Rresolution of Stresses in Rock Using Steriographic Projection

7/29/2019 Rresolution of Stresses in Rock Using Steriographic Projection

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I n t . J . R o c k M e c h . M i n i n g S c i . Vol. I, pp. 93-103. Pe rgamon Press 1963. Printed in Gre at Britain.

T H E R E S O L U T I O N O F S T R E S S E S I N R O C K U S I N GSTEREOGRAPHIC PROJECTION

R I C H A R D E . G O O D M A NDe partm ent o f Mineral T echno logy, Universi ty of Cali fornia, Berkeley

Abstract . This paper shows how to f ind the shear and norm al s t ress o n an arbit rar ily orientedplane o f weakness in a k now n three dimensional s t ress f ie ld by manipulat ions on a s te teog raphicproject ion. Firs t are fou nd graphical ly the direct ion angles o f the no rma l to the plane relevantto co-o rdinate axes in principal di rect ions . Kn ow ing the mag ni tude of the principal s t resses andthe direct ion cosines of the plane, the m agni tud e of the resultant s t ress across the plane iscalculated. Th e direction o f the resulta nt stress is fo un d by calculating its direction anglesand plotting these angles as small circles centered about the principal directions, the pointof intersection of the small circles being the stereographic pro jectio n of the resultant stress.The angle between th e resul tant s t ress and the plane o f weakness is read directly fro m thes tereographic project ion, facil ita ting calculat ion of the magn i tudes of the shear and nor ma ls t resses o f the plane. Finally, the direct ion of the m axim um shear s t ress in the plane is foundgraphical ly, ma king use of the fact tha t the no rma l s t ress, shear s t ress , and resultant s tressmu st be coplanar. Th e principles o f s tereographic project ion are reviewed and an example o fstress resolution is wo rked ou t.

1. I N T R O D U C T I O NA N AL Y SIS o f t h e s a f e t y o f a r o c k m a s s u n d e r a n e n g i n e e r i n g l o a d r e q u i r e s t h a t w e d e f in et h e t h r e e - d i m e n s i o n a l s t a te o f st r es s i n t h e r o c k , d e f i n e a c r i t e r i o n f o r f a i lu r e i n t h r e ed i m e n s i o n s , a n d c o m p a r e t h e t w o . S t e r e o g r a p h i c p r o j e c t i o n f a c il it a te s t h e m a t h e m a t i c si n v o l v e d i n s t re s s a n a ly s i s b y m a k i n g i t c o n v e n i e n t t o c h o o s e p r i n c i p a l d i r e ct i o n s a s r e fe r e n c ea x e s a n d b y a l l o w i n g o n e t o r e a d d i r e ct i o n a n g l e s d i r e ct l y w i t h o u t c o m p u t a t i o n . F u r t h e r -m o r e , s t e r e o g r a p h i c p r o j e c t i o n f a c i l it a t e s t h e s t a t e m e n t o f f a i lu r e c r i te r i a in t h r e e d i m e n s i o n sf o r r o c k c o n t a i n i n g p l a n e s o f w e a k n e s s. T h i s p a p e r d i sc u s se s i n d e ta i l a n e x a m p l e o f s tr e ssr e s o l u t i o n i n a r o c k w i t h a p l a n e o f w e a k n e s s .

R o c k d if fe r s f r o m m a n y e n g i n e e r i n g m a t e ri a l s i n t h a t i t is t r a v e r se d b y f l aw s a n d w e a k -n e s se s t h a t c a n n o t b e a v o i d e d . T h e c o m p l e t e d e s c r i p t io n o f a r o c k m a s s f o r e n g i n e e r in gp u r p o s e s m u s t i n c l u d e s u c h e le m e n t s . I n w o r k i n g w i t h s u c h m a t e r i a ls a s s o il s, w h i c h h a v en o t e n s i le s t r e n g t h i n a n y d i r e c t i o n , i t is c u s t o m a r y t o c o n f i n e a n a l y s i s t o t h e p l a n e t h a tp r o v i d e s t h e m o s t s e v e re c o m b i n a t i o n o f t w o o f t h e th r e e s p a t ia l e le m e n t s. F o r e x a m p l e ,l a n d s l i d e a n a l y s e s c o n s i d e r a c y l i n d r i c a l s l i d i n g s u r f a c e , t h e a x i s o f w h i c h p a r a l le l s t h ec o n t o u r . I n m o s t p r o b l e m s i n v o l v i n g r o c k , a t h r e e - d i m e n s i o n a l a n a l y si s s h o u l d b e a t te m p t e d .

F o r t u n a t e l y s e v er a l m e t h o d s o f p r o j e c t i o n c a n a l l o w v i su a l i z at i o n o f t h e t h r e e - d i m e n s i o n a le l e m e n t s o n a s in g l e t w o - d i m e n s i o n a l b a s e. T h e m o s t w e l l k n o w n o f th e s e is s t e r e o g r a p h i cp r o j e c t i o n , w h i c h i s i n w i d e u s e b y s t r u c t u r a l g e o l o g i s t s a n d c r y s t a l l o g r a p h e r s . T h e p r i n c i p l e so f s t e r e o g r a p h i c p r o j e c t i o n w i ll b e r ev i e w e d b e l o w . T h e r e a d e r i s r e f er r e d f o r m o r e c o m p l e t ed i s c u s s i o n o f t h e p r o p e r t i e s o f s t e r e o g r a p h i c p r o j e c t i o n t o t h e r e f e r e n c e s li s te d a t th e e n d o ft h i s p a p e r .

2 . P R I N C I P L E S O F S T E R E O G R A P H I C P R O J E C T I O NT h e s t e r e o g r a p h i c p r o j e c t i o n o f a g e o m e t r i c a l e l e m e n t i s a p l o t o f it s s p h e r i c a l p r o j e c t i o n

9 3


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9 4 R I C H A R D E . G O O D M A No n t o a c o m m o n p l a n e. A s p h e r ic a l p r o j e c t io n i s si m p l y t h e in t e r s e c t io n o f t h e e l e m e n t w i t h ac o n t a i n i n g s p h e r e c e n te r e d o n a p o i n t o f t h e e le m e n t . T h e p r o j e c t io n p l a n e i s c h o s e n t o b eh o r i z o n t a l . T h e s p h e r i c al p r o j e c t i o n s o f a h o r i z o n t a l a n d a n i n c l i n e d p l a n e a r e s h o w n i nF i g . l ( a ) .

I n s t e re o g r a p h i c p r o j e c t i o n , o n l y o n e o f t h e t w o h e m i s p h e r e s o f t h e s p h e ri c al p r o j e c t i o ni s u s e d . S t r u c t u r a l g e o l o g i s t s p r e f e r t o u s e t h e l o w e r h e m i s p h e r e w h i l e c r y s t a l l o g r a p h e r sw o r k w i t h t h e u p p e r. W e w i ll a d o p t t h e u s e o f t h e u p p e r h em i s p h e r e . T h e m a n n e r i n w h i c ht h e s p h e r i c a l p r o j e c t i o n i s p r o j e c t e d s t e r e o g r a p h i c a l l y i s s h o w n i n F i g . l ( b ) .

C O N T A I N I N G SPHERE

SPHERICALPROJECTIONOF INCLINEDPLANE/ / /

! ~ SPHERICALP R O J E C T /O NO FHORIZONTALP L A N

FIG. l(a). Sph erical projection.

CT/ON LINE

S P H E R I C A LP R O J E C T I O NOF 1NCLINED

L A N E

R O d E C T / O N ". . . . . \ / I , IE R E O G R A P H I CW i / / O J E C T I O N" \ ~ ~ // / OF NCL INEDS T EREO GRA PH/ C \ / P L A N E .P R O d E C T / ONO F H O R I Z O N T ALPLA NE, "'POINT OF

P R O J E C T I O NF l o . l ( b ) . S t e r e o g r a p h i c p r o j e c t i o n .

B y c o n s t r u c t i n g t h e s t e r e o g r a p h i c p r o j e c t i o n o f a s y s t e m o f p l a n e s p a r a l l e l t o a g i v e na x i s , a S t e r e o n e t i s p r e p a r e d . A 2 S t e r e o n e t i s s h o w n i n F i g . 2 ; a s p a r e c o p y i s a l s o i n c l u d e dw i t h t h i s is s u e . T h i s w i l l b e u s e f u l in f o l l o w i n g t h e t e x t a n d m a y b e m o u n t e d o n c a r d b o a r d ,w i t h a t h u m b t a c k s t i c k i n g u p a t t h e c e n t e r .


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T H E R E S O L U T I O N O F S TR ES SE S I N R O C K U S I N G S T E R E O G R A P H I C P R O J E C T I O N 9 5F i g u r e 3 s h o w s t h e s t e r e o g r a p h i c p r o j e c t i o n o f a p l a n e a n d i ts n o r m a l , a n d o f t h e p l u n g e

a n d p i t c h o f a li n e in t h e p l a n e . T h i s d r a w i n g i s o f a l o w e r h e m i s p h e r e p l o t b u t t h e r e s u l t sf o r t h e u p p e r h e m i s p h e r e a r e s i m i la r . T h e o n l y d i f fe r e n c e i s t h a t t h e n o r m a l ( o r p o l e ) t o ap l a n e p l o t s i n t h e d i p d i r e c t i o n f r o m t h e c e n t e r o f t h e n e t w h e n t h e u p p e r h e m i s p h e r e isu s e d , w h e r e a s t h e p o l e a p p e a r s o n t h e o p p o s i t e s i d e w h e n t h e l o w e r h e m i s p h e r e i s u s e d .

F i g u r e 4 i s t h e s t e r e o g r a p h i c p r o j e c t i o n o f a p l a n e d i p p i n g 3 0 t o t h e S 6 0 E ( s t r i k i n gN 3 0 E ). T h e p l u n g e a n d b e a r i n g o f a li n e i n t h i s p l a n e w h i c h p i t c h e s 4 0 f r o m t h e N E ise q u a l t o 1 9 t o N 6 6 E . T h e s e v a l u e s a re r e a d d i r e c t l y f r o m t h e n e t . N o t e t h a t u s i n g t h eu p p e r h e m i s p h e r e , th e b e a r i n g a n d p l u n g e o f a l in e p lo t a c ro s s t h e n e t f r o m t h e i r m a pp o s i t i o n s .

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- - I 0 S I 0 - "F IG . 2 . T w o d e g r e e M e r i d i a n S t e r e o n e t.

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9 6 R I C H A R D E . G O O D M A N

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F IG . 3 . S t e r e o g r a p h i c p r o j e c t io n o f c o m p o n e n t s o f a d i p p i n g s t r a t u m ( a f t er D O N N a n d SHIMER, 2 ] ) .

E P L A / V EI P P I N G/II ~ - I V O R M A L T O Ji 1 P L . / v e /

OF LI NE .tx~ 1 6 " \ \ o /IMEASUR~O " ~ . ~ j /DOW/V FROM 66 ~AlE OR UPF R O M S W )

F I G . 4 . S t e r e o g r a p h i c p r o j e c t i o n o f a p l a n e .


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T H E R E S O L U T I O N O F S TR ES SE S I N R O C K U S I N G S T E R E O G R A P H I C P R O J E C T I O N 9 7Projection of cones: small circles

A u s e f u l p r o p e r t y o f s t e r e o g r a p h i c p r o j e c t i o n i s t h a t a s m a l l c i r c le o n a s p h e r i c a l p r o j e c t i o np l o t s a s a s m a l l c i r c l e i n s t e r e o g r a p h i c p r o j e c t i o n . A s m a l l c i r c l e i n s p h e r i c a l p r o j e c t i o n i sg e n e r a t e d b y a c o n e , t h e a p e x o f w h i c h c o i n c i d e s w i t h t h e c e n t e r o f t h e p r o j e c t io n s p h e r e.F o r e x a m p l e , t h e lo c u s o f l in e s e q u ia n g l e f r o m t h e n o r m a l t o a p l a n e i s a c o n e t h a t i n te r s e c tst h e p r o j e c t i o n s p h e r e i n a s m a l l c ir c le . T h e s t e r e o g r a p h i c p r o j e c t i o n o f t h i s l o c u s is li k e w i s e ac i r c le ( s ee p r o o f i n [1 ]) . T h e p l o t t i n g o f s u c h a l o c u s i s d e m o n s t r a t e d i n F i g . 5 . I n t h isF i g u r e t h e l o c u s o f l in e s 3 0 f r o m t h e n o r m a l t o a p l a n e d i p p i n g 5 6 t o t h e N 5 3 E i sr e q u i r e d . F i r s t t h e n o r m a l N i s p l o t t e d b y r o t a t i n g t h e p o i n t B ( w h i c h i s 5 3 E a s t o f N o r t h )t o t h e e a s t - w e s t d i a m e t e r w h e r e t h e i n c l i n a ti o n o f t h e n o r m a l , 3 4 is la i d o f f f r o m t h ec i r c u m f e r e n c e o f th e n e t . T w o d i a m e t e r p o i n t s o f t he r e q u i r e d s m a l l c ir c le a r e d e f in e d b yp o i n t s A a n d C , f o u n d b y m a r k i n g o f f p o i n t s 3 0 t o e i t h e r s id e o f N . T h e c e n t e r o f t h e p r o -j e c t e d c i r c l e is n o t , i n g e n e r a l , c o i n c i d e n t w i t h t h e a x i s o f t h e c o n i c a l l o c u s .

lq

STEREOGRAPHIC PLOT OF LOCUS OF LINES 30 FROM NORMAL TO PLANE DIPPING 5 6 TO N 53E.

F IG . 5 . C o n s t r u c t i o n o f a s m a l l c i rc l e l y i n g e n t i re l y w i t h i n u p p e r h e m i s p h e r e .

I f o n e o f t h e o t h e r s m a l l c i r c le s li e s p a r t l y o f f t h e n e t , i t m a y b e c o n s t r u c t e d b y r o t a t i n gt h e a x i s o f t h e c o n e t o t h e h o r i z o n t a l a n d t h e n t o t h e N o r S p o s i t i o n o f t h e n e t . I n t h i sp o s i t i o n t h e s m a l l c i r c l e s o n t h e s t e r e o n e t s e r v e a s g u i d e s f o r d r a w i n g t h e r e q u i r e d s m a l lc i r c l e . F i n a l l y , r o t a t e t h e s ma l l c i r c l e b a c k t o i t s i n i t i a l p o s i t i o n .

A m e t h o d o f c o n s t ru c t i n g s m a l l ci rc le s o f a n y d i a m e t e r w h i c h d o e s n o t r e q u i r e t r a c i n gf r o m t h e n e t i s i l l u s tr a t e d i n t h e e x a m p l e b e l o w a n d i n F i g . 6. T h e j u s t i f ic a t i o n f o r t h isc o n s t r u c t i o n i s p r e s e n t e d i n [1 a n d 3 ].Required--to f i n d s m a l l c i r c le o f r a d i u s 7 0 a b o u t a l i n e s e g m e n t ( A ) w h i c h p l u n g e s6 5 t o t h e S 3 0 W .Procedure--Rotate A t o e q u a t o r i a l d i a m e t e r . E x t e n d e q u a t o r i a l d i a m e t e r b e y o n d n e t .M a r k p o i n t P o v e r S o u t h p o s i t i o n . D r a w l in e PA m e e t i n g o u t e r c i r c l e ( p r i m i t i v e c i r c l e ) o fn e t a t A ' . F r o m A ' m e a s u r e o f f 7 0 a r c s in e i th e r d i r e c ti o n m a r k i n g p o i n t s R a n d Q . L o c a t e


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98 R I C H A R D E . G O O D M A NR ' a s i n t e r s e c t io n o f P R a n d t h e e q u a t o r i a l d i a m e t e r . S i m i l a r l y , l o c a t e Q ' a s t h e i n t e r s e c t i o no f P Q a n d t h e e q u a t o r i a l d i a m e t e r . B i s e c t R ' Q ' t o f i n d X . T h e r e q u i r e d s m a l l c i r c l e h a sc e n t e r X a n d r a d i u s X R ' .

P a r t o f th e s m a l l c ir c l e d r a w n l ie s o u t si d e t h e p r i m i t i v e g r e a t c i r c l e o f t h e n e t ( o u t e r c i r c le ) ;i .e . t h e l o w e r p o r t i o n o f t h e c o n e d r o p s b e l o w t h e h o r i z o n t a l . H o w e v e r , s in c e t h e v e r t i c e s o fa l l a n g l e s l ie a t t h e c e n t e r o f th e p r o j e c t i o n s p h e r e , t h e l o c u s o f l in e s e q u i a n g l e f r o m ag i v e n l i n e i s i n r e a l i t y a d o u b l e c o n e . A n y l i n e s e g me n t l y i n g i n t h i s c o n e w h i c h f a l l s b e l o wt h e h o r i z o n t a l o n o n e s i d e o f t h e p r o j e c t i o n s p h e r e , w i ll ri se a b o v e t h e h o r i z o n t a l o n t h eo t h e r s i de o f t h e s p h e r e .

JjFIG. 6. Construction of small circles; general case.

T h e p r o b l e m is t o c o n s t r u c t t h e p r o j e c t i o n o n t h e u p p e r h e m i s p h e r e o f t h e l o c i o f li n e sm a k i n g a n a n g l e o f 7 0 w i th a l in e s e g m e n t d i r e c t e d S 3 0 W a n d p l u n g i n g a t 6 5 . O n e e n do f t h i s li n e is a t A ; t h e o t h e r e n d o f t h e l i n e i n t e r s e c t s t h e p r o j e c t i o n s p h e r e i n t h e l o w e rh e m i s p h e r e a n d h e n c e i ts s t e r e o g r a p h i c p r o j e c t i o n l ie s o u t s id e o f t h e p r i m i t i v e c i rc l e , a t , 4.T h e p o s i t i o n o f ,4 c a n b e f o u n d a s t h e i n t e r se c t i o n o f th e e q u a t o r i a l d i a m e t e r w i t h a l in ed r a w n 9 0 f r o m P A p a s s i n g t h r o u g h P . T h e c o n s t r u c t i o n o f a s m a l l c ir c l e o f r a d i u s 7 0 a b o u t ` 4 i s d o n e e x a c t l y a s f o r t h e c o n i c a l c e n t e r A . L i n e P ` 4 i n t e r e s e c t s t h e p r o j e c t i o ns p h e r e a t ` 4 '. A n g u l a r d i s t a n c e s o f 70 e i t h e r d i r e c t i o n f r o m ` 4 ' a r e l a i d o f f t o g i v e / ~ a n d 0 .R a y s P R a n d P 0 i n te r s e ct th e e q u a t o r i a l d i a m e t e r a t _~' a n d ~ ' . T h e c e n t e r o f th e r e q u i re dc i r c le i s a t t h e b i s e c t o r o f , R ' Q ' a n d i ts r a d i u s i s R ' O ' ( F i g. 6 ) .

N o t e , i t i s n o t n e c e s s a r y to c o n s t r u c t p o i n t , 4; w h a t i s r e q u i r e d i s A ' w h i c h m u s t l ied i a m e t r i c a l l y o p p o s i t e A ' . A l s o i t m u s t b e n o t e d t h a t t h e r e i s n o t h i n g w r o n g t h e o r e t i c a l l yw i t h w o r k i n g o u t s i d e o f t h e p r i m i t i v e c i rc l e. S m a l l ci r c le s o u t s i d e o f t h e p r i m i t i v e m a y ,h o w e v e r , p l o t a n i n f in i te d i s t a n c e f r o m t h e c e n t e r o f th e n e t . ( C o n s i d e r t h e p r o j e c t i o n o n t h e


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THE RESOLUTION OF STRESSES 1N ROCK USING STEREOGRAPHIC PROJECTION 99upper hemisphere o f a line directed vertically downward.) Hence it is convenient to restrictthe region of projection to the area confined within the primitive.Angles between lines

The angle between two lines is foun d by rotat ing the overlay until the points which are theprojections of the two lines lie along the same grea t circle, i.e. until they are in the same plane.The angle is read directly as the difference in the pitch angles of the two lines in the comm onplane. Fo r example, the angle between lines (I) plungi ng 30 N 40 W and (2) plunging 70 to the north 20 E is 52 as shown in Fig. 7. The angle is read by coun ting small circles alongthe common great circle.

N

FIG. 7. Measurement of angles between lines.

A line is fixed in space by its direction angles, the angle it makes with a given orthogonaset of axes. The direction cosines of a line are the cosines of its direction angles. If x, y,and z are the co-ordinate axes, and the direction angles are a to the x axis,/3 to the y axisand 7 to the z axis, then the direction cosines of the line are 1 = cos a, m = cos/3, n = cos Y-Magnitude of the resultant stress across any plane

The resultant stress across any plane is given byR 2 = o1212 --~ o2 2m 2 @ r82n (1)

if: the direction of ol coincides with the x direction; o~. is in the y direction; and cr3 is in thez direction (i.e. 1 = cos ~ where ct is the angle to the ol direct ion; m = cos/3 where/3 is theangle to cr2 direction; and n = cos 7 where y is the angle to the 03 direction).

In real problems in rock mechanics, ~rl, ~rz and ~'3 are not in general horizontal and vertical.For example, the stresses in the ground near a deep valley are in general inclined; oneprincipal stress is more nearly perpendicular to the walls of the valley than vertical. Thegeneral problem of stress analysis in rock mechanics therefore requires us to be able todefine the stress across arbitrarily oriented planes un der an arbitr arily inclined stress field.The method of solving such a problem, using equation 1, is illustrated as follows:


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10 0 R I C H A R D E . G O O D M A N

E xa m p l e . Wha t is the resu ltant stress across a plane which strikes S 45 E a nd dips 70 to the SW ? The stresses in the reg ion are as follows: ai d ips 38 to N 9 E; c,z dips 36 tothe S 45 E; and aa dips 32 to the S 70 W. O '1 = 10, 0"2 = 5 , 0 "3 = = 3 kips/inL

In Fig. 8, the stresses and the normal to the plane are plotted on the upper hemisphereprojectio n. The reader ca n satisfy himself that these stresses constitut e a right-h andedorthogonal set by rotating any two of the three points to a common plane (to the same

I

\ ' i " / " ' " /

O - j 5 7 0 " W J 2 O ' z N 9 E J 7 0 -~ S 4 6 " E 3 6 *

FIG. 8. Stereographic plot of principal directions and plane across which stress is to be resolved.

great circle) and then reading the angles: (1) between the two points in the plane; and (2)between the plane and the third p oint. A system of stresses is right han ded if in going fromal to crz to cra one proceeds in clockwise fashion. This is true for both the upper and lowerhemisphere projections.

In Fig. 8 ~rl makes angles of:

Thus:

36 with the negative directi on of ol123 with the negati ve dir ecti on o f a,~

76 with the negat ive di rect ion of a,_,a= 144 fl = 104 y= 57 1 = -0"81 , m = -0. 24, n = +0. 54I z ---- 0.65 , m z = 0.06, n z = 0.29

(check I z = m e + n 2 = 1.00)R ~ --= 100 (0.65) + 25 (0.06) .'-- 9 (0.29) Ki ps /i n2R z =6 5 .0 + 1 .5 - - 2 .6R = V'69"1R = 8.3 Ki ps/ in2


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THE RESOLUTION OF STRESSES IN ROCK USING STEREOGRAPHIC PROJECTION 101

T h e d i r e c ti o n o f t h e r e s u l ta n t s t r e s s a c r o s s a n y p l a n eThe direction of R can be obtained by calculating its direction cosines, using the following

relation:

For the previous example:I n = 10 t ~_081 0.9758.3

Thus"mR = -- -

5 / (-0 .24). . . . . . . . . . . 0 .1448"3

SOl

R z R y R zI n = - ~ , m n = , n n = --R

Rx = trll, R v = ~2m, R z = ~an

a l lI n - - Rt r ~ n

mR = --R

txannR~--- ~-L J

(2)

3 (0.54)_ +0.195na -- 8.3

aR = 167 = 180 - 13fin = 98 = 180 -- 827n = 79 = 180 - 10 1Fr om the negative ends of ~1, ~2, and era (which are what we must plot using the uppe r

hemisphere for the given data) construct small circles of radii given above using the methoddescribed earlier. In Fig. 9, ~rl, ~2, and aa have been replo tted and small circles are plottedof radii: 13 about - a l ; 82 about - a 2; and 101 about -a a. These intersect to give Rwhich is thus an upward directed stress. There is no a p r i o r i reason why R must be directedupwards; it can in general be in any direction across the plane. If in fact it were a downwarddirected stress, no intersection of three small circles would be found in the upper hemisphere.However, by drawing the opposite ends of the lower hemisphere portions of the conicalloci an intersection would be found.

R e s o l u t i o n o f t h e r e s u l t a n t s t r e s s in t o n o r m a l a n d s h e a r s t r e s sIt is useful to resolve the resultant stress across any plane into normal and shear com-ponents. Since the normal stress must lie in the direction of N,

% ---- R cos 3an d

Tmax = R sin 3where 3 is the angle between the direction of R and the normal to the plane.


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1 02 R I C H A R D E . G O O D M A NT h e a n g l e 8 c a n b e r e a d f r o m t h e s t e re o n e t . I n F i g . 1 0, i n w h i c h t h e l o c a t i o n s o f R a n d N

h a v e b e e n r e p l o t t e d , ~ i s r e a d d i r e c t l y a s 2 6 .

N

F IG . 9 . C o n s t r u c t i o n o f s m a l l c i r c le s t o f i n d d i r e c t i o n o f r e s u l t a n t s t r e s s .

P L A N E O NW H I C H A N A L Y , .~ S I ~ ,o F s t r e s s e s '/s aehva / J

F IG . 1 0 . S t e r e o g r a p h i c c o n s t r u c t i o n t o f i n d d i re c t i on o f m a x i m u m s h e a r s t r e s s in p l a n e .

c ,n = 8 "3 c o s 2 6 = 7 . 4 K i p s / i n z" r m a x = 8 "3 s i n 2 6 = 3 - 6 K i p s / i n 2

T o c o m p l e t e t h e d e s c r i p t i o n o f t h e s t a t e o f st r e ss in t h e p l a n e , w e r e q u i r e t h e d i r e c t i o no f "rm ax . W e c a n m a k e u s e o f t h e f a c t t h a t ~ n , " r m a x , a n d R m u s t b e c o p l a n a r , i . e. " r m a x m u s tb e o n t h e p l a n e o f ~rn a n d R . F u r t h e r m o r e 7 m u s t b e i n th e p l a n e o n w h i c h i t ac t s . T h e g r e a t


11/11

THE RESOLUTION OF STRESSES IN RO CK US ING STEREOGRA PHIC PROJECTION 103c i r c le p r o j e c t i o n s o f t h e s e t w o p l a n e s c a n b e t r a c e d f r o m t h e s t e r e o n e t a s in F i g . 1 0. T h ei n t e r s e c t i o n o f t h e s e g r e a t c i r c l e s i s t h e d i r e c t i o n o f r m ax . T h e m o s t u s e f u l r e p r e s e n t a t i o no f t h is d i r e c t i o n i s t h e p i t c h o f ~ , i n t h e p l a n e o n w h i c h i t a c t s w h i c h e q u al s :4 8 ,~ u p f r o m t h es o u t h - e a s t .

T h e c o m p l e t e d e s c r i p t i o n o f th e s t r e s s e s a c t i n g o n t h e p l a n e i s s h o w n i n F i g . 1 I .

\ FIG. 11. Com ple te des cr ipt ion of the st resses ac t ing on the plan e of Fig . 8 .

R E F E R E N C E S1. PHILLIPSF. C. All Introduction to Crystallography, Chap . 2 . Me thod o f Pro j ec t i on , L ongm ans , Greenand Co. (1949) .2 . Do~q WILLIAM L. an d SHINIER JOHN A. Graphic Methods in Structural Geology, A p p l e t o n - C e n t u r yCrofts (1958).3. TER~'rRA P. a nd C.ODD L. W . Crystallometry, Longmans (1961) .

Rresolution of Stresses in Rock Using Steriographic Projection

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