《外文资料--Buckling of short tanks due to》由会员分享，可在线阅读，更多相关《外文资料--Buckling of short tanks due to（9页珍藏版）》请在金锄头文库上搜索。
1、ELSEVIER Plh S0141-0296(97)00109-0 Engineering Stm.mre. Vol. 20, No. 8, pp. 752 760, 1998 (cj 1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0141-0296/98 $19.00 + 0.(X Buckling of short tanks due to hurricanes Fernando G. Flores Departamento de Estructuras, Universidad Nacio
2、nal de C6rdoba and CONICET, Casilla de Correos 916, 5000 C6rdoba, Argentina Luis A. Godoy Civil Engineering Department, University of Puerto Rico at Mayaguez, Mayaguez, PR 00681-5000, USA (Received June 1996; revised version accepted April 1997) Buckling of thin-walled, short tanks, under severe win
3、d conditions, is investigated using numerical methods. The shells are representa- tive of cylindrical tanks that failed during hurricane Marilyn in the Caribbean islands in 1995, with radius/thickness ratio of 1900 and radius/height ratio of 5. Several models are employed to study instability: bifur
4、cation buckling from a linear fundamental path, nonlinear analysis of imperfect tank, and dynamic response. Wind is modeled using several pressure distributions in the circumfer- ential direction, and the results are compared with those due to axisymmetric pressure. The results show that for the pre
5、sent very thin-walled short tanks, bifurcation buckling produces good esti- mates of the critical state; however, the structure is imperfection sensitive and this load is reduced in the order of 30-50%. 1998 Elsevier Science Ltd. Keywords: thin-walled structures, wind load, buckling, nonlinear behav
6、ior, finite elements, cylindrical shell, dynamic stability 1. Introduction The instability of liquid storage tanks under wind loads is of considerable interest in many parts of the world. Recent examples of failure of such circular metal tanks for storage of oil and water occurred in the US Virgin I
7、slands due to Hurricane Hugo (St Croix in 1989) and Hurricane Marilyn (St Thomas in 1995). In both cases there is a clear evidence that failure was due to buckling of the shell, with the tanks being either empty or with a low level of liquid t. Those are short tanks with large diameters, so that the
8、 ratio between radius and height of the cylinder is of the order of 5, with radius to thickness ratio as high as 1900. Figure 1 shows one such tank in St Thomas, in which buckling of the upper part of the shell can be observed. The loss of the top closure of one tank is illustrated in Figure 2. Larg
9、e diameter thin-walled steel tanks under wind load- ing have been studied by several researchers, but with dif- ferent emphasis and range of applications from those con- sidered in this paper. Early studies of cylindrical shells under wind loads were due to Wang and Billington 2 using an analytical
10、formulation, while most recent investigations in this field use finite element models for the shell 3. Work in the 1960s include References 4 and 5 with poor agree- Figure 1 Buckled tank in St Thomas during hurricane Marilyn (Photograph by L. Godoy) ment between theoretical and experimental results
11、7. Wind tunnel experiments were reported in References 6 and 7 on small scale models. The dimensions tested in Reference 6 included values of 250 - r/t - 500 with height to radius ratio 0.25- h/r - 2.50; while those of interest in Reference 7 had 1030 r/t - 2960 and 0.33 - h/r - 2.4. 752 Buckling of
12、 short tanks due to hurricanes: F. G. F/ores and L. A. Godoy Table 1 Fourier coefficients of wind pressure variation Ref. 21 Ref. 22 co 0.2765 0.387 Cl -o.3419 -0.338 -0.5418 -o.533 -0.3872 -0.471 -0.0525 -0.166 0.0771 0.066 0.0039 0.055 -0.0341 - 753 Figure2 Closure of a tank in St Thomas (Photogra
13、ph by L. Godoy) Test results for long shells under wind loads have been reported in References 8-11, for which nonlinearity of the fundamental path is very important. Cylinders under wind load were studied in Reference 12 using finite elements, but for shells with r/t= 100 and h/r= 2. Other numerica
14、l results are reported 3,H. Notice that it is not just a matter of extending results from one range of shells to another, because the mechanics of behavior varies with the geometry of the shell: tor example, snap buckling is reported in Figure 1 of Reference 7, while very different buckling modes ar
15、e show: in our Figure 1. There is a number of factors that need to be taken into account to model the stability behavior of the shell. First, is this a dominantly dynamic problem, or else can it be tackled by static buckling considerations? Second, is it necessary to take large displacements of the
16、structure into account, or can it be studied using linear bifurcation buck- ling analysis? Third, what is the influence of simplifying assumptions concerning tlae pressure distribution on the failure load predicted? Finally, what is the influence of the liquid stored in the tank on the buckling resp
17、onse? In this paper we attempt to address the above questions by means of numerical experiments on one shell configur- ation and using several models of load and structure. The studies are carried out using finite elements which produce a two-dimensional discretization of the shell, and also by ring
18、 elements that take advantage of the axisymmetric nat- ure of the structure. The conclusions are only valid for short tanks, which are typical of those found in the Caribbean islands, and for taller tanks it is expected that the mechanics of behavior and consequently the tools required for the analy
19、sis may be different. 2. Case studied We consider, in this section, the geometry and load for the circular cylindrical steel tanks of Figures 1 and 2, for which buckling was observed following hurricane Marilyn in Sep- tember 1995. The radius of the tank considered is 19 m, with height 7.6 m and thi
20、ckness of 1 cm. To carry out the computations, the data of the material was assumed as E= 2.06 108 kN/m 2, t,= 0.3, p = 7800 kg/m 3 and yield stress % = 2.156 105 kN/m 2. The wind pressure is here assumed with a circumferential variation given by 7 p = A , ci cos(/0) i-O (1) The Fourier coefficients
21、 are given in Table 1 and the press- ure variation is plotted in Figure 3a; A is a scalar parameter used to increase the load pressure; and the meridian of inci- dence of the wind is assumed to occur at 0 = 0. There is a significant set of data for wind pressures on large cooling tower shells, as re
22、flected, for example, in References 15-20. For short tanks, on the other hand the authors could not find experimental data available and this paper relies on pressures developed for other shell geo- metries. In the absence of more detailed experimental evi- dence from real hurricanes on short shells
23、, two sets of coef- ficients have been used for the analysis. First, those currently employed in the analysis of cooling towers 2, and second, an earlier wind distribution on cylinders from Reference 22. However, it will be shown in this paper that the actual pressure distribution in the circumferen
24、ce does not significantly affect the buckling results. The wind pressure was assumed to follow different dis- (a) (D 0_ (b) 1 0.5 0 -0.5 -1 -1.5 E ,E- ._m -i- A.CI 0 30 60 90 120 150 180 Theta angle deg 8 ! 7 - constant ASCE23 . i 6 linear . 5 4 3 2 1 0 0 r ,. 0,5 Pressure Figure3 Wind pressure vari
25、ation: (a) circumferential; (b) in height 754 Buckling of short tanks due to hurricanes: F. G. Flores and L. A. Godoy tributions in height, shown in Figure 3b. First a constant unit pressure in the vertical direction; second, a pressure following Reference 23 with a maximum value on top of 1.12; and
26、 third, similar to the latter, but with a linear vari- ation in height from 0 to 1.03 at 4.6 m. All this was done to investigate the incidence of the pressure distribution in height on the buckling load. Preliminary wind tunnel results for the shells considered in this work indicate pressure distrib
27、utions similar to those given by equation ( 1 ). Furthermore, the present results indi- cate that the actual pressure distribution is not so critical for a very thin shell like the one we consider here. Finally, the present studies assume a deterministic pressure distri- bution. Random loading with
28、different spatial correlation has been considered 27. A detailed discussion about the modeling of wind effects on structures may be found in Reference 24. 3. Bifurcation buckling from a linear fundamental path 3.1. Coupled analysis The first model to be presented is a bifurcation analysis from a lin
29、ear fundamental path. This model leads to an eigenvalue problem; however, several levels of approxi- mation can be introduced according to the way in which the wind pressure is modeled. The results are based on a computer code called ALREF presented by the authors in Reference 12 and based on the po
30、st-buckling theory for dis- crete structural system 25,26. As usual, in this type of struc- ture, this code uses ring elements 3, but can handle bifur- cations from non-axisymmetric pressure distributions such as those associated to the wind loads defined in equation (1). This is a semi-analytical f
31、inite element, with Fourier series in the circumferential direction and quintic/cubic interpolation of displacements (quintic poly- nomial for out-of-plane displacements and cubic for in- plane displacements). The critical state is evaluated from a linear eigenvalue problem, in which it is necessary
32、 to include a set of har- monic (Fourier) components. The procedure employed is as follows. First, the bifurcation analysis is carried out for axisym- metric stresses, and from the results, a set of harmonic components is identified for which the lowest values of critical loads are obtained. In this
33、 problem the values of critical loads for the relevant range of harmonic compo- nents are plotted in Figure 4 (worst loaded meridian, or E Z o .) 5 4 , 3 2 1 0 15 I I I I I I I I I WLM o WSM -+ - - ._ - - 0 - . 4 .-+ . 4) .-O .O I I I I I I I I I 17 19 21 23 25 27 29 31 33 35 Harmonic Figure 4 Criti
34、cal loads for different harmonic modes for a tank with axisymmetric stresses WLM). The lowest value occurs for harmonic component n=22, for which A=l.817kN/m 2. Fifteen harmonic components are included in the analysis: those for 17- n- -40 -60 Figure 8 K / .-v. L,. -80 2 (b) 0.6 6 8 10 12 14 Displac
35、ement cm 0.2 00- 7 : . . . . . . . . -0.2 -0.4 -0.6 0 1 2 3 4 5 6 Time sec Phase chart and energy values for imperfect cylinder Buckling of short tank.,; due to hurricanes: F. G. F/ores and L. A. Godoy 757 puted for each increment in displacements. The results for the quasi-perfect cylinder are show
36、n in Figure 9a. Here the static buckling load is Ac and occurs for Ac = 2.191 kN/m 2 with a small critical displacement UA = 1.6 cm. Following the static critical state there is a descending path with unstable equilibrium states. The energy -becomes zero along this path at a displacement UA = 15.5 c
37、m, for which the load on the descending: path is AD = 1.690 kN/m 2. This latter value is the approximate dynamic buckling load under step loading. The components V and W are also shown in the same figure. Although this criterion is entirely adequate for one DOF system, it only represents a lower bou
38、nd approximation for multiple DOF systems. In the present case such lower bound leads to Ao/Ac = 0.77. Next we employ the total potential energy criterion for the imperfect shell. The maximum load attainable by the system under static load is Ac = 1.621 kN/m 2 with a critical displacement UA = 8.0 c
39、m The energy is computed and found to be zero for UA = 20.1 cm. For such displacement the load on the unstable branch of equilibrium has AD= 1.410kN/m 2. The ratio between the lower bound dynamic load and the static critical load is in this case AD/Ac = 0.87. Notice that the two boundaries are very
40、close in the imperfect shell, while they become apart in the quasi- perfect shell. The above study based on the total potential energy cri- terion shows very conservative lower bounds to dynamic buckling under step loads for almost perfect shells. The bounds have closer agreement with the dynamic cr
41、iterion of Budiansky and Roth for imperfect shells. Numerical values of those limits are reported in Table 2. The present dynamic results indicate that for the perfect structure there is no influence of inertia efects; for the (a) 2.5 2.0 t o 1.5 ._1 1.0 , 0.5 - LU 0.0 -0.5 (b) 2.5 Lc L -V (W*Vlml n
42、 I I r , , I 0 5 10 15 20 25 UAcm 2.0 10 1.5 o, ! 1.0 ._e 0.5 =m C LU 0.0 -0.5 -1.0 Figure 9 , , T , I r I I I i 0 5 10 15 20 25 UAcm Budiansky-Roth criteria for case (b) and (c) Table 2 Upper and lower bounds for dynamic step loads Lower bound Upper bound Ratio Case AD Ac AD/Ac (b) 1.690 2.191 0.77
43、 (c) 1.410 1.621 0.87 Table3 Critical loads for different liquid levels Water level Critical load Normalized load 0 2.207 1.000 H/6 2.211 1.002 H/3 2.264 1.025 H/2 2.510 1.137 2H/3 3.116 1.411 5H/6 4.769 2.159 H 11.377 5.153 imperfect structure the influence is more marked, but lead- ing to much the
44、 same conclusions as the static analysis, thus showing that inertia effects do not play an important role in the maximum carrying capacity of these short shells. 6. Influence of the liquid stored in the tank It is expected that the presence of water or other liquid stored in the tank should have a s
45、tabilizing effect. The prevalent idea is that these tanks should have a liquid level of about one third of their height H in order to have a better buckling behavior. To investigate the influence of the liquid stored, a model based on the code ALREF has been used. An important feature of ALREF in th
46、is case is that it allows two separated loads acting on the structure: one fixed load (the water pressure) and another load (wind) to be increased until bifurcation buckling occurs. The results for several cases of liquid level are presented in Table 3: for liquid level H/6 there is a negligible inc
47、rease in the load. For H/3 the increase is of only 2.5%, while for H/2 the pressure is increased by 14% with respect to the empty tank. Water height greater than H/2 create higher circumferential tensions and raise buckling loads further, as seen from the results in Table 3. Finally, practicing engi
48、neers sometimes model the water influence by decreasing the actual height of the tank. Again ALREF was used to model this situation and investigate if such study produces reasonable results. For different values of effective heights of the shell, the values of buckling load are shown in Table 4. Suc
49、h values should be compared with those of Table 3: for example, for a water level of H/2, Table 3 indicates a buckling load equal to 1.137 times the value for the empty tank. By shortening the effective height of the tank to H/2, and without any water inside, buckling Table 4 Normalized critical loa
50、ds for different cylinder heights Cylinder height Clamped edge Hinged edge H 1.000 0.995 5/6 H 1.184 1.175 2/3 H 1.518 1.500 1/2 H 2.286 2.228 758 Buckling of short tanks due to hurricanes: F. G. Flores and L. A. Godoy is predicted to be 2.286 in Table 4. The boundary condition does not modify drama
51、tically this value: for a simply- supported shell, the critical load is 2.228. A decrease in the height of the shell produces an increase in the buckling load; however, this increase bears no relation to the influence of the liquid stored. 7. Conclusions The numerical results presented in this paper
52、 support the evidence that buckling occurred for the wind velocities esti- mated during hurricane Marilyn in 1995 on the island of St Thomas, and hurricane Hugo in 1989 on the island of St Croix. Of course, there are many uncertainties about the actual wind speed at the location of the tanks, near t
53、he air- port of St Thomas, where anemometers were operating. The locations of the tanks varied: some were on the top of a small hill; others were located on the scarpments; while a third group was on rather flat land. In general, there was no shielding to the tanks. The buckling mode observed in the
54、 real tanks represents advanced deformations in the post-buckling path, while the computations model the initial stages of buckling. How- ever, there are close similarities between the mode detected for several tanks in St Thomas and those computed using our codes ALREF and ALPHA. The models show th
55、at instability is mainly driven by static effects, so that the inertia of the shell and dynamic effects are not very important. Furthermore, it was shown that for the class of tanks investigated, with ratios of diam- eter to height of the order of 5, buckling occurs in the form of a bifurcation with
56、 small displacements at the critical state. Post-critical states are unstable, and lead to a decreas- ing post-critical equilibrium path in the load-displacement space. The maximum load that the shell attains is a function of the geometric imperfection present, and for imperfec- tions with amplitude
57、 of the order of the thickness the maximum load is about 70% of the bifurcation load. The presence of a liquid inside the tank has a stabilizing effect, however, the increase in the bifurcation load for a liquid level of half of the height is of only 14%. Acknowledgement This work was carried out at
58、 National University of C6rdoba and at the University of Puerto Rico. Work at the Group of Numerical Methods in Mechanics at the Univer- sity of C6rdoba, was supported by grants from CONICET (the Science Research Council of Argentina) and CON- ICOR (the Science Research Council of the Province of C6
59、rdoba). Work at the University of Puerto Rico was sup- ported by a grant from NSF-EPSCOR. The second author is indebted to the Civil Engineering Department of the Uni- versity of Puerto Rico at Mayagtiez and to FEMA for the support to travel to St Thomas in September 1995. References 1 Godoy, L., Fl
60、ores, F., Elaskar, S. and Zapata, R. Comportamiento no lineal de silos y tanques frente a acciones de viento (in Spanish), in Lecciones del Huracdn Marilyn, Mayagiiez, Puerto Rico, 1996 2 Wang, Y. and Billington, D. P. Buckling of cylindrical shells by wind pressures, J. Engng Mech. Div. ASCE 1974,
61、100, 1005-1023 3 Gould, P. L. Finite element analysis of shells of revolution, Pitman, Marshfield, MA, 1985 4 Holownia, B. P. Buckling of cylindrical shells under wind loading, Struct. and Mater. Note Australian Department of Supply, 1964, 292 5 Langhaar, H. L. and Miller, R, E. Buckling of an elast
62、ic isotropic cylindrical shell subjected to wind pressure, in Proc. Syrup. Theory of Shells, University of Houston, Texas, 1967 6 Prabhu, K. S., Gopalacharyulu, S. and Johns, D. J. Design criteria for stability of cylindrical shells subjected to wind loading, HMSO, London 7 Megson, T. H. G., Harrop,
63、 J. and Miller, M. N. The stability of large diameter thin-walled tanks subjected to wind loading, in Stability of plate and shell structures, Dubas, P. and Vandepitte, D. (eds), Ghent University, Ghent, 1987, pp 529-538 8 Uematsu, Y. and Uchiyama, K. Deflection and buckling behavior of thin, circul
64、ar cylindrical shells under wind loads, ,L Wind Engng Indust. Aerodvnam. 1985, 18, 245-261 9 Johns, D. J. Wind-induced static instability of cylindrical shell, J. Wind Engng lndust. Aerodynam. 1983, 13, 261-270 10 Greiner, R. and Derler, P. Effect of imperfections on wind-loaded cylindrical shells,
65、Thin-Walled Struct. 1995, 23, 271-281 11 Resinger, F. and Greiner, R. Buckling of wind-loaded cylindrical shell- application to unstiffened and ring-stiffened tanks, in Buck- ling of shells, Ramm, E. (ed.), Springer, Berlin, 1982, pp 217-281 12 Flores, F. G. and Godoy, L. A. Instability of shells of
66、 revolution using ALREF: studies lbr wind loaded shells, in Buckling of shells structures, on land, bt the sea and in the air, Elsevier, London, 1991, pp 213-222 13 Saal, H. and Schrufer, W. Stability of wind-loaded shells of cylindri- cal tanks with unrestrained upper edge, in Buckling of shells st
67、ruc- tures on land, in the sea and in the air, Jullien, J. F. (ed.), Elsevier, London, 1991 14 Moy, S. S. J. Wind loads on tall, thin-walled structures, in Finite element applications to thin-walled structures, Bull, J. (ed.), Chapter 8, Elsevier, London, 1990, pp 225-286 15 Basu, P. K. and Gould, P
68、. L. Cooling towers using measured wind data, J. Engng Mech. Div. ASCE 1980, 103, 579-600 16 Sollenberger, N. J., Scanlan, R. H. and Billington, D. P. Wind load- ing and response of cooling towers, J. Engng Mech. Div. ASCE 1980, 103, 600-621 17 Armitt, J. Wind loading on cooling towers, ./. Engng Me
69、ch. Div. ASCE 1980, 103, 623-641 18 Niemann, H. J. Wind effects in cooling towers shells, J. Engng Mech. Div. ASCE 1980, 103, 643-661 19 Shu, W. and Wanda, L. Gust factor for hyperbolic cooling towers on soils, J. Engng Struct. 1991, 13 (I), 21-26 20 Flaga, A. Quasi-static computational approach to
70、wind-load on coo- ling towers in conditions of aerodynamic interference towers on soils, J. Engng Struct. 1991, 13 (4), 317-328 21 ACI-ASCE Committee 334, Reinforced concrete cooling tower shells - practice and commentary, ACI 334,2R,91, American Concrete Institute, New York, 1991 22 Rish, R. F. For
71、ces in cylindrical shells due to wind, in Proc. Inst. Civil Engineers, Vol. 36, 1967, pp 791-803 23 American Society of Civil Engineers, Minimum design loads for buildings and other structures, ASCE, New York, 1995 24 Gould, P. L. and Abu-Sitta, S. H. Dynamic response of structures to wind and earth
72、quake loading, Wiley, New York, 1980 25 Flores, F. G. and Godoy, L. A. Elastic post-buckling analysis via finite elements and perturbation techniques. Part I: formulation, Int. J. Num. Meth. Engng 1992, 33, 1777-1794 26 Flores, F. G. and Godoy, L. A. Elastic post-buckling analysis via finite element
73、s and perturbation techniques. Part II: application to shells of revolution, Int. J. Num. Meth. Engng 1993, 36, 331-354 27 Riera, J. D. and Ambrosini, R. D. Analysis of structures subjected to random loading using transfer matrix or numerical integration methods, Engng Struct. 1992, 14(3), 176-179 2
74、8 Flores, F. G. and Ofiate, E. New assumed strain triangles for non- linear shell analysis, Comput. Mech. 1995, 17(1-2), 107-114 29 Simo, J. C. and Fox, D. D. On stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization, Corn- put. Meth. Appl. Mech. Engng 1989
75、, 72, 267-304 30 Simo, J. C., Fox, D. D. and Rifai, M. S. On stress resultant geometri- cally exact shell model. Part III: computational aspects of the nonlin- ear theory, Comput. Meth. AppL Mech. Engng 1990, 79, 21-70 31 Ofiate, E., Zienkiewicz, O. C., Suarez, B. and Taylor, R. L. A meth- odology f
76、or deriving shear-constrained Reissner-Mindlin plate elements, Int. J. Num. Meth. Engng 1992, 33, 345-367 32 Simo, J. C. and Kennedy, J. G. On stress resultant geometrically exact shell model. Part V, nonlinear plasticity: formulation and inte- Buckling of short tanks due to hurricanes: F. G. Flores
77、 and L. A. Godoy 759 gration algorithms, Compvt. Meth. Appl. Mech. Engng 1992, 96, 133-171 33 Simo, J. C. and Tarnow, N. A new energy and momentum conserv- ing algorithm for the non-ltinear dynamics of shells, Int. J. Num. Meth. Engng 1994, 37, 2527-2549 34 Flores, F. G. ALPHA: a static/dynamic impl
78、icit finite element pro- gram users manual, National University of C6rdoba, C6rdoba, Argentina, 1996 35 Wriggers, P. and Simo, J. C. A general procedure for the direct computation of turning and bifurcation points, Int. J. Num. Meth. in Engng 1990, 30, 155-176 36 Hoff, N. J. and Bruce, V. C. Dynamic
79、 analysis of the buckling of laterally loaded flat arches, Q. Math. Phys. 1954, 32, 276-278 37 Simitses, G. J. Dynamic buckling of suddenly loaded structures, Springer, New York, 1990 38 Brewer, A. T. and Godoy, I_,. A. Dynamic buckling of discrete struc- tural systems under combined step and static
80、 loads, Nonlinear Dynam. 1996, 9, 249-264 39 Budiansky, B. and Roth, R. S. Axisymmetric dynamic buckling of clamped shallow sperical shells, in Collected papers on stability of shell structures, NASA TN-1510, 1962 40 Simitses, G. J., Kounadis, A. N. and Giri, J. Dynamic buckling of simple frames und
81、er a step load, ASCE J. EM Division 1979, 105, 896-900 41 Hsu, C. S. Stability of shallow arches against snap-though under timewise step loads, Int. J. Engng Sci. 1968, 4, 31-39 42 Kounadis, A.N. and Raftoyiannis, I. Dynamic stability criteria of nonlinear elastic damped/undamped systems under step
82、loading, AIAA 1990, 28(7), 1217-1223 Next we define a = qp,.q, y, = ,.t b = qp,.t, Note that a, is the first fundamental form of the mid-sur- face and b,e is the second fundamental form provided the director t is normal to the mid-surface. The following Lag- rangian strain tensors can be obtained wi
83、th covariant components referred to the dual undeformed basis a ,a2,a 3 El I El2 er = E21 E22 LE3 i E32 la, - a o, I I = a , L % il ! o o o El3 E23 E33 al2-al2 Yl - 1 a22 - a2 )/2 - 2 2 - :2 0 Appendix The shell configuration in three dimensions is defined by the position of the mid-surface p and th
84、e direction of the pseudo-normal (director t) both as functions of two curvi- linear coordinates (,) defined on a two-dimensional domain . Vector t defines the direction of the fiber across the thickness which remains straight during deformations (generalized Kirchhoff hypothesis). The thickness h i
85、s assumed constant along the deformation, while drilling rotations of the director are not considered in the formu- lation thus leading to a five parameter shell theory (three displacements and two rotations). With this notation, the position of any point of the shell x can be described, as usual, f
86、rom the corresponding point on the mid surface and the distance to it () as: x = + 3t 3 e l-h/2, +h/2 The independence of the mid-surface configuration and the director field leads to transverse shear deformable elements with C continuity. A total Lagrangian formulation is adopted with objective str
87、ain and stress measures (Green- Lagrange strains and second Piola-Kirchhoff stresses). A brief summary of the energy formulation follows, but the interested reader should consider the original sourc- es29,3o.32,33. A local covariant basis can be defined in both unde- formed and deformed configuratio
88、ns a,a,a - #9,q92,1 (undeformed) al,a2,a3 - tp, l,q,z,t (deformed) where the derivatives are denoted in the form (_), = a(_)/3 The local deformation gradients relative to the unde- formed configuration q,t are defined by (et = 1,2) f = , a + t ,a3 and g = t, a where et are the mid-surface membrane a
89、nd shear strains and eg are the curvature changes of the mid-surface. As in the three-dimensional theory, nominal stress result- ants n,m can be found as work-conjugate to the defor- mation gradients f,g in the sense that the stress power = n : f + m : . For an elastic material = IV, where W (f,g) =
90、 if (ee,%) is a stored energy function invariant under superposed rigid motions. It can be shown that n = OfW = ffi + gffa T and m = OgW = frh where ,lh are the second Piola-Kirchhoff stress result- ants obtained as partial derivative of the stored internal energy l = OefW and lil = 0%17V A quadrati
91、c stored energy function is assumed of the form 1 1 if= er C:ef+%:D:% where C and D are the standard fourth-order constant ten- sors for elastic shells. To compute the kinetic energy K we first define on the reference (undeformed) configuration h_ _h Apo= f pd 3 Ipo= f p(3)2 d3 2 2 as the time indep
92、endent nominal surface density and nomi- nal rotational inertia functions of the shell. Under a motion with velocity ,t the linear and angular momentum of the shell are simply 760 Buckling of short tanks due to hurricanes: F. G. Flores and L. A. Godoy 1 =Aoo j = l, oi The Hamiltonean function H defined as the sum of the kin- etic and potential energies, takes the form and the kinetic energy can then be evaluated as 1 Ioi.i di f H = K + / if (ef, eg) d,/+ Vext (,t) d :A where Vext(q,t) is the potential energy loads. of the external
