Drilled Shaft in Rock Analysis and Design_Part3

Axial load capacity of drilled shafts in rock

215

In LRFD, the ultimate (factored) axial capacity of a drilled shaft can be calculated using the expression for reinforced concrete columns: (6.1) where is the capacity reduction (resistance) factor=0.75 for spiral columns and 0.70 for horizontally tied columns (ACI, 1995); Qu is the nominal (computed) structural capacity; β is the eccentricity factor=0.85 for spiral columns and 0.80 for tied columns; is the specified minimum concrete strength; Ac is the cross-sectional area of the concrete; fy is the yield strength of the longitudinal reinforcing steel; and As is the crosssectional area of the longitudinal reinforcing steel. The Standard Specifications for Highway Bridges adopted by the American Association of State Highway and Transportation Officials (AASHTO, 1989) stipulates a minimum shaft diameter of 18 inches, with shaft sizing in 6-inch increments. Where the potential for lateral loading is not significant, drilled shafts need to be reinforced for axial loads only. The design of longitudinal and spiral reinforcement should conform to the requirements of reinforced compression members.

Table 6.1 Allowable concrete stresses for drilled shafts (after ASCE, 1993). Uniform axial compression Confined

0.33f′c

Unconfined

0.27f′c

Uniform axial tension

0

Bending (extreme fiber) Compression

0.40f′c

Tension

0

Note: f′c is the specified minimum concrete strength.

6.3 CAPACITY OF DRILLED SHAFTS RELATED TO ROCK Assuming that the shaft itself is strong enough, its load capacity depends on the capacity of the rock to accept without distress the loads transmitted from the shaft. The required area of shaft-rock interface (i.e., the size of drilled shaft) depends on this factor. The ultimate axial load of a drilled shaft related to rock, Qu, consists of the ultimate side shear load, Qus, and the ultimate end bearing load, Qub (see Fig. 6.1): Qu=Qus+Qub (6.2)

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216

The ultimate side shear load and the ultimate end bearing load are respectively calculated as the average side shear resistance multiplied by the shaft side surface area and as the end bearing resistance multiplied by the shaft bottom area, i.e.

Fig. 6.1 Axially loaded drilled shaft. Qus=πBLτmax (6.3) (6.4) where L and B are respectively the length and diameter of the shaft; and τmax and qmax are respectively the average side shear resistance and the end bearing resistance. The ultimate side shear resistance and the end bearing resistance are usually determined based on local experience and building codes, empirical relations, or field load tests. Methods based on local experience and building codes and empirical relations are discussed in this chapter. The methods for conducting field load tests and interpretation of test results will be discussed in Chapter 12. 6.3.1 Side shear resistance The shear resistance mobilized at the shaft-rock interface is affected by many factors. These include the shaft roughness, strength and deformation properties of the concrete

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and the rock mass, geometry of the shaft, and initial stresses in the ground. The effect of shaft roughness is emphasized by most investigators and considered in a number of empirical relations for estimating the side shear resistance. (a) Correlation with SPT N value Standard Penetration Tests (SPT) are often carried out in weak or weathered rock. Table 6.2 shows the measured side shear resistances of drilled shafts and their corresponding SPT N values in weathered sedimentary rocks. It can be seen that the τmax/N ratio is generally smaller than 2.0 except the case reported by Toh et al. (1989). We can also see that the τmax/N ratio tends to decrease as N increases.

Table 6.2 Side shear resistance and SPT N values in weathered sedimentary rock. Rock

SPT N values (blows/0.3 m)

τmax (kPa)

τmax/N (kPa)

Reference

Highly weathered siltstone

230

>195– 226

>0.87– 1.0

Buttling (1986)

Highly weathered siltstone, silty sandstone and shale

100–180

100– 320

1.0–1.8

Chang and Wong (1987)

Very dense clayey/sandy silt to highly weathered siltstone

110–127

80–125 0.63– 1.14

Highly to moderately weathered siltstone

200–375

340

0.9–1.7

Completely to partly weathered interbedded sandstone, siltstone and shale/mudstone

100–150 150–200

– –

1.2–3.7 0.6–2.3

Toh et al. (1989)

Highly to moderately fragmented siltstone/shale

400–1000

300– 800

0.5–0.8

Radhakrishnan and Leung (1989)

Highly weathered sandy shale

150–200

120– 140

0.8–0.7

Moh et al. (1993)

Slightly weathered sandy shale and sandstone

375–430

240– 280

ave. 0.65

Buttling and Lam (1988)

(b) Empirical relations between side shear resistance and unconfined compressive strength of intact rock Empirical relations between the side shear resistance and the unconfined compressive strength of rock have been proposed by many researchers. The form of these empirical relations can be generalized as τmax= ασcβ (6.5)

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218

where τmax is the side shear resistance; σc is the unconfined compressive strength of the intact rock (if the intact rock is stronger than the shaft concrete, σc of the concrete is used); and α and β are empirical factors. The empirical factors proposed by a number of researchers have been summarized by O’Neill et al. (1996) and are shown in Table 6.3. Most of these empirical relations were developed for specific and limited data sets, which may have correlated well with the proposed equations. However, O’Neill et al. (1996) compared the first nine empirical relations listed in Table 6.3 with an international database of 137 pile load tests in intermediate-strength rock and concluded that none of the methods could be considered a satisfactory predictor for the database. Kulhawy and Phoon (1993) developed a relatively extensive load test database for drilled shafts in soil and rock and presented their data both for individual shaft load tests and as site-averaged data. The results are shown in Figures 6.2 and 6.3, in terms of adhesion factor, σc, versus normalized shear strength, cu/pa or σc/2pa (assuming cu≈ σc/2), where pa is atmospheric pressure (≈0.1 MPa). It should be noted that Kulhawy and

Table 6.3 Empirical factors a and β for side shear resistance (modified from O’Neill et al., 1996). Design method

α

β

Horvath and Kenney (1979)

0.21

0.50

Carter and Kulhawy (1988)

0.20

0.50

Williams et al. (1980)

0.44

0.36

Rowe and Armitage (1984)

0.40

0.57

Rosenberg and Journeaux (1976)

0.34

0.51

Reynolds and Kaderbek (1980)

0.30

1.00

Gupton and Logan (1984)

0.20

1.00

Reese and O’Neill (1987)

0.15

1.00

Toh et al. (1989)

0.25

1.00

Meigh and Wolshi (1979)

0.22

0.60

Horvath (1982)

0.20–0.30

0.50

Phoon (1993) defined αc as the ratio of the side shear resistance τmax to the undrained shear strength cu. Understandably, the results of individual load tests show considerably greater scatter than the site-averaged data. On the basis of the site-averaged data, Kulhawy and Phoon (1993) proposed the following relations for drilled shafts in rock: (6.6a) (6.6b)

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(6.6c) Equation (6.6) can be rewritten in a general form as (6.7) This leads to a general expression for the side shear resistance τmax=Ψ[paσc/2pa]−0.5 (6.8) It is very important to note that the empirical relations given in Equations (6.6b) and (6.6c) are bounds to site-averaged data, and do not necessarily represent bounds to individual shaft behavior. The coefficient of determination (r2) is approximately 0.71 for the site-averaged data, but is only 0.46 for the individual data, reflecting the much greater variability of the individual test results (Seidel & Haberfield, 1995).

Fig. 6.2 Adhesion factor αc(=τmax/0.5σc) versus normalized shear strength for site-averaged data (after Kulhawy & Phoon, 1993).

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220

Fig. 6.3 Adhesion factor αc(=τmax/0.5σc) versus normalized shear strength for individual test data (after Kulhawy & Phoon, 1993). (c) Empirical relations considering roughness of shaft wall The roughness of the shaft wall is an important factor controlling the development of side shear resistance. Depending on the type of drilling technique and the hardness of the rock, a drilled shaft will have a certain degree of roughness. Research has shown that the benefits gained from increasing the roughness of a shaft wall can be quite significant, both in terms of peak and residual shear resistance. Studies by Williams et al. (1980) and others showed that smooth-sided shafts exhibit a brittle type of failure, while shafts having an adequate roughness exhibit ductile failure. Williams and Pells (1981) suggested that rough shafts generate a locked-in normal stress such that there is practically no distinguishing difference between peak and residual side shear resistance. Classifications have been developed so that roughness can be quantified. One such classification proposed by Pells et al. (1980) is based on the size and frequency of grooves in the shaft wall (see Table 6.4). Based on this classification, Rowe and Armitage (1987b) proposed the following relation for shafts with different roughness: τmax=0.45(σc)0.5 for shafts with roughness R1, R2 or R3 (6.9a) τmax=0.60(σc)0.5 for shafts with roughness R4 (6.9b) where both τmax and σc are in MPa.

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Horvath et al. (1980) also developed a relation from model shaft behavior using various roughness profiles. They found that as shaft profiles go from smooth to rough, the roughness factor increases significantly, as does the peak side shear resistance. These findings were confirmed in a later study by Horvath et al. (1983), and the following equation was proposed for the roughness factor (RF): (6.10) where hm is the average roughness (asperity) height of the shaft; Lt is the total travel length along the shaft wall profile; R is the nominal radius of the shaft; and L is the nominal length of the shaft (see Fig. 6.4). Using Equation (6.10), the following relation was developed between the side shear resistance and RF: τmax= 0.8σc(RF)0.45 (6.11) Kodikara et al. (1992) developed a rational model for predicting the relationship of τmax to σc based on a specific definition of interface roughness, initial normal stress on the interface and the stiffness of the rock during interface dilation. The parameters needed to define interface roughness in the model are also shown in Figure 6.4. The model accounts for variability in asperity height and angularity, assuming clean, triangular interface discontinuities. Figure 6.5 shows the predicted adhesion factor, α(=τmax/σc), for Melbourne Mudstone with the range of parameters and roughnesses as given in Table 6.5. The adhesion factor is presented as a function of Em/σc, σc/σn and the degree of roughness, where Em is the elastic modulus of the rock mass and σn is the initial normal stress on the shaft-rock interface. It can be seen that the adhesion factor is affected not only by the interface roughness, but also by Em/σc and σc/σn.

Table 6.4 Roughness classes after Pells et al. (1980). Roughness Class

Description

R1

Straight, smooth-sided shaft, grooves or indentation less than 1.00 mm deep

R2

Grooves of depth 1–4 mm, width greater than 2 mm, at spacing 50 to 200 mm.

R3

Grooves of depth 4–10 mm, width greater than 5 mm, at spacing 50 to 200 mm.

R4

Grooves or undulations of depth greater than 10 mm, width greater than 10 mm, at spacing 50 to 200 mm.

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Fig. 6.4 Parameters for defining shaft wall roughness (after Horvath et al., 1980 and Kodikara et al., 1992).

Axial load capacity of drilled shafts in rock

223

Fig. 6.5 Simplified design charts for adhesion factor α(=τmax/σc) for Melbourne Mudstone (after Kodikara et al., 1992).

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224

Table 6.5 Definition of borehole roughness and range of parameters for Melbourne Mudstone (after Kodikara et al., 1992). Range of values for shafts in Melbourne Mudstone Parameter

Smooth

im(degrees)

10–12

isd(degrees) hm(mm)

Medium

Rough 12–17

17–30

2–4

4–6

6–8

1–4

4–20

20–80

hsd/hm

0.35

B(m)

0.5–2.0

σc(MPa)

0.5–10.0

σn(MPa)

50–500

Em(MPa)

50–500

Notes: 1) Refer to Figure 6.4 for the definitions of im, isd, hm and hsi 2) B=diameter of the shaft. 3) σc=unconfined compressive strength of the intact rock. 4) σn=initial normal stress on the shaft-rock interface. 5) Em=deformation modulus of the rock mass.

Seidel and Collingwood (2001) introduced a nondimensional factor called Shaft Resistance Coefficient (SRC) to reflect the influence of shaft roughness and other factors on the shaft side shear resistance. The SRC is defined as follows: (6.12) where hm is the mean roughness height (either assessed directly by estimation or measurement, or computed as the product of asperity length, la, and the sine of the mean asperity angle); B is the shaft diameter; ηc is the construction method reduction factor as shown in Table 6.6; n is the ratio of rock mass modulus to the unconfined compressive strength of the rock (Em/σc), known as the modulus ratio; and ν is the Poisson’s ratio of the rock. Using SRC, Seidel and Collingwood (2001) have created shaft resistance charts as shown in Figures 6.6 and 6.7. These charts are based on results of a parametric study using a computer program called ROCKET. To develop these charts, the intact rock strength parameters were related to the unconfined compressive strength using the HoekBrown strength criteria described in Chapter 4. Mohr-Coulomb strength parameters adopted in the analyses were determined after the method of Hoek (1990) using the unconfined compressive strength of the rock and appropriate values of parameters s and m.

Axial load capacity of drilled shafts in rock

225

(d) Estimation of roughness height of shaft wall Application of the empirical relations considering shaft wall roughness in design requires estimation of likely shaft wall roughness height. A small number of studies have produced actual roughness profiles which enable quantitative analysis. Detailed studies have been carried out into shafts in Melbourne Mudstone (Williams, 1980; Holden, 1984; Kodikara et al., 1992; Baycan, 1996). The results show that shaft wall roughness in this low- to medium-strength argillaceous rock can vary considerably and appears to be influenced by rock discontinuities, drilling techniques, and rate of advance. Shaft wall roughness profiles in medium-strength shale were also recorded by Horvath et al. (1983), but most of their shafts were artificially roughened by grooving. O’Neill & Hassan (1994) and O’Neill et al., (1996) recorded measurements of roughness profiles of shafts in clay shale, argillite and sandstone.

Table 6.6 Indicative construction method reduction factor ηc (after Seidel & Collingwood, 2001). Construction method

ηc

Construction without drilling fluid Best construction practice and high level of construction control

1.0

(e.g., shaft sidewalls free of smear and remoulded rock) Poor construction practice or low-quality construction control (e.g.,

0.3–0.9

smear or remoulded rock present on shaft sidewalls) Construction under bentonite slurry Best construction practice and high level of construction control

0.7–0.9

Poor construction practice or low-quality construction control

0.3–0.6

Construction under polymer slurry Best construction practice and high level of construction control

0.9–1.0

Poor construction practice or low-quality construction control

0.8

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226

Fig. 6.6 Adhesion factor α(=τmax/σc) versus σc (after Seidel & Collingwood, 2001).

Fig. 6.7 Adhesion factor α(=τmax/σc) versus SRC (after Seidel & Collingwood, 2001).

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227

Based on roughness heights back-calculated from load tests on shafts in rock, Seidel and Collingwood (2001) developed the effective roughness height versus the unconfined compressive strength plot as shown in Figure 6.8. The back-calculations were conducted using Equation (6.12) and assuming ηc=1.0. In the case of a shaft for which the concreterock interface is clean and unbounded, the roughness height back-calculated assuming ηc=1.0 should provide a reasonable estimate of the roughness height magnitude. However, if the shaft resistance is adversely influenced by construction procedures, the roughness height would be underestimated if ηc is assumed to be 1. Example 6.1 A drilled shaft of diameter 1.0 m is to be socketed 3.0 meters in rock. The rock properties are as follows: Unconfined compressive strength of intact rock, σc=15.0 MPa Deformation modulus of intact rock, Er=10.6 GPa RQD=76

Determine the side shear resistance.

Fig. 6.8 Effective roughness height versus σc (after Seidel & Collingwood, 2001).

Drilled shafts in rock

228

Solution: Method of Kulhawy and Phoon (1993)—Equations (6.6) to (6.8) Lower bound τmax=1.0[paσc/2]0.5=1.0[0.1×15.0/2]0.5=0.87 MPa Upper bound τmax=3.0[paσc/2]0.5=3.0[0.1×15.0/2]0.5=2.60 MPa Method of Seidel and Collingwood (2001) From Figure 6.8, the mean roughness height hm=1.64 mm (lower bound) and 6.19 mm (upper bound). Using Equation (4.24), the rock mass modulus: αE=0.0231(RQD)−1.32=0.297 Em=αEEr=0.297×10.6=3.15 GPa The modulus ratio n=Em/σc=210. The Poisson’s ratio of the rock is simply assumed to be ν=0.25. Using ηc=1.0, SRC can be obtained from Equation (6.12) as:

From Figure 6.6, the adhesion factor a can be obtained as α=0.102 (lower bound) α=0.225 (upper bound) So the side shear resistance can be obtained as τmax=ασc=0.102×15.0=1.53 MPa (lower bound) τmax=ασc=0.225×15.0=3.37 MPa (upper bound) The results show that the shaft wall roughness (reflected by the roughness height) has a great effect on the side shear resistance. (e) Factors affecting side shear resistance As stated above, the shaft wall roughness, which is an important factor controlling the development of side shear resistance, has been studied extensively. Other factors such as the discontinuities in the rock mass and the shaft geometry have also been studied by some researchers. Williams et al. (1980) suggested that the existence of discontinuities in

Axial load capacity of drilled shafts in rock

229

the rock mass reduces the side shear resistance by reducing the normal stiffness of the rock mass. They developed the following empirical relation that considers the effect of discontinuities on the side shear resistance: τmax=αwβwσc (6.13) where αw is a reduction factor reflecting the strength of the rock, as shown in Figure 6.9; and βw is the ratio of side shear resistance of jointed rock mass to side shear resistance of intact rock. βw is a function of modulus reduction factor, j, as shown in Figure 6.10, in which βw=f(j), j=Em/Er (6.14) where Em is the elastic modulus of the rock mass; and Er is the elastic modulus of the intact rock. When the rock mass is such that the discontinuities are tightly closed and seatns are infrequent, βw is essentially equal to 1.0. Comparing Equation (6.13) with Equation (6.5), it can be seen that αwβw is just the adhesion factor, a, for β=1. Since αw is derived from field test data, the effect of discontinuities is already included in αw. If αw is multiplied by βw which is obtained from laboratory tests (Williams et al., 1980), the effect of discontinuities will be considered twice. So Equation (6.13) may be too conservative. Pabon and Nelson (1993) studied the effect of soft horizontal seams on the behavior of laboratory model shafts. The study included four instrumented model shafts in manufactured rock, three of which have soft seams. They concluded that a soft seam significantly reduces the normal interface stresses generated in the rock layer overlying it. Consequently the side shear resistance of shafts in rock with soft seams is much lower than that of shafts in intact rock. The effect of shaft geometry on side shear resistance was studied by Williams and Pells (1981). They tested 15 shafts in Melbourne Mudstone, with diameters ranging from 335 mm to 1580 mm, and 27 shafts in Hawkesbury Sandstone, with diameters ranging from 64 mm to 710 mm. The results of these tests indicated that the shaft length, L, does not have a discernible effect on the side shear resistance. They argued that the interface dilation creates a locked-in normal stress with the result that the shear displacement behavior exhibits virtually no peak or residual behavior. They also reported that the shaft diameter has a negligible effect on the side shear resistance. On the other hand, tests by Horvath et al. (1983) indicated that the side shear resistance decreases as the shaft diameter increases. Williams and Pells (1981) explained this phenomenon by referring to the theory of expansion of an infinite cylindrical cavity, which suggests that cylinders with smaller diameters develop higher normal stresses for a given absolute value of dilation. However, they offered no physical explanation why the shaft diameter does not affect their own test results.

Drilled shafts in rock

230

Fig. 6.9 Side shear resistance reduction factor αw [Equation (6.13)] (after Williams & Pells, 1981). 6.3.2 End bearing resistance (a) End bearing behavior of drilled shafts The typical bearing capacity failure modes for rock masses depend on discontinuity spacing with respect to foundation width (or diameter), discontinuity orientation, discontinuity condition (open or closed), and rock type. Table 6.7 illustrates typical failure modes according to rock mass conditions (ASCE, 1996). Prototype failure modes may actually consist of a combination of modes. The failure modes shown in Table 6.7 are for foundations with the base at or close to the ground surface. The depth of shaft embedment may change the end bearing failure modes of drilled shafts. As shown in Figure 6.11, when the base of the shaft is at or close to the ground surface, a wedge type of failure is developed and the shaft undergoes both vertical settlement and rotation. When the depth of embedment is greater than twice the diameter of the shaft, a punching type of failure occurs and a truncated conical plug of fractured rock is formed below the base (Williams et al., 1980).

Axial load capacity of drilled shafts in rock

231

Fig. 6.10 Side shear resistance reduction factor βw [Equation (6.13)] (after Williams & Pells, 1981). In a study by Johnston and Choi (1985), stereo photogrammetric techniques were used to study the process of failure of a model pile socketed into simulated rock. As shown in Figure 6.12, the study suggests that failure progresses from initial radial cracking to a fan shaped wedge. These observations were compared to typical load displacement curve where four points are identified as: 1) at the end of elastic deformation; 2) a little before major yielding; 3) a little after major yielding; and 4) failure. (b) End bearing resistance based on local experience and codes Peck et al. (1974) suggested a correlation between the allowable bearing pressure and RQD for footings supported on level surfaces in competent rock (Fig. 6.13). This correlation can be used as a first crude step in determination of the end bearing resistance of drilled shafts in rock. It need be noted that this correlation is intended only for unweathered jointed rock where the discontinuities are generally tight. If the value of allowable pressure exceeds the unconfined compressive strength of intact rock, the allowable pressure is taken as the unconfined compressive strength. In Hong Kong design practice, for large diameter drilled shafts in granitic and volcanic rocks, the allowable end bearing resistance may be used as specified in Table 6.8. The presumptive end bearing resistance values range from 3.0 to 7.5 MPa, depending

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232

on the rock category which is defined in terms of the rock decomposition grade, strength and total core recovery.

Table 6.7 Typical bearing capacity failure modes associated with various rock mass conditions (after ASCE, 1996). Rock mass conditions Joint dip

Joint spacing

Failure Illustration

Mode Brittle rock: Local shear failure caused by localized brittle fracture

N/A

s»B

Ductile rock: General shear failure along well defined failure surfaces

Open joints: Compressive failure of individual rock columns. Near vertical joint set(s)

70° 10 feet apart

100

Very Good

Tightly interlocking, 85 undisturbed rock

Nms(4) A

B

C

D

E

500

95–100

3.8

4.3

5.0

5.2

6.1

100

90–95

1.4

1.6

1.9

2.0

2.3

Axial load capacity of drilled shafts in rock

241

with rough unweathered discontinuities spaced 3 to 10 feet apart Good

Fresh to slightly weathered rock, slightly disturbed with discontinuities spaced 3 to 10 feet apart

65

10

75–90

0.28

Fair

Rock with several sets of moderately weathered discontinuities spaced 1 to 3 feet apart

44

1

50–75

0.049 0.056 0.066 0.069 0.081

Poor

Rock with numerous 23 weathered discontinuities spaced 1 to 20 inches apart with some gouge

0.1

25–50

0.015 0.016 0.019 0.020 0.024

0.01

1.7 Glos and Briggs (1983)

Drilled shafts in rock

248

shaley, RQD=74% 19 Sandstone, horizontally bedded, shaley, with some coal stringers, RQD=88%

610

16.9

9.26

13.1

1.41

>1.7 Glos and Briggs (1983)

20 Mudstone, highly weathered

300

2.01

0.65

6.4

9.8

6.4

Williams (1980)

21 Mudstone, highly weathered

300

1

0.67

7

10.5

5.7

Williams (1980)

22 Mudstone, moderately weathered

1000

15.5

2.68

5.9

2.2

1.1

Williams (1980)

23 Mudstone, moderately weathered

1000

15.5

2.45

6.6

2.7

0.7

Williams (1980)

24 Mudstone, moderately weathered

1000

15.5

2.45

7

2.9

0.6

Williams (1980)

25 Mudstone, moderately weathered

1000

15.5

2.68

6.7

2.5

0.7

Williams (1980)

26 Mudstone, moderately weathered

600

1.8

1.93

9.2

4.8

14.1 Williams (1980)

27 Mudstone, moderately weathered

1000

3

1.4

7.1

5

10.9 Williams (1980)

No. Rock description

Diameter Depth σc qmax Nc= Sb/Ba Reference B (mm) to (MPa) (MPa) qmax/σc (%) base L (m)

28 Shale

**

**

34

28

0.82

**

Thorne (1980)d

29 Sandstone

**

**

12.5

14

1.12

**

Thorne (1980)d

30 Sandstone, fresh, defect free

**

**

27.5

50

1.82

**

Thorne (1980)d

31 Shale, occational

**

**

55

27.8

0.51

**

Thorne (1980)d

Axial load capacity of drilled shafts in rock

249

recemented moisture fractures and thin mud seams, intact core lengths 75 to 250 mm 32 Clayshale

740

7.24

1.42

5.68

4

~8.8 Aurora and Reese (1977)

33 Clayshale

790

7.29

1.42

5.11

3.6

~8.9 Aurora and Reese (1977)

34 Clayshale

750

7.31

1.42

6.11

4.3

~6.0 Aurora and Reese (1977)

35 Clayshale

890

7.63

0.62

2.64

4.25

~6.6 Aurora and Reese (1977)

36 Siltstone, medium hard, fragmented

705

7.3

9

13.1

1.46

~12.0 Radhakrishnan and Leung (1989)

37 Marl, intact, RQD=100%

1200

18.5

0.9

5.3

5.89

**

Carrubba (1997)

38 Diabase Breccia, highly fractured, RQD=10%

1200

19

15.0

8.9

0.59

**

Carrubba (1997)

39 Limestone, intact, RQD=100%

1200

13.5

2.5

8.9

3.56

**

Carrubba (1997)

a

Sb is the shaft base displacement at qmax. Gypsum mixed with cement is used as pseudo-rock in centrifuge tests. The and depths are the equivalent prototype dimensions corresponding to 40 g in the centrifuge tests. The equivalent prototype depths to the shaft base range 4.04 m to 4.35 m with an average of 4.20 m. c Till is not a rock. It is used here because its σc is comparable to that of some rocks. d These tests were not conducted by Thorne (1980). He only reported the data other references b

capacity. Unfortunately, relevant information on this factor is unavailable for most of the cases in Table 6.14. 3. The conditions below the base of the shaft also influence the end bearing capacity. If the base of the drilled hole cannot be cleaned, little or no end bearing support will be developed. For all the test shafts in Table 6.14, the base of the drilled hole was cleaned. 4. Different methods are used to separate the side shear resistance from the end bearing capacity in load tests. 5. Clearly it would be interesting to have a relatively narrowly defined shaft base displacement which one can associate with the end bearing capacity. However, the values of sb/B in Table 6.14 indicate that the base displacement at qmax ranges from 0.6 to 20% of the shaft diameter, i.e., 6 to 210 mm. It is thus difficult to say at this

Drilled shafts in rock

250

point what typical base displacements at qmax are. [For comparison, the displacement at ultimate side shear resistance is smaller; examination of more than 50 loaddisplacement curves for large-diameter drilled shafts showed that an average displacement of only 5 mm was necessary to reach initial failure of side shear resistance (Horvath et al., 1983)].

Fig. 6.18 qmax versus σc (after Zhang & Einstein, 1998a). All the load test data in Table 6.14 are plotted in Figure 6.18. A log-log plot is used. It can be seen that there is a strong relation between qmax and σc. Using linear regression, the relationship between qmax and σc is as follows: qmax=4.83(σc)0.51 (6.26) The coefficient of determination, r2, is 0.81. Example 6.2 A drilled shaft of diameter 1.0 m is to be socketed 3.0 meters in siltstone. The rock properties are as follows: Unconfined compressive strength of intact rock, σc=15.0 MPa

Axial load capacity of drilled shafts in rock

251

The rock mass is heavily jointed and the average discontinuity spacing near the base of the shaft is 0.5 m The discontinuities are moderately weathered and filled with debris with thickness of 3 mm Deformation modulus of intact rock, Er=10.6 GPa RQD=45

Determine the end bearing resistance. Solution: Method of AASHTO (1989)—Equation (6.20) From Table 2.8, the rock is classified as Type B. From Table 6.11, the rock quality is classified as Fair and the value of Nms is 0.056. Using Equation (6.20), the end bearing resistance can be obtained as qmax=Nmsσc=0.056×15.0=0.84 MPa Method of Zhang & Einstein (1998a)—Equations (6.22) & (6.23) From Table 2.8, the rock is classified as Type B. From Table 6.12, the rock quality is classified as Fair and the values of s and mb are respectively 10−4 and 0.2. Assuming that the effective unit weight of the rock mass is 13.0 kN/m3 and ignoring the weight of the soil above the rock, the end bearing resistance can be obtained from Equations (6.22) and (6.23) as

Method of CGS (1985)—Equation (6.25) Empirical factor Ksp=[3+s/B]/[10(1+300g/s)0.5]=(3+0.5/1.0)/[10(1+300×0.003/0.5)0.5] =0.21 Depth factor D=1+0.4(L/B)=1+0.4(3.0/1.0)=2.2.

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252

The end bearing resistance can be calculated from Equation (6.25) as

Method of Zhang and Einstein (1998)—Equation (6.26) The end bearing resistance can be simply calculated from Equation (6.26) as

The results clearly show the wide range of the estimated end bearing capacity from different methods. It is therefore important not to rely on a single method when estimating the end bearing capacity.

6.4 CAPACITY OF DMLLED SHAFT GROUPS In many cases, drilled shaft foundations will consist not of a single drilled shaft, but of a group of drilled shafts. The drilled shafts in a group and the soil/rock between them interact in a very complex fashion, and the axial capacity of the group may not be equal to the axial capacity of a single isolated drilled shaft multiplied by the number of shafts. One way to account for the interaction is to use the group efficiency factor η, which is expressed as: (6.27) where QuG is the ultimate axial load of a drilled shaft group; N is the number of drilled shafts in the group; and Qu is the ultimate axial load of a single isolated drilled shaft, which can be determined using the methods described in Section 6.3. The group efficiency for axial load capacity depends on many factors, including the following: •The number, length, diameter, arrangement and spacing of the drilled shafts. •The load transfer mode (side shear versus end bearing). •The elapsed time since the drilled shafts were installed. •The rock type. Katzenbach et al. (1998) studied the group efficiency of a large drilled shaft group in rock. For the 300 m high Commerzbank tower in Frankfurt am Main, 111 drilled shafts are used to transfer the building load through the relatively weak Frankfurt Clay to the stiffer underlying Frankfurt Limestone. Of the 111 drilled shafts, 30 were instrumented

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253

and monitored during the 2-year construction period. The measurements give a detailed view into the interaction between the drilled shafts in the group. Figure 6.19 shows the variation of the group efficiency factor with the shaft head settlement. At service loads of the building the value of the group efficiency factor is about 60%. When drilled shafts are closely spaced, the shafts in a group may tend to form a “group block” that behaves like a giant, short shaft (see Fig. 6.20). In this case, the bearing capacity of the drilled shaft group can be obtained in a similar fashion to that for a single isolated drilled shaft, by means of Equation (6.2), but now taking the shaft base area as the block base area and the shaft side surface area as the block surface area. It should be noted that the deformation required to mobilize the base capacity of the block will be larger than that required for a single isolated shaft.

6.5 UPLIFT CAPACITY In many cases, drilled shafts in rock may be required to resist uplift forces. Examples are drilled shaft foundations for structures subjected to large overturning moments such as tall chimneys, transmission lines, and highway sign posts. Drilled shafts through expansive soils and socketed into rock may also subject to uplift forces due to the swelling of the soil. Drilled shafts can be designed to resist uplift forces either by enlarging or belling the base, or by developing sufficient side shear resistance. Belling the base of a shaft is common in soils, but this can be an expensive and difficult operation in rock. Moreover, since large side shear resistance can be developed in drilled shafts socketed into rock, it is usually more economical to deepen the socket than to construct a shorter, belled socket. For drilled shafts subject to uplift forces, it is important to check the structural capacity of the shaft. This can be done using the methods presented in Section 6.1. The ultimate uplift resistance of a straight-sided drilled shaft related to rock can be determined by Quu=πBLτmax+Ws (6.28) where Quu is the ultimate uplift resistance; L and B are respectively the length and diameter of the shaft; τmax is the average side shear resistance along the shaft; and Ws is the weight of the shaft.

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254

Fig. 6.19 Variation of group efficiency factor with shaft head settlement (after Katzenbach et al., 1998).

Fig. 6.20 Treating the drilled shaft group as a group block. Uplift loading does not produce the same stress conditions in the shaft or rock mass as those produced by compression loading. Compression loading compresses the shaft,

Axial load capacity of drilled shafts in rock

255

causing outward radial straining in the concrete (positive Poisson effect), which results in higher frictional stresses at the interface with the rock mass; simultaneously it adds total vertical stress to the rock mass around the shaft through the process of load transfer, which consequently adds strength to rock masses that drain during loading. Uplift loading, however, produces radial contraction of the concrete (negative Poisson effect) and reduces the total vertical stresses in the rock mass around the shaft. Because of the different stress conditions, the average side shear resistance for uplift loading should usually be lower than that for compression loading.

Fig. 6.21 Measured side shear resistance from compression tests and pull-out tests. Figure 6.21 shows the variation of measured side shear resistance with the unconfined compressive strength of intact rock respectively from the compression load tests and the pull-out load tests. The data are collected from the published literature. We can see that the measured side shear resistances from the pull-out load tests are about the same as or even higher than those from the compression load tests. One of the reasons for this might be that the pull-out test shafts have rougher wall surfaces than the compression test shafts. However, we are not sure about this at this point since no information on the wall roughness is available for most of the test shafts shown in Figure 6.21.

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256

For preliminary design, the side shear resistance for uplift loading can be simply taken to be the same as that for compression loading and estimated using the methods presented in Section 6.3.1. Where vertical drilled shafts are arranged in closely-spaced groups the uplift resistance of the complete group may not be equal to the sum of the resistance of the individual shafts. This is because, at ultimate-load conditions, the block of rock enclosed by the shafts may be lifted. The uplift resistance of the block of rock may be determined by (see Fig. 6.20) (6.29) where QuuG is the total ultimate uplift resistance of the shaft group; B1 and B2 are respectively the overall length and width of the group (see Fig. 6.20); and WB is the combined weight of the block of rock enclosed by the shaft group plus the weight of the shafts.

7 Axial deformation of drilled shafts in rock 7.1 INTRODUCTION Predicting the axial load-displacement response of drilled shafts is in some cases as important as, or possibly more critical than, predicting the ultimate bearing capacity. Many methods are available for predicting the axial displacement of drilled shafts in rock. While the most reliable means for predicting the axial displacement of drilled shafts is probably to carry out an axial loading test of the prototype shaft (which will be discussed in Chapter 12), theoretical analyses may also be usefully employed. The main three theoretical methods used to predict the axial load-displacement response of drilled shafts in rock are the load-transfer (t-z) method, the continuum approach and the finite element method. The general load-displacement curve for a drilled shaft under axial loading can be simply illustrated in Figure 7.1. The whole curve can be described in three stages: 1. As load is first applied to the head of the shaft, a small amount of displacement occurs which induces the mobilization of side shear resistance from head to base. During this initial period, the shaft behaves essentially in a linear manner, and the displacement can be computed using the theory of elasticity. This linear behavior is illustrated in Figure 7.1 as the line OA. The side shear stress along the shaft is smaller than the ultimate side shear resistance (Fig. 7.2a). 2. As load is increased to point A in Figure 7.1, the shear stress at some point along the interface will reach the ultimate side shear resistance (Fig. 7.2b), and the shaft-rock ‘bound’ will begin to rupture and relative displacement (slip) will occur between the shaft and the surrounding rock. As the loading is increased further (beyond point A), this process will continue along the shaft, more of the shaft will slip, and a greater proportion of the applied load will be transferred to the end of the shaft (Fig. 7.2c). If loading is continued, eventually the side shear stress everywhere will reach the ultimate side shear resistance and the entire shaft will slip (point B in Fig. 7.1). 3. Beyond point B, a greater proportion of the total axial load will be transmitted directly to the end of the shaft. When both side shear resistance and end bearing resistance are fully mobilized (point C), any increase of load may produce significant displacement. This indicates that the ultimate bearing capacity of the drilled shaft has been reached.

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Fig. 7.1 Generalized load-displacement curves for drilled shafts under compressive loading. 7.2 LOAD-TRANSFER (t-z CURVE) METHOD The load-transfer method models the reaction of soil/rock surrounding the shaft using localized springs: a series of springs along the shaft (the t-z or τ-w curves) and a spring at the tip or bottom of the shaft (the q-w curve). τ is the local load transfer or side shear resistance developed at displacement w, q is the base resistance developed at displacement w, and w is the displacement of the shaft at the location of a spring. The physical drilled shaft is also represented by a number of blocks connected by springs to indicate that there will be compression of the drilled shaft due to the applied compressive load. The mechanical model is shown in Figure 7.3. The displacement of the shaft at any depth z can be expressed by the following differential equation: (7.1)

where Ep is the composite Young’s modulus of the shaft (considering the contribution of both concrete and reinforcing steel); A and B are respectively the cross-sectional area and diameter of the shaft; w is the displacement of the shaft at depth z; and τ is the side shear resistance developed at displacement w at depth z. Equation (7.1) can be solved analytically or numerically depending on the τ-w and q-w curves (linear or nonlinear), which is discussed in the sections below.

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260

7.2.1 Linear analysis For linear analysis, the relationship between τ and w at any depth z and that between q and w are assumed to be linear, i.e.,

Fig. 7.2 Shear stress at different values of applied load (QA is the applied load corresponding to point A in Fig. 7.1). (7.2a)

Axial deformation of drilled shafts in rock

261

(7.2b) where ks and kb are spring constants respectively of the side springs and the base spring. Substitution Equation (7.2a) into Equation (7.1) gives (7.3)

where

Fig. 7.3 Load-transfer (t-z curve) model of axially loaded drilled shaft. (7.4)

The general solution to Equation (7.3) is

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262

(7.5) where C1 and C2 are integration constants. The axial force at any depth is proportional to the first derivative of the displacement with respect to depth: (7.6) If a load Qt is applied at the top of the shaft (z=0) and the force transferred to the base of the shaft (z=L) is Qb, we have, from Equation (7.6), (7.7a) (7.7b) From Equations (7.2b) and (7.5), We have (7.8) Solving Equations (7.7) and (7.8), constants C1 and C2 can be obtained as (7.9a)

(7.9a)

The displacement at the top of the shaft (z=0) is then obtained from Equations (7.5) and (7.9) as

7.2.2 Nonlinear analysis In general, the τ-w and q-w curves are nonlinear. In this case, a convenient way to solve differential Equation (7.1) is to use the finite difference method (Desai & Christian, 1977). Computer programs can be easily written to do the computations. The main issue for the nonlinear analysis is the determination of the τ-w and q-w curves. There are several techniques for determining the load transfer curves in soils (Vijayvergiya, 1977; Kraft et al., 1981; Castelli et al., 1992) and rock masses (Baguelin et

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al., 1982; O’Neill & Hassan, 1994). However, research has not advanced to the point that the load transfer curves (τ-w and q-w curves) can be determined for all conditions with confidence (O’Neill & Reese, 1999). Construction practices and the particular response of a given formation to drilling and concreting will affect the load transfer curves. For major projects, therefore, it is advisable to measure the load transfer curves using fullscale loading tests of instrumented shafts. Chapter 12 will show how to obtain the experimental load transfer curves from the results of an axial loading test of an instrumented shaft. Based on measured load displacement curves, Carrubba (1997) conducted numerical analyses to evaluate the side shear resistance and the end bearing capacity and obtained the load transfer curves for five rock-socketed shafts. The model is based on a hyperbolic transfer function approach and solves the equilibrium of the shaft by means of finite element discretization. The interaction at the shaft-soil and shaft-rock interfaces is described by the following function (7.11) where f(z) is the mobilized resistance along a shaft portion (τ) or at the shaft base (q); and w(z) is the corresponding displacement (see Fig. 7.3). In the transfer function, parameters a and b represent the reciprocals of initial slope and limit strength, respectively: (7.12a) (7.12b) where flim is the end bearing capacity (qmax) in rock or the side shear resistance in soil or rock (τmax). Numerical analyses are carried out by selecting three transfer functions for each shaft: one representative of overall friction in soil, one for overall friction in rock, and the last one for end bearing resistance in rock. The friction transfer functions in soils, once selected, are maintained constant throughout the analyses. Transfer function parameters for rock, both along the shaft and at the base, are first estimated and then modified with an iterative process until the actual load displacement curve is reproduced. Figure 7.4 shows the comparison between the test results and the numerical simulations for the shaft in marl. Since the side and base strengths are not mobilized at the same time and the numerical model used cannot simulate this event, two different ideal shaft behaviors are examined. The first neglects the base reaction; the second takes into account the contemporary mobilization of side and base resistances from the beginning of the test. The rock properties and the transfer function parameters obtained for the five rocksocketed shafts are shown in Table 7.1. O’Neill and Hassan (1994) proposed an interim criterion for a hyperbolic τ-w curve in most types of rock until better solutions become accepted:

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(7.13)

where B is the diameter of the shaft; and Em is the deformation modulus of the rock mass. This model is based on the fact that the interface asperity pattern is regular and the asperities are rigid, even though in most cases the interface asperity pattern is not regular, some degree of smear exists, and asperities are deformable, which results in ductile, progressive failure among asperities. Equation (7.13) is a special form of Equation (7.11) with a=2.5B/Em.

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Fig. 7.4 Comparison between test results and numerical simulations for the drilled shaft in Marl. Curve a neglects base reaction; curve b takes into account cotemporary mobilization of side and base resistances (after Carrubba, 1997). Table 7.1 Rock properties and transfer function parameters (Carrubba, 1997). Rock type Marl

σc (MPa) 0.90

RQD (%)

Em (MPa)

Shaft side in rock

Base in rock

1/b (MPa)

1/b (MPa)

1/a (MN/m3)

1/a (MN/m3)

100

200a

0.14

100

5.30

220

b

0.49

70

8.90

300

Diabasic Breccia

15.00

10

200

Gypsum

6.00

60

2,000a

0.47

200

–

–

50

a

1.20

500

–

–

b

0.40

500

8.90

3,000

Diabase Limestone a b

40.00 2.50

100

10,000 500

From compression tests on specimens From plate bearing tests

The q-w curve is usually assumed to have an initial elastic response given by

where Eb and νb are respectively the deformation modulus and Poisson’s ratio of the rock below the shaft base. Nonlinear response is usually assumed to initiate between 1/3 and 1/2 of qmax. This response can be simply modeled using an equation similar to Equation (7.13).

7.3 CONTINUUM APPROACH The continuum approach assumes the soil/rock to be a continuum. Mattes and Poulos (1969) are among the first to investigate the load-displacement behavior of rock-socketed shafts by integration of Mindlin’s equations. Carter and Kulhawy (1988) provide a set of approximate analytical solutions to predict the load-displacement response of drilled shafts in rock by modifying the solutions of Randolph and Wroth (1978) for piles in soil.

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The majority of the theoretical continuum solutions for predicting the displacement of drilled shafts in rock, however, have been developed using finite element analyses (e.g., Osterberg & Gill, 1973; Pells & Turner, 1979; Donald et al., 1980; Rowe & Armitage, 1987a). Most of the techniques proposed for calculating the vertical displacements of drilled shafts in rock are based on the theory of elasticity. It has been usual to assume that the drilled shaft is essentially an elastic inclusion within the surrounding rock mass and that no slip occurs at the interface between the shaft and the rock mass, although the solutions of Rowe and Armitage (1987a) and Carter and Kulhawy (1988) can consider the possibility of slip. 7.3.1 Linear continuum approach (a) Solutions based on finite element results As stated in Chapter 6, axially loaded drilled shafts in rock are designed to transfer structural loads in one of the following three ways (CGS, 1985): 1. Through side shear only; 2. Through end bearing only; 3. Through the combination of side shear and end bearing. The following presents the elastic solutions based on the finite element results for estimating the axial deformation of the above three types of shafts. Side shear only shaft Based on finite element analysis, Pells and Turner (1979) presented the following general equation for calculating the axial deformation of side shear only shafts in a single elastic half space: (7.15) where wt is the axial deformation of the shaft at the rock surface; Qt is the applied load at the top of the shaft; Em is the deformation modulus of the rock mass; B is the diameter of the shaft; and I is the axial deformation influence factor given in Figure 7.5. The values of I given in Figure 7.5 have been calculated for a Poisson’s ratio of 0.25. It has been found that variations in the Poisson’s ratio in the range 0.1–0.3 for the rock mass and 0.15–0.3 for the concrete have little effect on the influence factors. The values of the influence factor shown in Figure 7.5 are for drilled shafts that are fully bonded from the rock surface. In many cases, the drilled shaft is recessed by casing the upper part of the drilled hole or for conditions where the shaft passes through a layer of soil or weathered rock where little or no side shear resistance will be developed. Recessment of the shaft will result in a decrease in axial deformation of the shaft at the head of the socket. This reduction can be expressed in terms of a reduction factor RF such that the axial deformation of the shaft at the ground surface is given by

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(7.16)

Fig. 7.5 Axial deformation influence factors for side shear only drilled shafts (after Pells & Turner, 1979). where Qt is the applied load at the top of the shaft; D and Bl are respectively the length and diameter of the recessed shaft; Ep is the composite Young’s modulus of the shaft (considering contributions of both concrete and reinforcing steel); RF is a reduction factor for the effect of recessment; B is the diameter of the socketed shaft; Em is the deformation modulus of the rock mass; and I is the influence factor for shaft with no recessment (see Fig. 7.5). The first portion of Equation (7.16) simply represents the elastic compression of the shaft over the length D. The second portion of Equation (7.16) gives the axial deformation of the socketed portion of the shaft. The reduction factor RF is given in Figure 7.6 for a range of situations.

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Fig. 7.6 Reduction factors for calculation of axial deformation of recessed drilled shafts (after Pells & Turner, 1979). End bearing only shaft

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An end bearing only shaft can be considered a shaft that is wholly recessed (See Fig. 7.7). The axial deformation of an end bearing only shaft at the ground surface consists of the elastic compression of the shaft and the axial deformation of the shaft base: (7.17)

where Qt is the applied load at the top of the shaft; D and Bl are respectively the length and diameter of the shaft; Ep is the composite Young’s modulus of the shaft (considering contributions of both concrete and reinforcing steel); Em and νm are respectively the deformation modulus and Poisson’s ratio of the rock mass; Cd is the shape and rigidity factor equal to 0.85 for a flexible footing and 0.79 for a rigid footing; and RF′ is a reduction factor for an end bearing only shaft as shown in Figure 7.7. The axial deformation of the shaft base is calculated in a similar manner to that of a footing on the surface. However, because the rock mass below the base of the shaft is more confined than surface rock mass, the axial deformation of the shaft base will be smaller than that of a footing at the surface. The effect of this confinement if accounted for by applying the reduction factor RF′ to the deformation equation as shown in Equation (7.17). The value of the reduction factor depends on the ratio of the shaft length D to the shaft diameter B1, and the relative stiffness of the shaft and the rock mass. Figure 7.7 shows the values of the reduction factor RF′ obtained by Pells and Turner (1979). Side shear and end bearing shaft For side shear and ending bearing shafts, the axial deformation at the rock surface can be calculated using Equation (7.15). Considering the interaction between the side shear and end bearing, the influence factors given in Figure 7.8 should be used. These factors have been developed for elastic behavior without slip along the side walls by Rowe and Armitage (1987a).

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Fig. 7.7 Reduction factors for calculation of axial deformation of end bearing only drilled shafts (after Pells & Turner, 1979). Comparison of Figure 7.8(a) (for Eb/Em=1) with Figure 7.5 shows that the influence factor for a side shear and end bearing shaft is smaller than that for a side shear only shaft, which demonstrates that a shaft with both side shear and end bearing will settle less than a shaft with side shear only. Figure 7.9 shows the percentage of the load carried in the end bearing. (b) Analytical solutions of Carter and Kulhawy (1988) Carter and Kulhawy (1988) provide a set of approximate analytical solutions to predict the load-displacement response of drilled shafts in rock. Two layers of rock mass as shown in Figure 7.10 are considered in the solutions. The solutions are for a shaft without slip or with full slip. The following presents the solution for a shaft without slip while the solution for a shaft with full slip will be presented in Section 7.3.2.

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Under an applied axial load, the displacements in the rock mass are predominantly vertical, and the load is transferred from the shaft to the rock mass by vertical shear stresses acting on the cylindrical interface, with little change in vertical normal stress in the rock mass (except near the base of the shaft). The pattern of deformation around the shaft may be visualized as an infinite number of concentric cylinders sliding inside each other (Randolph & Wroth, 1978). Randolph and Wroth (1978) have shown that, for this type of behavior, the displacement of the shaft w may be described adequately in terms of hyperbolic sine and cosine functions of depth z below the surface, as given below: w=A1 sinh(µz)+A2 cosh(µz) (7.18) in which, A1 and A2 are constants which can be determined from the boundary conditions of the problem. The constant µ is given by (7.19) where ζ=ln[2.5(1−νm)L/R]; R=B/2 is the radius of the shaft; λ=Ep/Gm; Ep is the Young’s modulus of the shaft; Gm=Em/[2(1+νm)] is the shear modulus of the rock mass surrounding the shaft; and Em and νm are respectively the deformation modulus and Poisson’s ratio of the rock mass surrounding the shaft. For side shear and end bearing shafts as shown in Figure 7.10(a), the shaft base can be approximated as a punch acting on the surface of an elastic half-space with Young’s modulus Eb and Poisson’s ratio νb. Using the standard solutions for the displacement of a rigid punch resting on an elastic half-space as the boundary condition at the base of the shaft, the elastic displacement at the head of the shaft can be obtained by (Randolph & Wroth, 1978): (7.20)

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Fig. 7.8 Axial deformation influence factors for side shear and end bearing drilled shafts (after Rowe & Armitage, 1987a).

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Fig. 7.9 Load distribution curves for side shear and end bearing drilled shafts (after Rowe & Armitage, 1987a).

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274

where ξ=Gb/Gm; Gb=Eb/[2(1+νb)] is the shear modulus of the rock mass below the shaft base; and Eb and νb are respectively the deformation modulus and Poisson’s ratio of the rock mass below the shaft base. The proportion of the applied load transmitted to the shaft base is (7.21)

For side shear only shafts as shown in Figure 7.10(b), the boundary condition at the shaft base is one of zero axial stress. For this case, the elastic displacement at the head of the shaft can be obtained by

Fig. 7.10 Axially loaded drilled shafts in rock (after Carter & Kulhawy, 1988).

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275

Fig. 7.11 Comparison of analytical solution with finite element solution for predicting axial elastic displacement (after Carter & Kulhawy, 1988). (7.22)

The solution given by Equations (7.20) and (7.22) are in general agreement with the finite element solutions by Pells and Turner (1979) and Rowe and Armitage (1987a) as presented in last sections (Fig. 7.11). Example 7.1 A drilled shaft of 3.0 meters long and 1.0 meter in diameter is to be installed in siltstone. The rock properties are as follows: Unconfined compressive strength of intact rock, σc=15.0 MPa Deformation modulus of intact rock, Er=10.6 GPa RQD=70

Determine the settlement of the drilled shaft at a work load of 10.0 MN.

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276

Solution: For simplicity, the Young’s modulus of the drilled shaft is simply assumed to be Ep=30 GPa. The Poisson’s ratio of 0.25 is selected for both the drilled shaft and the rock. Using Equation (4.24), the rock mass modulus: αE=0.0231×70−1.32=0.297 Em=0.297×10.6=3.15Gpa Using solutions based on finite element method L/B=3.0/1.0=3.0 Ep/Em=30/3.15=9.52 If the drilled shaft is side shear resistance only (i.e., the shaft base cannot be cleaned), from Figure 7.5, the axial deformation influence factor is I=0.462. Using Equation (7.15), the settlement of the drilled shaft at the rock surface is

If the drilled shaft has both side shear and end bearing resistance, from Figure 7.8, the axial deformation influence factor is I=0.417 for Eb/Em=1.0. Using Equation (7.15), the settlement of the drilled shaft at the rock surface is

From Figure 7.9, it can be seen that about 15% of the load is transmitted to the shaft base. Using analytical solutions of Carter and Kulhawy (1988)

Axial deformation of drilled shafts in rock

277

ξ=Gb/Gm=1.0 for Eb/Em=1.0 If the drilled shaft is side shear resistance only (i.e., the shaft base cannot be cleaned), the settlement of the drilled shaft at the rock surface can be calculated from Equation (7.22) as

If the drilled shaft has both side shear and end bearing resistance, the settlement of the drilled shaft at the rock surface can be calculated from Equation (7.20) as

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278

The percentage of the load transmitted to the shaft base can be calculated from Equation (7.21) as

The results from the solutions based on the finite element method are in good agreement with those from the analytical solutions of Carter & Kulhawy (1988). 7.3.2 Nonlinear continuum approach (a) Solutions based on finite element results Rowe and Armitage (1987a) performed an elastic-plastic finite element analysis that accounts for slip along the interface based on the technique developed by Rowe and Pells (1980). Two layers of rock are considered in the analyses. The interface behavior is established in terms of the Coulomb failure criterion. The roughness of the interface is modeled implicitly through the use of an angle of interface dilatancy that produces additional normal stress on the interface as the shaft deflects vertically due to the applied load. The contribution of the interface dilatancy commences once slip occurs at the interface. The results of this study are presented in three sets of design charts respectively for Eb/Em=0.5, 1.0 and 2.0. Although the analysis is carried out considering the behavior of a cohesive-frictional-dilative interface, the design charts are developed only for nondilative-cohesive interfaces. The procedure for using the design charts is described in Rowe and Armitage (1987b). (b) Analytical solutions of Carter and Kulhawy (1988) The case of slip along the entire length of the shaft has also been considered in detail by Carter and Kulhawy (1988). For this case, the shear strength of the interface is given by the Coulomb criterion: (7.23)

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279

where c is the interface cohesion; is the interface friction angle; and σr is the radial stress acting on the interface. As relative displacement (slip) occurs, the interface may dilate, and it is assumed that the displacement components follow the dilation law: (7.24) where ∆u and ∆w are the relative shear and normal displacements of the shaft-rock interface; and ψ is the angle of dilation defined by Davis (1968). To determine the radial displacements at the interface, the procedure suggested by Goodman (1980) and Kulhawy and Goodman (1987) is followed, in which conditions of plane strain are assumed, as an approximation, independently in the rock mass and in the slipping shaft. The rock mass is considered to be linear elastic, even after full slip has taken place, and the shaft is considered to be an elastic column. These assumptions, together with the dilatancy law, allow one to derive an expression for the variation of vertical stress in the compressible shaft. The distribution of the shear stress acting on the shaft can then be calculated from equilibrium conditions, and the vertical displacement can be determined as function of depth z by treating the shaft as a simple elastic column. The ‘full slip’ solution for the displacement of the shaft head is derived as (7.25)

in which F3=a1(λ1BC3−λ2BC4)−4a3 (7.26) (7.27) C3,4=D3,4/(D4−D3) (7.28) (7.29)

(7.30)

(7.31)

Drilled shafts in rock

280

(7.32) a1=(1+νm)ς+a2 (7.33) (7.34)

(7.35)

All other parameters in Equations (7.25) to (7.35) are as defined before. The adequacy of the closed-form expressions is demonstrated by comparing them with the finite element solution of Rowe and Armitage (1987a, b). The overall agreement between the closedform solutions and the finite element results is good (Fig. 7.12). It must be noted that the closed-form solutions of Carter and Kulhawy (1988) just consider “no slip” (presented in Section 7.3.1) and “full slip” conditions. They cannot predict the load-displacement response between the occurrence of first slip and full slip of the shaft. However, the finite element results indicate that the progression of slip along the shaft takes place over a relatively small interval of displacement. Therefore it seems reasonable, at least for most practical cases, to ignore the small region of the curves corresponding to the progressive slip and to assume that the load-displacement relationship is bilinear, with the slope of the initial portion given by Equation (7.20) and the slip portion by Equation (7.25) (Carter & Kulhawy, 1988).

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Fig. 7.12 Axial displacement of a drilled shaft in rock considering full slip (after Carter & Kulhawy, 1988). 7.4 FINITE ELEMENT METHOD (FEM) The finite element method is probably the most powerfiil and the most widely used numerical method currently available to engineers. Suitable elements can be used to simulate not only linearly elastic materials, but also nonlinear materials with different failure criteria, including rock discontinuities and shaft-rock interfaces (see Sections 4.3.4 and 4.4.3 for discussion of joint elements). However, the finite element method is time consuming and needs sophisticated soil or rock constitutive relations whose parameters are often difficult if not impossible in design practice to obtain. Therefore, the finite element method is, in general, used for analysis of important structures and for generation of parametric solutions for the load-displacement relations of axially loaded drilled shafts, such as the charts presented in Sections 7.3.1 and 7.3.2. Typical of many geotechnical problems, the analysis of drilled shafts in rock involves an unbounded domain. It is a common practice in finite element modeling of these problems to truncate the finite element mesh at a distance deemed far enough so as not to influence the near field solutions. These truncations are usually determined by trail and

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error until an acceptable solution is obtained. Such a method places a heavy demand on computer resources, both memory and time, as solutions for the far field which are of no interest are generated as well. In the last decade or so, several methods have been developed to model unbounded domains. Of these methods, the use of infinite elements with finite elements appears to be the most popular. Leong and Randolph (1994) successfully used finite elements and infinite elements in the modeling of axially loaded shafts in rock.

7.5 DRILLED SHAFT GROUPS Numerous methods exist for analyzing axially loaded pile groups in soil (Poulos, 2001), some of which can be applied to drilled shaft groups in rock and are briefly described in the following. 7.5.1 Settlement ratio method In the settlement ratio method, the group settlement is related to the single-shaft settlement as follows: (7.36) where wtG is the settlement of the shaft group; wtav is the settlement of a single shaft at the average load of a shaft in the group; and Rw is the settlement ratio. wtav can be estimated using the methods presented in the previous sections or from the results of load test on a prototype drilled shaft. Theoretical values of Rw for various pile groups in soil have been presented by Poulos and Davis (1980) and Butterfield and Douglas (1981). A particularly useful approximation for the settlement ratio has been derived by Fleming et al. (1992): (7.37) where n is the number of piles in the group; and e is an exponent depending on pile spacing, pile proportions, relative pile stiffness and the variation of soil modulus with depth. For typical pile proportions and pile spacings, Poulos (1989) suggested the following approximate values: e≈0.5 for piles in clay, and e≈0.33 for piles in sand. For drilled shafts in rock, the e values suggested by Poulos (1989) for soils may be used for the very preliminary design. For the final design of major projects, it is desirable, when feasible, to conduct axial load tests on groups of two or more drilled shafts in rock in order to confirm the e values of Poulos (1989) or to derive new, site-specific values. 7.5.2 Equivalent pier method The equivalent pier method, frequently used for pile groups in soils, treats the pile group as an equivalent pier consisting of the piles and the soil between them (Poulos & Davis,

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1980; Randolph, 1994). For closely spaced drilled shafts in rock, the shaft group may also be analyzed using the equivalent pier method. Consider the drilled shaft group as an equivalent pier (Fig. 7.13), the diameter of the equivalent pier Beq can be taken as (Randolph, 1994).

Fig. 7.13 Equivalent pier method treating drilled shaft group as a group block. (7.38) where Ag is the plan area of the drilled shaft group as a block. Deformation modulus of the equivalent pier Eeq is then calculated as Eeq=Em+(Ep−Em)Apt/Ag (7.39) where Ep is the Young’s modulus of the drilled shafts; Em is the deformation modulus of the rock mass; and Apt is the total cross-sectional area of the drilled shafts in the group. The load-settlement response of the equivalent pier can be calculated using the solutions as described in the previous sections for the response of a single drilled shaft. Based on the equivalent pier method and the load-transfer (t-z curve) approach, Castelli and Maugeri (2002) presented a simplified nonlinear analysis for settlement prediction of pile groups in soil. To take into account the group action due to pile-soilpile interaction, load-transfer functions are modified to relate the behavior of a single pile to that of a pile group. The bearing capacity of the equivalent pier can be evaluated using the procedure in Section 6.4. The initial stiffness of the equivalent pier is estimated by

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(7.40)

where Kgi is the initial stiffness of the equivalent pier and β is an empirical parameter. To take into account the increase of pile group head settlements with respect to the case of a single pile, the following expression is used (7.41)

where wg is the average settlement of the equivalent pier and ε is an empirical parameter. The empirical parameters β and ε can be derived on the basis of numerical analysis of field tests. Castelli and Maugeri (2002) derived values of 0.30 and 0.15 respectively for β and ε based on analysis of field test piles and pile groups in soils. For drilled shafts in rock, similar values of β and ε can be obtained from field tests of shafts and shaft groups. 7.5.3 Finite element method (FEM) The finite element method has been used to analyze axially loaded pile groups in soil by simplifying the group to an equivalent plane strain or axisymmetric system. If necessary, it can also be used to analyze drilled shaft groups in rock.

8 Lateral load capacity of drilled shafts in rock 8.1 INTRODUCTION In the design of drilled shafts subjected to lateral forces, two criteria must be satisfied: first, an adequate factor of safety against ultimate failure, second, an acceptable deflection at working loads. This chapter discusses the prediction of ultimate load of drilled shafts and drilled shaft groups. The calculation of lateral deflection will be discussed in Chapter 9. As the axial load capacity, the lateral load capacity of a drilled shaft in rock is determined by the smaller of the two values: the structural strength of the shaft itself, and the ability of the rock to support the loads transferred by the shaft.

8.2 CAPACITY OF DRILLED SHAFTS RELATED TO REINFORCED CONCRETE The structural capacity of a drilled shaft under lateral loading is controlled by the bending capacity and the shear capacity. The bending capacity is usually checked by considering the interaction between axial load and bending moment. Figure 8.1 shows the normalized axial load-moment intersection diagrams for fy=10f′c and fy=15f′c, where fy is the yield strength of the longitudinal reinforcing steel and f′c is the specified minimum concrete strength. The factored axial load ΣγiQi is normalized by dividing by the factored nominal axial capacity

, where γi is the load factor for axial load i, Qi is the nominal value

is the resistance factor for the nominal (computed) structural axial of axial load i, and load capacity Qu. The factored moment ΣγmMm is similarly normalized by dividing by the factored nominal moment capacity

, where γm is the load factor for moment m, Mm

is the nominal value of moment m, and is the resistance factor for the nominal (computed) structural moment capacity Mu. The factored axial capacity is estimated from Equation (6.1). Normalized axial load-moment interaction diagrams may be developed for any fy/f′c ratios and cage diameters other than 0.6B. With the axial load-moment interaction diagrams available, the structural capacity can be checked as follows: 1. Estimate the combined axial load ΣγiQi. 2. Compute the factored nominal axial capacity

from Equation (6.1).

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Fig. 8.1 Normalized axial loadmoment interaction diagrams for drilled shafts for (a)fy=10f′c and (b)fy=15f′c.

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Table 8.1 Nominal structural moment capacity Mu of drilled shaft. Mu/f′cBAg As/Ag

fy−10f′c

fy=15f′c

0.01

0.037

0.050

0.02

0.067

0.092

0.03

0.088

0.119

0.04

0.107

0.147

0.05

0.126

0.172

0.06

0.144

0.197

0.07

0.161

0.208

0.08

0.176

0.244

As is the cross-sectional area of the longitudinal reinforcing steel Ag is the gross cross-sectional area of the shaft fy is the yield strength of the longitudinal reinforcing steel f′c is the specified minimum concrete strength B is the diameter of the shaft

3. Estimate the factored (required) moment ΣγmMm. 4. Estimate the nominal structural moment capacity Mu of the drilled shaft. This may be based on complete analysis, or it may be obtained from design aids such as Table 8.1. 5. Compute the factored nominal moment capacity reinforced concrete.

where

for

6. Determine the ratios , and and with these values locate an appropriate point on the axial load-moment interaction diagram. If the point falls inside the area defined by the interaction curve, the shaft capacity is adequate. If this is not the case, the shaft size should be increased and the analysis repeated until the shaft capacity is adequate. The factored nominal shear capacity of a drilled shaft without special shear reinforcement can be calculated by (O’Neill & Reese, 1999): (8.1) where is the capacity reduction (resistance) factor for shear=0.85; Vu is the nominal (computed) shear resistance; νc is the limiting concrete shear stress; and Av is the area of the shaft cross section that is effective in resisting shear, which can be taken as B(0.5B+0.5756rls) for a circular drilled shaft, where r1s is the radius of the ring formed by the centroids of the longitudinal reinforcing steels. The limiting concrete shear stress νc can be evaluated from:

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(8.2a) (8.2b)

where both f′c and νc are in kPa. If the factored shear load is greater than the factored nominal shear resistance determined above, two options are available. The first and simplest solution is to increase the shaft diameter to increase the shear capacity. The second alternative is to provide properly designed shear reinforcement (O’Neill & Reese, 1999). O’Neill and Reese (1999) provide detailed discussion and examples on checking the structural load capacity of drilled shafts.

8.3 CAPACITY OF DRILLED SHAFTS RELATED TO ROCK 8.3.1 Method of Carter and Kulhawy (1992) Carter and Kulhawy (1992) presented a method to determine the lateral load capacity of drilled shafts related to rock. When a lateral load is applied at the rock surface, the rock mass immediately in front of the shaft will be subject to zero vertical stress, while horizontal stress is applied by the leading face of the shaft. Ultimately, the horizontal stress may reach the uniaxial compressive strength of the rock mass and, with further increase in the lateral load, the horizontal stress may decrease as the rock mass softens during postpeak deformation. Large lateral deformations may be required for the rock mass at depth to exert a maximum reaction stress on the leading face of the shaft. Therefore, Carter and Kulhawy (1992) assumed that the reaction stress at the rock mass surface, in the limiting case of loading of the shaft, is zero or very nearly zero as a result of the postpeak softening. Along the sides of the shaft, some shearing resistance may be mobilized. The shearing resistance varies along the perimeter and the average can be chosen as τmax/2, where τmax is likely to be approximately the same as the maximum unit side resistance under axial compression. Therefore, at the rock surface, the ultimate force per unit length resisting the lateral loading is Bτmax. At greater depth, Carter and Kulhawy (1992) assume that the stress in front of the shaft increases from the initial in situ horizontal stress to the limit stress, pL, reached during the expansion of a long cylindrical cavity, i.e., a plane strain condition will apply. Behind the shaft, the horizontal stress will decrease, and after tensile rupture of the bond between the concrete and the rock mass, the horizontal stress will reduce to zero. At the sides of the shaft, some shearing resistance may also be mobilized. Therefore, at depth, the ultimate force per unit length resisting the lateral loading is B(pL+τmax). To determine the depth at which the limit stress is mobilized, the result of Randolph and Houslby (1984) in a cohesive material is adopted, i.e., the depth is about three times shaft diameter. Therefore, the distribution of ultimate force per unit length resisting the shaft is as shown in Figure 8.2.

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The ultimate lateral force that may be applied can be obtained from the horizontal equilibrium as: For L3B (8.3b) where τmax is the shearing resistance along the sides of the shaft, which is assumed to be the same as the maximum side resistance under axial loading; pL is the limit stress reached during the expansion of a long cylindrical cavity. Closed-form solutions have been found for the limit stresses developed during the expansion of a long cylindrical cavity in an elasto-plastic, cohesive-frictional, dilatant material (Carter et al., 1986). This limit stress pL can be determined from the following parametric equation in the nondimensional quantity ρ (Carter et al., 1986): (8.4)

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290

with (8.5) in which (8.6)

(8.7)

(8.8)

(8.9) (8.10) (8.11) (8.12) (8.13) (814) and σhi is the initial in situ horizontal stress; Gm is the elastic shear modulus; νm is the Poisson’s ratio; cm is the cohesion intercept; φm is the friction angle; and ψm is the dilation angle, all of the rock mass. The rock mass is assumed to obey the Coulomb failure criterion, and dilatancy accompanies yielding according to the following flow rule (8.15)

in which dε1p and dε3p are the major and minor principal plastic strain increments, respectively. For convenience, solutions for the limit pressures pL have been plotted in Figure 8.3 for selected values of νm, φm, ψm. The central vertical axis on each plot

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291

indicates the ratio of the plastic radius at the limit condition R to the cavity radius a. These charts may be used by entering with a value of Gm/(σhi+cmcotφm) and working clockwise around the figure, determining in turn values of R/a, then ρL=(pL+cmcotφm)/(σR+ cmcotφm), and thus, determining the limit pressure pL. 8.3.2 Method of Zhang et al. (2000) Zhang et al. (2000) presented an approximate method for calculating the ultimate lateral resistance of drilled shafts in rock. As shown in Figure 8.4(a), the total reaction of the rock mass consists of two parts: the side shear resistance and the front normal resistance. So the ultimate resistance pult can be estimated by (Briaud & Smith, 1983; Carter & Kulhawy, 1992): (8.16)

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Fig. 8.3 Limit solution for expansion of cylindrical cavity (after Carter & Kulhawy, 1992)

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Fig. 8.4 (a) Components of rock mass resistance; and (b) Calculation of normal limit stress pL (after Zhang et al., 2000).

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Fig. 8.5 Distribution of ultimate lateral resistance with depth (after Zhang et al., 2000). where B is the diameter of the shaft; τmax is the maximum shearing resistance along the sides of the shaft; and pL is the normal limit resistance. For simplicity, τmax is assumed to be the same as the maximum side resistance under axial loading and can be determined using the methods presented in Chapter 6. To determine the normal limit stress pL, the strength criterion for rock masses developed by Hoek and Brown (1980, 1988) is used. Assuming that the minor principal effective stress σ′3 is the effective overburden pressure γ′z and the limit normal stress pL is the major principal effective stress σ′1 [see Fig. 8.4(b)], we have, from Equation (4.64), the following (8.17)

where γ′ is the effective unit weight of the rock mass; z is the depth from the rock mass surface; and mb, s and a are rock mass parameters as described in Chapter 4. With the distribution of pult along the depth determined, the ultimate lateral load that may be applied can be approximated by (see Fig. 8.5) (8.18)

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Fig. 8.6 Comparison of estimated pult and that from field shaft tests (after Cho, 2002). Cho (2002) used the method of Zhang et al. (2000) to estimate pult of field test shafts in rock. The estimated values agree well with the field test data (see Fig. 8.6). Example 8.1 A drilled shaft of diameter 1.0 m is to be installed 3.0 meters in siltstone. The rock properties are as follows: Unconfined compresive strength of intact rock, σc=15.0 Mpa RMR=55

Determine the ultimate lateral load capacity of the shaft. Solution: From Table 4.5, mi=9 for siltstone.

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Using Equation (4.68),

Using Equation (6.8) and choosing Ψ=1.0 (lower bound), the side shear resistance can be obtained as

Assuming that the effective unit weight of the rock mass is 13.0 kN/m3, the limit normal stress pL can be obtained from Equation (8.17) as follows

Using Equation (8.18), the ultimate lateral load capacity can be obtained as

It need be noted that the ultimate lateral load capacity obtained above does not consider the moment equilibrium of the shaft. The structural strength of the shaft should also be checked when using the ultimate lateral load capacity obtained above in design.

8.4 CAPACITY OF DRILLED SHAFT GROUPS The ultimate lateral load capacity of a drilled shaft group can be calculated in a similar way to calculating the axial load capacity of a drilled shaft group, i.e. (8.19) where HultG is the ultimate lateral load of a drilled shaft group; N is the number of drilled shafts in the group; Hult is the ultimate lateral load of a single isolated drilled shaft; and α is the group efficiency factor. Table 8.2 lists the values of α recommended by the American Association of State Highway and Transportation Officials (AASHTO, 1989).

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Equation (8.19) applies when the head boundary conditions of the single shaft and the shaft group are the same. If the head boundary conditions of the single shaft and the shaft group are different (e.g., the single shaft has a free-head boundary condition while the shaft group has a flxed-head boundary condition because of the cap), a modification factor should be added to Equation (8.19) to account for the difference in head boundary conditions (Frechette et al., 2002): HultG=αNHultR (8.20) where R is the modification factor to account for the difference in head boundary conditions. For the case of a single shaft at a free-head boundary condition and a shaft group at a fixed-head boundary condition, Frechette et al. (2001) recommended a R value of 2.2 based on five case studies while Matlock and Foo (1976) recommended a R value of 2.0 based on a single case study (Frechette et al., 2002). If the drilled shafts are closely spaced, the drilled shaft group can be represented by a group block and its ultimate load can be calculated using the methods described in Section 8.3 by treating the group block as a big single shaft.

8.5 DISCONTINUUM METHOD In Sections 8.3 and 8.4, the rock mass is treated as a continuum. Since most rocks contain discontinuities, drilled shafts may fail due to the sliding of the rock blocks or wedges along discontinuities (see Fig. 8.7). In such cases, the lateral resistance is only provided by the shear resistance along the discontinuities and the weight of wedge bounded by the shaft and the discontinuities. Obviously, the rock mass need be treated as a discontinuous medium in order to obtain the lateral resistance provided by the wedges.

Table 8.2 Group efficiency factor a recommended by AASHTO (1989). Center-to-Center Shaft Spacing in Direction of Loading

Group Efficiency Factor α

3B

0.25

4B

0.40

6B

0.70

8B

1.00

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Figure 8.7 Sliding of rock blocks due to laterally loaded shaft. To (1999) developed a discontinuum method for determining the lateral load capacity of drilled shafts in a jointed rock mass containing two or three discontinuity sets. The method consists of two parts: a kinematic analysis and a kinetic analysis. In the kinematic analysis, Goodman’s block theory (Goodman & Shi, 1985) is extended to analyze the movability of a combination of blocks laterally loaded by a drilled shaft. Based on the extended theory, a 2-dimensional (2D) graphical method was developed to select the possible combinations of movable blocks. This 2D graphical method can be easily implemented with CAD programs such as AutoCAD or with spreadsheet programs such as Excel. In the kinetic analysis, the stability of each kinematically selected movable combination of blocks or wedges is analyzed with the limit equilibrium approach. This analysis, similar to slope stability analysis, considers the axial and lateral forces exerted by the drilled shaft in addition to the weight of the wedge and the shearing resistance along the discontinuities. From the stability analysis, simple analytical relations were developed to solve for the lateral load capacity of the drilled shaft. The lateral load capacity can also be obtained by analyzing the load-displacement response of a drilled shaft using the discrete element method (DEM) as described in Section 9.5.

9 Lateral deformation of drilled shafts in rock 9.1 INTRODUCTION For drilled shafts in rock to resist lateral loads, the design criterion in the majority of cases is not the ultimate lateral capacity of the shafts, but the maximum deflection of the shafts. Predicting the deformation of laterally loaded drilled shafts is, therefore, the most important aspect in designing drilled shafts to withstand lateral loads. To date, it has been customary practice to adopt the techniques developed for laterally loaded piles in soil (Poulos, 1971a, b, 1972; Banerjee & Davies, 1978; Randolph, 1981) to solve the problem of drilled shafts in rock under lateral loading (Amir, 1986; Gabr, 1993; Wyllie, 1999). However, the solutions for laterally loaded piles in soil do not cover all cases for laterally loaded drilled shafts in rock in practice (Carter & Kulhawy, 1992). Carter & Kulhawy (1992), therefore, developed a method for predicting the deformation of laterally loaded drilled shafts in rock. This method treats the rock mass as an elastic continuum and has been found to give reasonable results of predicted deflections only at low load levels (20–30% capacity). At higher load levels, the predicted displacements are too small (DiGioia & Rojas-Gonzalez, 1993). Reese (1997) developed a p-y curve method for analyzing drilled shafts in rock under lateral loading. The major advantage of the p-y curve approach lies in its ability to simulate the nonlinearity and nonhomogeneity of the rock mass surrounding the drilled shaft. However, since it represents the rock mass as a series of springs acting along the length of the shaft, the p-y curve approach ignores the interaction between different parts of the rock mass. Also, the p-y curve approach uses empirically derived spring constants that are not measurable material properties. Advances in computer technology have made it possible to analyze laterally loaded piles using three-dimensional (3D) finite element (FE) models. p-y curves (Hoit et al., 1997) or sophisticated constitutive relations (Wakai et al., 1999) are usually used to represent the soil or rock behavior in the 3D FE analyses. However, p-y curves have the limitations as described above. As for sophisticated soil or rock constitutive relations, it is often difficult if not impossible in design practice to obtain the parameters in the constitutive relations. Zhang et al. (2000) developed a nonlinear continuum method for analyzing laterally loaded drilled shafts in rock. The method can consider drilled shafts in a continuum consisting of a soil layer overlying a rock mass layer. The deformation modulus of the soil is assumed to vary linearly with depth while the deformation modulus of the rock mass is assumed to vary linearly with depth and then stay constant below the shaft tip. The effect of soil and/or rock mass yielding on the behavior of shafts is considered by assuming that the soil and/or rock mass behaves linearly elastically at small strain levels and yields when the soil and/or rock mass reaction force exceeds the ultimate resistance.

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9.2 SUBGRADE-REACTION (p-y CURVE) APPROACH Treating the rock as a series of springs along the length of the shaft (see Fig. 9.1), the behavior of the shaft under lateral load can be obtained by solving the following differential equation (Reese, 1997)

where Q is the axial load on the shaft; y is the lateral deflection of the shaft at a point z along the length of the shaft; p is the lateral reaction of the rock; EpIp is the flexural rigidity of the shaft; and W is the distributed horizontal load along the length of the shaft. Equation (9.1) is the standard beam-column equation where the values of EpIp may change along the length of the shaft and may also be a function of the bending moment. The equation (a) allows a distributed load to be placed along the upper portion of a shaft; (b) can be used to investigate the axial load at which a shaft will buckle; and (c) can deal with a layered profile of soil or rock (Reese, 1997). Computer programs, such as COM624P and LPILE, are available to solve equation (9.1) efficiently. COM624P (version 2.0 and higher) and LPILEPLUS can also consider the variation of EpIp with the bending moment (see O’Neill & Reese, 1999 for the detailed procedure). To solve Equation (9.1), boundary conditions at the top and bottom of the shaft also need be considered. For example, the applied shear and moment at the shaft head can be specified, and the shear and moment at the base of the shaft can be taken to be zero if the shaft is long. For short shafts, a base boundary condition can be specified that allows for the imposition of a shear reaction on the base as a function of lateral base deflection. Full or partial head restraint can also be specified. Other formula that are used in the analysis are (9.2)

(9.3)

(9.4) where V, M and S are respectively the transverse shear, bending moment and deflection slope of the drilled shaft. The major difference between various methods lies in the determination of the variation of p with y or the p-y curve, which are described below.

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Fig. 9.1 Subgrade-reaction (p-y curve) model of laterally loaded drilled shafts. 9.2.1 Linear analysis For linear analysis, the relationship between rock reaction p and shaft deflection y at any point along the shaft is assumed to be linear, i.e., (9.5) where kh is the coefficient of subgrade reaction, in the unit of force/length3; and B is the width or diameter of the shaft. Substituting Equation (9.5) into Equation (9.1) and neglecting the influence of Q and W, the governing equation for the deflection of a laterally loaded shaft with constant EpIp can be simplified as (9.6)

Solutions to the above equation may be obtained analytically as well as numerically with a computer program. The analysis of the load-displacement behavior of a drilled shaft also requires knowledge of the variation of kh along the shaft. A number of distributions of kh along the

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depth have been employed by different investigators, which can be described by the following general expression proposed by Bowles (1996): kh=Ah+Bhzn (9.7) where Ah, Bh and n are empirical constants which can be determined for a particular site by working backward from the results of lateral shaft load tests. If the rock is considered homogeneous with a constant kh down the length of the drilled shaft, the deflection u (both y and u are used to denote lateral deflection in this book) and rotation θ at the ground level due to applied load H and moment M can be calculated by (9.8a)

(9.8b)

where Lc is the critical length given by (9.9)

It should be noted that Equation (9.8) is applicable only to flexible shafts, i.e., shafts longer than their critical length defined by Equation (9.9). For non-flexible shafts, solutions in closed-form expressions or in the form of charts are also available (Tomlinson, 1977; Reese & Van Impe, 2001). 9.2.2 Nonlinear analysis In general, the relationship between rock reaction p and shaft deflection y at any point along a shaft is nonlinear. Kubo (1965) used the following nonlinear relationship for soil between reaction p, deflection y, and depth z: p=kzmyn (9.10) where k, m, and n are experimentally determined coefficients. Equation (9.10) can also be used for rock if the corresponding coefficients k, m, and n can be determined. Since Matlock (1970) developed a method for deriving the variation of p with y, or the p-y curves, for soft clay, based on field test results, a number of methods for deriving p-y curves for different soils have been developed. Some of them are listed below (for details, the reader can refer to the listed references): 1.API RP2A (1982) or Reese et al. (1974) method for sand.

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2. Bogard and Matlock (1980) method for sand. 3. API RP2A (1991) or O’Neill and Murchison (1983) method for sand. 4. API RP2A (1982) or Matlock (1970) method for soft clay. 5. API RP2A (1982) or Reese et al. (1975) method for stiff clay. 6. Integrated method for clay by Gazioglu and O’Neill (1984). 7. Pressuremeter methods for all soils (Robertson et al., 1982, 1986; Briaud & Smith, 1983; Briaud, 1986). The method, developed by Reese (1997), specifically for calculating the p-y curves for rock is described in the following section. 9.2.5 p-y curves for rock Reese (1997) presented a p-y curve method for analyzing laterally loaded drilled shafts in rock. The concepts and procedures for constructing the p-y curves for rock are as follows (Reese, 1997): (1) The secondary structure of rock, related to joints, cracks, inclusions, fractures, and any other zones of weakness, can strongly influence the behavior of the rock and thus need be taken into account when applying the method described in this section. (2) The p-y curves for rock and the bending stiffness E0Ip for the shaft must both reflect nonlinear behavior in order to predict loadings at failure. (3) The initial slope Kmi of the p-y curves must be predicted because small lateral deflections of shafts in rock can result in resistances of large magnitudes. For a given value of compressive strength, Kmi is assumed to increase with depth below the ground surface. (4) The modulus of the rock Em, for correlation with Kmi, may be taken from the initial slope of a pressuremeter curve. Alternatively, the correlations presented in Chapter 4 can be used to determine Em. (5) The ultimate resistance pult for the p-y curves will rarely, if ever, be developed in practice, but the prediction of pult is necessary in order to reflect nonlinear behavior. (6) The component of the strength of rock from unit weight is considered to be small in comparison to that from compressive strength, and therefore the weight of rock is ignored. (7) The compressive strength σc of the intact rock for computing pult may be obtained from tests of intact specimens. (8) The assumption is made that fracturing will occur at the surface of the rock under small deflections; therefore, the compressive strength of intact rock specimens is reduced by multiplication by αm to account for fracturing. The value of αm is assumed to be 1/3 for RQD of 100 and to increase linearly to unity at RQD of zero. If RQD is zero, the compressive strength may be obtained directly from a pressuremeter curve. (a) Calculation of ultimate resistance pult of rock The following expressions are used for calculating the ultimate resistance pult of rock

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(9.11a)

(9.11b) where B is the diameter of the shaft; zm is the depth below the rock surface; σc is the unconfined compressive strength of the intact rock; and αm is the strength reduction factor considering that fracturing will occur at the surface of the rock under small deflections and thus reducing the resistance of the rock. (b) Calculation of the slope of initial portion of p-y curves The slope of the initial portion of p-y curves, kmi, is estimated by Kmi=kmiEm (9.12) where Em is the modulus of the rock (mass); and kmi is a dimensionless constant which can be determined by (9.13a) kmi=500 zm≥3B (9.13b) Equation (9.13) is developed from experimental data and reflect the assumption that the presence of the rock surface has a similar effect on kmi, as was shown for the ultimate resistance pult. (c) Calculation of p-y curves Referring to Figure 9.2, the p-y curve consists of three portions. The initial and the third portions are straight-lines and the second portion is a curve. The three portions can be expressed by First Portion: p=Kmiy; y≤yA (9.14a) (9.14b) Third Portion p=pult (9.14c)

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in which ym=kmB, where km is a constant, ranging from 0.0005 to 0.00005, that serves to establish overall stiffness of curves. The value of yA is found by solving the intersection of Equations (9.14a) and (9.14b), and is shown by (9.15)

Fig. 9.2 Sketch of p-y curve for rock (after Reese, 1997). (d) Comments The equations described above for constructing the p-y curves for rock are based on limited data and should be used with caution. An adequate factor of safety should be employed in all cases; preferably, field tests should be undertaken on full-sized shafts with appropriate instrumentation. If the rock contains joints that are filled with weak soil, the selection of strength and stiffness must be site-specific and will require a comprehensive geotechnical investigation. In those cases, the application of the method presented in this section should proceed with even more caution than normal (Reese, 1997). Cho et al. (2001) conducted lateral load tests on two drilled shafts embedded in weathered Piedmont rock. These shafts were instrumented with inclinometers and strain gauges. The field data obtained from the instrumented shafts were used to backcalculate the p-y curves. A comparison of the back-calculated p-y curves with the p-y curves predicted using the method of Reese (1997) shows that the method of Reese (1997) significantly overestimates the resistance of the weathered rock.

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9.3 CONTINUUM APPROACH The continuum approach assumes the soil and rock to be a continuum. Numerical solutions were developed by assuming that the soil and rock are ideally elastic, first with the boundary element method (Poulos, 1971a, b, 1972; Banerjee & Davies, 1978) and second with the finite element method (Randolph, 1981). Most of these elastic solutions were presented in the form of charts. Randolph (1981) published approximate but convenient closed-form expressions for the response of flexible piles to lateral loading. Considering the fact that the closed-form expressions of Randolph (1981) for the lateral response of flexible piles in soils may not cover the ranges of material and geometric parameters encountered in drilled shafts in rock, Carter and Kulhawy (1992) expanded the solutions by Randolph (1981). The solutions of Carter and Kulhawy (1992) give a reasonable agreement between measured and predicted displacements for drilled shafts in rock at low load levels (20–30% capacity). At higher load levels, however, the predicted displacements are too small (DiGioia & Rojas-Gonzalez, 1993). Zhang et al. (2000) developed a nonlinear continuum approach for the analysis of laterally loaded drilled shafts in rock. The approach can consider the effect of soil and/or rock mass yielding on the behavior of shafts. 9.3.1 Linear continuum approach (a) Approach of Poulos (1971a, b, 1972) and Poulos and Davis (1980) By modeling the soil as an elastic continuum and idealizing the pile as an infinitely thin strip of the same width and bending rigidity as the prototype pile, Poulos (1971a, b, 1972) and Poulos and Davis (1980) obtained the solutions for laterally loaded piles using the boundary element method. The solutions are presented in the form of charts and can be used to predict the deflection of drilled shafts in rock. For a free head drilled shaft, the lateral deflection u and rotation θ under lateral force H and overturning moment M at ground surface are given by (9.16a)

(9.16b)

where L is the length of the shaft; EmL is the deformation modulus of the rock mass at the level of shaft tip; and IuH, IuM, IθH and IθM (note that IuM=IθH) are deflection and rotation influence factors which are a function of the drilled shaft flexibility factor KR and the rock mass non-homogeneity η: (9.17)

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(9.18) where Em0 is the deformation modulus of the rock mass at the ground surface. A homogeneous rock mass is represented by η=1, whereas η=0 represents a rock mass with zero modulus at the surface. The deflection and rotation influence factors are plotted in Figures 9.3 to 9.5 for values of η of 0 and 1. If the shaft is partially embedded, the deflection of the free-standing portion due to shaft rotation and bending can be added to the groundline deflection to obtain the deflection at the shaft head. If the drilled shaft is fixed-headed, the horizontal deflection can be obtained by putting θ =0 in Equation (9.16b) and substituting for the obtained moment in Equation (9.16a), as (9.19)

Fig. 9.3 Deflection influence factor IuH (after Poulos & Davis, 1980)

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Fig. 9.4 Deflection and rotation influence factors IuM and IθH (after Poulos & Davis, 1980)

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Fig. 9.5 Rotation influence factor IθM (after Poulos & Davis, 1980). For a single raking shaft, Poulos and Madhav (1971) have shown that the force acting on the shaft head may be resolved into axial and normal components and the shaft then treated as a vertical shaft subjected to these forces and the applied moment. (b) Approach of Randolph (1981) and Carter and Kulhawy (1992) Randolph (1981) conducted a parametric study of the response of laterally loaded piles embedded in an elastic soil continuum. The study was conducted using the finite element method and the results were fitted with closed-form expressions from which the lateral response of piles may be readily calculated. Considering the fact that the closedform expressions for the lateral response of flexible piles in soils may not cover the ranges of material and geometric parameters encountered in drilled shafts in rock, Carter and Kulhawy (1992) expanded the solutions by Randolph (1981). The expressions were derived from the results of finite element studies of the behavior of laterally loaded drilled shafts in rock. For a drilled shaft wholly embedded in rock [Fig. 9.6(a)], the shaft response can be calculated in the following way (Carter & Kulhawy, 1992): (1) The shaft is considered flexible when (9.20)

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where Ee is the effective Young’s modulus of the shaft (9.21)

in which B and EpIp are respectively the diameter and flexural rigidity of the shaft; and G* is the equivalent shear modulus of the rock mass (9.22) in which Gm and νm are respectively the shear modulus and Poisson’s ratio of the rock mass. The shaft response can then be obtained by the closed-form expressions suggested by Randolph (1981), i.e., (9.23a)

(9.23b)

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Fig. 9.6 (a) Drilled shaft wholly embedded in rock; and (b) Drilled shaft embedded in soil and rock. (2) The shaft is considered rigid when (9.24)

The shaft response can then be obtained by the following closed-form expressions (9.25a)

(9.25b)

(3) The shaft can be described as having intermediate stiffness whenever the slenderness ratio is bounded approximately as follows

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(9.26)

The finite element results show that the displacements for an intermediate case exceed the maximum of the predictions for corresponding rigid and flexible shafts by no more than about 25%, and often by much less. For simplicity, it is suggested that the shaft displacement in the intermediate case be taken as 1.25 times the maximum of either: (a) The predicted response of a rigid shaft with the same slenderness ratio L/B as the actual shaft; or (b) the predicted response of a flexible shaft with the same modulus ratio (Ee/G*) as the actual shaft. Values calculated in this way should, in most cases, be slightly larger than those given by the more rigorous finite element analysis for a shaft of intennediate stiffness. If there exists a layer of soil overlying rock as shown in Figure 9.6(b), Carter and Kulhawy (1992) assume that the complete distribution of soil reaction on the shaft is known and that the socket provides the majority of resistance to the lateral load or moment. The groundline horizontal displacement u and rotation θ can then be determined after structural decomposition of the shaft and its loading, as shown in Figure 9.7. To determine the distribution of the soil reaction, they simply assume that the limiting condition is reached at all points along the shaft, from the ground surface to the interface with the underlying rock mass, and then use the reaction distribution suggested by Broms (1964a, b). For shafts through cohesive soils (Fig. 9.8), the lateral displacement uAO and rotation θAO of point A relative to point O are given by (9.27a)

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Fig. 9.7 Consideration of soil reaction: (a) Loading and displaced shape; and (b) Decomposition of loading (after Carter & Kulhawy, 1992). (9.27b) where Ls is the thickness of the soil layer; and su is the undrained shear strength of the soil. The shear force Ho and bending moment Mo at point O are determined by HO=H−9su(Ls−1.5B)B (9.28a) 2 MO=M−4.5su(Ls−1.5B) B+HLs (9.28b) The contribution to the groundline displacement from the loading transmitted to the rock mass can then be computed by analyzing a fully rock-socketed shaft of embedded length L, subject to horizontal force HO and moment MO applied at the level of the rock mass. For shafts through cohesionless soils (Fig. 9.9), the lateral displacement uAO and rotation θAO of point A relative to point O are given by (9.29a) (9.29b)

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where γ′ is the effective unit weight of the soil; and Kp is the Rankine passive earth pressure coefficient. The shear force HO and bending moment MO at point O are determined by (9.30a)

Fig. 9.8 Idealized loading of socketed shaft through cohesive soil (after Carter & Kulhawy, 1992).

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Fig. 9.9 Idealized loading of socketed shaft through cohesionless soil (after Carter & Kulhawy, 1992). (9.30b) Example 9.1 A drilled shaft of diameter 1.0 m is to be installed 3.0 meters in siltstone. The rock properties are as follows: Unconfined compressive strength of intact rock, σc=15.0 Mpa Deformation modulus of intact rock Er=10.6 GPa RQD=70

Determine the lateral displacement and rotation of the drilled shaft at the groundline by a horizontal force of 2.6 MN at 2.5 m above the groundline. Solution:

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For simplicity, the Young’s modulus of the drilled shaft is simply assumed to be Ep=30 GPa. A Poisson’s ratio of 0.25 is selected for both the drilled shaft and the rock. The flexural rigidity of the shaft is

Using Equation (4.24), the deformation modulus of the rock mass is αE=0.0231×70−1.32=0.297 Em=0.297×10.6=3.15 Gpa and the shear modulus of the rock mass is Gm=3.15/(1+0.25)=1.26 Gpa Using Equation (9.22), the equivalent shear modulus of the rock mass is G*=1.26×(1+3×0.25/4)=1.50 Gpa Since

the shaft is considered flexible and the lateral displacement and rotation of the drilled shaft at the groundline can be obtained from Equation (9.23) as

9.3.2 Nonlinear continuum approach Poulos and Davis (1980) presented an approximate nonlinear approach for calculating the deflection of laterally loaded piles in soil. This approach uses the elastic solutions presented in the last section, but introduces yield factors. The yield factors are a function of relative flexibility and load level and allow for the increased deflection and rotation of a pile due to the onset of local yielding of the soil adjacent to the pile. This approach can also be used to calculate the nonlinear deflection of drilled shafts in rock.

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For a drilled shaft subjected to a lateral load H at an eccentricity of e above the groundline, the groundline deflection u and rotation θ can be expressed as follows: (1) Uniform modulus with depth, i.e., η=1.0

(9.31a) (9.31b) (2) Linearly increasing modulus with depth, i.e., η=0

(9.32a) (9.32b) where uelastic and θelastic are respectively deflection and rotation from elastic solutions as described in the previous section; and Fu, Fθ, F′u, and Fθ are yield deflection and rotation factors which can be found from Poulos and Davis (1980). The yield factors are functions of a dimensionless load level H/Hu, where Hu is the ultimate lateral load capacity of the equivalent rigid shaft and can be estimated using the methods presented in Chapter 8. Zhang et al. (2000) developed a nonlinear continuum approach for the analysis of laterally loaded drilled shafts in rock. This approach adopts and extends the basic idea of Sun’s (1994) work on laterally loaded piles in soil. Sun’s model treats soil as a homogeneous elastic continuum with a constant Young’s modulus, which may apply to stiff clay, and it does not consider yielding of the soil. In the nonlinear approach developed by Zhang et al. (2000), drilled shafts in a soil and rock mass continuum (see Fig. 9.10) are considered, and the effect of soil and/or rock mass yielding on the behavior of shafts is included. For simplicity, the shaft is assumed to be elastic, while the soil/rock mass can be either elastic or elasto-plastic. It is, nevertheless, possible to also check whether the shaft concrete will yield or not using standard concrete design methods, as will be briefly mentioned later. (a) Method of analysis—elastic behavior Governing equations of shaft and soil/rock mass system Consider a drilled shaft of length L, radius R and flexural rigidity EpIp, embedded within a

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Fig. 9.10 (a) Shaft and soil/rock mass system; (b) Coordinate system and displacement components; and (c) Shear force V(z) and moment M(z) acting on shaft at z (after Zhang et al., 2000). soil/rock mass system (Fig. 9.10). The deformation modulus of the soil varies linearly from Es1 at the ground surface to Es2 at the soil/rock mass interface. The deformation modulus of the rock mass varies linearly from Em1 at the soil/rock mass interface to Em2 at the shaft tip and stays constant below the shaft tip. For convenience of presentation, nonuniformity indices defined by (9.33) (9.34) are introduced. The increase of the deformation moduli of the soil and the rock mass with depth, z, can then be expressed, respectively, by (9.35a)

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(9.35b) Em=Em2 (z>Ls+L) (9.35c) By adopting the basic idea of Sun (1994), the displacements usm, νsm and usm of the soil and/or rock mass can be approximated by separable functions of the cylindrical coordinates r, θ and z as (9.36a) (9.36b) wsm(r,θ,z)=0 (9.36c) is a where u(z) is the displacement of the shaft as a function of depth; and dimensionless function representing the variation of displacements of the soil and/or rock mass in the r-direction. For the displacements of Equation (9.36), the governing equations for the shaft can be obtained as (9.37a)

(9.37b)

with boundary conditions (9.38a)

(9.38b)

(9.38c) us−um=0 (z=Ls) (9.38d)

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(9.38e) (9.38f)

(9.38g)

(9.38h) (9.38i) where us and um are the displacement components u of the shaft in the soil and in the rock mass, respectively; and ts, ks and tm, km are parameters that can be expressed as (9.39a)

(9.39b)

(9.39c)

(9.39d) where m1 and m2 are parameters describing the behavior of the elastic foundations, which can be obtained by (9.40a) (9.40b)

Function

can be obtained by solving the following equation

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(9.41) where γ is a nondimensional parameter that can be expressed as (9.42)

and

The solution to Equation (9.41) that satisfies the unit condition at the finite condition at and m2 can then be expressed as (Sun, 1994)

can be obtained and the parameters m1 (9.43a)

(9.43b) where K0( ) is the modified Bessel function of the second kind of zero order; and K1( ) is the modified Bessel function of the second kind of first-order. The shear force V(z) acting on the shaft (see Fig. 9.10) can be obtained by (9.44a)

(9.44b) and the bending moment M(z) acting on the shaft (see Fig. 9.10) can be obtained by (9.45a)

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(9.45b)

The governing differential equations and the shear force V(z) and bending moment M(z) are solved using the classical finite difference method as described below. At this point it is also possible to check if the shaft concrete yields (recall that the basic assumption is non-yielding concrete). This can be done using the calculated shear force and moment together with the axial force on the shaft and using standard concrete design methods. Finite difference model The classical finite difference method (Desai & Christian, 1977) is employed to solve the governing differential Equation (9.37). By dividing the shaft in the soil into Ns equal segments (see Fig. 9.11) and using the central difference operator, for an interior node i (i= 0, 1, 2,…, Ns), the following equation is obtained:

Fig. 9.11 Dividing shaft into segments for finite-difference analysis, and estimating reaction force p of soil and rock from shear force V (after Zhang et al., 2000).

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(9.46)

where (9.47)

in which hs=Ls/Ns. Similarly, by dividing the shaft in the rock mass into Nm equal segments (see Fig. 9.11), the following equation is obtained for an interior nodey j(j=0, 1, 2,…, Nm): (9.48)

where (9.49)

in which hm=L/Nm. Equations (9.46) and (9.48) can be written recursively for each point i=0, 1, 2,…, Ns and j=0, 1, 2,…, Nm(see Fig. 9.11), resulting in a set of simultaneous equations in u. To solve the set of equations the boundary conditions must be introduced. By incorporating the boundaiy conditions expressed by Equation (9.38), the following finite difference equations can be obtained: at z=0 (9.50a)

(9.50b)

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−us(−1)−us(1)=0 (fixed-head) (9.50c) at z=Ls (9.50d) (9.50e) (9.50f)

(9.50g)

at z=Ls+L (9.50h) (9.50i)

The set of equations [Equations (9.46) and (9.48)] is modified by introducing the boundary conditions given in Equation (9.50). The resulting equations are solved simultaneously for u by using the Gaussian elimination procedure. After the shaft displacement u is obtained, the shear force V acting on the shaft can be obtained from Equation (9.44) as: (9.51a)

(9.51b)