Yield-point loading curves for circular plates

Int. J. mech. Sei. Pergamon Press. 1969. Vol. 11, pp. 455-459. Printed in Great Britain

YIELD-POINT LOADING CURVES FOR CIRCULAR PLATES J.

B.

HADDOW

University of Alberta (Received 18 November 1968, and i n revised f o r m 7 J a n u a r y 1969)

Summary--Yield-point loading curves relating the intensities of uniformly distributed transverse load and outer edge tensile load are obtained for simply supported annular and circular plates.

r , O, Z Or, 0"0, O'z, Trz

~o

~,~

Vr, vz er, es, ez

H A B

M. Mo N . No Me = ao H2/4 No = ao H

NOTATION cylindrical polar co-ordinates non-zero components of stress for axial symmetry uniaxial yield stress components of middle-surface velocity velocity components strain-rate components plate thickness outer radius inner radius principal moments components of tensile force yield moment yield force INTRODUCTION

UPPER a n d lower b o u n d s for y i e l d - p o i n t loading curves relating t h e intensities o f u n i f o r m t r a n s v e r s e l o a d a n d edge load for a n n u l a r p l a t e s w i t h v a r i o u s s u p p o r t conditions h a v e b e e n o b t a i n e d b y H o d g e a n d S a n k a r a n a r a y a n a n . 1 I n this p a p e r e x a c t y i e l d - p o i n t loading curves, b a s e d on t h e usual a s s u m p t i o n s of t h e plastic t h e o r y of p l a t e s 2 a n d t h e T r e s c a yield condition, are o b t a i n e d for s i m p l y s u p p o r t e d a n n u l a r plates w i t h stress-free inner edges a n d for s i m p l y c o n n e c t e d circular plates. S t a t i c a l l y admissible stress fields a n d r e l a t e d k i n e m a t i c a l l y admissible v e l o c i t y fields which satisfy t h e T r e s c a yield condition a n d a s s o c i a t e d flow rule r e s p e c t i v e l y are o b t a i n e d . C o n s e q u e n t l y t h e corres p o n d i n g y i e l d - p o i n t loads are exact. ANNULAR

PLATE

Let r, 0, z be cylindrical polar co-ordinates and let the plate occupy the region -- H I 2 <~z <~HI2, B <~r ~ A with a uniform tensile force of intensity T acting at r = A and a

uniform transverse load of intensity P acting in the z direction. A possible statically admissible stress field has the tensile force taken by the region -- S ~
455

= 0

f

)

(1)

456

J.B.

HADDOW

w h e r e t h e p r i m e s d e n o t e d i f f e r e n t i a t i o n w i t h r e s p e c t t o r, a n d t h e c o n ( h t l o n s ~: = m a x ( I X ~ !, i X 0 I, l N , - 5 " 0

I ) < 28~o =

2SNo/H t

(2)

M = m a x ( [ M ~ J, I M o I, [ M r - M e I ) < - M o ( 1 - 4 S 2 / H 2 ) ~ w h m h follow f r o m t h e T r e s c a yield c o n d i t i o n . I t follows f r o m e q u a t i o n (2) t h a t M

+

~< 1.

(3)

Since e q u a t i o n (3) is b a s e d o n a s t a t i c a l l y a d m i s s i b l e d i s t r i b u t i o n of stress t h e c o r r e s p o n d i n g s u r f a c e i n f o u r - d i m e n s i o n a l (Mr, M e , N r , No) s p a c e lies o n or w i t h i n t h e yield surface. P r a g e r a n d O n a t a h a v e o b t a i n e d t h e yield surface for shells of r e v o l u t i o n a n d e q u a t i o n (3) w i t h t h e e q u a l i t y coincides w i t h p a r t of t h i s surface. I t is c o n v e n i e n t t o i n t r o d u c e t h e following n o n - d i m e n s i o n a l q u a n t i t i e s : r

p =-~,

p = PAe/6Mo,

n - - - N / N o,

m=M/M

b= B/A,

o

t = T / N o.

I f P = 0, t h e g e n e r a l i z e d stresses a n d m i d d l e surface velocltms* of a c o m p l e t e s o l u t i o n are

no=

1,

nr = 1 - - ( 1 - t ) / p ,

t = to = ( l - - b )

(4)

v~ = c > 0 , v~ = 0 (5) w h e r e C is a c o n s t a n t . Also ff T = 0 t h e g e n e r a l i z e d stresses a n d m i d d l e - s u r f a c e velocities of a c o m p l e t e s o l u t i o n axe m o = 1, m r = l + 3 p b ~ - p p 2 - ( l + 3 p b 2 - p ) / p I P = po(l+b-2b2)

-1

i

(6)

V~ = 0, V~ = D ( 1 - - p ) > 0 (7) w h e r e D is a c o n s t a n t . I f i t is a s s u m e d t h a t a, a n d ~,~ m a y b e n e g l e c t e d in t h e yield c o n d i t i o n a n d %~ b u t n o t a, c a n b e o b t a i n e d f r o m e q u i h b r i u m c o n d i t i o n s t h e stress field c o r r e s p o n d i n g t o e q u a t i o n (6) is (~0 =

O'oSgnz

or

a0[1 + 3pb ~ _ p p 2 _ (1 + 3pb 2 - p ) / p ] s g n z t

~-r.

--~

0-

(8)

(tzt-H/2)

a n d , f r o m e q u a t i o n (7) a n d t h e u s u a l k i n e m a t m a l a s s u m p t i o n s of p l a t e t h e o r y , t h e corres p o n d i n g s t r a i n - r a t e s are er = O, e o = z D / ( r A ) = - e ~ . I f 0 < t < t 0 = (1 - b) t h e g e n e r a l i z e d s t r e s s field t h a t is o b t a i n e d b y c o m b i n i n g nr a n d no o f e q u a t i o n (4) m u l t i p l i e d b y t( 1 - b) a n d e q u a t i o n (6) m u l t i p l i e d b y [1 - t2/( 1 - b) ~] satisfies t h e e q u i l i b r i u m e q u a t i o n s (1) a n d t h e yield c o n d i t i o n (3). T h i s g e n e r a l i z e d stress field h a s t h e t e n s i l e force t a k e n b y t h e region --S ~
(9)

a n d S < z <<.H / 2 t h e stresses a r e g i v e n b y e q u a t i o n s (8) a n d for O"0 ~

O"0

a~ = a o ( l - - b / p ) 3Pa° ( p - ~ )

(10) (S-H/2'~

* T h e v e l o c i t y field is of course n o t u n i q u e .

Yield-point loading curves for circular plates

457

I t follows from the lower-bound theorem of limit analysis t h a t

[ - ( ~ - b ) ] t' p -- ( l + b - 2 b ~ ) -1 [1

(11)

is a lower bound for the yield-polnt loading curve. Equation (11) is identical to the upperbound curve o b t a ~ e d b y i o d g e and Sankaranarayanan,1 consequently it is exact. This m a y be shown otherwise. According to the flow rule associated with the Tresca yield condition the strain-rate components associated with the stresses ar and a0 of the statically admissible stress field given b y equations (8) and (10) satisfy the conditions in Table 1. TABLE 1 Region

Stresses

Strain -rates

-H/2<~z< -S -S<~z<~S S
- a 0 < a r ~ < 0 , ao = - s o

er = O, eo<~O er = O, eo>~O, ez = - e o er = O, eo>~O

O<~ar
A kinematically admissible velocity field that satisfies the above conditions on er and e0 can be found. The middle-surface velocity field Vz= D(1-r/A

),

D>0,

V r = C,

C>0

which is a combination of fields (5) and (7) results in velocities vr =

-~

+ C,

v~ =

Dz2 2A

CZ+ D ( 1 - p )

(12)

and strain-rates Dz co= C/r+~-~ =-e~.

er = O,

These strain-rates satisfy the conditions of Table 1 if D C

A S

and using equation (9) this requirement becomes D

,(1-b)A = ~ ~ .

(13)

I t follows that the statically admissible stress field given b y equations (8) and (10) has a n associated kinematically admissible velocity field given b y equations (12) and (13), consequently equation (11) is exact.

SIMPLY

CONNECTED

PLATE

The generalized stresses and middle-surface velocities for T = 0 have been obtained by Hopkins and Prager, 4 and are mr = 1 - p ~, m = V~ = D ( 1 - p ) ,

1,

Vr = 0,

p=

1

(14)

D>0.

(15)

The corresponding stress field is ~ 0 ~- ~o S ~

Z /

ar

a0(1 _ p2) sgn z /

(16)

458

J . B . HADDO~,V

a n d t h e v e l o c i t y a n d strain r a t e fields are

Dz

Dz 2

v~ = ~ - ,

vz - - - W ~ - A + D ( 1 - p )

(17)

Dz er ~

O,

e° ~- r ~

- ~ - - ez"

T h e strain-rates e0 a n d e~ b e c o m e infimte at r = A. This is permissible since the m a t e r i a l is inviseid. I f P = 0 t h e generalized stresses are nr = n 0 =

1,

t = to = 1.

(18)

T h e following v e l o c i t y field is kinernatically admissible a n d is associated w i t h the stress field e x c e p t at r -- r 0 where there is a d i s c o n t i n u i t y of %, v~ = V ~ = C > 0 ,

%<.r<~A

v,=---, r

•

Cr

vr = v,

= - -

v z = - - - - ,

TO ~

O<<.r<~r

o

(19)

~

,gO

T h e radius % can be m a d e v a n i s h i n g l y small a n d for t h e limiting case r 0 = 0 t h e v e l o c i t y field is associated w i t h t h e stress field for all r. T h e strain rates er and e~ t h e n b e c o m e infinite a t r = 0 b u t this a g a i n is permissible since t h e m a t e r i a l is inviseid. I f 0 < t < t0 = 1 the generalized stress field t h a t is o b t a i n e d b y c o m b i n i n g t h e field (14) m u l t i p l i e d b y 1 - t ~ a n d n r a n d n o of (18) m u l t i p l i e d b y t satisfies t h e equilibrium e q u a t i o n s (1) a n d t h e yield c o n d i t i o n (3). This statically admissible generalized stress field corresponds to a statically admissible stress field w i t h t h e tensile force t a k e n b y t h e region - S ~ < z ~ < S where 2S = tiT-/. F o r regions -H/2<<.z< - S , S
O"0

an

a0

"rr~

~

"~

1 "

(20)

p( [z ]- HI2)

I t follows f r o m t h e lower-bound t h e o r e m of limit analysis t h a t p =

1-t

2

(21)

is a lower b o u n d for t h e yield-point loading curve. B y proceeding in t h e s a m e m a n n e r as for t h e a n n u l a r plate it m a y be shown t h a t t h e v e l o c i t y field v,--

_~

Dz ~ Cz v~=--~-~+-7+D(1-p)

+C,

which is o b t a i n e d b y c o m b i n i n g e q u a t i o n s (15) a n d (19), w i t h r 0 = 0, is associated w i t h the stress field g i v e n b y e q u a t i o n s (16) a n d (20) if

D/C = 2A/tH. C o n s q u e n t l y e q u a t i o n (21) is exact•

CONCLUDING

REMARKS

The type of stress field considered in this paper can be used to obtain lower bounds for yield-point loading curves for uniformly loaded annular and simply c o n n e c t e d c i r c u l a r p l a t e s w i t h c l a m p e d e d g e s . I f t h e e d g e s a r e c l a m p e d i t is n o t p o s s i b l e t o o b t a i n a n a s s o c i a t e d k i n e m a t i c a l l y a d m i s s i b l e v e l o c i t y field.

Yield-point loading curves for circular plates REFERENCES 1. 2. 3. 4.

P. H. E. H.

31

G. HODOE and R. SANKA~A_NAI~AYANAI%J. Mech. Phys. Solids 8, 153 (1960). G. HoPKr~s, Proc. R. Soc. London, 241A, 153 (1957). T. ONAT and W. P~AGE~, Proc. R. Netherlands Acad. Sci. B57, 534 (1954}. G. HOPKINS and W. PaAGER, J. Mech. Phys. Solids 2, 1 (1953).

459