calibration of side, tip, and total resistance factors for load and resistance factor design of...

38

Transportation Research Record: Journal of the Transportation Research Board, No. 2310, Transportation Research Board of the National Academies, Washington, D.C., 2012, pp. 38–48.DOI: 10.3141/2310-05

Louisiana Transportation Research Center, Louisiana State University, 4101 Gourrier Avenue, Baton Rouge, LA 70808. Corresponding author: M. Y. Abu-Farsakh, [email protected].

The first AASHTO LRFD specifications were introduced in 1994, at which time the proposed resistance factors for deep foundations had been obtained by fitting to the allowable stress design, because of the lack of a good load test database (2). Although this calibration approach took full advantage of past successful design experience, it could not provide the benefits of a reliability-based design approach. Many researchers have been working to develop a reasonable approach to implement the LRFD methodology for bridge substructure design and to determine the appropriate resistance factors for different regional soil conditions (3–10).

Drilled shafts have been widely used to support individual bridge columns. The total resistance of axially loaded drilled shafts consists of side resistance and tip resistance. Three resistance factors—tip, side, and total—need to be calibrated to achieve a reliable LRFD of drilled shafts. Because of either the limited number of drilled shaft load tests or the low quality of the load tests, there have been limited successes in the calibration of the separated resistance factors for drilled shafts. Paikowsky et al. (4) calibrated the resistance factors for drilled shafts through the use of a database developed by the University of Florida, FHWA, and O’Neill et al. (11). Resistance factors for total nominal resistance and side resistance were cali-brated for drilled shafts in different types of soil, and the effects of construction methods were considered. The side resistance factor calibrated in the Paikowsky et al. study was for the side friction of drilled shafts whose resistances were primarily provided by side friction. Because the bearing components were not separated, the study provided a total resistance factor. To demonstrate the change in the drilled shaft design method in the AASHTO LRFD specifications (12, 13), Allen recalibrated the resistance factors for drilled shafts through the use of the databases reported in previous literature by fitting to the allowable stress design, as well as by using the Monte Carlo simulation method (14). The databases used were from NCHRP Reports 507 (4) and 343 (15). Yang et al. (3) calibrated the resistance factor for side resistance estimated through the O’Neill and Reese method (12) on the basis of data from 19 Osterberg cell (O-cell) tests in Colorado, Kansas, and Missouri. A side resistance factor of 0.69 was obtained from the Yang et al. calibration. On the basis of the top-down test data from drilled shafts collected in NCHRP Project 24-17, Liang and Li (16) calibrated the total resistance factors of drilled shafts, designed using the O’Neill and Reese method (12), through the Monte Carlo simulation approach. The authors found resistance factors of 0.45 for clay, 0.50 for sand, and 0.35 for mixed soil.

The geotechnical research team at the Louisiana Transportation Research Center collected high-quality load test data from drilled shafts in Louisiana and Mississippi. The effect of the interpretation criterion on the total resistance factor had been evaluated in a previous

Calibration of Side, Tip, and Total Resistance Factors for Load and Resistance Factor Design of Drilled Shafts

Murad Y. Abu-Farsakh, Xinbao Yu, and Zhongjie Zhang

This paper presents a reliability-based analysis for the calibration of side, tip, and total resistance factors for axially loaded drilled shafts. Twenty-two drilled shafts that had been tested with the Osterberg cell (O-cell) method were collected from project archives at the Mississippi Depart-ment of Transportation and the Louisiana Department of Transportation and Development. This database was carefully selected to represent the typical subsurface soil conditions and design practice in Louisiana. The prediction of the load–settlement curves of the drilled shafts from soil borings was determined with the FHWA O’Neill and Reese design method and the SHAFT 6.0 computer program. The interpreted and predicted axial nominal resistances for the drilled shafts were deter-mined with the FHWA 5% shaft diameter settlement criterion from predicted load–settlement curves. The measured nominal side, tip, and total axial resistances for drilled shafts were determined from the O-cell measurements. Statistical analyses were first performed to compare the predicted nominal axial resistances and the measured nominal resistances for the drilled shafts. In general, the selected design method under-estimated the measured drilled shaft resistances. The first-order reliability and the Monte Carlo simulation methods were selected to determine the side, tip, and total resistance factors under the Strength I limit at the target reliability index of 3.0. The resulting calibrated side, tip, and total resistance factors were 0.3, 0.6, and 0.7, respectively.

Modern design of civil engineering structures requires an accurate characterization of load and resistance uncertainties. The conventional allowable stress design, also called the working stress design, incor-porated all the uncertainties associated with loads and resistances together through the use of a safety factor. Load and resistance factor design (LRFD) separates the load and resistance uncertainties on the basis of the limit state design methodology. Rigorous reliability approaches have been used to calibrate load and resistance factors for engineering infrastructures, for which uncertainties can be more quantitatively defined (1). These approaches can result in compatible design reliability between the superstructure and the substructure. As a result, a more efficient, and probably more economical, design can be achieved by adopting the LRFD approach. Therefore, the LRFD methodology has gained popularity and has become the mandatory design method for all bridge projects funded by FHWA.

Abu-Farsakh, Yu, and Zhang 39

study (17). The main objective of this study was to calibrate the tip (ϕtip), side (ϕside), and total (ϕtotal) resistance factors for the LRFD of axially loaded drilled shafts. Twenty-two drilled shaft cases that were tested with the O-cell method and met or closely met the FHWA settlement criteria were selected for this study. The FHWA method suggested by O’Neill and Reese was adopted for the design of the drilled shafts (12). The nominal resistance of the drilled shafts was determined by the FHWA 5%-B settlement interpretation criterion, in which B is the diameter of the drilled shaft. Statistical analyses were first conducted to evaluate the FHWA design method for the prediction of the measured drilled shaft resistance (12). Reliability analyses were then conducted on the collected database through the first order reliability method (FORM) and the Monte Carlo simulation approach to calibrate resistance factors (ϕtip, ϕside, and ϕtotal).

Background

The design of drilled shaft foundations depends on the resistance of shafts (R) and the predicted structural loads (Q). Variables R and Q both have various sources and levels of uncertainty and can be treated as random variables following certain probability distributions. The resulting limit state function (g) can be defined as follows:

g R Q R Q, ( )( ) = − 1

where R is the resistance of a given structure and is a random variable and Q is the applied load and is also a random variable.

If R and Q follow a normal or lognormal distribution, then the limit state function g also follows a normal or lognormal distribution, as shown in Figure 1 (which shows a normal distribution). The probabil-ity of failure (i.e., the resistance being less than the load) is shown as a shaded area in Figure 1. The probability of failure (pf) can be defined as

p p g R Q p R Qf = ( ) ≤[ ] = ≤[ ], ( )0 2

As shown in Figure 1, the probability of failure depends on the mean and standard deviation of the load and the resistance. For a

normal distribution of g values, the probability of failure can be equated explicitly to the value of the reliability index β = ug/σg, where ug is the mean value of g and σg is the standard deviation of g. The relationship between the probability of failure and the reliability index (β) can be calculated with the following function:

pf = − ( )1 3Φ β ( )

where Φ is the standard normal cumulative function.Also, if the load and resistance values are normally distributed

and the limit state function is linear, then β can be determined from the following relation:

βµ µ

σ σ=

−

+R Q

R Q2 2

4( )

where µR and µQ are the mean of the resistance and the load, respec-tively, and σR and σQ are the standard deviation of the resistance and the load, respectively.

If the load and resistance distributions are both lognormal and the limit state function is a product of random variables, then β can be calculated using a closed-form solution, reported by Nowak (10) and by Withiam et al. (18), as follows:

β =+( ) +( )

+( ) +(ln

ln

R Q Q R

Q R

1 1

1 1

2 2

2 2

CV CV

CV CV )) ( )5

where

R– = mean value of the resistance R;

Q– = mean value of the load Q; and

CVR and CVQ = coefficients of variation for the resistance and load values, respectively.

The Strength I limit state function for LRFD implies the following (19):

γ φ φi ni n i niQ R R≤ = ∑∑ ( )6

where

γi = load factor applicable to a specific load, Qni = specific nominal load, Rn = total nominal resistance available, Rni = resistance component, ϕ = total resistance factor, and ϕi = resistance factor for each resistance component.

The main objective of LRFD calibration is to determine a resistance factor so that Equation 7 is always fulfilled for the target reliability index (βT). Thus, through the combination of Equations 1 and 6, the limit state LRFD equation is as follows:

g R Q Q R Q Ri ni n i ni i ni, ( )( ) = − = −∑∑ ∑γ φ γ φ 7

drilled Shaft load teSt dataBaSe

An extensive search of the Louisiana Department of Transportation and Development’s archives was conducted to collect all the available drilled shaft test data in Louisiana. Only 16 drilled shaft test cases

Pf = probability of failure

g = R – Q

βσg f(g) = probability density of g

0

ug

FIGURE 1 Probability of failure and reliability index in reliability-based design (b = reliability index, s = standard deviation, and u = mean) (18).

40 Transportation Research Record 2310

were available in the state. These drilled shafts were constructed by means of the wet method. Of these load tests, only seven cases had been tested by the O-cell method and met or almost met the FHWA settlement 5%-B criterion. Because of the limited number of avail-able cases in Louisiana, it was not possible to perform a statistical reliability analysis of the drilled shafts. Therefore, the geotechni-cal research team at the Louisiana Transportation Research Center decided to search for more drilled shaft cases in the neighboring states of Mississippi and Texas. Fifty drilled shaft test cases were collected from the Mississippi Department of Transportation and no cases from Texas. These 50 drilled shafts had been constructed with either the wet method or the casing method. Of the 50 cases provided by the Mississippi Department of Transportation, 26 were selected on the basis of an initial screening that identified cases with subsurface soil conditions similar to Louisiana soils. Only 15 of the selected cases met the FHWA settlement criterion and were there-fore included in the database for the statistical reliability analysis. The combined drilled shaft test database contained 22 cases, summarized in Table 1, that best represented the typical subsurface soil condition in Louisiana. The approximate geographical locations of the drilled shafts in the final database are shown in state maps in Figure 2. The diameter of the drilled shafts in the database ranged from 0.61 m to 1.83 m, and the lengths ranged from 6 m to 42.1 m. All 15 of the

drilled shaft cases from Mississippi and the seven drilled shaft cases from Louisiana had been tested with the O-cell test.

geotechnical conditions

The geotechnical conditions of the drilled shaft tests were also collected and categorized. The collected information included the soil boring profile, the moisture content, the liquid limit, the plastic limit, the shear strength for clay, and the standard penetration test value for sand. The soils encountered in the investigated database included silty clay, clay, sand, clayey sand, and gravel. Most of the soil strata were not uniform and contained interlayers, as described in Table 1. Typical geotechnical data obtained for a tested drilled shaft are depicted in Figure 3.

Measured drilled Shaft resistance from o-cell Shaft test

Because of its low cost and convenience over conventional top-down static load tests, the O-cell test has been widely used in the United States to determine the resistance of piles and drilled shafts. Unlike the top-down load test, loads in an O-cell test are applied at the bottom or near the bottom of the drilled shafts through a preinstalled hydraulic cell. During an O-cell load test, the shaft above the cell moves upward, and the shaft below the cell moves downward. As a result, the side friction and the end bearings can both be measured from the O-cell test, as shown in Figure 4. The upward load shown in the figure is the net upward load (the O-cell–measured upward load minus the buoyant weight of the drilled shaft). An equivalent top-down curve can be constructed from the two component curves to investi-gate the total drilled shaft resistance (17). The additional elastic com-pression of the shaft attributable to the extra load applied at the shaft top should be included in the equivalent top-down curve. According to the comparison study available in the literature, the O-cell method has a result very close to that of the traditional top-down method in terms of measurement of the top-down load–settlement curve (20).

Separation of resistance components

In this study, a data set of 22 drilled shaft cases tested by O-cells in mixed soils were collected from the project libraries. The FHWA interpretation criterion (5%-B settlement) has been widely acknowl-edged and used to interpret the ultimate axial capacity of drilled shafts (4, 16, 21). Statistical analysis showed that the FHWA’s 5%-B cri-terion produced the closest and most consistent ultimate resistance with the mean value of the capacities determined through seven methods, which has been further confirmed and used by Zhang et al. (21) and Liang and Li (16). In this study, the interpreted measured resistances of the drilled shafts were also determined by the FHWA interpretation criterion. As described in Table 1, only a few drilled shafts were in sand- or clay-type soils. Most of the soils in the drilled shaft sites were mixed soils. So the soil type for this database can be classified as mixed soil. The resistance factors developed in this study were the total, side, and tip resistance factors for the mixed soil.

The interpreted measured total resistance of the drilled shafts can be determined from the equivalent top-down settlement curve (the circled line in Figure 5a) by means of the FHWA interpretation criterion. In Figure 5a, the predicted total resistance was calculated

TABLE 1 Characteristics of Investigated Drilled Shafts

Location B (m) L (m) Soil Type

Louisiana

East Baton Rouge 0.91 16.5 Clayey silt, silty clay, silty sand base

Ouachita 1.68 23.2 Silty sand, sand, silt, silt base

Calcasieu 1.83 26.5 Stiff clay, silt, stiff clay base

Winn 0.75 7.2 Sand, clay, dense sand base

Winn 0.75 6.0 Sand, clay, clay base

East Baton Rouge 0.76 15.2 Silt, clay, sandy silt base

Beauregard 1.68 12.4 Clay, silt, silty sand base

Mississippi

Union 1.37 15.2 Sand

Union 1.22 22.3 Sand with clay interlayer, sand base

Washington 1.22 37.5 Silt clay and sand, sand base



Washington 1.68 28.7 Sand, gravel base



Washington 1.22 29.6 Sand with clay interlayer, sand base


Lee 1.22 11.6 Silty clay, sand base

Forrest 1.83 14.6 Silty sand and sand with gravel

Perry 1.37 19.5 Sand, gravels and silty clay, silty clay base

Wayne 1.22 19.5 Sand and clay, clay base

Madison 0.61 13.7 Clay

Note: B = shaft diameter; L = the shaft length.


by using the O’Neill and Reese method (12), as is described in the next section. The O-cell test results can provide side friction and tip resis-tance separately, as shown in Figure 4. The side friction is the net upward force that equals the friction resistance, as in a top-down load test based on O-cell test assumptions. The interpreted side resis-tance or tip resistance is determined from the measured curves from O-cell tests at a settlement of 5%-B minus elastic compres-sion. The elastic compression can be calculated or measured from the plots that are available in load test reports. Only the side friction or the tip resistance needs to be determined by such an approach. Once one resistance component is estimated, the other component can be determined as being the difference between the total resistance and the known component. Usually, the component with the larger displacement is preferred to determine the component resistance at the 5%-B settlement. This can help to minimize the possible errors introduced by the extrapolation of load–settlement curves, if needed. The interpretation results are presented in a later section.

fhWa deSign Method for drilled ShaftS

As mentioned earlier, all the collected drilled shaft data contained shaft dimensions, in-situ soil profiles, elevations, and strength parameters. These data made it possible to predict the load–settlement curves

using the FHWA (O’Neill and Reese) method (12) and the SHAFT software program (22).

The nominal ultimate axial resistance (Rp-u) of a drilled shaft consists of the end-bearing resistance (Rp-b) and the skin frictional resistance (Rp-s). The nominal ultimate drilled shaft resistance can be calculated with the following equation:

R R R q A f Ap u p b p s b b i sii

n

- - -= + = +=∑i

1

8( )

where

qb = unit tip bearing resistance, Ab = cross-section area of the drilled shaft base, fi = average unit skin friction of the soil layer i, Asi = area of the drilled shaft that interfaces with layer i, and n = number of soil layers along the drilled shaft.

Table 2 presents a summary of the method suggested by O’Neill and Reese for different soil conditions (12). Details of the application of the equation can be found in the O’Neill and Reese study (12).

The load–settlement behavior of a drilled shaft under short-term compression loading can be calculated with the normalized relations proposed by O’Neill and Reese (12). The side friction resistance

FIGURE 2 Approximate locations of investigated drilled shafts.

(a) (b)


0 1 2 3 4 5 6Load (MN)

-60

-40

-20

0

20

40

60

80

Upw

ard

Top

of B

otto

mO

-cel

l Mov

emen

t (m

m)

Dow

nwar

d B

otto

m o

f Bot

tom

O-c

ell M

ovem

ent (

mm

)

0 2 6 84 10Load (MN)

50

40

30

20

10

0

Set

tlem

ent (

mm

)

(a) (b)

FIGURE 4 O-cell test results: (a) O-cell settlement curves and (b) equivalent top-down settlement curve.

FIGURE 3 Typical summary of geotechnical data for tested shaft (BR = brown; GR = gray; CL = clay; TR = trace; SA = sand; SI = silt; mc = moisture content; LL = liquid limit; PL = plastic limit; Su = undrained shear strength; SPT = standard penetration test).


(Rp-s) developed for each layer i at a specific settlement can be calculated with the ratio of the average deflection along the sides of a drilled shaft (ws) to the shaft diameter (B). The same procedure can also be applied to calculate the tip resistance developed at a specific settlement (Rb-s).

The total developed load (RT) at a specific settlement can be calculated as follows:

R R RT p b p s= ( ) + ( )- -developed developed ( )9

The load–settlement behavior was calculated with SHAFT 6.0 (22). An example of a predicted load–settlement curve for a drilled shaft of 1.52-m diameter is shown in Figure 5. The mea-sured top-down settlement curves include the elastic compression of drilled shafts. Side resistance and tip resistance at a settlement of 5%-B can be easily determined from the SHAFT results, as shown in Figure 5b. The interpretation results are presented in a later section.

caliBration of reSiStance factorS

Theoretically, the estimation of the probability of failure, in which Equation 8 is used for the limit state, should be based on measure-ments of the structural loads applied to the drilled shafts and on the shaft resistance for an adequate number of nominally similar structures in similar geotechnical conditions. Then, the statistical parameters needed to estimate the probability of failure can be taken from these measurements. However, such measurements are difficult to obtain in practice. Available databases generally include measured values of load and resistance from multiple locations within a given structure or from multiple case histories. Therefore, a baseline comparison is needed. The nominal prediction of a load or resistance value can be used as this baseline. As expected, a pre-dicted value is not usually equal to the corresponding measured value because of the variation in material properties and model errors. The ratio of the measured value to the predicted value, termed as bias (λ), is used to evaluate the statistical distributions of the

0 4321Load (MN)

100

80

60

40

20

0

Set

tlem

ent (

mm

)

Predicted ResistanceMeasured Resistance

5% of the shaft diameterShaft diameter = 1.52 m

Rp Rm

0 2 4 6Load (MN)

-200

-100

0

100

200

Set

tlem

ent (

mm

)

Interpreted side resistance (Rp_side)

Interpreted tip resistance (Rp_tip)

5%B

5%B

Shaft diameter = 1.52 m

(a) (b)

0 4321Load (MN)

100

80

60

40

20

0

Set

tlem

ent (

mm

)

Predicted ResistanceMeasured Resistance

5% of the shaft diameterShaft diameter = 1.52 m

Rp Rm

0 2 4 6Load (MN)

-200

-100

0

100

200

Set

tlem

ent (

mm

)

Interpreted side resistance (Rp_side)

Interpreted tip resistance (Rp_tip)

5%B

5%B

Shaft diameter = 1.52 m

(a) (b)

FIGURE 5 Example of load–settlement analysis through SHAFT 6.0: (a) measured and predicted total resistance and (b) predicted resistance components versus settlement.

TABLE 2 Drilled Shaft Design Methods (12)

Soil ConditionResistance Component Equations Parameters

Cohesive Skin friction fsz = αzSuz α: shear strength reduction factor

R f dAp s sz

L

- = ∫0

Suz: undrained shear strength at depth zfsz: ultimate load transfer in skin friction at zdA: differential area of the shaftL: penetration depth of the drilled shaft below ground surface

End bearing qb = NcSub, Sub ≤ 0.25 MPa Nc: bearing capacity factorNc = 1.33 (ln Ir + 1) Sub: average undrained shear strength of the clay between the base and a

IE

Srs

ub

=3

depth of 2BIr: rigidity index of the soilEs: Young’s modulus of the soil

Cohesionless Skin friction fsz = β σ′z ≤ 190 kPa σ′z: vertical effective stress in soil at depth z

R dAp s z

L

- = ′∫βσ0

z: depth below the ground surface

β = 1.5 − 0.135z0.5

End bearing qb = 0.0575 NSPT (MPa) NSPT: average blow count from the zone between the base and a depth of 2B


nondimensional load and resistance by which the probability of failure can be calculated (23).

Statistical analyses

The measured resistance contributions of the investigated drilled shafts are plotted in Figure 6a. The average contribution of the side resistance to the total resistance was about 52%. The side resistance and the tip resistance both contributed significantly to the total resistance. However, the SHAFT software program significantly underestimated the tip resistance, as indicated in Figure 6b. The majority of the total resistance (76%) came from the side resistance. The tip resistance only contributed 24% of the total resistance. The interpreted measured resistances were compared with the inter-preted predicted resistances, as shown in Figure 7. The interpreted predicted total resistance was 84% of the measured total resistance (Figure 7c). This underestimation was even lower for the tip resis-tance, as indicated by the predicted resistance being only 38% of the measured resistance (Figure 7a). The accuracy of the side resistance estimation (Figure 7b), 122%, was better than that of the tip resistance. The prediction of the side resistance was also less scattered than the prediction of the tip resistance. The lower contribution of the measured side resistance could have been partly caused by the side resistance from the shaft below the O-cell that is counted toward the tip resistance (e.g., the O-cell results shown in Figure 4a).

A statistical analysis of the interpreted resistance was conducted to evaluate the statistical characteristics of the nominal drilled shaft resistance of the different components. The corresponding resistance bias factors (λR) for each resistance component and the total resistance were determined; the resistance bias factors are the average of the ratio of the measured resistance to the predicted resistance (Rm/Rp). The maximum, minimum, mean (µ), and coefficient of variation of the bias were calculated and summarized in the table below. The resistance components had a larger variation than did the total resistance. The prediction of the tip resistance was the most conservative, as the model bias factor was the largest of the three values. The prediction of total

resistance had the highest precision, as evidenced by its having the lowest CV.

Statistic Tip Side Total

Maximum 5.38 1.47 1.81Minimum 0.79 0.43 0.70Mean 2.56 0.84 1.20CV 0.47 0.37 0.28

Figure 8 presents the histogram and the normal and lognormal distributions of the bias of each resistance component. As shown in the figure, the lognormal distribution matched the histogram of bias better than the normal distribution did; therefore, the lognormal distribution was adopted in the reliability calibration analysis.

lrfd calibration

Several reliability analysis methods are available in the literature for the LRFD calibration of resistance factors. Among the most widely used methods are the FORM and the Monte Carlo simulation method. Both methods can be used to calibrate the resistance factors for each resistance component.

In this study, the Strength I limit state was considered. The limit state function (g) can be rewritten as

g Q Q R Q Q Rn n= + − = + −γ γ φ γ γ φLL LL DL DL LL LL DL DL tip -tipp -− φs n sR( )10

where

γLL and γDL = live and dead load factors, respectively; QLL and QDL = nominal live and dead loads, respectively; Rn-tip = tip nominal resistance; Rn-s = side nominal resistance; ϕtip = tip resistance factor; and ϕs = side resistance factor.

Many researchers, such as Zhang et al. (24), Kim et al. (25), McVay et al. (26), Abu-Farsakh and Titi (27), and Yoon et al. (28) have used the load statistics and load factors from the AASHTO

5 10 15 20

Shaft Number

0

20

40

60

80

100

Mea

sure

d R

esis

tanc

e C

ontr

ibut

ion

(%)

Tip resistance (48%)

Side resistance (52%)

5 10 15 20

Shaft Number

0

20

40

60

80

100

Pre

dict

ed R

esis

tanc

e C

ontr

ibut

ion

(%)

Tip resistance (24%)

Side resistance (76%)

(a) (b)

FIGURE 6 Contribution of side and tip resistance: (a) measured resistance components and (b) predicted resistance components.


LRFD specifications, which were originally recommended by Nowak (10). Both live and dead loads were assumed to be lognormally distributed. In this study, the load statistics and factors from the AASHTO LRFD specifications were also adopted, as follows (13):

• λDL = 1.08, CVDL = 0.13, and γDL = 1.25 and• λLL = 1.15, CVLL = 0.18, and γLL = 1.75.

where CVDL and CVLL are the coefficient of variation values for the dead load and the live load, respectively, and λDL and λLL are biases. QDL/QLL is the dead load to live load ratio, which is dependent on the bridge span length (29). However, it has been found that the calibrated resistance factor (ϕ) is insensitive to a QDL/QLL ratio greater than 3.0 (5, 30). In this study, a QDL/QLL value of 3.0 was used for the LRFD calibration of the resistance factors of the drilled shafts. This QDL/QLL ratio was also used in the previous studies conducted by Barker et al. (15) and Allen (14).

The side resistance and tip resistance are determined at the same settlement, at the shaft top. The bias of side resistance and tip resis-

tance are considered to be independent variables. Therefore, the resistance factor from the side and the tip can be calibrated separately using the FORM and the Monte Carlo simulation method.

First-Order Reliability Method

The reliability method proposed by Hasofer and Lind (31) and its subsequent generalizations to handle non-Gaussian correlated random variables is commonly called the First-Order Reliability Method (FORM). Hasofer and Lind proved mathematically that the nearest distance of the limit state function from the origin of a stan-dard Gaussian space is an invariant measure of reliability (31). A modified reliability index was proposed on the basis of this finding; the modified reliability index does not have the variance problem of Cornell’s index and retains the practical second moment simplifica-tion. The Rackwitz and Fiessler algorithm provides a practical and computationally efficient method to compute this reliability index with no restriction on the number of random variables (32, 33).

0 200 400 600 800 1000 1200 1400

Measured Drilled Shaft Resistance, Rm (tons)

0

200

400

600

800

1000

1200

1400

Pre

dict

ed D

rille

d S

haft

Res

ista

nce,

Rp

(ton

s)Tip resistance

Perfec

t fit

Rfit = 0.38 * Rm

Measured Drilled Shaft Resistance, Rm (tons)

Perfec

t fit

0 400 800 1200 16000

400

800

1200

1600

Pre

dict

ed D

rille

d S

haft

Res

ista

nce,

Rp

(ton

s)

Side resistanceRfit = 1.22 * Rm

(a) (b)

0 500 1000 1500 2000 2500

Measured Drilled Shaft Resistance, Rm (ton)

0

500

1000

1500

2000

2500

Pre

dict

ed D

rille

d S

haft

Res

ista

nce,

Rp

(ton

)

FHWA 5%BRfit = 0.84 * Rm

Perfec

t fit

(c)

FIGURE 7 Interpreted measured and predicted resistance of drilled shafts: (a) tip, (b) side, and (c) total.


The FORM evaluates the limit state function at a point known as the “design point” instead of the mean values. The design point is a point on the failure surface g = 0. Because the design point is gener-ally not known a priori, an iteration technique must be used to solve the reliability index. The detailed FORM procedure can be found in Nowak and Collins (34). Only information on the means and stan-dard deviations of the resistances and loads was needed; detailed information on the type of distribution for each random variable was not needed. In this study, the reliability index was calculated using the Rackwitz and Fiessler algorithm, following the procedure recommended in Transportation Research Circular E-C079 (1).

Monte Carlo Simulation

The Monte Carlo simulation method employs the generation of ran-dom numbers and is used to solve deterministic problems. Random values of biases of loads and resistances are generated according to basic statistical parameters [i.e., mean (µ), CV, and an assigned

distribution, such as the lognormal distribution used in this study]. The random values are then combined to form a limit state function (g) according to a predetermined combination, such as g = ϕR − γQ. From the definition of failure (e.g., g < 0), the number of failure simulations is counted, and the probability of failure is therefore determined. The reliability index can be calculated from the known probability of failure.

In this study, the random numbers of load and resistance biases with lognormal distributions were generated with a MATLAB code. The limit state function that needed to be evaluated is shown in Equation 11.

g Q

QQ Q

QR=

−− +

LL

LL DLDL

LLLL LL DL DL

DL

LL

γ γ

φλ γ λ γ λ

( )11

where λR, λLL, and λDL are biases, which are generated random num-bers that follow a lognormal distribution. The value of ϕ is iterated until the desired reliability is reached.

0 1 2 3

Rm/Rp

0

10

20

30

40

Pro

babi

lity

(%)

Total resistanceLognormaldistributionNormaldistribution

0 2 64

Rm/Rp

0

5

10

15

20

25

Pro

babi

lity

(%)

Tip resistanceLognormaldistributionNormaldistribution

0 1 2 3

Rm/Rp

0

10

20

30

40

Side resistanceLognormal distributionNormal distribution

50

Pro

babi

lity

(%)

(a) (b)

(c)

FIGURE 8 Histograms of bias for different resistance components: (a) tip, (b) side, and (c) total.


The required number of Monte Carlo trials is based on achiev-ing a particular confidence level for a specified number of random variables and is not affected by the variability of the random variables (30, 35, 36). Through the procedure described by Harr, the number of Monte Carlo trials required for a confidence level of 90% is approx-imately 4,500 (35). For the probabilistic calculations reported in this paper, Monte Carlo simulations with 50,000 trials were conducted. Many researchers have suggested a required reliability index (β) for the deep foundations of between 2.33 and 3.0. In this study, a target reliability index of 3.0, in accordance with AASHTO, was used in the calibration of the resistance factors of the drilled shafts.

Calibration Results

The calculated resistance factors for each resistance component and the total resistance obtained at the target reliability index of 3.0 are as follows:

Resistance Tip Side Total

FORM 0.70 0.30 0.60Monte Carlo 0.70 0.30 0.60

Both methods gave the same resistance factors when rounded to the nearest 0.05. Figure 9 presents the resistance factors calibrated only by the Monte Carlo simulation, as both calibration methods generated close results. The tip resistance factor of 0.7 is much higher than the side resistance factor of 0.3. This is consistent with the fact that the prediction of side resistance is unconservative and has a larger variance than the tip resistance. Figure 9 presents the variation in the resistance factors as a function of the reliability index to allow the resistance factors to be determined at different reliability indices.

Liang and Li proposed a total resistance factor of 0.35 for mixed soils on the basis of 65 conventional top-down compression load tests of drilled shafts collected in NCHRP Project 24-17 (16). The resistance factor in mixed soils determined by Liang and Li is less than the value of 0.60 proposed in this study (16). This difference might be a result of the difference in soil conditions. AASHTO recommends side resistance factors of 0.45 in clay (α method) and 0.55 (β method) in sand and tip resistance factors of 0.40 in clay and 0.50 in sand (19). In comparison to AASHTO, the tip resistance factor in this study is higher and the side resistance factor is lower. The calibrated side resistance factor in AASHTO was based on the mea-sured total resistance in the friction type of drilled shafts, in which the total resistance was treated as side resistance. This assumption

caused a higher side resistance factor and could explain the higher resistance found in AASHTO than in the current study, besides the causes attributable to the difference in soil conditions and construction methods.

SuMMary and concluSionS

In this study, a database of 22 O-cell drilled shaft load tests were collected and evaluated for the separate resistance components of the drilled shafts. For each drilled shaft, the load–settlement behavior was estimated using the FHWA (O’Neill and Reese) method through the SHAFT program. The FHWA 5%-B settlement criterion was used to determine the nominal resistances (tip, side, and total) of each drilled shaft.

The results of this analysis showed that the FHWA (O’Neill and Reese) method underestimated the measured total drilled shaft resistance by an average of 16%. The prediction of the tip resistance was much more conservative than that of the side resistance.

Design input parameters for the loads were adopted from the AASHTO LRFD specifications for bridge substructures. Tip, side, and total resistance factors for drilled shafts were calibrated using FORM and the Monte Carlo simulation method. In the calibration, the lognormal distribution of the bias of each resistance component was assumed. On the basis of the reliability calibration results, a total resistance factor (ϕtotal) of 0.6, a side resistance factor (ϕside) of 0.30, and a tip resistance factor (ϕtip) of 0.70 for mixed soils were recommended for Louisiana soils.

acknoWledgMentS

This research project was funded by the Louisiana Transportation Research Center (LTRC) and the Louisiana Department of Trans-portation and Development. The authors acknowledge the support of Mark Morvant at LTRC.

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0.2

0.3

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The Foundations of Bridges and Other Structures Committee peer-reviewed this paper.

calibration of side, tip, and total resistance factors for load and resistance factor design of...

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