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Sloshing Effects on the Seismic Design of Horizontal-Cylindrical and Spherical Industrial Vessels Article  in  Journal of Pressure Vessel Technology · August 2006 DOI: 10.1115/1.2217965

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Lazaros A. Patkas Manolis A. Platyrrachos Department of Mechanical & Industrial Engineering, University of Thessaly, 38334 Volos, Greece

1

Sloshing Effects on the Seismic Design of Horizontal-Cylindrical and Spherical Industrial Vessels The present paper investigates sloshing effects on the earthquake design of horizontalcylindrical and spherical industrial vessels. Assuming small-amplitude free-surface elevation, a linearized sloshing problem is obtained, and its solution provides sloshing frequencies, modes, and masses. Based on an “impulsive-convective” decomposition of the container-fluid motion, an efficient methodology is proposed for the calculation of seismic force. The methodology gives rise to appropriate spring-mass mechanical models, which represent sloshing effects on the container-fluid system in an elegant and simple manner. Special issues, such as the deformability of horizontal-cylindrical containers or the flexibility of spherical vessel supports, are also taken into account. The proposed methodology can be used to calculate the seismic force, in the framework of liquid container earthquake design, and extends the current design practice for vertical cylindrical tanks stated in existing seismic design specifications (e.g., API Standard 650 and Eurocode 8). The methodology is illustrated in three design examples. 关DOI: 10.1115/1.2217965兴

Introduction

Cylindrical and spherical tanks or pressure vessels are common in chemical plants or refineries. They are employed as storage vessels for liquefied petroleum gas 共LPG兲, liquid propane, propylene, and LNG, and their seismic design and, in particular, the calculation of the maximum sloshing force due to earthquake constitutes a crucial issue for their structural integrity under earthquake loads. Sloshing has been considered as a typical linear eigenvalue problem in terms of a fluid velocity potential function, representing the small-amplitude free vibrations of the surface of an ideal liquid inside a motionless container. In many engineering applications, in addition to sloshing frequencies, hydrodynamic pressures and forces due to sloshing need to be calculated. In particular, earthquake-induced sloshing has been recognized as an important issue toward safeguarding the structural integrity of liquid storage tanks or vessels, and has been the subject of numerous analytical, numerical, and experimental works. The pioneering work of Housner 关1兴 presented a solution for the hydrodynamic effects in nondeformable vertical cylinders and rectangles, splitting the solution in two parts, namely the “impulsive” and the “convective” motion. This concept has constituted the basis for the API 650 standard provisions 共Appendix E兲 for vertical cylindrical tanks 关2兴. Veletsos and Yang 关3兴, Haroun and Housner 关4兴, and Haroun 关5兴 have extended this formulation to include the effects of shell deformation, and its interaction with hydrodynamic effects, whereas notable experimental investigations were reported in 关6,7兴. More recently, uplifting of unanchored tanks, as well as soil-structure interaction effects, have been studied extensively in 关8–14兴. Some of the above-mentioned works have constituted the basis for the seismic design provisions for vertical cylindrical tanks in Annex A of Eurocode 8—part 4.3 关15兴. The reader is referred to the review paper of Rammerstorfer et al. 关16兴 for a 1 Corresponding author. Contributed by the Pressure Vessels and Piping Division of ASME for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received May 4, 2005; final manuscript received September 12, 2005. Review conducted by Rudolph J. Scavuzzo. Paper presented at the 2004 ASME Pressure Vessels and Piping Division Conference 共PVP2004兲, July 25, 2004–July 29,2004, San Diego, California, USA.

328 / Vol. 128, AUGUST 2006

thorough presentation and a concise literature review of liquid storage tank response under seismic loads, including fluidstructure and soil-structure interaction effects. The above studies focused on the seismic response of verticalcylindrical tanks. On the other hand, horizontal-cylindrical and spherical vessels 共Fig. 1兲 have significant industrial applications 共e.g., in chemical plants and refineries兲, and their sloshing response under strong seismic events is of particular interest for a reliable estimate of the total horizontal force and the corresponding overturning moment. Nevertheless, the amount of works concerning sloshing in horizontal-cylindrical and spherical containers is quite limited, compared with the large number of publications on vertical cylinders. Note that the API 650 provisions for seismic design 关2兴 refer exclusively to vertical cylinders, whereas in 关15,17兴 little attention is given to horizontal cylinders and spheres. Budiansky 关18兴 was the first to calculate sloshing frequencies and hydrodynamic forces in spheres and two-dimensional circular canals under transverse excitation, describing the flow field through a set of integral equations, which was solved numerically. Experimental measurements of sloshing in spheres and horizontal cylinders have been reported in 关19兴 and, more recently, in 关20兴, whereas notable semianalytical and numerical works on the calculation of sloshing frequencies in those geometries have been presented in 关21–26兴. Recently, Patkas and Karamanos 关27兴 reported on externally induced sloshing calculations in spheres and horizontal cylinders under transverse excitation, using a variational formulation. The particular case of sloshing in horizontal cylinders under longitudinal excitation has received less attention. Kobayashi et al. 关20兴, based on experimental measurements, demonstrated that hydrodynamic forces in horizontal cylinders under longitudinal excitation can be calculated quite accurately replacing the cylindrical vessel by an “equivalent rectangular container.” This concept has been demonstrated numerically in 关28兴, using a finite element formulation. The present study is aimed at proposing an efficient and unified methodology for earthquake-induced sloshing analysis in vessels of horizontal-cylindrical and spherical shape, under horizontal ground motion. The method is based on a modal analysis that provides sloshing frequencies and the corresponding sloshing masses, so that the liquid response and total seismic force are readily calculated. Considering an “impulsive-convective” decom-

Copyright © 2006 by ASME

Transactions of the ASME

⌽U = X˙共t兲x

共4兲

the “sloshing motion” potential ⌽S, expressed as ⬁

兺

⌽S共x,y,z,t兲 =

Y˙ n共t兲⌿n共x,y,z兲

共5兲

n=1,2,3,. . .

Fig. 1 Industrial vessels „horizontal cylinder and sphere…

position of the liquid motion, simplified mechanical models are proposed to approximate the response. In horizontal cylindrical vessels, excitation in both the transverse and longitudinal direction is considered. The deformation effects of long horizontal cylinders are also investigated, through a simplified mechanical model. For the particular case of a horizontal cylinder under longitudinal excitation, its equivalence with an appropriate rectangular container is demonstrated. In addition, the flexibility effects of the support system in spherical containers are taken into account. The proposed methodology offers an elegant and efficient tool for calculating the seismic force in the course of a seismic design procedure, and extends the current design practice, stated in API 650 关2兴 and Eurocode 8 关15兴 for vertical cylindrical tanks.

2

where Y n共t兲 are generalized coordinates and the dot denotes derivative with respect to time. The choice of spatial functions ⌿n共x , y , z兲 is an important step for the efficient solution of the sloshing problem. Quite often, those functions are chosen as the eigenmodes of the corresponding free-vibration problem 共obtained by setting X˙ = 0 in Eq. 共2兲, so that ⌽U = 0兲, referred to as “sloshing modes.” In such a case, functions ⌿n共x , y , z兲 are mutually orthogonal and the generalized coordinates Y n共t兲 are computed from a series of uncoupled linear equations Y¨ n + 2␰n␻nY˙ n + ␻2nY n = − ␥nX¨

ⵜ 2⌽ =

⳵ 2⌽ ⳵ 2⌽ ⳵ 2⌽ + + =0 ⳵x2 ⳵ y 2 ⳵z2

in the fluid domain ⍀

F=−␳

B1

兺

Ln =

冕

␳⌿ndB1

共8兲

B1

Using the following change of variables

共2兲

an =

Yn ␥n

or

un = an + X

共3兲

where ex the unit vector in the x direction and n is the outward normal vector at B1. The unknown potential can be decomposed in two parts, and the “uniform motion” potential ⌽U

n = 1,2,3, . . .

共9兲

n = 1,2,3, . . .

共10兲

the liquid motion equations 共6兲 become a¨n + 2␰n␻na˙n + ␻2nan = − X¨

at the free surface 〉2

共7兲

where ML is the total liquid mass, and Ln depend on the container geometry, i.e., on the spatial functions ⌿n共x , y , z兲

at the “ wet ” surface of the vessel wall 〉1

⳵⌽ ⳵ 2⌽ =0 +g ⳵y ⳵t2

冕

⬁

⳵⌽ LnY¨ n 共ex · n兲dB1 = − MLX¨ − ⳵t n=1,2,3,. . .

共1兲

subjected to the following boundary conditions 共Fig. 1兲

⳵⌽ ˙ = X共ex · n兲 ⳵n

共6兲

where ␻n is the sloshing frequency of the nth mode, ␥n expresses the contribution of the external excitation to the specific mode, and a damping ratio ␰n is introduced to express dissipation effects. The liquid motion is associated with hydrodynamic pressures p共x , y , z , t兲, calculated directly from ⌽ through the Bernoulli ˙ 兲 and the total hydrodynamic force at the conequation 共p = −␳⌽ tainer wall is obtained through an appropriate integration of those pressures on the “wet” surface of the container in the direction of the earthquake excitation. Thus, the total force F is the sum of the “uniform motion” force FU

General Approach

Assuming ideal fluid conditions and small-amplitude elevation of the free surface, the liquid motion in a undeformed 共rigid兲 container under horizontal excitation X共t兲 in the x direction 共Fig. 2兲 is described by the flow potential ⌽共x , y , z , t兲, so that the liquid velocity is the gradient of ⌽ 共u = ⵜ⌽兲, which satisfies the Laplace equation,

n = 1,2,3, . . .

or equivalently, u¨n + 2␰n␻n共u˙n − X˙兲 + ␻2n共un − X兲 = 0

n = 1,2,3, . . .

共11兲

Equation 共10兲 expresses the liquid motion with respect to the container, and Eq. 共11兲 expresses the total liquid motion 共including the motion of the container兲. Furthermore, the force in Eq. 共7兲 becomes ⬁

F=−

兺

Mnca¨n − MLX¨

共12兲

Mncu¨n − MIX¨

共13兲

n=1,2,3,. . .

or equivalently ⬁

F=−

兺

n=1,2,3,. . .

where Mnc = ␥nLn. Equation 共13兲 shows that F consists of an “impulsive” force F⌱ FI = − MIX¨ , Fig. 2 Liquid container under external excitation

Journal of Pressure Vessel Technology

共14兲

where AUGUST 2006, Vol. 128 / 329

Table 1 Variation of the first three „two-dimensional… sloshing frequencies with respect to the liquid height in a horizontal cylinder

Fig. 3 Configuration container

of

a

horizontal-cylindrical

liquid

⬁

兺

MI = ML −

Mnc ,

共15兲

n=1,2,3,. . .

e

␻21R / g

␻22R / g

␻23R / g

␻24R / g

−1.0 −0.9 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 0.9 0.95 1.0

1.0 1.0209 1.0438 1.0970 1.1627 1.2461 1.3557 1.5075 1.7346 2.1237 3.0215 4.3115 6.1519 ¯

6.0 5.6462 5.3550 4.9370 4.6987 4.6067 4.6511 4.8509 5.2768 6.1395 8.3152 11.5515 16.2222 ¯

15.0 12.4355 10.7704 9.0076 8.1988 7.8537 7.8199 8.0783 8.7221 10.0815 13.5640 18.7696 26.2969 ¯

28.0 23.324 16.494 13.041 11.656 11.074 10.972 11.293 12.157 14.016 18.810 25.990 36.386 ¯

and a “convective” force FC ⬁

兺

FC = −

Mncu¨n

共16兲

n=1,2,3,. . .

Furthermore, the total liquid mass ML can be considered as the sum of: • •

the “impulsive” mass MI, which represents the mass that “follows” the container motion X共t兲, the “convective” or “sloshing” masses Mnc 共n = 1 , 2 , 3 , . . . 兲, which correspond to liquid motion due to free-surface elevation 共convective motion兲. Each sloshing mass represents the participation of the corresponding mode on the entire response; it is expected that the Mnc / ML ratio decreases monotonically with increasing values of n.

The sum of all the convective masses is referred to as “convective mass” ⬁

兺

MC =

Mnc

共17兲

n=1,2,3,. . .

Note that in the course of a seismic design procedure, the impulsive force should comprise the inertia force of the container’s shell mass MSH, so that in Eqs. 共13兲–共15兲 MI = 共ML + MSH兲 −

兺M

nc

= 共ML + MSH兲 − MC

共18兲

n

and the sum of MSH and ML is the total moving mass MT of the container-liquid system. In rectangular and vertical-cylindrical liquid storage tanks, assuming nondeformable walls, analytical expressions for the sloshing frequencies ␻n and modes ⌿n共x , y , z兲 can be obtained 共e.g., 关19兴兲, and the calculation of sloshing masses Mnc is straightforward. On the other hand, such analytical expressions for the sloshing frequencies and the corresponding modes for horizontalcylindrical and spherical vessels do not exist, and should be computed numerically. A high-accuracy computation of ␻n and ⌿n共x , y , z兲 in horizontal cylindrical and spherical vessels, as well as the corresponding sloshing masses Mnc, has been conducted in 关27兴, through a semianalytical variational formulation that uses a series expansion of the sloshing potential in terms of harmonic functions. Results from that work are directly employed in the next two sections.

3

Horizontal Cylindrical Vessels

In this section, the response of horizontal-cylindrical liquid vessels under horizontal seismic excitation is examined in both the transverse and longitudinal directions, x and z, respectively 共Fig. 3兲. The free surface of the liquid may be at any height 共0 ⬍ H 330 / Vol. 128, AUGUST 2006

⬍ 2R兲. Sloshing frequencies, modes, and masses have been computed elsewhere through a variational formulation 关27兴, under the assumption that the horizontal cylindrical container is nondeformable. Note that industrial vessels are quite thick to resist high levels of internal pressure and, therefore, they remain practically undeformed. However, for a long cylinder, supported near the two ends, the cylinder may deform as a beam, and deformation effects can be taken into account using a simplified fluid-vessel interaction formulation, as described in the Appendix. 3.1 Nondeformable Horizontal Cylinders Under Transverse Excitation. The flow potential for a nondeformable horizontal cylindrical vessel under transverse excitation 共x axis in Fig. 3兲 can be expressed in the form of Eq. 共6兲. Under this type of excitation, the problem is two-dimensional 共independent of z兲, and a unit length in the z direction is considered. Therefore, the fluid domain ⍀ can be considered as a circular sector of unit length in the z direction. The sloshing frequencies ␻n and the corresponding modal functions ⌿n共x , y兲 are computed numerically, using the variational form of the boundary value problem 共1兲–共3兲, and applying Galerkin’s through harmonic bounded functions, as well as constant-strain triangular finite elements, to obtain the discretized equations of motion. Subsequently, using standard modal analysis, the sloshing frequencies and modes are calculated, and the sloshing masses are obtained using the methodology described in Sec. 2. More details on the numerical formulation and solution can be found in 关27兴. In Table 1, numerical results are presented in dimensionless form for the first three sloshing frequencies 共␭n = ␻2nR / g , n = 1 , 2 , 3兲 for different liquid levels. The dimensionless parameter e is equal to H / R − 1 where H is the liquid depth 共0 艋 H 艋 2R兲. The results are in very good agreement with those computed numerically in 关25兴, as well as with experimental results, reported in 关19兴. The sloshing frequencies are also depicted in Fig. 4. Furthermore, the sloshing mass ratios M1C / ML, M2C / ML, 兺MnC / ML, MI / ML that correspond to the above sloshing frequencies computed through the methodology outlined in Sec. 2, are shown in Table 2 and graphically in Fig. 5. The tabulated values show that when the cylindrical container is nearly full, the entire mass responds “impulsively,” i.e., it follows the motion of the container, and that sloshing effects are negligible. On the other hand, when the container is nearly empty, the impulsive mass is almost zero and sloshing dominates liquid response. For the cases where the liquid height has an intermediate value 共cases of practical interest兲, the total liquid mass is divided in an “impulsive” mass and the “convected” or “sloshing” masses. An important observation is that sloshing mass ratios corresponding to higher modes 共n 艌 2兲 are considerably smaller than the first sloshing mass ratio. Therefore, estimates of the total sloshing force that employ only Transactions of the ASME

Fig. 4 Variation of the first four sloshing frequencies of a twodimensional circular container with respect to liquid height „␭n = ␻2nR / g…, in a two-dimensional circular container

the first sloshing mode 共mass兲 are quite accurate for practical engineering purposes. Regarding the direction of forces FC and FI, since all pressures on the container’s wall are in the radial direction, the resultant forces always pass through the center of the container. Based on the force equation 共13兲 and the equations of motion 共10兲 or 共11兲, and considering only the first sloshing mode for simplicity, a simple spring-mass mechanical model can be proposed, as shown in Fig. 6, which illustrates the motion of the fluid-container system. In this model, y 2 = X共t兲 represents the motion of the external source, and y 1 = u1共t兲 expresses the motion of the liquid mass associated with sloshing. The total liquid mass ML is split in two parts m1 and m2, which correspond to y 1 and y 2, and express the so-called “convective” 共or “sloshing”兲 motion and “impulsive” motion, respectively, a concept introduced in 关1兴. For the particular case of a half-full cylindrical container, m1 ⯝ 0.6ML and m2 ⯝ 0.4ML. It is also straightforward to develop more elaborate mechanical models, which include an appropriate

Fig. 5 Variation of sloshing and impulsive mass ratios mass ratios M1C / ML, M2C / ML, ⌺MkC / ML, MI / ML with respect to liquid height in a two-dimensional circular container

number of spring-mass oscillators, to account for higher modes, describing more accurately the sloshing response of the liquid. 3.2 Effects of Wall Deformation in Long Cylinders Under Transverse Excitation. Industrial horizontal-cylindrical vessels are rather thick, with radius-to-thickness ratio less than 80, to resist high internal pressure. When such vessels are relatively short 共L / R 艋 10兲, it is reasonable to assume that the vessel wall is nondeformable 共rigid兲, as assumed in Sec. 3.1. On the other hand, when the cylindrical vessel is rather long, supported at two locations, the wall may deform. Assuming a beam-type deformation while its cross section remains practically circular 共undeformed兲 it is possible to develop a simple methodology to simulate the coupled response of the liquid-vessel system, as described in the Appendix. In such a case, the motion of the cylindrical container is directly determined by the cylinder axis motion, which can be decomposed in the motion of the supports Xg共t兲, independent of z coordinate, and the beam-like motion due to container deformation, described by a function y共z , t兲 as shown in Fig. 7

Table 2 Variation of the first three sloshing masses with respect to the liquid height in a horizontal cylinder. The values of ML refer to liquid only, excluding the mass of the container. e

M1C ML

M2C ML

M3C ML

M4C ML

兺MkC ML

MI ML

−1.0 −0.95 −0.9 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 0.9 0.95 1.0

1.0 0.97994 0.95974 0.91895 0.83566 0.74989 0.66125 0.56916 0.47276 0.37077 0.26115 0.14032 0.07361 0.03793 0.0

0.0 0.000154 0.000577 0.002010 0.006113 0.010585 0.014636 0.017797 0.019669 0.019790 0.017534 0.011849 0.007076 0.003948 0.0

0.0 0.0000006 0.0000079 0.0000821 0.0006170 0.0016235 0.0028551 0.0040544 0.0050013 0.0054773 0.0052114 0.0037700 0.0023414 0.0013389 0.0

0.0 0.0000000 0.0000004 0.0000115 0.0001524 0.0004958 0.0009891 0.0015392 0.0020323 0.0023499 0.0023401 0.0017673 0.0011258 0.0006539 0.0

1.0 0.98009 0.96032 0.92105 0.84264 0.76310 0.68110 0.59547 0.50407 0.40753 0.30165 0.18100 0.09045 0.05940 0.0

0.0 0.01991 0.03968 0.07895 0.15736 0.23690 0.31890 0.40543 0.49593 0.59247 0.69835 0.81900 0.90955 0.94060 1.0

Journal of Pressure Vessel Technology

AUGUST 2006, Vol. 128 / 331

Fig. 6 Mechanical model representing sloshing in liquid storage vessels Fig. 8 Mechanical model representing sloshing in a deformable horizontal cylindrical container

X共z,t兲 = Xg共t兲 + y共z,t兲

共19兲

Using an admissible function ␺共z兲, considering only the first sloshing mode 共i.e., n = 1兲, and assuming a straightforward threedimensional generalization of the two-dimensional solution of the nondeformed cylinder, the response of the coupled fluid-vessel system under earthquake excitation can be approximated in a simple and elegant manner through the equivalent mechanical model shown in Fig. 8. The formulation and the development of the model is described in detail in the Appendix. The mechanical model of Fig. 8 indicates that the response is governed by two frequencies, namely the fundamental sloshing frequency ␻1, which represents sloshing liquid motion and is directly obtained from Table 1 共or Fig. 4兲, and the “deformationimpulsive” frequency ␻⌬, which represents the motion due to container deformation. Furthermore, the model indicates that the total mass of the system MT is divided in three parts: a convective mass M1C, which represents the low-frequency sloshing motion of the liquid, an “impulsive” mass MI* that follows exactly the motion of the container, and a “deformation-impulsive” mass M*⌬ expressing container’s deformation 共a high-frequency motion兲. Masses M1C, M*⌬, and MI* are defined in the Appendix. Thus, the total horizontal force becomes F = − M1Cy¨ 1 − M⌬* y¨ 2 − MI*X¨g

共20兲

3.3 Horizontal Cylinders Under Longitudinal Excitation. There exist no analytical or numerical results for externally induced sloshing in a horizontal cylinder under excitation Z共t兲 in the longitudinal direction z 共Fig. 3兲. However, it has been shown experimentally 关20兴 that the sloshing response is similar to the sloshing response of an equivalent rectangular container, which has the same free-surface dimensions with the cylindrical vessel, and contains the same liquid volume. Papaspyrou et al. 关29兴 presented a numerical proof of this equivalence for the particular case of halffull horizontal cylinders, comparing the corresponding sloshing frequencies. Recently, this equivalence has been demonstrated for

Fig. 7 Beam-type deformation of a long horizontal cylinder

332 / Vol. 128, AUGUST 2006

horizontal cylinders with arbitrary liquid depth, using a numerical finite element formulation 关28兴. More specifically, setting the external excitation equal to zero 共Z共t兲 = 0兲 then ⌽U = 0, so that ⌽S = ⌽ and the sloshing frequencies are determined by the solution of a free-vibration 共eigen-value兲 problem. Solutions for the sloshing potential ⌽ are sought in the following form ⬁

⌽=

兺

˜␸ p共x,y,t兲cos共k pz兲

共21兲

p=1,2,3,. . .

where k p = 共2p − 1兲␲ / L, p = 1 , 2 , 3 , . . ., so that the boundary conditions at z = 0 and z = L are directly satisfied, and the solution is antisymmetric with respect to the z = L / 2 plane. Subsequently, using the variational form of the problem and applying Galerkin discretization in functions ␸ p in terms of triangular finite ele共p兲 共p兲 ments, the sloshing frequencies ␻n are calculated 共␻n the nth frequency of the pth longitudinal mode兲, as described in detail in 关28兴. The numerical values for the sloshing frequencies are very close to the the experimental values reported in 关19兴. Furthermore, in Fig. 9, the sloshing frequencies, computed through the finite element analysis, are compared with the corresponding values from the equivalent rectangular container. The cross section of the equivalent rectangular container is shown schematically in Fig. 10. The liquid height Heq of the equivalent rectangle can be easily computed in terms of the cylinder diameter D = 2R and the liquid height H in the cylinder 共0 艋 H 艋 2R兲, using simple geometric considerations

Fig. 9 Variation of the fundamental sloshing frequency ␻1„1… in a horizontal cylinder †longitudinal excitation‡, with respect to the liquid depth; finite element results versus equivalent rectangle predictions

Transactions of the ASME

Fig. 10 Equivalent rectangular container concept to approximate the sloshing response of horizontal cylinders under longitudinal excitation

冋

sin−1共H/R − 1兲 + ␲/2 Heq 1 共H/R − 1兲 + = 冑共H/R兲共2 − H/R兲 2 R

册

共22兲

According to sloshing solutions in rectangular containers 关19兴, the following closed-form expressions can be used to calculate the sloshing frequencies and the corresponding convective masses

冋

˜ 2pR 共2p − 1兲␲R ␻ 共2p − 1兲␲Heq = tanh g L L

册

p = 1,2,3, . . . 共23兲

˜ M 8 tanh关共2p − 1兲␲Heq/L兴 pc = 共Heq/L兲␲2共2p − 1兲3 ML

p = 1,2,3, . . .

共24兲 ⌽S = ␸共r, ␪,t兲cos ␺

where Heq is given by Eq. 共22兲, and the total force is ⬁

F = − MTZ¨ −

兺

˜ d¨ M pc p

共25兲

p=1,2,3,. . .

where dn are calculated from the corresponding equations of motion ˜ pd˙ p + ␻ ˜ 2pd p = − Z¨ d¨ p + 2␰ p␻

p = 1,2,3, . . .

The dependence of normalized sloshing frequencies

共26兲 ˜ 2pR / g兲 共␻

˜ / M 共p = 1 , 2 , 3 , . . . 兲 on the liquid depth within the cylinand M pc L drical container are depicted in Figs. 11 and 12. Figure 13 shows the dependence of the dominant sloshing frequency ␻共1兲 1 on the aspect ratio L / R. In particular, for long containers with large values of aspect ratio L / R 共corresponding to Heq / L ⬍ 0.1兲 the rectangular container can be considered “shallow” and the following expressions for the sloshing frequencies and masses can be used

冉 冊

˜ 2pL ␻ Heq = 共2p − 1兲2␲2 g L ˜ M 8 pc = ML ␲2共2p − 1兲

p = 1,2,3, . . .

p = 1,2,3, . . .

Fig. 11 Variation of the first three sloshing frequencies of a horizontal cylinder under longitudinal excitation, with respect to the liquid height

共29兲

where coordinates r , ␪ , ␺ are defined in Fig. 14. Subsequently, considering the variational form of the boundary-value problem and assuming a discretization of ␸ in terms of harmonic functions, the differential equations of motion are obtained, and sloshing frequencies, modes, and masses are computed, as described in 关27兴. In Table 3 and Fig. 15, numerical results are presented for the normalized sloshing frequencies 共␭n = ␻2nR / g兲 of a spherical vessel in terms of the liquid height parameter 共e兲. The results are in very good agreement with those from other numerical solutions 关25兴,

共27兲

共28兲

Equations 共27兲 and 共28兲 and are directly obtained from Eqs. 共23兲 and 共24兲 and considering Heq / L → 0. Spring-mass mechanical models for rectangular containers have been presented in 关1,19兴.

4

Spherical Vessels

In the following, the sloshing response of spherical vessels under horizontal seismic excitation is examined, in terms of sloshing frequencies and masses, employing the seminumerical results obtained in 关27兴. Finally, the effects of support flexibility on the overall seismic response are also investigated. 4.1 Nondeformable Spherical Vessel Under Horizontal Excitation. Due to the nature of the external excitation, solution for the sloshing potential ⌽S is sought in the following form Journal of Pressure Vessel Technology

Fig. 12 Variation of the first three sloshing masses of a horizontal cylinder under longitudinal excitation, with respect to the liquid height

AUGUST 2006, Vol. 128 / 333

Table 3 Variation of the first three sloshing frequencies with respect to the liquid height in a spherical vessel e

␻21R / g

␻22R / g

␻23R / g

␻24R / g

−1.0 −0.9 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 0.9 0.95 1.0

1.0 1.0347 1.0723 1.1583 1.2625 1.3924 1.5602 1.7882 2.1232 2.6864 3.9595 5.7615 8.3121 ¯

7.0 6.5638 6.2008 5.6742 5.3683 5.2406 5.2756 5.4930 5.9729 6.9574 9.4551 13.1776 18.5527 ¯

17.0 13.8911 11.8764 9.8543 8.9418 8.5509 8.5045 8.7793 9.4763 10.9566 14.7598 20.4520 28.6891 ¯

31.0 26.757 17.032 13.866 12.421 11.800 11.684 12.021 12.938 14.918 20.033 27.702 38.816 ¯

冋 Fig. 13 Variation of the first sloshing mass of a horizontal cylinder under longitudinal excitation, with respect to the liquid height and the aspect ratio of the container

as well as with experimental results reported in 关19兴. The corresponding sloshing mass ratios M1C / ML , M2C / ML , 兺MnC / ML , MI / ML are shown in Table 4 and in Fig. 16 in terms of the liquid height. The values are similar to those depicted in Table 2 and Fig. 5 for horizontal cylinders. Higher mode masses 共n 艌 2兲 are generally much smaller than the first sloshing mass ratio and, therefore, they may be neglected in seismic force calculations. Furthermore, the spring-mass mechanical model, shown in Fig. 6, also applies for the spherical vessel. Finally, forces FC and FI always pass through the center of the spherical container.

M1C

0

0

MI

册冋 册 冋 y¨ 1

y¨ 2

+

KC

− KC

− KC KC + Kbs

册冉冋 册 冋 册 冊 y1

y2

−

1 1

Xg = 0 共32兲

KC = ␻21M1C

where and MI = MT − M1C. Equation 共32兲 corresponds to a standard two-DOF mechanical system, shown in Fig. 18, representing the coupled motion of the liquid-container system. Support system stiffness Kbs should comprise the stiffness of the columns and the stiffness of the X-braces, whereas damping can be considered introducing a term proportional to the first time derivatives of the unknown functions. Conducting a standard eigenvalue analysis of the two-DOF mechanical system described in Eq. 共32兲, and using the fact that in the majority of practical applications, KC Ⰶ Kbs, one readily results in the following approximate expressions for the two natural frequencies of the system

␻共21兲 = ␻21 =

KC M1C

共33兲

4.2 Effects of Support Flexibility. Industrial spherical vessels are supported by a structural system with vertical legs and X-braces 共Fig. 17兲, which deforms with the sphere motion. Assuming elastic behavior of the support system, the total horizontal force including the inertia force of the container wall, should be equilibrated by the force of the support system, so that F = Kbs共X − Xg兲

共30兲

where Kbs is the elastic stiffness of the support system, X共t兲 is the motion of the elevated container, and Xg共t兲 is the ground motion. Subsequently, setting y 1 = a1共t兲 + X共t兲

and

y 2 = X共t兲

共31兲

and using Eq. 共10兲 for n = 1 共first sloshing mode only兲, and Eqs. 共30兲 and 共31兲, the following system of equations is obtained

Fig. 14 Configuration of a spherical container

334 / Vol. 128, AUGUST 2006

Fig. 15 Variation of the first four sloshing frequencies of a two-dimensional circular container with respect to liquid height „␭n = ␻2nR / g…, in a spherical container

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Table 4 Variation of the first three sloshing masses with respect to the liquid height in a spherical vessel. The values of ML refer to liquid only, excluding the mass of the container. e

M1C ML

M2C ML

M3C ML

M4C ML

兺MkC ML

MI ML

−1.0 −0.95 −0.9 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 0.9 0.95 1.0

1.0 0.98315 0.96594 0.93038 0.85437 0.77117 0.67990 0.57969 0.46981 0.35009 0.22222 0.09363 0.03655 0.01364 0.0

0.0 0.000102 0.000387 0.001374 0.004341 0.007850 0.011396 0.014576 0.016874 0.017526 0.015419 0.009185 0.004387 0.001851 0.0

0.0 0.0000005 0.0000064 0.0000667 0.0005155 0.0013969 0.0025337 0.0037169 0.0047195 0.0052523 0.0048997 0.0031036 0.0015438 0.0006696 0.0

0.0 0.0000000 0.0000003 0.0000104 0.0001406 0.0004640 0.0009401 0.0014976 0.0020190 0.0023532 0.0022806 0.0014989 0.0007635 0.0003361 0.0

1.0 0.98326 0.96634 0.93184 0.85947 0.78136 0.69619 0.60594 0.49844 0.38440 0.26162 0.12608 0.05586 0.01810 0.0

0.0 0.01674 0.03366 0.06816 0.14053 0.21864 0.30381 0.39406 0.50156 0.61560 0.73838 0.87392 0.94414 0.98190 1.0

␻共22兲 ⬟ ␻I2 =

Kbs MI

共34兲

and that the masses corresponding to ␻1 and ␻1 are M1C and MI, respectively, so that the total force is F = − M1Cy¨ 1 − MIy¨ 2

5

共35兲

Seismic Design Applications

In the following, the calculation of the seismic design force is described using the models presented in the previous sections. The methodology is illustrated in three design examples, which refer to typical cylindrical and spherical pressure vessels. 5.1 Combination of Convective and Impulsive Forces. An important issue toward the calculation of the seismic design force is the calculation of the maximum convective force FC,max. Using only the first mode 共n = 1兲 in Eq. 共16兲, then

共36兲

FC,max = M1Cu¨1,max

If higher modes are used, then an appropriate combination of the maximum modal values Fnc,max = Mncu¨n,max is necessary. Nevertheless, numerical calculations of hydrodynamic forces in horizontal cylinders and spheres 关27兴 showed that consideration of the first mode only may provide a very accurate prediction of the convective force. Furthermore, in the course of a design procedure, the u¨1,max value in Eq. 共36兲 is replaced by the corresponding spectral value SA. Subsequently, the maximum convective force FC,max and the maximum impulsive force 共37兲

FI,max = MIAg

should be combined to provide the seismic design force FD 共Ag is the peak ground acceleration兲. For simplicity, the seismic design force can be calculated through the SRSS combination method, as suggested by Eurocode 8 FD = 冑共FC,max兲2 + 共MIAg兲2

共38兲

Alternatively, the sum of the impulsive and convective maximum values, as suggested in API 650 关2兴, can be employed. 5.2 Horizontal-Cylindrical Vessel I. A horizontal-cylindrical steel vessel containing liquid with density ␳ = 600 kgr/ m3, is 6 m long, with external diameter and thickness equal to 2 m and 2 cm, respectively. Due to the low value of the container’s aspect ratio 共L / R = 6兲, deformation under transverse excitation can be neglected. For design purposes, the following elastic design spectrum is used 0 艋 T 艋 0.15 s:

冋

SA共T兲 = Ag 1 +

0.15 s 艋 T 艋 0.60 s:

Fig. 16 Variation of sloshing and impulsive mass ratios mass ratios M1C / ML, M2C / ML, ⌺MkC / ML, MI / ML with respect to liquid height in a spherical container

Journal of Pressure Vessel Technology

T 共2.5␩ − 1兲 0.15

SA共T兲 = 2.5Ag␩

册

共39兲 共40兲

Fig. 17 Deformation and equilibrium of support system in spherical containers

AUGUST 2006, Vol. 128 / 335

Fig. 18 Mechanical model representing sloshing in an elevated spherical container

T 艌 0.6 s:

SA共T兲 = 1.5Ag␩

1 T

共41兲

where T is the natural period of the oscillator in seconds, and ␩ is a damping coefficient, given in terms of the damping ratio ␰ by

␩ = 冑7/共2 + ␰兲

共42兲

The above spectrum is taken directly from Eurocode 8 关30兴, ENV 1998-1-1 共CEN 1994兲, paragraph 4.2.2, assuming subsoil class B, and importance factor equal to 1, and it is shown graphically in Fig. 19 for different values of damping. First, consider the cylinder to be half-full and excited in the transverse direction, with a peak ground acceleration Ag = 0.24 g. The liquid mass and the container’s shell mass are ML = 5.43 ⫻ 103 kg, and MSH = 5.82⫻ 103 kg, respectively 共density of steel is 7800 kg/ m3兲. From Table 2 共or equivalently, Fig. 5兲, MC = 0.595ML = 3.23⫻ 103 kg, and therefore, MI = ML + MSH − MC = 8.02⫻ 103 kg. From Table 1 共or equivalently, Fig. 4兲, ␻21R / g = 1.356, so that ␻1 = 3.684 rad/ s, the first natural period is T1 = 1.71 s and 关from Eq. 共41兲兴 the corresponding spectral acceleration value for 1% damping is SA = 3.163 m / s2. It is assumed that the entire convective mass MC corresponds to this fundamental frequency. Therefore, using Eqs. 共36兲 and 共37兲, the maximum convective and impulsive forces are computed equal to 10.22 and 18.89 kN, respectively. In addition, using Eq. 共38兲, the seismic design force is equal to 29.11 kN. Using the same methodology, the impulsive, convective, and total seismic forces for different liquid levels are shown in the last three columns of Table 5. Com-

Fig. 19 Elastic design spectrum „normalized…, from Eurocode 8 for soil class B, importance factor equal to 1, and two different damping ration „0.5% and 2%…

parison of the impulsive and convective forces shows that the former is significantly larger, due to the mass of the container, which constitutes a substantial part of the impulsive mass, and because of the small values of ␻1, which correspond to relatively small spectral values SA共T兲. The half-full cylinder 共e = 0兲 is also examined under longitudinal excitation with Ag = 0.24 g. For e = 0, one obtains Heq / R = 0.785. Using Eqs. 共23兲 and 共24兲, and assuming 1% damping, one obtains the sloshing frequencies, periods, spectral values, and the corresponding masses, depicted in Table 6. The impulsive mass is 3 ˜ =M +M −⌺ M ˜ M I L SH p pc = 6.664⫻ 10 kg. Using those values, the design convective forces F1C,maxF2C,max, and F3C,max are 5.09, 1.04, and 0.36 kN, respectively, and the impulsive force FI,max is 15.69 kN. Finally, using a SRSS combination of the above four

Table 5 Impulsive, convective, and total force in the nondeformable horizontal cylinder of Design Example I for different positions of the liquid surface

e

␻1 共rad/s兲

ML 共103 kg兲

MT 共103 kg兲

MC 共103 kg兲

MI 共103 kg兲

SA 共T兲 共m / s2兲

M IA g 共kN兲

M CS A 共kN兲

FD 共kN兲

−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8

3.232 3.314 3.412 3.532 3.684 3.885 4.167 4.611 5.500

0.565 1.55 2.74 4.06 5.43 6.81 8.12 9.32 10.30

6.39 7.37 8.56 9.88 11.25 12.63 13.94 15.14 16.12

0.521 1.30 2.09 2.76 3.23 3.43 3.31 2.81 1.86

5.87 6.06 6.47 7.11 8.02 9.19 10.63 12.33 14.25

2.78 2.85 2.93 3.03 3.16 3.34 3.58 3.96 4.72

13.81 14.28 15.23 16.75 18.89 21.65 25.03 29.02 33.56

1.45 3.71 6.13 8.38 10.22 11.44 11.84 11.12 8.80

13.89 14.75 16.42 18.73 21.47 24.49 27.69 31.08 34.70

Table 6 Sloshing frequencies, masses, and spectral values for cylindrical vessel I, under longitudinal excitation p

˜ 2pR / g ␻

˜ p 共rad/s兲 ␻

T p 共s兲

SAp 共m / s2兲

˜ /M M pc L

˜ 共kg兲 M pc

1 2 3

0.204 1.325 2.533

1.422 3.624 5.011

4.419 1.734 1.254

1.221 3.111 4.301

0.768 0.062 0.015

4.170⫻ 103 0.335⫻ 103 0083⫻ 103

336 / Vol. 128, AUGUST 2006

Transactions of the ASME

Fig. 20 Configuration for half-full long cylindrical pressure vessel

values, one obtains a design force FD equal to 16.53 kN, whereas consideration of the first mode only results in a seismic design force FD equal to 16.49 kN. Clearly, consideration of the first sloshing mass only provides very good results. 5.3 Design Example II: Horizontal-Cylindrical Vessel II. An 18-m-long horizontal-cylindrical steel vessel is considered, which contains liquid with density ␳ = 600 kg/ m3, with external diameter and thickness equal to 2 m and 2 cm, respectively. The vessel configuration is schematically shown in Fig. 20. Due to the relatively high value of the container’s aspect ratio L / R, deformation under transverse excitation should be taken into consideration. A finite element analysis of the empty vessel, using standard shell elements, has shown that a beam-type mode is the dominant mode for external excitation in the transverse direction. For design purposes, the elastic design spectrum defined in Eqs. 共39兲–共41兲 is used. The container’s mass per unit length is mSH = MSH / L = 970.3 kg/ m and the liquid mass per unit length is mL = 905 kg/ m. To describe the deformation of the container, the following assumed function is considered

␺共z兲 = sin

␲z − 0.342 L

共43兲

so that ␺ = 0 at z = 2 m and z = 16 m. Using Eqs. 共A12兲 and 共A13兲 for half-full container, the values of Table 7 are calculated. Furthermore, using Eq. 共A28兲, one finds the values of MI* and M*⌬ equal to 12.56⫻ 103 kg and 11.51⫻ 103 kg, respectively, as well as the value of Kb = 106,980 kN/ m from Eq. 共A14兲. For halffull container, ␻21R / g = 1.356, so that ␻1 = 3.684 rad/ s, the first natural period is T1 = 1.71 s and 共from Eq. 共41兲兲 the corresponding spectral value for 1% damping is SA = 3.163 m / s2. The “deformation” frequency ␻⌬ is computed from Eq. 共A16兲 equal to 156.5 rad/ s, corresponding to a period of 0.040 s. Therefore, the convective, impulsive, and deformation forces are 30.66, 29.56, and 43.75 kN. Using a SRSS combination of the maximum forces, the seismic design force is equal to 61.06 kN. 5.4 Spherical Pressure Vessel. A steel pressure vessel of spherical shape 共Fig. 21兲 has diameter D = 21,216 mm and thickness t = 43 mm, and contains propylene 共density ␳ = 553 kg/ m3兲. The mass of the empty vessel is MSH = 0.472⫻ 106 kg. The vessel is supported by 12 tubular 쏗1160 mm⫻ 60 mm legs with effective height hL = 9.2 m and X braces of rectangular section 共250 mm⫻ 35 mm兲 and effective length Lb = 8.2 m, inclined 60 deg with respect to the ground. An approximate expression for calculating the support stiffness is

Table 7 Masses and generalized masses for horizontal cylinder II Convective masses 共103 kg兲 MC MC M C⬘

9.693 2.856 1.759

Total masses 共103 kg兲 MT MT M T⬘

33.757 9.945 6.127

Journal of Pressure Vessel Technology

Impulsive masses 共103 kg兲 MI MI M I⬘

24.064 7.089 4.368

Fig. 21 Spherical vessel of design example

NL

Kbs =

兺 j=1

12EIL hL3

Nb

+

兺 k=1

EAb cos2 ak cos2 ␤ Lb

共44兲

where hL and IL are the height and the moment of inertia of the columns, respectively, Ab and Lb are the cross-sectional area and the length of the braces, NL is the number of vertical columns 共legs兲, Nb is the number of braces in tension, ␣k is the horizontal angle between each brace 共k兲 and the earthquake excitation, and ␤ is the brace inclination with respect to the ground. In addition, fixed-fixed conditions for the vertical legs have been assumed. Using Eq. 共44兲, the stiffness of the support system is estimated Kbs = 1.56⫻ 106 kN/ m. If the vessel is half-full 共e = 0兲, the liquid mass is ML = 1.366 ⫻ 106 kg so that the total moving mass is MT = 1.838⫻ 106 kg. From Table 4, the convective mass is MC = 0.828⫻ 106 kg and the impulsive mass is MI = MT − MC = 1.010⫻ 106 kg. Using Table 3, the first convective eigen-period T1C is found equal to 5.22 s, and using Eq. 共34兲 the “impulsive” eigenperiod TI is 0.16 s. Using the earthquake spectrum defined by Eqs. 共39兲–共41兲, the corresponding spectral values for the impulsive and the first convective period are SAI共TI兲 = 7.786 m / s2 共for 2% damping兲 and SA1共TC兲 = 1.033 m / s2 共for 1% damping兲. Therefore, the seismic design shear force FD is computed using a SRSS combination FD = 冑共MCSA1兲2 + 共MISAI兲2 = 7.91 MN

共45兲

The force passes through the sphere centroid, located at a height 14.32 m from the ground. Thus, the overturning moment with respect to the ground level is 113.3 MN m. Finally, the last three columns of Table 8 show the maximum convective force, the impulsive force, and the total design force for different liquid levels within the container, using the same methodology.

6

Conclusions

A methodology is presented for calculating the total horizontal seismic force in horizontal-cylindrical and spherical vessels, including the effects of sloshing. The proposed methodology is based mainly on a “convective”—“impulsive” decomposition of the liquid-vessel motion, and a semianalytical solution of sloshing in nondeformable containers, presented by the authors in a previous publication. In all cases, equivalent mechanical 共spring-mass兲 models are proposed, which represent the response of the liquidvessel system. In both horizontal cylinders and spheres, due to the fact that the first convective mass is significantly larger than higher mode masses, the convective force can be estimated using the first AUGUST 2006, Vol. 128 / 337

Table 8 Impulsive, convective, and total force in the sphere of Design Example III for different positions of the liquid surface

e

␻I 共rad/s兲

␻l 共rad/s兲

ML 共106 kg兲

MT 共106 kg兲

MC 共106 kg兲

MI 共106 kg兲

SI 共T兲 共m / s2兲

SA 共T兲 共m / s2兲

M IS I 共MN兲

M CS A 共MN兲

FD 共MN兲

−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8

57.17 55.20 50.95 45.18 39.30 33.87 29.52 26.16 23.64

1.00 1.04 1.08 1.14 1.20 1.29 1.40 1.58 1.92

0.076 0.284 0.590 0.962 1.366 1.770 2.141 2.448 2.655

0.549 0.756 1.062 1.434 1.838 2.242 2.614 2.920 3.128

0.071 0.244 0.461 0.669 0.828 0.882 0.823 0.640 0.335

0.477 0.512 0.601 0.764 1.010 1.360 1.790 2.279 2.792

6.334 6.476 6.821 7.390 7.786 7.786 7.786 7.786 7.786

0.857 0.890 0.930 0.976 1.033 1.106 1.206 1.356 1.646

3.023 3.315 4.099 5.647 7.866 10.588 13.941 17.747 21.743

0.061 0.217 0.429 0.654 0.855 0.976 0.992 0.868 0.551

3.023 3.322 4.122 5.685 7.912 10.633 13.976 17.768 21.750

sloshing mode only. Inclusion of higher modes may not increase significantly the accuracy of the results. For the particular case of long horizontal cylinders under transverse excitation, deformation of the container’s wall may affect the response. Assuming a beamtype deformation, it is possible to estimate those deformation effects, through a simplified liquid-container interaction model. Furthermore, the response of horizontal cylinders under longitudinal excitation can be examined quite accurately using the concept of “equivalent rectangular container.” Finally, in the case of spherical containers, the deformation effects of the support system are taken into account through a simplified 2DOF model. The above issues are illustrated in three typical design examples. Due to the fact that sloshing is a low-frequency motion, the corresponding spectral values are small, and the impulsive part of the total seismic force dominates the response. The proposed methodology offers a simple and efficient tool to calculate seismic forces in horizontal-cylindrical and spherical vessels, it is compatible with the corresponding methodology in existing specifications for vertical liquid storage tanks 共e.g., 关2,15兴兲, and can be used for the seismic design of industrial vessels.

Acknowledgment This work was partially supported by the Earthquake Planning and Protection Organization 共E.P.P.O.兲, Athens, Greece. The authors would also like to thank Mrs. Sotiria Houliara, Graduate Student at the University of Thessaly, for her assistance in the finite element analysis of the horizontal-cylindrical vessel.

Nomenclature an ⫽ generalized sloshing coordinate ag , a ⫽ generalized sloshing coordinates due to ground motion and vessel deformation ¯a1共z , t兲 ⫽ ag + ␺a Ag ⫽ peak ground acceleration d p ⫽ generalized sloshing coordinate 共equivalent rectangle兲 f共z , t兲 ⫽ force per unit length along a horizontal cylinder F ⫽ total horizontal force FI FC ⫽ impulsive and convective forces FD ⫽ design seismic force H , Heq ⫽ liquid height and liquid height of equivalent rectangle k p ⫽ 共2p − 1兲␲ / L KC ⫽ ␻21M1C Kb ⫽ generalized stiffness defined in Eq. 共A14兲 Kbs ⫽ support stiffness in spherical vessels L ⫽ length of horizontal cylinder ML ⫽ liquid mass MI , MC ⫽ impulsive and convective masses MSH ⫽ empty vessel mass MT ⫽ MSH + ML 338 / Vol. 128, AUGUST 2006

Mnc ⫽ convective mass of nth sloshing mode M*⌬ , MI* ⫽ deformation and impulsive mass in deformable horizontal cylinders 共Appendix兲 ˜ ˜ MI , Mnc ⫽ impulsive mass and nth mode convective mass 共equivalent rectangle兲 M C M C⬘ M TM T⬘ ⫽ generalized masses, defined in Eqs. 共A12兲 and 共A13兲 MI ⫽ MT − MC M I⬘ ⫽ M T⬘ − M C⬘ r , ␪ , z ⫽ cylindrical coordinates r , ␪ , ␺ ⫽ spherical coordinates R ⫽ radius of cylinder or sphere SA(T) ⫽ spectral acceleration value un ⫽ an + X ug ⫽ ag + Xg Xg ⫽ ground displacement X, ⫽ container total displacement y共z , t兲 ⫽ container deformation displacement y 1 , y 2 ⫽ degrees of freedom in equivalent mechanical models Z ⫽ ground displacement in the longitudinal direction of a cylinder ␤ ⫽ deformation shape parameter, depending on the assumed shape function ␺ 共Eq. 共A24兲兲 ⌬ ⫽ generalized deformation parameter of a ⌬T ⫽ ␤⌬ + Xg ␩ ⫽ damping parameter for design spectrum ␭n ⫽ ␻2nR / g ␰n ⫽ damping ratio of nth mode ˜␸ p共x , y , t兲 ⫽ two-dimensional sloshing potential corresponding to the pth longitudinal mode ⌽ ⫽ liquid potential ⌽S , ⌽U ⫽ sloshing and uniform motion potential ␺共z兲 ⫽ assumed shape function for vessel deformation ⌿n ⫽ mode shape of the nth sloshing mode ␻n ⫽ nth sloshing frequency ␻共p兲 n ⫽ nth sloshing frequency the pth longitudinal mode ˜ 2p ⫽ sloshing frequency of pth mode 共equivalent ␻ rectangle兲 ␻⌬ ⫽ natural frequency of sphere due to support flexibility

Appendix: Simplified Fluid-Vessel Interaction for Long Cylinders Under Transverse Excitation It is possible to estimate the effects of wall deformation on the fluid-vessel system response, using a simplified formulation. This method has been introduced for half-full cylinders in 关31兴, and herein, it is reformulated for the case of arbitrary liquid height, within the cylindrical container. In addition, an equivalent meTransactions of the ASME

chanical two-DOF model is developed for the case of earthquake 共high-frequency兲 excitation, which illustrates the coupled liquidvessel response in an elegant manner. Industrial horizontal-cylindrical vessels are rather thick, with radius-to-thickness ratio less than 80, to resist high levels of internal pressure. When such vessels are relatively short 共L / R 艋 12兲, a common case in practice, it is reasonable to assume that the vessel wall is nondeformable 共rigid兲, a basic assumption of the above-described methodology. In longer cylindrical vessels, wall deformation occurs that may affect the seismic response. As a first approximation and for the sake of simplicity, a beam-type deformation of the container is considered, neglecting local 共shell-type兲 modes, while the cylinder cross section remains circular 共undeformed兲 due to its significant thickness. Thus, the motion of the cylindrical container is directly determined by the motion of the cylinder axis, which is decomposed in two parts 共Fig. 7兲, the motion of the supports Xg共t兲, independent of z coordinate, and the motion due to the deformation of the container described by a function y共z , t兲, as expressed in Eq. 共19兲. Furthermore, an admissible function ␺共z兲 is considered for the beam-type deformation of the cylindrical container in Eq. 共19兲 y共z,t兲 = ␺共z兲⌬共t兲

共A1兲

so that the deformable vessel becomes a generalized singledegree-of-freedom system. In such a case the vessel undergoes a nonuniform motion with respect to z coordinate and the sloshing solution is three-dimensional. Nevertheless, in the majority of practical applications, the principal sloshing frequency ␻1 is significantly smaller than the fundamental frequency of the container and, therefore, it can be assumed that the two-dimensional sloshing solution for the nondeformable container described in the previous section and expressed in the form of Eq. 共10兲 and 共11兲, is still valid for every cross section. More specifically, for the cross section corresponding to coordinate z, the following equation is considered for the first mode 共n = 1兲 which is analogous to Eq. 共10兲

⳵2¯a1 ⳵¯a1 ⳵ 2X + ␻21¯a1 = − 2 2 + 2 ␰ 1␻ 1 ⳵t ⳵t ⳵t

共A2兲

where ¯a1 = ¯a1共z , t兲, ␻1 is the first sloshing frequency and ␰1 is the corresponding damping ratio. Equation 共A2兲 contains two unknowns, namely the sloshing motion ¯a1 = ¯a1共z , t兲 and the motion of the container ⌬共t兲, and, therefore, the dynamic interaction between the liquid motion and the deformed vessel should be considered. The force per unit length f along the cylinder is given by an equation analogous to Eq. 共12兲 f共z,t兲 = − mC

⳵2¯a1 ⳵ 2X 2 − mT 2 ⳵t ⳵t

共A3兲

where mC = M1C / L and mT = MT / L are the convective and total mass per unit length, respectively, and the value of M1C is given in Fig. 4. Note that both mC and mT are constant along the cylinder. From Eqs. 共19兲 and 共A1兲, the unknown sloshing function ¯a1共z , t兲 is the sum of two parts, one corresponding to the ground motion ag共t兲 and the other ␺共z兲a共t兲 corresponding to the vessel motion relative to the ground motion ¯a1共z,t兲 = ag共t兲 + ␺共z兲a共t兲

共A4兲

so that the unknown functions ag共t兲 and a共t兲 satisfy a¨g + 2␰1␻1a˙g + ␻21ag = − X¨g

共A5兲

¨ a¨ + 2␰1␻1a˙ + ␻21a = − ⌬

共A6兲

In the above equations, a共t兲 expresses the effects of wall deformation on sloshing, and the dot denotes derivative with respect to time. Furthermore, from Eqs. 共A3兲–共A6兲, the total lateral force per unit length of the cylinder at cross-section z is Journal of Pressure Vessel Technology

f共z,t兲 = − mCa¨g − mTX¨g − mC␺共z兲a¨ − mT␺共z兲⌬¨

共A7兲

Equilibrium of the beam-like container requires that EI

冉 冊

⳵4y = f共z,t兲 ⳵z4

共A8兲

where EI is the bending stiffness of the beam-like cylinder. Using an admissible trial function w共z兲, the weak form of the above equilibrium equation is obtained

冕

L

EIy ⬙共z,t兲w⬙共z兲dz =

0

冕

L

f Tw共z兲dz

共A9兲

0

where 共 兲⬙ denotes double differentiation with respect to z. Approximating the trial function as follow w共z兲 = Aw␺共z兲

共A10兲

where Aw is an arbitrary number, using Eqs. 共A7兲, 共A9兲, and 共A10兲, and introducing a term proportional to ⌬˙ to account for structural damping, one obtains M Ca¨g + M TX¨g + M C⬘ a¨ + M T⬘ ⌬¨ + Cb⌬˙ + Kb⌬ = 0

共A11兲

where the “generalized masses” M C, M T, M C⬘ , and M T⬘ , and the “generalized bending stiffness” Kb are given by M C = mC

冕

L

␺共z兲dz

M T = mT

0

M C⬘ = mC

冕

冕

L

␺共z兲dz

共A12兲

␺2共z兲dz

共A13兲

0

L

M T⬘ = mT

␺2共z兲dz

0

冕

L

0

Kb = EI

冕

L

␺⬙2共z兲dz

共A14兲

0

and Cb is a damping coefficient. Equations 共A5兲, 共A6兲, and 共A11兲 are the equations of motion of the coupled liquid-vessel system, expressing the dynamic interaction between the liquid and the vessel. Setting X共t兲 = 0, the three natural frequencies ␻共i兲 of the undamped coupled fluid-vessel system 共␰S = Cb = 0兲 can be calculated. The first natural frequency ␻共1兲 is directly calculated from the first equation of motion 共A5兲 equal to ␻1. Furthermore, using the fact that in practical applications ␻21 Ⰶ Kb / M T⬘ , the following simple expressions can be obtained for ␻共2兲 and ␻共3兲

␻共22兲 ⯝ ␻21 ␻共23兲 ⬟ ␻⌬2 ⯝

Kb M T⬘ − M C⬘

共A15兲

=

Kb M I⬘

共A16兲

where M I⬘ = M T⬘ − M C⬘ in Eq. 共A16兲 can be regarded as a “generalized impulsive mass,” analogous to MI in Eq. 共15兲. Following the definition of ␻⌬, which expresses the motion of the generalized impulsive mass M I⬘, the damping coefficient can be computed as follows Cb = 2␰b␻⌬M I⬘

共A17兲

where ␰b is the structural damping ratio. The total force on the vessel wall 共including the inertia force of the container兲 is computed as follows AUGUST 2006, Vol. 128 / 339

F=

冕

L

f共z,t兲dz = − MC共a¨g + X¨g兲 − MIX¨g − M C共a¨ + ⌬¨ 兲 − M I⌬¨

0

共A18兲 where MI = MT − MC and M I = M T − M C. A simplification of the above formulation is possible assuming that the contribution of container deformation on sloshing a共t兲 is negligible. Therefore from Eq. 共A6兲, a¨ + ⌬¨ = 0, and the force equation 共A7兲 becomes F ⯝ − MC共a¨g + X¨g兲 − MIX¨g − M I⌬¨

共A19兲

On the other hand, Eq. 共A11兲, which expresses the motion of the container, can be written as follows M I⬘⌬¨ + Cb⌬˙ + Kb⌬ = − M C共a¨g + X¨g兲 − M IX¨g

共A20兲

Furthermore, in the case of high-frequency excitation Xg共t兲, one can easily show that the amplitude of a¨g + X¨g is very small when compared with the amplitude of X¨g. Therefore, the above equation of motion can be written approximately as follows ¨ +C ⌬ ˙ ¨ M I⬘⌬ b + K b⌬ = − M IX g

共A21兲

Subsequently, setting ug = ag + Xg

共A22兲

⌬T = ␤⌬ + Xg

共A23兲

where

冕 冕

L

␤=

␺2 dz

0

共A24兲

L

␺ dz

0

the equations of motion 共A5兲 and 共A21兲 and the total force equation 共A19兲 become u¨g + ␻21共ug − Xg兲 = 0

共A25兲

¨ + ␻2 共⌬ − X 兲 = 0 ⌬ T T g ⌬

共A26兲

F = − MCu¨g − M ⌬* ⌬¨ T − M I*X¨g

共A27兲

where M⌬* =

M I2 M I⬘

and

MI* = MI −

M I2 M I⬘

共A28兲

Obviously M⌬* + MI* + MC = MT

共A29兲

Equation 共A29兲 shows that the total mass of the liquid-container MT system is divided in three parts. The first part is a convective mass MC, which represents the sloshing motion of the liquid, the second part is an “impulsive” mass M*⌬ that expresses container’s deformation, and the third part is another “impulsive” mass MI* that follows the motion of the container. Furthermore, Eqs. 共A25兲–共A29兲 motivate the equivalent mechanical model shown in Fig. 8 with parameters y 1 = ug共t兲, y 2 = ⌬T共t兲, y 3 = Xg共t兲.

References 关1兴 Housner, G. W., 1957, “Dynamic Pressures on Accelerated Fluid Containers,” Bull. Seismol. Soc. Am., 47, pp. 15–35.

340 / Vol. 128, AUGUST 2006

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