SHRINK RING NEAR A FLANGE ON A HEATED CYLINDRICAL SHELL

A. G. Zgurovskii and I. I. Mariners UDC 629.7.023.024

It was shown earlier [I] that the shape and size chosen for the cross section of shrink rings can be such as to minimize the thermal stresses in the rings and the bending stresses in the enclosed shell in the case of a shell heated by an internal gas flow. Here, thesafety factors of the ring -- henceforth referred to as the "optimum" ring -- and shell will be the same for the given shell design and heating conditions. Most cyclically loaded shells have flanges on their ends. These flanges are "cold rigid" rings and have a cross-sectional area several times greater than the cross-sectional area of theoptimumrings. The bending stresses in the shell under such flanges when heated are much greater than the stresses under the op- timum ring and are the main reason for shell failure.

The goal of the present study is the prove that it is possible to reduce these stresses by installing a ring -- henceforth referred to as a "damping" ring-- near the flange at a dis- tance that will ensure a stress level during shell bending under the flange no higher than the stress level attained under the optimum ring. Thus, we can make the safety factor of the shell under the flange equal to the safety factor of the shell under the optimum ring. It is also very desirable that the damping ring be structurally identical to the optimum ring. Thus, in the problem that will be formulated we will seek to determine the proper distance from the flange to the damping ring, while the main condition for ensuring equality of the stresses during bending of the shell under the optimum ring and flange will be equality of the shell deflections during heating near the flange and optimum ring for the moment of greatest load- ing of the structure by thermal stresses. This situation will obviously be possible only when the sum of these deflections is no greater than the increase in the radius of the free heated shell. Here, we write the main condition for equality of the deflections in the form

w{ = ~, (i)

where wf is the shell deflection near the flange with a damping ring; w c is the deflection of the shell under the optimum ring.

Figure la-c illustrates the deflection of the heated shell near the flange and optimum ring, respectively, and the deflection of the shell with the damping ring. It follows from the results in the figure that the below equation is valid in the elastic region for the shell:

-r .............. ~ ..,

~.r C

Fig. i. Deflection of a reinforced shell during heating.

Kiev. Translated from Problemy Prochnosti, No. 7, pp. 70-72, July, 1986. ticle submitted September 17, 1984.

Original ar-

0039-2319/86/1807-0935512,50 ~ 1987 Plenum Publishing Corporation 935

'd.r

i i i i

A,i'

Fig. 2 Fig. 3

Fig. 2. The function flZd. r = f(Wc/&R).

Fig. 3. Damping ring on the exhaust pipe of the engine of an An- 22 airplane.

AR = ~f + w~ + w~, (2)

where AR is the increase in the radius of the free shell with heating; w'f is the deflection of the shell at the distance ~d.r/2 from the flange caused by its compression by the cold rigid flange" w' is the deflection of the shell at the distance Zd.r/2 from the damping ring

' C due to the installation of the optimum ring as the damping ring; Zd. r is the sought distance between the flange and damping ring.

The below formula gives the shell deflection near the cross section acted upon by the load p on the shell from the ring, this load being distributed uniformly about the circum- ference [2]

~_ P~R 2 2E? ~ (~x). (3)

Here, R is the radius of the shell; E c is the elastic modulus of the shell material at the corresponding temperature; h is the thickness of the shell; ~(~x) =e-~xcos~x+e-~xsin~x, where x is the distance from the plane of action of the load p to the cross section in question;

4 3 ( 1 - ~=),

p is the Poisson's ratio of the shell material.

According to (3), in the case of equal distances of the flange and damping ring from the sites of application of loads distributed about the circumference, the deflections of the shell are proportional to the maximum deflections in these shell sections:

w~ w~ mc = AR (4)

Equations (2) and (4) are obviously also valid in the elastic region in the section lo- cated the distance ~d.r/2 from the flange for the shell with the damping ring. The simultane- ous solution of these equations will be

m c I-- molAR (5)

m c 1+ wclAR

Since the damping ring deflects the shell on the flange side by the amount w' c < Wc, the force p which produces stresses in the ring is smaller than the corresponding force from the optimum ring. Thus, given an equal cross-sectional area, the stresses in the damping ring and the stresses created in the bending of the shell under the ring are lower than in the op- timum ring. It also follows from (3) that the main condition of the stated problem can ac- tually be written as equality of the deflections (i).

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With allowance for (3) and the fact that x = O,(p( !Sx)~ l Eq. (5) changes to the form

1 -P ~ - ' c / ~ R "

at the site of the damping ring,

(6)

Tables of values of the functions~p =/(~x) in [2, 3] can be used to easily determine the

value of Bx corresponding to the resulting value of ~(~)and, thus, the sought distance be-

tween the flange and damping ring that will satisfy the main condition of the problem. The value of B~d. r can also be obtained from the graph in Fig. 2. It is evident that the follow- ing approximate formula can be used with sufficient accuracy for --0,2~mc/AR~0,9

~/d.r = 0.85 ~ 0.7mJAR. (7)

The quantities Wc, AR, and p, regarded here as known quantities, can be determined from the formulas in [4].

For heated shells, it is possible to have the case Wc/AR < 0.5, i.e., when conditions (I) and (2) are not satisfied. If the stresses during bending of the shell under the flange [i, 5] are greater than the allowable stresses, two damping rings can be installed. The distances of these rings from the flange can be determined from the formula

/, = (8) , I + ~ c / ( A R - - ~'c)

Then the r a t i o Wc/AR in Eq, (7) must be r e p l a c e d by Wc/(AR -- Wc). For most heated s h e l l s -- i n - c l u d i n g the exhaus t p ipes o f a i r c r a f t eng ines -- the goal s t a t e d i n the problem can be a t t a i n e d by i n s t a l l i n g a s i n g l e damping r i n g ( F i g , 3 ) ,

The effectiveness of installing a damping ring on the exhaust pipes of an An-22 airplane was established from the resu]ts of operation of the plane before and after the installation. Analysis of the data for all planes shows that the frequency of malfunctioning of an exhaust pipe with a damping ring per flight was reduced by a factor of 4.7, while pipe costs per half- hour of flying time were reduced by a factor of 3.9.

Thus, the installation of a damping ring on a shell heated by an internal gas flow makes it possible to reduce thermal stresses in the shell due to the effect of the cold rigid flange to the level of the stresses caused by shrink rings.

The installation of a damping ring on the exhaust pipe of an aircraft engine, in which the life of the pipe is determined mainly by the startup load [4, 5], is one of the simplest and most effective methods of increasing pipe life.

LITERATURE CITED

i. A. G. Zgurovskii, "Design of cylindrical shells heated by an internal gas flow," Probl.

Prochn., No. 12, 108-110 (1985). 2. A. I. Lur'e, Statics of Thin-Walled Elastic Shells, Gostekhizdat, Moscow-Leningrad (1947). 3. S. P. Timoshenko, Strength of Materials [Russian translation], Vol. 2, Nauka, Moscow

(1965) 4. A. G. Zgurovskii, A. I, Kashchuk, and I. I. Mariners, "Fracture of shrink rings on a re-

inforced cylindrical shell cyclically heated by an internal gas flow," Probl. Prochn.,

No. 7, 109-113 (1984). 5. A. G. Zgurovskii, "Buckling and loss of stability of a reinforced cylindrical shell heated

by an internal gas flow of variable thermodynamic parameters," Probl. Prochn., No. II,

108-112 (1984).

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