fhwa drilled shafts; construction procedures and lrfd design methods app-c

FHWA-NHI-10-016 C – Commentary on Axial Resistance Drilled Shaft Reference Manual C-1 May 2010

APPENDIX CCOMMENTARY ON METHODS FOR COMPUTING NOMINAL AXIAL RESISTANCE OF

DRILLED SHAFTS

C.1 SIDE RESISTANCE IN COHESIONLESS GEOMATERIALS

In Chapter 13, the nominal side resistance of a drilled shaft in cohesionless soil is modeled as the frictional resistance that can be developed over a cylindrical surface at the soil-shaft interface, given by:

� �� tanzBzBR hSNSN f C-1

in which RSN = nominal side resistance, B = shaft diameter, �z = thickness of the soil layer over which resistance is calculated, and fSN = nominal unit shearing resistance,�'h = horizontal effective stress, and �= effective stress angle of friction for the soil-shaft interface. Figure C-1 depicts a segment of drilled shaft and the resulting unit shearing resistance developed along the interface. The horizontal effective stress acts as a normal stress at the interface, and tan � is equivalent to a sliding coefficient of friction. Horizontal effective stress is expressed in terms of vertical effective stress (�'v) and the coefficient of horizontal soil stress (K = �'h/�'v), resulting in the following expression:

� �� tanKzBR vSN C-2

The last two terms in Equation C-2 often are grouped as follows:

��tan� C-3

and

fSN = �'v � C-4

in which � = side resistance coefficient and fSN = nominal unit side resistance. In terms of the coefficient � total side resistance for a cohesionless soil layer is then given by:

� �vSN zBR �� C-5

Two approaches for evaluating the coefficient � have been used in U.S. practice. In one approach, trends of � versus depth (z) determined from field load tests are used to develop empirical relationships between � and z. O’Neill and Hassan (1994) refer to this as the “depth-dependent � method” and this is the basis of the equations given in the previous version of this manual (O’Neill and Reese, 1999) as well as the current AASHTO LRFD Bridge Design Specifications (2007). A more fundamental approach is to evaluate � in terms of K and ��Each approach is reviewed, followed by a discussion on the relative merits of each.


��'h = K �'v

fSN = �'h tan �

Qc

Figure C-1 Frictional Shear Model of Drilled Shaft Side Resistance in Cohesionless Soil

The depth-dependent � method is given by the following expressions:

For sandy soils:

z135.05.1� � for N60 > 15 0.25 < � < 1.2 C-6

For gravelly sands and gravels:

� � 75.0z06.00.2 �� for N60 > 15 0.25 < � < 1.8 C-7

For all cohesionless soils:

� �z135.05.115N

� 60 � for N60 < 15 C-8

Unit side resistance calculated by the above expressions is limited to an upper bound value of 4,000 psf unless higher values are shown to be valid by load tests. This value is not a theoretical limit, but was reported to be the largest value measured when Equation C-6 was first proposed by O’Neill and Reese (1978). In the AASHTO (2007) LRFD specifications a resistance factor of 0.55 is adopted for use with the above expressions for � based on the recommendation of Allen (2005).

Rollins et al. (2005) proposed the following modified form of the depth-dependent � method for gravels (> 50 percent gravel size)

� �ze 085.04.3� �� 0.25 < � < 3.0 C-9


Rollins et al. note that almost all of the gravels in the database used to establish Equation C-9 exhibited N-values greater than 25, with values up to 100. According to the authors, Equation C-9 therefore is not applicable to low blow count gravels but would apply to gravels with N60 > 50.

Equations C-6 through C-9 are used to assign nominal values to the coefficient � solely as a function of depth (z). A basic premise of the depth-dependent approach, as described by O’Neill and Hassan (1994), is that drilled shaft construction disturbs the soil, reducing its density and allowing relaxation of horizontal stress. It is further assumed that disturbance can reduce the soil friction angle to a lower-bound value corresponding to the critical state void ratio (O’Neill and Reese, 1988 and 1999). Based on this reasoning, detailed evaluations of in-situ strength and state of stress are not warranted because the in-situ properties are changed by construction and the changes cannot be predicted reliably, in particular if the designer does not know ahead of time what construction methods will be used. Instead, near lower-bound values of � back-calculated from field load tests are assumed to provide a conservative approximation of unit side resistance. Using this model, the nominal value of � in a cohesionless soil layer with N60 = 15 is assumed to have the same value of � as a cohesionless soil layer at the same depth but with N60 = 50.

The load test results used by O’Neill and Hassan (1994) to develop Equations C-6 through C-8 are shown in Figure C-2 as back-calculated values of � versus depth. The trend line labeled “Improved lower-bound design relation N > 15” corresponds to Equation C-6. For a full discussion of the data and trend lines shown in Figure C-2 the reader should consult O’Neill and Hassan (1994). It can be observed that �exhibits a wide range of values at shallow depths, and there is a general trend of decreasing � with increasing depth. This trend, which is reflected in Equations C-6 through C-9, can be attributed to higher values of K near the surface, where many soil deposits are overconsolidated. Overconsolidation occurs in response to mechanical processes (burial and subsequent erosion) but also from fluctuations in the water table, capillary rise, desiccation, and aging. The effect of preconsolidation is to increase the in-situ horizontal stress and, therefore, �. With increasing depth, most soil deposits trend toward a normally consolidated state, a lower value of K, and therefore a lower value of �� From Equation C-6, � is assumed to reach a constant minimum value of 0.25 below a depth of 85 ft, corresponding approximately to a normally consolidated value of K for a friction angle of 22 degrees. While the depth-dependent � method has been found to provide conservative estimates of nominal side resistance for most soil profiles, its failure to account explicitly for the in-situ state of stress and the soil shear strength imposes a limitation on designers to model properly the mechanisms of soil-structure interaction that control side resistance.

The more fundamental approach, as presented for example by Kulhawy (1991), Mayne and Harris (1993), Chen and Kulhawy (2002), Kulhawy and Chen (2007), and others, is to evaluate separately the parameters that are lumped into the coefficient �. From Equation C-3, these are the interface friction angle (�) and the coefficient of horizontal soil stress (K). For concrete cast in place against soil, as in a drilled shaft, the interface is assumed to be rough and � can be taken equal to the effective stress friction angle of the soil:

�� C-10

The value of soil friction angle can be determined through correlation to common in-situ tests such as SPT N-values or CPT cone resistance as presented in Chapter 3. When SPT results are available, the recommended correlation for sands and gravels is:

� �� 601log2.95.27 N�� C-11

Equations C-6 through C-9 are used to assign nominal values to the coefficient � s solely as a function of Equations C-6 through C-9 are used to assign nominal values to the coefficient � solely as a function of depth (z). A basic premise of the depth-dependent approach, as described by O’Neill and Hassan (1994), depth (z). A basic premise of the depth-dependent approach, as described by O’Neill and Hassan (1994), is that drilled shaft construction disturbs the soil, reducing its density and allowing relaxation of il, reducing its density and allowing relaxation of horizontal stress. It is further assumed that disturbance can reduce the soil friction angle to a lower-bound urbance can reduce the soil friction angle to a lower-bound value corresponding to the critical state void ratio (O’Neill and Reese, 1988 and 1999). Based on this value corresponding to the critical state void ratio (O’Neill and Reese, 1988 and 1999). Based on this reasoning, detailed evaluations of in-situ strength and state of stress are not warranted because the in-situ reasoning, detailed evaluations of in-situ strength and state of stress arproperties are changed by construction and the changes cannot be predicted reliably, in particular if the properties are changed by construction and the changes cannot be predicted reliably, in particular if the designer does not know ahead of time what construction methods will be used. Instead, near lower-bound designer dvalues of

r doe� b

does not know ahead of time what constructi back-calculated from field load tests are assumed to provide a conservative approximation of back-calculated from field load tests are assumed

unit side resistance. Using this model, the nominal value of med t of � i

ed to provide a conservative approximation of in a cohesionless soil layer with N60 = 15 is unit side resistance. Using this mod

assumed to have the same value of ode

of � adel, the nominal value of in a cohesionless soil layer with N

as a cohesionless soil layer at the same depth but with N60 = 50.

The load test results used by O’Neill and Hassan (1994) to develop Equations C-6 through C-8 are shown The load test results used by O’Neill and Hin Figure C-2 as back-calculated values of

d Has of � v

Hassan (1994) to develop Equations C-6 through C-8 are shown versus depth. The trend line labeled “Improved lower-bound in Figure C-2 as back-calculated values of � versus depth. The trend line labeled “Improved lower-bound

design relation N > 15” corresponds to Equation C-6. For a full discussion of the data and trend lines design relation N > 15” corresponds to Equation C-6. shown in Figure C-2 the reader should consult O’Neill and Hassan (1994). It can be observed that �shown in Figure C-2 the reader should consult O’Neexhibits a wide range of values at shallow depths, and there is a general trend of decreasing � w with exhibits a wide range of values at shallow depths, and there is a general trend of decreasing �increasing depth. This trend, which is reflected in Equations C-6 through C-9, can be attributed to higher increasing depth. This trend, which is reflected in Equations C-6 through C-9, can be attributed to higher values of K near the surface, where many soil deposits are overconsolidated. Overconsolidation occurs in values of K near the surface, where many soil deposiresponse to mechanical processes (burial and subsequent erosion) but also from fluctuations in the water response to mechanical processes (burial and subsequent erosion) but also from fluctuations in the water table, capillary rise, desiccation, and aging. The effect of preconsolidation is to increase the in-situ table, capillary rise, desiccation, and aging. The effect of preconshorizontal stress and, therefore, �. With increasing depth, most soil deposits trend toward a normally . With increasing depth, most soiconsolidated state, a lower value of K, and therefore a lower value of of

soil d of �� F

il deposits trend toward From Equation C-6,

ard a� i

rd a normally is assumed � From Equation C-6, �

to reach a constant minimum value of 0.25 below a depth of 85 ft, corresponding approximately to a to reach a constant minimum value of 0.25 below a depth of 85 ft, corresponding approximanormally consolidated value of K for a friction angle of 22 degrees. While the depth-dependent

imate� mately to a

method normally consolidated value of K for a friction angle of 22 degrees. While the depth-dependent �has been found to provide conservative estimates of nominal side resistance for most soil profiles, its has been found to provide conservative estimates of nominal side resistance for most soil profiles, its failure to account explicitly for the in-situ state of stress and the soil shear strength imposes a limitation offailure to account explicitly for the in-situ state of stress and the soil shear strength imposes a limitatio ofon designers to model properly the mechanisms of soil-structure interaction that control side resistance.


Figure C-2 Variations of � with Depth (O’Neill and Hassan, 1994)

The value of K, coefficient of horizontal stress, is a function of the in-situ (at-rest) value, Ko, and changes in horizontal stress that occur in response to drilled shaft construction, which can be expressed in terms of the ratio K/Ko. In this approach, the coefficient beta can be expressed as:

��

��

�� tan

KKK

oo C-12

To apply this approach it is necessary to select values of Ko and construction-related changes in terms of K/Ko. As described by Chen and Kulhawy (2002), early studies suggested that K/Ko be taken as 1 for dry construction, 5/6 for casing construction, 2/3 for slurry construction, and 11/12 for combined dry/casing construction. However, back-analyses of field load tests using the approach described herein by Chen and Kulhawy (2002) suggest these values are overly conservative and that K/Ko is close to 1 for properly constructed shafts. As a first-order approximation it will be assumed that K/Ko = 1, and therefore the operative value of K equals the in-situ value Ko. For simple virgin loading-unloading of “normal soils” that are not cemented, the Ko value increases with overconsolidation ratio (OCR) according to (Kulhawy and Mayne, 1982):

Ko = (1 – sin �') OCRsin �' < Kp C-13

v

pOCR��

C-14

where �'p = effective vertical preconsolidation stress. In Equation C-13, Ko is limited to an upper bound value equal to the Rankine coefficient of passive earth pressure, Kp. A variety of methods have been proposed for evaluation of either Ko or �'p by correlations with in-situ test measurements. For a practical


estimate based on the most commonly used in-situ test (SPT) the following correlation is suggested by Mayne (2007):

� �m60

a

p N47.0p

��

C-15

where m = 0.6 for clean quartzitic sands and m = 0.8 for silty sands to sandy silts (e.g., Piedmont residual soils). Kulhawy and Chen (2007) suggest the following correlation provides a good fit for gravelly soils:

60a

p N15.0p

��

C-16

Substituting Equations C-12 through C-15 into Equation C-11 leads to the following approximation of �for cohesionless soils:

� � ��

��

�

��

��

tanKtansin1 p

sin

v

p C-17

where �'p is estimated by Equation C-15 for sandy soils and by Equation C-16 for gravelly soils. The calculated value of � is substituted into Equation C-5 for determination of nominal side resistance RSN.The advantage of this approach is that it allows the designer to account for site-specific variations in horizontal stress and soil strength in a rational manner. The principal limitation to this approach is that in-situ stress and soil strength are determined through correlation to N-values, and therefore are subjected to all sources of error and variability associated with the SPT. Furthermore, this method, in terms of the equations presented above, has not been evaluated for calibration of resistance factors using the procedures required for incorporation into the AASHTO LRFD specifications. However, a simple calibration to allowable stress design (historical practice), assuming a factor of safety = 2.0, yields a resistance factor of 0.45. This value is recommended until additional calibration studies are conducted.

The database used by Chen and Kulhawy (2002) included 100 axial load tests on drilled shafts at 53 sites. Figure C-3 shows the back-calculated values of � versus depth for all 100 tests. The general trend is similar to Figure C-2, showing high values of � and large scatter at shallow depths and decreasing values of � and less scatter with increasing depth, converging to the normally consolidated range at depths greater than 100 ft (30 meters) or so. Both uplift (54 tests) and compression (46 tests) are included. The results demonstrate that � values are essentially the same for uplift and compression, varying by less than 4 percent. Test shaft depths ranged from 4.5 ft to 200 ft and diameters ranged from less than 1 ft to 6.5 ft. The range of depth to diameter ratios (L/B) is 2.5 to 56.4. Soil types at the test sites are dominated by sands and range from gravelly sand to sand to silty sand. A few cemented sand sites are included.

All load test results were evaluated in a consistent manner to establish nominal resistances using the L1-L2interpretation described by Hirany and Kulhawy (1989, 2002). The basic concept is depicted graphically in terms of a normalized load versus normalized displacement curve as shown in Figure C-4. The elastic limit is defined at L1, and occurs on average at a normalized displacement of �0.4%. This is followed by a nonlinear yield region. The end of this region is denoted by L2, the interpreted failure load, which is defined as the point where a final linear region begins. On average, data from compression load tests show that the point L2 occurs at a normalized displacement of �4% (as shown in Figure C-4). For uplift,


L2 occurs at an average value of absolute displacement of 0.5 inch. For each compression load test, nominal resistance was established graphically by the load at 4% normalized displacement. For uplift tests, nominal resistance was taken as the load at 0.5 inch upward displacement. According to its authors, attributes of the L1-L2 interpretation for drilled shafts are: independence of scale and individual judgment; does not involve extrapolation of the measured load-displacement curve; accounts for foundation diameter; and considers the shape of the load-displacement curve. Further background on this method, is given in the two references cited above.

Figure C-3 Variation of Measured � with Depth (Chen and Kulhawy, 2002)

Of the 100 load tests considered by Chen and Kulhawy (2002), 58 tests were accompanied by data that were deemed sufficient to make predictions of � on the basis of soil properties. At the majority of sites, soil properties were characterized by correlations with SPT N-values. An updated analysis of the 2002 data, with additional data from load tests on shafts in gravel and cobbles, is given by Kulhawy and Chen (2007). For each test accompanied by suitable information on soil properties, the coefficient � was predicted using Equation C-12 and �' was evaluated by Equation C-11. The soil profile along the shaft was divided into several layers and average Ko and �' values were evaluated at the mid-depth of each layer. These were used to calculate a � value for each layer and then weighted averages were used to calculate an average � over the length of the shaft. Figure C-5 shows a comparison between the ratio of values of � from load test measurements (�m) to the predicted values (�p) versus normalized depth (depth/diameter). Analyses of these data give a mean �m/�p = 1.16. These results suggest that the analysis model yields predictions of side resistance to a level of accuracy and reliability that is acceptable for geotechnical practice. Furthermore, the results are consistent over a range of cohesionless soil profiles including sand, gravel, and cobbles, provided the soil parameters are evaluated properly.


Figure C-4 Average Normalized Load-displacement Curve that Forms the Basis of Load Test Interpretation for Compression (Chen and Kulhawy, 2002).

Note: DS = drilled shaft foundations; PIF = pressure injected footings

Figure C-5 Ratio of Measured to Predicted � Values Versus Depth Ratio for Drilled Foundations in Cohesionless Soils (Kulhawy and Chen, 2007)


The approach described above is adaptable to other in-situ methods that allow measurement of horizontal soil stress and its variation with depth, such as pressuremeter test (PMT) and flate plate dilatometer test (DMT). The measured horizontal effective stress is then used directly in Equation C-1 to evaluate nominal side resistance. When cone penetration test (CPT) results are available, the following expression given by Mayne (2007) provides a direct estimate of OCR from measured cone tip resistance values:

� �

��

��

�

��

��

�

�

!

"

��

��

� ��

��

��

�

27.0sin1

31.0

a

v

22.0

a

t

psin1

p192.0

OCR

q

C-18

in which qt = cone tip resistance, �'v = vertical effective stress, and pa = atmospheric pressure in the same units as qt and �'v. The OCR value is then substituted into Equation C-13 for an estimate of K, which is used to evaluate � and nominal side resistance. Equation C-18 was derived empirically from statistical evaluations on 26 different series of CPT calibration chamber tests. Cohesionless soils used in the tests were primarily quartz and feldspar sands with OCR values ranging from 1 to 15.

In Equation C-17, it is assumed that no change in horizontal stress, and therefore no change in K, occurs as a result of construction. Analysis of load test data demonstrates this assumption is valid for dry, slurry (wet-hole), and casing methods of construction with minimal sidewall disturbance, proper handling of slurry and casing, and prompt placement of concrete (Chen and Kulhawy, 2002). However, when these aspects of construction quality are not controlled properly, the coefficient K can be reduced to 2/3 of its initial in-situ value (Ko). Judgment and accurate knowledge of field realities are therefore needed to assess the applicability of the design equations to individual projects. The recommended approach is to take the necessary measures that will assure the highest quality of construction, thereby justifying the use of the design equations presented above. When there is reason to believe quality construction cannot be achieved, the drilled shaft designer has the option to apply reduced values of K and/or �' for computing side resistance.

Discussion

The recommendation given herein to adopt the �-method with separate evaluation of K and ��rationalmethod), versus the depth-dependent �-method, represents a departure from previous FHWA practice and current AASHTO specifications (2007). Justification for these revisions is based on both theoretical and empirical arguments. First, it was recognized very early that a frictional shear model of side resistance, as expressed by Equation C-2, provides the proper effective stress analysis for side resistance of deep foundations in cohesionless materials. For example, see Tomlinson (1963), Vesic (1977), O’Neill and Reese (1978), Kulhawy et al. (1983), or Turner and Kulhawy (1990). However, a lack of data from load tests on drilled shafts in cohesionless soils limited the development and verification of specific methods for proper assessment of the parameters K and tan � needed to apply this model to drilled shaft design. The depth-dependent � method was introduced by O’Neill and Reese (1978) as an interim, empirical approach that would provide conservative estimates of side resistance given the uncertainties associated with construction effects and the limited data available at the time. The database of load tests against which the depth-dependent � values were evaluated consisted of only two load tests in sand and 18 tests in “mixed” soil profiles of sand and clay (Reese and O’Neill, 1988). Since that time, a significant amount of additional information available from load tests has made it possible to move beyond depth-dependent empirical equations for � and into the realm of methods based on the correct theoretical model. The


resulting research published over the past 20 years demonstrates the validity of applying an analysis model that incorporates careful geotechnical evaluation of the soil parameters that determine side resistance as expressed by Equation C-2. This research has consisted of careful studies involving analysis of data from full-scale load tests and also through development of improved correlations between in-situ tests, in particular SPT and CPT measurements, and horizontal stress in soils. The reader is referred to the following references that present research supporting this approach: Kulhawy (1991), Mayne and Harris (1993), O’Neill and Hassan (1994), Chen and Kulhawy (2002), and Kulhawy and Chen (2007).

The practice of lumping K and � into a single parameter (�) and then evaluating � solely as a function of depth neglects the influence of geology, material type, and stress history. Use of this method restricts the ability of a foundation engineer to design a drilled shaft on the basis of site-specific ground conditions. While this approach may have been warranted in the past as a result of construction-related uncertainties and insufficient data, compelling evidence now exists to demonstrate that these factors can be taken into account in engineering practice. As stated by O’Neill and Hassan (1994), the rational method is “clearly superior to the depth dependent � method from a soil mechanics perspective” and “should give more accurate values for �” than the depth-dependent � method.

A further important reason for adopting the rational �-method is that the previous version of this manual (O’Neill and Reese, 1999) adopted this approach for cohesionless materials with N60 > 50 (cohesionless IGM). This created a discrepancy between the design equations recommended for shafts in cohesionless soils and the method for cohesionless IGM. Adoption of a single approach therefore provides a unified design model for all cohesionless geomaterials.

For strictly illustrative purposes (not for design), Figure C-6 shows curves of � versus depth as calculated for three cases: rational method with N60 = 15, rational method with N60 = 50, and the depth-dependent beta method for sand with N60 > 15 (Equation C-6). For the rational method, � at shallow depths is limited to the value corresponding to a depth of 7.5 ft, which corresponds approximately to a vertical effective stress of �900 psf. At lower confining stress, the correlations for effective stress friction angle and preconsolidation stress have not been validated and it would be prudent to limit � to the values corresponding to this depth. The unit weight used to calculate � values in Figure C-6 is assumed to be 120 pcf and constant with depth, and no water table effects are considered. The figure serves to illustrate the restriction imposed by the depth-dependent method (dashed line) on a designer’s ability to assign nominal values of side resistance to cohesionless soil layers. For soils with relatively high N-values and quality construction, overly conservative estimates of side resistance will result, diminishing the cost advantages of drilled shaft foundations. The rational method correctly provides the designer with a tool to assign higher values of side resistance to layers exhibiting higher N-values, and lower nominal side resistance values to layers exhibiting lower N-values. This approach leads to designs that are both more cost-effective and more reliable by accounting for site-specific ground conditions. The curves for �within the range of 15 < N60 < 50 also provide a much improved match to the distribution of � versus depth as illustrated in Figure C-2 and Figure C-3.

Application of the rational approach for evaluation of nominal side resistance in cohesionless soils can be summarized by the following steps. For each cohesionless geomaterial layer:

# Establish mean value of N60 and mean vertical effective stress �'v# Establish �' by correlation to N60 and �'v# Establish �'p by correlation to N60 by Equation C-15 if sand; by Equation C-16 if gravel # Calculate Ko using estimated values of �'p, �', and �'v (Equation C-13) # Calculate �by Equation C-17 and average nominal unit side resistance fSN by Equation C-4 # Calculate nominal side resistance RSN by Equation C-1


0

10

20

30

40

50

60

70

80

90

100

0 0.5 1 1.5 2 2.5 3 3.5

Beta Coefficient, �

Dep

th (f

t)

N60 = 50

N60 = 15All N60 > 15

Figure C-6 Comparison of � Values Computed by Rational Method and Depth-Dependent Method

C.2 BASE RESISTANCE BY BEARING CAPACITY ANALYSIS

Bearing capacity theory provides a theoretical framework for evaluating the base (tip) resistance of deep foundations in soil and rock. The degree to which bearing capacity analyses are applicable depends upon the extent to which the assumed base conditions correspond to actual field realities. Experience and observations from load tests suggest that bearing capacity theory provides reliable estimates of base resistance for shafts bearing on cohesive soils. The method recommended in Chapter 13 for nominal base resistance of shafts in cohesive soils (Equation 13-18) is based on bearing capacity analysis. For shafts bearing on cohesionless soils, experience shows that full mobilization of the theoretical bearing capacity generally requires downward displacements that average approximately 10 percent of base diameter. Load test results also show higher variability in base resistance in cohesionless soils, possibly due to disturbance of the soil caused by stress release, seepage, and drilling. Nevertheless, when the base load-displacement behavior is accounted for properly, bearing capacity theory provides a useful tool for evaluating drilled shaft strength and service limit states. In this Appendix, bearing capacity equations applicable to drilled shafts in cohesionless and cohesive soils are presented in greater detail than in Chapter 13, along with recommendations for evaluating the various parameters.

Bearing capacity equations have not been applied widely to the design of drilled shafts in rock. The empirical approach presented in Chapter 13, based on strength of intact rock, is recommended for routine applications. However, recent advancements in characterization of rock mass strength, for example use of Geological Strength Index (GSI) for fractured rock mass, make it possible to formulate analytical expressions for base resistance from bearing capacity theory. Future improvements in base resistance predictions are likely when considered within the framework of bearing capacity theory, as described herein.


Nominal base resistance is the product of the nominal unit base resistance (qBN) and the cross-sectional area of the shaft base (Abase), or:

BN

2

baseBNBN q4

�BAqR C-19

When bearing capacity theory is applied, the nominal unit base resistance is taken as the ultimate bearing capacity (qult) of the soil or rock beneath the shaft base. The general form of the solution to the bearing capacity equation is given by:

qrqdqsq�r�d�s�crcdcscult ��qN��N�B0.5��cNq �� C-20

in which c = soil cohesion, $ = soil unit weight, q = vertical stress at the shaft base elevation (%$i�zi), Nc,N$, and Nq = bearing capacity factors, and the & terms are modifications factors to account for foundation shape (s), depth (d), and rigidity (r). The first subscript indicates the term in Equation C-20 to which the & factor applies.

The bearing capacity factors are functions of soil friction angle �, as follows:

��

�� tan�2

q e245tanN C-21

� � �� cot1NN qc (as �� 0, Nc = 5.14) C-22

� � �� tan1N2N q� C-23

For a circular cross section (drilled shaft), expressions for the modification factors are given in TABLE C-1. Application of Equation C-20 to drilled shaft base resistance is considered for the three cases of drained loading in soil, undrained loading in soil, and rock.

C.2.1 Drained Loading

Fully-drained response can be assumed for shafts bearing on cohesionless soils or for shafts bearing on cohesive soils where the long-term condition may be critical. The latter case generally applies to heavily overconsolidated cohesive soils. For drained loading, the problem is analyzed in terms of effective stress and it is usually assumed that the effective stress cohesion c' is zero. For these assumptions, and considering values for the modification factors as given in TABLE C-1, Equation C-20 reduces to the following:


qrqdqsq�r�ult ��Nq�N�B0.3q � C-24

in which $ = average effective unit weight, N$ and Nq are functions of soil effective stress friction angle (�'), q = vertical effective stress at the base elevation, and the remaining & terms are evaluated from Table C-1. The soil properties (�', $ , and Irr) are average values over the depth interval extending from the base of the shaft to one shaft diameter below the base.

TABLE C-1 MODIFICATION FACTORS FOR CIRCULAR FOUNDATIONS (KULHAWY 1991)

Modification Symbol Value

&csc

qN

N�1

&$s 0.6

Shape

&qs 1 + tan �

&cd ��

� !

" ��

�'

'tan

1

c

qdqd N

&$d 1

Depth

&qd � � ��

� !

"��

��

BD12 tan

180sin1tan21 ��

&cr ��

� !

" ��

�'

'tan

1

c

qrqr N

&$r &qr

Rigidity

&qr � � � �()*

+,-

��

� !

"�

��

��sin1

logsin07.3tan8.3exp 10 rrI

To evaluate the rigidity modification factors, the soil rigidity index (Ir) must be evaluated and compared to the critical rigidity index (Vesic 1975). For drained loading and c' = 0, the soil rigidity index is given by:

� � �tanq�(12EI

ad

dr �� C-25

in which Ed = soil drained modulus, .d = drained Poisson’s ratio, and aq = average vertical effective stress from the base to one diameter below the base. Drained modulus can be estimated on the basis of relative density or correlated to in-situ test results for cohesionless soils (Chapter 3). For cohesive soils, drained modulus must be evaluated from consolidated-drained (CD) triaxial compression tests on undisturbed samples. Poisson’s ratio of granular soils ranges from 0.1 to 0.4 and can be estimated from (Kulhawy, 1991):


.d = 0.1 + 0.3 �rel C-26

in which the relative friction angle (�rel) ranges from 0 to 1 and is given by:

��

�

254525

rel�

�� C-27

Equation C-27 has not been validated for calcareous or cemented soils. The reduced rigidity index utilized in the expressions given in Table C-1 accounts for volumetric strain and is given by (Vesic 1975):

�I1II

r

rrr �

C-28

in which � = volumetric strain, which can be estimated by:

aa

rel

/pq10.005�

�� C-29

where pa = atmospheric pressure in the same units as aq . Equation C-29 also has not been validated for cemented or calcareous soils.

The reduced rigidity index (Irr) is compared to the critical rigidity index (Irc) given by (Vesic 1975):

��

� !

"��

��

�2�45cot2.85exp0.5I rc

� C-30

If Irr > Irc, the soil beneath the base behaves as a rigid-plastic material resulting in general shear failure mode and the rigidity modification factors are equal to 1.0. If Irr < Irc, the assumption of rigid-plastic behavior is not satisfied and local or punching shear failure mode is likely. In this case, the rigidity modification factors will be less than 1.0 and should be evaluated from the expressions given in Table C-1.

C.2.2 Undrained Loading

Undrained bearing capacity is evaluated in terms of total stress with � = 0 and c = undrained shear strength (su). The bearing capacity factors become: Nc = 5.14, N$ = 0, and Nq = 1.0. For � = 0 the modification factors applied to the third term in Equation C-20 (&qs, &qd, &qr) are equal to 1.0, and the shape factor &cs = 1.2. Equation C-20 then becomes:

q��c6.17q crcduult � C-31


in which q = total vertical stress at the base elevation (= %$i�zi) and $ indicates soil total unit weight. The remaining modification factors are given by the expressions in Table C-2.

TABLE C-2 MODIFICATION FACTORS FOR CIRCULAR FOUNDATION, � = 0

Modification Symbol Value

Depth &cd ��

� !

"��

��

BD1tan

18033.01 �

Rigidity &cr rI10log60.044.0 �

The undrained rigidity index is required to evaluate the rigidity modification factor, and is given by:

� � u

u

uu

ur c3

Ec�12

EI �

C-32

in which Eu = soil undrained modulus and .u = undrained Poisson’s ratio which is taken as 0.5 for saturated cohesive soils. Considering that zero volume change occurs during undrained loading, the rigidity index is not reduced to account for volume change. The rigidity index is compared to the critical rigidity index, which by Equation C-30 is Irc = 8.64. If Ir > Irc, general shear failure mode is assumed and &cr = 1.0. If Ir < Irc, local or punching shear is assumed and &cr is less than one and is computed by the expression given in Table C-2.

Examination of the depth factor &cd as given in Table C-2 shows that as the depth to diameter ratio of a shaft (D/B) goes to infinity, &cd reaches an upper bound value of 1.52, and Equation C-31 becomes:

q�s9.37q cruult � C-33

Practically, when D/B reaches 5, the first term in Equation C-33 is approximately equal to 9. Furthermore, if general shear failure mode occurs, which is often the case when the shaft is bearing on stiff clay (su > 2,000 psf), then &cr = 1. Equation C-33 forms the basis of the recommendation given in Chapter 13 (see Equation 13-18) for approximation of undrained bearing capacity as:

qBN = N*c su C-34

with values of the bearing capacity factor N*c given in Table 13-2 and ranging from 6.5 to 9.0 depending

on the range of undrained shear strength of the cohesive soil beneath the base of the shaft (O’Neill and Reese 1999). The recommendations given in Chapter 13 are suitable for routine design. More refined analyses can be conducted by application of Equation C-33 with evaluation of the various factors as described above. In most cases the computed nominal base resistance will not differ significantly.