- Email: [email protected]
Discussion on the paper Ultimate bearing capacity at the tip of a pile in rock—Part 1: theory
International Journal of Rock Mechanics & Mining Sciences 40 (2003) 601–604
Discussion
Discussion on the paper Ultimate bearing capacity at the tip of a pile in rock—Part 1: theory William G. Pariseau* Department of Mining Engineering, University of Utah, 315 WC Browning Building, Salt Lake City, UT 84112-0113, USA Accepted 16 February 2003
1. Introduction The authors of the subject paper (International Journal of Rock Mechanics and Mining Sciences 2002;39(7):833–46) obtain estimates of upper bounds to bearing capacity of an end-loaded pile after a detailed development and analysis of the problem within the context of traditional plasticity theory and a twodimensional specialization to slipline theory. The problem is essentially one of a flat, cylindrical indenter penetrating a rigid-plastic medium and is based on stress equations of equilibrium and a yield condition. In case of axial symmetry, rather than plane strain or plane stress, some assumption about the intermediate principal stress is made. The resulting system then consists of three equations in three unknowns and leads to upper bounds of collapse loads after integration along the sliplines that are also the stress characteristics of the system. The authors use the popular Hoek–Brown (HB) failure criterion. Because HB implies two distinct shear strengths for a given mean normal stress, restriction of HB is necessary to avoid physically meaningless conclusions. While I suspect that most authors using HB tacitly adopt the needed restriction, perhaps an open discussion of HB and two shear strengths, associated envelopes and a fix-up with alternatives, is in order.
unconfined compressive strength, material constant and scale factor, respectively. The scale factor varies between 0 and 1 and is an empirical adjustment for the difference between intact rock (s ¼ 1) and rock mass unconfined compressive strengths. Rock mass unconfined compressive strength is Cos ¼ Co Os: The material constant m may be expressed in terms of the unconfined compressive and tensile strength (To ) by performing thought experiments and observing that the tensile stress at failure is negative and numerically equal to tensile strength. Thus, m¼
Co To 1 ¼r ; r To Co
ð2Þ
where intact rock (s ¼ 1) is assumed for the discussion here. Many rock types have a ratio of compressive to tensile strength (r) between 10 and 20, so the constant m is approximately equal to this ratio. If rock mass tensile strength is scaled in the same way as unconfined compressive strength, then Tos ¼ To Os and m ¼ Osðr 1=rÞ which is larger than laboratory m: Transformation of HB to a normal stress–shear stress representation involves the definitions, s1 þ s3 2Co s 1 s3 tm ¼ 2Co
sm ¼
s1 ¼ Co ðsm þ tm Þ; ð3Þ s3 ¼ Co ðsm tm Þ;
2. Two shear strengths from HB Consider HB in one of its earlier forms [1] s1 ¼ s3 þ ½mCo s3 þ sCo2 1=2 ;
ð1Þ
where compression is positive and s1 ; s3 ; Co ; s and m are major principal stress, minor principal stress, *Tel.: +1-801-581-5164; fax: +1-801-585-5410. E-mail address: [email protected] (W.G. Pariseau).
of dimensionless mean normal stress and maximum shear stress, respectively. Thus, after transformation to these new variables, HB has the form
m2 tm þ ¼ 8
2 2 1 1 ð16msm Þ þ ðm2 þ 16sÞ; 8 8
ð4Þ
that is a parabola when plotted in a normal stress–shear stress plane. After division of Eq. (4) by ðm=8Þ2 ; the
1365-1609/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S1365-1609(03)00030-3
W.G. Pariseau / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 601–604
602
results are the primed dimensionless forms ðt þ 1Þ2 ¼ 2s0m þ c; s0m ¼
sm ; m=8
t0m ¼
c¼1þ
3. Envelopes and circles
tm ; m=8
16s : m2
ð5a; cÞ
Eq. (5a) is clearly a parabola when plotted in a normal stress–shear stress plane and is essentially that presented by Serrano and Olalla. However, the bold line plots of HB by Serrano and Olalla appear to be parabolas incorrectly centered on the normal stress axis, e.g., their plots in Figs. 3 and 4. Fig. 1 here shows a true plot of Eq. (5a). The lack of symmetry with respect to the normal stress axis is a cause for concern because of the physical implications. A consequence of the lack of symmetry is two values of shear strength for each value of mean normal stress, a physically improbable event. Solving Eq. (5a) for the maximum (dimensionless) shear stress at failure results in two numerically different shear strengths. Thus, t0m ¼ 17½2s0m þ c1=2 :
ð6Þ
For example, in case of intact rock (s ¼ 1) and a reasonable ratio of unconfined compressive to tensile strength of 10, c ¼ 1:16: The dimensionless strengths of rock in pure shear ðsm ¼ 0Þ implied by HB are then 0.077 and 2.077. These values are the intercepts of the HB parabola in Fig. 1.
A circle is also plotted in Fig. 1 (an ‘‘HB’’ circle) that has a center at (0, 1) and a radius of c1=2 ¼ 1:077: This circle is not a Mohr’s circle representing a stress state at failure. The equation for this circle in dimensionless form consistent with the previous notation is ðs0 s0m Þ2 þ ðt0 þ 1Þ2 ¼ ðt0m þ 1Þ2 ;
ð7Þ
where points on the circle have the primed but not subscripted dimensionless normal and shear stress values. A family of such HB circles would have centers along the line of maximum shear stress equal to 1, as may be inferred from Fig. 1. Mohr circles must have centers along the normal stress axis (maximum shear stress equal to zero). One such circle is shown in Fig. 1 for a dimensionless mean normal stress of unity. This circle represents failure for the portion of HB in the upper (positive) half plane. A reversal of the sign of the maximum shear stress would require enlarging the circle to intersect the HB parabola in the lower (negative) half plane. This situation shows graphically the violation of conventional wisdom that states the sign of the shear stress has no importance to failure. The HB parabola has an envelope of HB circles that may be found from Eq. (5a) or Eqs. (6) and (7) in the usual way (e.g., [2]). The selected two equations contain two parameters (mean normal stress and maximum shear stress). Use of Eq. (5a) or Eq. (6) in Eq. (7) to eliminate the maximum shear stress results in ðs0 s0m Þ2 þ ðt0 þ 1Þ2 ¼ 2s0m þ c:
ð8Þ
The partial derivative of Eq. (8) with respect to the remaining parameter, the mean normal stress, shows that ðs0 s0m Þ ¼ 1:
ð9Þ
Elimination of the parameter between Eqs. (8) and (9) gives the equation of the envelope of HB circles. Thus, ðt0 þ 1Þ2 ¼ 2s0 þ c þ 1:
ð10Þ
The envelope of HB circles is also a parabola that has the same form as HB. The procedure may be generalized by using any function f ðs0m Þ that gives the radius of a circle relative to the considered horizontal line of centers in the normal stress–shear stress plane. The general result for the envelope of considered circles in normalized dimensionless variables used previously is ðt0 þ 1Þ2 ¼ ðf Þ2 ½1 ðf 0 Þ2 ;
Fig. 1. Hoek–Brown failure criterion plotted in a dimensionless mean normal stress.
ð11Þ
where the prime on f means derivative with respect to its argument. In Eq. (11), f is considered to be a function of the normal stress s0 : This equation shows that an envelope exists only when jf 0 jo1: When an envelope exists, the slopes of HB (Eq. (5a)) and the envelope of
W.G. Pariseau / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 601–604
HB circles (Eq. (10)) are related by sinðfÞ ¼ tanðcÞ;
Imposition of the absolute value condition gives ð12Þ
where f is the inclination of the HB envelope to the normal stress axis and c is the inclination of HB to the normal stress axis. This result is given by Hill [3] for conventional Mohr’s circles and failure criteria that are symmetric with respect to the normal stress axes. The qualification (Eq. (12)) indicates that an envelope exists provided jcjop=4: Interestingly, in case of HB, this limit occurs at the normal stress axis intercept ðt0m ¼ 0Þ: If consideration of HB and Mohr circles representing failure is restricted to the upper half plane (positive shear stress), then a partial envelope equation may be found by a similar procedure. In this case, 1 ðs0 s0m Þ ¼ 1 : ð13Þ ½2s0m þ c1=2 Solution for the mean normal stress s0m in terms of the shear stress s0 requires solution of a cubic equation: ½ðs0 s0m Þ þ 12 ð2s0m þ cÞ ¼ 1:
603
ð14Þ
When the proper solution is obtained, back substitution into Eq. (8) eliminates sm and gives the envelope of Mohr circles associated with HB in the upper half plane. Although an algebraic solution is possible in principle, the effort seems unwarranted for the simple reason that a failure envelope is not needed in application to problems of engineering interest. Indeed, a failure criterion expressed in terms of principal stresses is much more practical for computation because only the principal stresses need to be calculated. A second computation of failure plane orientations and associated normal and shear stresses is avoided.
4. A fix for HB and alternatives The physically questionable implications of HB and any other failure criterion expressed in terms of principal stresses that turn out not to be symmetric with respect to the normal stress axis after transformation, e.g., [4,5], can be avoided by restricting applicability to positive shear stress values and then reflecting the restricted portion of the considered criterion about the mean normal stress axis. The mean normal stress is restricted to values greater than the normal stress axis intercept, while reflection is achieved by simply using the absolute value of shear stress. In case of HB, this ‘‘fix’’ has the restriction 1c s0m X : ð15Þ 2 This restriction also corresponds to the range of HB in the upper half plane over which an envelope exists. In the plane of s3 ; s1 ; this restricts s3 to values greater than Co =mE To :
jt0m j
¼ 1 þ ½2s0m þ c1=2 :
ð16Þ
Inspection of Eq. (16) shows that a lesser value of mean normal stress than one satisfying Eq. (15) leads to an imaginary number in Eq. (16) and is physically meaningless. A plot of the restricted HB given by Eq. (16) is shown in Fig. 1. Restricted HB has a sharp nose as does the wellknown Mohr–Coulomb (MC) criterion that has a dimensionless form ðCo =To Þ 1 .m t0m ¼ as0m þ b; a ¼ ; ðCo =To Þ þ 1 8 . 1 m b¼ : ð17a; cÞ ðCo =To Þ þ 1 8 The MC criterion (17a) is plotted in Fig. 1 using the same numerical values in the plot of HB. In this regard, when s0m ¼ 0:5; the actual mean normal stress roughly equals the one-half unconfined compressive strength. In fact, sm ¼ ðm=8ÞCo ¼ ð9:9=16ÞCo ¼ 0:62Co and the major principal stress is about equal to the unconfined compressive strength. Thus, over a range of mean normal stress from zero to Co, there is little to choose between problematic HB and the well-known MC criteria. The difference at s0m ¼ 1 is less than 3%. This range is very practical because unsupported rock excavations are unconfined; wall failure is essentially at unconfined compressive strength. Even when confined by support or reinforcement, the confining pressure introduced is usually small compared with rock strength. Comparisons of yield zones about circular tunnels obtained from MC and HB by Sofianos and Halakatevakis [6] show differences of about 10% or less. Over an extended range of mean normal stress or ‘‘confining pressure’’, experience with test data leads one to expect a decrease in slope of the failure criterion plot, a decrease in the angle of internal friction, f: A better fit than MC may then be possibly obtained by a nonlinear form jtm jn ¼ asm þ b;
ð18Þ
where the dimensionless maximum shear stress and mean normal stress are defined in Eq. (3), a and b are dimensionless material constants, and n is an exponent [7]. Solution for the maximum shear stress gives tm ¼ 7½asm þ b1=n ;
ð19Þ
where a positive root is implied. The dimensionless constants are given by n n n n 1 1 1 1 1 þ 2 2r r 2 2r a ¼ ; b ¼ ; ð20Þ 1 1 1 1þ þ 2 2r 2r where the ratio r ¼ Co =To :
604
W.G. Pariseau / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 601–604
Fig. 2. N-type failure criteria plot. MC: n ¼ 1; parabola; n ¼ 2; superparabola; n ¼ 10; sub-parabola; n ¼ 15; hb=restricted HB. Co=unconfined compressive strength.
The sharp nose of MC, restricted HB and the n-type sub-parabolas raises an issue regarding a physically suitable shape of failure criteria plots in the region of tensile stress. This is an old issue, but one that seems to be overlooked in recent times. Various suggestions for rounding the nose of sharp-nosed failure criteria have been made (e.g., [8]), but since there are few test data in this important region, the choice seems to be one of personal taste and mathematical convenience. In any case, rounding produces slopes greater than 1 and thus a region where an envelope does not exist. For these reasons, failure envelopes are incidental features to mathematically simple and physically appropriate criteria. In case of anisotropy, and in three dimensions, the situation becomes more complex and therefore demanding of greater care in selecting candidate functions for failure criteria.
References When n ¼ 1; Eqs. (18) and (19) reduce to MC. When n ¼ 2; a parabola is obtained that is symmetric with respect to the normal stress axis. Super-parabolas are obtained when n > 2; while sub-parabolas are obtained for 1ono2: Sub-parabolas have pointed noses. As the exponent n becomes very large, a tension cut-off appears and the slope tends to zero. The dimensionless nature of the plots in Fig. 2 results in all graphs passing through the point (0.5, 0.5), that is, ða=2 þ bÞ1=n ¼ 0:5: Restricted HB is also shown in Fig. 2. The plots suggest that a suitable choice of exponent n for a sub-parabola, n-type criterion, would result in a close match to HB. The advantage of an n-type criterion is automatic symmetry with respect to the mean normal stress axis; no restriction is necessary.
[1] Hoek E, Brown ET. Underground excavations in rock. London: The Institution of Mining and Metallurgy, 1980. p. 140. [2] Ford LR. Differential equations. New York: McGraw-Hill, 1955. p. 231–3. [3] Hill R. The mathematical theory of plasticity. Oxford: Clarendon Press, 1950. p. 295. [4] Price NJ. Fault and joint development in brittle and semi-brittle rock. Oxford: Pergamon, 1966. p. 26. [5] Bieniawski ZT. Estimating the strength of rock masses. J S Afr Inst Min Metall 1974;74:312–20. [6] Sofianos AI, Halakatevakis N. Equivalent tunneling Mohr– Coulomb strength parameters for given Hoek–Brown ones. Int J Rock Mech Min Sci 2002;39(1):131–7. [7] Pariseau WG. Post-yield mechanics of rock and soil. Mineral Industries, vol. 36(8). The Pennsylvania State University, University Park, 1967. p. 1–6. [8] Nadai A. Theory of flow and fracture of solids, vol. II. New York: McGraw-Hill, 1963. p. 476–88.
Discussion
Discussion on the paper Ultimate bearing capacity at the tip of a pile in rock—Part 1: theory William G. Pariseau* Department of Mining Engineering, University of Utah, 315 WC Browning Building, Salt Lake City, UT 84112-0113, USA Accepted 16 February 2003
1. Introduction The authors of the subject paper (International Journal of Rock Mechanics and Mining Sciences 2002;39(7):833–46) obtain estimates of upper bounds to bearing capacity of an end-loaded pile after a detailed development and analysis of the problem within the context of traditional plasticity theory and a twodimensional specialization to slipline theory. The problem is essentially one of a flat, cylindrical indenter penetrating a rigid-plastic medium and is based on stress equations of equilibrium and a yield condition. In case of axial symmetry, rather than plane strain or plane stress, some assumption about the intermediate principal stress is made. The resulting system then consists of three equations in three unknowns and leads to upper bounds of collapse loads after integration along the sliplines that are also the stress characteristics of the system. The authors use the popular Hoek–Brown (HB) failure criterion. Because HB implies two distinct shear strengths for a given mean normal stress, restriction of HB is necessary to avoid physically meaningless conclusions. While I suspect that most authors using HB tacitly adopt the needed restriction, perhaps an open discussion of HB and two shear strengths, associated envelopes and a fix-up with alternatives, is in order.
unconfined compressive strength, material constant and scale factor, respectively. The scale factor varies between 0 and 1 and is an empirical adjustment for the difference between intact rock (s ¼ 1) and rock mass unconfined compressive strengths. Rock mass unconfined compressive strength is Cos ¼ Co Os: The material constant m may be expressed in terms of the unconfined compressive and tensile strength (To ) by performing thought experiments and observing that the tensile stress at failure is negative and numerically equal to tensile strength. Thus, m¼
Co To 1 ¼r ; r To Co
ð2Þ
where intact rock (s ¼ 1) is assumed for the discussion here. Many rock types have a ratio of compressive to tensile strength (r) between 10 and 20, so the constant m is approximately equal to this ratio. If rock mass tensile strength is scaled in the same way as unconfined compressive strength, then Tos ¼ To Os and m ¼ Osðr 1=rÞ which is larger than laboratory m: Transformation of HB to a normal stress–shear stress representation involves the definitions, s1 þ s3 2Co s 1 s3 tm ¼ 2Co
sm ¼
s1 ¼ Co ðsm þ tm Þ; ð3Þ s3 ¼ Co ðsm tm Þ;
2. Two shear strengths from HB Consider HB in one of its earlier forms [1] s1 ¼ s3 þ ½mCo s3 þ sCo2 1=2 ;
ð1Þ
where compression is positive and s1 ; s3 ; Co ; s and m are major principal stress, minor principal stress, *Tel.: +1-801-581-5164; fax: +1-801-585-5410. E-mail address: [email protected] (W.G. Pariseau).
of dimensionless mean normal stress and maximum shear stress, respectively. Thus, after transformation to these new variables, HB has the form
m2 tm þ ¼ 8
2 2 1 1 ð16msm Þ þ ðm2 þ 16sÞ; 8 8
ð4Þ
that is a parabola when plotted in a normal stress–shear stress plane. After division of Eq. (4) by ðm=8Þ2 ; the
1365-1609/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S1365-1609(03)00030-3
W.G. Pariseau / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 601–604
602
results are the primed dimensionless forms ðt þ 1Þ2 ¼ 2s0m þ c; s0m ¼
sm ; m=8
t0m ¼
c¼1þ
3. Envelopes and circles
tm ; m=8
16s : m2
ð5a; cÞ
Eq. (5a) is clearly a parabola when plotted in a normal stress–shear stress plane and is essentially that presented by Serrano and Olalla. However, the bold line plots of HB by Serrano and Olalla appear to be parabolas incorrectly centered on the normal stress axis, e.g., their plots in Figs. 3 and 4. Fig. 1 here shows a true plot of Eq. (5a). The lack of symmetry with respect to the normal stress axis is a cause for concern because of the physical implications. A consequence of the lack of symmetry is two values of shear strength for each value of mean normal stress, a physically improbable event. Solving Eq. (5a) for the maximum (dimensionless) shear stress at failure results in two numerically different shear strengths. Thus, t0m ¼ 17½2s0m þ c1=2 :
ð6Þ
For example, in case of intact rock (s ¼ 1) and a reasonable ratio of unconfined compressive to tensile strength of 10, c ¼ 1:16: The dimensionless strengths of rock in pure shear ðsm ¼ 0Þ implied by HB are then 0.077 and 2.077. These values are the intercepts of the HB parabola in Fig. 1.
A circle is also plotted in Fig. 1 (an ‘‘HB’’ circle) that has a center at (0, 1) and a radius of c1=2 ¼ 1:077: This circle is not a Mohr’s circle representing a stress state at failure. The equation for this circle in dimensionless form consistent with the previous notation is ðs0 s0m Þ2 þ ðt0 þ 1Þ2 ¼ ðt0m þ 1Þ2 ;
ð7Þ
where points on the circle have the primed but not subscripted dimensionless normal and shear stress values. A family of such HB circles would have centers along the line of maximum shear stress equal to 1, as may be inferred from Fig. 1. Mohr circles must have centers along the normal stress axis (maximum shear stress equal to zero). One such circle is shown in Fig. 1 for a dimensionless mean normal stress of unity. This circle represents failure for the portion of HB in the upper (positive) half plane. A reversal of the sign of the maximum shear stress would require enlarging the circle to intersect the HB parabola in the lower (negative) half plane. This situation shows graphically the violation of conventional wisdom that states the sign of the shear stress has no importance to failure. The HB parabola has an envelope of HB circles that may be found from Eq. (5a) or Eqs. (6) and (7) in the usual way (e.g., [2]). The selected two equations contain two parameters (mean normal stress and maximum shear stress). Use of Eq. (5a) or Eq. (6) in Eq. (7) to eliminate the maximum shear stress results in ðs0 s0m Þ2 þ ðt0 þ 1Þ2 ¼ 2s0m þ c:
ð8Þ
The partial derivative of Eq. (8) with respect to the remaining parameter, the mean normal stress, shows that ðs0 s0m Þ ¼ 1:
ð9Þ
Elimination of the parameter between Eqs. (8) and (9) gives the equation of the envelope of HB circles. Thus, ðt0 þ 1Þ2 ¼ 2s0 þ c þ 1:
ð10Þ
The envelope of HB circles is also a parabola that has the same form as HB. The procedure may be generalized by using any function f ðs0m Þ that gives the radius of a circle relative to the considered horizontal line of centers in the normal stress–shear stress plane. The general result for the envelope of considered circles in normalized dimensionless variables used previously is ðt0 þ 1Þ2 ¼ ðf Þ2 ½1 ðf 0 Þ2 ;
Fig. 1. Hoek–Brown failure criterion plotted in a dimensionless mean normal stress.
ð11Þ
where the prime on f means derivative with respect to its argument. In Eq. (11), f is considered to be a function of the normal stress s0 : This equation shows that an envelope exists only when jf 0 jo1: When an envelope exists, the slopes of HB (Eq. (5a)) and the envelope of
W.G. Pariseau / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 601–604
HB circles (Eq. (10)) are related by sinðfÞ ¼ tanðcÞ;
Imposition of the absolute value condition gives ð12Þ
where f is the inclination of the HB envelope to the normal stress axis and c is the inclination of HB to the normal stress axis. This result is given by Hill [3] for conventional Mohr’s circles and failure criteria that are symmetric with respect to the normal stress axes. The qualification (Eq. (12)) indicates that an envelope exists provided jcjop=4: Interestingly, in case of HB, this limit occurs at the normal stress axis intercept ðt0m ¼ 0Þ: If consideration of HB and Mohr circles representing failure is restricted to the upper half plane (positive shear stress), then a partial envelope equation may be found by a similar procedure. In this case, 1 ðs0 s0m Þ ¼ 1 : ð13Þ ½2s0m þ c1=2 Solution for the mean normal stress s0m in terms of the shear stress s0 requires solution of a cubic equation: ½ðs0 s0m Þ þ 12 ð2s0m þ cÞ ¼ 1:
603
ð14Þ
When the proper solution is obtained, back substitution into Eq. (8) eliminates sm and gives the envelope of Mohr circles associated with HB in the upper half plane. Although an algebraic solution is possible in principle, the effort seems unwarranted for the simple reason that a failure envelope is not needed in application to problems of engineering interest. Indeed, a failure criterion expressed in terms of principal stresses is much more practical for computation because only the principal stresses need to be calculated. A second computation of failure plane orientations and associated normal and shear stresses is avoided.
4. A fix for HB and alternatives The physically questionable implications of HB and any other failure criterion expressed in terms of principal stresses that turn out not to be symmetric with respect to the normal stress axis after transformation, e.g., [4,5], can be avoided by restricting applicability to positive shear stress values and then reflecting the restricted portion of the considered criterion about the mean normal stress axis. The mean normal stress is restricted to values greater than the normal stress axis intercept, while reflection is achieved by simply using the absolute value of shear stress. In case of HB, this ‘‘fix’’ has the restriction 1c s0m X : ð15Þ 2 This restriction also corresponds to the range of HB in the upper half plane over which an envelope exists. In the plane of s3 ; s1 ; this restricts s3 to values greater than Co =mE To :
jt0m j
¼ 1 þ ½2s0m þ c1=2 :
ð16Þ
Inspection of Eq. (16) shows that a lesser value of mean normal stress than one satisfying Eq. (15) leads to an imaginary number in Eq. (16) and is physically meaningless. A plot of the restricted HB given by Eq. (16) is shown in Fig. 1. Restricted HB has a sharp nose as does the wellknown Mohr–Coulomb (MC) criterion that has a dimensionless form ðCo =To Þ 1 .m t0m ¼ as0m þ b; a ¼ ; ðCo =To Þ þ 1 8 . 1 m b¼ : ð17a; cÞ ðCo =To Þ þ 1 8 The MC criterion (17a) is plotted in Fig. 1 using the same numerical values in the plot of HB. In this regard, when s0m ¼ 0:5; the actual mean normal stress roughly equals the one-half unconfined compressive strength. In fact, sm ¼ ðm=8ÞCo ¼ ð9:9=16ÞCo ¼ 0:62Co and the major principal stress is about equal to the unconfined compressive strength. Thus, over a range of mean normal stress from zero to Co, there is little to choose between problematic HB and the well-known MC criteria. The difference at s0m ¼ 1 is less than 3%. This range is very practical because unsupported rock excavations are unconfined; wall failure is essentially at unconfined compressive strength. Even when confined by support or reinforcement, the confining pressure introduced is usually small compared with rock strength. Comparisons of yield zones about circular tunnels obtained from MC and HB by Sofianos and Halakatevakis [6] show differences of about 10% or less. Over an extended range of mean normal stress or ‘‘confining pressure’’, experience with test data leads one to expect a decrease in slope of the failure criterion plot, a decrease in the angle of internal friction, f: A better fit than MC may then be possibly obtained by a nonlinear form jtm jn ¼ asm þ b;
ð18Þ
where the dimensionless maximum shear stress and mean normal stress are defined in Eq. (3), a and b are dimensionless material constants, and n is an exponent [7]. Solution for the maximum shear stress gives tm ¼ 7½asm þ b1=n ;
ð19Þ
where a positive root is implied. The dimensionless constants are given by n n n n 1 1 1 1 1 þ 2 2r r 2 2r a ¼ ; b ¼ ; ð20Þ 1 1 1 1þ þ 2 2r 2r where the ratio r ¼ Co =To :
604
W.G. Pariseau / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 601–604
Fig. 2. N-type failure criteria plot. MC: n ¼ 1; parabola; n ¼ 2; superparabola; n ¼ 10; sub-parabola; n ¼ 15; hb=restricted HB. Co=unconfined compressive strength.
The sharp nose of MC, restricted HB and the n-type sub-parabolas raises an issue regarding a physically suitable shape of failure criteria plots in the region of tensile stress. This is an old issue, but one that seems to be overlooked in recent times. Various suggestions for rounding the nose of sharp-nosed failure criteria have been made (e.g., [8]), but since there are few test data in this important region, the choice seems to be one of personal taste and mathematical convenience. In any case, rounding produces slopes greater than 1 and thus a region where an envelope does not exist. For these reasons, failure envelopes are incidental features to mathematically simple and physically appropriate criteria. In case of anisotropy, and in three dimensions, the situation becomes more complex and therefore demanding of greater care in selecting candidate functions for failure criteria.
References When n ¼ 1; Eqs. (18) and (19) reduce to MC. When n ¼ 2; a parabola is obtained that is symmetric with respect to the normal stress axis. Super-parabolas are obtained when n > 2; while sub-parabolas are obtained for 1ono2: Sub-parabolas have pointed noses. As the exponent n becomes very large, a tension cut-off appears and the slope tends to zero. The dimensionless nature of the plots in Fig. 2 results in all graphs passing through the point (0.5, 0.5), that is, ða=2 þ bÞ1=n ¼ 0:5: Restricted HB is also shown in Fig. 2. The plots suggest that a suitable choice of exponent n for a sub-parabola, n-type criterion, would result in a close match to HB. The advantage of an n-type criterion is automatic symmetry with respect to the mean normal stress axis; no restriction is necessary.
[1] Hoek E, Brown ET. Underground excavations in rock. London: The Institution of Mining and Metallurgy, 1980. p. 140. [2] Ford LR. Differential equations. New York: McGraw-Hill, 1955. p. 231–3. [3] Hill R. The mathematical theory of plasticity. Oxford: Clarendon Press, 1950. p. 295. [4] Price NJ. Fault and joint development in brittle and semi-brittle rock. Oxford: Pergamon, 1966. p. 26. [5] Bieniawski ZT. Estimating the strength of rock masses. J S Afr Inst Min Metall 1974;74:312–20. [6] Sofianos AI, Halakatevakis N. Equivalent tunneling Mohr– Coulomb strength parameters for given Hoek–Brown ones. Int J Rock Mech Min Sci 2002;39(1):131–7. [7] Pariseau WG. Post-yield mechanics of rock and soil. Mineral Industries, vol. 36(8). The Pennsylvania State University, University Park, 1967. p. 1–6. [8] Nadai A. Theory of flow and fracture of solids, vol. II. New York: McGraw-Hill, 1963. p. 476–88.