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Buckling of cylindrical shells under static and dynamic shear loading
Engineering Structures 22 (2000) 535–543 www.elsevier.com/locate/engstruct
Buckling of cylindrical shells under static and dynamic shear loading G. Michel *, A. Limam, J.F. Jullien URGC Structures, INSA LYON, 69621 Villeurbanne, France Received 21 November 1997; received in revised form 13 November 1998; accepted 13 November 1998
Abstract The aim of this study is cylindrical shell behavior under seismic loading. To this objective, static and dynamic tests and a vibration analysis were performed. At the same time, Finite Element simulations enabled us to understand the effect of frequency excitation. When the frequency excitation is close to first eigen frequency, a coupling between buckling and vibration modes occurs if the load level is equal to 70% of the critical static load. This induced instability leads to large deformations and a great decrease in shell stiffness. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Cylindrical shell; Buckling; Dynamic shear loading; Coupled instability
Nomenclature M: R: L: t: E: : 0.2: Fcr: ␦cr: u0: 0: 1: : Pd: Ps:
Mass at the top of the shell Radius of the shell Length of the shell Thickness of the shell Young’s modulus Poisson’s coefficient Yield stress Static buckling load Static buckling displacement Displacement amplitude Frequency associated with buckling mode Dominant frequency of seismic excitation Dynamic amplification factor Dynamic buckling load Static buckling load
1. Introduction Cylindrical shell buckling under seismic loading has been a subject of great interest since the latest earthquakes. These cylindrical shells are often filled with liquid, so fluid–structure interactions and sloshing prob* Corresponding author. Tel.: ⫹ 33-4-72-43-83-33; fax: ⫹ 33-472-43-85-23; e-mail: [email protected]
lems have priority for many researchers. In our case we do not take into account the fluid action, we are only interested in empty cylindrical shell behavior under a shear dynamic load. Static cylindrical shell shear buckling was first investigated by Lundquist [1] in the thirties. Galletly and Blachut [2] also made experimental plastic shear buckling tests. However, the first dynamic shear buckling tests were carried out by Kokubo [3] at the end of the eighties. Experimental dynamic tests have been done with a shaking table and dynamic buckling was not well defined. They observed a sudden change in the deformation mode similar to the case of the static buckling beyond a certain limit of the acceleration of the shaking table. Buckling was determined by the visual inspection of the cylinder, a change in the strain wave and by sound generation. At the initiation of buckling the occurrence of a change in sound was induced by the normal displacement w of the shell which corresponded to the shell vibration mode at higher frequency. Comparing Mx¨ with the static buckling load and ␦x with the displacement of the stability limit the initiation of buckling is nearly the same as that of static buckling, regardless of the frequency of the shaking table. In many studies, a shaking table is used and the load is assimilated to the inertial load Mx¨, taking also into account the nonlinear effect with the intermediary of a correction factor .
0141-0296/00/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 1 - 0 2 9 6 ( 9 8 ) 0 0 1 3 2 - 1
536
G. Michel et al. / Engineering Structures 22 (2000) 535–543
(0) with a Rayleigh method, then to calculate the ratio between predominant seismic frequency (1) and (0). The dynamic factor amplification ( ⫽ Pd/Ps) can be found on the curve. But many questions still remain for empty cylindrical shells subjected to shear load: —does the critical dynamic load depend on frequency excitation? —what is the effect of geometrical imperfections and preloading on eigen frequency? —does coupling between the first beam and the buckling modes occur?
Fig. 1.
Dynamic design rule proposed by Combescure [5].
Kokubo et al. [4] demonstrated the feeble effect of an initial imperfection on the buckling load compared to a cylindrical shell under compression or external pressure. A rule (Fig. 1) for seismic design, in the case of a cylindrical shell subjected to radial dynamic pressure, was proposed by Combescure [5]. First, he proposed calculating the buckling mode and the associated frequency
Fig. 2.
Firstly, static tests, for reference, were conducted. Secondly, we carried out a numerical vibration analysis of the whole structure, in which we studied top mass, geometrical imperfection and shear load influences. Then, dynamic tests were done and compared with the Finite Element simulations and the static tests results.
2. Experimental machine test For our study, the radius–shell thickness ratio is equal to 450 and the length–radius ratio is 1. We have used an electroplating nickel shell so that there will be very little initial imperfection (the thickness is equal to 270
Experimental test machine.
G. Michel et al. / Engineering Structures 22 (2000) 535–543
m) with material property close to 316L steel. At each extremity, the shell is connected to rigid inserts to provide a boundary condition close to an embedment. The material properties are measured for each specimen and we take the mean values ( ⫽ 0.3, E ⫽ 180 000 Mpa, 0.2 ⫽ 450 000 Mpa) for finite element simulations. A special dynamic buckling test machine has been designed (Fig. 2). In our tests, the bottom of the shell is clamped to a rigid base plate and the top is connected (with a rigid plate) to a horizontal hydraulic ram which applies a controlled displacement load. A constant tensile load, simulating the effect of the self-weight of the shell in the case of a hanging shell, may be applied with another hydraulic ram. An automatic system enables us to measure the initial imperfection and the buckling deformations during static tests. A laser sensor is used for the radial displacement measurement. Shear load, tensile load, displacement and accelerations are measured at the top of the shell. A special internally gauged load sensor, linked to the ram and the rigid top plate, enables us to measure the load without the inertia mass effect during dynamic tests. A high speed CCD camera is used for the visualization of buckling and vibration modes during high frequency tests.
Fig. 3.
537
First eigen mode (80 Hz).
calculations, we use the modified RIKS method. For dynamic simulations, we do not use modal analysis but we make a direct integration with an implicit operator. We also take into account a small initial imperfection, affine to the buckling mode, during dynamic simulations.
3. Finite element codes We use two different finite element codes for test simulations: —the first one, INCA code of the CASTEM system developed at the CEA [6], we use a quasi axisymmetric shell element (COMU). This element is a two node flat MINDLIN shell element; at each node the geometry is defined by the r, z coordinates and an initial non-axisymmetric displacement. The initial imperfection is given on a set of Fourier circumferencial harmonics defined by the user. The displacement response is decomposed in the Fourier series and the plasticity is evaluated on discrete points around the circumference. In order to take into account the added mass of the thick top plate in dynamic tests, the top plate is meshed. In dynamic analysis, we make an incremental calculation, also taking into account a small initial imperfection ( ⬍ 10% of the thickness) affine to the buckling mode in order to simulate the post-buckling behavior. —in the second one, ABAQUS [7], we use a 3D modeling. We choose the isoparametric S8R5 shell element. It is an 8 nodes doubly curved thin shell element with reduced integration and five degrees of freedom per node. In order to simulate the post-buckling behavior, during static nonlinear
4. Vibration analysis We perform a shell vibration analysis in order to predict the dynamic behavior of the structure. Eigen modes and eigen frequency are defined by shell dimensions [8]. The first eigen mode (Fig. 3) is a beam mode (shear mode). If we take into account the rigid top plate mass (30 kg), the corresponding eigen frequency is equal to 80 Hz. Higher eigen modes (shell modes) (Fig. 4) are not influenced by this added mass. Parametric calcu-
Fig. 4.
Shell mode (mode 12) without shear load (930 Hz).
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G. Michel et al. / Engineering Structures 22 (2000) 535–543
Table 1 Eigen frequency analysis, effect of initial imperfection amplitude Imperfection No mass amplitude (m)
Rigid plate mass 38 kg
Mode type
0
10
20
100
200
300
500
1000
231 433 587 842 943 1013 1100 1200 1315 1503 1515 1520 1570 1640
80 188 473 839 943 1013 1100 1200 1315 1503 1515 1520 1570 1640
80 188 473 839 944 1011 1098 1202 1320 1497 1511 1515 1560 1630
79 188 473 837 975 1028 1113 1218 1356 1505 1517 1530 1653 1842
78 188 473 834 1035 1067 1156 1268 1442 1519 1529 1579 1611 1741
77 187 473 831 1079 1152 1210 1341 1514 1530 1551 1646 1720 1830
74 187 473 825 1125 1223 1389 1485 1550 1602 1662 1810 1843
67.1 184 473 810 1226 1400 1433 1637 1803 1822 1954 1958 2015
Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 14
lations are made in order to understand the effect of an initial imperfection (affine to the buckling mode) on eigen modes and the effect of an imposed shear displacement. In the case of an imperfection amplitude analysis (Table 1) we notice that: —eigen frequencies of mode 1 type are not modified by an initial imperfection. —eigen frequencies of the imperfect shell mode types which contain an initial imperfection increase with the amplitude imperfection but these changes are small for an initial imperfection amplitude smaller than the shell thickness. —the mode shapes are all coupled with the deformation associated with the initial imperfection.
Shear Torsion Extension Bending Mode (1, 14) Mode (1, 15) Mode (1, 16) Mode (1, 17) Mode (1, 18) Mode (2,15) Mode (2, 14) Mode (2, 16) Mode (2, 17) Mode (2, 18)
the buckling mode does not influence the vibration modes and eigen frequencies in our case. The second parameter in this vibration analysis is a shear displacement imposed before the eigen frequency calculation: —the first eigen frequency (beam mode) does not change (Fig. 5) whatever the amplitude displacement was. —all shell mode frequencies decrease, the mode shapes change. When the imposed displacement is close to the static critical one, the first shell mode is similar to the buckling mode (Fig. 6) and the first shell frequency is close to 0. The eigen frequency modification, induced by imposed displacement, is not as great as the stiffened
So, we can conclude that initial imperfection affine to
Fig. 5. Shell mode (411 Hz) with an imposed shear displacement of 80% of the critical one.
Fig. 6. tude.
Eigen frequency evolution with displacement imposed ampli-
G. Michel et al. / Engineering Structures 22 (2000) 535–543
539
shell under axial compression [9] because there is a multimodal buckling mode. Nevertheless, for an imposed displacement equal to 97% of the static critical one, the first shell frequency is equal to the shear mode frequency. 5. Static results Experimental buckling tests are conducted under a controlled displacement load. The load displacement curve shows bilinear behavior with stable post critical behavior (Fig. 7). The feeble initial imperfection effect for shear loading can explain stable post-buckling behavior, Koiter [10] links post critical stability to initial imperfection sensitivity. The buckling mode (Fig. 8) is representative of a shear buckling mode with inclined wrinkles in each side of the shell. There is a multimodal buckling mode (mode 10 to 21), with a predominance of modes 14 to 18. The tension load has little effect on the buckling load, on the other hand it contributes to an increase in stability in the post critical area. After a test, small residual deformations affine to the buckling mode still remain; these plastic deformation amplitudes are less than 20 m. If we carry out cyclic buckling tests (up to 120% of the critical displacement) these deformations do not increase and the buckling load is almost the same as the first one: we observe an elastic global behavior and the plastic deformations are just limited to the top of the waves. There is also a smoother load-displacement curve during cyclic tests (Fig. 9). The small residual deformations affine to the buckling mode, induced by the first cycle, explain this behavior. For simulations, we obtain material characteristics by tensile tests and the shell thickness is precisely measured.
Fig. 7. Static load-displacement curve.
Fig. 8.
Static experimental buckling mode.
Fig. 9.
Cyclic loading.
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G. Michel et al. / Engineering Structures 22 (2000) 535–543
Fig. 10.
INCA simulations, effect of initial imperfection amplitude.
With INCA we use 20 axisymmetric COMU elements in order to mesh the shell. Each rigid insert is modelised with 1 element and the rigid top plate with 8 elements. We choose a modal decomposition with modes 1 and 10 to 20 in order to have the better response in the case of linear buckling analysis. For the nonlinear cases it is necessary to take into account an initial imperfection affine to the first buckling mode in order to simulate the post buckling behavior. The parametric INCA simulations (Fig. 10) on the effect of an initial imperfection (affine to the buckling mode) show that shear buckling is less imperfection sensitive than axial compression or pressure loading. With ABAQUS, 1750 elements S8R5 are necessary to mesh half (vertical symmetry) of the shell, the rigid top plate and rigid inserts. RIKS calculations (Fig. 11)
enable us to find a correct buckling mode (Fig. 12) and simulate the post critical behavior without initial imperfection. Finite Element simulations (INCA or ABAQUS) enable us to find a buckling load equal to 95% of the experimental one.
Fig. 11. ABAQUS non-linear simulation RIKS method without initial imperfection.
Fig. 12. Buckling mode ABAQUS simulation, RIKS method, without initial imperfection.
6. Dynamic results During dynamic tests, the displacement amplitude is fixed (Eq. (1)) and a sweep frequency is done. u ⫽ u0 sin (t)
(1)
First, the displacement amplitude u0 is equal to 10% of the static critical one. For each excitation frequency (1 to 100 Hz) several cycles are done. We are able to
G. Michel et al. / Engineering Structures 22 (2000) 535–543
Fig. 15. Fig. 13. Shear load and radial displacement evolutions according to the excitation frequency (u0 ⫽ 0.1␦cr) FFT signals.
plot the shear load, one point radial displacement and the shear displacement versus the frequency excitation (Fig. 13). When we reach 73 Hz we notice that shear load decreases and radial displacement increases. For frequencies between 75 and 80 Hz the load level increases and for higher frequencies the load level remains weak. Load drop is not an inertial load effect because it is not as regular as the ram load drops. The transfer function (u0/F) (Fig. 14) confirms this behavior. A second test is performed on the same specimen with a displacement amplitude equal to 70% of the static critical one (cyclic loading does not affect the shell behavior). We notice a higher load drop when first beam eigen frequency is reached but the radial displacement ampli-
Fig. 14.
Ratio u0/F.
541
Shear load and radial displacement evolutions according to excitation frequency (u0 ⫽ 0.9␦cr) FFT signals.
fication is higher (Fig. 15). The ratio u0/F (Fig. 15) is similar to the vibration test one. The dynamic test visualizations enable us to understand the shell behavior and explain the load curves: —for low load level there is always a beam mode during sweep frequency. So, there is just a classical vibration response with a weak load decrease when the first beam eigen frequency is reached. —with a load level equal to 70% of the static critical load, the critical geometry appears to be a coupled mode between the buckling mode and the shell eigen mode (Figs. 16 and 17). This happens only for excitation frequencies close to the first beam eigen frequency and consequently the shell stiffness becomes very low. The measurement of residual deformation (Fig. 18)
Fig. 16. Coupled instability mode: excitation frequency ⫽ 80 Hz, u ⫽ 0.7u0cr.
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G. Michel et al. / Engineering Structures 22 (2000) 535–543
never reaches the static critical one so this imposed load is too small for buckling to occur. So, we can plot the minimum input load level necessary for buckling to occur for each frequency (Fig. 19), this curve is nearly the same as the experimental shear load curve (Fig. 15).
Fig. 17. Coupled instability mode: excitation frequency ⫽ 80 Hz, u ⫽ 0.7u0cr.
Fig. 19. Minimum load necessary to obtain buckling in load control calculation.
Fig. 18. Load evolution at each cycle INCA dynamic load control simulation.
shows that a modal wave (mode 14) remains on the shell in addition to the buckling mode. This modal wave is due to coupled instability which produces local plastic deformations. In finite element simulations displacement or load control calculation are both available. In the case of a displacement control simulation we can only calculate the reaction load. It is necessary for there to be a displacement amplitude equal to the critical static one to obtain the shell buckling, whatever the excitation frequency is. We cannot calculate the applied load at the top of the shell in the same way as the measured load during the experimental test. In load control calculation, a load associated to an excitation frequency is applied at the top of the shell and we observe the time evolution of displacements and of the reaction load. Sometimes the top displacement increases at each cycle and the critical displacement is reached after 5 cycles (Fig. 18). For other imposed loads, the top displacement remains constant at each cycle and
The deformations, when coupled instability occurs, are the same as experimental ones with a mixed buckling and vibration mode (Fig. 20).
Fig. 20.
Finite Element simulation (load control, f ⫽ 80 Hz).
G. Michel et al. / Engineering Structures 22 (2000) 535–543
7. Conclusion This study on dynamic instabilities enables us to understand better cylindrical shell behavior under shear dynamic loading. Shear buckling insensitivity to geometrical imperfection is confirmed. The Finite Element vibration analysis underlines the effect of a shear preload on eigen modes and eigen frequencies. Dynamic tests and Finite Element simulations clearly exhibit that a coupled instability between shell eigen mode and buckling mode can occur, and in this case large deformations are generated. In this case, the necessary critical input load is only equal to 70% of the critical static one. Consequently, it is very important to take into account two parameters for dynamic instability studies: the frequency excitation and the load level. Another significant parameter is the closeness of two eigenmodes.
Acknowledgements Authors express their thanks to E.D.F Septen (Electricite´ de France, Service Etudes Projets Thermiques et Nucleaires), NOVATOME and C.E.A. (Commisseriat a` l’Energie Atomique) for their financial support and for their technical contribution to this study.
543
References [1] Lundquist EE. Strength tests of thin-walled duralumin cylinders in combined transverse shear and bending. NACA Technical note No. 523, 1935. [2] Galletly GD, Blachut J. Plastic buckling of short vertical cylindrical shells subjected to horizontal edge shear loads. Journal of Pressure Vessel Technology 1985;107:101–6. [3] Kokubo K, Nakamura N, Madokoro M, Sakurai A. Buckling behaviors of short cylindrical shells under dynamic loads. SMIRT 9 1987;E:167–72. [4] Kokubo K, Nagashima H, Takayanagi M, Mochizuki A. Analysis of shear buckling of cylindrical shells. JSME International Journal Series A 1995;153:305–17. [5] Combescure A. Buckling under time-dependent loads. Rapport interne CEA DEMT/90-043, 1989. [6] Combescure A. Contribution a` l’e´tude de la stabilite´ des coques minces sous chargements complexes dans INCA. Rapport interne CEA DMT/94-450, 1994. [7] Hibbit, Karlsson, Sorensen. ABAQUS theory manual, 1995. [8] Liew KM, Hung KC, Lim KM. Vibration of stress-free hollow cylinders of arbitrary cross section. ASME Journal of Applied Mechanics 1995;62(3):718–824. [9] Singer J, Weller T, Abramovich H. The influence of initial imperfections on the buckling of stiffened cylindrical shells under combined loading. In: Jullien JF, editor. Buckling of shell structures on land, in the sea and in the air. Amsterdam: Elsevier Applied Science, 1991:1–10 [10] Koiter WT. A consistent first approximation in general theory of buckling of structures. IUTAM Symposium, Harvard University, June 1974, 133–47.
Buckling of cylindrical shells under static and dynamic shear loading G. Michel *, A. Limam, J.F. Jullien URGC Structures, INSA LYON, 69621 Villeurbanne, France Received 21 November 1997; received in revised form 13 November 1998; accepted 13 November 1998
Abstract The aim of this study is cylindrical shell behavior under seismic loading. To this objective, static and dynamic tests and a vibration analysis were performed. At the same time, Finite Element simulations enabled us to understand the effect of frequency excitation. When the frequency excitation is close to first eigen frequency, a coupling between buckling and vibration modes occurs if the load level is equal to 70% of the critical static load. This induced instability leads to large deformations and a great decrease in shell stiffness. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Cylindrical shell; Buckling; Dynamic shear loading; Coupled instability
Nomenclature M: R: L: t: E: : 0.2: Fcr: ␦cr: u0: 0: 1: : Pd: Ps:
Mass at the top of the shell Radius of the shell Length of the shell Thickness of the shell Young’s modulus Poisson’s coefficient Yield stress Static buckling load Static buckling displacement Displacement amplitude Frequency associated with buckling mode Dominant frequency of seismic excitation Dynamic amplification factor Dynamic buckling load Static buckling load
1. Introduction Cylindrical shell buckling under seismic loading has been a subject of great interest since the latest earthquakes. These cylindrical shells are often filled with liquid, so fluid–structure interactions and sloshing prob* Corresponding author. Tel.: ⫹ 33-4-72-43-83-33; fax: ⫹ 33-472-43-85-23; e-mail: [email protected]
lems have priority for many researchers. In our case we do not take into account the fluid action, we are only interested in empty cylindrical shell behavior under a shear dynamic load. Static cylindrical shell shear buckling was first investigated by Lundquist [1] in the thirties. Galletly and Blachut [2] also made experimental plastic shear buckling tests. However, the first dynamic shear buckling tests were carried out by Kokubo [3] at the end of the eighties. Experimental dynamic tests have been done with a shaking table and dynamic buckling was not well defined. They observed a sudden change in the deformation mode similar to the case of the static buckling beyond a certain limit of the acceleration of the shaking table. Buckling was determined by the visual inspection of the cylinder, a change in the strain wave and by sound generation. At the initiation of buckling the occurrence of a change in sound was induced by the normal displacement w of the shell which corresponded to the shell vibration mode at higher frequency. Comparing Mx¨ with the static buckling load and ␦x with the displacement of the stability limit the initiation of buckling is nearly the same as that of static buckling, regardless of the frequency of the shaking table. In many studies, a shaking table is used and the load is assimilated to the inertial load Mx¨, taking also into account the nonlinear effect with the intermediary of a correction factor .
0141-0296/00/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 1 - 0 2 9 6 ( 9 8 ) 0 0 1 3 2 - 1
536
G. Michel et al. / Engineering Structures 22 (2000) 535–543
(0) with a Rayleigh method, then to calculate the ratio between predominant seismic frequency (1) and (0). The dynamic factor amplification ( ⫽ Pd/Ps) can be found on the curve. But many questions still remain for empty cylindrical shells subjected to shear load: —does the critical dynamic load depend on frequency excitation? —what is the effect of geometrical imperfections and preloading on eigen frequency? —does coupling between the first beam and the buckling modes occur?
Fig. 1.
Dynamic design rule proposed by Combescure [5].
Kokubo et al. [4] demonstrated the feeble effect of an initial imperfection on the buckling load compared to a cylindrical shell under compression or external pressure. A rule (Fig. 1) for seismic design, in the case of a cylindrical shell subjected to radial dynamic pressure, was proposed by Combescure [5]. First, he proposed calculating the buckling mode and the associated frequency
Fig. 2.
Firstly, static tests, for reference, were conducted. Secondly, we carried out a numerical vibration analysis of the whole structure, in which we studied top mass, geometrical imperfection and shear load influences. Then, dynamic tests were done and compared with the Finite Element simulations and the static tests results.
2. Experimental machine test For our study, the radius–shell thickness ratio is equal to 450 and the length–radius ratio is 1. We have used an electroplating nickel shell so that there will be very little initial imperfection (the thickness is equal to 270
Experimental test machine.
G. Michel et al. / Engineering Structures 22 (2000) 535–543
m) with material property close to 316L steel. At each extremity, the shell is connected to rigid inserts to provide a boundary condition close to an embedment. The material properties are measured for each specimen and we take the mean values ( ⫽ 0.3, E ⫽ 180 000 Mpa, 0.2 ⫽ 450 000 Mpa) for finite element simulations. A special dynamic buckling test machine has been designed (Fig. 2). In our tests, the bottom of the shell is clamped to a rigid base plate and the top is connected (with a rigid plate) to a horizontal hydraulic ram which applies a controlled displacement load. A constant tensile load, simulating the effect of the self-weight of the shell in the case of a hanging shell, may be applied with another hydraulic ram. An automatic system enables us to measure the initial imperfection and the buckling deformations during static tests. A laser sensor is used for the radial displacement measurement. Shear load, tensile load, displacement and accelerations are measured at the top of the shell. A special internally gauged load sensor, linked to the ram and the rigid top plate, enables us to measure the load without the inertia mass effect during dynamic tests. A high speed CCD camera is used for the visualization of buckling and vibration modes during high frequency tests.
Fig. 3.
537
First eigen mode (80 Hz).
calculations, we use the modified RIKS method. For dynamic simulations, we do not use modal analysis but we make a direct integration with an implicit operator. We also take into account a small initial imperfection, affine to the buckling mode, during dynamic simulations.
3. Finite element codes We use two different finite element codes for test simulations: —the first one, INCA code of the CASTEM system developed at the CEA [6], we use a quasi axisymmetric shell element (COMU). This element is a two node flat MINDLIN shell element; at each node the geometry is defined by the r, z coordinates and an initial non-axisymmetric displacement. The initial imperfection is given on a set of Fourier circumferencial harmonics defined by the user. The displacement response is decomposed in the Fourier series and the plasticity is evaluated on discrete points around the circumference. In order to take into account the added mass of the thick top plate in dynamic tests, the top plate is meshed. In dynamic analysis, we make an incremental calculation, also taking into account a small initial imperfection ( ⬍ 10% of the thickness) affine to the buckling mode in order to simulate the post-buckling behavior. —in the second one, ABAQUS [7], we use a 3D modeling. We choose the isoparametric S8R5 shell element. It is an 8 nodes doubly curved thin shell element with reduced integration and five degrees of freedom per node. In order to simulate the post-buckling behavior, during static nonlinear
4. Vibration analysis We perform a shell vibration analysis in order to predict the dynamic behavior of the structure. Eigen modes and eigen frequency are defined by shell dimensions [8]. The first eigen mode (Fig. 3) is a beam mode (shear mode). If we take into account the rigid top plate mass (30 kg), the corresponding eigen frequency is equal to 80 Hz. Higher eigen modes (shell modes) (Fig. 4) are not influenced by this added mass. Parametric calcu-
Fig. 4.
Shell mode (mode 12) without shear load (930 Hz).
538
G. Michel et al. / Engineering Structures 22 (2000) 535–543
Table 1 Eigen frequency analysis, effect of initial imperfection amplitude Imperfection No mass amplitude (m)
Rigid plate mass 38 kg
Mode type
0
10
20
100
200
300
500
1000
231 433 587 842 943 1013 1100 1200 1315 1503 1515 1520 1570 1640
80 188 473 839 943 1013 1100 1200 1315 1503 1515 1520 1570 1640
80 188 473 839 944 1011 1098 1202 1320 1497 1511 1515 1560 1630
79 188 473 837 975 1028 1113 1218 1356 1505 1517 1530 1653 1842
78 188 473 834 1035 1067 1156 1268 1442 1519 1529 1579 1611 1741
77 187 473 831 1079 1152 1210 1341 1514 1530 1551 1646 1720 1830
74 187 473 825 1125 1223 1389 1485 1550 1602 1662 1810 1843
67.1 184 473 810 1226 1400 1433 1637 1803 1822 1954 1958 2015
Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 14
lations are made in order to understand the effect of an initial imperfection (affine to the buckling mode) on eigen modes and the effect of an imposed shear displacement. In the case of an imperfection amplitude analysis (Table 1) we notice that: —eigen frequencies of mode 1 type are not modified by an initial imperfection. —eigen frequencies of the imperfect shell mode types which contain an initial imperfection increase with the amplitude imperfection but these changes are small for an initial imperfection amplitude smaller than the shell thickness. —the mode shapes are all coupled with the deformation associated with the initial imperfection.
Shear Torsion Extension Bending Mode (1, 14) Mode (1, 15) Mode (1, 16) Mode (1, 17) Mode (1, 18) Mode (2,15) Mode (2, 14) Mode (2, 16) Mode (2, 17) Mode (2, 18)
the buckling mode does not influence the vibration modes and eigen frequencies in our case. The second parameter in this vibration analysis is a shear displacement imposed before the eigen frequency calculation: —the first eigen frequency (beam mode) does not change (Fig. 5) whatever the amplitude displacement was. —all shell mode frequencies decrease, the mode shapes change. When the imposed displacement is close to the static critical one, the first shell mode is similar to the buckling mode (Fig. 6) and the first shell frequency is close to 0. The eigen frequency modification, induced by imposed displacement, is not as great as the stiffened
So, we can conclude that initial imperfection affine to
Fig. 5. Shell mode (411 Hz) with an imposed shear displacement of 80% of the critical one.
Fig. 6. tude.
Eigen frequency evolution with displacement imposed ampli-
G. Michel et al. / Engineering Structures 22 (2000) 535–543
539
shell under axial compression [9] because there is a multimodal buckling mode. Nevertheless, for an imposed displacement equal to 97% of the static critical one, the first shell frequency is equal to the shear mode frequency. 5. Static results Experimental buckling tests are conducted under a controlled displacement load. The load displacement curve shows bilinear behavior with stable post critical behavior (Fig. 7). The feeble initial imperfection effect for shear loading can explain stable post-buckling behavior, Koiter [10] links post critical stability to initial imperfection sensitivity. The buckling mode (Fig. 8) is representative of a shear buckling mode with inclined wrinkles in each side of the shell. There is a multimodal buckling mode (mode 10 to 21), with a predominance of modes 14 to 18. The tension load has little effect on the buckling load, on the other hand it contributes to an increase in stability in the post critical area. After a test, small residual deformations affine to the buckling mode still remain; these plastic deformation amplitudes are less than 20 m. If we carry out cyclic buckling tests (up to 120% of the critical displacement) these deformations do not increase and the buckling load is almost the same as the first one: we observe an elastic global behavior and the plastic deformations are just limited to the top of the waves. There is also a smoother load-displacement curve during cyclic tests (Fig. 9). The small residual deformations affine to the buckling mode, induced by the first cycle, explain this behavior. For simulations, we obtain material characteristics by tensile tests and the shell thickness is precisely measured.
Fig. 7. Static load-displacement curve.
Fig. 8.
Static experimental buckling mode.
Fig. 9.
Cyclic loading.
540
G. Michel et al. / Engineering Structures 22 (2000) 535–543
Fig. 10.
INCA simulations, effect of initial imperfection amplitude.
With INCA we use 20 axisymmetric COMU elements in order to mesh the shell. Each rigid insert is modelised with 1 element and the rigid top plate with 8 elements. We choose a modal decomposition with modes 1 and 10 to 20 in order to have the better response in the case of linear buckling analysis. For the nonlinear cases it is necessary to take into account an initial imperfection affine to the first buckling mode in order to simulate the post buckling behavior. The parametric INCA simulations (Fig. 10) on the effect of an initial imperfection (affine to the buckling mode) show that shear buckling is less imperfection sensitive than axial compression or pressure loading. With ABAQUS, 1750 elements S8R5 are necessary to mesh half (vertical symmetry) of the shell, the rigid top plate and rigid inserts. RIKS calculations (Fig. 11)
enable us to find a correct buckling mode (Fig. 12) and simulate the post critical behavior without initial imperfection. Finite Element simulations (INCA or ABAQUS) enable us to find a buckling load equal to 95% of the experimental one.
Fig. 11. ABAQUS non-linear simulation RIKS method without initial imperfection.
Fig. 12. Buckling mode ABAQUS simulation, RIKS method, without initial imperfection.
6. Dynamic results During dynamic tests, the displacement amplitude is fixed (Eq. (1)) and a sweep frequency is done. u ⫽ u0 sin (t)
(1)
First, the displacement amplitude u0 is equal to 10% of the static critical one. For each excitation frequency (1 to 100 Hz) several cycles are done. We are able to
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Fig. 15. Fig. 13. Shear load and radial displacement evolutions according to the excitation frequency (u0 ⫽ 0.1␦cr) FFT signals.
plot the shear load, one point radial displacement and the shear displacement versus the frequency excitation (Fig. 13). When we reach 73 Hz we notice that shear load decreases and radial displacement increases. For frequencies between 75 and 80 Hz the load level increases and for higher frequencies the load level remains weak. Load drop is not an inertial load effect because it is not as regular as the ram load drops. The transfer function (u0/F) (Fig. 14) confirms this behavior. A second test is performed on the same specimen with a displacement amplitude equal to 70% of the static critical one (cyclic loading does not affect the shell behavior). We notice a higher load drop when first beam eigen frequency is reached but the radial displacement ampli-
Fig. 14.
Ratio u0/F.
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Shear load and radial displacement evolutions according to excitation frequency (u0 ⫽ 0.9␦cr) FFT signals.
fication is higher (Fig. 15). The ratio u0/F (Fig. 15) is similar to the vibration test one. The dynamic test visualizations enable us to understand the shell behavior and explain the load curves: —for low load level there is always a beam mode during sweep frequency. So, there is just a classical vibration response with a weak load decrease when the first beam eigen frequency is reached. —with a load level equal to 70% of the static critical load, the critical geometry appears to be a coupled mode between the buckling mode and the shell eigen mode (Figs. 16 and 17). This happens only for excitation frequencies close to the first beam eigen frequency and consequently the shell stiffness becomes very low. The measurement of residual deformation (Fig. 18)
Fig. 16. Coupled instability mode: excitation frequency ⫽ 80 Hz, u ⫽ 0.7u0cr.
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never reaches the static critical one so this imposed load is too small for buckling to occur. So, we can plot the minimum input load level necessary for buckling to occur for each frequency (Fig. 19), this curve is nearly the same as the experimental shear load curve (Fig. 15).
Fig. 17. Coupled instability mode: excitation frequency ⫽ 80 Hz, u ⫽ 0.7u0cr.
Fig. 19. Minimum load necessary to obtain buckling in load control calculation.
Fig. 18. Load evolution at each cycle INCA dynamic load control simulation.
shows that a modal wave (mode 14) remains on the shell in addition to the buckling mode. This modal wave is due to coupled instability which produces local plastic deformations. In finite element simulations displacement or load control calculation are both available. In the case of a displacement control simulation we can only calculate the reaction load. It is necessary for there to be a displacement amplitude equal to the critical static one to obtain the shell buckling, whatever the excitation frequency is. We cannot calculate the applied load at the top of the shell in the same way as the measured load during the experimental test. In load control calculation, a load associated to an excitation frequency is applied at the top of the shell and we observe the time evolution of displacements and of the reaction load. Sometimes the top displacement increases at each cycle and the critical displacement is reached after 5 cycles (Fig. 18). For other imposed loads, the top displacement remains constant at each cycle and
The deformations, when coupled instability occurs, are the same as experimental ones with a mixed buckling and vibration mode (Fig. 20).
Fig. 20.
Finite Element simulation (load control, f ⫽ 80 Hz).
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7. Conclusion This study on dynamic instabilities enables us to understand better cylindrical shell behavior under shear dynamic loading. Shear buckling insensitivity to geometrical imperfection is confirmed. The Finite Element vibration analysis underlines the effect of a shear preload on eigen modes and eigen frequencies. Dynamic tests and Finite Element simulations clearly exhibit that a coupled instability between shell eigen mode and buckling mode can occur, and in this case large deformations are generated. In this case, the necessary critical input load is only equal to 70% of the critical static one. Consequently, it is very important to take into account two parameters for dynamic instability studies: the frequency excitation and the load level. Another significant parameter is the closeness of two eigenmodes.
Acknowledgements Authors express their thanks to E.D.F Septen (Electricite´ de France, Service Etudes Projets Thermiques et Nucleaires), NOVATOME and C.E.A. (Commisseriat a` l’Energie Atomique) for their financial support and for their technical contribution to this study.
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