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FREE VIBRATION ANALYSIS OF CYLINDRICAL TANKS PARTIALLY FILLED WITH LIQUID P. B. G¸ N. R. S. S. R Pontifı´ cia Universidade Cato´lica—PUC/Rio, Department of Civil Engineering, Rio de Janeiro—CEP 22453-900, Brazil (Received 16 February 1995, and in final form 15 January 1996) A simple but effective modal solution based on the underlying ideas of the hierarchical finite element method is presented for evaluating the free vibration characteristics of vertical, thin, circular cylindrical shells, partially or completely filled with liquid and subjected to any variationally consistent set of boundary conditions on the lower and upper boundaries. Effects of static liquid pressure, in-plane inertias and liquid free surface motions are taken into account. The solution of the shell problem is obtained through a procedure in which Sanders’ shell equations are transformed into a new system of first order ordinary differential equations which are solved by the Galerkin error-minimization procedure. The system variables are those quantities which appear in the boundary conditions on a rotationally symmetric edge of a cylindrical shell. The liquid is taken as non-viscous and incompressible, and the coupling between the deformable shell and this medium is taken into account. The solution for the liquid velocity potential is assumed as a sum of two sets of linear combinations of suitable harmonic functions which satisfy Laplace equation and the relevant boundary conditions. This procedure leads to a determinantal equation for the determination of the shell and liquid natural frequencies and the associated mode shapes. Application of the method to a few selected cases and comparisons of the numerical results with those obtained by other theories and from experiments are found to be good and demonstrate the effectiveness and accuracy of the present methodology. 7 1996 Academic Press Limited

1. INTRODUCTION

The dynamic behaviour of liquid-filled cylindrical shells is a problem of widespread industrial relevance, as most cylindrical shells are utilized as containment vessels or tanks for the storage of liquids. The combined effects of inertial loading and hydrostatic pressure on the shell wall, the effects of the free surface motions in shells partially filled with liquid and the coupling between fluid and shell wall motions may affect significantly the dynamic behaviour and stability of fluid-filled shells. The significant influence of the contained liquid on the structural behaviour of cylindrical tanks is corroborated by a large collection of experimental results found in the literature [1–8]. To design and assess the safety of these structures under different boundary and loading conditions, it is of primary importance to determine their natural frequencies and associated mode shapes. Various methods for the determination of the vibration characteristics of cylindrical tanks have been suggested in the literature. They range from simple assumptions concerning the deformations of the tank and its interaction with the liquid [9–12] to complex finite and boundary element formulations [13–15]. Accurate semi-analytical solutions have been obtained by the use of the Galerkin or Rayleigh–Ritz error-minimization procedures [3, 4, 9, 11, 12, 16–19]. Apart from frequent simplifications on shell and fluid equations and related boundary 429 0022–460X/96/330429 + 16 $18.00/0

7 1996 Academic Press Limited

430

. . ¸ . . . .

conditions, these modal solutions converge to the exact values provided that a sufficient number of terms is used in the analysis. However, a review of the literature also reveals that the majority of these works is concerned with particular sets of boundary conditions for the shell [9–12, 16–19]. This is no doubt due to the traditional difficulties involved in trying to solve coupled partial differential equations of motion and, at the same time, satisfy arbitrary sets of boundary conditions at both edges. Although various papers have been written on the free vibrations of circular cylindrical shells in a vacuum with arbitrary sets of boundary conditions at each end [20–22], no such kind of analysis appears to have been consistently carried out for partially liquid-filled cylindrical shells. The purpose of the present work is to develop a general formulation for the problem, based on the underlying ideas of the hierarchical finite element method [23, 24], which allows one to obtain the free vibration frequencies and vibration modes of vertical cylindrical shells partially or completely filled with liquid and subjected to any variationally consistent set of boundary conditions on the upper and lower boundaries. The vibrations of the shell are examined by using Sanders’ shell theory [25]. The fluid is treated as non-viscous and incompressible and its motion is assumed to be irrotational. In order to solve the problem, the partial differential equations of motion in terms of the shell displacements are transformed into a system of first order ordinary differential equations in terms of the shell displacements and stress resultants and a solution based on the use of linear and trigonometric shape functions is derived. These system variables are those variables which describe the natural and geometric boundary conditions on a rotationally symmetric edge of a cylindrical shell. The fluid velocity potential is expanded in terms of harmonic functions which satisfy the Laplace equation term by term and the relevant boundary conditions. To check the validity and accuracy of the present methodology, solutions of selected problems are presented and compared with those obtained by other theories and available experimental results. 2. BASIC THEORY

2.1. Consider a thin-walled circular cylindrical shell of length L, radius R and thickness h. The shell is assumed to be made of an elastic material with Young’s modulus E, Poisson ratio n and mass density rc . The bottom of the shell is assumed to be flat and rigid. The radial, circumferential and axial co-ordinates are denoted by, respectively, r¯ , u and x, and the corresponding displacements on the shell middle surface are in turn denoted by W, V and U, as shown in Figure 1. The tank is filled to a height H with a non-viscous, incompressible and irrotational liquid of mass density rF , Time is denoted by t. According to the non-linear shell theory of Sanders [25], the dynamic behaviour of elastic circular cylinders can be described by the following equations: l{lu,j + l 2dw,j2 + n[v,u + w + d(w,u − v)2 ]},j + a[lv,j + u,u + 2dlw,j (w,u − v)],u +ab[lw,ju + (u,u − 3lv,j )/4],u − g 2u¨ = 0,

(1a)

[v,u + w + d(w,u − v) + n(lu,j + l dw )],u + al[lv,j + u,u + 2dlw,j (w,u − v)],j 2

2

2 ,j

+2aldw,j [lv,j + u,u + 2dlw,j (w,u − v)] − 3abl[lw,ju + (u,u − 3lv,j )/4],j +2d(w,u − v)[v,u + w + d(w,u − v)2 + n(lu,j + l 2dw,j2 )] −b[w,uu − v,u + nl 2w,jj ],u − g 2v¨ = 0,

(1b)

431

g w¨ − p¯ + [v,u + w + d(w,u − v) + n(lu,j + l dw )] 2

2

2

2 ,j

−{2dl2w,j [lu,j + dl 2w,j2 + n(v,u + w) + dn(w,u − v)2 ]},j −{2d(w,u − v)[v,u + w + d(w,u − v)2 + n(lu,j + dl 2w,j2 )]},u −{2dal(w,u − v)[lv,j + u,u + 2dlw,j (w,u − v)]},j −{2dalw,j [lv,j + u,u + 2dlw,j (w,u − v)]},u + bl 2[l 2w,jj + n(w,uu − v,u )],jj +b[w,uu − v,u + nl 2w,jj ],uu + 4abl[lw,ju + (u,u − 3lv,j )/4],ju = 0.

(1c)

Solutions of the preceding system must satisfy the following natural or geometric boundary conditions at j = 0 and j = 1 [25]: Natural

Geometric

lu,j + dl w + n[v,u + w + d(w,u − v) ] = n* 2

2 ,j

2

a[lv,j + u,u + 2ldw,j (w,u − v)] − 3ab[lw,ju + (u,u − 3lv,j )/4] = t*

or

u = u*,

or

v = v*,

2ldw,j [lu,j + dl w + n(v,u + w) + dn(w,u − v) ] 2

2 ,j

2

+2da(w,u − v)[lv,j + u,u + 2dlw,j (w,u − v)] −4ab[lw,ju + (u,u − 3lv,j )/4],u − bl[l 2w,jj + n(w,uu − v,u )],j = q*

or w = w*,

−b[l 2w,jj + n(w,uu − v,u )] = m*

or

− w,j = s*. (2)

In the foregoing the following non-dimensional quantities have been introduced: j = x/L,

r = r¯ /R,

l = R/L,

u = U/h,

v = V/h,

w = W/h,

n = Nx R/Ch,

q = Qx R/Ch,

d = h/2R,

a = (1 − n)/2,

g 2 = rc R 2(1 − n 2 )/E, t = Nxu R/Ch,

b = h2/12R 2,

p¯ = (1 − n 2 )p/4Ed 2,

m = Mx R 2b/Dh.

(3)

Here C = Eh/(1 − n 2 ), D = Eh 3/12(1 − n 2 ) and Nx , Qx , Nxu and Mx are the shell stress resultants per unit length at an edge j = constant [25] and p is the fluid pressure. The definitions of the stress resultants, stress–strain and strain–displacement relations can be found in references [25, 26]. Now the equations for the static deformation as well as the free-vibration analysis of a cylindrical shell will be derived.

Figure 1. The co-ordinate system, displacements and geometry.

. . ¸ . . . .

432

One of the purposes of the present paper is to present a general method for evaluating the free vibration characteristics of a circular cylindrical shell with classical boundary conditions of any type. In order to solve the stated boundary value problem, Sanders’ partial differential equations will be converted into a set of first order ordinary differential equations having as independent variable the axial co-ordinate j. The order of the system of equations (1) is eight with respect to j, and consequently it is possible to reduce equations (1) to a set of eight first order differential equations involving only eight unknowns [27, 28]. Here, the most appropriate set of fundamental variables are those quantities which denote the natural and geometric boundary conditions at an edge j = constant. These are the displacements u, v and w, the rotation s and the boundary forces n, t, q and m. The relationship between the generalized stress resultants n, t, q and m and the displacements are given by equations (2). The variables associated with the vibrations of the pre-stressed shell consist of an axisymmetric static pre-stress field V0 (j) = {u0 , n0 , v0 , t0 , w0 , q0 , s0 , m0 }

(4)

which occurs prior to excitation, plus an additional field V 1 (j, u, t) = {uˆ1 , nˆ1 , vˆ1 , t 1 , wˆ1 , qˆ1 , sˆ 1 , mˆ1 }

(5)

resulting from the excitation. For the free harmonic vibrations of a complete cylindrical shell it is appropriate to represent these additional variables, upon assuming separability with respect to the three independent variables, j, u and t, as V 1 (j, u, t) = {u1 (j) cos (ku), n1 (j) cos (ku), v1 (j) sin (ku), t1 (j) sin (ku), w1 (j) cos (ku), q1 (j) cos (ku), s1 (j) cos (ku), m1 (j) cos (ku)} cos (vt),

(6)

where k is the circumferential wavenumber and v is the natural frequency. Using equations (1) and (2), eliminating the dependence of all the shell variables upon the co-ordinate u and the time variable t by means of expansions (6), and retaining only linear terms in the incremental quantities, one obtains, after various algebraic manipulations, the following set of eight first order ordinary differential equations in terms of the eight incremental variables [26]: u1,j − (n1 − nkv1 − nw1 )/l + 2dls0 s1 = 0, n1,j −

$

%

abk 16k (3b − 4) u + t1 − 16ks1 − 24ds0 (v1 + kw1 ) + V 2u1 = 0, 9b + 4 l 1 lab

(7b)

4 k(3b − 4) 12bk 8d t + u − s + s (v + kw1 ) = 0, al(9b + 4) 1 l(9b + 4) 1 9b + 4 1 9b + 4 0 1

(7c)

v1,j − t1,j −

(7a)

nk (n 2 − 1) 2 2dn (n1 + m1 ) + [k (b + 1)v1 + k(bk 2 + 1)w1 ] − n (v + kw1 ) l l l 0 1 −

24abdlk 36abd 2l 2 24abdk 8d s0 s1 − s (v + kw1 ) + su s0 t1 − 9b + 4 9b + 4 0 1 9b + 4 0 1 9b + 4

+

(n 2 − 1) 2dw0 (v1 + kw1 ) + V 2v1 = 0, l

(7d)

w1,j + s1 = 0,

(7e)

433

n (n − 1) nk 2dnk q1,j − n1 + [k(bk 2 + 1)v1 + (bk 4 + 1)w1 ] − m1 − n0 (v1 + kw1 ) l l l l 2

−

2

24abdlk 2 36abd 2lk 2 36abd 2lk 2 2 8dk s 0 s1 − s0 v 1 − sw s0 t1 − 9b + 4 9b + 4 9b + 4 0 1 9b + 4 24abdk 2 (n 2 − 1) p¯ s0 u1 + 2dkw0 (v1 + kw1 ) + F + V2w1 = 0, 9b + 4 l l

+

(7f)

s1,j + (nk/l 2 )v1 + (nk 2/l 2 )w1 − (m1 /bl 2 ) = 0, 2

m1,j +

(7g)

2

16abk 12bk q 16abk t − 1− u + s − 2ds0 n1 − 2dn0 s1 l(9b + 4) 1 l(9b + 4) 1 l 9b + 4 1 −

24abdk 24abdk 2 s 0 v1 − s w = 0. 9b + 4 9b + 4 0 1

(7h)

Here V2 = g 2v 2/l and p¯F is the hydrodynamic pressure. The associated natural or geometric boundary conditions applicable to Sanders’ theory reduce to prescribing the following quantities at the ends of the cylindrical shell: n1 = 0

or u1 = 0,

t1 = 0

or

q1 = 0

or w1 = 0,

m1 = 0

v1 = 0,

or

s1 = 0.

(8)

Thus, the desired system of eight first order differential equations, in which the effects of a pre-stress state are included, has been obtained which, together with the boundary conditions on the two rotationally symmetric edges of the cylindrical shell, constitute a two-point boundary value problem. In order to solve this problem, one must obtain the axisymmetric solution of the shell subjected to the static liquid pressure. The equations for the static case are obtained from the foregoing as follows (v0 = t0 = 0): u0,j − (n0 − nw0 )/l = 0,

n0,j = 0,

q0,j − [nn0 + (1 − n )w0 − p¯h ]/l = 0, 2

w0,j + s0 = 0, 2

s0,j − m0 /(bl ) = 0,

(9a–c)

m0,j − q0 /l = 0.

(9d–f)

Here the hydrostatic pressure ph = [(1 − n 2 )rF gR 2L/Eh 2 ][(H/L) − j] for j E H/L, while the boundary conditions are given by n0 = n* 0

or

u0 = u* 0 ,

q0 = q* or 0

w0 = w* 0 ,

m0 = m* or 0

s0 = s* 0 . (10)

With the aim of applying the Galerkin procedure, the eight axial mode functions associated with the fundamental variables are represented by ND

u1 (j) = s U1i fi ,

ND

n1 (j) = s N1i fi ,

i=1

ND

w1 (j) = s W1i fi ,

i=1

ND

i=1

q1 (j) = s Q1i fi ,

i=1

i=1

ND

v1 (j) = s V1i fi , ND

s1 (j) = s S1i fi , i=1

ND

t1 (j) = s T1i fi , i=1

ND

m1 (j) = s M1i fi ,

(11)

i=1

where {U1i , N1i , V1i , T1i , W1i , Q1i , S1i , M1i }, i = 1, ND, are the unknown parameters. The trial functions fi (j) are given by f1 (j) = (1 − j),

f2 (j) = j,

fi (j) = sin [(i − 2)pj],

i = 3, ND.

(12)

The first two linear functions are used to enforce the boundary conditions at j = 0 and j = 1. All the higher order trial functions are zero at each end of the shell and represent

. . ¸ . . . .

434

simply additive refinements of the displacement and stress fields. The imposition of boundary conditions follows the standard procedures used in the classical finite element method [29]. A similar modal solution is used to describe the axisymmetric static variables (4): NS

NS

u0 (j) = s U0i fi ,

n0 (j) = s N0i fi ,

i=1

NS

w0 (j) = s W0i fi ,

i=1

NS

i=1

NS

q0 (j) = s Q0i fi ,

s0 (j) = s S0i fi ,

i=1

NS

m0 (j) = s M0i fi .

i=1

(13)

i=1

Here {U0i , N0i , W0i , Q0i , S0i , M0i }, i = 1, NS, are the modal amplitudes of the axisymmetric state. The selected set of fundamental variables together with the proposed modal solutions allows one to satisfy all natural and geometric boundary conditions exactly and obtain all displacements and internal forces simultaneously and with the same degree of accuracy. Furthermore, no additional manipulation needs to be carried out after the solution is completed, since all displacement and stress resultants which are generally of interest in the analyses of shells can be readily obtained from the eight fundamental variables. Making use of the expansions (13) and taking the functions fi (j) as the weighting functions, one obtains, by applying the Galerkin error-minimization procedure to equations (9), the following set of algebraic equations: NS

s i=1

6

U0i I3ij −

7

N0i I2ij n + W0i I2ij = 0, l l

NS

s N0i I3ij = 0, i=1

NS

s {W0i I3ij + S0i I2ij } = 0, i=1

(14a–c) NS

s i=1

6

7

n (1 − n 2 ) (1 − n 2 )rF gR 2L Q0i I3ij − N0i I2ij − W0i I2ij = − I1j , l l Eh 2

NS

s i=1

6

S0i I3ij −

7

M0i I2 = 0, bl 2 ij

6

NS

s i=1

M0i I3ij −

7

Q0i I2ij = 0, l

(14d)

(14e, f)

where j = 1, NS, I1j =

g

H/L

0

[(H/L) − j]fi dj,

I2ij =

g

1

fi fj dj,

0

I3ij =

g

1

fi,j fj dj.

(15)

0

Accumulating the modal contributions and deleting any row/column of the matrices associated with a prescribed boundary condition leads to an assembled modal system of 6(NS − 1) equations of the form [K AX]{a AX} = {p AX},

(16)

where [KAX] is the coefficient matrix, {p AX} is the load vector and {aAX} is the vector of the unknown modal amplitudes. The substitution of expressions (11) into equations (7), the use of the solution for the

435

basic linear static state, and the application of the Galerkin method yield the modal equations for the pre-stressed shell: ND

s

i=1

ND

s

i=1

6

6

U1i I3ij −

N1i I3ij −

+

ND

s

i=1

6

ND

s

i=1

6

Tli I3ij −

(17a)

16abk 2 k(3b − 4) 16abk 2 U1i I2ij − S I2 T1i I2ij + l(9b + 4) 9b + 4 1i ij l(9b + 4)

7

24abdk 24abdk 2 S0 V1i + S W + V 2U1i I2ij = 0, 9b + 4 9b + 4 0 1i

V1i I3ij −

+

7

N1i I2ij nk n + V1i I2ij + W1i I2ij + 2dlS0 S1i = 0, l l l

(17b)

4 k(3b − 4) 12bk T I2 + U I2 − S I2 al(4 + 9b) 1i ij l(4 + 9b) 1i ij 4 + 9b 1i ij

7

8d 8dk S V + S W = 0, 9b + 4 0 1i 9b + 4 0 1i

(17c)

nk k 2(n 2 − 1)(b + 1) k(n 2 − 1)(bk2 + 1) N1i I2ij + V1i I2ij + W1i I2ij l l l

−

nk 2dn 2dnk 8d 24abdk Mli I2ij − N 0 V1i − N 0 W1i − S T − S (lS − U1i ) l l l 9b + 4 0 1i 9b + 4 0 1i

−

36abd 2l 2 36abd 2lk 2 2d(n 2 − 1) S0 V1i − S0 W1i + W 0 V1i 9b + 4 9b + 4 l

− 2dk

7

(n 2 − 1) W 0 W1i + V 2V1i I2ij = 0, l

(17d)

ND

s {W1i I3ij + S1i I2ij } = 0,

(17e)

i=1

ND

s

i=1

6

n k(n 2 − 1)(bk 2 + 1) (n 2 − 1)(bk4 + 1) Q1i I3ij − N1i I2ij + V1i I2ij + Wli I2ij l l l −

nk 2dnk 24abdk 2 8dk S (lS − U1i ) − M1i I2ij − N 0 (V1i + kW1i ) − S T l l 9b + 4 0 1i 9b + 4 0 1i

−

36abd 2lk 2 2dk(n 2 − 1) 2dk 2(n 2 − 1) S0 (V1i + kW1i ) + W 0 V1i + W 0 W1i 9b + 4 l l

+ V 2W1i I2ij } +

g

H/L

0

p¯F f dj = 0, l j

(17f)

. . ¸ . . . .

436 ND

s

i=1

ND

s

i=1

6

M1i I3ij +

−

6

S1i I3ij +

7

nk nk 2 I2 W1i I2ij − M1i 2ij = 0, 2 V1iI2ij + l2 bl l

(17g)

I2 16abk 2 12bk U I2 + T I2 − Q1i ij l(9b + 4) 1i ij l(9b + 4) 1i ij l

7

24abdk 24abdk 2 16abk 2 S1i I2ij − 2dS0 N1i − 2dN 0 S1i − S0 V1i − S W = 0, 9b + 4 9b + 4 9b + 4 0 1i (17h)

Here j = 1, ND, and NS

S0 = s S0k I4ijk ,

NS

NS

W 0 = s W0k I4ijk ,

k=1

N 0 = s N0k I4ijk ,

k=1

NS

k=1

NS

S20 = s s S0k S0l I5ijkl ,

(18)

k=1 l=1

I4ijk =

g

1

0

(fi fj fk ) dj,

I5ijkl =

g

1

(fi fj fk fl ) dj.

(19)

0

The eigenvalue problem is obtained by superimposing the building blocks obtained from equations (17) and constraining the Fourier coefficients of the linear functions (f1 and f2 ) in such a way as to satisfy the prescribed boundary conditions. The eigenvalue matrix can be written as {([K] + [Kg]) − V 2[M]}{a} = {0},

(20)

where [K] is the ‘‘stiffness’’ matrix, [M] is the mass matrix, [Kg] is the geometric matrix and {a} is the vector of the vibration amplitudes. The numerical values of the elements of the matrices appearing in equations (16) and (20) are calculated by evaluating the integrals (15) and (19). Here, symbolic computing was used to evaluate and tabulate the exact values of every possible product of hierarchical shape functions and its derivatives used in this work, reducing significantly the numerical effort involved in generating the required matrices. The solution of the resulting eigenvalue problem gives the vibration modes of a pre-stressed shell in a vacuum and the corresponding natural frequencies. The present formulation could also be used to evaluate buckling loads and modes of cylindrical shells as well as the relationship between the load parameter and the shell frequencies. As one can observe, inertia terms appear in only three of the eight equations (7). To take advantage of the inherent sparseness of the mass matrix, a condensation procedure is used, reducing greatly the dimension of the resulting eigenvalue problem. The method developed here is applicable to two-point boundary value problem which is governed by m first order ordinary differential equations with m/2 boundary conditions prescribed at each end of the interval.

437

2.2. The irrotational motion of an incompressible and non-viscous fluid can be described by a velocity potential f(x, r¯ , u, t) [14, 30]. This potential function must satisfy the Laplace equation which can be written in non-dimensional form as l 2(1 2F/1j 2 ) + (1 2F/1r 2 ) + (1/r)(1F/1r) + (1/r 2 )(1 2F/1u 2 ) = 0,

(21)

where F = (g 2f)/R 2. The dynamic fluid pressure acting on the shell surface is obtained from the equation p¯F = −(rF /rc ) (1/4d2 )(1F/1t).

(22)

At the liquid boundary the following conditions have to be satisfied: at j = 0, at j = H/L,

(1F/1j) = 0;

(1 2F/1t 2 ) + (g/L)(1F/1j) = 0; (1F/1r) = 2g 2d(1w/1t).

at r = 1,

(23) (24) (25)

Furthermore, for a fluid-filled shell, the following restriction must be imposed lim (1F/1r) = 0.

(26)

r:0

The boundary conditions suggest seeking the solution for the velocity potential in the form NMF

F = s Am cos (zm j)Ik (zm r) cos (ku) sin (vt) m=1

NMS

+ s Bn v cosh (an j)Jk (an r) cos (ku) sin (vt),

(27)

n=1

where zm = [((2m − 1)/2)(pL/H)], Jk and Ik are, respectively, the kth order Bessel function and modified Bessel function of the first kind and an are the roots of the equation [31] dJk (an r)/dr =r = 1 = 12 [Jk − 1 (an ) − Jk + 1 (an )] = 0.

(28)

This solution is obtained by the superposition of two complementary boundary value problems. The first expression is the solution of a boundary value problem with homogeneous boundary conditions prescribed at j = 0 (equation (23)) and j = H/L (p¯F = 0) and a non-homogeneous boundary condition (equation (25)) prescribed at r = 1. The second expression is the solution for sloshing liquids in a rigid tank. Thus, the modal solution (27) already satisfies the Laplace equation and boundary condition (23). Substituting equations (6), (11) and (27) into boundary condition (25), one obtains NMF

ND

m=1

j=1

s Am cos (zm j)I'k (zm ) = −2vg 2d s W1j fj (j),

(29)

where [31] I'k (zm ) = dIk (zm r)/dr =r = 1 = zm Ik − 1 (zm ) − kIk (zm ).

(30)

. . ¸ . . . .

438

Using cos (zp j) as a weighting function and applying the Galerkin method, one obtains the modal amplitudes Am as a function of the modal amplitudes W1j , Am = −

2vg 2d ND s I6 W , I'k (zm )Icmm j = 1 jm 1j

(31)

where Icmm =

g

H/L

cos2 (zm j) dj,

I6jm =

0

g

H/L

cos (zm j)fj (j) dj.

(32)

0

Substitution of the potential function (27) into the sloshing condition (24) yields NMF

g z sin (zm H/L)Ik (zm r) Lv m

− s Am m=1

NMS

+ s Bn n=1

NMS g an sinh (an H/L)Jk (an r) − s Bn v 2 cosh (an H/L)Jk (an r) = 0. (33) L n=1

Finally, substituting equation (31) in equation (33), using the function Jk (ap r) as the weighting function and applying the Galerkin method, one obtains the following system of homogeneous linear equations in terms of the modal amplitudes Bn and W1j : ND 2gd NMF zm g NMS s sin (zm H/L)IJmp s I6mj W1j + 2 s an sinh (an H/L)JJnp Bn L m = 1 I'k (zm )Icmm Lg n = 1 j=1

− V2

l NMS s cosh (an H/L)JJnp Bn = 0, g4 n = 1

p = 1, NMS,

(34)

g

(35)

where JJkp = 2p

g

1

Jk (ak r)Jk (ap r)r dr,

0

IJmp = 2p

1

Ik (zm r)Jk (ap r)r dr.

0

For the evaluation of the Bessel functions and modified Bessel functions, the polynomial expansions presented by Abramowitz and Stegun [32] were used. The integrals involving products of Bessel functions for which there are no closed form solutions were computed by using the Newton–Cotes formula [32] with 400 values over the range of integration. With equation (22) the hydrodynamic pressure p¯F takes the form p¯F = p* F cos (ku) cos (vt)

(36)

where p* F =

rF V 2l rc 2d

6

NMF

s cos (zm j)

m=1

7

Ik (zm ) NM 1 NMS s I6mj W1j − 2 s cosh (an j)Jk (an )Bn . I'k (zm )Icmm j = 1 2g d n = 1

(37)

Substituting the last expression in equation (17f), one obtains the additional terms necessary for free vibration analysis of fluid-filled tanks.

439

T 1 Comparison of natural frequencies (Hz) for an empty cylindrical shell

Mode, k

Lowest natural frequencies ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV Experiment, Sanders exact, Present reference [6] reference [33] method, ND = 12

7 8 9 10 11 12 13 14 15

318 278 290 334 362 418 478 550 626

305·32 281·37 288·28 317·51 362·22 417·96 482·23 553·67 631·59

305·22 281·31 288·24 317·49 362·20 417·94 482·22 553·67 631·58

The resulting eigenvalue problem can be written in the form det

6$

[K] [K1 ]

% $

[0] [M] + [M2 ] − V2 [Kslo ] [0]

%7

[M1 ] [Mslo ]

= 0.

(38)

The coefficient matrices [K] and [M] consist of the coefficients obtained from the shell equations (7), the submatrices [Kslo ], [K1 ] and [Mslo ] are given by equation (34) and [M1 ] and [M2 ] are the added mass matrices derived from equation (37). This eigenvalue problem is of order 8(ND − 1) + NMS. Due to the sparseness of the mass matrix, before attempting to establish the eigenvalues corresponding to the shell and fluid natural frequencies, a condensation procedure is used, leading to an eigenvalue problem of much lower dimension.

3. NUMERICAL TESTS

To check the validity and accuracy of the present methodology, empty and fluid-filled shells under different boundary conditions are analyzed and the numerical results are compared either with experimental values or with values obtained by other analytical methods found in literature. As a first example, the lowest natural frequencies of an empty cylindrical shell with simply supported ends were calculated and compared with the analytical solution derived by Dym, using Sanders’ shell theory (equation (9) in reference [33]), and the experimental results obtained by Gasser [6] for an empty steel tank. The geometrical data and physical properties for the steel tank are L = 410·0 mm, R = 301·5 mm, h = 1·0 mm, E = 2·1 × 1011 N/m2, n = 0·30, rc = 7·85 × 103 kg/m3, Z = 534·5, where Z = L 2z1 − n 2/Rh is the Batdorf geometrical parameter. When the shell is simply supported at both ends the boundary conditions are n = v = w = m = 0,

j = 0, 1.

(39)

The results are shown in Table 1, in which k denotes the circumferential wavenumber. As may be seen, the present results agree quite well with the exact solution and the experimental results. In the calculations, 12 terms in equations (11) were taken into consideration. To test the convergence of the present methodology the lowest natural

. . ¸ . . . .

440

T 2 Convergence of the present solution for the lowest natural frequencies (Hz) Mode k

Sanders exact

7 14

305·32 553·67

Present method ZXXXXXXXXXXXXXXCXXXXXXXXXXXXXXV ND = 4 ND = 6 ND = 8 ND = 10 ND = 12 302·52 553·38

304·57 553·60

305·01 553·65

305·16 553·66

305·22 553·67

frequencies were evaluated for an increasing number of terms in equations (11) for k = 7 and k = 14 and these are compared in Table 2 with the exact solution of Sanders [33]. A remarkably good convergence of the present procedure may be observed. Note that for the problem analyzed here, taking ND = 4 would be enough to achieve convergence with an error less than 1%, which is within engineering accuracy. The number of terms necessary to achieve convergence depends on the prescribed boundary conditions. To verify the present methodology in the presence of fluid loading, the experimental results obtained by Gasser [6] for the previous shell completely filled with water (rF = 1·0 × 103 kg/m3, H = L) are compared with the present results and results obtained by the approximate analytical solution of Gonc¸alves and Batista [12]. These results are shown in Table 3 for selected values of the circumferential parameter k. The present results were obtained by taking ND = NMF = 12 and NMS = 1. Again, the agreement between the present results and the experimental natural frequencies is very good and the largest error is less than 5·3%. The agreement between the analytical results and those of the present model is also very good, demonstrating the accuracy of the present methodology. To provide additional comparison of the method described here, some calculations to test the theory in the case of partially filled clamped–free cylindrical containers are presented here. The numerical parameters used for these studies are L = 113·9 mm, R = 100·0 mm, h = 0·25 mm, H = variable, E = 5·56 × 109 N/m2, n = 0·30, 3 3 3 3 rc = 1·405 × 10 kg/m , rF = 1·0 × 10 kg/m , Z = 500, and the boundary conditions adopted are u=v=w=s=0

at

j = 0,

n=t=q=m=0

at

j = 1.

(40)

This shell, a water filled clamped–free shell made of polyester film, was previously studied both numerically and experimentally by Chiba, Yamaki and Tani [17, 18, 5]. To compare the results with experiment, the liquid depth was varied such that the fractional filling, H/L, took the values H/L = 0, 1/4, 1/2, 3/4 and 1, and the corresponding T 3 Measured and computed frequencies of cylinder filled with water Lowest natural frequencies (Hz) Mode k

Experiment, reference [6]

Approximate method, reference [12]

Present method

8 9 10 11 12 13

120 124 146 182 214 254

118 124 144 171 204 243

119 128 146 173 206 245

441

Figure 2. Influence of the circumferential wavenumber and H/L ratio on the shell frequencies. Experimental results of Chiba et al. (reference [5]): r, first mode, x, second mode; t, third mode, H/L values: (a) 0·0; (b) 0·25; (c) 0·50; (d) 0·75; (e) 1·0.

frequencies and vibration modes were computed. These data were obtained by taking ND = NMF = 30 and NMS = 3. In all cases analyzed here these numbers of terms were sufficient to attain convergence to within 1%. The three lowest shell frequencies are shown in Figure 2 as a function of the circumferential wavenumber k. Also shown in Figure 2 are the experimental results obtained by Chiba et al. These experimental results were taken from the figures in reference [5] and the accuracy of the values is as good as direct reading of these figures permitted. For each liquid height, the numerical results agree well with the experimental results. It can be observed that the shell frequencies decrease rapidly with increasing H/L. This lowering effect is mainly due to an increase in kinetic energy without a corresponding increase in the strain energy of the shell–liquid system. To see the variation of the mode of vibration with the liquid depth, axial distributions of the vibration amplitude were determined for k = 8. The results are illustrated in Figure 3. The maximum vibration amplitude is taken as unity. As one can see, the effect of the contained liquid on the mode shape is most significant when H/L = 1/4, 1/2 and 3/4.

. . ¸ . . . .

442

Figure 3. Effects of the filling ratio H/L on the axial vibration mode shapes. k = 8. ——, first mode; - - - - -, second mode; — - — - —, third mode.

On the other hand, the effects of the filling ratio H/L on the sloshing frequencies are negligible, especially for high values of k. The fundamental sloshing frequencies are listed in Table 4 and compared with the sloshing frequencies for a rigid tank (values between brackets). If the fluid container is considered to be rigid, the first term in equation (34) vanishes and one obtains from the resulting equation the following analytical expression for the lowest sloshing frequency f1s = (1/2p)z(ga1 /L) tanh (a1 (H/L)),

(41)

in which f1s is in Hz and a1 is the lowest root of equation (28). As observed, there is little influence of the flexibility of the tank on the lower sloshing frequencies. In these examples the contribution of the pre-stress state to the overall response of the system is found to be negligible, but this may not be the case for other tank geometries used in industry.

5. CONCLUSIONS

A relatively simple—yet effective—methodology has been proposed for evaluating the free vibration characteristics of a fluid-filled cylindrical shell with classical boundary T 4 Sloshing frequencies (Hz); sloshing frequencies for liquid in a rigid tank appear in brackets Mode, k 3 5 7

H/L ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV 0·25 0·50 0·75 1·00 2·6766 (2·6731) 3·5922 (3·5874) 4·2663 (4·2606)

2·9819 (2·9780) 3·7345 (3·7295) 4·3244 (4·3186)

3·0210 (3·0174) 3·7403 (3·7353) 4·3252 (4·3194)

3·0263 (3·0222) 3·7406 (3·7356) 4·3252 (4·3194)

443

conditions of any type. To examine the vibrations of the shell a set of first order differential equations in terms of the shell displacements and stress resultants, based on Sanders’ shell theory, was deduced and a simple modal solution based on the use of linear and trigonometric shape functions was derived. This method, in which the shell is treated as a single element, offers distinct computational advantages over other numerical solution methods such as the classical FEM, avoiding continuity and discretization problems, and can be easily coded and used for the solution of not only vibration problems but also equilibrium and buckling problems of cylindrical shells under various combinations of boundary conditions and loading. Another advantage of the present formulation is that it allows one to obtain all displacement and stress resultants simultaneously and with the same degree of accuracy. By using a consistent shell theory, by choosing appropriate system variables and devising a modal solution that satisfies any set of boundary and continuity conditions for the shell and fluid medium and, finally, by taking into account the effects due to static deformations, in-plane inertias and liquid free surface motions, the present formulation overcomes many of the shortcomings encountered in previous studies for shells in a vacuum or partially filled with liquid. Hence the present formulation is capable of furnishing a consistent solution for a wide range of fluid containers used in the engineering industry. Application of the method to a few selected cases and comparisons of the numerical results with those obtained by other theories and from experiments are found to be good and show the validity and accuracy of the present methodology. ACKNOWLEDGMENT

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