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A statistical theory of the polyaxial compressive strength of materials
Int. J. Rock Mech. Min. Sci.
Vol. 9, pp. 617-624. Pergamon Press 1972. Printed in Great Britain
A STATISTICAL THEORY OF THE POLYAXIAL COMPRESSIVE STRENGTH OF MATERIALS N. LUNDBORG Swedish Detonic Research Foundation, Vinterviken, Stockholm, Sweden (Received 5 November
1971)
Abstract--Weibull's statistical theory of strength, extended to compressive stresses, is used in calculating the influence of the intermediate principal stress on the strength. The effective shear stress re = I~.1 --t"r. is calculated and integrated over the solid angle where z. > 0, and the probability of rupture in polyaxial compression is calculated. The results are in good agreement with experimental results. INTRODUCTION
THE influence of the intermediate principal stress (or2) on the strength of brittle materials has long been a matter of discussion. Experimental work [1-5] shows an increase in strength with increasing u2, but up to this time no satisfactory explanation has been presented. The most commonly used theory, that of Mohr-Coulomb, does not even show an effect of o2. WmBOLS and COOK [6] recently presented an interesting theory, where an energy criterion is used, which reproduces the increase of strength with cr2 found experimentally. In the present work, the idea used by WEmULL [7] to explain the volume effect and polyaxial stress effect on the tensile strength of materials is applied. The theory is used in the compressive state of stress and an effective shear stress is used, defined in the same way as by Wiebols and Cook. STATISTICAL THEORY O F STRENGTH
The conventional representation of Mohr-Coulomb's theory of strength is shown in Fig. 1. Rupture will occur when ~-, in a certain point and at a certain angle (/3) reaches the envelope (a). If instead, we introduce an effective shear stress z e that appears over a certain angle we may state that the probability of rupture is a function of ~e and the size of the angle. ¢al
-%
o-3 FIG. 1. Stress relations in the r, e plane. 617 ROCK 9/5--D
o-1 %
618
N. LUNDBORG
I f we lower the line (a) by the a m o u n t (~-o) to the line (b) in Fig. 1 we see t h a t the shear stress (T,) in an angle ~2-a~ exceeds the frictional stress (/~0.n). Let us define the effective shear stress re = lrnl --/~an which is positive in that angle.*
0-3
o"I
FIG. 2. Unit sphere with principal stress axes.
SHEAR AND N O R M A L STRESSES IN THE POLYAXIAL STRESS FIELD
I n the p o l y a x i a l stress state we can calculate the stresses in a plane, whose n o r m a l (OP Fig. 2) has the direction cosines 1, m a n d n to the 0.1, ~2 a n d o 3 axes, respectively. These are f o u n d to be 0.n = 12crl -[- m20.2 -~ n20.3
(1)
r n 2 ~-~ 120"12 -{- m20"22 - ~ //20"32 - - o ' n 2
(2)
12 + m 2 + n 2 = 1.
(3)
~'e = l~'nl --/*on.
(4)
1 -----cos~ cosA m = cosct sinA n = sinct.
(5)
and where By definition F r o m Fig. 2 we get the relation
By using e q u a t i o n s (1)-(5) we c a n n o w calculate the solid angle (£2) where ~e > 0. T h e result is shown in Fig. 3 with 0.1 = 8 kb, 0.3 = 1 k b a n d / z = 1. I n the s h a d e d a r e a o f the angle n o r u p t u r e can t a k e place for any value o f or2 between gl a n d cr3. F r o m Fig. 3 it m a y be seen t h a t the solid angle at first decreases with increasing o2, reaches a m i n i m u m , a n d then a g a i n increases when g3 < g2 < 0.1- A s in the p l a n e case we m a y assume t h a t the p r o b a b i l i t y o f r u p t u r e is a f u n c t i o n o f "r e > 0 a n d the solid angle $2. * The effective shear stress ze, may have other expressions.
POLYAXIAL COMPRESSIVE STRENGTH OF MATERIALS
619
oz -3
"~._ Cr2= o] = 8 kb
0-:%= i kb
FIG. 3. Solid angles for re > 0 at different crz when al = 8 kb, a3 = 1 kb and g = 1. THE PROBABILITY FUNCTION Weibull has put the probability o f rupture in tension in the f o r m :
S(x)
= 1 - - exp ( - - k X ) *
(6)
x = f~M dt~
(7)
and k and M are constants for a given material. I f we n o w p u t X = ~ : ' dO
(8)
where
where the integration is carried out over the solid angle where ¢~ > 0, we m a y in the same way get the probability o f rupture in the compressive state o f stress. M and k m a y have other values than those f o u n d f r o m tensile fracture. By using experimental values o f uniaxial compression, say those for 50 per cent rupture, we will find the value o f k X f r o m equation (6). I f then tt and M are known, we can calculate the strength at any combination o f the stresses by letting X be constant. This is easily done by using a computer. Figure 4 shows the variation in strength with a 2 when a z = 0,/~ = 1 and M = 2. F o r M -----2 the result coincides with that f r o m WmBOLS and COOK [6]. We find that their treatment is a special case in the present general theory. RELATION BETWEEN STRENGTH,/.L and M Determination o f a relation between/~ and M can be made using the uniaxial and biaxial compressive strength. In the case when az =: ~ = 0, and or2 ---- % or c,2 = or1 we get f r o m equations (1)-(4)
~,. = 12~1 % = l.ntr 1
(9)
and ~'e =
~rll(n--ld).
* Other probabilistic models, such as the normal distribution, may be used instead of the Weibull distribution.
N. LUNDBORG
620
FIG. 4. Strength variation with o2 when c+ = 0, p = 1 and M = 2.
From Fig. 3 it may be seen that this state of stress is held constant when rotating around either the u1(u2 = CJ~)or u3(02 = ul) axis. Using equations (5) and (9) this gives for X = 0
where tana
= p is constant.
When X is held constant as stated above, equations (8) and (10) give “I2
s
[COSCC (sina - +OSCC)]~sina da.
M -zzz
cc? n/2
--
[COSa
(sina
-
pcosa)lM
(11) cosa
da
a, where ulo and uzo are the uniaxial and biaxial compressive strengths, respectively. It can be shown (see Appendix) that equation (11) gives M
() 020 -
=
VT1
+
p2) +
(12)
p.
010
This is a simple and useful relation for the determination of one of the constants when the other three are known. Figure 5 shows the variation of strength with u2 for different M-values when TV= 1. We see that the influence of u2 decreases with increasing M, and for M = COwe will have _-_..lr_ :_ ___^^___C __.:41. LL,. ,.,,.,“:,,., X.6-L.. P_*,,-_h +Lpnrtr resuiLs m agrcemcm W~LLL gut: ~~ilssl~al IVLVIII-~~UIVIIIV LLL~“LJ. THE p-VALUE
AS A FUNCTION
OF THE NORMAL
STRESS
In the literature the p-value is mostly considered independent of the normal stress. If p is defined from the simple relation Tf = pu,,, p is not a constant at high normal stresses where
POLYAXIAL COMPRESSIVE STRENGTH OF MATERIALS
621
I
o
I crz/% FIG. 5. Strength variation with % at different M when ~ = 1 and oa = 0.
~'s increases only slowly with %. In this region many expressions are given that all show a decreasing/z-value with increasing ~n. The author [8] has used the relation /~ = #zo/(1 d- (/Zo%/r=))
(13)
which is derived from experimental results./z o is a constant and ~-~ the friction stress when % tends to infinity. By using a computer, the expression (13) may easily be used instead of a constant tz-value. COMPARISON BETWEEN EXPERIMENTAL AND CALCULATED STRENGTH HosrdNs [1 ] and AKAI and MORI [2] among others have measured the strength of trachyte and sandstone in the polyaxial state of stress. Hoskins' values are plotted in Fig. 6. By using the strength relation ~1 = f ( % ) when ~2 ---- % the/z-value is found to be /~ ~ 0.9/(1 q- 0-075 %)
(14)
and by using the uniaxial and biaxial strength together with the relation (12) we get M ~ 1.2.
(15)
Using these values we can calculate the strength, which is shown as full lines in Fig. 6. Figure 7 shows in the same way the comparison between calculated and experimental result from Akai and Mori on sandstone. The calculated values are in good agreement with the experimental results. COMBINED TENSILE AND COMPRESSIVE STRESSES If we extend the calculations to combined tensile and compressive stresses, assuming there are some friction stresses which still decrease when % < 0, we can calculate the strength also in this region. We may then assume that the calculation will hold when, according to GRlrrrrrI [9] or1 + 3 % > 0, and then let % be constant when ~I tends to zero. Under this assumption it is possible to find the ratio between the uniaxial compressive and tensile strength. From the values of Hoskins in the compressive region we find, in this way, the ratio 9.85.
622
N. L U N D B O R G %(kb) 0-69(")
%=%
5
•
"¢/
',
]
F
J
]
I
I
2
3
4
5
6
crz (kb)
FIo. 6. Comparison between calculated strength and Hosg.[NS' [1] experimental values on trachyte. The symbols indicate different ~3 used by Hoskins corresponding to the calculated lines in the figure.
/ 0
~
I
I
I
I
I
8
3
4
5
o-~(kb) Fro. 7. Comparison between calculated strength and experimental values from AKAI and MORt [2] on sandstone.
POLYAXIAL COMPRESSIVE S T R E N G T H OF MATERIALS
623
It is also possible to let the friction stress be zero when % < 0, and then the above-mentioned ratio becomes 7.40. More experimental results are, however, needed to prove the validity of the calculations in the case of combined stresses. The calculations in this work were carried out on a H P table-top calculator 9100 B. Acknowledgements--This work has been carried out at the Swedish Detonic Research Foundation, with financial support from the Swedish Board for Technical Development, Nitro Nobel AB and Atlas Copco AB. The author would like to express his thanks to Dr P. A. PERSSONand Mr FINN OUCnTERLONYfor useful discussions and critical examination of the manuscript.
REFERENCES 1. HOSKINSE. R. The failure of thick-walled hollow cylinders of isotropic rock. Int. J. Rock Mech. Min. Sci. 6, 99-125 (1969). 2. AKAI K. and MoRt H. Study on the failure mechanism of a sandstone under combined compressive stresses. Proc. Sac. cir. Engrs No. 147, 11-24 (1967). 3. MAZArZrI B. B. The Effect of the Intermediate Principal Stress on the Strength of Rock, Thesis, Georgia Institute of Technology (1967). 4. HOJEM J. P. M. and COOK N. G. W. The design and construction of a triaxial and polyaxial cell for testing rock materials. S. Afr. mech. Engr 18, 57-61 (1968). 5. HANDINJ., HEARD H. C. and MAGOUIRKJ. N. Effects of the intermediate principal stress on the failure of limestone, dolomite and glass at different temperatures and strain rates. J. geophys. Res. 72, 611-640 (1967). 6. WIEaOLSG. A. and COOk N. G. W. An energy criterion for the strength of rock in polyaxial compression. Int. J. Rock Mech. Min. Sci. 5, 529-549 (1968). 7. WEmULL W. A statistical theory of the strength of materials. Proc. R. Sw. Acad. Engng Sci. 151, 5-45 (1939). 8. LUNDaOaG N. Strength of rock-like materials. Int. J. Rock Mech. Min. Sci. 5, 427-454 (1968). 9. GRJrrlTH A. A. The Theory of Rupture, Proceedings of the First International Congress on Applied Mechanics, Delft, pp. 55-63 (1924).
APPENDIX Deduction of the Relation Between Strength, tL and M By using equation (10) and integrating the expression: ,,/2
.4 = f C~d~ = G1Mf [cos~(sin~ - - ~cos~)lu d~
(A.1)
O-o
we find the shaded area A, in Fig. AI. By the substitution 7/"
O. o
4
~-
a=_+
+x
(A.2)
equation (A.1) gives
A ~
GI.M
'~T
£to
4
2 f
--(4--?)
[ cos
+ ~°° + x
) cos (-~ , + ~°°. - x
dr.
(A.3)
624
N. L U N D B O R G
o-3
O'1,
/
~
o-2
FIG. A I . Unit sphere with principal stress axes as in Fig. 2. As the limits o f the integral are symmetric around x = 0, and the integrand is an even function of x, we find the centre o f gravity of the integrated area A, to be at x = 0, i.e.
~=~
7r
~o
+-~.
N o w by using Guldin's rule the integrals in equation (11) are simply obtained by rotating the area in Fig. A1 around the al and a3 axes, respectively, so that equation (11) will be
0"20 ~
(4rr + cos( + o) sin
1 + sinao
cosoo
(A.4)
Putting sin %
COS % we finally get
if20) M
= V 0 + ~2) + t,.
(A.5)
Vol. 9, pp. 617-624. Pergamon Press 1972. Printed in Great Britain
A STATISTICAL THEORY OF THE POLYAXIAL COMPRESSIVE STRENGTH OF MATERIALS N. LUNDBORG Swedish Detonic Research Foundation, Vinterviken, Stockholm, Sweden (Received 5 November
1971)
Abstract--Weibull's statistical theory of strength, extended to compressive stresses, is used in calculating the influence of the intermediate principal stress on the strength. The effective shear stress re = I~.1 --t"r. is calculated and integrated over the solid angle where z. > 0, and the probability of rupture in polyaxial compression is calculated. The results are in good agreement with experimental results. INTRODUCTION
THE influence of the intermediate principal stress (or2) on the strength of brittle materials has long been a matter of discussion. Experimental work [1-5] shows an increase in strength with increasing u2, but up to this time no satisfactory explanation has been presented. The most commonly used theory, that of Mohr-Coulomb, does not even show an effect of o2. WmBOLS and COOK [6] recently presented an interesting theory, where an energy criterion is used, which reproduces the increase of strength with cr2 found experimentally. In the present work, the idea used by WEmULL [7] to explain the volume effect and polyaxial stress effect on the tensile strength of materials is applied. The theory is used in the compressive state of stress and an effective shear stress is used, defined in the same way as by Wiebols and Cook. STATISTICAL THEORY O F STRENGTH
The conventional representation of Mohr-Coulomb's theory of strength is shown in Fig. 1. Rupture will occur when ~-, in a certain point and at a certain angle (/3) reaches the envelope (a). If instead, we introduce an effective shear stress z e that appears over a certain angle we may state that the probability of rupture is a function of ~e and the size of the angle. ¢al
-%
o-3 FIG. 1. Stress relations in the r, e plane. 617 ROCK 9/5--D
o-1 %
618
N. LUNDBORG
I f we lower the line (a) by the a m o u n t (~-o) to the line (b) in Fig. 1 we see t h a t the shear stress (T,) in an angle ~2-a~ exceeds the frictional stress (/~0.n). Let us define the effective shear stress re = lrnl --/~an which is positive in that angle.*
0-3
o"I
FIG. 2. Unit sphere with principal stress axes.
SHEAR AND N O R M A L STRESSES IN THE POLYAXIAL STRESS FIELD
I n the p o l y a x i a l stress state we can calculate the stresses in a plane, whose n o r m a l (OP Fig. 2) has the direction cosines 1, m a n d n to the 0.1, ~2 a n d o 3 axes, respectively. These are f o u n d to be 0.n = 12crl -[- m20.2 -~ n20.3
(1)
r n 2 ~-~ 120"12 -{- m20"22 - ~ //20"32 - - o ' n 2
(2)
12 + m 2 + n 2 = 1.
(3)
~'e = l~'nl --/*on.
(4)
1 -----cos~ cosA m = cosct sinA n = sinct.
(5)
and where By definition F r o m Fig. 2 we get the relation
By using e q u a t i o n s (1)-(5) we c a n n o w calculate the solid angle (£2) where ~e > 0. T h e result is shown in Fig. 3 with 0.1 = 8 kb, 0.3 = 1 k b a n d / z = 1. I n the s h a d e d a r e a o f the angle n o r u p t u r e can t a k e place for any value o f or2 between gl a n d cr3. F r o m Fig. 3 it m a y be seen t h a t the solid angle at first decreases with increasing o2, reaches a m i n i m u m , a n d then a g a i n increases when g3 < g2 < 0.1- A s in the p l a n e case we m a y assume t h a t the p r o b a b i l i t y o f r u p t u r e is a f u n c t i o n o f "r e > 0 a n d the solid angle $2. * The effective shear stress ze, may have other expressions.
POLYAXIAL COMPRESSIVE STRENGTH OF MATERIALS
619
oz -3
"~._ Cr2= o] = 8 kb
0-:%= i kb
FIG. 3. Solid angles for re > 0 at different crz when al = 8 kb, a3 = 1 kb and g = 1. THE PROBABILITY FUNCTION Weibull has put the probability o f rupture in tension in the f o r m :
S(x)
= 1 - - exp ( - - k X ) *
(6)
x = f~M dt~
(7)
and k and M are constants for a given material. I f we n o w p u t X = ~ : ' dO
(8)
where
where the integration is carried out over the solid angle where ¢~ > 0, we m a y in the same way get the probability o f rupture in the compressive state o f stress. M and k m a y have other values than those f o u n d f r o m tensile fracture. By using experimental values o f uniaxial compression, say those for 50 per cent rupture, we will find the value o f k X f r o m equation (6). I f then tt and M are known, we can calculate the strength at any combination o f the stresses by letting X be constant. This is easily done by using a computer. Figure 4 shows the variation in strength with a 2 when a z = 0,/~ = 1 and M = 2. F o r M -----2 the result coincides with that f r o m WmBOLS and COOK [6]. We find that their treatment is a special case in the present general theory. RELATION BETWEEN STRENGTH,/.L and M Determination o f a relation between/~ and M can be made using the uniaxial and biaxial compressive strength. In the case when az =: ~ = 0, and or2 ---- % or c,2 = or1 we get f r o m equations (1)-(4)
~,. = 12~1 % = l.ntr 1
(9)
and ~'e =
~rll(n--ld).
* Other probabilistic models, such as the normal distribution, may be used instead of the Weibull distribution.
N. LUNDBORG
620
FIG. 4. Strength variation with o2 when c+ = 0, p = 1 and M = 2.
From Fig. 3 it may be seen that this state of stress is held constant when rotating around either the u1(u2 = CJ~)or u3(02 = ul) axis. Using equations (5) and (9) this gives for X = 0
where tana
= p is constant.
When X is held constant as stated above, equations (8) and (10) give “I2
s
[COSCC (sina - +OSCC)]~sina da.
M -zzz
cc? n/2
--
[COSa
(sina
-
pcosa)lM
(11) cosa
da
a, where ulo and uzo are the uniaxial and biaxial compressive strengths, respectively. It can be shown (see Appendix) that equation (11) gives M
() 020 -
=
VT1
+
p2) +
(12)
p.
010
This is a simple and useful relation for the determination of one of the constants when the other three are known. Figure 5 shows the variation of strength with u2 for different M-values when TV= 1. We see that the influence of u2 decreases with increasing M, and for M = COwe will have _-_..lr_ :_ ___^^___C __.:41. LL,. ,.,,.,“:,,., X.6-L.. P_*,,-_h +Lpnrtr resuiLs m agrcemcm W~LLL gut: ~~ilssl~al IVLVIII-~~UIVIIIV LLL~“LJ. THE p-VALUE
AS A FUNCTION
OF THE NORMAL
STRESS
In the literature the p-value is mostly considered independent of the normal stress. If p is defined from the simple relation Tf = pu,,, p is not a constant at high normal stresses where
POLYAXIAL COMPRESSIVE STRENGTH OF MATERIALS
621
I
o
I crz/% FIG. 5. Strength variation with % at different M when ~ = 1 and oa = 0.
~'s increases only slowly with %. In this region many expressions are given that all show a decreasing/z-value with increasing ~n. The author [8] has used the relation /~ = #zo/(1 d- (/Zo%/r=))
(13)
which is derived from experimental results./z o is a constant and ~-~ the friction stress when % tends to infinity. By using a computer, the expression (13) may easily be used instead of a constant tz-value. COMPARISON BETWEEN EXPERIMENTAL AND CALCULATED STRENGTH HosrdNs [1 ] and AKAI and MORI [2] among others have measured the strength of trachyte and sandstone in the polyaxial state of stress. Hoskins' values are plotted in Fig. 6. By using the strength relation ~1 = f ( % ) when ~2 ---- % the/z-value is found to be /~ ~ 0.9/(1 q- 0-075 %)
(14)
and by using the uniaxial and biaxial strength together with the relation (12) we get M ~ 1.2.
(15)
Using these values we can calculate the strength, which is shown as full lines in Fig. 6. Figure 7 shows in the same way the comparison between calculated and experimental result from Akai and Mori on sandstone. The calculated values are in good agreement with the experimental results. COMBINED TENSILE AND COMPRESSIVE STRESSES If we extend the calculations to combined tensile and compressive stresses, assuming there are some friction stresses which still decrease when % < 0, we can calculate the strength also in this region. We may then assume that the calculation will hold when, according to GRlrrrrrI [9] or1 + 3 % > 0, and then let % be constant when ~I tends to zero. Under this assumption it is possible to find the ratio between the uniaxial compressive and tensile strength. From the values of Hoskins in the compressive region we find, in this way, the ratio 9.85.
622
N. L U N D B O R G %(kb) 0-69(")
%=%
5
•
"¢/
',
]
F
J
]
I
I
2
3
4
5
6
crz (kb)
FIo. 6. Comparison between calculated strength and Hosg.[NS' [1] experimental values on trachyte. The symbols indicate different ~3 used by Hoskins corresponding to the calculated lines in the figure.
/ 0
~
I
I
I
I
I
8
3
4
5
o-~(kb) Fro. 7. Comparison between calculated strength and experimental values from AKAI and MORt [2] on sandstone.
POLYAXIAL COMPRESSIVE S T R E N G T H OF MATERIALS
623
It is also possible to let the friction stress be zero when % < 0, and then the above-mentioned ratio becomes 7.40. More experimental results are, however, needed to prove the validity of the calculations in the case of combined stresses. The calculations in this work were carried out on a H P table-top calculator 9100 B. Acknowledgements--This work has been carried out at the Swedish Detonic Research Foundation, with financial support from the Swedish Board for Technical Development, Nitro Nobel AB and Atlas Copco AB. The author would like to express his thanks to Dr P. A. PERSSONand Mr FINN OUCnTERLONYfor useful discussions and critical examination of the manuscript.
REFERENCES 1. HOSKINSE. R. The failure of thick-walled hollow cylinders of isotropic rock. Int. J. Rock Mech. Min. Sci. 6, 99-125 (1969). 2. AKAI K. and MoRt H. Study on the failure mechanism of a sandstone under combined compressive stresses. Proc. Sac. cir. Engrs No. 147, 11-24 (1967). 3. MAZArZrI B. B. The Effect of the Intermediate Principal Stress on the Strength of Rock, Thesis, Georgia Institute of Technology (1967). 4. HOJEM J. P. M. and COOK N. G. W. The design and construction of a triaxial and polyaxial cell for testing rock materials. S. Afr. mech. Engr 18, 57-61 (1968). 5. HANDINJ., HEARD H. C. and MAGOUIRKJ. N. Effects of the intermediate principal stress on the failure of limestone, dolomite and glass at different temperatures and strain rates. J. geophys. Res. 72, 611-640 (1967). 6. WIEaOLSG. A. and COOk N. G. W. An energy criterion for the strength of rock in polyaxial compression. Int. J. Rock Mech. Min. Sci. 5, 529-549 (1968). 7. WEmULL W. A statistical theory of the strength of materials. Proc. R. Sw. Acad. Engng Sci. 151, 5-45 (1939). 8. LUNDaOaG N. Strength of rock-like materials. Int. J. Rock Mech. Min. Sci. 5, 427-454 (1968). 9. GRJrrlTH A. A. The Theory of Rupture, Proceedings of the First International Congress on Applied Mechanics, Delft, pp. 55-63 (1924).
APPENDIX Deduction of the Relation Between Strength, tL and M By using equation (10) and integrating the expression: ,,/2
.4 = f C~d~ = G1Mf [cos~(sin~ - - ~cos~)lu d~
(A.1)
O-o
we find the shaded area A, in Fig. AI. By the substitution 7/"
O. o
4
~-
a=_+
+x
(A.2)
equation (A.1) gives
A ~
GI.M
'~T
£to
4
2 f
--(4--?)
[ cos
+ ~°° + x
) cos (-~ , + ~°°. - x
dr.
(A.3)
624
N. L U N D B O R G
o-3
O'1,
/
~
o-2
FIG. A I . Unit sphere with principal stress axes as in Fig. 2. As the limits o f the integral are symmetric around x = 0, and the integrand is an even function of x, we find the centre o f gravity of the integrated area A, to be at x = 0, i.e.
~=~
7r
~o
+-~.
N o w by using Guldin's rule the integrals in equation (11) are simply obtained by rotating the area in Fig. A1 around the al and a3 axes, respectively, so that equation (11) will be
0"20 ~
(4rr + cos( + o) sin
1 + sinao
cosoo
(A.4)
Putting sin %
COS % we finally get
if20) M
= V 0 + ~2) + t,.
(A.5)