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Dynamic buckling of cylindrical stringer stiﬀened shells R. Yaﬀe, H. Abramovich

*

Faculty of Aerospace Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel

Abstract The dynamic buckling of cylindrical stringer stiﬀened shells was investigated both numerically and experimentally. A new criterion to deﬁne the numerical ‘‘dynamic’’ buckling load was developed yielding consistent results. The ADINA ﬁnite element code was applied to simulate the static and dynamic buckling loads of the shells. It was shown numerically that when the period of the applied loading (half-wave sine) equals half the lowest natural period of the shell, there is a slight drop in the dynamic load ampliﬁcation factor (DLF). The DLF is deﬁned, as the ratio of the dynamic buckling to the static buckling of the shell. This factor drops below unity, when the ratio of the given sound speed in solids, c, to the velocity developed axially due to the applied dynamic loading, approaches unity. It means that, for this particular loading period, the dynamic buckling load would be lower than the static one. It was shown numerically that the shape of the loading period, half-wave sine, a shape encountered during the tests, as well as the initial geometric imperfections have a great inﬂuence on the dynamic buckling of the shells. The relatively simple test set-up design to cause a shell to buckle dynamically did not fulﬁll our expectations. Although, the process leading to eventually the dynamic buckling of the shell worked properly, still no test results were obtained to form a sound experimental database for this phenomenon. Based on the numerical predictions, correct guidelines were formulated for better test procedures to be applied in future tests, which will be reported in due time. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: ADINA code; Dynamic buckling; Impact; Stringer stiﬀened shells; Tests; DLF

1. Introduction The need to design structures that have to withstand time dependent dynamic loads, sometimes quite severe, and thus may be susceptible to ‘‘dynamic buckling’’, also referred to in literature as dynamic stability or dynamic instability is the driver behind the increasing interest in this phenomenon. A few excellent monographs treating the various aspects and issues of the dynamic instability can be found for instance in Refs. [1–5]. Though many classes of problems and many physical phenomena are encompassed by the term dynamic buckling, the term dynamic buckling has in particular been assigned in the

* Corresponding author. Tel.: +972-4-829-2308; fax: +972-4829-3193. E-mail addresses: [email protected] (R. Yaﬀe), [email protected] (H. Abramovich).

literature to two essentially diﬀerent phenomena (see Refs. [2,5]). One is associated with the response of a structure to the action of oscillating loads, namely, vibration buckling, where the transverse vibrational deﬂection becomes unacceptably large at critical combinations of load amplitude, load frequency and damping. For the case of a column, it can be shown that for sufﬁciently small values of the axial force, vibration buckling (resonance) will occur when the loading frequency is twice the lowest natural bending frequency of the column. This column behavior is often called ‘‘vibration buckling’’ or ‘‘parametric resonance’’ and was extensively studied (see Refs. [1,3,6]). The second phenomenon, which is the subject of the present study, relates to the behavior 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 pulse, i.e. dynamic buckling under impact loads (see Refs. [3,5] for detailed

0045-7949/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0045-7949(02)00417-0

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reviews). It was shown that the permissible load intensity is load duration dependent which reveals a special feature of pulse buckling, namely, the load determines the mode of buckling. Consequently, whereas in static buckling the buckling mode is known and maximum safe load is determined, in pulse buckling the load amplitude is prescribed and dictates the buckling modes, thus determining the maximum safe duration of its application. For the last three decades many researchers have extensively studied the phenomenon of pulse buckling experimentally and theoretically with main emphasis on crashworthiness, energy absorption and nuclear containment structures (see for example Refs. [7–10]). Basic structures like, bars, plates and shells were investigated and various dynamic buckling criteria were applied (see for example Refs. [11–15]). The dynamic buckling of metal and laminated composite columns and plates has been studied intensively at the Aerospace Structures Laboratory, Faculty of Aerospace Engineering, Technion, I.I.T., Haifa, Israel for the last two decades (see for example Refs. [4,5,16–20]). The research included both analytical and experimental studies yielding a very favorable agreement between test and analytical predictions. It was shown, that in general the tested structures experience dynamic buckling loads larger than the relevant static buckling loads, provided their duration is very short. The dynamic load ampliﬁcation factor (DLF), deﬁned, as the ratio of the dynamic buckling to the static buckling of the structure was greater than unity. However, for those cases with loading frequency close to the lowest bending frequency of the structure, one would ﬁnd a DLF less than unity, namely 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. To check the applicability of this important conclusion to more complex structures it was decided to extend the research to stiﬀened and composite laminated shells. In the present study, the buckling of aluminum cylindrical stringer stiﬀened shells under axial dynamic applied loading is investigated, both numerically and experimentally. Like in the previous studies, the Hutchinson– Budiansky dynamic buckling criterion [21] is applied in this study to determine the DLF. According to this criterion, which in principle resembles the one for the static buckling case, the maximum response of a structure Rmax (in our case the lateral displacement) vs. a load parameter k (the axial load) is plotted, and the critical load parameter kcr is deﬁned as the middle, more or less, of the narrow k over which Rmax rises steeply. One should note that although this analogy with static buckling is not quite complete since the loading and the response are time dependent, the above dynamic buckling criterion yielded consistent and reliable results when applied both experimentally and analytically and therefore was implemented throughout this study.

A ﬁnite element model for a shell was built using the ADINA [22] code to simulate the experimental case used throughout the present test series. The shell is impacted through the upper end plate and pairs of strain gages being glued circumferentially at the mid-height measure its response. The amplitude of the impact was governed by the height of the dropping mass, while the duration depends on the ratio of striking mass to the upper end plate mass. During the tests, the shape of the duration almost resembled the one of a half-sine and therefore numerical predictions were computed for a rectangular shape and half-wave sine. A parametric investigation was then performed to ﬁnd the dynamic buckling loads of the impacted shells and their respective loading durations to yield a DLF less than unity.

2. The ﬁnite element model The ﬁnite element model employed in the ADINA code, consisted of a quarter of the shell, as can be seen in Fig. 1. The boundary conditions applied on the curved edges were fully clamped, with one edge ‘‘allowed’’ to move axially, while on the straight edges, symmetry conditions were applied. The stringers were modeled as beams using the Isobeam element. The skin of the shell was modeled using Shell elements [22–24]. Between every two stringers two shell elements were employed. The total number of elements on the circumference was 42, with 18 elements distributed axially. The applied load had a shape of half-wave sine having a length of T seconds, with the loads being modeled as concentrated ones at the nodes of the elements on the boundaries. The initial geometric imperfection shape used to calculate the buckling loads due to imperfections, had the shape of the calculated static buckling mode. The critical ampliﬁcation criterion, gmax , was used to deﬁne both the static and dynamic buckling loads. The value used for gmax was 10 (gmax ¼ zmax =w0 , zmax ða lateral displacementÞ P 0:3 mm, almost the value of the shell

Fig. 1. The ADINA ﬁnite element model.

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Table 1 Dimensions, material properties, buckling loads and lowest bending frequencies of a typical tested shell Shell

KR-4

Material YoungÕs modulus, E [Mpa] Density, q [kg/mm2 ] Thickness, h [mm] Length, a [mm] Radius, R [mm] Stringers details Boundary conditions Theoretic buckling load [N] (ADINA) Theoretic buckling load [N] (with imperfection of w0 =h ¼ 0:1) (ADINA) Experimental buckling load [N] Theoretical lowest frequency [Hz] (without imperfection) (ADINA) Experimental lowest frequency [Hz] Velocity of sound in aluminum, c [m/s]

Aluminum E ¼ 75, t ¼ 0:3 2730 0.242 200 120.12 84 stringers, height ½mm ¼ 1:505, width ½mm ¼ 0:900 Clamped–Clamped 46,700 n=m ¼ 10=1 40,300 n=m ¼ 10=1 36,000 n=m ¼ 8=1 653.4 n=m ¼ 10=1 584 n=m ¼ 8=1 5240

thickness, h ¼ 0:242 mm (see Table 1) and w0 is the amplitude of the initial geometric imperfection). For the dynamic cases, we have used the Dynamic-Direct-Integration, employing the Newmark method with an automatic time-step [23,24].

3. Test specimens and experimental set-up The test specimens and experimental set-up are described in details in Ref. [25]. For clarity, the main points of the experimental process are summarized again. The dimensions and properties of one typical shell, KR-4 (see Fig. 2), an aluminum externally stiﬀened stringered cylindrical one, are shown in Table 1. The shell was clamped to two loading plates and placed under the loading tube, in which a dropping mass would impact the upper loading plate of the shell (see Ref. [25]). To measure the response of the shell, three pairs of strain gages were bonded face to face on each side of the circumferential mid-plane of the shell (120° apart). The responses of the strain gages are stored on a PC using a data logger for the dynamic strain acquisition (see Fig. 3). To monitor the test and to measure online the actual period of the striking mass, as experienced by the tested shell, a memory scope was added to the test set up. The data stored on the PC is used to calculate the bending and compression strains related to the dynamic load. By plotting a graph of the bending strains vs. the compression strains and applying the Hutchinson–Budiansky dynamic buckling criterion [21], the dynamic buckling strain of the tested shell can be evaluated. The length of the loading period (see Fig. 3(c)), which has a shape similar to a half-sine, can be altered by increasing/decreasing the dropping mass, by

Fig. 2. A typical specimen, shell KR-4.

changing the mass of the upper loading plate (see a discussion on this topic further on) and by adding a rubber layer on top of the upper loading plate. A schematic view of the experimental procedure and test set-up is presented in Fig. 3(a)–(e). Prior to the dynamic tests, two preliminary tests are performed: (a) First the shell is vibrated or alternatively lightly impacted to obtain its lowest bending frequency, fb , yielding the period of natural vibration, Tb .

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Fig. 3. A schematic view of the experimental procedure and test set-up: (a) the shape of the compression strains, (b) the shape of the bending strains, (c) the bending strains vs. compression strains––determination of buckling, (d) the data acquisition system, (e) the specimen.

T ¼ 1120 ls. Next the response of the strain gages are plotted as compression vs. bending strains, as shown in Fig. 5(a) and (b) for a period of Tav ¼ 1200 ls and Tav ¼ 1120 ls, respectively, as compared to 0.5 Tlowest frequency ¼ 765 ls (half the period of the lowest bending frequency of the KR-4 shell). The main problem encountered during the tests was the relatively large ﬂuctuations in the impact period as recorded by the strain gages for various dropping heights, yielding a relatively large error, combined with the inability to tune the impact

(b) Secondly, a nondestructive static buckling is performed to obtain the static buckling load, and using the bonded pairs of strain gages to get the relevant static buckling strain ecr-st . This test is terminated after incipient elastic buckling would occur, to ensure that no plastic strains are experienced. Typical results, as recorded by the three pairs of strain gages, are shown in Fig. 4, for a loading period of

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period to ﬁt the region of half the lowest experimental period (2Timpact =Texp 1), this in contradiction to our previous experience with ﬂat plates (see Refs. [20,25]).

4. Results and conclusions The numerical and experimental results for a typical shell, KR-4, are presented in Table 1. The linear numerical buckling load, as calculated using ADINA code, was found to be 46,700 N (see Table 1 and Fig. 6), while the nonlinear buckling load, calculated with an imperfection amplitude of w0 =h ¼ 0:1 (the shape taken was the mode of the linear buckling load), was 40,300 N (see Table 1 and Fig. 7). The mode shape of the buckling

Fig. 5. Compression vs. bending strains––typical results (shell KR-4): (a) Tav ¼ 1200 ls, (b) Tav ¼ 1120 ls. Fig. 7. The mode shape of the nonlinear buckling load (with initial geometric imperfection of w0 =h ¼ 0:1 (Pcr ¼ 40,300 N).

Fig. 6. The mode shape of the linear buckling load (Pcr ¼ 46,700 N).

Fig. 8. The lowest bending mode f ¼ 653:4 Hz (shell KR-4).

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with geometric imperfections is used as initial geometric imperfections for the calculation of the dynamic buckling loads. The lowest natural frequency was found at 653.4 Hz with 10 circumferential waves and half axial wave (see Fig. 8). Fig. 9(a)–(e) present the shape of modes for higher frequencies of 2522, 5462, 5515, 5546

and 5734 Hz, respectively. Fig. 10(a)–(e) show typical results of the buckling modes at various periods of the dynamic loading and their respectively dynamic buckling loads (T ¼ 2500 ls with Pcr ¼ 43,700 N, T ¼ 1000 ls with Pcr ¼ 70,600 N, T ¼ 400 ls with Pcr ¼ 75,600 N, T ¼ 80 ls with Pcr ¼ 25,200 N and T ¼ 30 ls with

Fig. 9. Mode shapes for higher frequencies: (a) f ¼ 2522 Hz, (b) f ¼ 5462 Hz, (c) f ¼ 5515 Hz, (d) f ¼ 5546 Hz, (e) f ¼ 5734 Hz.

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Pcr ¼ 117,600 N). Comparing the dynamic buckling modes with the modes at high frequencies, reveals a striking similarity in their shapes. This in contradiction with the static buckling mode shape (see Fig. 6) or the lowest natural mode shape (see Fig. 8).

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Next, graphs of the DLF vs. the period length of the dynamic loading were generated. The DLF is deﬁned, as the ratio of the dynamic buckling to the static buckling of the shell. Fig. 11(a) presents the DLF vs. the nondimensional period, T 0 , deﬁned as T 0 ¼ cT =2a, where T is

Fig. 10. Buckling modes at various periods of the dynamic loading and their respectively dynamic buckling loads: (a) T ¼ 2500 ls with Pcr ¼ 43,700 N, (b) T ¼ 1000 ls with Pcr ¼ 70,600 N, (c) T ¼ 400 ls with Pcr ¼ 75,600 N, (d) T ¼ 80 ls with Pcr ¼ 25,200 N, (e) T ¼ 30 ls with Pcr ¼ 117,600 N.

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the period of the dynamic loading, a is the length of the shell and c is the speed of the sound in the shell (see Table 1 for values of the various constants). Fig. 11(b) presents the DLF vs. the nondimensional period, T 0 , for lower values of T 0 . The same data is presented in Fig. 12(a) and (b), in a traditional manner (see Ref. [25]), namely DLF vs. the nondimensional period, s, deﬁned as s ¼ 2T =Tb , where, as before, T is the period of the dynamic loading and Tb is the period of the lowest natural bending frequency of the shell. As shown in earlier studies (see Refs. [20,25]), we get a slight drop in the DLF at s ¼ 1, meaning those cases with loading frequency close to the lowest bending frequency of the shell. However this drop is not below unity, as was shown in the cases of columns and plates (see Ref. [20]). For T 0 > 7:5, we get buckling modes with a large number of circumferential waves and a small number of axially waves, resembling the static buckling mode shape, and thus yielding a DLF P 1. In the region where, T 0 < 2:4 we get buckling modes with a large number of circumferential waves with no axially waves, resembling the dynamic buckling mode shape, which brings the DLF below unity (DLF < 1). A DLF less than unity, namely, the dynamic buckling load is smaller than the static one, means that taking the static buckling load as the design point for dynamic problems might be misleading. Unlike our experience with columns and plates (see Ref. [20]), for cylindrical stiﬀened shells, a

DLF < 1 occurs at T 0 ¼ 1, and not at s ¼ 1. This can be explained as follows: the half-sine wave shape of the dynamic loading applied axially to the shell is able to excite mainly those frequencies above its own frequency. When the dynamic buckling mode shape would be similar to a vibration mode, a DLF less than unity is expected to appear. For the present cylindrical stringer stiﬀened shell, this phenomenon will happened at low periods, namely at high natural frequencies, due to the fact that their respectively mode shapes are similar to the dynamic buckling mode shape. The relatively simple test set-up design to cause a shell to buckle dynamically, did not fulﬁll our expectations. Although, the process leading to eventually the dynamic buckling of the shell worked properly, still no test results were obtained to form a sound experimental database for this phenomenon. Based on the present numerical predictions, the correct region of the loading period was found for better test procedures to be applied in future tests, which will be reported in due time.

Acknowledgements The authors would like to thank the German Israeli Foundation (GIF) for its generous support for this research. They also would like to thank Mr. A. Grunwald for designing, preparing and performing the tests.

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