CALIBRATION OF SIDE, TIP, AND TOTAL RESISTANCE FACTORS FOR LRFD …docs.trb.org/prp/12-2262.pdf · · 2012-01-03Abu-Farsakh, Yu, and Zhang 2 2 CALIBRATION OF SIDE, TIP, AND TOTAL

Yu, Abu-Farsakh, and Zhang 1

CALIBRATION OF SIDE, TIP, AND TOTAL RESISTANCE FACTORS FOR LRFD OF

DRILLED SHAFTS

Murad Y. Abu-Farsakh, Ph.D., P.E. (Corresponding Author) Associate Professor, Research

Louisiana Transportation Research Center Louisiana State University

4101 Gourrier Avenue, Baton Rouge, LA 70808 Phone: 225-767-9147, e-mail: [email protected]

Xinbao Yu, Ph.D.

Research Associate Louisiana Transportation Research Center

Louisiana State University 4101 Gourrier Avenue, Baton Rouge, LA 70808

Phone: 225-767-9194, E-mail: [email protected]

And

Zhongjie Zhang, Ph.D., P.E. Pavement and Geotechnical Engineering Administrator

Louisiana Transportation Research Center 4101 Gourrier Avenue, Baton Rouge, LA 70808

e-mail: [email protected]

Submitted to: 91th Transportation Research Board Annual Meeting

January 2012 Washington, D.C.

TRB 2012 Annual Meeting Paper revised from original submittal.

2

Abu-Farsakh, Yu, and Zhang 2

CALIBRATION OF SIDE, TIP, AND TOTAL RESISTANCE FACTORS FOR LRFD OF DRILLED SHAFTS

ABSTRACT

This paper presents a reliability based analyses for calibrations of side, tip, and total resistance factors (φ) for axially-loaded drilled shafts. A total of 22 drilled shafts tested using O-cell were collected from Louisiana Department of Transportation and Development (LADOTD) and Mississippi Department of Transportation (MDOT) project archives. This database was selected carefully to represent the typical subsurface soil conditions and design practice in Louisiana. Prediction of load-settlement curves of drilled shafts from soil borings was determined using the FHWA O’Neill and Reese design method via SHAFT 6.0 computer program. The interpreted predicted drilled shaft axial nominal resistances were determined using the FHWA 5% shaft diameter settlement criterion from predicted load-settlement curves. The measured nominal drilled shaft side, tip, and total axial resistances were determined from the Osterberg cell (O-cell) measurements. Statistical analyses were first performed to compare the predicted nominal drilled shaft axial resistances and the measured nominal resistances. In general, the selected design method underestimates the measured drilled shaft resistances. The First Order Reliability Method (FORM) and the Monte Carlo simulation (MCS) method were selected to determine the side, tip, and total resistance factors under strength I limit state at the target reliability index (βT) of 3.0. The resulted calibrated side (φside), tip (φtip), and total resistance factors (φtotal) are 0.3, 0.6, and 0.7, respectively.


3


INTRODUCTION Modern design of civil engineering structures requires accurate characterization of the load and resistance uncertainties. The conventional allowable stress design (ASD), also called working stress design (WSD), incorporates all uncertainties associated with loads and resistances together using the factor of safety. On the other hand, the load and resistance factor design (LRFD) separates the load and resistance uncertainties based on the limit state design methodology. Rigorous reliability approaches have been used to calibrate load and resistance factors for engineering infrastructures, where 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 design approach. Therefore, the LRFD methodology has gained its popularity and has become the mandatory design method for all bridge projects funded by the Federal Highway Administration (FHWA).

The first AASHTO LRFD specifications (2) were introduced in 1994, where the proposed resistance factors for deep foundations were obtained by fitting to ASD due to lack of a good load test database. Although this calibration approach can take full advantage of past successful design experience, it cannot 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, 4, 5, 6, 7, 8, 9, and 10).

Drilled shafts have been widely used to support individual columns of bridges. The total

resistance of axially loaded drilled shafts consists of side resistance and tip resistance. Three resistance factors, total, side, and tip, need to be calibrated for reliable LRFD design of drilled shafts. Due to either the limited number of drilled shaft load tests or the low quality of the load tests, there are limited successes of calibration of the separated resistance factors for drilled shafts. Paikowsky et al. (4) calibrated the resistance factors for drilled shafts based on database developed by the University of Florida, the FHWA, and O’Neill et al. (11). Resistance factors for total nominal resistance and side resistance were calibrated for drilled shafts in different types of soils considering the effects of construction methods. The side resistance factor calibrated in their study was for side friction of drilled shafts whose resistances are primarily provided by side friction. Without separating the bearing components, it is still a “total” resistance factor. To demonstrate the change of drilled shaft design method (12) in the AASHTO LRFD specifications (13), Allen (14) recalibrated the resistance factors for drilled shafts based on the databases reported in previous literature (4 and 15) by fitting to ASD as well as using the Monte Carlo Simulation (MCS) method. Yang et al. (16) calibrated the resistance factor for side resistance estimated by the O’Neill and Reese method (12) based on 19 Osterberg cell (O-cell) test data in Kansas, Colorado, and Missouri. A side resistance factor of 0.69 was obtained from their calibration. Based on the top-down test data of drilled shaft collected in the NCHRP Project 24-17, Liang and Li (17) calibrated total resistance factors of drilled shafts designed using O’Neill and Reese method (12) via the MCS approach. They found the resistance factors of 0.45 for clay, 0.50 for sand, and 0.35 for mixed soil.


4


The geotechnical research team at the Louisiana Transportation Research Center (LTRC) collected high quality load test data of drilled shafts from Louisiana and Mississippi states. The effect of interpretation criterion on the total resistance factor was completed in a previous study (18). The main objective of this study was to calibrate side (φside), tip (φtip), and total (φtotal) resistance factors for LRFD design of axially loaded drilled shafts. A total of 22 drilled shaft cases tested by O-cells that met or closely met the FHWA settlement criteria were selected in this study. The FHWA method suggested by O’Neill and Reese (12) was also adopted for the design of drilled shafts. The nominal resistance of drilled shafts was determined using the FHWA 5%B (B is the diameter of drilled shaft) settlement interpretation criterion. Statistical analysis was first conducted to evaluate the FHWA design method (12) for predicting the measured drilled shaft resistance. Reliability analyses were then conducted on the collected database using the FORM and the MCS approach to calibrate resistance factors (φtotal, φside, and φtip). Background The design of drilled shaft foundations depends on the resistance of shafts (R) and predicted structure loads (Q). Both variables R and Q have various sources and levels of uncertainty and can be treated as random variables following certain probability distributions. The resulted limit state function, g, can be defined as follows: g (R, Q) = R – Q (1) where R is the resistance of a given structure, which is a random variable, and Q is the applied load, which is also a random variable.

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

[ ] Q]p[R0Q)g(R,pfp ≤=≤= (2)

As shown in Figure 1, the probability of failure depends on the mean and standard deviation of the load and resistance. For a normal distribution of g values, the probability of failure can be equated explicitly to the value of 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 using the following function. ( )βΦ−= 1p f (4)

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:

2

Q2

R

QR

σσ

µµβ

+

−= (5)


5


FIGURE 1 Probability of failure and reliability index in reliability-based design (19).

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

If both the load and resistance distributions are lognormal and the limit state function is a product of random variables, then β can be calculated using a closed-form solution reported by Withiam et al. (19) and Nowak (10) as follows:

( ) ( )[ ]

( )( )[ ]22

22

11ln

1/1/ln

RQ

RQ

COVCOV

COVCOVQR

++

++=β (6)

where, R is the mean value of the resistance R, and Q is the mean value of the load Q; COVR and COVQ are the coefficients of variation for the resistance and load values, respectively.

The strength limit state I function for LRFD design implies the following (20):

∑∑ φ=φ≤ niinnii RRQγ (7) where, γi is the load factor applicable to specific load, Qni is the specific nominal load, Rn is the total nominal resistance available, Rni is the resistance component, φ is the total resistance factor, and φi is the resistance factor for each resistance component.

The main objective of LRFD calibration is to determine a resistance factor so that equation (8) is always fulfilled for the targeted reliability index (βT). Thus, combining equations (1) and (7), the limit state LRFD design equation is shown as follows:

∑∑∑ φ−=φ−= niiniinnii RQγRQγQ)g(R, (8)

Pf = probability of failure

g = R - Q

βσg f(g) = probability density of g

0

ug


6


DRILLED SHAFT LOAD TEST DATABASE An extensive search was conducted on the LADOTD’s archives to collect all available drilled shaft test data in Louisiana. Only 16 drilled shaft test cases were available in LA. These drilled shafts were constructed using the wet method. Among these load tests, only seven cases were tested using the O-cell and met or closely met the FHWA settlement 5%B criterion. Due to the limited number of available cases in Louisiana, it is not possible to perform statistical reliability analysis of drilled shafts. Therefore, the geotechnical research team at LTRC decided to search for more drilled shaft cases in neighboring states, i.e., Mississippi and Texas. The authors were able to collect 50 drilled shaft test cases from Mississippi Department of Transportation (MSDOT); with no case from Texas. These drilled shafts were constructed using either the wet method or casing method. Among the 50 cases provided by MSDOT, 26 cases were selected based on initial screening to identify cases with subsurface soil conditions similar to Louisiana soils. Only 15 of the initial selected cases that met the FHWA settlement criterion were chosen and included in the database for the statistical reliability analysis. The combined drilled shaft test database then had 22 cases as summarized in Table 1 that best represent the typical subsurface soil condition occurred in LA. The geographical locations of drilled shafts in the final selected database are approximately shown in state maps in Figure 2. The diameter of drilled shafts in the database ranges from 0.61 m to 1.83 m and the length ranges from 6 m to 42.1 m. All the 15 drilled shaft cases from MS and seven drilled shaft cases from LA were tested using the O-cell test. Geotechnical Conditions Geotechnical conditions of drilled shaft tests were also collected and categorized. The collected information includes soil boring profile, moisture content, liquid limit, plastic limit, shear strength for clay, and SPT 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 interlayer 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 Due to its low-cost and convenience over conventional top-down static load tests, O-cell tests have 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 drilled shafts via 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, both side friction and end bearings can be measured from the O-cell test as shown in Figure 4. The upward load shown in the figure was the net upward load (the O-cell measured upward load minus buoyant weight of the drilled shaft). An equivalent top-down curve can be constructed from the two component curves to investigate the total drilled shafts resistance (18). Additional elastic compression of the shaft due to extra load applied at the shaft top should be included in the constructed equivalent top-down curve. According to the comparison study available in literature (21), the O-cell method has a very close result as the traditional top-down method in terms of measurement of the top-down load-settlement curve.


7


TABLE 1 Summary of the Characteristics of the Investigated Drilled Shafts

Location B (m) L (m) Soil Type

E. Baton Rouge, LA 0.91 16.5 Clayey silt, silty clay, silty sand base Ouachita, LA 1.68 23.2 Silty sand, sand, silt, silt base Calcasieu, LA 1.83 26.5 Stiff clay, silt, stiff clay base

Winn, LA 0.75 7.2 Sand, clay, dense sand base Winn, LA 0.75 6.0 Sand, clay, clay base

E. Baton Rouge, LA 0.76 15.2 Silt, clay, sandy silt base Beauregard, LA 1.68 12.4 Clay, silt, silty sand base

Union, MS 1.37 15.2 sand Union, MS 1.22 22.3 Sand with clay interlayer, sand base

Washington, MS 1.22 37.5 Silt clay and sand, sand base Washington, MS 1.22 42.1 Silt clay and sand, sand base Washington, MS 1.22 36.3 Silt clay and sand, sand base Washington, MS 1.68 28.7 Sand, gravel base Washington, MS 1.22 29.3 Sand, gravel base Washington, MS 1.22 25.0 Sand, gravel base Washington, MS 1.22 29.6 Sand with clay interlayer, sand base Washington, MS 1.22 25.0 Sand, gravel base

Lee, MS 1.22 11.6 Silty clay, sand base Forrest, MS 1.83 14.6 Silty sand and sand with gravel Perry, MS 1.37 19.5 Sand, gravels and silty clay, silty clay base

Wayne, MS 1.22 19.5 Sand and clay, clay base Madison, MS 0.61 13.7 Clay

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


8


FIGURE 2 Approximate locations of the investigated drilled shafts.

FIGURE 3 Typical summary of geotechnical data for a tested shaft.

26

24

22

20

18

16

14

12

10

8

6

4

2

00 20 40 60 80 100

mc, LL, PL

50 100SPT (N)

26

24

22

20

18

16

14

12

10

8

6

4

2

050 100 150 200 250 300

Su (kPa)Soil Description

26

24

22

20

18

16

14

12

10

8

6

4

2

0

Dep

th(m

)

mcLLPL

SPTSu

BR ELASTIC SI W/TR SABR CL

BR & GR LEAN CL

BR SI W/TR SA

GR LEAN CL W/TR SA

RED BR LEAN CL

BR ELASTIC SIBR LEAN CL W/TR SA

BR SI SA

LT STR BR SI

BR SA W/TR OX

BR CL

BR SI SA


9


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

Separation of Resistance Components In this study, a dataset 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 acknowledged and used for interpreting the ultimate axial capacity of drilled shafts (4, 17, and 22). Statistical analysis showed that the FHWA’s 5%B criterion produced the closest and most consistent ultimate resistance with the mean value of the capacities determined by seven different methods, which has been further confirmed and used by Zhang et al. (22) and Liang and Li (17). In this study, the interpreted measured resistances of drilled shafts were also determined using the FHWA interpretation criterion. As described in Table 1, only few drilled shafts are in sand or clay type of soil. Most of soils in drilled shaft sites are mixed type of soils. So the soil type for this database can be classified as mixed soil. The resistance factors developed in this study are the total, side, and tip resistance factor for the mixed soil condition. The interpreted measured total resistance of drilled shafts can be determined from the equivalent top-down settlement curve (circled line in Figure 5a) using the FHWA interpretation criterion. In Figure 5a, the predicted total resistance was calculated using O’Neill and Reese (12) method as will be described in next section. The O-cell test results can provide side friction and tip resistance separately as shown in Figure 4. The side friction is the net upward force which equals the friction resistance as in a top-down load test based on O-cell test assumptions. The interpreted side resistance or tip resistance is determined from the measured curves from O-cell tests at settlement of 5%B minus elastic compression. The elastic compression can be calculated or measured from the plots that are available in load test reports. Only the side friction or tip resistance needs to be determined by such an approach. Once one resistance component is estimated, the other component can be determined as the difference between the total resistance and the known component. Usually, the component with the larger displacement was preferred to

0 1 2 3 4 5 6Load (MN)

-60

-40

-20

0

20

40

60

80

Upw

ard

Top

ofB

otto

mO

-cel

lMov

emen

t(m

m)

Dow

nwar

dB

otto

mof

Bot

tom

O-c

ellM

ovem

ent(

mm

)

0 2 4 6 8 10Load (MN)

50

40

30

20

10

0

Set

tlem

ent(

mm

)


10


determine the component resistance at 5%B settlement. This can help to minimize the possible errors induced by extrapolation of load-settlement curves if needed. The interpretation results will be presented in later section.

a) Measured and Predicted total resistance b) Predicted resistance components vs. settlement

FIGURE 5 An example of load-settlement analysis using SHAFT 6.0. FHWA DESIGN METHOD FOR DRILLED SHAFTS As mentioned earlier, all collected drilled shaft data contain shaft dimensions, in-situ soil profiles, elevations, and strength parameters. This makes it possible to predict the load-settlement curves using the FHWA (O’Neill and Reese, 12) method via SHAFT program (23).

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 then be calculated using the following equation:

∑=

+=+=n

1isiibbs-pb-pu-p Af.AqRRR (9)

where, qb is the unit tip bearing resistance, Ab is the cross-section area of the drilled shaft base, fi is the average unit skin friction of the soil layer i, Asi is the area of the drilled shaft area interfacing with layer i, and n is the number of soil layers along the drilled shaft. Table 2 presents a summary of the method suggested by O’Neill and Reese (12) for different soil conditions. Details of application of the equation can be found in the reference (12).

The load-settlement behavior of a drilled shaft under short-term compression loading can be calculated using the normalized relations proposed by O’Neill and Reese (12). The side friction resistance (Rp-s) developed for each layer i at a specific settlement can be calculated using 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).

0 1 2 3 4Load (MN)

100

80

60

40

20

0

Settl

emen

t(m

m)

--,-- Predicted Resistance--(-- Measured 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 = 1.52m


11


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

RT = Rp-b (developed) + Rp-s (developed) (10)

In this study, the load-settlement behavior was calculated using the program SHAFT 5.0

(23). An example of a predicted load-settlement curve for a drilled shaft of 1.52-m diameter is shown in Figure 5. The measured top-down settlement curves include elastic compression of drilled shafts. Side resistance and tip resistance at settlement of 5%B can be easily determined from SHAFT results as shown in Figure 5b. The interpretation results will be presented in a latter section.

TABLE 2 Summary of Drilled Shaft Design Methods Proposed by O’Neill and Reese (12) Soil Condition Resistance

Component Equations Parameters

Cohesive Soil

Skin Friction uzzsz Sαf =

dAf RL

0 szs-p ∫=

α: shear strength reduction factor Suz: undrained shear strength at depth z fsz: ultimate load transfer in skin friction at z dA: differential area of the shaft L: penetration depth of the drilled shaft below ground surface

End Bearing

MPa25.0S ,SNq ububcb ≤=

1) I(ln 1.33N rc +=

ub

sr 3S

EI =

Nc: bearing capacity factor Sub: average undrained shear strength of the clay between the base and a depth of 2B Ir: rigidity index of the soil Es: Young’s modulus of the soil

Cohesionless Soil

Skin Friction

kPa190σ'βf zSZ ≤=

∫=L

0zs-p dAσ'βR

0.5z0.1351.5β −=

zσ' : vertical effective stress in soil at depth zz: depth below the ground surface

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

CALIBRATION OF RESISTANCE FACTORS Theoretically, the estimation of the probability of failure using equation (8) for the limit state should be based on measurements of applied structure loads on drilled shafts and shaft resistance for adequate number of nominally similar structures in similar geotechnical conditions. Then, statistical parameters needed to estimate the probability of failure can be taken from these measurements. However, such measurements are difficult to obtain in engineering practices. 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 load or resistance value can be used as this base line As expected, a predicted value is usually not equal to the corresponding measured value due to variation of 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 nondimensional load and resistance by which the probability of failure can be calculated (24).


12


Statistical Analyses

The measured resistance contributions of the investigated drilled shafts are plotted in Figure 6a. The average contribution of side resistance to the total resistance is about 52%. Both side resistance and tip resistance contribute significantly to the total resistance. However, the SHAFT program significantly underestimates the tip resistance as indicated in Figure 6b. The majority of total resistance (76%) comes from side resistance. Tip resistance only contributes 24% of total resistance. The interpreted measured resistances are compared to the interpreted predicted resistances as shown in Figure 7. The interpreted predicted total resistance is 84% of the measured total resistance (Figure 7c). This underestimation is even lower for tip resistance as indicated with predicted resistance only 38% of the measured resistance (Figure 7a). The accuracy of side resistance estimation (Figure 7 b), 122%, is better than that of tip resistance. The prediction of side resistance is also less scattered than the prediction of tip resistance. The lower contribution of measured side resistance could be partly caused by the side resistance from the shaft below O-cell that is counted toward the tip resistance (e.g., O-cell results shown in Figure 4a).

A statistical analysis was conducted on the interpreted resistance to evaluate the statistical characteristics of the nominal drilled shaft resistance of different components. The corresponding resistance bias factors (λR) for each resistance component and the total resistance, which are the average of the ratio of the measured resistance to the predicted resistance (Rm/Rp), were determined. The maximum, minimum, mean (µ), and COV of the bias were calculated and summarized in Table 3. It can be observed that resistance components have larger variation compared to the total resistance. The prediction of tip resistance is the most conservative as the model bias factor is the largest among the three values. The prediction of total resistance has the highest precision as evidenced by its lowest coefficient of variation.

(a) Measured resistance components (b) Predicted resistance components

FIGURE 6 Contribution of side and tip resistance.

5 10 15 20Shaft number

0

20

40

60

80

100

Mea

sure

dre

sist

ance

cont

ribut

ion(

%)

Tip resistance (48%)

Side resistance (52%)

5 10 15 20Shaft number

0

20

40

60

80

100

Pre

dict

edre

sist

ance

cont

ribut

ion(

%)

Tip resistance (24%)

Side resistance (76%)


13


(a) Tip resistance (b) Side resistance

(c) Total resistance

FIGURE 7 Interpreted measured and predicted resistance of drilled shafts

TABLE 3 Summary of Bias for Drilled Shafts Statistics Tip Side Total

Max. 5.38 1.47 1.81 Min. 0.79 0.43 0.70 Mean 2.56 0.84 1.20 COV 0.47 0.37 0.28

Figures 8a-8c present the histogram and the normal and lognormal distributions of the

bias of each resistance component. As shown in the figures, the lognormal distribution matches the histogram of bias better than the normal distribution; therefore, the lognormal distribution was adopted in the reliability calibration analysis.

0 200 400 600 800 1000 1200 1400Measured Drilled Shaft Resistance, Rm (tons)

0

200

400

600

800

1000

1200

1400P

redi

cted

Dril

led

Shaf

tRes

ista

nce,

Rp

(tons

)

Tip resistanceRfit = 0.38 * Rm

(a)

0 400 800 1200 1600Measured Drilled Shaft Resistance, Rm (tons)

0

400

800

1200

1600

Pre

dict

edD

rille

dSh

aftR

esis

tanc

e,R

p(to

ns)

Side resistanceRfit = 1.22 * Rm

(b)

0 500 1000 1500 2000 2500Measured Drilled Shaft Resistance, Rm (ton)

0

500

1000

1500

2000

2500

Pred

icte

dD

rille

dSh

aftR

esis

tanc

e,R

p(to

n)

FHWA 5%BRfit = 0.84 * Rm


14


(a) Tip resistance (b) Side resistance

(c) Total resistance

FIGURE 8 Histograms of bias for different resistance components.

LRFD Calibration Several different reliability analysis methods are available in literature for LRFD calibration of resistance factors. Among the most widely used methods are the FORM and the MCS 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 and the limit state function, g, can be re-written as: s-ntip-nn RRRg stipDLDLLLLLDLDLLLLL QQQQ φ−φ−γ+γ=φ−γ+γ= (11) where, γLL and γDL are the live and dead load factors, QLL and QDL are the nominal live and dead loads, Rn-tip is the tip nominal resistance, Rn-s is the side nominal resistance, φtip is the tip resistance factor, φs is the side resistance factor, and the other symbols are the same as defined in previous sections.

0 2 4 6Rm/Rp

0

5

10

15

20

25Pr

obab

ility(

%)

Tip resistanceLognormalDistributionNormalDistribution

0 1 2 3Rm/Rp

0

10

20

30

40

50

Prob

abilit

y(%

)

0 1 2 3Rm/Rp

0

10

20

30

40

Pro

babi

lity(

%)

Total resistanceLognormal DistributionNormal Distribution


15


Many researchers, such as Zhang et al. (25), Kim et al. (26), McVay et al. (27), Abu-Farsakh and Titi (28), and Yoon et al. (29) used the load statistics and the load factors from the AASHTO LRFD Specifications, which was originally recommended by Nowak (10). Both live and dead loads were assumed to be lognormally distributed. In this study, load statistics and factors from the AASHTO LRFD specifications (13) were also adopted as follows:

λDL = 1.08 COVDL = 0.13 γDL = 1.25

λLL = 1.15 COVLL = 0.18 γLL = 1.75

where, COVDL and COVLL are the coefficient of variation values for dead load and live load, respectively, and γDL and γLL are the load factors for dead load and live load, respectively. QDL/QLL is the dead load to live load ratio, which is dependent on the bridge span length (30). However, it was found that the calibrated resistance factor (φ) is insensitive to the QDL/QLL ratio greater than 3.0 (5 and 35). In this study, a QDL/QLL value of 3.0 is used for LRFD calibration of the resistance factors of 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 resistance are considered as independent variables. Therefore, the resistance factor from side and tip can be calibrated separately using the FORM and the MCS methods.

FORM The reliability method proposed by Hasofer & Lind (31) and its subsequent generalizations to handle non-Gaussian correlated random variables commonly is called the First Order Reliability Method. Hasofer and Lind (31) proved mathematically that the nearest distance of the limit state function from the origin of a standard Gaussian space is an invariant measure of reliability. A modified reliability index was proposed based on this finding, which does not have the variance problem as Cornell's index while retaining the practical second-moment simplification. Rackwitz and Fiessler algorithm (32) provides a practical and computationally efficient recipe for computing this reliability index with no restriction on the number of random variables (33).

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. Since the design point is generally not known a priori, an iteration technique must be used to solve the reliability index. Detailed procedure regarding the FORM can be found in Nowak and Collins (34). Only information on the means and standard deviations of the resistances and loads are needed while detailed information on the type of distribution for each random variable was not needed. In this study, the reliability index was calculated using Rackwitz & Fiessler algorithm following the procedure recommended in Transportation Research Circular E-C079 (1). MCS MCS method employs generation of random numbers and is used for solving deterministic problems. Random values of bias of loads and resistances are generated according to basic


16


statistical parameters. i.e., mean (µ), COV and an assigned distribution, e.g. lognormal distribution in this study. The random values are then combined to form a limit state function, g, according to a pre-determined combination such as g = φR - γQ. Based on the definition of failure, e.g., g < 0, the number of failure simulations is counted and the probability of failure is therefore determined. Reliability index can be calculated from known probability of failure.

In this study, the random numbers of load and resistance bias with lognormal distribution were generated using a MATLAB code. The limit state function needs to be evaluated is shown in equation (12).

λγ+λγλ

φ

γ−γ

=LL

DLDLDLLLLLR

LL

DLDLLL

LL QQ-

QQ

Qg (12)

where, λR, λLL, and λDL are bias, which are generated random numbers following log-normal distribution. The value of φ is iterated until the desire reliability is reached.

The required number of Monte Carlo trials is based upon achieving a particular confidence level for a specified number of random variables and is not affected by the variability of the random variables (35, 36, and 37). Using the procedure described by Harr (36), the number of Monte Carlo trials required for a confidence level of 90% is approximately 4,500. 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 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 resistance factors of drilled shafts.

Calibration Results The calculated resistance factors for each resistance component and total resistance obtained at target reliability index of 3.0 are presented in Table 4. It is interesting to notice that both methods gave the same resistance factors after rounded off to the nearest 0.05. Figure 9 presents calibrated resistance factors only by MCS as both calibration methods generate 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 changes of resistance factors along with reliability index for the convenience of determination of resistance factors at different reliability index.

TABLE 4 Calibrated Resistance Factors Resistance Tip Side Total

FORM 0.70 0.30 0.60 Monte Carlo 0.70 0.30 0.60


17


FIGURE 9 Calibrated resistance factors for each resistance component by MCS.

Liang and Li (17) proposed a total resistance factor of 0.35 for mixed soils based on 65 conventional top-down compression load tests of drilled shafts collected in the NCHRP project 24-17. Resistance factor in mixed soils by Liang and Li (17) are less than the proposed value of 0.60 in this study. This might be due to the difference of soil conditions. AASHTO (38) 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. In comparison with 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 measured total resistance in friction type of drilled shafts where the total resistance was treated as side resistance. This assumption causes a higher side resistance factor. This could explain the higher resistance in AASHTO than the current study, besides the causes due to the difference in soil conditions and construction methods.

SUMMARY AND CONCLUSIONS In this study, a database of 22 O-cell drilled shafts load tests were collected and evaluated for truly separated resistance components of drilled shafts. For each drilled shaft, the load-settlement behavior was estimated using the FHWA (O’Neill and Reese) method via 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 analyses showed that the FHWA (O’Neill and Reese) method underestimates the measured total drilled shaft resistance by an average of 16%. The prediction of tip resistance is much more conservative than that of side resistance.

Design input parameters for loads were adopted from the AASHTO LRFD design

specifications for bridge substructure. Tip, side, and total resistance factors for drilled shafts were calibrated using FORM and MCS methods. In the calibration, the lognormal distribution of bias of each resistance component was assumed. Based on the reliability calibration results, total

0 1 2 3 4Reliability index (βΤ)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Res

ista

nce

fact

or( φ

)Total ResistanceTip ResistanceSide Resistance

0.32

0.60

0.72


18


resistance factor (φtotal) of 0.6, side resistance factor (φside) of 0.30, and tip resistance factor (φtip) of 0.70 for mixed soils are recommended for Louisiana soils. ACKNOWLEDGEMENTS This research project is funded by the Louisiana Transportation Research Center (LTRC Project No. 07-2GT) and Louisiana Department of Transportation and Development (State Project No. 736-99-1408). The authors gratefully acknowledge the support of Mark Morvant at LTRC. REFERENCES

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Documents

CALIBRATION OF SIDE, TIP, AND TOTAL RESISTANCE FACTORS FOR LRFD …docs.trb.org/prp/12-2262.pdf · · 2012-01-03Abu-Farsakh, Yu, and Zhang 2 2 CALIBRATION OF SIDE, TIP, AND TOTAL