performance based design of laterally loaded drilled shafts · performance based design of...

Performance Based Design of

Laterally Loaded Drilled Shafts

Dr. Robert Liang & Dr. Haijian Fan

Department of Civil Engineering

The University of Akron

outlines

1. problem statement

– uncertainties in deep foundation design

– deficiencies of ASD and LRFD

2. research method

– performance based design

– reliability method

3. computer-based tools and results

4. summary and conclusions

source of uncertainties

• soil properties and construction materials

• loads

• model errors

• design criteria

• construction

soil properties

Phoon, K.K. and Kulhawy, F. “Characterization of geotechnical

variability” Canadian Geotechnical Journal

concrete compressive strength

ACI Committee 214. Evaluation of Strength Test Results of Concrete.

load variability

• Dead load, COV = 8% to 10%

• Live load, COV = 18% to 88%

Nowak, A. S., and Collins, K. R. (2000). Reliability of Structures.

New York: McGraw-Hill.

model errors

• load transfer curves

– p-y curves

– t-z curves

– q-w curves

• elastic modeling of piles

– concrete behavior

– steel behavior

design criteria

• specify the allowable displacements

• foundations may not fail when the actual

displacements are greater than the

allowable displacements

How to consider the

uncertainties in the design ?

ASD

LRFD

Level I: Deterministic Methods

• Allowable Stress Design

• Advantages:

– Easy to implement

– Conceptually simple

nn

RQ

FS

Level II: Semi-probabilistic Methods

• Load and Resistance Factor Design (LRFD)

η:load modifier; γ: load factor; ϕ: resistance factor

• Advantages:

– Consider uncertainties of R and Q

– A consistent format as ASD

i i i i iQ R

resistance factors

AASHTO LRFD Bridge Design Specifications

comments on resistance factors

1. underlying assumptions unknown to

users

2. resistance factors only available to

predefined reliability index

3. service limit check is still deterministic

4. system reliability is not considered

LRFD

implementation issue

Example

Axially Loaded Piles

• Ultimate capacity

• LRFD requires that

1 1 2 2 3 3R R R Q

1 2 3uR R R R

Q

R1

R2

R3

uR Q

or

The soil stratifications change!

Q Q Q

R1

R2

R3

1 2 3uR R R R

The soil stratifications change!

Ru

Ru

1 2 3uR R R R

Q V M

Q V M

Q V M

laterally loaded piles

observations

1. Soil stratifications differ from site to site.

2. The mean and/or variance of each soil

stratification would differ from site to site.

3. The same resistance factor is applied in

LRFD, regardless of soil stratifications.

CONCLUSION

A uniform level of safety cannot be

achieved in the LRFD framework,

when LRFD is implemented

independently of soil stratifications.

Reference

Ching, J.Y., Phoon, K.K., Chen, J.R. and Park, J.H.

(2013). “Robustness of constant load and resistance

factor design factors for drilled shafts in multiple strata.”

Journal of Geotechnical and Geoenvironmental

Engineering, Vol. 139(7):1104-1114.

PROBLEM STATEMENT

How to accommodate the

uncertainties in deep foundation

design?

an alternative to LRFD

in

reliability based design

computer-based reliability tools

Reliability-Based Design

• How to define the failure event ?

– design criteria

• How to calculate the failure probability?

– reliability analysis

Performance Based Design

(PBD) • design criteria

displacements

• parameter variability

random variable modeling

random field modeling for soil properties

mean, variance and correlation function

• Monte Carlo statistical methods

regular MCS

importance sampling

research methods

design criteria

• axial movement

• lateral deflection

• angular distortion

dep

th

soil property

random field

Q

sectional

stiffness, EA

•••

•••

toe spring

zn

tn

z2

t2

z1

t1

zn-1

tn-1

•••

w

q

side springs

t-z model

M V

Q

yn

pn

y2

p2

y1

p1

yn-1

pn-1

dep

th

soil property

random field

p-y method

design criteria in PBD

0 500 1000 15000

20

40

60

80

100

120

Sett

lem

en

t, m

m

Load, KN

random field

• distribution type

• mean

• variance

• correlation function

random

variable

correlation function

• To describe how points are correlated

spatially, a correlation function is required.

2

, exp

realizations of 1-D random fields -4

-2

0

2

4X

(z)

= 0.1

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

z

X(z

)

= 4

generate random fields

• Fast Fourier Transform (FFT)

• Covariance Matrix Decomposition (CMD)

• Local Averaging Subdivision (LAS)

Fenton, G.A. and Griffiths, D.V. (2008). Risk Assessment in Geotechnical

Engineering.

model errors

• load transfer curves

– p-y curves

– t-z curves

– q-w curves

• modeling of pile material

– concrete behavior

– steel behavior

accounting for model error

• The prediction most of the time deviates

from the measurement.

m pY e Y

Ym: The measurement

Yp: The prediction by using a model

e: Bias Factor, A Random Number that is applied to correct the

prediction

e may be a normal or lognormal variate.

In this study, N is assumed to be lognormal.

multiple failure modes

• structural failures

– bending moment, m

– shear, v

– axial force (ignored)

• performance-related failures

– lateral deflection, δ

– vertical movement, w

– angular distortion, α

system reliability

• structural failure

• performance-related failure

• system failure

struct M VP F P F F

perf wP F P F F F

system struct perfP F P F F

Tools: Computer Codes

1. Laterally loaded piles

2. Axially loaded piles

3. XPILE

inputs to computer codes

1. variability model of soil properties (µ,σ,θ)

2. variability model of concrete and steel

properties (probability distribution, µ,σ)

3. variability model of allowable

displacements

4. variability model of external loads

– live load component

– dead load component

5. uncertainty of load transfer curves

reliability analysis

• first order reliability method

• Monte Carlo simulation (MCS)

Deficiency of FORM

• For concave g(x’)

• For convex g(x’)

• For linear g(x’)

fP

fP

fP

Probability of Failure

• definition of failure

• mathematical formulation

– f: failure domain

– i[∙]: indicator function

– g: limit state function

– x: a input vector; f(x) is the pdf of x.

( 0) 0f

F

P P G f d I G f d x x x x

0G

0 2 4 6 8 10

x 104

0

0.5

1

1.5

2

2.5

3x 10

-3

Number of samples

Pro

ba

bil

ity

of

fail

ure

1

1Failure

n

f i

i

P In

comments on MCS

• Mathematically simple to evaluate Pf

• Unbiased estimate of Pf

• But computationally demanding

• Limit state function

• X1 ~ standard normal variable

• X2 ~ standard normal variable

• Calculate P (G ≤ 0) using MCS

example of MCS

1 2 1 2, 3 2 12G x x x x

• Limit state function

• Reliability index

• Design point

FORM Solution

2 2

123.3282

3 2

1 2 1 2, 3 2 12G x x x x

* 2.7692,1.8462x

1 4.3704E 04fP

when you use

Monte Carlo simulations…

-6

-4

-2

0

2

4

6

-6 -1 4

Failure Domain

G < 0

Safe Domain

G >0

Limit State

G = 0

x2

x1

1

1Failure

n

f i

i

P In

observation

The crude MCS method has

low computational efficiency.

• Draw samples from a different function

• Importance Sampling Quotient

implement importance Sampling

0f

fP I G h d

h

xx x

x

1

10

n

IS i

i

fP I G

n h

x

x

fR

h

x

x

• crude MCS

• importance sampling

difference between mcs and is

,

1

10

n

f MCS i

i

P I Gn

,

1

10

n

f IS i

i

fP I G

n h

x

x

when you implement importance sampling method,

you will get…

-6

-4

-2

0

2

4

6

-6 -1 4

Failure Region

G < 0 Safe Region

G >0

Limit State

G = 0

x2

x1

Importance Sampling

0 200 400 600 800 10000

1

2

3

4

5

6

7

8

x 10-4

Number of Samples

Pro

bab

ilit

y o

f F

ailu

re

Importance SamplingP

IS = 4.4617E-04

COV = 6.17%n = 1000

Exact Solution

soil variability model

how to determine

μ, σ and θ

How to determine the variability

model of soil properties

• Standard penetration test (SPT) data is

used

• Bayesian approach

• Markov Chain Monte Carlo

Standard Penetration Test

Corrected N value

ER: Energy efficiency

N0

N1

N2

60 1 260

ERN N N

1 2N N N

Empirical correlations

• To convert SPT data to soil properties

0.72

600.29u aS N p

1 60

0

tan

12.2 20.3a

N

p

reference

Braja Das. Principles of Foundation Engineering, seventh edition. page 84 & 88.

Assumptions

1. Each N value is considered a unit in

Bayesian approach;

2. The mean, standard deviation and

correlation length do NOT change in the

unit;

3. Soil properties are lognormally

distributed.

Bayesian Approach

1. λ = (µ,σ,θ) are the parameters of interest;

X : log ( soil properties );

c : normalizing constant

2. L(λ|X) : Likelihood function

3. f′(λ) and f″ (λ) are prior and posterior PDF.

|f c L f X

observational data X

• Invoke Assumption #1.

0.72

600.29u aS N p

1 60

0

tan

12.2 20.3a

N

p

60 2 , 1,260

i

ERN N i

N0

N1

N2

likelihood function L

• Invoke Assumption #2 and #3

2 22

1 1| exp

2 12 1

TL

X X R X

1

1

R

ρ: correlation coefficient

Flow Chart

Example 1: Lateral Loading

Resource: Brown, D. A., Turner, J. P. and Casetelli, R. J. (2010). “Drilled shaft:

construction procedures and LRFD design methods.” No. FHWA-NHI-10-016, Federal

Highway Administration. page 12-27

Service load: H=111.2 kN (25 kips) and M=677.9 kN∙m (500 kip-ft.)

original design outcome

• shear = 25 kips and moment = 500 kip-ft

• D = 1.22 m (4 ft) and L = 6.10 m (20 ft)

• reinforced by 12 #11 bars with cover of 3

in.

parametric study

parameter probability

distribution mean cov θ

Su lognormal 103.4 KPa 40% varied

ε50 lognormal 0.005 20% varied

γ' lognormal 19.0 KN/m3 4% varied

conversion

Su: 103.4 Kpa = 15 psi

UW: 19 KN/m3 = 0.07 pci

Parametric study

soil mean cov θ

Su 103.4 KPa 40% varied

ε50 0.005 20% 1.0 m

γ' 19.0 KN/m3 4% 1.0 m

Parametric Study

soil properties mean cov θ

Su 103.4 kpa 40% varied

ε50 0.005 20% 1.0 m

γ' 19.0 kn/m3 4% 1.0 m

Effects of Spatial Variability of Su

-4

-2

0

2

4

X(z

)

= 0.1

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

z

X(z

)

= 4

Example 2

importance sampling

parameters

• axial load = 1300 KN

• allowable settlement = 25 mm

• D = 1.10 m and L = 8.0 m

• soil type: clay

– average Su = 100 KPa, lognormal distributed

– COV = 20%

– correlation length θ = 1 m

crude Monte Carlo simulation

0 0.5 1 1.5 2 2.5 3

x 105

0

0.5

1

1.5

2

2.5

3x 10

-3

Number of Samples

Pro

ba

bil

ity

of

Fa

ilu

re

Regular MCSp

f = 0.0013

COV = 5.24%

= 3.022

Crude MCSP

f,MCS = 0.0013

COV = 5.24%

= 3.022

importance sampling

don’t wow…

0 200 400 600 800 10000

0.5

1

1.5

2

2.5

3x 10

-3

Number of Samples

Pro

ba

bil

ity

of

Fa

ilu

re

Importance samplingP

f,IS = 0.0012

COV = 6.13%

= 3.035

FORMP

f,FORM = 0.0016

= 2.939

Example 3

real-world project

outlines of the example

1. general description

2. use of markov chain monte carlo to

determine (µ,σ,θ)

3. use of kriging to estimate unknown

parameters

4. reliability analysis and design

subsurface investigation

2 to 12 inches of topsoil

8.5 to 26.5 ft of hard gray silt and clay

10.5 to 14.3 ft of silt and clay

15.5 ft of very stiff to hard brown and gray clay

18 to 30 ft of very stiff to hard clay, silty clay and silt

Drilled Shaft Design

• Tolerable displacements

– 1 inch for lateral deflection

– 1 inch for vertical movement

• Axial load

– Dead load = 351.4 KN ()

– Live load = 631.6 KN

• Lateral load = 162.7 KN

• Load eccentricity = 1.69 m

SPT Data

MCMC

3

4

5

6

0

0.2

0.4

0.6

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.5

1

No. of Iterations

Statistical Analysis

Partial Results

Elevation

(m)

x

Soil

Strength

μ σ ρ θ (m) μx σx

256.5 ϕ′ 39.987 3.688 0.111 0.536 0.489 40.222 4.473

255.8 Su 218.740 5.360 0.285 0.392 0.326 221.55 64.375

255.0 Su 276.494 5.605 0.233 0.359 0.298 279.32 65.931

257.4 Su 93.621 4.549 0.502 0.663 0.742 107.20 57.353

correlation length θ

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

Cohesive SoilsSample Size = 18Minimum = 0.288 mMaximum = 0.751 mMean = 0.448 mStandard Deviation = 0.149 m

, m

Cu

mu

lati

ve

Dis

trib

uti

on

Fu

nct

ion

0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

Granular SoilsSample Size = 27Minimum = 0.312 mMaximum = 0.771 mMean = 0.516 mStandard Deviation = 0.130 m

, m

Cu

mu

lati

ve D

istr

ibu

tio

n F

un

cti

on

Kriging

Tj is an observation and lj is the corresponding

distance between Tj and the interested location.

0

1

m

j j

j

T w T

1

, 1,2, ,m

v v

j j i

i

w l l j m

Soil Stratifications 0

5

10

15

20

25

30

35

Dep

th,

m

BR1 BR2 Interpreted

Cohesive

Granular

Interpreted Soil Profile

Elevation(m) x μx

253.0 ϕ′ 36.2

251.0 ϕ′ 36.2

251.0 Su 199.5

250.5 Su 199.4

250.0 Su 199.2

248.9 Su 199.0

248.9 ϕ′ 35.8

Interpreted soil profile

Soil Layer

BR1 BR2 Interpreted

σx θ (m) σx θ (m) σx θ (m)

#1 57.35 0.74 53.73 0.75 56.01 0.75

#2 4.47 0.49 4.49 0.51 4.48 0.50

#3 65.15 0.31 56.35 0.52 61.90 0.39

#4 4.38 0.56 4.27 0.58 4.34 0.57

#5 59.63 0.46 60.09 0.43 59.80 0.45

#6 4.58 0.46 4.55 0.46 4.57 0.46

#7 60.19 0.43 62.10 0.36 60.90 0.41

#8 4.65 0.45 4.60 0.45 4.63 0.45

Reliability Analysis Variable Distribution Mean COV Remarks

f'c Lognormal 31 MPa 7% ACI (2002)

fy Lognormal 415 MPa 5% Mirza and MacGregor (1979)

Es Lognormal 200 GPa 5% Mirza and MacGregor (1979)

epy Lognormal 1 10% (Assumed)

etz Lognormal 1 10% (Assumed)

eqw Lognormal 1 10% (Assumed)

da,δ Lognormal 2.54 cm 10% Zhang and Ng (2005)

da,w Lognormal 2.54 cm 10% Zhang and Ng (2005)

da,ψ — — — —

VD Lognormal 162.72 KN 25% Nowak and Collins (2000)

VL — — — —

QD Lognormal 351.39 KN 10% Nowak and Collins (2000)

QL Lognormal 631.62 KN 50% Nowak and Collins (2000)

old design

1. Diameter = 42 in.

2. Length = 83 ft.

3. Reinforcement: 20 #11 bars, cover

thickness = 3 in.

reliability analysis of original design

0 2 4 6 8 10

x 104

0

1

2

3

4

5

6x 10

-3

Number of Simulations

Pro

ba

bil

ity

of

Fa

ilu

re

Pf = 0.0028

Pf,

= 0.0028

Pf,w

= 0

new design

design parameters old design

diameter 42 in

length 83 ft

rebar 20 #11 bars

cover thickness 3 in

new design

42 in

55 ft

22 #11 bars

3 in

reliability analysis of new design

0 2 4 6 8 10

x 104

0

0.5

1

1.5x 10

-3

Number of Simulations

Pro

ba

bil

ity

of

Fa

ilu

re

Pf = 5.1E-4

Pf,

= 7.0E-5

Pf,w

= 4.4E-4

Target

factor of safety

• Old design: 5.215

• New design: 3.311

computer program XPILE

Summary and Conclusions

1. Performance-Based Design (PBD) is

developed for deep foundation design.

2. Soil properties are modeled as random

fields.

3. Probability of failure is sensitive to θ.

4. A few computer codes were developed.

5. System reliability should be considered if

multiple failure modes exist.

Recommendations for future research

• computer-based reliability tool for pile group;


• statistics of tolerable displacements

dT: tolerable displacement

d: actual displacement

TG d d


• to develop guidance of determining the

variability of external loads.

Q V M


• to calibrate model errors

– p-y curves

– t-z curves and

– q-w curves

Thank you!

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