic stability of cylindrical shells under step loading. J Appl Mech 1975; 42(1):190–4. [6] Budiansky, B., Roth, R.S. Axisymmetric dynamic buckling of clamped shallow spherical shells. Collected Papers on Instability of Shell Structures, NASA. TN-D-1510 1962. 597–606 [7] Simitses GJ. Dynamic stability of suddenly loaded structures. Berlin: Springer; 1990. [8] Lindberg HE. Dynamic pulse buckling of imperfection-sensitive shells. J Appl Mech 1991;58:743–8. [9] Gilat R, Feldman E, Aboudi J. Axisymmetric response of nonlinearly elastic cylindrical shells to dynamic axial loads. Int J Impact Eng 1993;13(4):545–54. [10] Mustafa B, Al-Hassani STS, Reid SR. Axisymmetric dynamic buckling of submerged cylindrical shells. Comput Struct 1993;47(3):399–405. [11] Pegg NG. Dynamic pulse buckling of cylinders of various a/h ratios. Comput Struct 1991;39(1/2):173–83. [12] Pegg NG. A numerical study of dynamic pulse buckling of ring-stiffened cylinders. Comput Struct 1992; 44(6):1205–14. [13] Abramovich H, Grunwald A. Stability of axially impacted composite plates. Compos Struct 1995;32:151–8. [14] Cui S, Cheong HK, Hao H. Experimental study of dynamic buckling of plates under fluid–solid slamming. Int J Impact Eng 1999;22:675–91. [15] Zimcik DG, Tennyson RC. Stability of circular cylindrical shells under transient axial impulsive loading. AIAA J 1980;18(6):691–9. [16] Yaffe R, Abramovich H. Dynamic buckling of cylindrical stringer stiffened shells. Comput Struct 2003;81: 1031–9. [17] Schokker A, Sridharan S, Kasagi A. Dynamic buckling of composite shells. Comput Struct 1996;59(1): 43–53. [18] Sridharan S, Kasagi A. On the buckling and collapse of moderately thick composite cylinders under hydrostatic pressure. Composites B 1997;28B:583–96. [19] Tabiei A, Tanov R, Simitses GJ. Numerical simulation of cylindrical laminated shells under impulsive lateral pressure. AIAA J 1999;37(5):629–33. [20] Tanov R, Tabiei A. Static and dynamic buckling of laminated composite shells. Proceedings of the fifth international LS-DYNA users conference, South Field, MI; Sept. 1998. [21] Shaw D, Shen YL, Tsai P. Dynamic buckling of an imperfect composite circular cylindrical shell. Comput Struct 1993;48(3):467–72. [22] Eslami MR, Shariyat M, Shakeri M. Layerwise theory for dynamic buckling and postbuckling of laminated composite cylindrical shells. AIAA J 1998;36(10):1874–82. [23] Lam KY, Ng TY. Dynamic stability analysis of laminated composite cylindrical shells subjected to conservative periodic axial loads. Compos Part B 1998;29B:769–85. [24] Argento A, Scott RA. Dynamic instability of layered anisotropic circular cylindrical shells, part I: theoretical development. J Sound Vib 1993;162(2):311–22. [25] Argento A, Scott RA. Dynamic instability of layered anisotropic circular cylindrical shells, part II: numerical results. J Sound Vib 1993;162(2):323–32. [26] Liaw DG, Yang TY. Symmetric and asymmetric dynamic buckling of laminated thin shells with the effect of imperfection and damping. J Compos Mater 1990;24:188–207. [27] Huyan X, Simitses GJ. Dynamic buckling of imperfect cylindrical shells under axial compression and bending moment. AIAA J 1997;35(8):1404–12. [28] Bisagni C. Experimental buckling of thin composite cylinders in compression. AIAA J 1999;37(2):276–8.
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[29] Bisagni C. Numerical analysis and experimental correlation of composite shell buckling and post-buckling. Compos Part B: Eng 2000;31(8):655–67. [30] Bisagni C, Cordisco P. An experimental investigation into the buckling and post-buckling of CFRP shells under combined axial and torsion loading. Compos Struct 2003;64(4):391–402. [31] ABAQUS Manuals, Version 5.7, Hibbitt: Karlsson and Sorensen, Inc.; 1997. [32] Bathe KJ. Finite element procedures in engineering analysis. Englewoodcliff, NJ: Prentice-Hall; 1982. [33] Bisagni C. Instabilita` e comportamento post-critico di gusci in materiale composito. Ph.D. Thesis. Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano. 1997. [in Italian] [34] Riks E. The application of Newton’s method to the problem of elastic stability. J Appl Mech 1972;39: 1060–5. [35] Riks E. An incremental approach to the solution of snapping and buckling problems. Int J Solids Struct 1979;15:529–51. [36] Bisagni C, Zimmermann R., Buckling of axially compressed fiber composite cylindrical shells due to impulsive loading. In: Proceedings of European Conference on Spacecraft Structures, Materials and Mechanical Testing, Braunschweig, Germany; 1998. 557–562.
Dynamic buckling of fiber composite shells under impulsive axial compression Chiara Bisagni* Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La Masa 34, 20156 Milano, Italy Received 19 December 2003; accepted 27 July 2004 Available online 22 October 2004
Abstract The paper deals with dynamic buckling due to impulsive loading of thin-walled carbon fiber reinforced plastics (CFRP) shell structures under axial compression. The approach adopted is based on the equations of motion, which are numerically solved using a finite element code (ABAQUS/Explicit) and using numerical models validated by experimental static buckling tests. To study the influence of the load duration, the time history of impulsive loading is varied and the corresponding dynamic buckling loads are related to the quasi-static buckling loads. To analyse the sensitivity to geometric imperfections, the initial geometric imperfections, measured experimentally on the internal surface of real shells, are introduced in the numerical models. It is shown numerically that the initial geometric imperfections as well as the duration of the loading period have a great influence on the dynamic buckling of the shells. For short time duration, the dynamic buckling loads are larger than the static ones. By increasing the load duration, the dynamic buckling loads decrease quickly and get significantly smaller than the static loads. Since the common practice is to assume that dynamic bucking loads are higher than the static ones, which means that static design is safe, careful design is recommended. Indeed, taking the static buckling load as the design point for dynamic problems might be misleading. q 2004 Elsevier Ltd. All rights reserved. Keywords: Dynamic buckling; Finite element analysis; Fiber composite cylindrical shells; Impulsive axial compression; Geometric imperfections
* Tel.: C39 02 2399 8390; fax: C39 02 2399 8334. E-mail address: [email protected]. 0263-8231/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2004.07.012
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1. Introduction The stability of composite cylindrical shells is of fundamental interest in aircraft and missile design as thin-walled cylindrical shells have constituted primary structural parts for many years and because the composite materials are nowadays extremely attractive due to their considerable strength-to-weight ratio. They are subjected to various loading both static and dynamic. The investigations in the past two decades were primarily confined to the static stability. This effort has led to a reasonably good understanding of the response of the composite shells to static loading, including the effects of initial geometric imperfections and lamina stacking sequence. However, dynamic buckling is not yet clearly defined. It relates to the behaviour of the structures subjected to pulse loads, and represents the loss of stability or the deformation of a structure to unbounded amplitudes as the result of a transient response to an applied impact loads. Analytical work on dynamic buckling problems for axially loaded cylindrical shells was first carried out by Volmir [1], who obtained solutions of a two degrees of freedom system using Galerkin’s method. Then Coppa and Nash [2] and Roth and Klosner [3] used the potential energy method to study a limited degrees of freedom system, Karagiozova and Jones [4] studied a multi-degrees of freedom model for dynamic buckling of an elastic–plastic structure, while Tamura and Babcock [5] analysed the dynamic buckling of cylindrical shells with geometric imperfections, applying the Budiansky–Roth criterion [6]. More recently, Simitses [7] investigated the effect of static preloading on the dynamic critical loads; Lindberg [8] summarised and demonstrated the range of applicability of two dynamic buckling criteria choosing as example a cylindrical shell under dynamic pulse axial loads from nearly impulsive loads to step loads of infinite duration; Gilat et al. [9] analysed the axisymmetric response of nonlinearly elastic cylindrical shells subjected to dynamic axial loads using an incremental formulation. Finite element analyses were carried out by Mustafa et al. [10] for the prediction of the dynamic buckling response of thick tubes of various lengths, incorporating an idealised geometric imperfection in the radius, and by Pegg, who investigated the dynamic pulse buckling of infinite cylinders of various radius to thickness ratios [11] and of ring-stiffened cylinders [12], using the code ADINA. Very few experimental data can be found in literature, regarding dynamic buckling. Abramovich and Grunwald [13] performed an experimental test series on laminated composite plates, while Cui et al. [14] investigated the elastic–plastic dynamic buckling properties of rectangular mild steel plates under in-plane fluid–solid slamming. Zimcik and Tennyson [15] investigated experimentally the dynamic response of thin-walled circular cylindrical shells (with and without controlled initial shape imperfections) to transient dynamic square-pulse loading of varying time duration. Yaffe and Abramovich [16] investigated, both numerically and experimentally, the buckling of aluminium cylindrical stringer stiffened shells under axial dynamic applied loading. But, until now, no test results were obtained to form a sound experimental database for this phenomenon. There are then few works concerned with dynamic buckling of composite cylindrical shells and they do not cover all the fundamental aspects of the composite shells.
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Schokker et al. [17] investigated the dynamic instability of interior ring stiffened composite shells under hydrostatic pressure, using a potential energy function and then Sridharan and Kasagi [18] established imperfection sensitivity of typical ring-stiffened shells and the interaction of overall and local instabilities always under hydrostatic pressure. Also Tabiei and coworkers [19,20] studied the critical suddenly applied lateral pressure for cylindrical laminated shells with and without statically applied preloading. They compared explicit vs implicit finite element solutions and used the shell volume instead of displacement to assess buckling. Shaw et al. [21] investigated the dynamic stability of an imperfect fiber reinforced composite cylindrical shell under torsional impulsive load in the form of a step function. They used Donnell-von Ka´rma´n type nonlinear equations, and their attention is concentrated to compare two kinds of criteria to determine the critical dynamic loads. Regarding the composite cylindrical shell structure behaviour at axial loading, Eslami et al. [22] proposed a general layerwise theory that may be employed for dynamic buckling analysis of imperfect multilayered composite circular cylindrical shells, while Lam and Ng [23] studied the dynamic stability of thin laminated cylindrical shells under combined static and periodic forces using Love’s theory for thin shells. Argento and Scott presented a theoretical development [24] and numerical results [25] for the parametric resonance of layered anisotropic circular cylindrical shells. The shell’s ends are clamped and subjected to axial loading consisting of a static part and a harmonic part. Liaw and Yang [26] used a thin shell finite element to study the dynamic buckling behaviour of a class of thin shell problems, including axially compressed, laminated, asymmetrically imperfect, cylindrical shells. Huyan and Simitses [27] considered the problem of dynamic buckling of geometrically imperfect cylindrical shells under axial compression and bending moment and employed the finite element code ANSYS to generate dynamic responses and the equations of motion approach to determine dynamic critical loads. The objective of the present paper is to investigate the dynamic buckling due to impulsive loading of thin-walled CFRP shells structures, using a commercially available finite element code. Indeed, the numerical finite element approach allows investigation of dynamic buckling of many complex cases to which the simpler theories are not fully applicable. Cylindrical shells dynamically loaded by axial compression are investigated, because they exhibit almost all the features, which may influence buckling behaviour, and which can bring about trouble with computational prediction. Two four-ply CFRP laminated cylindrical shells, with the same material properties and geometric dimensions of real specimens already used for experimental static buckling tests [28–30] are investigated. The contribution of this paper to the state of the art is summarised by the following points: 1. Three different types of finite element analyses are compared to study the static buckling: eigenvalue analysis, non-linear Riks method and dynamic analysis. 2. The time history of impulsive loading with finite duration and constant in magnitude is varied, and the corresponding dynamic buckling loads are related to the static values.
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3. The effect of lamina stacking sequence on buckling load of cylindrical shells and the sensitivity to geometric imperfections are analysed for static and dynamic buckling loads. 4. Static and dynamic deformations are investigated for the considered laminated cylinders. Because the mode shapes are unknown, selected small portion of shells with appropriate boundary conditions cannot be used for the analyses and consequently the whole cylindrical shells are considered. In any case, the present work gives a contribution to study the problems related to the behaviour of laminated cylindrical shells subjected to dynamic loading, but a lot of experimental work still has to be performed to understand completely these phenomena.
2. Fiber composite cylindrical shells Two four-ply laminated cylindrical shells, with the same material properties and geometric dimensions of real specimens already used for experimental static buckling tests performed by the same author [28–30], are investigated. The cylindrical shells are made of carbon fiber reinforced plastics (CFRP) and are characterised by a mean radius of 350 mm, a length of 700 mm and a thickness of 1.32 mm, with a nominal radius-to-thickness ratio of 265. Two different stacking sequences are investigated: [08/458/K458/08] for the first shell and [458/K458]S for the second one, where 08 corresponds to the axial direction of the shells and where the stacking sequences is taken from outside to inside. Each ply is 0.33 mm thick and has the material properties reported in Table 1.
3. Methodology The buckling analysis of the fiber composite shells is carried out using two commercially available finite element codes: ABAQUS/Standard and ABAQUS/Explicit. ABAQUS [31] is a general purpose finite element program with linear static, dynamic and non-linear analysis capabilities. When solving dynamic problems, it uses two basic methods, the implicit and explicit integration operators, implemented in ABAQUS/ Standard and ABAQUS/Explicit respectively. The static buckling is investigated by eigenvalue analysis, non-linear static method, and dynamic analysis [32]. At first, the results of the three different types of finite element analysis are compared and the initial geometric imperfections are not considered, then Table 1 Mechanical properties of the CFRP ply Elastic modulus E11 (N/mm2)
Elastic modulus E22 (N/mm2)
Shear modulus G12 (N/mm2)
Poisson’s ratio n12
Density (kg/m3)
Thickness (mm)
52,000
52,000
2350
0.302
1320
0.33
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the geometric imperfections are accounted for in the finite element model to evaluate their effects on critical loads. Thereafter the dynamic buckling loads are estimated for impulsive loading using the equations of motion approach, numerically solved by the explicit integration operator implemented in ABAQUS/Explicit. The loads are suddenly applied with finite duration and constant in magnitude. Different values of load duration are considered and the corresponding dynamic buckling loads are related to the static ones.
4. Finite element model In the present study, the finite element model considers complete cylinders for the two four-ply laminated cylindrical shells. Indeed, neither a quarter of cylinder nor half cylinder are considered, so as not to influence the buckling load values and the buckling shapes, imposing symmetric conditions. The cylindrical specimens are modelled by 4-nodes shell elements. This is due primarily to the fact that ABAQUS/Explicit does not present 8-nodes elements and so 4-nodes elements are chosen even in ABAQUS/Standard to compare results obtained using equal elements in both codes. Consequently, S4R elements are used. They are 4-nodes reduced-integration shell elements with a large-strain formulation [31]. A convergence analysis was performed [29], and then a model containing 11,440 elements (220 elements in the circumferential direction and 52 elements in the axial direction) and 11,660 nodes, each having six degrees of freedom, has been chosen because it represents a reasonable choice between good accuracy of results and computational time. The boundary conditions are taken equal to those ones that are applied during the experimental buckling tests [30]: all three translations and three rotations are suppressed at the bottom side, while only the axial translation is free at the top side. The axial compression is transferred to uniform nodal forces along the circumference.
5. Static buckling The static buckling is investigated by eigenvalue analysis, non-linear static method, and dynamic analysis for the two fiber composite cylindrical shells. At first, the results of the three different types of finite element analyses are compared and the initial geometric imperfections are not considered, then the geometric imperfections are accounted for in the finite element model to evaluate their effects on critical loads. 5.1. Eigenvalue analysis performed using ABAQUS/Standard The linear buckling analysis predicts the theoretical buckling strength (the bifurcation point) of an ideal linear elastic structure. The buckling load computed using this method represents generally the upper limit of the buckling strength of the structure and is unconservative. The advantages of this method over the non-linear buckling analysis are
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that it is computationally much simpler and it serves as a guideline to carry out more accurate non-linear analyses. However, for structures with significant non-linearities, the actual buckling load and the buckling mode may be considerably different from those predicted by a linear analysis. In the finite element analysis, the problem of linear buckling is addressed by including the effect of the differential stiffness to the linear stiffness matrix. The differential stiffness results from including higher order terms of the strain–displacement relationships. From a physical standpoint, the differential stiffness matrix represents the linear approximation of softening the linear stiffness matrix in the case of compressive axial load, and stiffening the linear stiffness matrix in the case of tensile load. The differential stiffness is a function of element type, geometry and applied loads. Hence, it is also called the geometric stiffness matrix. Mathematically, linear buckling analysis of a structure requires the solution of the eigenvalue problem: ð½K C l½Kd ÞfFg Z f0g where [K] is the linear stiffness matrix, [Kd] the differential stiffness matrix, and l and {F} denote the associated eigenvalue and eigenvector, respectively. In terms of buckling analysis, the eigenvector {F} represents the buckling mode shape and the associated eigenvalue l indicates the multiple of [Kd] needed to make singular the equation, that is to cause buckling. In carrying out a linear buckling analysis the deflections have to be small, the element stresses have to remain elastic, the distribution of the internal element forces due to the applied loads have to remain constant and the follower force effect has not to be included in the differential stiffness. To perform eigenvalue analysis, the finite element code ABAQUS/Standard is used. The results of the eigenvalue analysis are reported in Table 2, where they are compared to the analytical solution, described in detail in [33]. The static buckling modes of the two shells, obtained by the eigenvalue analysis, are represented in Fig. 1. The buckling shapes for the CFRP shells depend on the stacking sequences. Indeed the cylinder with lay-up orientation [08/458/K458/08] exhibits a diamond shape with 14 circumferential waves and seven axial semi-waves, while the cylinder with lay-up orientation [458/K458]s exhibits an axisymmetric shape with 11 axial semi-waves.
Table 2 Comparison among buckling loads, obtained using different FE analyses
Analytical solution ABAQUS/Standard: eigenvalue analysis ABAQUS/Standard: RiKs analysis ABAQUS/Explicit: dynamic analysis
Buckling load (kN) cylinder [08/458/K458/08]
Buckling load (kN) cylinder [458/K458]S
240.00 248.60 251.16 244.28
118.58 120.56 119.13 119.03
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Fig. 1. Static buckling mode: cylinder [08/458/K458/08] and cylinder [458/K458]s.
5.2. Non-linear analysis performed using ABAQUS/Standard In linear mechanics of deformable bodies, displacements are proportional to loads. The essence of buckling, however, is a disproportionate increase in displacements resulting from a small increase in load. Consequently, buckling analysis is a subtopic of non-linear rather than linear mechanics. Even though the stress–strain relations remain linear, during buckling, the structure experiences large deformations and its changing geometric configuration causes the structure to respond non-linearly. Hence, though it is more time consuming and computationally more expensive, a geometric non-linear analysis is more accurate. The non-linear buckling analysis in ABAQUS/Standard is performed using the modified Riks procedure to solve the non-linear governing equations and activates automatic bisection of load increments [31,34,35]. In the analysis, fine loads increments are used as the loads approach the expected critical buckling load, because the basic approach in a non-linear buckling analysis is to constantly increment the applied loads until the solution begins to diverge. For this reason, a preliminary linear buckling analysis is useful as a predictor of the buckling load. The results of the non-linear analysis are presented in Table 2 with differences from the analytical solution and the eigenvalue analysis of about 5%. 5.3. Quasi-static analysis performed using ABAQUS/Explicit The finite element code ABAQUS/Explicit performs dynamic analysis, using a Lagrangian formulation and integrating the equations of motion in time explicitly by means of central differences. It can be also used for static analysis simulating the dynamics of a slow compression test and allowing for the investigation of the load vs. displacement curve [29]. The result of the quasi-static analysis is reported in Table 2, where it is compared with the previous results. The values obtained performing different types of finite element
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analyses are in good agreement. Indeed the difference with the values obtained by other methods is within 3%. Considering that the cylinder is sensitive to geometrical imperfections and hence also sensitive to the methodology, the former results are satisfactory. Therefore, a dynamic analysis using an explicit integration operator can be used very efficiently to study the buckling behaviour of cylindrical shells including laminated shells under axial compression loads. 5.4. Effect of initial geometric imperfections on critical loads Owing to the significant discrepancy between analytically predicted buckling loads and the ones observed in experiments, it is of great technical importance to clarify the effect of initial imperfections on the buckling of circular cylindrical shells, so as to compare later the results for the dynamic analyses. For this purpose, the midsurface of the cylinder under investigation is regarded to be the reference surface where initial imperfections are applied. A cylindrical coordinate system is employed, and the nodal coordinate in the radial direction is taken as rZRCw8, where R is the radius of the cylinder and w8 is the amplitude of the imperfection shape. The data of the geometric imperfections, measured before the tests on the internal surface of the real specimens using a laser scanning system [28–30], are so introduced into the finite element models. The measured initial geometric imperfections present a mean value of imperfection amplitude equal to 0.084 mm, that corresponds to a ratio between imperfection amplitude and shell thickness equal to 0.063. The numerical results are compared to the experimental values of buckling loads obtained during static axial compression buckling tests performed using a loading rig under position control [28–30]. The values, obtained using the non-linear Riks method (ABAQUS/Standard) and a slow dynamic analysis (ABAQUS/Explicit), are reported in Table 3 for a cylinder [08/458/K458/08]. These numerical buckling loads are compared to the experimentally measured one, and the differences result of about 15–20%. The ratio between the buckling loads obtained for the cylinders with imperfections and those of the cylinders without imperfections, known as knock-down factor, is an index of the sensitivity of the buckling load with respect to the considered geometric imperfections. It is possible to find out the high sensitivity of the cylindrical shells buckling load to the geometric imperfections: the buckling load is around 85% of that one of a shell without imperfections. The post-buckling deformation shows nine circumferential waves and two axial waves, like the one observed in the experimental tests (Fig. 2). Table 3 Numerical-experimental correlation for cylinder [08/458/K458/08] Buckling load (kN) Experimental test Riks analysis Dynamic analysis
163.46 187.11 198.17
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Fig. 2. Experimental and numerical post-buckling static deformation.
6. Dynamic buckling analysis Dynamic buckling due to axial load with finite duration is here investigated for the two fiber composite cylindrical shells using the code ABAQUS/Explicit. The effect of laminate lay-up on buckling load and the sensitivity to geometric imperfections are analysed. The time history of impulsive loading with finite duration and constant in magnitude is varied, and the corresponding dynamic buckling loads are related to the quasi-static buckling loads. 6.1. Methodology for dynamic critical conditions There are three methods used by the various investigators in estimating critical conditions for dynamically loaded elastic systems [1,7,24] Their concepts and details are briefly described as follows. 6.1.1. Total energy-phase plane approach Critical conditions are related to characteristics of the system phase plane, and the emphasis is on establishing sufficiency conditions for stability (lower bounds) and sufficiency conditions for instability (upper bounds). For a one-degree-of-freedom system, the energy balance equation is plotted in the phase plane for various values of the load parameter. When load parameters are small, the curve is a closed one and the motion is bounded and, therefore, it is stable. As the load parameter increases, a value is reached at which an escaping motion is possible and the system is in critical condition. 6.1.2. Total potential energy approach Critical conditions are related to characteristics of the system total potential. Through this approach, also, the lower and upper bounds of critical conditions are established. The energy balance equation is used and the total potential energy is plotted versus the system generalised coordinate for various values of the load parameter in this approach. When escaping motion is possible, a critical condition exists.
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6.1.3. Equation of motion approach—Budiansky–Roth criterion The equations of motion are solved for various values of the load parameter, thus obtaining system responses. The load parameter, at which there exists a large change in responses, is called critical. This approach is easily adapted to computational methods such as finite element codes and, therefore, it has become very popular in applications. In addition, it has met with success when applied to systems that exhibit violent buckling (limit point and bifurcation point with unstable post-buckling branch) under static loads. Cylindrical shells are typical violent buckling configurations, hence, this criterion is employed in the present work. In the dynamic analysis, the basic equations of motion of a cylindrical shell are _ C ½Kfug Z fFðtÞg ½Mfu€ g C ½Cfug where [M] is the mass matrix, [C] the damping matrix, [K] the stiffness matrix including _ the nodal velocity vector, fug € the non-linearity, {u} the nodal displacement vector, fug nodal acceleration vector and {F(t)} the load vector. The equations of motion are solved for various values of the loading and the value at which there is a significant jump in the response is assumed critical. When monitoring the system response through displacements of selected points for small values of the loading parameter, small oscillations are observed, the amplitudes of which gradually increases as the loading is increased. When the loading reaches its critical value, the maximum amplitude experiences a large jump. Therefore, implementation of this criterion requires to solve the equations of motion for different values of the loading parameter, then plot the displacement amplitude versus loading curve from which the critical loading value is determined.
6.2. Finite element analyses The critical conditions for dynamic buckling are so estimated by a sudden large change in dynamic responses, according to the Budiansky–Roth criterion [6]. Consequently, to get the dynamic critical load of a cylindrical shell subjected to suddenly applied axial compression with finite duration, it is necessary to solve the equations of motion for several load parameter values, so as to obtain the dynamic responses. The dynamic critical load is determined as the lowest load at which there is a large sudden change in transient response. At first some analyses were performed to compare the dynamic analyses performed with the implict method implemented in ABAQUS/Standard and with the explicit method implemented in ABAQUS/Explicit. The results show that both operators basically predict the same behaviour, but the explicit method ABAQUS/Explicit requires only a fraction of the CPU time used by the implicit scheme. Indeed, in the explicit dynamics procedure used by ABAQUS/Explicit, the use of small increments, dictated by the stability limit, is advantageous for dynamic buckling analysis, because it allows the solution to proceed without iterations and without requiring tangent stiffness matrices to be formed. Even if the analysis may take an extremely large
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number of increments, each increment is relatively inexpensive, resulting in an economical solution. Thus, all future investigations considering a dynamic analysis are performed using ABAQUS/Explicit. The axial compression is transferred to uniform nodal forces along the circumference and the load is suddenly applied with finite duration and constant in magnitude. Five different time durations are analysed: TZ1, 2.5, 5, 10, and 15 ms. The analyses are carried out with and without initial imperfections and with a mesh of the whole cylindrical shell of 220!52 4-nodes elements. To be able to get good results from the analyses, the points for which displacements are to be monitored are to be carefully chosen, otherwise the plots produced may be rather obscure and confusing. Instead of monitoring the radial displacements of several points from the midsection of the cylinder, where the largest displacement values were expected, the axial displacement is considered in this study. Fig. 3 presents the axial displacement of a point of the loaded side of the laminated cylinder [08/458/K458/08] without initial imperfection and subjected to suddenly applied axial compression of finite duration TZ10 ms with three load magnitudes. The curve corresponding to 132 kN is a regular response and the cylinder vibrates about its equilibrium position. From 132 to 134 kN, there is a great change between the two responses. This is the buckling condition and the load average equal to 133 kN is regarded as the dynamic buckling load of the shell. Using this criterion to estimate the dynamic buckling loads, dynamic analyses are performed for different time durations T.
Fig. 3. Axial displacement for load duration TZ10 ms—cylinder [08/458/K458/08].
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Table 4 Dynamic buckling loads versus load duration—Cylinder [08/458/K458/08] Load duration T (ms)
Dynamic buckling load (kN) (without imperfection)
Ratio of dynamic to static load (without imperfection)
Dynamic buckling load (kN) (with imperfection)
Ratio of dynamic to static load (with imperfection)
1 2.5 5 10 15
350 215 155 133 120
1.41 0.86 0.62 0.53 0.48
230 144 125 117 108
1.16 0.72 0.63 0.59 0.54
6.3. Results Dynamic buckling due to axial load with finite duration was investigated for the two fiber composite cylindrical shells, using the code ABAQUS/Explicit. At first, the cylinders are analysed without geometric imperfections, then the geometric imperfections measured on the real shells are introduced. The results of the analyses are reported for the cylinder [08/458/K458/08] and for the cylinder [458/K458]S in Tables 4 and 5, respectively, where they are also compared to the static buckling loads. The dynamic buckling loads versus load duration are then reported in Fig. 4 for the two lay-ups. It may be observed that the dynamic critical loads of a cylinder decrease quickly with increasing load duration and converge to a certain value. This value is defined as the critical load for the load case of infinite duration. The diagrams in Fig. 4 indicate also that shell stiffness increases for a short duration of loading. Indeed, for short time duration, the dynamic buckling loads are higher than the static buckling loads. As already shown by Zimcik and Tennyson [15] and by Eslami et al. [22], it is believed that these phenomena are due to the stress wave vibration between the impacted and fixed ends of the shell. This phenomenon is more evident in the case of the laminated cylinder [08/458/K458/08]. With increasing load duration, the dynamic buckling loads decrease quickly and get significantly smaller than the static ones. The present investigation gives smaller ratios of the dynamic buckling loads to the static ones in comparison with the results obtained by Huyan and Simitses [27], but the results are in good agreement to the ones obtained by Eslami et al. [22] and already performed by the author and Zimmermann on different composite cylindrical shells [36]. Table 5 Dynamic buckling loads versus load duration—Cylinder [458/K458]S Load duration T (ms)
Dynamic buckling load (kN) (without imperfection)
Ratio of dynamic to static load (without imperfection)
Dynamic buckling load (kN) (with imperfection)
Ratio of dynamic to static load (with imperfection)
1 2.5 5 10 15
135 118 102 85 81
1.12 0.98 0.85 0.71 0.67
120 107 93 70 66
1.03 0.92 0.79 0.60 0.57
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Fig. 4. Dynamic buckling loads versus duration: cylinders [08/458/K458/08] and cylinder [458/K458]s.
So in general the structures experience dynamic buckling loads larger than the static buckling loads, provided their duration is very short. If their duration is longer, the dynamic buckling load is smaller than the static one, which means that taking the static buckling load as the design point for dynamic problems might be misleading. The ratio between the dynamic buckling load and the static buckling load is of practical interest for the designer, since it provides a direct indication of the load carrying capacity of the structural elements exposed to rapidly applied loads relative to almost statically applied loads. Increasing the length of the loading period leads to a decrease in the ratio.
Fig. 5. Dynamic deformation—Cylinder [458/K458]s—Load duration TZ2.5 ms.
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Furthermore, the dynamic buckling loads, like the static ones, decrease with increasing imperfection amplitude. The sensitivity to initial geometric imperfection depends on the lay-up of the laminated shells. The imperfection amplitude has virtually the same effect on static critical load as on the dynamic critical load of infinite duration. Fig. 5 reports the dynamic deformation of laminated cylinder [458/K458]S for a load duration T equal to 2.5 ms.
7. Conclusion Dynamic buckling due to impulsive loading of thin-walled CFRP cylindrical shells has been investigated. The equations of motion approach, which associates the dynamic buckling of a structure with the state at which a small change in the magnitude of the loading leads to a large change in the structure response, has been successfully applied using the finite element code ABAQUS/Explicit. It has allowed to calculate the dynamic buckling loads for different values of load duration in a reliable and easy way. The dynamic buckling loads result strongly dependent on the load duration. Indeed for short time duration, the dynamic buckling loads are higher than the static ones, while by increasing the load duration, the dynamic buckling loads decrease quickly and get significantly smaller than the static loads. The initial geometric imperfections, measured on the internal surface of the real shells using a laser scanning system, have been introduced in the numerical models. The dynamic buckling loads, like the static ones, decrease because of the geometric imperfection and the laminate set-up of the shells influences this behaviour. The present study brings an important conclusion for design purposes. Since the common practice is to assume that dynamic bucking loads are higher than the static ones, which means that static design is safe, careful design is recommended. Indeed, taking the static buckling load as the design point for dynamic problems might be misleading. These conclusions are based on a numerical study, that will be confirmed by further computations and by experimental dynamic buckling tests with different values of load duration so to generate widely applicable design criteria for dynamic buckling of composite shells. For this aim work is currently under way on performing experimental dynamic buckling tests by means of a deceleration horizontal sled normally used for crash tests. The load intensity and the length of the loading period can be altered by changing the impact velocity and by increasing or decreasing the mass of the sled. The results of the tests will be reported in due time.
Acknowledgements The author wishes to thank Dr Rolf Zimmermann of DLR (German Aerospace Center), Institut fu¨r Strukturmechanik, Braunschweig, Germany, for having introduced to the interesting field of dynamic buckling of composite shells, through a research supported by the European Union under a Marie Curie Grant inside the programme Training and Mobility of Researcher (Contract No. ERBFMBI CT97 2465).
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