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The plastic bending of transversely anisotropic circular plates
Int. J . mech. Sel. P e r g a m o n Press. 1970. Vol. 12, pp. 109-112. P r i n t e d m Great Britain
THE PLASTIC B E N D I N G OF T R A N S V E R S E L Y ANISOTROPIC CIRCULAR PLATES S. TANVIR WASTI Civil Engineering Department, Middle East Technical University, Ankara, Turkey (Received 29 May 1969)
Summary--A yield criterion for transversely anisotropm solids under plane stress conditions is obtained as an extension of the Tresca hexagon for the motropic case. Solutions to the plastic bending problems of transversely anisotropic circular plates are given. NOTATION k
coefficient of yield stress in the thickness direction
Mr plate radial moment plate circumferential moment fully plastic in-plane moment dimensionless radial moment iT dimensionless circumferential moment mo A radius of plate r dimensionless plate radius p dimensionless uniform load radial curvature rate circumferential curvature rate W vertical velocity F total punch load ] dimensionless punch load in-plane yield stress of material ~0 {Yl ~ ~Y2 principal stresses in plane stress P dimensionless r~gime boundary dimensionless punch radms Mo Mo
INTRODUCTION
THE PLASTIC behaviour of orthotropic materials has been discussed b y Hill. 1 For the particular case of transverse anisotropy where the yield stress in the direction o f the axis of s y m m e t r y is greater t h a n in a n y direction perpendicular t o this axis, the yield criterion is the isotropic Mises ellipse elongated along its m a j o r axis a n d shortened along its m i n o r axis. N o true c o u n t e r p a r t for the Tresca yield h e x a g o n in a n i s o t r o p y is available. V e n k a t r a m a n a n d S a n k a r a n a r a y a n a n 2 h a v e used an irregular h e x a g o n to t r e a t o r t h o t r o p i c cylindrical shells. YIELD CRITERION The Tresca yield criterion for isotropic solids can be considered as an approximation to that of von Mises. For the plane stress case in which the direction of zero normal stress coincides with the axis of symmetry of a transversely anisotropie solid, a suitable hexagon A B C D E F is readily constructed by elongating the isotropic Tresca hexagon A'B'C'D'E'F" along the line representing equal biaxial stress as shown in Fig. 1. Thus the hexagon 109
i I0
S. T A N V I R W A S T I
A B C D E F r e p r e s e n t s t h e yield c r i t e r i o n for a m a t e r i a l t h a t h a s a n i n - p l a n e yield stress d e n o t e d b y a0 a n d a yield stress in t h e t h i c k n e s s d i r e c t i o n of ka0 w h e r e k > 1. T h e c r i t e r i o n is r e a d i l y a p p l i c a b l e to p r o b l e m s of h m i t analysis, a n d r e d u c e s to t h e T r e s c a h e x a g o n w h e n t h e a n i s o t r o p y is v a n i s h m g l y small.
il~
----
C
k
---I~.
~ e-- 1 ---t- B I I
°
D'
=
i
i
dl/do
I ~
Trescahexagonfor isotroplcmaterial
E
Elongatedhexagonfor transversety anlsotrop=c materla[
Fro. 1. Postulated yield criterion for transversely anisotropic material. APPLICATION
TO ANISOTROPIC
CIRCULAR
PLATES
F o r c i r c u l a r p l a t e s of t h e a b o v e - d e f i n e d a n i s o t r o p i c m a t e r i a l t h e yield locus r e f e r r e d to t h e m r a n d me axes h a s t h e s h a p e of t h e e l o n g a t e d h e x a g o n A B C D E F m Fig. 2 w i t h sides given by the equations: 9~, 0 ~
-J-]t,
?n r -- ?n O ~
± 1,
w h e r e mr = 3 I ~ / M o, me = Me~Me a n d M o is t h e flllly plastic i n - p l a n e m o m e n t . S i m p l y supported anisotropic circular plate
T h e s o l u t i o n for t h e isotropic p l a t e u n d e r u n i f o r m l y d i s t r i b u t e d load c a n b e f o u n d in H e d g e . a C o n s i d e r i n g t h e s a m e p l a t e n o w to b e a n i s o t r o p i c in t h e t h i c k n e s s direction, it is clear t h a t p o i n t B in Fig. 2 r e p r e s e n t s t h e stress s t a t e a t t h e c e n t r e of t h e p l a t e , w h e r e mr = m0 -- k, a n d p o i n t C' r e p r e s e n t s t h e stress s t a t e a t t h e b o u n d a r y , w h e r e m~ = O. T h u s t h e r e are t w o rdgimes for t h e p l a t e , g i v e n b y B C a n d CC'. P o i n t C c o r r e s p o n d s to t h e v a l u e p of t h e d i m e n s i o n l e s s p l a t e r a d i u s r. T h e e q m l i b r i u m e q u a t i o n is d dr (rm~) - m o = - 3pr ~
a n d is i n t e g r a t e d for 0 < r < p a n d m o = k. T h e b o u n d a r y c o n d i t i o n s a r e mr(0 ) = ma(0 ) a n d m ~ ( p ) = k - 1 for p < r < 1; also m , - ~ l o = - 1. F u r t h e r b o u n d a r y c o n d i t i o n s a r e mr(p) = k - 1 a n d my(l) = 0. T h e following t r a n s c e n d e n t a l e q u a t i o n for p is o b t a i n e d 3p-lnp
= 2k+l.
F o r i s o t r o p y , k = l, a n d t h e e q u a t i o n is satisfied b y t h e f a m i l i a r r e s u l t p = 1.
T h e p l a s t i c b e n d i n g of t r a n s v e r s e l y a n i s o t r o p i c c i r c u l a r p l a t e s
111
F o r k = 2, t h e e q u a t i o n b e c o m e s 3p - In p - 5 = 0, w h e n c e p = 1.876, w h i c h i n c i d e n t a l l y coincides w i t h t h e r e s u l t for a c l a m p e d isotropie p l a t o u n d e r u n i f o r m l y d i s t r i b u t e d loading. In order that the solution be considered complete, a kinematically admissible velocity field w(r) c o m p a t i b l e w i t h t h e stress field m u s t b e f o u n d . F r o m Fig. 2, r4gime B C r e q u i r e s t h a t t h e c u r v a t u r e r a t e c o m p o n e n t ~r b e zero a n d Ka positive. B u t Kr is g i v e n b y d ~ zb kr = - dr----~ = 0. I n t e g r a t i n g , t h e v e l o c i t y o n side B C is g i v e n b y ~b = - A r ÷ B w h e r e A a n d B are c o n s t a n t s . T h u s t h e r e c i p i e n t v e l o c i t y field for r~gime B C (0 < r < p) is conical.
rn~rnBply
T
S
supported plate
/ F
Fro. 2. Stress profiles for t r a n s v e r s e l y a n i s o t r o p i c circular plates. F o r r6gime CC' t h e s t r a i n - r a t e v e c t o r is p e r p e n d i c u l a r to CC' a n d h e n c e ka = - k r . P u t t i n g k0 = - ( l / r ) (d~b/dr) a n d Kr = - d ~ b / d r ~, one o b t a i n s ~b = C l n r a n d t h u s t h e i n c i p i e n t v e l o c i t y field for r6gime CC" is l o g a r i t h m i c . O n l y t w o o f t h e c o n s t a n t s A, B a n d C c a n b e d e t e r m i n e d f r o m t h e c o n d i t i o n s t h a t W a n d d~b/dr are c o n t i n u o u s a t r = p. Clamped anisotropic circular plate T h e c l a m p e d a n i s o t r o p i e p l a t e u n d e r u n i f o r m l o a d also h a s a stress profile o n t w o sides of t h e e l o n g a t e d h e x a g o n , i.e. B C a n d C D i n Fig. 2. T h e d i s c o n t i n u i t y a t t h e c l a m p e d edge r = 1 m u s t c o r r e s p o n d to a s t r a i n - r a t e v e c t o r for w h i c h t h e r a t i o kr/ko t e n d s t o - ~ , a n d t h i s c o n d i t i o n is satisfied b y p o i n t D. H e n c e t h e stress profile for t h e c l a m p e d p l a t e covers tlle r a n g e B C D , p o i n t C c o r r e s p o n d i n g a g a i n to t h e v a l u e p o f t h e d i m e n s i o n l e s s p l a t e r a d i u s r. T h e e q u i l i b r i u m e q u a t i o n is i n t e g r a t e d for t h i s case for 0 < r < p a n d me -- k. T h e b o u n d a r y c o n d i t i o n s a r e m~(0) = ma(0 ) a n d mr(p) = / c - 1 for p < r < 1 a n d m ~ - m a = - 1 . F u r t h e r b o u n d a r y c o n d i t i o n s are mr(p) = k - - 1 a n d mr(1 ) ------k. T h e following t r a n s c e n d e n t a l e q u a t i o n for p is o b t a i n e d i n t h i s case, 3p-lnp
- 4 k ÷ 1.
T h e s u b s t i t u t i o n k -- 1 yields t h e e q u a t i o n 3p - In p - 5 --- 0 for t h e isotropic case. T h e i n c i p i e n t v e l o c i t y field c o m p r i s e s a cone a n d a l o g a r i t h m i c p a r t as in t h e case of t h e s i m p l y s u p p o r t e d p l a t e , a n d h e n c e t h e s o l u t i o n is c o m p l e t e .
112
S. TANVII~ WASTI
S i m p l y supported plate loaded by a circular punch The transversely anlsotropm simply supported circular plate is now considered under the action of a centered rigid circular p u n c h of radi~rs ~A, where A is the radius of the plate and 0 < ~ < 1. The isotropic case has been t r e a t e d by Hedge. 3 Only the p a r t of t h e plate where the dimensionless radius r as greater t h a n ~ can deform. There are three r6gimes; under t h e punch, m r = m o = k if the p u n c h force is considered as a line load around its circumference. Thus the whole regmn u n d e r the p u n c h is in r6glme B. F o r ~ < r < p and p < r < 1 there are two r6glmes as before and the e q u i h b r m m e q u a t i o n is g,ven b y d dr (rm~) - m o = - f , where F is the total load exerted by the p u n c h a n ( I f = F/2~rMo. The integration is earmed out for ~ < r < p and m o = /~'. The b o u n d a r y conditions are mr(~) = k and mr(p) = k - 1 for p < r < 1 and m ~ - m o = - 1. F u r t h e r b o u n d a r y conditions are m~(p) = t - 1 and mr(l) = 0. The two e q u a t m n s o b t m n e d are D 1-k f = o ' - ~ and f - - i = l n p " E l i m i n a t i n g p,
k - 1 (1 _1~ = exp ~ f/. F o r ~ = 0, the solution reduces to the ease of a c o n c e n t r a t e d load a n d f = 1 as m the ease of the lsotropic plate. F o r finite values of ~ higher values o f f are obtained. The velocity field is again a cone corresponding to side B C and a logarithmic part corresponding to CC" in Fig. 2. The portion of the plate u n d e r the p u n c h m o v e s s t r m g h t down and there is a hinge circle at r = ~ corresponding to poit)t B. CONCLUSIONS F o r / c > 1 i t is o b s e r v e d t h a t t h e c a r r y i n g c a p a c i t y o f t r a n s v e r s e l y a n i s o t r o p i c p l a t e s i n c r e a s e s . F o r i n s t a n c e , t h e s i m p l y s u p p o r t e d c i r c u l a r p l a t e , w h e n k = 2, gives a load value at collapse equal to that of a clamped isotropic plate Experiments need to be carried out to check the results; in addition, post-yield behaviour
of transversely
the
anisotropic plates should be studied.
Aclcnowledgement--The a u t h o r is indebted to his colleague Dr. J. C h a k r a b a r t y for valuable exchanges of opinion. REFERENCES 1. R. HILL, The ~']lathemattcal Tl~eory of Plast~ctty, Chapter X I I . Oxford U m v e r m t y Press (1950). 2. B. VE~KATRAMAN and R. SANKARANARAYANAN,J . ~'rankl~n Inst. 278, 183 (1964). 3. P. G. ttODOE, JR., Plastic A n a l y s m of Str,uctures, Chapter 10. McGraw-Hall, New Y o r k (1959).
THE PLASTIC B E N D I N G OF T R A N S V E R S E L Y ANISOTROPIC CIRCULAR PLATES S. TANVIR WASTI Civil Engineering Department, Middle East Technical University, Ankara, Turkey (Received 29 May 1969)
Summary--A yield criterion for transversely anisotropm solids under plane stress conditions is obtained as an extension of the Tresca hexagon for the motropic case. Solutions to the plastic bending problems of transversely anisotropic circular plates are given. NOTATION k
coefficient of yield stress in the thickness direction
Mr plate radial moment plate circumferential moment fully plastic in-plane moment dimensionless radial moment iT dimensionless circumferential moment mo A radius of plate r dimensionless plate radius p dimensionless uniform load radial curvature rate circumferential curvature rate W vertical velocity F total punch load ] dimensionless punch load in-plane yield stress of material ~0 {Yl ~ ~Y2 principal stresses in plane stress P dimensionless r~gime boundary dimensionless punch radms Mo Mo
INTRODUCTION
THE PLASTIC behaviour of orthotropic materials has been discussed b y Hill. 1 For the particular case of transverse anisotropy where the yield stress in the direction o f the axis of s y m m e t r y is greater t h a n in a n y direction perpendicular t o this axis, the yield criterion is the isotropic Mises ellipse elongated along its m a j o r axis a n d shortened along its m i n o r axis. N o true c o u n t e r p a r t for the Tresca yield h e x a g o n in a n i s o t r o p y is available. V e n k a t r a m a n a n d S a n k a r a n a r a y a n a n 2 h a v e used an irregular h e x a g o n to t r e a t o r t h o t r o p i c cylindrical shells. YIELD CRITERION The Tresca yield criterion for isotropic solids can be considered as an approximation to that of von Mises. For the plane stress case in which the direction of zero normal stress coincides with the axis of symmetry of a transversely anisotropie solid, a suitable hexagon A B C D E F is readily constructed by elongating the isotropic Tresca hexagon A'B'C'D'E'F" along the line representing equal biaxial stress as shown in Fig. 1. Thus the hexagon 109
i I0
S. T A N V I R W A S T I
A B C D E F r e p r e s e n t s t h e yield c r i t e r i o n for a m a t e r i a l t h a t h a s a n i n - p l a n e yield stress d e n o t e d b y a0 a n d a yield stress in t h e t h i c k n e s s d i r e c t i o n of ka0 w h e r e k > 1. T h e c r i t e r i o n is r e a d i l y a p p l i c a b l e to p r o b l e m s of h m i t analysis, a n d r e d u c e s to t h e T r e s c a h e x a g o n w h e n t h e a n i s o t r o p y is v a n i s h m g l y small.
il~
----
C
k
---I~.
~ e-- 1 ---t- B I I
°
D'
=
i
i
dl/do
I ~
Trescahexagonfor isotroplcmaterial
E
Elongatedhexagonfor transversety anlsotrop=c materla[
Fro. 1. Postulated yield criterion for transversely anisotropic material. APPLICATION
TO ANISOTROPIC
CIRCULAR
PLATES
F o r c i r c u l a r p l a t e s of t h e a b o v e - d e f i n e d a n i s o t r o p i c m a t e r i a l t h e yield locus r e f e r r e d to t h e m r a n d me axes h a s t h e s h a p e of t h e e l o n g a t e d h e x a g o n A B C D E F m Fig. 2 w i t h sides given by the equations: 9~, 0 ~
-J-]t,
?n r -- ?n O ~
± 1,
w h e r e mr = 3 I ~ / M o, me = Me~Me a n d M o is t h e flllly plastic i n - p l a n e m o m e n t . S i m p l y supported anisotropic circular plate
T h e s o l u t i o n for t h e isotropic p l a t e u n d e r u n i f o r m l y d i s t r i b u t e d load c a n b e f o u n d in H e d g e . a C o n s i d e r i n g t h e s a m e p l a t e n o w to b e a n i s o t r o p i c in t h e t h i c k n e s s direction, it is clear t h a t p o i n t B in Fig. 2 r e p r e s e n t s t h e stress s t a t e a t t h e c e n t r e of t h e p l a t e , w h e r e mr = m0 -- k, a n d p o i n t C' r e p r e s e n t s t h e stress s t a t e a t t h e b o u n d a r y , w h e r e m~ = O. T h u s t h e r e are t w o rdgimes for t h e p l a t e , g i v e n b y B C a n d CC'. P o i n t C c o r r e s p o n d s to t h e v a l u e p of t h e d i m e n s i o n l e s s p l a t e r a d i u s r. T h e e q m l i b r i u m e q u a t i o n is d dr (rm~) - m o = - 3pr ~
a n d is i n t e g r a t e d for 0 < r < p a n d m o = k. T h e b o u n d a r y c o n d i t i o n s a r e mr(0 ) = ma(0 ) a n d m ~ ( p ) = k - 1 for p < r < 1; also m , - ~ l o = - 1. F u r t h e r b o u n d a r y c o n d i t i o n s a r e mr(p) = k - 1 a n d my(l) = 0. T h e following t r a n s c e n d e n t a l e q u a t i o n for p is o b t a i n e d 3p-lnp
= 2k+l.
F o r i s o t r o p y , k = l, a n d t h e e q u a t i o n is satisfied b y t h e f a m i l i a r r e s u l t p = 1.
T h e p l a s t i c b e n d i n g of t r a n s v e r s e l y a n i s o t r o p i c c i r c u l a r p l a t e s
111
F o r k = 2, t h e e q u a t i o n b e c o m e s 3p - In p - 5 = 0, w h e n c e p = 1.876, w h i c h i n c i d e n t a l l y coincides w i t h t h e r e s u l t for a c l a m p e d isotropie p l a t o u n d e r u n i f o r m l y d i s t r i b u t e d loading. In order that the solution be considered complete, a kinematically admissible velocity field w(r) c o m p a t i b l e w i t h t h e stress field m u s t b e f o u n d . F r o m Fig. 2, r4gime B C r e q u i r e s t h a t t h e c u r v a t u r e r a t e c o m p o n e n t ~r b e zero a n d Ka positive. B u t Kr is g i v e n b y d ~ zb kr = - dr----~ = 0. I n t e g r a t i n g , t h e v e l o c i t y o n side B C is g i v e n b y ~b = - A r ÷ B w h e r e A a n d B are c o n s t a n t s . T h u s t h e r e c i p i e n t v e l o c i t y field for r~gime B C (0 < r < p) is conical.
rn~rnBply
T
S
supported plate
/ F
Fro. 2. Stress profiles for t r a n s v e r s e l y a n i s o t r o p i c circular plates. F o r r6gime CC' t h e s t r a i n - r a t e v e c t o r is p e r p e n d i c u l a r to CC' a n d h e n c e ka = - k r . P u t t i n g k0 = - ( l / r ) (d~b/dr) a n d Kr = - d ~ b / d r ~, one o b t a i n s ~b = C l n r a n d t h u s t h e i n c i p i e n t v e l o c i t y field for r6gime CC" is l o g a r i t h m i c . O n l y t w o o f t h e c o n s t a n t s A, B a n d C c a n b e d e t e r m i n e d f r o m t h e c o n d i t i o n s t h a t W a n d d~b/dr are c o n t i n u o u s a t r = p. Clamped anisotropic circular plate T h e c l a m p e d a n i s o t r o p i e p l a t e u n d e r u n i f o r m l o a d also h a s a stress profile o n t w o sides of t h e e l o n g a t e d h e x a g o n , i.e. B C a n d C D i n Fig. 2. T h e d i s c o n t i n u i t y a t t h e c l a m p e d edge r = 1 m u s t c o r r e s p o n d to a s t r a i n - r a t e v e c t o r for w h i c h t h e r a t i o kr/ko t e n d s t o - ~ , a n d t h i s c o n d i t i o n is satisfied b y p o i n t D. H e n c e t h e stress profile for t h e c l a m p e d p l a t e covers tlle r a n g e B C D , p o i n t C c o r r e s p o n d i n g a g a i n to t h e v a l u e p o f t h e d i m e n s i o n l e s s p l a t e r a d i u s r. T h e e q u i l i b r i u m e q u a t i o n is i n t e g r a t e d for t h i s case for 0 < r < p a n d me -- k. T h e b o u n d a r y c o n d i t i o n s a r e m~(0) = ma(0 ) a n d mr(p) = / c - 1 for p < r < 1 a n d m ~ - m a = - 1 . F u r t h e r b o u n d a r y c o n d i t i o n s are mr(p) = k - - 1 a n d mr(1 ) ------k. T h e following t r a n s c e n d e n t a l e q u a t i o n for p is o b t a i n e d i n t h i s case, 3p-lnp
- 4 k ÷ 1.
T h e s u b s t i t u t i o n k -- 1 yields t h e e q u a t i o n 3p - In p - 5 --- 0 for t h e isotropic case. T h e i n c i p i e n t v e l o c i t y field c o m p r i s e s a cone a n d a l o g a r i t h m i c p a r t as in t h e case of t h e s i m p l y s u p p o r t e d p l a t e , a n d h e n c e t h e s o l u t i o n is c o m p l e t e .
112
S. TANVII~ WASTI
S i m p l y supported plate loaded by a circular punch The transversely anlsotropm simply supported circular plate is now considered under the action of a centered rigid circular p u n c h of radi~rs ~A, where A is the radius of the plate and 0 < ~ < 1. The isotropic case has been t r e a t e d by Hedge. 3 Only the p a r t of t h e plate where the dimensionless radius r as greater t h a n ~ can deform. There are three r6gimes; under t h e punch, m r = m o = k if the p u n c h force is considered as a line load around its circumference. Thus the whole regmn u n d e r the p u n c h is in r6glme B. F o r ~ < r < p and p < r < 1 there are two r6glmes as before and the e q u i h b r m m e q u a t i o n is g,ven b y d dr (rm~) - m o = - f , where F is the total load exerted by the p u n c h a n ( I f = F/2~rMo. The integration is earmed out for ~ < r < p and m o = /~'. The b o u n d a r y conditions are mr(~) = k and mr(p) = k - 1 for p < r < 1 and m ~ - m o = - 1. F u r t h e r b o u n d a r y conditions are m~(p) = t - 1 and mr(l) = 0. The two e q u a t m n s o b t m n e d are D 1-k f = o ' - ~ and f - - i = l n p " E l i m i n a t i n g p,
k - 1 (1 _1~ = exp ~ f/. F o r ~ = 0, the solution reduces to the ease of a c o n c e n t r a t e d load a n d f = 1 as m the ease of the lsotropic plate. F o r finite values of ~ higher values o f f are obtained. The velocity field is again a cone corresponding to side B C and a logarithmic part corresponding to CC" in Fig. 2. The portion of the plate u n d e r the p u n c h m o v e s s t r m g h t down and there is a hinge circle at r = ~ corresponding to poit)t B. CONCLUSIONS F o r / c > 1 i t is o b s e r v e d t h a t t h e c a r r y i n g c a p a c i t y o f t r a n s v e r s e l y a n i s o t r o p i c p l a t e s i n c r e a s e s . F o r i n s t a n c e , t h e s i m p l y s u p p o r t e d c i r c u l a r p l a t e , w h e n k = 2, gives a load value at collapse equal to that of a clamped isotropic plate Experiments need to be carried out to check the results; in addition, post-yield behaviour
of transversely
the
anisotropic plates should be studied.
Aclcnowledgement--The a u t h o r is indebted to his colleague Dr. J. C h a k r a b a r t y for valuable exchanges of opinion. REFERENCES 1. R. HILL, The ~']lathemattcal Tl~eory of Plast~ctty, Chapter X I I . Oxford U m v e r m t y Press (1950). 2. B. VE~KATRAMAN and R. SANKARANARAYANAN,J . ~'rankl~n Inst. 278, 183 (1964). 3. P. G. ttODOE, JR., Plastic A n a l y s m of Str,uctures, Chapter 10. McGraw-Hall, New Y o r k (1959).