Dynamic behaviour of thin cylindrical shells subjected to impulsive bending moments along their tips

NUCLEAR ENGINEERING AND DESIGN 26 (1974) 263-273. © NORTH-HOLLAND PUBLISHING COMPANY

DYNAMIC BEHAVIOUR OF THIN CYLINDRICAL SHELLS SUBJECTED TO IMPULSIVE BENDING MOMENTS ALONG THEIR TIPS Shin-ichi SUZUKI

Department of Aeronautics, Nagoya University, Chikusa-ku, Nagoya, Japan Received 26 April 1973 The dynamic behaviour of thin cylindrical shells with concentrated masses at their tips is investigated when they are subjected to impulsive bending moments at their tips. The relationships between the dimensions of a cylinder, concentrated mass and maximum values of dynamic stresses are obtained. The fundamental equation of motion is solved by the Laplace transformation method. From the results of the theoretical analysis, it became evident that impulsive stresses become much larger than static ones. For the case where bending moments are applied statically, the fundamental equations are also derived, taking into account the effect of shearing force.

1. Introduction When nuclear reactors are overheated, control rods have to be inserted which collide with a grid plate. The author [ 1 ] has previously investigated the behaviour of dynamic stresses acting in the grid plate and has found that extraordinarily large dynamic stresses were induced. On the assumption that the side structures of reactors are rigid, an analysis was carried out when the plate was clamped along its outer edge. In this paper, attention is paid to the dynamic behaviour of side structures. It is, of course, desirable to consider the case where side structures are connected to grid plates. However, since it becomes very difficult to solve the problems, an analysis is made for the simplified mechanical model. That is, a uniformly distributed mass and spring are attached to one end of a thin cylindrical shell, clamped at the other end. The impulsive bending moment, which is assumed to be a step function with respect to time, is applied along the free end of the cylinder. The relationships between cylinder dimensions, spring constant, distributed mass and maximum stresses acting in the cylinder are obtained. Attention is paid to stresses in the longitudinal and circumferential directions. Solutions are found using the Laplace transformation method. Solutions are also obtained, when a bending moment is applied statically, taking into account shearing force, the effect of which is investigated.

2. Formulation of the problem and its ~olution The co-ordinates and dimensions of a cylinder are illustt'ated in fig. l(a). The cylinder is clamped at one end and a uniformly distributed mass m 0 is attached at the 9~aer end. Since displacement in the radial direction at the tip is restrained to some extent, because o f the existence of a grid plate, a distributed spring is attached (see fig. l(b)). The tip becomes free for k = 0 and is simply supported for k = oo. The case where a uniformly distributed impulsive bending moment M 0 is applied along the tip of a cylinder is analysed. M 0 is assumed to be a step function with respect to time, as illustrated in fig." l(c). The equation of motion o f a cylinder under axially symmetric conditions (see ref. [2]) is:

D(O4w/Ox 4) + (Eh/R2)w + ph(a2w/at 2) = 0.

(1)

264

S. Suzuki, Dynamic behaviour of thin cylindrical shells R+W

,,

°

(00

~Mo I

k .~ Mo ~

M°I

(c)

0

Fig. 1: (a) Dimensions and co-ordinates of thin cylindrical shell; (b) tip of cylindrical shell; (c) relationship between time and bending moment M o. Introducing dimensionless variables and the constants ~ = x / l , T = (t/12)(D/hp) ~ , and 474 = 12(1 - v 2) l 4 / R 2 h 2

gives ~4w/~4

+ 474w + ~ 2 w / ~ T 2 = 0.

(2)

The initial conditions for the cylinder are w = 3 w / a T = 0,

at T = 0.

(3)

Laplace transformation of eq. (2), with respect to T, gives, after replacing p by ip, w .... + (43'4 - p 2 ) w = 0,

(4)

where-the primes indicate differentiation with respect to ~. Bending moment M, shearing force Q, stresses o and o 0 in the longitudinal and circumferential directions, respectively, are given as follows: M = - ( D / 1 2 ) w '',

a = - ( D / 1 3 ) w ''',

o = - (El1 - v 2 ) ( z / 1 2 ) w '',

o o = E ( w / R - (v/1 - v Z ) ( z / 1 2 ) w " ) ,

(5) (6)

where z is the distance from the neutral axis. The equilibrium condition at the free end becomes Q + m 0 ( 3 2 w / O t 2) + kw = 0.

(6a)

Introducing dimensionless variables as mentioned above gives 3 3 w / 3 ~ 3 = r ( 3 2 w / 3 T 2) + n w ,

(6b)

S. Suzuki, Dynamic behaviour of thin cylindrical shells

265

where r = m o / l h o and n = kl3/D. Therefore, the transformed boundary conditions are w=w

"t

=0,

w" = m/ip,

att=0 w"' + rp2w-

n w = O,

att

1.

(7)

The following case~ will be considered: (1) 474 - p2 > 0; (2) 43'4 - p2 < 0. An analysis will be carried out for each case. 2.1. The case 4")'4 - p2 > 0

Equation (4) is written in the following form: w .... + 4 c ~ 4 w = 0 ,

(8)

where 474 - p2 = 4~4. From eq. (8), w = C 1 ch a t cos a t + C 2 ch a t sin a t + C 3 sh a t cos a t + C 4 sh a t sin a t ,

(9)

where C i are constants. With the aid of eq. (7), these constants can be determined, and w becomes w--/m = [ - ( 2 a 3 (sh a cos a - ch a sin a) + (rp 2 - n) sh a sin a ) (ch a t sin a t - sh a t cos a t )

+ (4Or3 ch a cos a + (rp 2 - n)(ch a sin a - sh a cos a)): sh a t sin a~] /2ipc~ 2 { 2 a 3 ( c h 2 a + cos 2 a) - ( r p 2 - n)(shc~ c h a - s i n a cos a ) ) .

(10)

The inversion theorem gives w / m = [ ( (ch 3' sin 3' - sh 3' cos 3') + n/273 sh 3' sin 7 ) (ch 7 t sin 7 t - sh 7~ cos 7 t )

+ { 2 c h 7 c o s 7 - (n/273)(ch 7 sin7 - s h 7 c o s 7 ) ) s h T t sinTt] /23 '2 ( ( ch2 7 + cos2 3') + ( n / 2 7 3 ) ( s h 7 c h 7 -- sin7 cos 3')) + ~

[ ( 2a 3 (sh a cos a - ch a sin a) + (rp 2 - n) shot sin a ) (ch a~ sin a t - sh a~ cos a t )

- ( 4 a 3 ch a cos a + (rp 2 - n)(ch a sin a - sh a cos a) ) sh a t sin odj] 4 a cos p T / p 2 × {3a2(ch2 a + cos 2 a) - ( r p 2 - n)(ch2 a - cos 2 a) + 2a3(1 +4r)(sh a chcx - s i n s c o s a ) ) ,

(11)

where a are the roots of eq. (12): 2 a 3 ( c h 2 a + cos 2 c0 - (rp 2 - n ) ( s h a c h a - sincz c o s a ) = 0. F r o m eq. (11), w" becomes

(12)

266

S. Suzuki, Dynamic behaviour of thin cylindrical shells w " / m = [ { ( c h 7 s i n 7 - shy cosT) + ( n / 2 7 3 ) s h 7 sin"r) ( c h T t s i n T t + shT~ cosT~)

+ ( 2 c h 7 cos7 - ( n / 2 7 3 ) ( c h 7 s i n 7 - s h 7 cos"f)} chT~ cos3,~] / { (ch 2 3' + cos 2 7) + ( n / 2 7 3 ) ( s h 7 ch 7 - sin 7 cos 7) ) + ~[{2c~3(sha coss-

c h a s i n a ) + (rp 2 - n) s h s s i n s ) ( c h a t sinc~ + s h s t coso.~j)

- {4a 3 c h a c o s s + (rp 2 - n ) ( c h c~ s i n s - shc~ c o s a ) } c h s t c o s a t ] × 4 ~ cos p T / ( p 2 / 2 s 2 ) { 3 s 2 ( c h 2

a + cos 2 c0 - (rp 2 - n ) ( c h 2 a - cos 2 a)

+ 2 s 3 (1 + 4r)(sh a ch a - sin s cos s ) }.

(13)

2.2. The case 474 - p2 < 0

E q u a t i o n (4) is written in the following form: --tttt

W

-- Ct4w = O,

(14)

where 474 - p2 = _ s 4. F r o m eq. (14), w = C 1 ch a t + C 2 sh s~j + C 3 cos t~t + C 4 sin ct/~,

(15)

where C i are constants. With the aid o f e q . (7), these constants can be determined, and w becomes w / m = [ - { s 3 ( c h o t + cos ct) + (rp 2

n)(shct - s i n s ) } (chctt - cosct~)

+ {s3(shot - s i n s ) + (rp 2 - n ) ( c h a - coso0} (shot t - s i n a i ) ] / 2 p s 2 { - a 3 (1 + c h a c o s s ) + (rp 2 - n ) ( c h s sino~ - s h s coscQ}.

(16)

The inversion t h e o r e m gives w/m = ~

[ - {a3 (ch ot + c o s s ) + (rp 2 - n ) ( s h s - sin a ) } ( c h a t - cosctt)

+ { s 3 ( s h a - sinct) + (rp 2 - n ) ( c h s - cos ct)) ( s h s t - sinott)] 2 s c o s p T /p2 {_ 3a2(1 + c h a cosct) + 2(rp 2 - n ) s h s s i n a + s 3 ( 1 + 4r)(chct s i n a - shot c o s s ) ) ,

(17)

where ~ are the roots of eq. (18): - s 3 ( 1 + c h a c o s s ) + ( r p 2 - n ) ( c h s s i n a - shot c o s s ) = 0. F r o m eq. (17), w " becomes

(18)

S. Suzuki, Dynamic behaviour of thin cylindrical shells

267

0.8

0.4

! I I t

0 I.O

Fig. 2. Distribution of o in the ~ direction.

wt'/m = ~ [ _ { at 3 (ch at + cos at) + (rp 2 - n) (sh at - sin at)) (ch o~ + cos c~j) + (at3(shat - sin at) + (rp 2 - n)(chat - c o s a ) ) (sh o~j + sinat~)] 2a c o s p T / ( p 2 / a t 2 ) ( _ 3at2(1 + chat cos at) + 2(rp 2 - n)shat s i n a + at3(1 + 4r)(ch at sinat - shat c o s a ) } .

(19)

The final forms o f w and w " are obtained by adding the results of cases (1) and (2). The first terms on the right-hand sides o f eqs. (11) and (13) are the values o f w and w " w h e n M 0 is applied statically. F r o m eq. (6), the value o f outer fibre stress o and o0, on the neutral axis can be expressed in the following form:

o(z=h/2)/(6Mo/hE)=w

"

/m,

Oo(Z=O)/(6Mo/h2)=(1-v

2

/ 3 ) r1 2 7 2 ( w / m ) .

(20)

3. Numerical analysis Numerical calculations have been carried out for several cases. The most important concern in engineering is the maximum values o f o 0 and o. It is necessary at first to investigate these values when bending moment M 0 is applied statically. Since the values o f w and w " under static load are expressed by the first terms of eqs. (11) and (13), respectively, the values o f outer fibre stress o and o 0 on the neutral axis can be obtained from eq. (20). The distribution of o values in the ~ direction is illustrated in fig. 2. Solid lines and d o t t e d lines indicate the values for n = 0 and co, respectively. These values decrease remarkably as the value o f ~ decreases. It is seen that the tendency of distributions for b o t h cases is very similar. The distribution o f o 0 values in the ~ direction is illustrated in fig. 3. Solid lines and dotted lines indicate the values for n = 0 and ~ , respectively. F o r n = O, these values are very large at ~ = 1 and decrease remarkably as the value o f ~ decreases. F o r n = ~ , these values increase considerably, reach extreme values and decrease gradually as the value of ~ decreases. Extreme values o f o 0 appear, in general, in the neighbourhood o f ~ = 1. The relationships between 7, o at ~ = 0, and o 0 at ~j = 1 are illustrated in fig. 4. The values of o, which are large where 3' is small, decrease remarkably as the 7 increases. On the other hand, o 0 increases as 3' increases and converges to some constant value when the 7 exceeds about 1.5.

S. Suzuki, Dynamic behaviour o f thin cylindrical shells

268

0,6-

0.6

0.4 t 0"3 I i \

i

0.2 -o.3 02

06

,e= o,

I0

Fig. 3. D i s t r i b u t i o n o f a 0 in t h e ~ d i r e c t i o n .

Fig. 4. The r e l a t i o n s h i p s b e t w e e n 7, (7 (~ = 0) a n d o0(~ = 1).

n=O

1.2

- -

£,I,.," ,\ -.\

7=5 {=l.O r=o,]

r=o.s

,.,,

', .Z-.

\

0.4

0

0.1

0.2

0.3

0.4

0.5

T Fig. 5. The r e l a t i o n s h i p s b e t w e e n t i m e a n d o 0 (~ : 1) (for the case 7 = 5 a n d n : 0).

In this connection, the relationships between 7, o and a 0 are illustrated by dotted lines for the case where 2~ shearing force is taken into consideration. The values o f h/R and K are taken as 0.1 and 7 , respectively• The effect o f shearing force for o, which is large for small 3' values, becomes very small when 3' exceeds 3. On the other hand, the effect for o0, which is small for small 3' values, becomes large when 3' exceeds 2. A theoretical analysis is made in the Appendix for the case where shearing force is taken into consideration. Next, we will investigate the cases where M 0 is applied impusively. As seen in eqs. (11) and (17), w is expressed in series form. Therefore, convergence** o f these series must be investigated. It becomes evident that 20, 30 and 40 terms for 3' = 3, 5 and 8 should be summed to obtain the solutions which give satisfactory results for practical purposes. The relationships between time and o 0 for n = 0 and n = oo are illustrated in figs. 5 and 6. In fig. 5, the solid line, dotted line and chain line indicate the values o f o 0 at ~ = 1 for r = 0.1,0.3 and 0.5, respectively. It can be * C o w p e r [3] d i s c u s s e d in d e t a i l the value o f K a n d o b t a i n e d a b e t t e r a p p r o x i m a t e value• H o w e v e r , in this p a p e r , t h e value of K

is taken as -~. ** In eq. (11), there is only one a for n = 0 and zero for n = ~. Therefore, we have only to investigate convergence of the series for the case of eq. (17).

S. Suzuki, Dynamic behaviour of thin cylindrical shells n =oO Y=5

0.2

•

--

n=0D

~=0.3

....

~=o.6

--'--

~=0.9

~0.4

.~-~.1~.

/

:'"

':

269

/. . . . . P ~

',.ix\.

--~

i

r'~.

0.2

.~...;.~ .-

o . l ~

o.2

0.2

0.6

1.0

T

-o.2 Fig. 6. The relationship between time and o 0 (for the case ~, = 5 andn = ~).

Fig. 7. Distribution of maximum values of o 0 in the ~ direction for n = ~. 1.2

1.2

nffiO

n=o

rffio.I

r=0.5

°S"

0.8 E

~--~--0.4

0

0.2

0.6

0

1.0

Fig. 8. Distribution of maximum values of o 0 in the ~ direction for n = 0 (for r = 0.5).

0.2

0.6

1.0

Fig. 9. Distribution of maximum values of a 0 in the ~ direction for n = 0 (for r = 0.1).

seen that the f r e q u e n c y b e c o m e s low as the value o f r increases. In fig. 6, the solid line, d o t t e d line and chain line indicate the cases for ~ = 0.3, 0.6 and 0.9, respectively. In these cases, there is n o effect by r. The oscillating pattern b e c o m e s very c o m p l i c a t e d in c o m p a r i s o n w i t h that in fig. 5. The distribution o f m a x i m u m value* o f % [4] in the ~ direction in the range 0 ~< T ~< ~ is illustrated in fig. 7 for n = ~ . As ~ decreases, these values increase remarkably, reach e x t r e m e values and afterwards decrease. As increases, the location where these e x t r e m e values appear approaches ~ = 1 closely. As shown in fig. 3, the values o f o 0 are very small in the range ~ < 0.5 w h e n M 0 is applied statically. O n the o t h e r hand, the m a x i m u m values

* Stress a is generally expressed as

a=fo+~,ificosPiT

0 < T<**.

Therefore, the value of o will be fo - Yilf/Ho
270

S. Suzuki, Dynamie behaviour of thin cylindrical shells

of o 0 are comparatively large* even in the same range o f ~. This phenomenon is worthwhile noting. The distribution of maximum values of o 0 in the ~ direction for n = 0 is illustrated in figs. 8 and 9. These are the cases for r = 0.5 and 0.1, respectively. The tendency/'or both cases is rather similar and their values decrease remarkably as ~ decreases. When dynamic behaviour o f structures subjected to impulsive loads is discussed, we have often used the concept o f dynamic load factor (DLF)**. But, it is not adequate to use a DLF in this paper, because this factor becomes infinite when the values of stresses become zero in some points under static loads. As seen in eq. (19), convergence becomes very slow for w". Therefore, outer fibre stress o under impulsive loads is not discussed in this paper.

4. Conclusions The following conclusions can be drawn from the theoretical results o f the present work: (1) When M 0 is applied at the tip statically, the values of outer fibre stress o for n = 0 and oo and o0 on the neutral axis for n = 0 decrease remarkably as ~ decreases. On the other hand, the values of o 0 for n = oo increase remarkably, reach extreme values and afterwards decrease as ~ decreases. Extreme values of o 0 appear, in general, in the neighbourhood of ~ = 1. (2) The effect of shearing force on stresses cannot be neglected for some particular cases. (3) When M 0 is applied impulsively, the maximum values of stresses are much larger than those under static loads. They indicate comparatively large values even for the cases where they are very small under static loads.

Appendix: The effects of shearing force o n stresses The effects of shearing force on o and o 0 are investigated when M 0 is applied along the tip of the cylinder statically. The slope of the cylinder in the ~ direction is d w / d x = 91 + 92,

(i)

where 91 is the slope by the ordinary beam theory and 92 by shearing force. Bending moment M and shearing force Q are expressed as M= - D(dgl/dX),

Q =KGh92,

(ii)

where K is the coefficient determined by the section. The basic equations governing the flexure of the cylinder may be written dQ/dx - ( E h / R Z ) w = O,

- dM/dx + Q = 0.

(iii)

Substituting eqs. (i) and (ii) into eq. (iii), introducing variables ~ = x / l and 0 = 191, and simplifying, gives 0' = w" - )tw,

)tO" + 43,4 (w' - 0 ) = 0.

(iv)

where the primes indicate differentiation with respect to ~, and where 43'4 = 12(1 - v2)14/R2h 2, a = ?.yKG, and )t = al2 /R 2

S. Suzuki, Dynamic behaviour of thin cylindrical shells

271

Eliminating 0 from eq. (iv), the f u n d a m e n t a l equation will be o b t a i n e d as follows: w .... -- ~kw" + 4 3 ` 4 w = 0 .

(v)

0 and 0' are expressed as 43`40 = ~ w " ' + (43` 4 - ~ 2 ) w ' ,

0' = w " - ~,w.

(vi)

The equilibrium condition at x = I is

K G h { ( d w / d x ) - ~Pl } + k w = 0.

(a)

At x = l a n d 0, - D ( d ~ l / d x ) = M0,

(b)

w = ~01 = 0.

(c)

Therefore, the b o u n d a r y conditions are w = O,

X w " ' + ( 4 3 `4 -- ) t 2 ) w ' = O,

w" - Xw = m,

- w " ' + Xw' + 43`4 n o w = 0,

at t = 0

at t = 1

(vii)

where m = - MOI2/D and n O = kR2/Elh. Since X2 - 163`4 < 0* in eq. (v), w will be expressed in the following form: w = C 1 c h a t cos15t + C 2 c h a t sin15~ + C 3 s h ~ t cos15t + C 4 s h a t sin 15~,

(viii)

where C i are constants and ~, = [3`2 + (X/2)]-~. F r o m eq. (viii), w " b e c o m e s W" = C 1 { (a 2 - f12)ch a~ cos 15~ - 2a15 sh a t sin 15~} + C 2 { ( a 2 - fl2)ch a~ sin 15t + 2a15 sh a~ cos 15~} + C 3 ( (a 2 _ 152) sh a~ cos 15t - 2a43 ch a~ sin 15t } + C4 ( ( a2 - 152) sh a t sin 15~ + 2a15 ch a t cos 15t }. With the aid of eq. (vii), C i will be determined. C 1 = O,

C 2 / ( m / B ) = (a 4 + 3154) (a(15 c h a sinl3 - a s h a cos15) + (no/215)(a2 + 152)2 s h a sin 15},

* h 2 - 163' 4 b e c o m e s ~2 _ 163"4 =

14/R4 ~ ( E j K G ) 2

_ 4 8 ( 1 - v2) (R/h) 2 }.

C a l c u l a t i n g the value on the a s s u m p t i o n o f v = 0.3 a n d K = ~-, h 2 _ 163, 4 = / a / R 4 (15.21 _

43.68(R2/h2).

Since R > h, h2 _ 163,4 b e c o m e s negative.

(ix)

S. Suzuki, Dynamic behaviour of thin cylindrical shells

272

C3/(m/B ) =

(3~ 4 +/34){/3(/3 c h a sinfl - ~ sho~ cosfi) + (n0/2~)(o~2 +/32)2 sho~ sinfi},

C4/(m/B ) = (o~2 +/32) [(a2 +/32) 2 c h a cosCJ + 2c~/3(a 2 - 3 2 ) s h a

sinfl

+ (n0/2o43)(o~2 +/32) ( _ o~(a4 + 3/34)cha sin/3 + fl(3a 4 +/34)sh oe cos/3)], B = 2otfl{/32 (3a 4 +/34)ch2 ot + a2(oe 4 + 3/34)cos 2/3) + n0(o~2 +/32)2 (/3(3oe4 +/34)sha c h a - oe(a 4 + 3/34) sin/3 cos/3}.

(x)

Substituting a =/3 = 3' (that is, X = 0) into eq. (x), constants will be obtained for the case where the effect of shearing force is neglected. For the particular cases n o = 0 and % constants become as follows: (1) for n o = 0, C 1 =0,

C2/(mlB )

-

o~4 + 3/34 2/3 (/3 c h a sin/3 - a shc~ cos/3), 3a4 +/34 (/3 c h a sin3 - a s h a cos/3), 2a

C3/(m/B ) -

C41(m/B ) - Or2 _ 2043 _+1~2 {(0~2 + /32)2 c h ~ COS/3 + 2~/3(a2 _/32) s h a sin/3}

(x~)

B =/32 (3a4 +/34)ch2 o~ + oe2 (Oe4 + 3/34)cos 2/3. (2) for n o = ~ , C 1 =0,

C2/(m/B)

-

a 4 + 334 sh a sin/3, 2/3

C3/(m/B) -

3~42a+/34 sht~ sin3,

C4/(m/B) -

0~4 + 3/34 c h a sin/3 + 3a4 +/3~4 shc~ cos/3, 2/3 2or

B =/3(3oe 4 +/34)sh ot c h a - q ( a 4 + 3/34)sin/3 cosfl.

(xii)

o 0 at z = 0, ~ = 1, and o at z = h/2, ~ = 0 are expressed in the following form:

o(z = hi2, ~ = O)/(6Mo/h2 ) = w"(~ = O)/m,

Notation

D E G

= E h 3 / 1 2 ( 1 - v 2) = Young's modulus = shear modulus

Oo(Z = O, ~ = 1)/(6Mo/h2 ) = - x/(1 - v2/3)23 '2 (w/m).

(xiii)

S. Suzuki, Dynamic behaviour o f thin cylindrical shells h

= thickness of cylinder

k K ! m0 m M0 M nO n

= = = = = = = = =

273

spring c o n s t a n t coefficient determined by section length of cylinder u n i f o r m l y d i s t r i b u t e d mass - MOI2/D u n i f o r m l y d i s t r i b u t e d impulsive b e n d i n g m o m e n t bending moment kR2/Elh kl 3/D

Q

= s h e a r i n g force

r R t T

= = = =

w

= deflection of cylinder

w 43, 4

= fow exp (-pT) dT = 12(1 - v 2 ) 1 4 / R 2 h 2

p o ao v

= density = stress in the l o n g i t u d i n a l d i r e c t i o n = stress in the c i r c u m f e r e n t i a l d i r e c t i o n = Poisson's ratio = x/l.

mo/lh p m e a n radius o f c y l i n d e r time (t/12)(D/hp){

References [1] S. Suzuki, Dynamic response of circular plates subjected to transverse impulsive loads, Ing. Arch. 40 (2) (1971) 131 -144. [2] S. Suzuki, Dynamic behaviour of visco-elastic thin cylindrical shell subjected to impulsive inner pressure, JSME Trans. 38 (313) (1972) 2201-2209. [3] G. Cowper, The shear coefficient in Timoshenko beam theory, Trans. Amer. Soc. Mech. Eng. E 33 (2) (1966) 335-340. [4] S. Suzuki, Dynamic elastic response of a ring to transient pressure loading, Trans. Amer. Soc. Mech. Eng. E 33 (2) (1966) 261-266.