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Ultimate bearing capacity at the tip of a pile in rock—part 1: theory
International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
Ultimate bearing capacity at the tip of a pile in rock—part 1: theory A. Serranoa, C. Olallab,* b
a ETSICCP, Universidad Polit!ecnica, Madrid, Spain Laboratorio de Geotecnia, Alfonso XII, 3 y 5, Cedex-MOPTMA, 28014 Madrid, Spain
Accepted 30 April 2002
Abstract This paper proposes a method for calculating the ultimate bearing capacity at the tip of a pile that is embedded in rock, according to the theory of plasticity. The non-linear failure criterion of Hoek and Brown (Underground excavation in rock, The Institution of Mining and Metallurgy, London, 1980) is used. The plastified area is analysed as a two-dimensional medium using the characteristic lines method generalised for non-linear failure criteria. The embedment is simulated using the hypothesis proposed by Meyerhof for piles in soils. The results obtained through the calculations are corrected by applying a shape factor in order to take into account the three-dimensional nature of the pile. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Pile tip; Bearing capacity; Failure criterion; Plasticity; Rock mechanics
1. Introduction The ultimate bearing capacity at the tip of a pile embedded into the ground is calculated using perfect plasticity theory. The case of a wall in plane strain is analysed. Then a shape factor is used to establish the ultimate bearing capacity for a pile. The pile, whose diameter is B; is embedded in a rock medium down to a depth of HR : The embedment ratio n ¼ HR =B (Fig. 1) is incorporated using the same theoretical basis as Meyerhof [1]. The embedment is replaced by a virtual ground surface sloping at angle a: The Meyerhof basis is considered to be one of the most reasonable of all the possible options. In classical soil mechanics, the Meyerhof hypothesis has been used on many occasions with a linear failure criterion. It is generally the case in practice that the pile is embedded only a few diameters, but with the theory that is developed here, the theoretical effect of greater embedment ratio values will be analysed. The weight of the triangle of rock above (the overburden pressure) is exerted on the virtual surface, together with the potential external loads that rest upon it. It is assumed that this weight is uniformly distributed throughout the virtual surface. *Corresponding author. Tel.: +34-91-335-7300; fax: +34-91-3357322. E-mail address: [email protected] (C. Olalla).
The entire numerical analysis is carried out in a dimensionless way. This amounts to assuming that the stress unit is the so-called ‘‘pressure modulus’’, or ‘‘strength modulus’’, b obtained from the Hoek and Brown failure criterion, which will be defined later on in this paper. So, all the stresses (s1 ; s3 ; p; qy) are dimensionless. That is to say, they are divided by b: The specific weight unit is the specific weight of the rock formation, gR : Working with dimensionless variables, the mathematical development is simplified and it makes the calculations universal in nature. The different variables and their respective units are retrieved at the end of the development.
2. Basic hypotheses and failure criterion 2.1. Basic hypotheses The following basic hypotheses are used: 1. The rock mass is a homogeneous and isotropic medium. The isotropy hypothesis can be treated in a similar manner as that achieved by Serrano and Olalla [2,3] for surface foundations overlying anisotropic blocks. 2. The rock medium fails when the stress reaches Hoek and Brown’s strength criterion. It is assumed that the
1365-1609/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 5 - 1 6 0 9 ( 0 2 ) 0 0 0 5 2 - 7
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Nomenclature B diameter of the pile (m) GSI geomechanical index by Hoek et al. (1997) HR ; HS depth that the pile is embedded into the ground (m); thickness of the soil (m) h hm ¼ m average overburden load in ground, dimenb sionless I Riemann’s invariant i1 angle of inclination of the loads for Boundary 1 m; s parameters of the Hoek and Brown criterion (1980) m0 Hoek and Brown parameter m for sound rock Nq ; Nc Prandtl’s load coefficients NbP ; Nb load factor for the ultimate bearing capacity of the pile; for the wall Ns ; Ns =m0 load factor that when multiplied by the unconfined compressive strength provides the ultimate bearing capacity of the pile; load factor rationalised in such a way that when it is multiplied by the unconfined compressive strength it provides the ultimate bearing capacity of the pile n ¼ HR =B; nL1 ; nL2 embedment ratio; embedment ratio limit value 1; embedment ratio limit value 2 p s1 þ s3 Lambe’s variable (dimensionless) p¼ ¼ b 2b Lambe’s variables for Boundary 1 p1 ; q1 ; p 2 ; q2 (dimensionless); for Boundary 2 (dimensionless) p s1 s3 Lambe’s variable (dimensionless) q¼ ¼ b 2b RMR Bieniawski’s geomechanical index s1 ; t 1 ; s2 ; t 2 normal stress and shear stress exerted on Boundary 1 (dimensionless); on Boundary 2 (dimensionless) sq ; sb shape factors
rock medium does not undergo strain hardening as a result of the deformation process. 3. The failure zone is obtained using the characteristic lines method. This theory was developed by Sokolovski with a linear failure criterion in the 1950s. 4. The characteristic lines for the stress field and the characteristic lines for the deformation increase field are the same. That is to say, it is assumed that the material is coaxial and has associated dilatancy. 5. The rock mass is weightless. This hypothesis does not constitute a major limitation, because the weight of the material moved during the failure process is very small when compared to the strength forces that are created on the rock failure surface.
a
angle of inclination for the assumed virtual failure surface m sc strength modulus for the rock mass. b¼ 8 Parameter that represents the Hoek and Brown criterion (kN/m2) e angle that forms the main major stress with the inclination of the ground gR ; gS specific weight of the rock; of the soil (kN/m3) m1 ; m2 angles that form families of characteristic lines 1 and 2, respectively, with the ground r; r1 ; r2 ; rm instantaneous angle of internal friction (general; for Boundary 1; for Boundary 2; average) s main major stress (dimensionless) sI ¼ I b s sIII ¼ III main minor stress (dimensionless) b unconfined compressive strength of the sc ; scm : scm intact rock (kN/m2); unconfined compressive strength of the rock mass (dimensionless); with dimensions (kN/m2) 8s tensile strength coefficient of the rock mass. z¼ 2 m Parameter that represents the Hoek and Brown criterion sv vertical load exerted on Boundary 1 (dimensionless) sh ultimate bearing capacity under two-dimensionality hypothesis (kN/m2) shp ultimate bearing capacity of the pile (kN/m2) t; sn shear stress and normal stress for Mohr’s envelope (dimensionless) c1 ; c2 angle of inclination of the main major stress for Boundary 1 (with respect to the vertical axis); angle of inclination of the main major stress for Boundary 2 (with respect to the vertical axis)
2.2. Failure criterion 2.2.1. The Hoek and Brown criterion (1980 and later amendments) The Hoek and Brown criterion [4–6] is as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sI sIII s ¼ m III þ s; ð2:1Þ sc sc where sI and sIII are the major and minor failure stresses, sc is the unconfined compressive strength of the intact rock, m and s are Hoek and Brown’s constants, which depend on the type of the rock and its fractured and weathered characteristics.
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
Hs
Soil
835
B
BOUNDARY 1
D
αv
α C
HR
Rock B O' M
O
A BOUNDARY 2
Fig. 1. Sketch of assumed failure.
This criterion is based on empirical concepts. It successfully simulates the main aspects that appear in rock mass failure, such as non-linearity with the stress levels; the dependence on the type of rock and the state of the mass; the ratio between the compression strength and the tensile strength; the way in which the angle of friction becomes lower as the confining stress increases, etc. Parameters m and s can be obtained using different procedures. However, the one that is probably most advisable involves obtaining them from the type of rock (m0 ) and the geomechanical index RMR, or the GSI by Hoek and Brown (1997) [6], using the following formulae: RMR 100 ; ð2:2Þ m ¼ m0 exp a RMR 100 ; ð2:3Þ b where a ¼ 28 and b ¼ 9; when the natural rock medium has not been artificially disturbed, e.g. through the use of explosives, Hoek and Brown [5]. s ¼ exp
2.2.2. Parametric formulae The above failure criterion formula by Hoek and Brown can be simplified and rationalised by using Lambe’s variables p ¼ ðsI þ sIII Þ=2 and q ¼ ðsI sIII Þ=2; as formulated by Serrano and Olalla [7], to give the following formula: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q p ¼ 2 þz þ11 ð2:4Þ b b which in dimensionless form with a pressure modulus b ðp ¼ p =b and q ¼ q =bÞ; becomes:
qþ1¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðp þ zÞ þ 1;
ð2:5Þ
where parameters b and z; which establish the rock strength, are based directly on Hoek and Brown’s m and s parameters as follows: m sc ; b¼ 8 8s z ¼ 2: ð2:6Þ m The dimensional parameter b serves to scale the failure criterion and all the stress variables. It is referred to as the ‘‘strength modulus’’. Parameter z represents the relative quality and strength of the rock mass. The type of rock and the extent to which it is fractured and disturbed are inherent to this parameter. In view of the fact that it indicates the contact point of Hoek and Brown’s envelope, for coordinates (t; s), at the axis of s; it measures the ‘‘dimensionless isotropic tensile strength’’ of the rock mass (q ¼ 0), so it could be referred to as the ‘‘tensile strength coefficient of the rock mass’’. From a rational perspective, the failure criterion is expressed in the following way: 2ðp þ zÞ ¼ q2 þ 2q
ð2:7Þ
which is Hoek and Brown’s strength law expressed in its most general and simplified form. The instantaneous angle of friction (r) is expressed using the following formula [7]: sin r ¼
dq 1 ¼ : dp 1 þ q
ð2:8Þ
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A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
Depending on the instantaneous angle of friction, Lambe’s variables (dimensionless), which follow Hoek and Brown’s criterion, are as follows: 1 sin r ; q¼ sin r cotan2 r p¼ z: ð2:9Þ 2 Mohr’s envelope for the failure circles, dimensionless, is expressed as a function of parameter r by means of the following equations: t ¼ q cos r; sn ¼ p q sin r:
α ε ψ α I ,1
α
1
h
α
C
V
α
II ,1
Ellipse of stress distribution (Boundary 1)
1 B O' 2
O
A
ð2:10Þ
When the failure criterion takes this form, the instantaneous angle of friction is expressed in the following way: dt tan r ¼ : ð2:11Þ dsn Note: As pointed out earlier, when this procedure is being mathematically developed, the stress variables (s; t; p; q; etc.) are used in dimensionless form. That is to say, divided by parameter b; they are indicated throughout this document without the asterisk. Thus for example, the unconfined p compressive strength value of the ffiffi rock mass ðscm ; scm ¼ sc sÞ in dimensionless form (scm ) is pffiffiffiffiffi scm ¼ scm =b ¼ 2 2z: ð2:12Þ
3. Calculation procedure 3.1. Introduction The calculation method is similar to the one used by Serrano and Olalla [7] to find the ultimate bearing capacity of surface foundations. The load coefficients Nb are obtained as a function of the different embedment ratios (n) and different overburden pressures ‘‘hm ’’. This amounts to a generalisation and an extension of the traditional load coefficients Nq and Nc by Prandtl. The ultimate bearing capacity sh under the twodimensional hypothesis matches the surface load case, normal to the plane of action (Boundary 2), with the exterior surface ascending with an angle a (Boundary 1), and a load (s1 ) for Boundary 1 with angle a sloping in a destabilising direction (Fig. 2). The starting point is an overburden pressure hm and an angle a that is known before the analysis is initiated. This angle is only valid for a certain embedment ratio (n) given by the relation (Fig. 1): HR DO n¼ ¼ : ð3:1Þ B OO0 The embedment ratio n is established later as a function of the instantaneous angles of friction r1 and
α
I ,2
Ellipse of stress distribution (Boundary 2)
Fig. 2. Failure wedge: force diagram and stress ellipses.
r2 for Boundaries 1 and 2, respectively (see Section 3.5), and the virtual angle a: The calculation procedure is divided into three stages: Stage 1: Mohr’s circle (passive) is established for Boundary 1, by knowing the load sv that is exerted on this boundary (see Section 3.2). Stage 2: The Riemann’s invariant Iðr1 Þ is established for Boundary 1 and it is transferred to Boundary 2 (see Section 3.3), which makes it possible to obtain the invariant Iðr2 Þ for Boundary 2. Stage 3: The invariant for Boundary 2, Iðr2 Þ; is used to determine Mohr’s circle in this Boundary 2, and this is used, in turn, to find the ultimate bearing capacity of the wall sh (see Section 3.4). In summary, this procedure is first used to obtain the ultimate bearing capacity sh of a wall embedded in rock when the overburden pressure is hm and the embedment ratio is n: The stress conditions for the whole plastified zone and the failure surface can be obtained as a byproduct of the calculation process. Subsequently, once the ultimate bearing capacity of the wall is known, i.e., under the plane strain hypotheses, the ultimate bearing capacity of the pile (shp ) is obtained by multiplying it by a shape factor sb ; which depends on the average instantaneous angle of friction (rm ), which is then established through r1 and r2 : 3.2. Boundary 1 3.2.1. External load Weight W of the OCD triangle (Fig. 1): W ¼ 12HR DC gR :
ð3:2aÞ
Vertical pressure (sv ) exerted on OC: sv ¼
W gR HR cos a : ¼ 2 OC
ð3:2bÞ
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
gR HR cos a : 2b
*
ð3:2cÞ
*
By way of simplification, the entire process is going to be calculated with the dimensionless parameter hm ; which stands for the average overburden pressure that is exerted upon the virtual surface OC: hm ¼
gR H R : 2b
In the first case (circles shown in Fig. 3), all the positions on the overburden circle are possible for the stresses on Boundary 1; it is possible to proceed in an ongoing way, depending on the embedment, from a ¼ 0 to p=2: The latter is the outermost position, where the failure bulb has closed upon the pile. In the second case, when a increases above a ¼ 0; an outermost position aL1 is reached (point TL1 in Fig. 3). The positions lying between TL1 and T’L1 cannot exist, because they are for strengths that exceed the failure criterion. However, the positions lying between T’L1 ; and 0, which are for potential stresses, stand for states that are physically unacceptable (Mohr’s circles with tensile strengths and active pressure state on Boundary 1). The cases of minor overburden circles are the most interesting from a practical viewpoint, because they coincide with most actual situations. The major overburden circles are consistent with extreme situations in which the rock masses are extremely fractured and the piles are deeply embedded. pffiffiffiffiffi The discriminated case with hm ¼ 2 2z needs a special study, which will be carried out later on in this paper.
ð3:3Þ
In the most general case, there is a ground pressure Hs gs exerted on the rock surface. The parameter hm to be considered (average overburden exerted upon OC) thus takes the following value: hm ¼
H R gR H s gs þ : 2b b
ð3:4Þ
The vertical stress sv exerted upon Boundary 1 is sv ¼ hm cos a:
pffiffiffiffiffi 2zÞ: pffiffiffiffiffi S circles for minor overburden pressure hm o2 p2z ffiffiffiffiffi; G circles for major overburden pressure hm > 2 2z:
rock mass expressed in dimensionless form ð2
Dimensionless vertical pressure sv exerted on OC: sv ¼
ð3:5Þ
The stress components sv ; in the direction of the plane that represents Boundary 1 (t1 ) and the one that runs perpendicular to it (s1 ), are as follows: s1 ¼ sv cos a ¼ hm cos2 a;
ð3:6aÞ
t1 ¼ sv sin a ¼ hm sin a cos a:
ð3:6bÞ
3.2.2. Mohr’s circle for Boundary 1 (a) Bases: The overburden pressure sv exerted upon a virtual surface OC and the stresses ‘‘f ’’ to which the failure surface is subjected balance the ultimate bearing capacity sh (Fig. 4). If this balance is broken, i.e. when failure occurs (Fig. 4), the velocity field is, in the case of the figure, essentially a counterclockwise rotational movement around the singular point 0 (maximum load).
In a diagram t; sn (see Fig. 3) the points that stand for the stress exerted upon Boundary 1 are located on a circle whose diameter is hm ; which will be referred to as the ‘‘overburden circle’’. As far as the Hoek and Brown strength criterion is concerned, there could be two types of overburden pressure, depending on whether hm is greater or smaller than the unconfined compressive strength scm of the
τ TL1 Hoek & Brown criterion
G TL2
P
TL1 '
S P
α
α L1
* σcm = σcm /β = 2
8 2ζ = m
t1
σ
t1 s1
O
837
hm
σcm
s1
hm
s ; Uniaxial Compressive Strength of the rock mass
Fig. 3. Overburden circles: major overburden circle (G). Minor overburden circle (S). Mohr’s circle for the unconfined compression (sc ).
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A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
σv
C
α
A
O
f B
(a) Hoek & Brown criterion
τ
P Stress at the Boundary
Overburden Stress circle
q
ρ
M
A
(b)
t
O
σ
G
α
hm
s
τ Hoek & Brown criterion T
Stress at the Boundary
P Overburden Stress circle
ρ
M
α (c)
O
t s
A
q
σ
S hm
Fig. 4. Boundary 1: (a) kinematics of failure, (b) major overburden circle, (c) minor overburden circle.
When the same pressure is exerted upon Boundary 1 (point M on the overburden circle) two Mohr’s circles can be plotted tangent to the strength curve: the minimum or active circle (A), and the maximum or passive circle (P) (Fig. 4). That figure shows the two overburden circle possibilities: major and minor. Mohr’s maximum circle (P) for Boundary 1 matches the counterclockwise rotation kinematics. That is to say, the stresses are in a passive state in relation to this boundary. (b) Angle a on the virtual plane and instantaneous angle of friction r1 : The following equation holds for Mohr’s
passive, circle (see Fig. 5) ðp1 s1 Þ2 þ t21 q21 ¼ 0;
ð3:7Þ
where p1 and q1 are variables of Lambe that verifies the Hoek and Brown criterion expressed either by formula (2.7) or by the parametric Eqs. (2.9), depending on the instantaneous angle of friction (r). Taking into account Eqs. (3.6) and (3.7), the angle a for virtual boundary 1 is expressed in the following way cos a ¼
p21 q21 p21 q21 ¼ : 2p1 hm h2m p21 ðp1 hm Þ2
ð3:8Þ
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
τ ,q
can be verified, Serrano and Olalla [7]:
M = Stress ay Boundary 1 T = Contact point with H & B envelope
IðrÞ þ c ¼ constant:
µ1 Pole
ε
t1
α σIII s1
ρ1
M
Dir σI 1
µ1
Iðr1 Þ þ C1 ¼ Iðr2 Þ þ C2 ¼ constant:
q1
2ε
ψ1
p
1
σI
ð3:13Þ
This relationship enables one to transfer the stress state from Boundary 1 to Boundary 2, because:
T
α cos hm
839
σ,p
ð3:14Þ
The inclination of the main major stress at Boundary 2, (c2 ), is zero; c2 ¼ 0; in view of the fact that the pile tip load is always vertical. The preceding equation allows r2 to be obtained through the inverse function of Riemann’s Invariant (I 1 ): r2 ¼ I 1 ½Iðr1 Þ þ c1 :
ð3:15Þ
3.4. Boundary 2 Fig. 5. Mohr’s circle for Boundary 1.
This equation links the angle of virtual inclination a; with the overburden pressure hm and with the instantaneous angle of friction for Boundary 1 (r1 ) in view of the fact that p1 and q1 are a function of r1 according to Hoek and Brown’s strength law, Eqs. (2.9). For a given value of hm ; if a is known, it is possible to determine r1 and vice versa (see Fig. 6). (c) Inclination c1 of the main major stress: The angle of inclination for the main major stress for Boundary 1 with respect to the vertical axis (c1 ) is (Fig. 5): p ð3:9Þ C1 ¼ þ a þ e: 2 This relation establishes the direction of the main major stress for Boundary 1 (Dir s1I ). It can also be deduced from Fig. 5 that angle e is expressed by t1 tan 2e ¼ : ð3:10Þ p1 s 1 In view of this, angle e and thus angle c1 can be expressed as a function of angle a and therefore as a function of angle r1 according to Eqs. (2.9), (2.10) and (3.8). The main major stress s1I ; forms angle 7m1 with the directions of the characteristic lines of the stress field, where: p r m1 ¼ 1 : ð3:11Þ 4 2
Once the angle of friction for Boundary 2 (r2 ) is known, Lambe’s variables that define Mohr’s circle can be obtained: p2 ¼
cotan2 r2 z; 2
ð3:16aÞ
q2 ¼
1 sin r2 : sin r2
ð3:16bÞ
The dimensionless vertical stress exerted upon Boundary 2, s2 ; in this particular case, the main stress, because c2 ¼ 0; is as follows: s2 ¼ p 2 þ q 2 :
ð3:17Þ sh ;
the The vertical stress with pressure dimensions ultimate bearing capacity that is the purpose of the calculations for the entire process is sh s2 ¼ b s2 ¼ bðNb zÞ;
ð3:18Þ
where Nb ¼ s2 þ z ¼ p2 þ q2 þ z:
ð3:19Þ
Nb is a new load coefficient that enables one to establish the ultimate bearing capacity of an embedment ‘‘wall’’. The load coefficient Nb is similar to the Prandtl’s coefficient, generally used in soil mechanics to calculate the ultimate bearing loads according to the Mohr– Coulomb criterion. Its value is N b ¼ s2 þ z ¼
cotan2 r2 1 sin r2 þ : 2 sin r2
ð3:20Þ
3.3. Transferring the invariant
3.5. The embedment ratio
Riemann’s invariant IðrÞ is a function of r; as defined by the following equation:
3.5.1. General case The embedment ratio (n) is (Fig. 1): HR sin a OC sin a OC OB OA ¼ ¼ n¼ : 2 OM 2 OB OA OM B
IðrÞ ¼ 12½cot r þ ln cotanðr=2Þ :
ð3:12Þ
Throughout the length of any characteristic line that surrounds the singular point O, the following relation
ð3:21Þ
840
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846 2
1
0.5
0.25
0.0894
0.04
0.01
Fi(movil)
0.001
Virtual Inclination Angle (α;º )
90 80 70
ζ = 0.001
60 50 40 30 20 10 0 0
10
20
(a)
30
40
50
60
70
80
90
Instantaneous Friction Angle (ρ ( 1; 0)
2
1
0.5
0.2828
0.1
0.05
0.01
0.001
fi(movil)
90
Virtual Inclination Angle (α;º )
80 70
ζ = 0.01
60 50 40 30 20 10 0 0
10
20
(b)
30
40
50
60
70
80
90
Instantaneous Friction Angle (ρ ( 1; º)
2
0.8944
0.5
0.25
0.1
0.05
Fi (movil)
0.001
Virtual Inclination Angle (α;º )
90 80
ζ = 0.1
70 60 50 40 30 20 10 0 0
(c)
10
20
30
40
50
60
70
80
90
Instantaneous Friction Angle (ρ ( 1; º)
Fig. 6. (a–c) Relation between r1 and a angles (z ¼ 0:001; 0.01 and 0.1), for different values of hm.
Each one of the fractions in brackets can be obtained from the equation that defines the characteristic line O0 ABC, which determines the limits of the plastified zone, Serrano and Olalla [7]. In the OBC triangle, the following is verified: OC sin 2m1 2 sin m1 cos m1 ¼ ¼ ; OB sinðm1 þ eÞ sinðm1 þ eÞ
ð3:22Þ
where m1 is the angle that the family of characteristic lines that surround the singular point O for Boundary 1 forms with the direction of the main major stress,
s1I : As has already been pointed out (Eq. (3.11)), its value is p r m1 ¼ 1 : 4 2 The value of e is expressed by Eq. (3.10). In Prandtl’s plastified radial zone, the following holds, Serrano and Olalla [7]: sffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi OB tan r1 1 sin r2 tan r1 sin m2 ¼ ¼ : ð3:23Þ OA tan r2 1 sin r1 tan r2 sin m1
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
Finally, in the plastified zone belonging to the OMA triangle: OA 1 ¼ : ð3:24Þ OM sin m2 Once worked out and simplified, the following holds: sffiffiffiffiffiffiffiffiffiffiffiffiffi cos m1 tan r1 : ð3:25Þ n ¼ sin a sinðm1 þ eÞ tan r2 This expression provides the embedment ratio n; as a function of the angle a of inclination for the virtual plane and the instantaneous internal angles of friction at Boundaries 1 and 2. Fig. 7 have been prepared, in which the relation between the virtual plane angle a and the embedment n; for different overburdens hm ; is shown for each value of the tensile strength coefficient z: A distinction is made between two types of curves in Fig. 7: pffiffiffiffiffi 1. Curves for the minor overburden hm o2 2z: All of them end at a singular point NS with co-ordinates a ¼ p=2; n ¼ nSL2 ðzÞ: pffiffiffiffiffi 2. Curves for a major overburden hm > 2 2z: These curves end on a limit curve nL1 : pThe ffiffiffiffiffi discriminated curve that is obtained for hm ¼ 2 2z ends at the extreme point of validity NG on the limit curve nL1 ; with the following co-ordinates: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 G 1 ð3:26aÞ 1 þ pffiffiffiffiffi; aL2 ¼ tan 2z n ¼ nG L2
nS ¼ L2 : 2
ð3:26bÞ
3.5.2. Analysis of the embedment (a) General considerations: It has been stated that there are basically two types of overburden: minor overburden circle (S) and major overburden circle (G). It has also been shown that for each one of them an embedment ratio limit value nsL2 and nL1 is reached. The way in which the actual depth of the embedment in rock HR affects both of the aforementioned cases will now be examined. pffiffiffiffiffi (b) Minor overburden circle hm o2 2z: The limiting embedment condition nSL2 is reached when Mohr’s circle for Boundary 1 stress coincides with Mohr’s circle for the unconfined compressive strength conditions. Under these circumstances the following is reached: sffiffiffiffiffiffiffiffiffiffiffiffiffi tan r1 ¼ nL2 ðzÞ: nSL2 ¼ cotan m1 ð3:27Þ tan r2 The limiting embedment condition nSL2 is a function that is exclusive to parameter z; which defines the state of the rock mass. Fig. 8 shows this relation nSL2 z for
841
the minor overburden circles where it is mathematically valid. At the limit situation the actual depth at which the pile is socketed into the rock, HRL2 ; is as follows, where B is the pile diameter: HRL2 ¼ nSL2 B:
ð3:28Þ
From a conceptual perspective, piles for which HR > nSL2 B ¼ HRL2 can be regarded as deep foundation piles. When analysing them, the following approach has to be (Fig. 9): (1) The limit condition nL2 is obtained from the value of z; through Fig. 8, and HRL2 is established, where HRL2 ¼ BnSL2 : (2) If HR > HRL2 ; one is faced with a deep foundation pile situation. The value of hm to be considered is as follows: HS gS þ HR HRL2 =2 gR hm ¼ : ð3:29Þ b (3) If the depth at which the piles are socketed into the rock is such that HR oHRL2 ; one is dealing with cases of foundations that are regarded conceptually as being semi-deep. Under these circumstances, the plastic bulb cannot develop to such an extent that it closes up on itself. The value of hm to be considered is the general value of Eq. (3.8) hm ¼
H S gS H R gR þ : b 2b
ð3:30Þ
In summary, there can be two types of pile foundations under minor overburden: deep (DL) and semi-deep (SL) (Fig. 9). pffiffiffiffiffi (c) Major overburden circles hm > 2 2z: In this case, the limiting state nL1 occurs when Mohr’s circle for Boundary 1 is tangent to Mohr’s envelope at the point where the latter cuts the overburden circle (point TL1 ; Fig. 3). The valid point between the two cuts is the one 0 with the greater stress TL1 and not TL1 : That is to say, the following is verified at this point: hm ¼
t2L1 þ s2L1 HðrL1 Þ sL1
ð3:31Þ
from which the following may be obtained sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos mL1 tan rL1 nL1 ¼ sin aL1 ¼ nL1 ðz; hm Þ: sinðmL1 þ eL1 Þ tan r2 ð3:32Þ The limiting embedment nL1 is now a function of hm as well as z: Fig. 8 shows nL2 and nL1 together. As is the case for the minor overburden, the piles can either be deep or semi-deep. The actual depth limit separating both types, for the pile embedment in the
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
842
0.001
0.005 0.25
0.01 1
0.05 nL2S
0.0894 nL2G
S n L2 N S
ζ= 0.001
ζ = 0,01 25
Embedment ratio (n)
25
Embedment ratio (n)
0.1 nL2G
0.5 nL2S
30
30
20
15
0.01 1
0.001 0.282
G
n L2
NG
10
0 0 (a)
15 S
n L2
NS
10 G
n L2
n L1
5
20
NG
5 n L1
0 10 20 30 40 50 60 70 80 Virtual inclination angle ( α )
90
0.001 10
0
10 20
0.01 nL2S
30
40
50
60
70
80
90
Virtual inclination angle ( α )
(b)
0.894
0.1 nL2G
30
ζ = 0.1
Embedment ratio (n)
25
20
15
10 S
n L2 5
NS
G
n L2 NG
0 0 (c)
10 20 30 40 50 60 70 80 Virtual inclination angle ( α )
90
Fig. 7. (a–c) Relation between the virtual angle of inclination a and the embedment ratio n (z ¼ 0:001; 0.01 and 0.1).
rock is HRL1 ¼ nL1 B: Then the following holds (Fig. 10): Deep foundation piles (DH): HR > HRL1 ¼ nL1 B:
Semi-deep foundation piles (SH): ð3:33Þ
HR oHRL1 ¼ nL1 B: However, in this case, the plastic bulb is not completely developed in either of the two circumstances. The overburden hm is too great, and when the boundary is sloping at a greater angle than the limit angle aL1 ; the
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846 0.1
0.01
0.001
0.0001
n(L2) s
843
n(L2) g
30
ζ= 0.001 (n) embedment ratio
25 20
ζ = 0.0001 15
nL2g
nL2s
ζ = 0.01
10 5
ζ = 0.1
nL1
0 0.01
0.1
1
(hm ) parameter
Fig. 8. Limit values for the embedment ratio n as a function of the overburden pressure hm for different values of parameter z:
MAJOR OVERBURDEN : h m> 2 2ζ B
ROCK
α L1
HR
H RL1= n L1B
DEEP PILE AND LARGE OVERBURDEN (DH)
B
ROCK
α < α L1
HR = nB
Fig. 9. Deep foundation and semi-deep foundation pile with minor overburden pressure.
kinematic conditions that can be seen in Fig. 4 are inverted. The design overburden pressure hm to be considered if HR > HRL1 ; is similar to the one for the preceding
SEMI-DEEP PILE AND LARGE OVERBURDEN (SH)
Fig. 10. Deep foundation and semi-deep foundation pile with major overburden pressure.
case: HS gS þ HR HRL1 =2 gR : hm ¼ b
ð3:34Þ
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
844
The value of hm must be obtained through successive iterations because HRL1 is now a function of hm ; as well as z: (d) Limit embedment curves: Fig. 8 shows the curves that give the limit value for the embedment ratio nL1 for different values of the constant z: The expression for this curve is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos m1L tan r1L nL1 ¼ sin a1L ¼ nL1 ðhm ; zÞ: sinðm1L þ e1L Þ tan r2 ð3:35Þ These curves plottedpinffiffiffiffiffico-ordinates (hm ; nL ) end at the point wherep hm ffiffiffiffiffi¼ 2 2z: That is to say, Eq. (3.26) is valid for hm > 2 2z: The geometrical location for all these end points amounts to representing the limit values nG L2 ; for major overburden cases, in view of the fact p that this location ffiffiffiffiffi has been reached through values hm > 2 2z: This limit geometrical location is expressed in the following way: sffiffiffiffiffiffiffiffiffiffiffiffiffi 1 tan r2 G ; ð3:36Þ nL2 ¼ cotan m1 2 tan r1 where (Eqs. (2.9), (2.11) and (3.15)) 1 pffiffiffiffiffi; r1 ¼ sin1 1 þ 2z m1 ¼
p r1 ; 4 2
r2 ¼ I nG L2
1
Eq. (3.25). It can be observed that pffiffiffiffiffithere is a discontinuity of function nL1 for hm ¼ 2 2z: (e) Example of application: If, for example, the parameter z representative for a rock mass is z ¼ 0:01; the limiting condition will be nL1 in Fig. 8 and if hm o0:282 it will bepffiffiffiffiffi nSL2 ; for all the piles whose overburden is hm > 2 2z ¼ 0:282: The four types of pile that are given as a function of the overburden (hm ) and the embedment ratio (n), can be seen in Fig. 11, with the value z ¼ 0:01:
4. Ultimate bearing capacity of a pile tip 4.1. Approach Once the ultimate bearing capacity has been obtained under the plane strain hypothesis, it is necessary to take into account the three-dimensional geometry of the pile. For this, a shape coefficient sb must be applied. What is proposed as a shape coefficient sb is a generalisation and adaptation to the Hoek and Brown criterion conditions, of the shape coefficient developed by De Beer [8] (sq ). It is assumed that the shape factor sb plays the role of the sq where sb ¼ 1 þ tan rm
½Iðr1 Þ þ p ;
ð4:1Þ
and rm is an average angle of friction.
pffiffiffiffiffi G ¼ nG L2 ðzÞ ¼ nL2 ð2 2zÞ:
ð3:37Þ
As has already been demonstrated, the limiting pffiffiffiffiffi embedment condition for hm o2 2z is (Eq. (3.27)) sffiffiffiffiffiffiffiffiffiffiffiffiffi tan r2 S ¼ 2nG nL2 ¼ cotan m1 L2 : tan r1 The limiting embedments nSL2 and nG L2 are only function of one variable z (or hm ), in viewpof ffiffiffiffiffi the fact that both are linked by theprelation h ¼ 2 2z: m ffiffiffiffiffi In summary, for hm o2 2z; nL1pis constant and its ffiffiffiffiffi value is nL1 ¼ nSL2 ; and for hm > 2 2z; nL1 is given by
4.2. Simplified way of obtaining the average angle The average angle of internal friction rm is the same as the angle for a virtual linear failure criterion and that would be jointly verified for the stresses at Boundaries 1 and 2 (see Figs. 1 and 12). Angle y is going to be used, i.e. the slope for the Hoek and Brown failure criterion in Lambe’s variables. The following holds: sin rm ¼ tan ym
ð4:2aÞ
Types of pile (for ζ= 0.01) 30
(n) Embedment Ratio
25
DL
20 15
DH ζ = 0.01
10 5 0 0.01
SL SH 0.1
ζ = 0.01 1
(hm) Parameter
Fig. 11. Different pile types (z ¼ 0:01) function of the embedment ratio (n) and the overburden pressure.
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
845
5. Summary and conclusions
Fig. 12. Concept of average angle of friction.
then 2 p2 p 1 ¼ 2 cotan ym ¼ 2 : sin rm q2 q 1
ð4:2bÞ
Bearing in mind the Hoek and Brown criterion, and taking into consideration Formulae 2.9: p2 p1 1 1 2 ¼ q2 þ q1 þ 2 ¼ þ ð4:2cÞ sin r1 sin r2 q2 q1 then the average angle of friction is established by the following (harmony average for the sines of angles r1 and r2 ); 2 1 1 ¼ þ : ð4:3Þ sin rm sin r1 sin r2 4.3. Result The ultimate bearing capacity at the tip of a pile (shp ), once the dimensionless factor has been dispensed with is shp ¼ bðNb zÞsb
ð4:4Þ
that is to say shp ¼ b NbP ;
Acknowledgements ð4:5Þ
where NbP the so-called ‘‘load factor’’ of the pile is NbP ¼ sb ðNb zÞ:
ð4:6Þ
For practical purposes and to allow for easier comparison with other theories, the ultimate bearing capacity can also be expressed in the following way: snP ¼ b NbP ¼ Ns sc :
m NbP : 8
! Munoz * The authors would like to thank Maria Jesus Mart!ın for her patience throughout the months that she has spent writing up the many different versions that led to this document. We would also like to thank Angel Carretero Avellaneda for the figures he has prepared with Autocad.
ð4:7Þ
Factor Ns incorporates all the parameters involved in this theory, and it is related to NbP through the following equation: Ns ¼
1. When applying the characteristic lines method, the ultimate bearing capacity for a pile tip embedded into the rock was calculated, under the assumption that the rock mass behaves according to the Hoek and Brown [4–6] failure criterion, and also applying the theoretical basis proposed by Meyerhof [1]. 2. The entire calculation procedure was carried out by means of the instantaneous angle of internal friction, r; which enabled to draw up a parametric formula of the failure criterion and the plastic phenomenon. This made it easy to solve the problem numerically. 3. The analysis of the overburden circle with respect to the Hoek and Brown failure criteria in a stress diagram (t; s), reveals that there are two different situations, depending on the relation that exists between the overburden pressure (hm ) and parameter z; which represents the strength and the state of the rock p mass: minor overburden pressure when ffiffiffiffiffi hm p2pffiffiffiffiffi 2z; and major overburden pressure when hm X2 2z: 4. In cases where there is minor overburden pressure, the failure bulb can develop completely (see Fig. 9). In cases where there is major overburden pressure, the failure bulb cannot develop completely because the overburden pressure prevents this from happening, inverting the failure kinematics (see Fig. 10). 5. The existence of two pile types is shown: deep foundation piles and semi-deep ones. In the former type, the failure bulb is completely contained within the rock mass. In the latter, the bulb reaches the outer surface of the rock. 6. In summary, four different types of situations can arise: deep piles with a minor overburden (DL) and major overburden (DH), and semi-deep piles with minor overburden (SL) and major overburden (SH).
ð4:8Þ
References [1] Meyerhof GG. The ultimate bearing capacity of foundations. G!eotechnique, London 1951;II(4):301–21. [2] Serrano A, Olalla C. Ultimate bearing capacity of an anisotropic rock mass, part I: basic modes of failure. Int J Rock Mech Min Sci 1998;35(3):301–24.
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[3] Serrano A, Olalla C. Ultimate bearing capacity of an anisotropic rock mass, part II: procedure for its determination. Int J Rock Mech Min Sci 1998;35(3):325–48. [4] Hoek E, Brown ET. Underground excavation in rock. London: The Institution of Mining and Metallurgy, 1980. [5] Hoek E, Brown ET. The Hoek–Brown failure criterion—a 1988 update. Proceedings of the Canadian Rock Mechanics Symposium, University of Toronto, Ontario, Canad!a, 1988. p. 31–8.
[6] Hoek E, Brown ET. Practical estimates of rock mass strength. Int J Rock Mech Min Sci 1997;34:1165–86. [7] Serrano A, Olalla C. Ultimate bearing capacity of rock masses. Int J Rock Mech Min Sci 1994;31:93–106. [8] de Beer EE. Experimental determination of the shape factors and the bearing capacity factors of sand. G!eotechnique 1970;20: 387–411.
Ultimate bearing capacity at the tip of a pile in rock—part 1: theory A. Serranoa, C. Olallab,* b
a ETSICCP, Universidad Polit!ecnica, Madrid, Spain Laboratorio de Geotecnia, Alfonso XII, 3 y 5, Cedex-MOPTMA, 28014 Madrid, Spain
Accepted 30 April 2002
Abstract This paper proposes a method for calculating the ultimate bearing capacity at the tip of a pile that is embedded in rock, according to the theory of plasticity. The non-linear failure criterion of Hoek and Brown (Underground excavation in rock, The Institution of Mining and Metallurgy, London, 1980) is used. The plastified area is analysed as a two-dimensional medium using the characteristic lines method generalised for non-linear failure criteria. The embedment is simulated using the hypothesis proposed by Meyerhof for piles in soils. The results obtained through the calculations are corrected by applying a shape factor in order to take into account the three-dimensional nature of the pile. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Pile tip; Bearing capacity; Failure criterion; Plasticity; Rock mechanics
1. Introduction The ultimate bearing capacity at the tip of a pile embedded into the ground is calculated using perfect plasticity theory. The case of a wall in plane strain is analysed. Then a shape factor is used to establish the ultimate bearing capacity for a pile. The pile, whose diameter is B; is embedded in a rock medium down to a depth of HR : The embedment ratio n ¼ HR =B (Fig. 1) is incorporated using the same theoretical basis as Meyerhof [1]. The embedment is replaced by a virtual ground surface sloping at angle a: The Meyerhof basis is considered to be one of the most reasonable of all the possible options. In classical soil mechanics, the Meyerhof hypothesis has been used on many occasions with a linear failure criterion. It is generally the case in practice that the pile is embedded only a few diameters, but with the theory that is developed here, the theoretical effect of greater embedment ratio values will be analysed. The weight of the triangle of rock above (the overburden pressure) is exerted on the virtual surface, together with the potential external loads that rest upon it. It is assumed that this weight is uniformly distributed throughout the virtual surface. *Corresponding author. Tel.: +34-91-335-7300; fax: +34-91-3357322. E-mail address: [email protected] (C. Olalla).
The entire numerical analysis is carried out in a dimensionless way. This amounts to assuming that the stress unit is the so-called ‘‘pressure modulus’’, or ‘‘strength modulus’’, b obtained from the Hoek and Brown failure criterion, which will be defined later on in this paper. So, all the stresses (s1 ; s3 ; p; qy) are dimensionless. That is to say, they are divided by b: The specific weight unit is the specific weight of the rock formation, gR : Working with dimensionless variables, the mathematical development is simplified and it makes the calculations universal in nature. The different variables and their respective units are retrieved at the end of the development.
2. Basic hypotheses and failure criterion 2.1. Basic hypotheses The following basic hypotheses are used: 1. The rock mass is a homogeneous and isotropic medium. The isotropy hypothesis can be treated in a similar manner as that achieved by Serrano and Olalla [2,3] for surface foundations overlying anisotropic blocks. 2. The rock medium fails when the stress reaches Hoek and Brown’s strength criterion. It is assumed that the
1365-1609/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 5 - 1 6 0 9 ( 0 2 ) 0 0 0 5 2 - 7
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Nomenclature B diameter of the pile (m) GSI geomechanical index by Hoek et al. (1997) HR ; HS depth that the pile is embedded into the ground (m); thickness of the soil (m) h hm ¼ m average overburden load in ground, dimenb sionless I Riemann’s invariant i1 angle of inclination of the loads for Boundary 1 m; s parameters of the Hoek and Brown criterion (1980) m0 Hoek and Brown parameter m for sound rock Nq ; Nc Prandtl’s load coefficients NbP ; Nb load factor for the ultimate bearing capacity of the pile; for the wall Ns ; Ns =m0 load factor that when multiplied by the unconfined compressive strength provides the ultimate bearing capacity of the pile; load factor rationalised in such a way that when it is multiplied by the unconfined compressive strength it provides the ultimate bearing capacity of the pile n ¼ HR =B; nL1 ; nL2 embedment ratio; embedment ratio limit value 1; embedment ratio limit value 2 p s1 þ s3 Lambe’s variable (dimensionless) p¼ ¼ b 2b Lambe’s variables for Boundary 1 p1 ; q1 ; p 2 ; q2 (dimensionless); for Boundary 2 (dimensionless) p s1 s3 Lambe’s variable (dimensionless) q¼ ¼ b 2b RMR Bieniawski’s geomechanical index s1 ; t 1 ; s2 ; t 2 normal stress and shear stress exerted on Boundary 1 (dimensionless); on Boundary 2 (dimensionless) sq ; sb shape factors
rock medium does not undergo strain hardening as a result of the deformation process. 3. The failure zone is obtained using the characteristic lines method. This theory was developed by Sokolovski with a linear failure criterion in the 1950s. 4. The characteristic lines for the stress field and the characteristic lines for the deformation increase field are the same. That is to say, it is assumed that the material is coaxial and has associated dilatancy. 5. The rock mass is weightless. This hypothesis does not constitute a major limitation, because the weight of the material moved during the failure process is very small when compared to the strength forces that are created on the rock failure surface.
a
angle of inclination for the assumed virtual failure surface m sc strength modulus for the rock mass. b¼ 8 Parameter that represents the Hoek and Brown criterion (kN/m2) e angle that forms the main major stress with the inclination of the ground gR ; gS specific weight of the rock; of the soil (kN/m3) m1 ; m2 angles that form families of characteristic lines 1 and 2, respectively, with the ground r; r1 ; r2 ; rm instantaneous angle of internal friction (general; for Boundary 1; for Boundary 2; average) s main major stress (dimensionless) sI ¼ I b s sIII ¼ III main minor stress (dimensionless) b unconfined compressive strength of the sc ; scm : scm intact rock (kN/m2); unconfined compressive strength of the rock mass (dimensionless); with dimensions (kN/m2) 8s tensile strength coefficient of the rock mass. z¼ 2 m Parameter that represents the Hoek and Brown criterion sv vertical load exerted on Boundary 1 (dimensionless) sh ultimate bearing capacity under two-dimensionality hypothesis (kN/m2) shp ultimate bearing capacity of the pile (kN/m2) t; sn shear stress and normal stress for Mohr’s envelope (dimensionless) c1 ; c2 angle of inclination of the main major stress for Boundary 1 (with respect to the vertical axis); angle of inclination of the main major stress for Boundary 2 (with respect to the vertical axis)
2.2. Failure criterion 2.2.1. The Hoek and Brown criterion (1980 and later amendments) The Hoek and Brown criterion [4–6] is as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sI sIII s ¼ m III þ s; ð2:1Þ sc sc where sI and sIII are the major and minor failure stresses, sc is the unconfined compressive strength of the intact rock, m and s are Hoek and Brown’s constants, which depend on the type of the rock and its fractured and weathered characteristics.
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
Hs
Soil
835
B
BOUNDARY 1
D
αv
α C
HR
Rock B O' M
O
A BOUNDARY 2
Fig. 1. Sketch of assumed failure.
This criterion is based on empirical concepts. It successfully simulates the main aspects that appear in rock mass failure, such as non-linearity with the stress levels; the dependence on the type of rock and the state of the mass; the ratio between the compression strength and the tensile strength; the way in which the angle of friction becomes lower as the confining stress increases, etc. Parameters m and s can be obtained using different procedures. However, the one that is probably most advisable involves obtaining them from the type of rock (m0 ) and the geomechanical index RMR, or the GSI by Hoek and Brown (1997) [6], using the following formulae: RMR 100 ; ð2:2Þ m ¼ m0 exp a RMR 100 ; ð2:3Þ b where a ¼ 28 and b ¼ 9; when the natural rock medium has not been artificially disturbed, e.g. through the use of explosives, Hoek and Brown [5]. s ¼ exp
2.2.2. Parametric formulae The above failure criterion formula by Hoek and Brown can be simplified and rationalised by using Lambe’s variables p ¼ ðsI þ sIII Þ=2 and q ¼ ðsI sIII Þ=2; as formulated by Serrano and Olalla [7], to give the following formula: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q p ¼ 2 þz þ11 ð2:4Þ b b which in dimensionless form with a pressure modulus b ðp ¼ p =b and q ¼ q =bÞ; becomes:
qþ1¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðp þ zÞ þ 1;
ð2:5Þ
where parameters b and z; which establish the rock strength, are based directly on Hoek and Brown’s m and s parameters as follows: m sc ; b¼ 8 8s z ¼ 2: ð2:6Þ m The dimensional parameter b serves to scale the failure criterion and all the stress variables. It is referred to as the ‘‘strength modulus’’. Parameter z represents the relative quality and strength of the rock mass. The type of rock and the extent to which it is fractured and disturbed are inherent to this parameter. In view of the fact that it indicates the contact point of Hoek and Brown’s envelope, for coordinates (t; s), at the axis of s; it measures the ‘‘dimensionless isotropic tensile strength’’ of the rock mass (q ¼ 0), so it could be referred to as the ‘‘tensile strength coefficient of the rock mass’’. From a rational perspective, the failure criterion is expressed in the following way: 2ðp þ zÞ ¼ q2 þ 2q
ð2:7Þ
which is Hoek and Brown’s strength law expressed in its most general and simplified form. The instantaneous angle of friction (r) is expressed using the following formula [7]: sin r ¼
dq 1 ¼ : dp 1 þ q
ð2:8Þ
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A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
Depending on the instantaneous angle of friction, Lambe’s variables (dimensionless), which follow Hoek and Brown’s criterion, are as follows: 1 sin r ; q¼ sin r cotan2 r p¼ z: ð2:9Þ 2 Mohr’s envelope for the failure circles, dimensionless, is expressed as a function of parameter r by means of the following equations: t ¼ q cos r; sn ¼ p q sin r:
α ε ψ α I ,1
α
1
h
α
C
V
α
II ,1
Ellipse of stress distribution (Boundary 1)
1 B O' 2
O
A
ð2:10Þ
When the failure criterion takes this form, the instantaneous angle of friction is expressed in the following way: dt tan r ¼ : ð2:11Þ dsn Note: As pointed out earlier, when this procedure is being mathematically developed, the stress variables (s; t; p; q; etc.) are used in dimensionless form. That is to say, divided by parameter b; they are indicated throughout this document without the asterisk. Thus for example, the unconfined p compressive strength value of the ffiffi rock mass ðscm ; scm ¼ sc sÞ in dimensionless form (scm ) is pffiffiffiffiffi scm ¼ scm =b ¼ 2 2z: ð2:12Þ
3. Calculation procedure 3.1. Introduction The calculation method is similar to the one used by Serrano and Olalla [7] to find the ultimate bearing capacity of surface foundations. The load coefficients Nb are obtained as a function of the different embedment ratios (n) and different overburden pressures ‘‘hm ’’. This amounts to a generalisation and an extension of the traditional load coefficients Nq and Nc by Prandtl. The ultimate bearing capacity sh under the twodimensional hypothesis matches the surface load case, normal to the plane of action (Boundary 2), with the exterior surface ascending with an angle a (Boundary 1), and a load (s1 ) for Boundary 1 with angle a sloping in a destabilising direction (Fig. 2). The starting point is an overburden pressure hm and an angle a that is known before the analysis is initiated. This angle is only valid for a certain embedment ratio (n) given by the relation (Fig. 1): HR DO n¼ ¼ : ð3:1Þ B OO0 The embedment ratio n is established later as a function of the instantaneous angles of friction r1 and
α
I ,2
Ellipse of stress distribution (Boundary 2)
Fig. 2. Failure wedge: force diagram and stress ellipses.
r2 for Boundaries 1 and 2, respectively (see Section 3.5), and the virtual angle a: The calculation procedure is divided into three stages: Stage 1: Mohr’s circle (passive) is established for Boundary 1, by knowing the load sv that is exerted on this boundary (see Section 3.2). Stage 2: The Riemann’s invariant Iðr1 Þ is established for Boundary 1 and it is transferred to Boundary 2 (see Section 3.3), which makes it possible to obtain the invariant Iðr2 Þ for Boundary 2. Stage 3: The invariant for Boundary 2, Iðr2 Þ; is used to determine Mohr’s circle in this Boundary 2, and this is used, in turn, to find the ultimate bearing capacity of the wall sh (see Section 3.4). In summary, this procedure is first used to obtain the ultimate bearing capacity sh of a wall embedded in rock when the overburden pressure is hm and the embedment ratio is n: The stress conditions for the whole plastified zone and the failure surface can be obtained as a byproduct of the calculation process. Subsequently, once the ultimate bearing capacity of the wall is known, i.e., under the plane strain hypotheses, the ultimate bearing capacity of the pile (shp ) is obtained by multiplying it by a shape factor sb ; which depends on the average instantaneous angle of friction (rm ), which is then established through r1 and r2 : 3.2. Boundary 1 3.2.1. External load Weight W of the OCD triangle (Fig. 1): W ¼ 12HR DC gR :
ð3:2aÞ
Vertical pressure (sv ) exerted on OC: sv ¼
W gR HR cos a : ¼ 2 OC
ð3:2bÞ
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
gR HR cos a : 2b
*
ð3:2cÞ
*
By way of simplification, the entire process is going to be calculated with the dimensionless parameter hm ; which stands for the average overburden pressure that is exerted upon the virtual surface OC: hm ¼
gR H R : 2b
In the first case (circles shown in Fig. 3), all the positions on the overburden circle are possible for the stresses on Boundary 1; it is possible to proceed in an ongoing way, depending on the embedment, from a ¼ 0 to p=2: The latter is the outermost position, where the failure bulb has closed upon the pile. In the second case, when a increases above a ¼ 0; an outermost position aL1 is reached (point TL1 in Fig. 3). The positions lying between TL1 and T’L1 cannot exist, because they are for strengths that exceed the failure criterion. However, the positions lying between T’L1 ; and 0, which are for potential stresses, stand for states that are physically unacceptable (Mohr’s circles with tensile strengths and active pressure state on Boundary 1). The cases of minor overburden circles are the most interesting from a practical viewpoint, because they coincide with most actual situations. The major overburden circles are consistent with extreme situations in which the rock masses are extremely fractured and the piles are deeply embedded. pffiffiffiffiffi The discriminated case with hm ¼ 2 2z needs a special study, which will be carried out later on in this paper.
ð3:3Þ
In the most general case, there is a ground pressure Hs gs exerted on the rock surface. The parameter hm to be considered (average overburden exerted upon OC) thus takes the following value: hm ¼
H R gR H s gs þ : 2b b
ð3:4Þ
The vertical stress sv exerted upon Boundary 1 is sv ¼ hm cos a:
pffiffiffiffiffi 2zÞ: pffiffiffiffiffi S circles for minor overburden pressure hm o2 p2z ffiffiffiffiffi; G circles for major overburden pressure hm > 2 2z:
rock mass expressed in dimensionless form ð2
Dimensionless vertical pressure sv exerted on OC: sv ¼
ð3:5Þ
The stress components sv ; in the direction of the plane that represents Boundary 1 (t1 ) and the one that runs perpendicular to it (s1 ), are as follows: s1 ¼ sv cos a ¼ hm cos2 a;
ð3:6aÞ
t1 ¼ sv sin a ¼ hm sin a cos a:
ð3:6bÞ
3.2.2. Mohr’s circle for Boundary 1 (a) Bases: The overburden pressure sv exerted upon a virtual surface OC and the stresses ‘‘f ’’ to which the failure surface is subjected balance the ultimate bearing capacity sh (Fig. 4). If this balance is broken, i.e. when failure occurs (Fig. 4), the velocity field is, in the case of the figure, essentially a counterclockwise rotational movement around the singular point 0 (maximum load).
In a diagram t; sn (see Fig. 3) the points that stand for the stress exerted upon Boundary 1 are located on a circle whose diameter is hm ; which will be referred to as the ‘‘overburden circle’’. As far as the Hoek and Brown strength criterion is concerned, there could be two types of overburden pressure, depending on whether hm is greater or smaller than the unconfined compressive strength scm of the
τ TL1 Hoek & Brown criterion
G TL2
P
TL1 '
S P
α
α L1
* σcm = σcm /β = 2
8 2ζ = m
t1
σ
t1 s1
O
837
hm
σcm
s1
hm
s ; Uniaxial Compressive Strength of the rock mass
Fig. 3. Overburden circles: major overburden circle (G). Minor overburden circle (S). Mohr’s circle for the unconfined compression (sc ).
838
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
σv
C
α
A
O
f B
(a) Hoek & Brown criterion
τ
P Stress at the Boundary
Overburden Stress circle
q
ρ
M
A
(b)
t
O
σ
G
α
hm
s
τ Hoek & Brown criterion T
Stress at the Boundary
P Overburden Stress circle
ρ
M
α (c)
O
t s
A
q
σ
S hm
Fig. 4. Boundary 1: (a) kinematics of failure, (b) major overburden circle, (c) minor overburden circle.
When the same pressure is exerted upon Boundary 1 (point M on the overburden circle) two Mohr’s circles can be plotted tangent to the strength curve: the minimum or active circle (A), and the maximum or passive circle (P) (Fig. 4). That figure shows the two overburden circle possibilities: major and minor. Mohr’s maximum circle (P) for Boundary 1 matches the counterclockwise rotation kinematics. That is to say, the stresses are in a passive state in relation to this boundary. (b) Angle a on the virtual plane and instantaneous angle of friction r1 : The following equation holds for Mohr’s
passive, circle (see Fig. 5) ðp1 s1 Þ2 þ t21 q21 ¼ 0;
ð3:7Þ
where p1 and q1 are variables of Lambe that verifies the Hoek and Brown criterion expressed either by formula (2.7) or by the parametric Eqs. (2.9), depending on the instantaneous angle of friction (r). Taking into account Eqs. (3.6) and (3.7), the angle a for virtual boundary 1 is expressed in the following way cos a ¼
p21 q21 p21 q21 ¼ : 2p1 hm h2m p21 ðp1 hm Þ2
ð3:8Þ
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
τ ,q
can be verified, Serrano and Olalla [7]:
M = Stress ay Boundary 1 T = Contact point with H & B envelope
IðrÞ þ c ¼ constant:
µ1 Pole
ε
t1
α σIII s1
ρ1
M
Dir σI 1
µ1
Iðr1 Þ þ C1 ¼ Iðr2 Þ þ C2 ¼ constant:
q1
2ε
ψ1
p
1
σI
ð3:13Þ
This relationship enables one to transfer the stress state from Boundary 1 to Boundary 2, because:
T
α cos hm
839
σ,p
ð3:14Þ
The inclination of the main major stress at Boundary 2, (c2 ), is zero; c2 ¼ 0; in view of the fact that the pile tip load is always vertical. The preceding equation allows r2 to be obtained through the inverse function of Riemann’s Invariant (I 1 ): r2 ¼ I 1 ½Iðr1 Þ þ c1 :
ð3:15Þ
3.4. Boundary 2 Fig. 5. Mohr’s circle for Boundary 1.
This equation links the angle of virtual inclination a; with the overburden pressure hm and with the instantaneous angle of friction for Boundary 1 (r1 ) in view of the fact that p1 and q1 are a function of r1 according to Hoek and Brown’s strength law, Eqs. (2.9). For a given value of hm ; if a is known, it is possible to determine r1 and vice versa (see Fig. 6). (c) Inclination c1 of the main major stress: The angle of inclination for the main major stress for Boundary 1 with respect to the vertical axis (c1 ) is (Fig. 5): p ð3:9Þ C1 ¼ þ a þ e: 2 This relation establishes the direction of the main major stress for Boundary 1 (Dir s1I ). It can also be deduced from Fig. 5 that angle e is expressed by t1 tan 2e ¼ : ð3:10Þ p1 s 1 In view of this, angle e and thus angle c1 can be expressed as a function of angle a and therefore as a function of angle r1 according to Eqs. (2.9), (2.10) and (3.8). The main major stress s1I ; forms angle 7m1 with the directions of the characteristic lines of the stress field, where: p r m1 ¼ 1 : ð3:11Þ 4 2
Once the angle of friction for Boundary 2 (r2 ) is known, Lambe’s variables that define Mohr’s circle can be obtained: p2 ¼
cotan2 r2 z; 2
ð3:16aÞ
q2 ¼
1 sin r2 : sin r2
ð3:16bÞ
The dimensionless vertical stress exerted upon Boundary 2, s2 ; in this particular case, the main stress, because c2 ¼ 0; is as follows: s2 ¼ p 2 þ q 2 :
ð3:17Þ sh ;
the The vertical stress with pressure dimensions ultimate bearing capacity that is the purpose of the calculations for the entire process is sh s2 ¼ b s2 ¼ bðNb zÞ;
ð3:18Þ
where Nb ¼ s2 þ z ¼ p2 þ q2 þ z:
ð3:19Þ
Nb is a new load coefficient that enables one to establish the ultimate bearing capacity of an embedment ‘‘wall’’. The load coefficient Nb is similar to the Prandtl’s coefficient, generally used in soil mechanics to calculate the ultimate bearing loads according to the Mohr– Coulomb criterion. Its value is N b ¼ s2 þ z ¼
cotan2 r2 1 sin r2 þ : 2 sin r2
ð3:20Þ
3.3. Transferring the invariant
3.5. The embedment ratio
Riemann’s invariant IðrÞ is a function of r; as defined by the following equation:
3.5.1. General case The embedment ratio (n) is (Fig. 1): HR sin a OC sin a OC OB OA ¼ ¼ n¼ : 2 OM 2 OB OA OM B
IðrÞ ¼ 12½cot r þ ln cotanðr=2Þ :
ð3:12Þ
Throughout the length of any characteristic line that surrounds the singular point O, the following relation
ð3:21Þ
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A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846 2
1
0.5
0.25
0.0894
0.04
0.01
Fi(movil)
0.001
Virtual Inclination Angle (α;º )
90 80 70
ζ = 0.001
60 50 40 30 20 10 0 0
10
20
(a)
30
40
50
60
70
80
90
Instantaneous Friction Angle (ρ ( 1; 0)
2
1
0.5
0.2828
0.1
0.05
0.01
0.001
fi(movil)
90
Virtual Inclination Angle (α;º )
80 70
ζ = 0.01
60 50 40 30 20 10 0 0
10
20
(b)
30
40
50
60
70
80
90
Instantaneous Friction Angle (ρ ( 1; º)
2
0.8944
0.5
0.25
0.1
0.05
Fi (movil)
0.001
Virtual Inclination Angle (α;º )
90 80
ζ = 0.1
70 60 50 40 30 20 10 0 0
(c)
10
20
30
40
50
60
70
80
90
Instantaneous Friction Angle (ρ ( 1; º)
Fig. 6. (a–c) Relation between r1 and a angles (z ¼ 0:001; 0.01 and 0.1), for different values of hm.
Each one of the fractions in brackets can be obtained from the equation that defines the characteristic line O0 ABC, which determines the limits of the plastified zone, Serrano and Olalla [7]. In the OBC triangle, the following is verified: OC sin 2m1 2 sin m1 cos m1 ¼ ¼ ; OB sinðm1 þ eÞ sinðm1 þ eÞ
ð3:22Þ
where m1 is the angle that the family of characteristic lines that surround the singular point O for Boundary 1 forms with the direction of the main major stress,
s1I : As has already been pointed out (Eq. (3.11)), its value is p r m1 ¼ 1 : 4 2 The value of e is expressed by Eq. (3.10). In Prandtl’s plastified radial zone, the following holds, Serrano and Olalla [7]: sffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi OB tan r1 1 sin r2 tan r1 sin m2 ¼ ¼ : ð3:23Þ OA tan r2 1 sin r1 tan r2 sin m1
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
Finally, in the plastified zone belonging to the OMA triangle: OA 1 ¼ : ð3:24Þ OM sin m2 Once worked out and simplified, the following holds: sffiffiffiffiffiffiffiffiffiffiffiffiffi cos m1 tan r1 : ð3:25Þ n ¼ sin a sinðm1 þ eÞ tan r2 This expression provides the embedment ratio n; as a function of the angle a of inclination for the virtual plane and the instantaneous internal angles of friction at Boundaries 1 and 2. Fig. 7 have been prepared, in which the relation between the virtual plane angle a and the embedment n; for different overburdens hm ; is shown for each value of the tensile strength coefficient z: A distinction is made between two types of curves in Fig. 7: pffiffiffiffiffi 1. Curves for the minor overburden hm o2 2z: All of them end at a singular point NS with co-ordinates a ¼ p=2; n ¼ nSL2 ðzÞ: pffiffiffiffiffi 2. Curves for a major overburden hm > 2 2z: These curves end on a limit curve nL1 : pThe ffiffiffiffiffi discriminated curve that is obtained for hm ¼ 2 2z ends at the extreme point of validity NG on the limit curve nL1 ; with the following co-ordinates: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 G 1 ð3:26aÞ 1 þ pffiffiffiffiffi; aL2 ¼ tan 2z n ¼ nG L2
nS ¼ L2 : 2
ð3:26bÞ
3.5.2. Analysis of the embedment (a) General considerations: It has been stated that there are basically two types of overburden: minor overburden circle (S) and major overburden circle (G). It has also been shown that for each one of them an embedment ratio limit value nsL2 and nL1 is reached. The way in which the actual depth of the embedment in rock HR affects both of the aforementioned cases will now be examined. pffiffiffiffiffi (b) Minor overburden circle hm o2 2z: The limiting embedment condition nSL2 is reached when Mohr’s circle for Boundary 1 stress coincides with Mohr’s circle for the unconfined compressive strength conditions. Under these circumstances the following is reached: sffiffiffiffiffiffiffiffiffiffiffiffiffi tan r1 ¼ nL2 ðzÞ: nSL2 ¼ cotan m1 ð3:27Þ tan r2 The limiting embedment condition nSL2 is a function that is exclusive to parameter z; which defines the state of the rock mass. Fig. 8 shows this relation nSL2 z for
841
the minor overburden circles where it is mathematically valid. At the limit situation the actual depth at which the pile is socketed into the rock, HRL2 ; is as follows, where B is the pile diameter: HRL2 ¼ nSL2 B:
ð3:28Þ
From a conceptual perspective, piles for which HR > nSL2 B ¼ HRL2 can be regarded as deep foundation piles. When analysing them, the following approach has to be (Fig. 9): (1) The limit condition nL2 is obtained from the value of z; through Fig. 8, and HRL2 is established, where HRL2 ¼ BnSL2 : (2) If HR > HRL2 ; one is faced with a deep foundation pile situation. The value of hm to be considered is as follows: HS gS þ HR HRL2 =2 gR hm ¼ : ð3:29Þ b (3) If the depth at which the piles are socketed into the rock is such that HR oHRL2 ; one is dealing with cases of foundations that are regarded conceptually as being semi-deep. Under these circumstances, the plastic bulb cannot develop to such an extent that it closes up on itself. The value of hm to be considered is the general value of Eq. (3.8) hm ¼
H S gS H R gR þ : b 2b
ð3:30Þ
In summary, there can be two types of pile foundations under minor overburden: deep (DL) and semi-deep (SL) (Fig. 9). pffiffiffiffiffi (c) Major overburden circles hm > 2 2z: In this case, the limiting state nL1 occurs when Mohr’s circle for Boundary 1 is tangent to Mohr’s envelope at the point where the latter cuts the overburden circle (point TL1 ; Fig. 3). The valid point between the two cuts is the one 0 with the greater stress TL1 and not TL1 : That is to say, the following is verified at this point: hm ¼
t2L1 þ s2L1 HðrL1 Þ sL1
ð3:31Þ
from which the following may be obtained sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos mL1 tan rL1 nL1 ¼ sin aL1 ¼ nL1 ðz; hm Þ: sinðmL1 þ eL1 Þ tan r2 ð3:32Þ The limiting embedment nL1 is now a function of hm as well as z: Fig. 8 shows nL2 and nL1 together. As is the case for the minor overburden, the piles can either be deep or semi-deep. The actual depth limit separating both types, for the pile embedment in the
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
842
0.001
0.005 0.25
0.01 1
0.05 nL2S
0.0894 nL2G
S n L2 N S
ζ= 0.001
ζ = 0,01 25
Embedment ratio (n)
25
Embedment ratio (n)
0.1 nL2G
0.5 nL2S
30
30
20
15
0.01 1
0.001 0.282
G
n L2
NG
10
0 0 (a)
15 S
n L2
NS
10 G
n L2
n L1
5
20
NG
5 n L1
0 10 20 30 40 50 60 70 80 Virtual inclination angle ( α )
90
0.001 10
0
10 20
0.01 nL2S
30
40
50
60
70
80
90
Virtual inclination angle ( α )
(b)
0.894
0.1 nL2G
30
ζ = 0.1
Embedment ratio (n)
25
20
15
10 S
n L2 5
NS
G
n L2 NG
0 0 (c)
10 20 30 40 50 60 70 80 Virtual inclination angle ( α )
90
Fig. 7. (a–c) Relation between the virtual angle of inclination a and the embedment ratio n (z ¼ 0:001; 0.01 and 0.1).
rock is HRL1 ¼ nL1 B: Then the following holds (Fig. 10): Deep foundation piles (DH): HR > HRL1 ¼ nL1 B:
Semi-deep foundation piles (SH): ð3:33Þ
HR oHRL1 ¼ nL1 B: However, in this case, the plastic bulb is not completely developed in either of the two circumstances. The overburden hm is too great, and when the boundary is sloping at a greater angle than the limit angle aL1 ; the
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846 0.1
0.01
0.001
0.0001
n(L2) s
843
n(L2) g
30
ζ= 0.001 (n) embedment ratio
25 20
ζ = 0.0001 15
nL2g
nL2s
ζ = 0.01
10 5
ζ = 0.1
nL1
0 0.01
0.1
1
(hm ) parameter
Fig. 8. Limit values for the embedment ratio n as a function of the overburden pressure hm for different values of parameter z:
MAJOR OVERBURDEN : h m> 2 2ζ B
ROCK
α L1
HR
H RL1= n L1B
DEEP PILE AND LARGE OVERBURDEN (DH)
B
ROCK
α < α L1
HR = nB
Fig. 9. Deep foundation and semi-deep foundation pile with minor overburden pressure.
kinematic conditions that can be seen in Fig. 4 are inverted. The design overburden pressure hm to be considered if HR > HRL1 ; is similar to the one for the preceding
SEMI-DEEP PILE AND LARGE OVERBURDEN (SH)
Fig. 10. Deep foundation and semi-deep foundation pile with major overburden pressure.
case: HS gS þ HR HRL1 =2 gR : hm ¼ b
ð3:34Þ
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
844
The value of hm must be obtained through successive iterations because HRL1 is now a function of hm ; as well as z: (d) Limit embedment curves: Fig. 8 shows the curves that give the limit value for the embedment ratio nL1 for different values of the constant z: The expression for this curve is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos m1L tan r1L nL1 ¼ sin a1L ¼ nL1 ðhm ; zÞ: sinðm1L þ e1L Þ tan r2 ð3:35Þ These curves plottedpinffiffiffiffiffico-ordinates (hm ; nL ) end at the point wherep hm ffiffiffiffiffi¼ 2 2z: That is to say, Eq. (3.26) is valid for hm > 2 2z: The geometrical location for all these end points amounts to representing the limit values nG L2 ; for major overburden cases, in view of the fact p that this location ffiffiffiffiffi has been reached through values hm > 2 2z: This limit geometrical location is expressed in the following way: sffiffiffiffiffiffiffiffiffiffiffiffiffi 1 tan r2 G ; ð3:36Þ nL2 ¼ cotan m1 2 tan r1 where (Eqs. (2.9), (2.11) and (3.15)) 1 pffiffiffiffiffi; r1 ¼ sin1 1 þ 2z m1 ¼
p r1 ; 4 2
r2 ¼ I nG L2
1
Eq. (3.25). It can be observed that pffiffiffiffiffithere is a discontinuity of function nL1 for hm ¼ 2 2z: (e) Example of application: If, for example, the parameter z representative for a rock mass is z ¼ 0:01; the limiting condition will be nL1 in Fig. 8 and if hm o0:282 it will bepffiffiffiffiffi nSL2 ; for all the piles whose overburden is hm > 2 2z ¼ 0:282: The four types of pile that are given as a function of the overburden (hm ) and the embedment ratio (n), can be seen in Fig. 11, with the value z ¼ 0:01:
4. Ultimate bearing capacity of a pile tip 4.1. Approach Once the ultimate bearing capacity has been obtained under the plane strain hypothesis, it is necessary to take into account the three-dimensional geometry of the pile. For this, a shape coefficient sb must be applied. What is proposed as a shape coefficient sb is a generalisation and adaptation to the Hoek and Brown criterion conditions, of the shape coefficient developed by De Beer [8] (sq ). It is assumed that the shape factor sb plays the role of the sq where sb ¼ 1 þ tan rm
½Iðr1 Þ þ p ;
ð4:1Þ
and rm is an average angle of friction.
pffiffiffiffiffi G ¼ nG L2 ðzÞ ¼ nL2 ð2 2zÞ:
ð3:37Þ
As has already been demonstrated, the limiting pffiffiffiffiffi embedment condition for hm o2 2z is (Eq. (3.27)) sffiffiffiffiffiffiffiffiffiffiffiffiffi tan r2 S ¼ 2nG nL2 ¼ cotan m1 L2 : tan r1 The limiting embedments nSL2 and nG L2 are only function of one variable z (or hm ), in viewpof ffiffiffiffiffi the fact that both are linked by theprelation h ¼ 2 2z: m ffiffiffiffiffi In summary, for hm o2 2z; nL1pis constant and its ffiffiffiffiffi value is nL1 ¼ nSL2 ; and for hm > 2 2z; nL1 is given by
4.2. Simplified way of obtaining the average angle The average angle of internal friction rm is the same as the angle for a virtual linear failure criterion and that would be jointly verified for the stresses at Boundaries 1 and 2 (see Figs. 1 and 12). Angle y is going to be used, i.e. the slope for the Hoek and Brown failure criterion in Lambe’s variables. The following holds: sin rm ¼ tan ym
ð4:2aÞ
Types of pile (for ζ= 0.01) 30
(n) Embedment Ratio
25
DL
20 15
DH ζ = 0.01
10 5 0 0.01
SL SH 0.1
ζ = 0.01 1
(hm) Parameter
Fig. 11. Different pile types (z ¼ 0:01) function of the embedment ratio (n) and the overburden pressure.
A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
845
5. Summary and conclusions
Fig. 12. Concept of average angle of friction.
then 2 p2 p 1 ¼ 2 cotan ym ¼ 2 : sin rm q2 q 1
ð4:2bÞ
Bearing in mind the Hoek and Brown criterion, and taking into consideration Formulae 2.9: p2 p1 1 1 2 ¼ q2 þ q1 þ 2 ¼ þ ð4:2cÞ sin r1 sin r2 q2 q1 then the average angle of friction is established by the following (harmony average for the sines of angles r1 and r2 ); 2 1 1 ¼ þ : ð4:3Þ sin rm sin r1 sin r2 4.3. Result The ultimate bearing capacity at the tip of a pile (shp ), once the dimensionless factor has been dispensed with is shp ¼ bðNb zÞsb
ð4:4Þ
that is to say shp ¼ b NbP ;
Acknowledgements ð4:5Þ
where NbP the so-called ‘‘load factor’’ of the pile is NbP ¼ sb ðNb zÞ:
ð4:6Þ
For practical purposes and to allow for easier comparison with other theories, the ultimate bearing capacity can also be expressed in the following way: snP ¼ b NbP ¼ Ns sc :
m NbP : 8
! Munoz * The authors would like to thank Maria Jesus Mart!ın for her patience throughout the months that she has spent writing up the many different versions that led to this document. We would also like to thank Angel Carretero Avellaneda for the figures he has prepared with Autocad.
ð4:7Þ
Factor Ns incorporates all the parameters involved in this theory, and it is related to NbP through the following equation: Ns ¼
1. When applying the characteristic lines method, the ultimate bearing capacity for a pile tip embedded into the rock was calculated, under the assumption that the rock mass behaves according to the Hoek and Brown [4–6] failure criterion, and also applying the theoretical basis proposed by Meyerhof [1]. 2. The entire calculation procedure was carried out by means of the instantaneous angle of internal friction, r; which enabled to draw up a parametric formula of the failure criterion and the plastic phenomenon. This made it easy to solve the problem numerically. 3. The analysis of the overburden circle with respect to the Hoek and Brown failure criteria in a stress diagram (t; s), reveals that there are two different situations, depending on the relation that exists between the overburden pressure (hm ) and parameter z; which represents the strength and the state of the rock p mass: minor overburden pressure when ffiffiffiffiffi hm p2pffiffiffiffiffi 2z; and major overburden pressure when hm X2 2z: 4. In cases where there is minor overburden pressure, the failure bulb can develop completely (see Fig. 9). In cases where there is major overburden pressure, the failure bulb cannot develop completely because the overburden pressure prevents this from happening, inverting the failure kinematics (see Fig. 10). 5. The existence of two pile types is shown: deep foundation piles and semi-deep ones. In the former type, the failure bulb is completely contained within the rock mass. In the latter, the bulb reaches the outer surface of the rock. 6. In summary, four different types of situations can arise: deep piles with a minor overburden (DL) and major overburden (DH), and semi-deep piles with minor overburden (SL) and major overburden (SH).
ð4:8Þ
References [1] Meyerhof GG. The ultimate bearing capacity of foundations. G!eotechnique, London 1951;II(4):301–21. [2] Serrano A, Olalla C. Ultimate bearing capacity of an anisotropic rock mass, part I: basic modes of failure. Int J Rock Mech Min Sci 1998;35(3):301–24.
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A. Serrano, C. Olalla / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 833–846
[3] Serrano A, Olalla C. Ultimate bearing capacity of an anisotropic rock mass, part II: procedure for its determination. Int J Rock Mech Min Sci 1998;35(3):325–48. [4] Hoek E, Brown ET. Underground excavation in rock. London: The Institution of Mining and Metallurgy, 1980. [5] Hoek E, Brown ET. The Hoek–Brown failure criterion—a 1988 update. Proceedings of the Canadian Rock Mechanics Symposium, University of Toronto, Ontario, Canad!a, 1988. p. 31–8.
[6] Hoek E, Brown ET. Practical estimates of rock mass strength. Int J Rock Mech Min Sci 1997;34:1165–86. [7] Serrano A, Olalla C. Ultimate bearing capacity of rock masses. Int J Rock Mech Min Sci 1994;31:93–106. [8] de Beer EE. Experimental determination of the shape factors and the bearing capacity factors of sand. G!eotechnique 1970;20: 387–411.