D E T E R M I N I N G T H E S T R E S S - S T R A I N S T A T E O F A

C I R C U L A R P L A T E R E I N F O R C E D W I T H R A D I A L R I B S

~.. S. U m a n s k i i a n d G. L . K a l i k h m a n UDC 539.4

The bending of c i rcu la r r ibbed plates, widely used in various construct ions, has been investigated by many authors [4-8, 10].

In the present paper we examine the bending of a disk res t ra ined on the inner per iphery and free on the outer, re inforced by equally spaced radial r ibs and loaded on the outer per iphery by uniformly d i s t r i - buted bending moments (Fig. la) or by t r ansve r se forces (Fig. lb).

Great difficulty is encountered in finding an exact solution of this problem. With a ra ther large num- ber of r ibs (m > 8), it may be solved with acceptable accuracy on the basis of the theory of struck~rally or thotropic c i rcu la r plates [5-7, 10]. Below we have presented an approximate solution of the problem by separa t ing the composite design into disk and r ibs, and the continuous interact ion between these is replaced by severa l equally spaced forces uniformly distr ibuted according to width of the r ibs, acting at right angles to the central surface of the disk. It is assumed that the r ibs a re in a uniaxial s t r e ss state, the disk In a biaxial s t r e s s state. This approach is s t r ic t ly appropriate only for symmet r ica l ly ar ranged r ibs re lat ive to the central surface of the disk (see Fig. 1). However, even when r ibs tliat are not very high a re a s y m - met r i ca l ly arranged, it is possible to p rese rve the same model, neglecting the tangential forces of in ter - action between r ibs and disk, since the bending moments due to these forces are small and cannot substan- tially affect the bending of the disk.

The unknown forces are determined f rom the equality of deflections of the disk and r ibs at the points the forces are applied. There is also another approach to substitution for the nature of interaction between r ibs and disk [1, 4, 8].

The f i rs t problem concerns bending of the disk by a concentrated force. Its solution is not difficult and has been discussed for other boundary conditions in a number of papers (such as [3, 9]).

In the present example, it is assumed that the unknown force is uniformly distributed along an a rc of the c i rcumference , the length of which is equal to the width of a r ib b, and it is fur ther assumed that the force function may be expanded along the c i rcumference in a Four ie r se r ies

q = ~ --~A- nO cosn~ , (1)

where r 0 is the radius of force distribution, and fl is the angular coordinate (see Fig. 1).

Having assumed that the polar axis lies along the axis of a rib, and having divided the imaginary disk into two par ts by cyl indrical sectioning r = r0, let us r ep resen t the function of disk deflection in the form

w, (r, ~) = A~,) o _~ A(O2o r~ + A(')~o lnr + A(')4o r~ Inr + (A~t~r + A(2~)r -I + A(at~ra + A~)r lnr) cos~

+tAU)r, q -A(~r-n " ACOr"+ 2 A(4~r-"+2)cosn~, (2)

where i = 1 r e f e r s to the outer zone of the disk and i = 2 r e f e r s to the inner part.

Kiev Polytechnic Institute. Trans la ted f rom Problemy l>rochnosti, No. 10, pp. 57-63, October, 1970. Original ar t ic le submitted December 15, 1969.

�9 1971 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced [or any purpose whatsoever without permission o[ the publisher. A copy o[ this article is available [rom the publisher for $15.00.

1023

a

Fig . 1. C o m p u t a t i o n a l s c h e m e of a r i b b e d d i s k with l oad ing on the o u t e r r i m by an a x i s y m m e t r i c a l load: by d i s t r i b u t e d r a d i a l m o m e n t s M 0 (a) and by f o r c e s Q0 (b).

A l l d i s p l a c e m e n t s and i n t e r n a l f o r c e s in t he d i s k a r e found f r o m w e l l - k n o w n f o r m u l a s of the e n g i n e e r i n g t h e o r y of th in e l a s t i c p l a t e s [3]:

O = OWar ' M , = - - D --07i- --}- ~ -7Ei- + r,--i~--~, ;

0~, Ow q_ O=w ~. M~j ---- - - D v ~ --b -TE;" r'ap' 1' Mp, = M p = - - D (1 - - v) ~./. i r_.y~_ , 0 0~ (3)

8 0 OMr[ ~ q , = o - ~ - V e w ; Q p = D T V~w; V, = Q, - - ra~ "

H e r e D is the c y l i n d r i c a l r i g i d i t y of the d i sk ; D = E h 3 / 1 2 ( 1 - v2)). [3) a r e d e t e r m i n e d f r o m the fo l lowing b o u n d a r y cond i t i ons :

1) a t the i n n e r r e s t r a i n e d r i m (r = rB)

2) a t the o u t e r f r e e r i m (r = rH)

3)

Mr. = Vr. -~- O;

at the con juga te c i r c u m f e r e n c e (r = r0)

A r b i t r a r y c o n s t a n t s in the func t ion w i ( r ,

(4)

(5)

w.-----:w., 0 = 0 B, M~ = M . , Q - - Q % = q . (6)

S u b s t i t u t i n g (2) and (3) in c o n d i t i o n s (4) - (6) , we o b t a i n a s y s t e m o f l i n e a r equa t i ons f o r d e t e r m i n i n g the a r b i t r a r y cons tan ts i n the f u n c t i o n s (2).

S ince the l o a d had been expanded in a F o u r i e r s e r i e s (1), t he i n d i c a t e d s y s t e m f a l l s i n t o n s y s t e m s o f the 8 th o r d e r , w h i c h have the f o l l o w i n g f o r m :

When n = 0 (2) _2 "'10/1(2) -1--- ~2o'1(2)rB2 -F --3oA (~) In r + A4o r In r = 0;

Am). A(2)r - t A (m ( 2 r In 0; 20zr -~- 30 B -~---40 r -J- r ) =

0) A(z) {3 = -~oA~ ~-21nr ~ - v ~ - 2 v ] n r ) 0;

A~,, = o;

A(2) _{_ A(2) ~ A O) A(2) 2 . n(1 ) A ( I ) _2 A(z ) In r e - - A4o r o m r o = O; IO 20%+ I n r o + In 0)_2,_ "'30 40 ro /o - - "'110 - - " - 2 0 [ 0 - - z't30

A(2)~ _L ~(2).--~ Am) --0)o r 0) 2In ro) 2oZro ' "'30"0 -J---40 ro( 1 - I -2 lnro) ~(l~ - i - - A2o " o - - ~3o ro - - A4o r o ( 1 + = 0 ;

A(2) t9 A (2) - 4- A (2) A (u (2 -b 2v) 20 ,- + 2v) - -3o ( l - - v ) ro 2 - - "'4o (3 + 2v lnro + 2In ro + v) - - - -~o

o) - A(u (3 + 2vln r o -F 2In r o ~- v) = 0; + A3o (1 - - v) r o 2__ 40

(7)

When n = 1

A(2) __ A([) R s 40 "'40 ~ - - 8 ~ O "

A(2)_ - - .4(2) --f (2)_3 ~ --(m_ In r a ---- 0; I I re -[- "q21 ra --[-- A31 re = /'t41 ra

A ( 2 ) (2)_-2 4_ A(2)3r2 A~2t ) (1 + In r . ) 0; I I A21 f8 = --31 n ~ =

A(' (2- - 2~) r : 3 + A~',' (6 + 2v ) r + A~',' (I + ~) q ' = 0; 21

(I) - - 4 A21 (2 - - 2v) r. -F A~'I)(6 -{- 2v) - - A~II ) (3 - - v) r . 2 = 0;

(2). A(2)_- I A(2) 3 (2) . A ( I ) a ( I ) - I _ _ A ( I ) 3 A( I ) | B r 0 0; A=1% -{- --21% - ~ - . ' t 3 1 r o - } - A 4 = r o l n . o - - r h l r o - - , = 2 1 , 0 ..3t %--,~41 ro =

A(2) (2)_-_~ .(2)~ 2 A (2) fl A (I) A H)--2 A(U.'~ -~ A (I) (] -~- |n ro) 0; t , - - A 2 1 % -t- ,a31 ~ r o -F --4t ,- -[- In %)-- -{- = " * l l " ' 2 1 /0 - - ' ~ v ; 0 - " ' 4 1

1024

0

- 2

- 4 / 5

2 5

>=- -, , /

a / b I

I0 15 n 0 5 I0 15

Fig . 2. Dependence of r e l a t i v e e r r o r 5 on the n u m b e r of m e m b e r s n of the e x p a n s i o n of the i n t e r a c t i n g s e r i e s in to a F o u r i e r s e r i e s (a) and the n u m b e r of po in t s k of a p p l i e d f o r c e : 1, 2) fo r the m o m e n t M T a t the o u t e r r i m of the d i sk , b e n e a t h the r i b s and b e t w e e n the r i b s r e - s p e c t i v e l y ; 3, 4) f o r d e f l e c t i o n of the o u t e r r i m of the d i sk , b e n e a t h the r i b s and b e t w e e n r i b s r e s p e c t i v e l y ; 5) fo r the m a x i m u m bend ing m o m e n t in the r i b .

A~2) t g - - 2v) - " + A ( e ) ( 6 - 4 - 2 V ) r o - k - A ( 2 ) ( l + v ) r o I AO)(2 2v) A0)(6.+ 2V)ro 0) 21 ~.-- ro 3 "-31 "'41 -21 ro 3 "'31 ~--- - - - - - - - - - A 4 1 ( l + v ) r o I O;

8A(2) A(2)9_-2 RA(l) + A(4',)2ro 2 R s "-31 - - "-41 -% ~"3] -~ ~ s i n - ~ - o �9

When n > 1

(8)

4n I}

O~ n + l (2) _ r--n+l A(l~ nr~-' ~ A~, ) nr-~ ~-' + AI2)3~ (n + . , r~ - - A4. ( . - - 2) ~ = 0 ;

- - n - - 2 n A(')n (n - - 1)(1 - - v) ~ - 2 + A~,)n (n + 1) (1 - - 'v) r. tn + A~ (n + 1) (n+2 - - vn + 2v)r,

_p " ( ' ) , n4~ tn - - I) (n - - 2 - - vn - - 2v) r~ -~ = 0;

A~ n 2 (n-I- I) (1 - - v) r -"-3 ~ A~ ) n (n + 1) (n - - vn - - A(,~ n ' (n - - 1 ) (1 - - v) ~ - 3 H- - - h

-(l) n (n - - 1) (n - - vn-4-4)r[n-l=O; - - 4 ) ~ - ' - - y A4n

A(2) (2) A(2) ~+2 A(2) - A~,~ ,, A(t) - A~,) ~-~2_ A ( ' %"42=0; (9) l n~ - I -A2nro " + 3. -t- 4nro " + 2 - i r o - - . - , ~ r o n - - 3. "'4.

.(2) n-~ (2) __-n-, + A(21 2) ro+~ A (2) - - A ~ - - A ~ " o (n + (n - - 2) n ~ - ' d i n /2?'0 3n - - " ' 4 n / . o n + I I n

+A(~) - - o) _ A ~ - = 0 ; --2n nro n l ~ A3" (n + 2) %+' + 4. r0 ~+1

a(2) n ( n + 1 ) ( 1 - - v ) r o "-2 A(2) (n + 1)(n +2) - - v n "-In A(2) n ( n - - 1)(1 - - v) ro - 2 + "'z~ "k- "'3.

J r 2v) ~o -F A(2)4n (n - - I ) ( n - - 2 - - v n - - 2v) r o " - - A(,~ n (n - - l ) ( 1 - - v) ro -'~

_,_o. A ( ' ) ( n + 1 ) ( n + 2 - - v n + 2 v ) " " ( ' ) ( n ~ l ) ( n - - 2 - - v n - - 2 v ) r - ~ o n = O ; - - A ~ 1)(1 - - v ) r 0 - - ' -3n ro~A4n 2n

A (2) 4 n ( n - - l)ro n-I - - A ( ~ ) 4 n ( n + 1)~ -I A~ r~on-I R3 A~ 4n(n -4- l ) ro - ' -4-'-,n "'3n - - ,n = - - n---h--ff~ sin r"-~'- �9

A f t e r the s y s t e m s (7)-(9) have been so lved , i t i s p o s s i b l e to d e t e r m i n e d e f l e c t i o n of the d i s k (2) a t any po in t due to a s i ng l e f o r c e , and, by u s i n g the t h e o r e m of m u t u a l d i s p l a c e m e n t , c o n s t r u c t a m a t r i x o f the c o e f f i c i e n t s of the e f fec t of d e f l e c t i o n s fo r a l l the unknown f o r c e s r e p l a c i n g the r i b s .

We s h a l l u s e the fo l lowing d e s i g n a t i o n s : j i s the po in t of the d i s k and r i b w h e r e the unknown s t r e s s of i n t e r a c t i o n b e t w e e n d i s k and r i b Xj i s app l i ed ; Wjm i s the d e f l e c t i o n of poin t j b e c a u s e of the f o r c e 7-m = 1; Wjp i s the d e f l e c t i o n of the g iven po in t b e c a u s e of t he l oad on the r i m ; 5jm i s the de f l e c t i on of the po in t of the r i b j b e c a u s e of the e f fec t of Xm = 1, d e t e r m i n e d fo r the c a n t i l e v e r . We ob ta in the s y s t e m of equa t ions

1 0 2 5

I;: k p = .V 6i,,,X., j = 1, 2, . . . , k, lm= WimX., + %~

r t l = l

(10)

w h e r e k i s the n u m b e r of unknown i n t e r a c t i n g f o r c e s b e t w e e n d i s k and r i b s . F r o m the s y s t e m (10) we f ind the i n d i c a t e d i n t e r a c t i n g f o r c e s , a f t e r which we u s e Eqs . (2) and (3) to c ompu te d i s p l a c e m e n t s and i n t e r n a l f o r c e s in the d i sk . The i n t e r n a l f o r c e s in a r i b a r e d e t e r m i n e d as n o r m a l l y fo r the a t t a c h e d c a n t i l e v e r .

C a l c u l a t i o n by the i n d i c a t e d m e t h o d invo lves l a b o r i o u s c o m p u t a t i o n s . In th i s connec t ion , a p r o g r a m was se t up on a M i n s k - 2 2 c o m p u t e r , p e r m i t t i n g d e t e r m i n a t i o n not only of the unknown f o r c e s but a l s o the be nd i ng m o m e n t in the r i b and a l s o the d e f l e c t i o n of the o u t e r r i m of the d i s k and the r a d i a l bend ing m o m e n t s on the i nne r r i m at s e v e r a l po in t s a long the c i r c u m f e r e n c e : u n d e r the r i b s , b e t w e e n the r i b s , and a t o n e - q u a r t e r the d i s t a n c e b e t w e e n r i b s . The p r o g r a m m a k e s i t p o s s i b l e to r a t e the d i s k u n d e r a r b i t r a r y r i m l oad i ng and wi th any n u m b e r of i d e n t i c a l equa l l y s p a c e d r i b s a r b i t r a r i l y a r r a n g e d . F o r t h i s i t i s n e c e s s a r y to s e t up a s u b p r o g r a m of expand ing the l oad in a F o u r i e r s e r i e s and of change in r i g i d i t y of the r i b s a long the r a d i u s .

R e p l a c e m e n t of the i n t e r a c t i o n b e t w e e n r i b s and d i s k by a f in i t e n u m b e r of f o r c e s k and p r e s e r v a t i o n of only s o m e m e m b e r s n of the e x p a n s i o n in the F o u r i e r s e r i e s l e a d to d e f i n i t e e r r o r s . D e t e r m i n a t i o n of e r r o r was m a d e on a d i s k wi th the fo l lowing d i m e n s i o n s ( see F ig . 1): r H = 100 cm, r B = 40 cm, h = 1 cm, b = 1 cm, H = 10 cm, and the n u m b e r of r i b s m = 4 ( d i s t r i b u t e d r a d i a l m o m e n t s of 1 k g - c m / c m w e r e a p - p l i ed on the o u t e r r a d i u s ) . To d e t e r m i n e the e f fec t of the va lue of n on the a c c u r a c y of t he so lu t ion , the r e l a t i v e e r r o r was c o m p u t e d f r o m the fo l lowing:

M(nmax) - - M(n ) W(nma x) - - W(n ) fin = �9 100% and (5= ---- �9 100%, (11)

M(nmax) W(nmax)

w h e r e M(n ) i s the r a d i a l bend ing m o m e n t at the i n n e r r i m of the d i s k o r the m a x i m u m bend ing m o m e n t in the r i b ; W(n ) i s the d e f l e c t i o n of the d i s k at the o u t e r r i m wi th the r e t e n t i o n of n m e m b e r s of t he e x p a n s i o n in to the s e r i e s (1); M(nmax) and W(nmax) a r e the s a m e wi th r e t e n t i o n of the m a x i m u m n u m b e r of m e m b e r s of the s e r i e s (1), for which s t r o n g c o n v e r g e n c e i s ob t a ined . F o r m o s t c a s e s i n v e s t i g a t e d , i t i s s u f f i c i e n t to adopt n m a x = 30.

w , c m

aolo % . . . . . . . . . . ~oos . i 2 o..oo6 ~ o ~ i

M r, kg-cmlcm a

1 f

M kg-cm

- . _ - - 2 - - - 1 = - _

0.4 ~ - . . , ~ . . ~ "qt -,L,. 40 45 30 135 B,deg 40 50 60 70 80 $Or, cm

c d

Fig . 3. S t r e s s - s t r a i n s t a t e o f a r i b b e d d i s k l o a d e d on the o u t e r r i m by u n i f o r m l y d i s t r i b u t e d m o m e n t s M 0 = 1 k g . c m / e m : a) eb~nge in d e - f l e c t i o n s of the d i s k in a c i r c u m f e r e n t i a l d i r e c t i o n f o r t he po in t s r H = 100 c m (I) and r a v e = 70 c m (2); b) change in d e f l e c t i o n s of the d i s k in a r a d i a l d i r e c t i o n b e n e a t h a r i b (1), m i d w a y b e t w e e n r i b s (2) and o n e - q u a r t e r the d i s t a n c e b e t w e e n r i b s (3); c) change in r a d i a l bend ing m o m e n t M r 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 at po in t s r B = 40 c m (1) and r a v e = 70 c m (2); d) change in r a d i a l bend ing m o m e n t M r in the r a d i a l d i r e c t i o n [1) b e n e a t h a r i b , 2) m i d w a y b e t w e e n r i b s , and 3) o n e - q u a r t e r t he d i s t a n c e be tw e e n r i b s ] and the bend ing m o m e n t in the r i b Mp (4). (Sol id l i n e s r e f e r to the r i b b e d d i s k , d a s h e d l i n e s to a d i s k wi thout r i b s . )

1026

W, cm

a4 --"" ~ 1 ,42

~ - - . . ~ / / ~ . ~ 0,2 _. _7~ . . . . ~ 7 , , _ ~ 7 ~ _ _ ~ _ ~ _ __

M r, kg-cm/cm

60 . ....>1, 2

,to __/_, _, ~ T / 7 . ~ _ _ / _ _ _ n ~

"/5 30 135 B, deg

~ J

M kg-cm

.,2<'-<

40 50 50 70 80 ,90 100r, cm

Fig . 4. S t r e s s - s t r a i n s t a t e of a r i b b e d d i s k l oaded on the o u t e r r i m by u n i f o r m l y d i s t r i b u t e d l oad Q0 = 1 k g / c m ( s y m b o l s the s a m e as in F ig . 3).

F i g u r e 2a shows the d e p e n d e n c e 5 M = f(n) and 5 w = f(n), f r o m which we s e e tha t su f f i c i en t a c c u r a c y fo r e n g i n e e r i n g c o m p u t a t i o n s i s o b t a i n e d by r e t a i n i n g 10-12 m e m b e r s of s e r i e s (1).

A n a l y s i s of the e f fec t of n u m b e r of po in t s k on the a c c u r a c y of the so lu t i on was m a d e in s i m i l a r f a s h - ion, fo r which i t was a l s o a s s u m e d tha t k m a x = 30. F i g u r e 2b shows g r a p h s of the dependen t r e l a t i o n s 5 M = f(k) and ~w = f(k). On the b a s i s of the r e s u l t s ob t a ined , the v a l u e adop ted fo r f u r t h e r c o m p u t a t i o n s was k = 8 -10 .

As an e x a m p l e , we have shown in F i g s . 3 and 4 the r e s u l t s of c o m p u t a t i o n s of the r i b b e d d i s k of the d i m e n s i o n s i n d i c a t e d above , a c t e d on by r i m l o a d s of M 0 = 1 k g - c m / c m and Q0 = 1 k g / c m r e s p e c t i v e l y .

F r o m the i n v e s t i g a t e d e x a m p l e s we s e e tha t a p p l i c a t i o n of the t h e o r y of s t r u c t u r a l l y o r t h o t r o p i c p l a t e s to c o m p u t a t i o n of d i s k s wi th a s m a l l n u m b e r of r i g i d r a d i a l r i b s m a y l e a d to c o n s i d e r a b l e e r r o r , which w a s a l s o no ted in [5]. Be low we p r o p o s e an e n g i n e e r i n g m e t h o d fo r c o m p u t a t i o n of d i s k s wi th a s m a l l n u m - b e r of r i b s hav ing any d e g r e e of r i g i d i t y .

T h e m e t h o d c o n s i s t s ch i e f ly of m a k i n g n u m e r o u s c o m p u t a t i o n s , by m e a n s of an ~.TsVM c o m p u t e r , of d i s k s wi th d i f f e r e n t n u m b e r s of r i b s hav ing d i f f e r e n t r i g i d i t y r e l a t i o n s .

D i s k s wi th c o m p a r a t i v e l y n a r r o w wid th w e r e c o n s i d e r e d : b / r B = 1 / 5 - 1 / 5 0 at r B / r H = 0.2, 0.4, and 0.6. It was e s t a b l i s h e d tha t t he s t r e s s s t a t e of the d i s k in th i s c a s e d e p e n d s v e r y i n s i g n i f i c a n t l y on change in r i b width wi th in the i n d i c a t e d l i m i t s a t a g iven m o m e n t of i n e r t i a of the r i b . Wi th in t h e s e l i m i t s we z~ight r e c o m m e n d u s e of an a p p r o x i m a t i o n m e t h o d of c o m p u t a t i o n , p e r f e c t l y s a t i s f a c t o r y fo r p r a c t i c a l a p p l i c a - t ion.

We i n t r o d u c e the fo l lowing t e r m s : t, the p a r a m e t e r of r e l a t i v e r i g i d i t y , t = I / h 3 r H (l = bH3/12 , the m o m e n t of i n e r t i a of a r ib ) ; K 1 = w " / w ; K 2 = w ' / w ~ ; K 3 = M p / M r ; K 4 = M ~ / M r , w h e r e w", w ' , and w r e - p r e s e n t d e f l e c t i o n of the d i s k at the o u t e r r i m b e t w e e n r i b s , be ne a th a r i b , and in the d i s k wi th no r i b s ,

! r e s p e c t i v e l y (in cm); M r and M r a r e m a x i m u m r a d i a l bend ing m o m e n t s at the i n n e r r i m of the r i b b e d d i s k and the d i s k wi thout r i b s ( k g - c m / c m ) ; and Mp is the g r e a t e s t bend ing m o m e n t in the r i b (kg -cm) . In F i g s . 5 and 6 we have shown g r a p h s of the d e p e n d e n c e of the c o e f f i c i e n t s K i - K 4 on the p a r a m e t e r t a t d i f f e r e n t v a l u e s of the r a t i o r B / r H and n u m b e r of r i b s m = 4, 6, and 8.

The g r a p h s p e r m i t us to d e t e r m i n e the m a x i m u m de f l ec t i on , the m a x i m u m bend ing m o m e n t in the d i s k and r i b s , and a l s o the d e g r e e of d e v i a t i o n of the s t r e s s - s t r a i n s t a t e f r o m ax ia l s y m m e t r y with u n i - f o r m l y d i s t r i b u t e d m o m e n t s o r t r a n s v e r s e f o r c e s on the o u t e r r i m .

A c o n t r o l c o m p u t a t i o n of the d i s k w a s m a d e a c c o r d i n g to the fo l lowing s c h e m e .

1. C o m p u t a t i o n s w e r e m a d e fo r r i g i d i t y of the r i b I = bH3/12 , the p a r a m e t e r t = 1 / h 3 r H , and the

r a t i o p = r B / r H.

2. F r o m g r a p h s ( F i g s . 5 o r 6), depend ing on the t y p e of load ing , we found the c o e f f i c i e n t s K t - K 4 fo r the g iven n u m b e r of r i b s .

3. We d e t e r m i n e d the m a x i m u m d e f l e c t i o n and m a x i m u m bend ing m o m e n t for a g iven p la te wi thout r i b s a c c o r d i n g to f o r m u l a s f r o m [3]:

1027

~8

O.5

~2

/5

8O

7O

60

5O

4O

3O

2O

I0

--...-.__

m=4 ~--

5 "

\ \

'. " . K: 4 " ~

- ~ , ~ { C ~- -_ .. I m=4

,,,~--~" Ka: ~ ~ . . . . / :7-

. , ;:---- ~ ~'~ m ! ,.--

301

.,-,=~..=

-2 - I 0 I @t -2 -1 0 / ~ t

Fig. 5 Fig. 6 Fig. 5. Dependence of the coef f ic ien t s K t -K 4 on the p a r a m e t e r s t, p, and the n u m b e r of r i b s m with u n i f o r m l y d i s t r i b u t e d r a d i a l m o m e n t s (dashed l i n e s , p = 0.2; so l id l i n e s , p = 0.4; d a s h e d - d o t t e d l i ne s , p = 0.6).

Fig. 6. Dependence of the coef f i c ien t s K~-K 4 on the p a r a m e t e r s t, p, and n u m b e r of r i b s m with u n i f o r m l y d i s t r i b u t e d t r a n s v e r s e fo rce s ( symbols as in Fig. 5).

a) with loading by m e a n s of the m o m e n t M 0 ( k g - c m / c m )

5 46 (1 - - Q2 _{_ 202 In [2) Mor~ W ~

1 . 3 + 0,70 = Eh 8 ' (12)

2Mo Mr = 1.3 + 0.702 ;

b) with the t r a n s v e r s e fo rce Q0 (kg/cm)

2~ (0.7169 - - 0:5648Q ~ - O. 152104 + 1.738Q I In 0 - - 1.129o. = In2 O) Qo raH w ~ 1 . 3 + 0.70= Eha '

(13) Mr Qor. 0.7 (02 - - 1) -{- 2.6 In 0

--~ 2 1,3 -{- 0.70 =

4. We found the p r i n c i p a l p a r a m e t e r s of the s t r e s s - s t r a i n s t a te of the d i sk

u/' = Klw; w' = K2w"; Mp = K3Mr; M~ =K4Mr. (14)

5. The m a x i m u m s t r e s s e s in the d i sk and r i b w e r e ca l cu l a t ed a c c o r d i n g to the f o r m u l a s

6M r 6Mp (15) 0 d= - - ~ ; ~p = bH= �9

As an example le t us c o n s i d e r the computa t ion for a d i sk with the fo l lowing d i m e n s i o n s : r H = 50 cm. r B =20 cm, h = 2 cm, H =18 cm, b = 2 cm, n u m b e r of r i b s m = 4 (with a d i s t r i b u t i v e t r a n s v e r s e fo rce Q0 = 20 k g / c m on the ou te r r i m ) .

1. We compute the m o m e n t of i n e r t i a of the r ib I = 972 cm 4, the p a r a m e t e r t = 1.215, l o g t = 0.085, and the r a t i o p = 0.4.

1028

o, kg/cm 2

o , 8 /2 76 gcm a

o, kg/cm 2

\ ~ _ _ .

200

0 4 8 b 72 ]6 ~cm

Fig. 7. Dependence of m a x i m u m s t r e s s e s in the d i sk (dashed l ines) and r i b s (sol id l ines) on d i m e n s i o n s of the r i b s .

2. F r o m the g raphs of Fig. 6 for p = 0.4, m = 4, and l o g t = 0.085 we f ind K 1 = 0.52, K 2 = 0.10, K 3 = 6 5 , a n d K 4 =0 .48 .

3. F r o m the f o r m u l a s of (13) we d e t e r m i n e w = 0.143 cm and M r = 1050 k g - c m / c m .

4. F r o m Eq. (14) we have

w" = 0.52.0.143 = 0.0745 cm; w' = 0.1,0.0745 = 0.00745 crn;

Mp = 65. 1050 = 68 200kg-cm; M~ = 0.48. 1050 = 505 kg-cm/cm.

5. The m a x i m u m s t r e s s e s in the d i sk and r i b a c c o r d i n g to Eq. (15) a r e

r d = 760 kg/cm 2 ; ap = 630 kg/cm 2.

The con t ro l compu ta t i ons of the d i sks made by m e a n s of the ~.TsVM compu te r showed that the e r r o r of ca l cu l a t i on f r o m Eq. (14) does not exceed 10-15%.

T h e r e i s p r a c t i c a l i m p o r t a n c e in the i nves t i ga t i on of the dependence of m a x i m u m s t r e s s e s in the d i sk and r i b on the height and width of r i b s with d i f fe ren t n u m b e r s of r i b s . Th i s i n v e s t i g a t i o n was made for a d i sk with r H = 100 cm, r B = 40 cm, h = 1 cm, and d i s t r i b u t i v e load of Q0 = 1 k g / c m . F i g u r e 7 shows the dependence of the m a x i m u m s t r e s s e s in the d i sk a d and the r i b Crp on he ight of a r i b H at d i f fe ren t t h i ck - n e s s for four (Fig. 7a) and s ix (Fig. 7b) r i b s . F r o m these g raphs i t a p p e a r s that r e i n f o r c e m e n t of the d~sk by tow and thin r i b s i s i n a d v i s a b l e , tt i s d e s i r a b l e to s e l e c t r i b d i m e n s i o n s such that a des ign of u n i f o r m s t r e n g t h is obta ined. In the i n v e s t i g a t e d example , u n i f o r m s t r e n g t h is ach ieved in a d i sk with four r i b s of b = 2 c m a n d H = 8 cm.

F o r computa t ion of d i sks with o ther d i m e n s i o n s , i t i s r e c o m m e n d e d that, u s i ng F igs . 5 or 6, g raphs s i m i l a r to those in Fig. 7 be c o n s t r u c t e d , and, a f t e r c o m p a r i n g r e s u l t s , d i m e n s i o n s of r i b s be se lec ted .

1.

2.

3.

4.

5.

6.

L I T E R A T U R E C I T E D

V. M. Agranov ich , D. V. Va inbe rg , and E. S. U ma nsk i i , "The bend ing of a c i r c u l a r p la te by t r a n s - v e r s e f o r c e s d i s t r i b u t e d in s t eps at poin ts of s e p a r a t e r a d i i , " in: The S t r e s s State of Wheels in Rol l ing Mills and Mining Equipment [in Russian], Izd. AN UkrSSR, Kiev (1959). I. A. Birger, B. F. Shorr, and R. M. Shneiderovich, Rating the Strength of Machine Parts [in Rus- sian], Mashgiz, Moscow (1959). D. V. Vainberg and E. D. Vainberg, Plates, Disks, and Wall Beams [in Russian], Gosstroiizdat, Kiev (1959). D. V. Vainberg, "Methods of rating circular ribbed plates," in: Calculation of Three-Dimensional Designs [in Russian], No. 5, Moscow (1959). A. N. Dukhovnyi, Criteria of the Axisymmetrical Character of the Strain State of Circular and Ring Plates Reinforced with Radial Ribs [in Russian], Trudy Vses. In-ta Gidromash., No. 31 (1962). A. N. Dukhovnyi, An Approximation Solution of the Problem of Bending Circular and Ring Plates Reinforced with Radial Ribs [in Russian], Trudy Vses. In-ta Gidromash., No. 30 (1962).

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7. A . N . Dukhovnyi, An Approximation Method of Determining S t resses when Bending Circular and Ring Plates Reinforced with Radial Ribs [in Russian], Trudy Vses. In-ta Gidromash. , No. 30 (1962).

8. K . A . Kitover, Thin Circular Plates [in Russian], Gosstroi izdat , Moscow (1953). 9. S . D . Ponomarev et al., Computations of Strength in Mechanical Engineering [in Russian], Vol. 2,

Mashgiz, Moscow (1958). 10. O. M. Rubach, ~The bending of c i rcu la r plates r e in fo rced with radial r i b s , " in. Collection of Works

of the Institute of Mechanical Engineering of the Academy of Sciences, UkrSSR, No. 20. Methods of Rating Wheels of Rolling Mills and Mining Equipment [in Russian], Izd. AN UkrSSR, Kiev (1955).

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