Dynamic behaviour of thin, ring-reinforced, cylindrical shells subjected to impulsive inner loads

Journal o f Sound and Vibration (1977) 51(4), 459-466

DYNAMIC BEHAVIOUR OF THIN, RING-REINFORCED, CYLINDRICAL SHELLS SUBJECTED TO IMPULSIVE INNER LOADS S.-I. SUZUKI Department of Aeronautics, Nagoya University, Nagoya, Japml (Received 20 October 1976)

Stress analysis is carried out for the case where a thin cylindrical shell reinforced with a ring at its mid-point is subjected to impulsive inner pressures. The relationships between the maximum dynamic stresses, the dimensions of the cylinder and the ring are obtained. The fundamental equation of motion of the cylinder is solved by the Laplace transformation method. From the results of theoretical analysis, it becomes evident that the range influenced by the reinforcing ring (on either side) is about 4 x / ' ~ in length (where R and h are mean radius and thickness of the cylinder) when pressures are applied statically, but this range becomes very wide when pressures are applied impulsively.

1. INTRODUCTION Many papers [1-6] have already been published on the dynamic behaviour of thin cylindrical shells subjected to inner impulsive loads or moving pressures and it has been found that very large stresses may be induced in comparison with those under static loads. In order to make dynamic stresses small, the cylinder is often reinforced with a ring. It is of interest for engineering purposes to investigate the effect of a reinforcing ring on the maximum dynamic stresses induced in the cylinder. In this paper, the case treated is that where the cylinder, whose ends are simply supported, is reinforced with the ring at its mid-point and subjected to step impulsive inner pressures. Donnell's equations are used. Since the transients involve high order modal contributions in an impact problem such as this, the effects [6] of shearing force and rotatory inertia on dynamic stresses will be large. But calculation becomes so complicated that the terms due to these factors are neglected in this paper. The relationships between the dimensions of the cylinder and reinforcing ring and the maximum dynamic stresses induced in the cylinder are obtained. The fundamental equation of motion of the cylinder is analyzed by the Laplace transformation method. (A list of symbols used in this paper is given in the Appendix.)

2. FUNDAMENTAL EQUATION AND BOUNDARY CONDITIONS Co-ordinates and dimensions of the cylinder are illustrated in Figure l(a). The cylinder is assumed to be simply supported at both ends and reinforced with a ring at its mid-point. Consider the case where the cylinder is subjected to a step impulsive inner pressure, qo, uniformly distributed over the entire curved surface (see Figure l(b)). The fundamental equation of motion of the cylinder becomes D 0~ w/Ox4 + (Eh/R 2) w + hp 02 wlOt 2 = qo,

459

(1)

460

S.-L SUZUKI R-I- w

r

%

(o) (b) Figure 1. (a) Co-ordinates and dimensions of a thin cylindrical shell; (b) relationship between qo and time. where w, R, h, E, p and v are the radial deflection, mean radius, thickness, Young's modulus, density and Poisson's ratio of the cylinder, repectively, and where D = E h 3 / 1 2 ( 1 - v2). Introducing dimensionless variables ~ = x~/12(1 - v2)/R2h 2 and T = ( t / R ) x / ~ gives (2)

a 4 w/O~ 4 + w + 02 w / a T 2 = q,

where q = qo R2/Eh. Consider the boundary conditions at ~ = 0. The mass of the ring is assumed to be concentrated at ~ = 0. The equilibrium condition at ~ = 0 gives Apo 02 Wo/at 2 + (AEo/R 2) Wo = -D(03 w~/Ox a - 03 w2lOx3),

(3)

where Wo, Eo,Po and A are the deflection, Young's modulus, density and sectional area of the ring, respectively, and where suffices 1 and 2 indicate the parts ~ > 0 and ~ < 0. Rewritten in terms of the dimensionless variables, equation (3) becomes (4)

a 02 Wo/aT 2 + bwo = 03 w2/O~ 3 - 03 w,[O~ 3,

where a = ~/12(1 - v2)/R2h2Apo/ph,

b = ~/12(1 - v2)/R2h2AEo/Eh.

(5)

The initial conditions are w = aw[OT = 0

at T = 0.

(6)

Laplace transformation with respect to T of equation (2) gives, after replacing p by ip, ~V" + (1 _p2) ~ = q/i?,

(7)

where ,P = .[~ w e - P r d T a n d primes indicate differentiation with respect to r In the same way, from equation (4), (b - ap 2) ~ o = ~

- ~1'.

(8)

In terms of the dimensionless number n = I ~ 1 2 ( I - v2)/R2h 2, the boundary conditions become ~t = r~ = 0

9 ~ = v~2= r~o,

a t ~ = n,

~2 = ~

~ = ff~ = O,

= 0

~"~ = r~,

at ~ = -n,

~"~ - if,"; = (b - ap 2) ~o

at ~ = 0.

(9)

3. SOLUTIONS The following two cases will be considered: (1) (1 _p2) > 0, (2) (1 _p2) < 0. Analysis will be carried out for each case. Case (1): (1 _pZ) > 0

Equation (7) can be written as ~ " + 4~* ~ = qlip,

(10)

DYNAMIC BEHAVIOUROF REINFORCED SHELLS

461

where (1 _ p 2 ) = 4~t4. F r o m e q u a t i o n (10), |~t = C t cosh a~ cos a~ + (72 cosh a~ sin a~ + Ca sinh ct~ cos ct~ + (74 sinh a~ sin a~ + Bt, +72 = Ct cosh a~ cos a~ - (72 cosh a~ sin a~ - Ca sinh ct~ cos a~ + (74 sinh a~ sin a~ + Bt,

(11)

where Bt = q/ip4~ + a n d the Cl are constants. W i t h the aid of e q u a t i o n (9), CI, C2, Ca a n d C+ are determined as

CI/Bt = { - c o s h an cos an - ml(sinh an cosh an - sin an cos an)}/ft, C2[Bt

= -Ca/Bt

=

mx(cosh an cos an - cosh 2 an + sin 2 an)if'l,

C+[Bt = {mt(sinh an cosh an + sin an cos an - sinh an cos an -

cosh an sin an) - sinh cm sin an}/ft,

fx = cosh 2 an - sin 2 ctn + m l ( s i n h an cosh an - sin an cos an), where

(12)

m t = (b - ap2)/8ct 3. F r o m equations (11) a n d (12), the inversion theorem gives wt/q = ~

1 I" / (1 +

e wr

Ft/fl)n---~-~-Z,,4dp,

(13)

t/

Br

where F1 = - {cosh an cos an + n h ( s i n h an cosh an - sin an cos an)} cosh a~ cos a~ + n h ( c o s h an cos an - cosh 2 an + sin 2 an) (cosh ct~ sin a~ - sinh a~ cos r i o + {nh(sinh cm cosh an + sin an cos an - sinh an cos an -

cosh an sin an) - sinh an sin an} sinh ct~ sin a~.

Therefore, wt becomes Iie lz~llV,~ Wt/q = [1 + 17 , I U I J m,=bl,V~"r ~ 2FI COS('k/T'Z"4~ T)/[-'~a (I

- 4ct+)]

x [2rot n(cosh = cm - cos z an) + {(3/8a +) (a - b) + ~a + 2n} (sinh an cosh an - sin an cos an)],

(14)

where the a I are the roots o f

A = 0.

(15)

It will easily be u n d e r s t o o d that a = 0 is n o t a singular p o i n t in e q u a t i o n (13).

Case (2): (1 _ p 2 ) < 0 E q u a t i o n (7) can be written as

+V" -

a + ~P = q / i p ,

(16)

where 1 _ p 2 = _ a +. F r o m e q u a t i o n (16), +~t = Ct cosh a~ + C2 sinh ar + C3 cos a~ + C+ sin ar - B2, ~2 = Ct c o s h a ~ - C2 sinh a~ + C3 cos a~ - C+ sin a~ - B2,

(17)

where B2 =q[ipa + a n d the Cl are constants. W i t h the aid o f e q u a t i o n (9), (71, C2, Ca a n d (74 are determined as

C1/B2 = {2 cos an + m2(sinh a n + sin an - 2 sinh an cos an)}/2f2, C2/B2 = -C4[B2 = m 2 ( - c o s h an - cos an + 2 cosh an cos an)/2f2, Ca/B2 = {2 cosh an + m2(2 cosh an sin an - sinh an - sin an)}/2f2, f2 = 2 cosh an cos cm + m2(cosh an sin an - sinh an cos an),

( l 8)

462

where

S.-I. SUZUKI

m2 = (b - ap2)/2~3. From

equations (17) and (18), the inversion theorem gives e trr

1

w,/q=-~n i

I(-1

+ F2/2f2) - ~ d p ,

(19)

Br

where F2 = {2 cos cm + m2(sinh ~n + sin c~n - 2 sinh ~n cos cm)) cosh c~ + + m2(cosh cm + cos cm - 2 cosh cm cos cm) (-sinh c~ + sin ~ ) + + {2 cosh cm + nh(2 cosh ~n sin cm - sinh ~n - sin ~n)} cos ~ . Therefore, wt becomes

wJq =

~ 2F2 cos(V~ + c~"T)/[~(I + ~4)] [2m2 n sinh ~n sin cm atI

+ {(3/2~ a) (a - b) - 89 - 2n} (cosh ~n sin cm - sinh cm cos ~n)],

(20)

where the cq are the roots o f A = O.

(21)

The final form of the deflection, w~, will be obtained by summing up the results of Cases (1) and (2). The value of the first term of equation (14) indicates the value o f w t for the case where qo is applied statically. The direct stresses ao and ar in the circumferential and longitudinal directions, respectively, are given as

ao=E

(R

vz aZw~ 1--v 2 ~ ] '

Ez 02w a ~ = - l _ v 2 ax2'

(22)

where z is the distance from the mid-plane of the cylinder wall.

4. NUMERICAL A N A L Y S I S Numerical calculations have been carried out for several cases. In order to study the effects of Eo/E and Po/Pon the dynamic stresses, it is necessary at first to investigate the values of a0 and a~ for the case where the pressure qo is applied statically. Since the deflection wl is given by the first term in equation (14), the values of a0 and a~ will be obtained by using equation (22). From the results of numerical calculations, the effect of n on the deflection almost vanishes for the case urn > 15. Calculations will be carried out for such a case. The distribution of a0 on the mid-plane of the cylinder wall along the ~ axis is illustrated in Figure 2 (a). The values o f % become almost constant in the range ~ > 7 for all values ofb. The distribution of outer fibre stress a~ along the ~ axis is illustrated in Figure 2(b). The values vary remarkably, but they become almost zero in the range ~ > 7. Therefore, it can be considered that the effects of reinforcement on ao and a~ lie within a distance of about 4V'R-h on either side, on the assumption of v = 0.3. The distributions of % and ar in the neighbourhood of ~ = n are not affected by the value of b for n > 15. The relationships between the values of~o and ar at ~ = 0 and b are illustrated in Figure 3. As the value ofb increases, the value o f % decreases and ar increases. Therefore, the value ofb must be chosen properly when the cylinder is reinforced. Next. consider the case where qo is applied impulsively. Since wt is expressed in the series fornl, convergence of the series in equations (14) and (20) must be investigated. Convergence becomes slow as the value of n increases and b decreases. From the results Of numerical

463

DYNAMIC BEHAVIOUR OF REINFORCED SttELLS 1"21

I o

I

(o)

-

8

I

,!/

~

I

I

....

~I ~

o

// ,It

(b) 0.8

o-4

I0

-L

3

6

n-6

n -3

n

Figure 2. Distribution of(a) Geand (b) erealong the ~ axis,

calculations, 45 and 40 terms are taken for the cases ofn = 40, b (= a) = 0 and 10, respectively, and here the accuracy of Gebecomes of the order of 10-3. On the contrary, convergence of the series for ar is slow in comparison with that ofao. In this case, twice the number of terms as for ao must be taken. =

I

=

I

0"8

9~.

o

0.4

IO

20 b

Figure 3. Relationships between b, tro a n d ar at ~ = O.

464

S.-I. S U Z U K I i

(o) 2

i

(b)

a=b=O =I0

I

2

o I

7

b~

or

A== i

I

I0

A I

20

I

I0

20

T

Figure 4. Relationships between (a) Tand ao at ~:= 0 and (b) Tand a~ at ~ = 0. The relationships between a0, the outer fibre stress ar at ~ = 0 and time are illustrated in Figures 4(a) and (b) for the cases n = 40, b ( - a) = 0, 10 and oo, respectively. The relationships between a, b, ~ and the maximum value of ao obtained above are illustrated in Figure 5(a). These values become almost constant in the range ~ > 7 for all values of a and b. The relationships between a, b, ~ and the maximum value of the outer fibre stress ar are illustrated in Figure 5(b). It will be noticed that the values, which become almost constant in the range ~ > 7, are not zero. Values of both a0 maxand a~ . . . . which are much larger than those in Figures 2(a) and (b), are influenced over a wide range by the reinforcement at = 0. But the ranges influenced most significantly are again of the order of about 4V'R-h. 4~..,,...

,

j

//

,

'

(a)

o=b=O 3

.

_co

b~

0

//

2

// (b)

o = b = 0 3

--

0

I

I

3

6

_ _ ~

//

I 34

37

40

Figure K. Distribution of (a) ae,~=, and (b) o'er,=, along the ~ axis..

D Y N A M I C BEHAVIOUR OF R E I N F O R C E D SHELLS I

t

1

5

6

9

465

E

I

\ 9

\

g E

Figure 6. Relationships between b, aom,x and ag=~,at ~ = O. ~ - - - , ao=,~ for a/b = 0-9.

, am~,for a/b = I'0; - - - , a,,~ for a/b = 1"1 ;

The relationships between a, b, a0ma~ and Crema~at ~ = 0 are illustrated in Figure 6. Solid lines, dashed lines and the chain line indicate the cases for a]b = 1.0, 1.1 and 0.9, respectively. There is almost no difference between the values of ar for a/b = 1"0 and 0.9. As will be seen in this figure, the value o f b must be chosen properly.

5. CONCLUSIONS The following conclusions can be drawn from the present results. (1) When the load is applied statically, the ranges influenced by the reinforcing ring (on either side) are about 4V'R-fi in length for both ao and ere. (2) When the load is applied impulsively, the maximum values of ao and ar are much larger than those under static load and the ranges influenced by the reinforcing ring become very wide. (3) As the value of b increases, the value of I '01.,ax decreases and I"el.,a increases. Therefore, the value of b must be chosen properly when the cylinder is reinforced.

REFERENCES 1. K. SCmrFNER and C. STEELE1971 American Institute o f Aeronautics and Astronautics Journal 9, 37-47. The cylindrical shell with an axisymmetric moving load. 2. S. TAgO 1965 American Society o f Civil Engineers 91, 97-122. Dynamic response of a tube under moving pressure. 3. S. TANG1967 American Society o f Civil Enghleers 93,239-256. Response of a finite tube to moving pressure. 4. P. BI~rrA 1963 Journal o f the A coustical Society o f America 34, 25-30. Transient response of a thin elastic cylindrical shell to a moving shock wave. 5. S. SuzuKi 1973 Ingenieur Archio 42, 69-79. Dynamic behaviour of thin cylindrical shells subjected to high-speed travelling inner pressures. 6. S. SUZUKI 1972 Zeitschriftfiir Angewandte Mathematik und Mechanik 52, 583-590. Effects of shearing force and rotatory inertia to dynamical behaviours of thin cylindrical shells subjected to impulsive inner pressures. 7. S. SUZtJKZ1966 Journal o f Applied Mechanics 33, 261-266. Dynamic elastic response of aring to transient pressure loading.

466

s.-L SUZUKt A P P E N D I X : LIST OF SYMBOLS

21 total length of cylinder

,z - - t ~ / v " ~

q =R2qo/Eh qo r t w Wo v p po tro trr A a

intensity of inner pressure = (t/R) V ' ~ time deflection of cylinder deflection of reinforcing ring =j-~o w e - p r d T Poisson's ratio =x~,'120 - v ~ / ~ ' - ~ density of cylinder density of reinforcing ring direct stress in the circumferential direction direct stress in the longitudinal direction sectional area ofreinforcingring = "~]2(I

- - v 2) A p o l P h

= ~q2(1 -- vZ)AEolEh~ Young's modulus of cylinder Eo Young's modulus of reinforcing ring h thickness of cylinder b E