Sloshing Effects in Half-Full Horizontal Cylindrical Vessels Under Longitudinal Excitation

Transcription

1 S. Paasyrou D. Valougeorgis Mem. ASME S. A. Karamanos Deartment of Mechanical Industrial Engineering, University of Thessaly, Volos 38334, Greece Sloshing Effects in Half-Full Horizontal Cylindrical Vessels Under Longitudinal Excitation A mathematical model is develoed for sloshing effects in half-full horizontal cylindrical vessels, under external excitation in the direction of the longitudinal vessel axis. In this geometry the roblem is not searable. The velocity otential is exressed in a double series form, after some mathematical maniulation the roblem results in a series of systems of ordinary linear differential equations, one for each longitudinal mode. For the articular case of harmonic motion, a semi-analytical solution is ossible. Keeing only the first two terms of the exansion in the transverse direction, an elegant formulation of remarkable accuracy is obtained. Hydrodynamic ressures forces are calculated for harmonic excitation for a real seismic motion event. Finally, the equivalence between the half-full horizontal cylinder an equivalent rectangle is demonstrated. DOI: 0.5/ Introduction The linearized sloshing roblem can be treated as an eigenvalue roblem, reresenting free fluid vibrations inside a container, rovides the sloshing frequencies modes. Under a secific external excitation it is converted into a transient roblem, reresenting fluid motion within the moving container. The sloshing solution deends strongly on the container shae. For nondeformable rectangular vertical-cylindrical containers the sloshing roblem for ideal fluids can be solved analytically, using searation of variables,, resulting in a set of uncouled equations, one for each sloshing mode. In articular, the case of vertical cylindrical vessels has been extensively investigated since the late 950s, mainly because of sace vehicle alications,,3, as well as for determining the seismic resonse of liquid storage tanks, 4,5. Based on this exact sloshing solution, other effects, such as tank wall deformation, 6,7, or soil-structure interaction, 7,8, can be considered in an aroximate manner. The recent ublication of Ibrahim et al. 9 offers a broad overview of sloshing dynamics with emhasis on vertical cylinders rectangles. Containers of different shae, such as horizontal cylinders or sheres have received much less attention. In these configurations exact analytical solutions may not be available the use of numerical methods becomes necessary, in addition to exerimental testing. Most of the work in horizontal cylinders is focused on the eigenvalue sloshing roblem, towards the comutation of sloshing frequencies the corresonding modes in the transverse longitudinal directions, 0 4. Exerimental measurements of sloshing frequencies in horizontal cylindrical vessels, were reorted by McCarty Stehens 5 Kana 6. The corresonding transient roblem in the transverse direction has been studied in an early work by Budiansky 7, while, recently, Kobayashi et al. 8 reorted exerimental measurements To whom corresondence should be addressed. Contributed by the Alied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for ublication in the ASME JOURNAL OF APPLIED ME- CHANICS. Manuscrit received by the ASME Alied Mechanics Division, Mar. 3, 003; final revision, July 5, 003. Associate Editor: D. A. Siginer. Discussion on the aer should be addressed to the Editor, Prof. Robert M. McMeeking, Journal of Alied Mechanics, Deartment of Mechanical Environmental Engineering University of California Santa Barbara, Santa Barbara, CA , will be acceted until four months after final ublication of the aer itself in the ASME JOURNAL OF APPLIED MECHANICS. for sloshing frequencies hydrodynamic forces in both directions, which were comared with analytical results from equivalent rectangular containers. Generally, sloshing analysis in horizontal cylindrical vessels filled u to an arbitrary height requires an aroximate or numerical solution. However, for the articular case of half-full horizontal cylinders an analytical solution is ossible. Evans Linton 9 resented such a solution for the eigenvalue-sloshing roblem free fluid vibrations in a nondeformable container, exing the velocity otential in terms of non-orthogonal bounded satial functions. The resent work is aimed at calculating the resonse in halffull horizontal cylindrical containers under any form of longitudinal excitation, extending the analytical eigenvalue formulation of Evans Linton 9. In articular, the objective of the aer is the solution of externally induced liquid sloshing in half full cylinders under longitudinal excitation, through a semi-analytical manner, without imlementing finite difference or finite element aroximations. The resent sloshing solution is divided in a uniform motion art, trivially obtained, a art related to sloshing, reresenting the relative fluid motion within the container. Using a series exansion of the velocity otential, the roblem reduces to a system of ordinary linear differential equations, which is solved numerically. For the case of harmonic excitation the formulation results in a system of algebraic equations yielding semi-analytical solutions of benchmark quality. Sloshing frequencies modes, hydrodynamic ressures the corresonding sloshing forces are comuted in a simle efficient manner. Dissiation effects are considered their influence is examined. Furthermore, considering the first two terms of the series in the transverse direction, an elegant simlified formulation is obtained, which yields quite accurate results. The case of harmonic excitation the resonse under a real seismic event are examined. The resonse of the half-full cylinder is comared with the resonse of an equivalent rectangular container, which has the same free-surface dimensions contains the same amount of liquid. It is shown analytically numerically that the equivalent rectangle can be used for aroximating the resonse of the halffull cylinder. Theoretical Formulation Solution. Problem Statement. The fluid is contained in a horizontal cylindrical vessel of radius R length L. In ractical industrial alications, horizontal cylinders are very thick to resist Journal of Alied Mechanics Coyright 004 by ASME MARCH 004, Vol. 7 Õ 55

2 velocity otential f corresonds to a rigid body motion of the fluid, which follows exactly the motion of the external excitation source, satisfies the Lalace equation the nonhomogeneous art of the kinematic conditions 6 at z0 zl. Thus the solution of the uniform motion is antisymmetric with resect to zl/ it is trivially obtained as f z,tżt z L. (8) Then the velocity otential, which reresents the relative motion of the fluid articles within the container due to sloshing, should satisfy the Lalace equation within the fluid region the following boundary conditions: t g r f, at /, rr, 0zL (9) t Fig. Configuration of half-full horizontal cylindrical container high levels of internal ressure therefore, in our formulation, the vessel is assumed rigid non deformable. Furthermore, the vessel is half full undergoes an arbitrary motion in the direction of the longitudinal axis z with dislacement Z(t), as shown in Fig.. The amlitude of the external excitation the resulting free surface elevation are assumed to be sufficiently small to allow linearization of the roblem. It is assumed that the fluid inside the container is inviscid the flow can be described by a velocity otential function (r,,z,t), which satisfies Lalace equation within the fluid volume: r r r r r z 0, rr, //, 0zL. () The velocity otential is subjected to the linearized dynamic kinematic free-surface conditions g0, t at /, rr, 0zL () 0, r t at /, rr, 0zL, (3) resectively, g is the gravitational constant (r,z,t) is the free-surface elevation. Combination of boundary conditions 3 leads to the mixed boundary condition t 0, r at /, rr, 0zL. (4) Moreover, should satisfy the kinematic conditions at the walls of the rigid container 0, r at rr, //, 0zL (5) Żt, z at z0, L, //, 0rR. (6) Subsequently, is decomosed in two arts r,,z,t f z,tr,,z,t (7) f (z,t) (r,,z,t) are the uniform motion velocity otential the otential related to sloshing, resectively. The 0, r at rr, //, 0zL (0) 0, at z0, L, //, 0rR. () z It is noted that the boundary value roblem stated above for the cylinder of semicircular cross section cannot be solved trivially by searation of variables.. Solution for the Potential Related to Sloshing. The solution of the roblem related to sloshing contains certain characteristics. Due to the nature of the external excitation the symmetry of the vessel geometry with resect to lanes zl/ 0, the above exression is anti-symmetric in terms of z longitudinal direction symmetric with resect to the 0 lane. Thus a general solution for the unknown function satisfying the Lalace equation is considered in a series form as r,,z,t,3,5 q ti n0 n n k rcosncosk z, rr, //, 0zL () q n (t) are unknown arbitrary time functions I n (x) are the modified Bessel functions of order n. The boundary conditions at z0 zl are satisfied rovided that k /L,,3,5,... Theexression of is rewritten in the form, 9, r,,z,t,3,5 q n n0 tcosni n k r q n tcosni n k rcosk z (3) searating odd even terms of the series in the transverse direction. Substituting the above exression for into boundary condition 0 alying the integral oerator 0 / cos(m)d, m 0,,..., the following relations between the even odd unknown time functions are obtained: q t m 0 I 0 k R m0 m/ I m k Rq m t,,3,5,... (4) 56 Õ Vol. 7, MARCH 004 Transactions of the ASME

3 t I n k mn R m0 mn/ mn/ mn I m k Rq m t, n0,,3,... (5) q n I(x) denotes the derivative of the modified Bessel function with resect to x. Next substituting Eqs. 8 3 for f (z,t) (r,,z,t) resectively into boundary condition 9 yields n q n,3,5 n0 ti n k r q n n g r I nk rcosk z z L Z t. (6) Alying the integral oerator 0L cos(k S z)dz, s,3,5,..., the double sum of Eq. 6 is converted into a system of equations n0 n q n ti n k rq n n g r I nk r 4 Z t,,3,5... (7) Lk Then, using the identity, 0, one obtains n0 ni n x x I nxi n x (8) n q n tq n t k g q nt k g I n k r 4 Z t, (9) Lk with q 0. Subsequently, imlementing the following exansion, 0, I 0 xi xi 4 xi 6 xi 8 x... (0) equating terms of equal order in n, the following exressions are obtained: q t k g 0 q t 4 Z t,,3,5,... () k L q n t k g q nt k g q nt 8 Z t, k L n0,,3,5,... () Finally by substituting Eqs. 4 5 into Eqs., resectively, the following infinite system of second-order ordinary differential equations is deduced for each mode : M q K q Z,3,5,... (3) M K are square matrices, is a vector, with elements M 0m m/ m I m k R I 0 k R M nm mn mn mn/ K k g, m0,,,...,,3,5,... (4) mn/ I m k R, I n k R n0, m0,,,...,,3,5,... (5) ] ] ] ] ] ],,3,5,... (6) 0 4 k L, n 8, n0,,3,5,... (7) k L q is the unknown vector with comonents q m (t), m 0,,,... Solution of vector Eq. 3 is erformed through a tyical time-marching numerical scheme, leads to the calculation of functions q m (t) their derivatives. For comuta- tional uroses a truncated series in Eq. 3 is considered (n N, N is the truncation size, a truncated system of ordinary differential equations is obtained, which is solved in terms of q n (t). Subsequently, Eqs. 4 5 are used to determine the even functions q n (t), so that the otential associated with sloshing is comletely defined. The elevation of the free surface is obtained by integrating the kinematic boundary condition 3 with resect to time: r,z,t,3,5 n0 q n t n I r n k rcosk z. (8).3 Energy Dissiation. To account for dissiation effects, a daming term roortional to the first derivative of the generalized coordinates is introduced in Eqs. 3, M q C q K q Z, (9) C is a square matrix, considered in the form of a Rayleigh daming matrix C 0 M K (30) 0 are constants. It is interesting to note that in revious works,,, a different formulation was roosed to model damed systems, which enables the use of otential theory introduces a daming term roortional to the otential related to sloshing in the dynamic boundary condition as g0, at /, rr, 0zL. (3) t The extra term reresents a force, which ooses article velocity, the roortionality constant is a viscosity coefficient. Equation 3 leads to the following mixed boundary condition at the free surface t t g r 0, at /, rr, 0zL. (3) If boundary condition 3, instead of 4, is imlemented in the formulation, it yields a system of ordinary differential equations identical to that of Eq. 9, with C M, a secial form of Raleigh daming. Journal of Alied Mechanics MARCH 004, Vol. 7 Õ 57

4 .4 Hydrodynamic Pressures Forces. Once the velocity otential associated with sloshing is calculated, the hydrodynamic ressure at any location can be comuted from the linearized Bernoulli equation Pr,,z,t t f t t. (33) On the right-h side of Eq. 33 the first term refers to the uniform motion, while the second term refers to sloshing effects. The ressure associated with sloshing at zl is P s,zl,3,5 q I n0, n n k rcosn. (34) The total horizontal force acting on the container is obtained by an aroriate integration of the ressure as FAPr,,z,te z "nda (35) A is the area of the two wet semicircular ends of the container (z0, zl), e z is the unit vector in the z direction Fig., n the outer unit vector normal to A. From Eqs , the total force F can be also exressed as the sum of the uniform motion force F U the force associated with sloshing F S f F U A t e z"nda F S A t e z"nda, (36) resectively, so that FF U F S. The uniform motion force is readily calculated as F U R L Z M L Z (37) M L R L (38) is the total liquid mass. The force related to sloshing at zl is calculated by integrating the ressure on the end section zl q F SL M L 0 S 0, L,3,5,... n,3,5,... S m,k 0 (n)/ 4 S n n, q n L (39) k I m k Rd. (40) The total force associated with sloshing F S is equal to twice the value of force F SL, due to the antisymmetry of the solution. It is interesting to note that from the series exansion in the transverse direction i.e., summation with resect to n, all odd terms (n,3,5...) contribute to the sloshing force, as from the even terms only the first term (n0) contributes. 3 Simlified Solution Mechanical Model It is ossible to develo a simlified version of the above formulation considering only the first two terms of the series exansion of the otential in the transverse direction truncation size N0 in Eq. 3. This yields elegant exressions of very good accuracy for frequencies, ressures forces, motivates the develoment of an equivalent mechanical model. 3. Simlified Sloshing Solution. Assuming that the otential related to sloshing is equal to,3,5 q I 0 0 k rq cos I k rcosk z (4) alying the boundary conditions, the following equations are obtained q I k R B 0 I 0 k Rq q,3,5,... (4) q gk B q 4L Z, (43) B which are analogous to Eqs. 4 3, resectively. To account for dissiation effects, a daming term may be introduced in each equation, roortional to the first derivative of the unknown time function: q q gk B q 4L Z (44) B which is an equation analogous to Eq. 9, is constant. This equation can also be written in the form of a linear oscillator equation q s q s q L I 0k R Z,3,5,... I k R (45) s gk I 0 k R (46) 4 I k R is the circular undamed sloshing frequency corresonding to the th mode obtained with the simlified methodology / s (47) is the corresonding daming ratio. Equation 47 can be emloyed to estimate the value of, if the daming ratio for a certain mode is somehow estimated e.g., exerimentally. Furthermore, the s value is an aroximation of the first sloshing frequency of the th mode, since only the first two terms of the series exansion in the transverse direction are emloyed, can be exressed in the following nondimensional form: s R g 4 k R I 0k R I k R (48) which indicates sloshing frequencies deend on the container asect ratio (L/R). The hydrodynamic ressure force associated with sloshing on the wall at zl are P s,zl,3,5 B I 0 k rcos I k rq (49) F SL M L,3,5 L B S 0, 4 S,q, (50) resectively, S m,k is defined in Eq. 40 B in Eq. 4. The force F SL is alied at a certain location on the symmetry axis x0 of the end section zl Fig.. Then the moment M SL of force F SL about the x-axis is 58 Õ Vol. 7, MARCH 004 Transactions of the ASME

5 Fig. Mechanical model aroximating the sloshing resonse of a half-full cylindrical container M SL AzL / yp s,zl da/ 0 R P s,zl r cos drd M L R,3,5 L 4B S 0, S, q (5) the distance y s of force F SL from the cross-section center is equal to the ratio M SL /F SL. The entire force associated with sloshing F S is the sum of the forces at z0 zl the total force F is the sum of F S F U the uniform motion force F U is given by Eq Equivalent Mechanical Model. The above simlified solution motivates the develoment of an equivalent mechanical model, which aroximates the liquid-container resonse. Introducing a new variable x (t) for each mode, x L I k R I 0 k R q (5) the corresonding equation of motion becomes ẍ P SP ẋ SP x Z. (53) Furthermore, the entire force associated with sloshing becomes F S,3,5 M 6 S S 0, F P S M P S ẍ (54),3,5 P S B, M L (55) F P S M P S ẍ (56) is the sloshing force corresonding to mode. In the above equation, M P S exresses the art of liquid mass associated with sloshing motion, which refers to mode. Equations corresonds to a series of oscillators, one for each longitudinal mode. Setting u x Z, (57) it is straightforward to rewrite Eqs in the form M S ü s M S u Ż s M S u Z0 (58) F,3,5,... M I M L M S ü M I Z (59),3,5,... M S P. (60) Based on Eqs , the liquid-container system is simulated through a series of linear oscillators as shown in Fig.. In this model, Z reresents the external source motion, u exresses the motion of the liquid mass associated with sloshing mode. Furthermore, the total liquid mass M L is slit in a art M I, which follows the motion of Z(t), exressing the so-called imulsive motion, a series of masses M S (,3,5,...), which corresond to u (t), exressing the so-called convective or sloshing motion, 3,4. The osition y S of each oscillator on the y-axis is calculated through their moments about the x-axis. It is readily obtained that S P y, 4B S S 0, (6) R 4 S, P B S 0, indicating that y S deends on the asect ratio (L/R) of the container. As an examle, for L/R3, the y S /R values for oscillators corresonding to modes, 3, 5 are 0.49, 0.379, 0.96, resectively. 4 Solution for Harmonic Excitation The mathematical formulation for arbitrary excitation is significantly simlified when the rigid container undergoes a harmonic motion ŻtUe it, (6) Journal of Alied Mechanics MARCH 004, Vol. 7 Õ 59

6 U is the velocity amlitude, is the angular frequency of the external excitation source. Assuming steady-state conditions, the velocity otentials become so that r,,z,t,3,5 n0 f z,t z L Ue it (63) q n ta n e it (64) a n cosni n k r a n cosni n k rcosk ze it (65) an equation analogous to Eq. 3. Alying the boundary conditions, the following infinite system of linear algebraic equations is obtained M K a U,,3,5,... (66) In the above system, the square matrix M, the diagonal matrix K the vector are given by Eqs. 4 7 a is the unknown vector with comonents a n, n0,,,... Similarly, the resonse of the damed system is obtained through the solution of the algebraic system M ic K a U (67) the matrix C is given by Eq. 30. Subsequently, the hydrodynamic ressures the force acting on the container can be comuted from Eqs , resectively. It is interesting to note that in the case of harmonic excitation the resent aroach may be considered as semi-analytical since the only comutational work required is the solution of a truncated linear algebraic system. If U0, Eqs. 66 reduce to a homogeneous system, identical to the one obtained in 9. An estimate of the externally induced sloshing effects on the resonse can be obtained from the added mass coefficient,, for each longitudinal mode, defined as C a Re F S (68) F U. On the other h, the dimensionless daming coefficient C Im F S F U (69) rovides a measure of dissiation effects when daming is included,. In the above exressions, Re Im denote the real the imaginary art of the F s /F U ratio, resectively. It is ossible to obtain an elegant closed-form analytical solution for harmonic excitation, if the velocity otential is aroximated only with the first two terms of the series exansion in the transverse direction (nn0 in Eq. 65. In such a case, becomes 4U r,,z,t,3,5 k L s I 0 k r B cos I k rcosk ze it (70) the force F s corresonding to sloshing mode is F s iu 4R L s S 0, I 0k R I k R S,e it. (7) Finally, the added mass coefficient the dimensionless daming coefficient for mode are C a 6 S 0, I 0k R I k R S, s (7) C 6 S 0, I 0k R I k R S, s 3 s, (73) resectively, / s is the ratio of the external frequency over the natural frequency of the oscillator. 5 Numerical Results Discussion The numerical results resented in this section are based on the solution of the truncated systems 67 9 for harmonic arbitrary excitation, resectively. The results for the eigenvalue roblem are based on the solution of system 67 with U0. 5. Sloshing Frequencies, Modes, Masses. The eigenvalue roblem is considered first, assuming no external excitation. The convergence rate the exected accuracy of the eigenvalues are demonstrated numerically, increasing the value of truncation size N (nn in Eq. 3. In Figs. 3a 3b the variation of the first three eigenvalues,, 3, for the first two longitudinal modes ( 3) in terms of the truncation size N for the case of zero dissiation is resented with L/R. In Fig. 3c the effect of the cylinder asect ratio L/R is shown, resenting similar results for L/R3. From the numerical oint-of-view, the results indicate that the convergence rate is quite raid, that faster convergence is obtained in lower sloshing frequencies. The required truncation size N to obtain accurate results u to three significant figures for the eigenvalues,, 3 in Fig. 3 is N3, N6, N0, resectively. The values of the comuted eigenfrequencies are in very good agreement with exerimental results, 6,7, analytical redictions, 9. It should be underlined that due to the nonorthogonality of the corresonding satial functions in the transverse direction, the sloshing frequency values j, j,,..., intheresent formulation deend on the truncation size of the series exansion N. Itis also imortant to note that for each longitudinal mode, more than one eigenvalues are obtained, deending uon the truncation size N. On the other h, when a series solution aroach is alied in rectangles, mutually orthogonal functions are emloyed, the corresonding sloshing modes are uncouled yielding one eigenvalue for each longitudinal mode see the Aendix. It is clarified that the sloshing roblem in a longitudinally excited horizontal cylinder is a three-dimensional roblem as the corresonding roblem in a rectangle is reduced in a two-dimensional roblem. In Fig. 4, the converged value of the first frequency for each mode (,3,5,7) is comared with the corresonding sloshing frequency rec of an equivalent rectangular container Eq. A6. The equivalent container has the same free surface dimensions LR, a deth h equal to R/4 so that it contains the same liquid volume with the horizontal cylinder. The very good comarison between rec is a good indication that this rectangular container can be used for aroximating sloshing effects in horizontal cylinders, in ractical engineering alications. Figure 4 also shows that the s values (,3,5,7), obtained from the simlified methodology Eq. 48, are fairly close to the corresonding values rec. Therefore, s offer a reasonable estimate of the converged value for each longitudinal mode. Note that the s values almost coincide for relatively long cylinders (L/R3). Mathematically, the equivalence of rec s values for long rectangles cylinders is 60 Õ Vol. 7, MARCH 004 Transactions of the ASME

7 Fig. 4 Dominant eigenvalues for each longitudinal mode various asect ratios of the container LÕR Fig. 3 Variation of sloshing frequencies with resect to the truncation size N readily roved by considering the limit of exressions 46 A6 as h/l 0 R/L 0, resectively. More secifically, s R lim R/L 0 g rec lim h/l 0 g k R 4 (74) R k R h R. (75) The two limits are equal for hr/4, which is exactly the deth of the equivalent rectangular container. In Fig. 5 sloshing modes corresonding to frequencies,, 3 () are deicted in terms of their free-surface elevation. Note that the mode corresonding to is quite similar to the mode corresonding to rec of an equivalent rectangular container. In Fig. 5a, the slight curvature of the free surface in the transverse direction offers a reasonable exlanation for the very small difference between the values of rec. 5. Hydrodynamic Forces Under Harmonic Longitudinal Excitation. Results under harmonic excitation are shown in terms of the added mass coefficient C a the dimensionless daming coefficient C v. The C a C v values () are lotted in Figs. 6a 6b, resectively, in terms of the normalized external excitation frequency ( R/g). Daming is considered in the form of Eq. 30 with 0 equal to 0, It should be noted that the C a C v values are converged values in the sense that they are obtained with an adequate truncation nn of Eq. 65. Figure 6a shows that for the case of zero daming, the resonse is characterized by large increases in the C a value in the vicinity of resonant frequencies. A larger value of truncation size N is required for convergence close to the resonant frequencies. There is a sign reversal in C a at each resonant frequency. When C a 0 the sloshing force F s is out-of-hase with the container s dislacement i.e., the uniform motion force F U ) resulting in a reduction of the total force amlitude. The extreme values of C a close to the resonant frequencies are significantly reduced when daming is resent, the resonant effect of the higher natural frequencies almost disaears. The large values of C a for a wide range of excitation frequencies indicate the significant effects of hydrodynamic sloshing on the overall resonse. Figure 6b resents the corresonding results for the dimensionless daming coefficient C v. The C v value exhibits a shar eak near the first resonant frequency, much smaller eaks for the higher resonant frequencies. When daming is increased, the eaks become smoother. Similar results have been obtained for higher longitudinal modes various asects ratios R/L. The converged C a values in the region of the dominant frequency are comared in Fig. 7 with those obtained from the simlified formulation Eq. 7 as well as with the corresonding values of the equivalent rectangular container Eq. A8. The comarison shows a very good agreement between the converged Journal of Alied Mechanics MARCH 004, Vol. 7 Õ 6

8 vessel has radius R m, length L m (L/R), liquid density 000 kgr/m 3 (g9.8 m/sec ). According to Eqs , for this half-full container, the sloshing masses for 3 are equal to 80% 8% of the liquid mass M L, as the imulsive mass M I is equal to 9% of M L. The linear system of Eqs. 9 is integrated in time (t0.0 sec) through a fourth-order Runge-Kutta scheme in Matlab rogramming. The deendence of the maximum value of the sloshing force F s,max on the truncation size is resented in Table for zero daming. It is interesting to note that consideration of few terms of the series in the transverse direction e.g., nn4) is adequate to rovide quite accurate results for engineering uroses. The results are very well comared with those from an equivalent rectangular container. Furthermore, for all longitudinal modes the results from the simlified formulation (N0) are very close to the converged results. Figures 8b, 8c, 8d show the uniform force F U, the force associated with sloshing F s the total force F, resectively for the cylindrical vessel under consideration, under the El Centro earthquake for 5% daming. The F s, F values are obtained with a truncation size N4, for six longitudinal modes (), the maximum total force is 5.87 kn at 6.83 sec. The results show that sloshing force counteracts the uniform motion force this is due to the fact that the dominant earthquake excitation frequencies are significantly larger than the dominant sloshing frequencies imlies that F s is out of hase with F U. Fig. 5 Eigenmodes in terms of free-surface elevation corresonding to the first three eigenfrequencies,, 3 of the first longitudinal mode Ä, with LÕRÄ C a values of the horizontal cylinder the C a values of the equivalent rectangle, as the simlified values are reasonably close. It is ossible to demonstrate analytically that for long cylinders (L ), the C a C v values of a half-full cylinder obtained through the simlified methodology coincide with those of the rectangular equivalent container. More secifically, comuting the limits of Eqs. 7, 73 A8, A9 considering hr/4, one obtains lim C a lim C a rec 8 R/L 0 h/l 0 (76) lim C v lim C 8 v rec 3 R/L 0 h/l 0. (77) 5.3 Results for Earthquake Ground Motion. The resonse of horizontal-cylindrical liquid containers under earthquake excitation is of articular imortance for the seismic analysis of ressure vessels used in refineries etrochemical industries. The efficiency of the roosed methodology to hle an arbitrary tye of external excitation is demonstrated calculating the resonse of a half-full horizontal cylindrical vessel, subjected to the El Centro 940 seismic ground motion Fig. 8a. The 6 Conclusions A mathematical model is develoed for the simle efficient solution of externally induced liquid sloshing in longitudinal excited half-full horizontal cylindrical containers. The velocity otential is slit in two arts, a uniform motion otential trivially obtained a otential associated with sloshing. In this configuration, the roblem formulation is not searable the general solution of the sloshing otential is written as a double series exansion of unknown time functions their associated satial functions, which are nonorthogonal in the transverse direction. The formulation reduces in a series of systems of linear differential equations one system for each longitudinal mode, which is truncated solved numerically. For the articular case of harmonic external source, a series of systems of linear algebraic equations, allowing for a semi-analytical solution. The convergence is generally raid a small truncation size N rovides very good results. The sloshing roblem is significantly simlified if only the first two terms of the series in the transverse direction (N0) are considered. This simlification yields quite accurate results in terms of sloshing frequencies forces, esecially for containers with relatively large asect ratio (L/R), motivates the develoment of an equivalent mechanical model to aroximate the sloshing resonse of the vessel. Subsequently, the resonse of a tyical half-full horizontal cylinder subjected to a real seismic event is examined, the hydrodynamic forces acting on the container are calculated. The results indicate that sloshing has a significant effect on the total force value. Finally, it is shown numerically analytically that sloshing frequencies, modes forces calculated for the half-full horizontal cylinder comare very well with those from an equivalent rectangular vessel, esecially for long cylinders. Therefore, the equivalent rectangle may be used for aroximating the halffull horizontal cylinder resonse for engineering uroses. Acknowledgments This work has been artially suorted by the Earthquake Planning & Protection Organization EPPO, Athens, Greece. 6 Õ Vol. 7, MARCH 004 Transactions of the ASME

9 Fig. 6 Converged values of C a C v in terms of external excitation frequency RÕg for LÕRÄ, Ä0, Ä Aendix Externally Induced Sloshing in Rectangular Containers. The exact analytical solution for the analysis of sloshing in rigid rectangular containers with viscous effects under horizontal excitation has been examined in early works, 3, is outlined for the sake of comleteness. Consider a rectangular container of dimensions Lb liquid height h, externally excited in the horizontal z-direction. The total otential satisfies the Lalace equation 0 within the liquid volume, the following boundary conditions: Żt, z at z0 zl (A) Journal of Alied Mechanics MARCH 004, Vol. 7 Õ 63

10 Table Maximum values of forces associated with sloshing in kn for different longitudinal modes no daming; half-full container RÄ m,lä m, gä9.8 mõsec, Ä000 kgrõm 3 N Equivalent rectangle , y at y0 (A) Fig. 7 Added mass coefficient C a for horizontal cylinder equivalent rectangle g 0, at yh. (A3) t y The solution is considered as a sum of an imulsive art f (z,t) given by Eq. 8 a sloshing art of the following form: y,z,t,3,5 ḋ k sinhk h coshk yhcosk z (A4) k /h,,3,5,....equation A4 is indeendent of x transverse coordinate because of symmetry, satisfies the Lalace equation, the kinematic boundary conditions at the tank walls. Enforcing the boundary condition on the free surface, using the orthogonality roerties of the trigonometric functions, introducing a daming term, the following set of uncouled equations of motion is obtained: d rec ḋ rec d Z,3,5,... (A5) rec k g tanhk h (A6) 4 tanhk h. (A7) If Z(t) is harmonic see Eq. 6, a closed-form exression can be obtained for the unkown time functions d (t) for the flow otential (y,z,t). In such a case, the dimensionless added mass daming coefficients become C a rec 8L h h tanhk 3 (A8) C rec 8L h tanhk h 3 3, (A9) resectively, / rec is the excitation frequency. Fig. 8 Resonse of a half-full cylindrical vessel subjected to the El Centro earthquake in its longitudinal direction for 5% daming. a El Centro ground motion source: htt:õõ b uniform motion force F U, c force associated with sloshing F s d total force F. References Currie, I. G., 974, Fundamentals Mechanics of Fluids, McGraw-Hill, New York, Cha. 6. Miles, J. W., 956, On the Sloshing of Liquid in a Cylindrical Tank, Reort No. AM6-5, The Ramo-Woodridge Cor., Guided Missile Research Div., GM- TR-8. 3 Abramson, H. N., 966, The Dynamic Behavior of Liquids in Moving Containers, Southwest Research Institute, NASA SP-06, Washington, DC. 4 Housner, G. W., 957, Dynamic Pressures on Accelerated Fluid Containers, Bull. Seismol. Soc. Am., 47, Veletsos, A. S., Yang, J. Y., 977, Earthquake Resonse of Liquid Stor- 64 Õ Vol. 7, MARCH 004 Transactions of the ASME

11 age Tanks, nd Engineering Mechanics Conference, ASCE, Raleigh, NC, Haroun, M. A., Housner, G. W., 98, Earthquake Resonse of Deformable Liquid Storage Tanks, ASME J. Al. Mech., 48, Rammerstorfer, F. G., Fisher, F. D., Scharf, K., 990, Storage Tanks Under Earthquake Loading, Al. Mech. Rev., 43, Veletsos, A. S., Tang, Y., 990, Soil-Structure Interaction Effects for Laterally Excited Liquid Storage Tanks, Earthquake Eng. Struct. Dyn., 9, Ibrahim, R. A., Pilichuk, V. N., Ikeda, T., 00, Recent Advances in Liquid Sloshing Dynamics, Al. Mech. Rev., 54, Moiseev, N. N., Petrov, A. A., 966, The Calculation of Free Oscillations of a Liquid in a Motionless Container, Adv. Al. Mech., 9, Fox, D. W., Kutler, J. R., 98, Uer Lower Bounds for Sloshing Frequencies by Intermediate Problems, J. Al. Math. Phys., 3, Fox, D. W., Kutler, J. R., 983, Sloshing Frequencies, J. Al. Math. Phys., 34, McIver, P., 989, Sloshing Frequencies for Cylindrical Sherical Containers Filled to an Arbitrary Deth, J. Fluid Mech., 0, McIver, P., McIver, M., 993, Sloshing Frequencies of Longitudinal Modes for a Liquid Contained in a Trough, J. Fluid Mech., 5, McCarthy, J. L., Stehens, D., 960, Investigation of the Natural Frequencies of Fluids in Sherical Cylindrical Tanks, Reort NASA TN D-5. 6 Kana, D. D., 979, Liquid Slosh Resonse in Horizontal Cylindrical Tank Under Seismic Excitation, Southwest Research Institute Reort, Project 0-938, San Antonio, TX. 7 Budiansky, B., 960, Sloshing of Liquids in Circular Canals Sherical Tanks, J. Aeros. Sci., 7, Kobayashi, N., Mieda, T., Shibata, H., Shinozaki, Y., 989, A Study of the Liquid Slosh Resonse in Horizontal Cylindrical Tanks, ASME J. Pressure Vessel Technol.,, Evans, D. V., Linton, C. M., 993, Sloshing Frequencies, Q. J. Mech. Al. Math., 46, Abramowitz, M., Stegun, I., 97, Hbook of Mathematical Functions, 0th Ed., Dover, New York. Faltinsen, O. M., 978, A Numerical Nonlinear Method of Sloshing in Tanks With Two-Dimensional Flow, J. Shi Res.,, Isaacson, M., Subbiach, K., 99, Earthquake-Induced Sloshing in a Rigid Circular Tank, Can. J. Civ. Eng., 8, Journal of Alied Mechanics MARCH 004, Vol. 7 Õ 65