BEHAVIOR VESSELS G.

OF

CYLINDRICAL

UNDER

PLASTIC

FLAT-BOTTOM STRAIN

G. P o p o v

UDC 62-222:539.4

In this paper the strength of the cylindrical p a r t of v e s s e l s is considered on the basis of the theory of limiting equilibrium [1] which a s s u m e s ideal plasticity of the m a t e r i a l . In calculating the limiting load using this theory, the elastic and elastoplastie stages in the work of the s t r u c t u r e can be neglected. The limiting load is independent of the Young' s modulus, and the m a t e r i a l is considered as a rigid-plastic material. The limiting load is determined by the relation [2]

~5'

(i)

where p is the ultimate elastoplastic strength; w2 = 12/D6; D is the mean diameter of the cylinder; 1 is the cylinder length; 6 is the wall thickness; and a T is the yield point of the m a t e r i a l . This expression is approximate since it is based on the substitution of a shell with homogeneous wall thickness by a t w o - l a y e r shell. This equation produces the limiting load which in the extreme case is 25% too high [3]. Equation (1) a s s u m e s that the bottom is thicker than the body wall, which is usually the case. The bottom of the v e s s e l m a y be considered as a c i r c u l a r plate loaded on the surface by a uniform p r e s s u r e and along the edge by the reaction moment (Fig. 1) [4]. The actual l o a d - c a r r y i n g ability of plates is much higher than that calculated from the theory of limiting equilibrium. The strength of plates is favorably affected by the change in their g e o m e t r y during loading. The deformation of the central plane can be neglected if the deflection is small in relation to the plate thickness; in the ease of l a r g e r deflections, however, the m e m b r a n e f o r c e s appear in the plate. The s t r e s s due to the m o m e n t produced at the beginning of loading gradually changes towards the z e r o - m o m e n t s t r e s s . The theory of limiting equilibrium neglects the effect of the changing g e o m e t r y . The f i r s t v e r y approximate methods of calculating the effect of considerable deflections during the plastic stage were given in papers [5, 6]. Later, Lepik [7] suggested a method for calculating plates loaded along the edges, the results of which are in a b e t t e r a g r e e m e n t with the experimental data. In the p r e s e n t paper we use this method in analyzing v e s s e l bottoms. Lepik [7] a s s u m e d in calculating plates with load-free edges, that the deflections are c o m m e n s u r able with the plate thickness but are small c o m p a r e d with its radius. The elastoplastic stage was neglected, i.e., the m a t e r i a l was a s s u m e d to be rigid-plastic. The investigation showed that in its plastic state the plate can be divided into two zones according to the distribution of s t r e s s e s (Fig. 1). In the central zone, 0 -< r -< rl, the position of the z e r o - s t r e s s layer is independent of radius. The distribution of s t r e s s e s over the layer thickness in the radial and the c i r c u m f e r e n t i a l directions is identical (Nr = N 0 , M r = M 0). The s t r e s s e s are equal to the yield point a T . In the second zone, located n e a r e r to the edge, the s t r e s s changes along the plate radius. The position of the z e r o - s t r e s s layer (coordinate z0) also changes. In this zone the plate d e f o r m s only in the annular direction while in the radial direction it r e m a i n s rigid (conical deformation shape). The central zone widens with i n c r e a s i n g deflection covering an increasing portion of the plate. At the same time the neutral layer m o v e s towards the external surface, i.e., the s t r e s s tends towards the z e r o - m o m e n t state.

T r a n s l a t e d f r o m Khimicheskoe i Neftyanoe Mashinostroenie, No. 5, pp. 10-12, May, 1967.

354

p, 3t m

3~

i

205

Pb:170

f

lOtS

L

t;bo=3t6 Pt= 1~5 0 ~r ~

i

f

f

tCr~

Mo

2

4

d

8

18 Wbo, mm

t

o

Fig. 1 Fig. I. Loading

diagram

Fig. 2 for the vessel bottom.

Fig. 20 Dependence of the deflection (i) in the central part of the body w b and (2) in the bottom center Wbo on pressure p. The dotted line is the calculated curve.

T h e c o o r d i n a t e of the zone b o u n d a r y a = r t / a and the d e f l e c t i o n at the p l a t e c e n t e r w 0 a r e c o r r e lated by

The c e n t e r d e f l e c t i o n w 0 and the load q a r e , in t u r n , l i n k e d b y the r e l a t i o n

q

The deflections and the position termined by the expressions:

%-h- ~~:') 1 -}- 3 h2 2 a ~3(1 of

the zero-stress

W I ~

W 0 --

(2 --

a) 2

'

layer for the first and second

q r 2 (2 - - a) 16wo% (! ~ ~)

zones

are de-

(4)

(5)

w2-= 2--~--~- 1 - -

(6) 2--a

z0.~= -- 2 - -

~

I

--

(7)

respectively. In c o n t r a s t to the c a s e i n v e s t i g a t e d i n [7], we have r e a c t i o n m o m e n t s a c t i n g a l o n g the b o t t o m edge (Fig. 1). I n v e s t i g a t i o n s show that in this c a s e the s t r e s s e d p l a t e has t h r e e r a t h e r than two z o n e s with d i f f e r e n t t y p e s of s t r a i n . To o b t a i n an e x a c t s o l u t i o n it is n e c e s s a r y to d e t e r m i n e the b o u n d a r i e s of t h e s e r e g i o n s : this r e n d e r s the p r o b l e m m u c h m o r e c o m p l i c a t e d . It t h e r e f o r e a p p e a r s d e s i r a b l e to u s e an a p p r o x i m a t e s o l u t i o n . It s h o u l d b e n o t e d that the n e g l e c t e d w o r k h a r d e n i n g of the m a t e r i a l p a r t i a l l y c o m p e n s a t e s the e r r o r i n t r o d u c e d b y the a p p r o x i m a t e s o l u t i o n . We u s e d the k i n e m a t i c m e t h o d [8] b a s e d on the e q u a t i o n o f t h e p r i n c i p l e of v i r t u a l w o r k , which can be e x p r e s s e d f o r the c a s e u n d e r c o n s i d e r a t i o n in the following a n a l y t i c a l f o r m :

355

r

q~ 2T: ~ w(r)rdr-- M12~al~il-- U=O.

(8)

0

where ql is the load on the plate w the edges of which are loaded by m o m e n t s . In (8) w(r) c h a r a c t e r i z e s the field of displacements of the s y s t e m being considered which also d e t e r m i n e s the angle of rotation y of the n o r m a l s to the basic contour and the quantity U which e x p r e s s e s the work of the internal f o r c e s in the system, and c h a r a c t e r i z e s the dissipation of energy during plastic strain. The approximate nature of this method is due to the fact that the actual field of displacements is unknown while the a s s u m e d field is in effect an approximation. The resulting load is also approximate and is l a r g e r than the actual load. The e r r o r d e c r e a s e s with the d e c r e a s i n g difference between the actual and a s s u m e d fields. In the problem being considered we assume that the field of displacements has been obtained by solving the problem of a plate with f r e e l y supported edges [7]. In this case U r e m a i n s unchanged and the clamping of the plate edges will be taken into account by a t e r m incorporating the m o m e n t M 1 which p e r f o r m s negative work during the rotation of the plate edge through angle Y. It should be noted that the solution of the problem of the limiting load for a plate with clamped edges by an analogous method, but without taking into account the effect of the changing g e o m e t r y of the plate, produces the following results. The accurate solution gives the value of the limiting load ~ = 2.82 crTh2/a 2, while the solution c a r r i e d out by the kinematic method, considering the field of displacements for a f r e e l y supported plate, gives the value = 3 ~Th2/a 2. The e r r o r is 6.6%. With the changing g e o m e t r y taken into account, it m a y be expected that the kinematic method involving the use of the field of displacements calculated from Eqs. (4) and (5) produces an e r r o r of the same o r d e r in calculating the plastic changes with the edge m o m e n t taken into account. Consider now (8). F r o m (5) we have

dw~

2 Wo

] 7 l = l ~ - r J = a (2 - - a)"

(9)

Taking the two zones of the displacement field into account Eq. (8) can be written as follows

q, 2~.

(i

w,rdr + ~ w~rdr~

"

r,

47:wo

J ~ -2 - - a M t ~ U =

O.

(lo)

A c c o r d i n g to [7]

q2~

w,rdr +

(11)

w~rdr - - U = O ,

where q is the load on the f r e e l y supported plate. By solving together Eqs. (10) and (11) we obtain

woM I

q~= q +

trl wlrdr + ~a (2--a) I~

(12)

i

After integration and some modifications (12) becomes

qt = q +

356

[2+a3 - - _ a2 [ 3

_

4 M, (2-a)2 a2. a~. q . . . 32(1--a)%w~

]

(13)

In engineering calculations it is m o s t convenient to take values and apply in turn Eqs. (2), (3), and

(13). The magnitude of the edge m o m e n t M 1 is determined by the r a t h e r complicated type of interaction between the bottom and the_cylindrical sections of the v e s s e l . The m o m e n t M 1 cannot be either l a r g e r than the limiting m o m e n t M = O-Th2/4 which the edge can r e s i s t in a plastic state, or l a r g e r than the l i m i ting m o m e n t M = CrT52/4 which can be r e s i s t e d by the cylindrical shell. Alongside with the edge m o m e n t , the plate and the shell c a r r y m e m b r a n e f o r c e s which, according to the t h e o r y of plasticity, reduce the limiting moment. In addition, in the plastic stage of loading, the i n c r e a s e of deflection i n c r e a s e s the m e m b r a n e f o r c e s and, consequently, the edge moment d e c r e a s e s . It is difficult to e x p r e s s this all in the f o r m of a calculation. The analysis of the experimental data shows that in p e r f o r m i n g the calculations it m a y be a s s u m e d that M 1 = O-T52/4 if the bottom thickness exceeds the shell thickness and M 1 = O-Th2/4 if this is not t]he case; it m a y also be a s s u m e d that during the entire loading p r o c e s s M 1 r e m a i n s constant. It has a l r e a d y been pointed out that a gradual transition from the m o m e n t state to the z e r o - m o m e n t state is a c h a r a c t e r i s t i c feature in the behavior of plates under load. The d e s c r i b e d method reflects this tendency. It can be shown that the relations obtained for w 0 = 0 and the r i g i d - p l a s t i c m a t e r i a l agree with the corresponding relations of the t h e o r y of limiting equilibrium which neglects the m e m b r a n e f o r c e s . On the other hand, with i n c r e a s i n g deflection, the load approaches a s y m m e t r i c a l l y the straight line q = 4 aThw0/a 2, r e p r e s e n t i n g the relationship between the load and deflection in the z e r o - m o m e n t t h e o r y [6]. We tested a n u m b e r of v e s s e l s made from a l o w - c a r b o n steel. The principal g e o m e t r i c a l relations v a r i e d in the following r a n g e s : D/h = 16-32; h / 6 = 1-2; and D/1 = 0.7-0.9. The tests involved the m e a s u r e m e n t of s t r e s s e s by means of r e s i s t a n c e wire, strain gauges and the m e a s u r e m e n t of deflections at various potcLts of the body and bottom. Figure 2 gives the experimental relations for the p r e s s u r e and deflection in the central p a r t of the body and in the center of the bottom; they were obtained in testing a v e s s e l with the following dimensions D =159mm;h=5mm;5 = 4 . 7 5 m m ; a n d l = 2 1 0 m m . The yield point of the m a t e r i a l was a T =28 kgf/ m m 2. The load axis shows the calculated values of the tensile strength for the elastic state PT and the elastoplastic ~ state. It should be noted that the slope of curves shows no m a r k e d change during the elastoplastic stage while at the end of this stage it begins to d e c r e a s e at a high rate. This suggests that the plastic deformations are m o r e constrained during the elastoplastic stage and develop at a higher rate at the end of the stage. The calculated value of the ultimate strength for the elastoplastic state is n e a r the section which shows a sharp reduction in the slope of the load v e r s u s deflection curve. It can thus be said that the results of tl~e analytical calculation on the basis of the limiting equilibrium theory are in s a t i s f a c t o r y a g r e e m e n t with the experimental data. However, with i n c r e a s i n g deflection, the curve i n c r e a s e s c o n s i derably beyond the limiting value, e s p e c i a l l y in the case of the deformation of the bottom which f i r s t b e c o m e s plastic but then continues to c a r r y the i n c r e a s i n g load. As the loading continues the body of the v e s s e l also b e c o m e s plastic. In the case of the bottom, the ability to r e s i s t an i n c r e a s i n g load after transition into the plastic state can be attributed to the effect of the work hardening of the m a t e r i a l and to changes in the g e o m e t r y ; in the case of the body it can be explained by the effect of work hardening only, since changes in the geom e t r y ; of the cylinder, i.e., the increase of the radius and the reduction of the wall thickness, reduce the strength. The dotted line in Fig. 2 gives the calculated load v e r s u s deflection curve which takes account the effect of the changing g e o m e t r y of the bottom by the method d e s c r i b e d above. The simplifications made in the calculations, i.e., the uniformity of the edge m o m e n t and the approximate kinematic method of its calculation, m u s t i n c r e a s e the calculated load. This condition is not satisfied for shells with thicker bottoms. The calculated curves for such v e s s e l s lie below the experimental c u r v e s . This contradiction can be attributed to the effect of the work hardening of the m a t e r i a l which is neglected by calculation. For the same deflections, the strain n e a r the surface of a thick plate is g r e a t e r than the strain of thin plates, and this r e s u l t s in a higher work hardening of the f o r m e r bottoms. Thus, the calculation e r r o r is c o m m e n s u r a b l e with the e r r o r introduced by work hardening. Both e r r o r s p a r t i a l l y cancel each other. This suggests the conclusion that a f u r t h e r increase in the a c c u r a c y of calculation will produce no s a t i s f a c t o r y results if the effect of work hardening is neglected.

357

Stress analysis showed that the maximum stress is reached in the cylindrical shell where this joins the bottom; thereupon the stress rapidly decreases with increasing distance from this zone. Adopting the theory of maximum shear stress, the ultimate elastic strength for the vessel being considered is 13.5 atm. Actually, the vessel did not i~il under a pressure of 300 arm. The experimental and analytical results obtained for other tested vessels are not given here since the general nature of the experimental curves and their agreement with the values calculated on the basis of the theory of limiting equilibrium are the same for all vessels. It should only be noted that a number of vessels were tested to failure. The bodies invariably failed in the central part, where stresses in the elastic state are much lower than stresses at the point where the cylindrical part of the vessel joins the bottom. Thus, the calculation for the elastic stage gives a picture both qualitatively as well as quantitatively on the behavior of vessels which is very different from their actual behavior. LITERATURE i.

2. 3.

4. 5. 6. 7. 8.

358

CITED

A. A. Gvozdev, Calculation of the Strength of a Structure by the Method of Limiting Equilibrium [in Russian], Stroizdat, Moscow (1959). P. G. Hodge, Calculation of Plastically Strained Structures [Russian translation], Mashgiz, Moscow

(1963). V. Ol'shak, Z. Mruz, and P. Pezhina, Present State of the Theory of Plasticity [in Russian], Mir, Moscow (1964). P. G. Hodgc, J. Mech. and Phys. Solid, 12, No. 7 (1964). A. R. Rzhanitsin, In: Investigations on the Problems of Building Mechanics and the Theory of Plasticity [in Russian], Gosstroiizdat, Moscow (1956). E. Onat and R. Haythornthwaite, J. Appl. Mech., 233, No. 1 (1956). Yu. R. Lepik, Izv. AN SSSR, OTN, Mekhanika i Mashinostroenie, No. 2 (1960). Yu. P. Rabotnov, Strength of Materials [in Russian], Fizmatgiz, Moscow (1962).