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International L'onference on Structural Failure - ;CSF' w7

30 - o: .larch 1387, Singapore

A FIRST ORDER SECOND MOMENT RELIABILITY ANALYSISOF SHORT PILE FOUNDATIONS ON EXPANSIVE SAILS

Elfatih M Ali

Lecturer

Civil Engineering Department

University Of Khartoum

Khartoum

Sudan

S UMM7IRY

Ahmed E Elhussein

Post-graduate Student

University Of Khartoum

Khartoum

Sudan

In this paper the First Order Second Moment approach(FOSM) was used to estimate the reliabilitf index of shortpile foundation on expansive soils.

Different material properties and .:ail-structureinteraction parameters were used in this method to estimater

~ the safety margin of the structure. An iterative proceduret was used to compute the reliability index and ner.ce the pro-

bability of failure was estimated.

1. INTRODUCTION

Expansive soils dominate many countries in Africaand other continents (1) These soils exhibit swellinctcharacteristics and are believed to b~: the cause of at least10 billion US dollars of damages eacY: year al_ cver the world(2}. The extremely high cost of damages coupled with uncert-ainties about loads acting on structures, so-~1 properties and

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analytical models used have necessitated the adoption of pro-

~abilistic methodolcgies for the analysis of foundations on

such structures.

P:.:`...bil~stic methods were used to study the behaviour

of shor' is-e foundations in expansive soils (3). Statistical

models :~ proposed to represent scil properties. Reliabil-

ity anaiy~is was carried out based :~n those statistical dist-

ribution models and the soil-structure interaction model pre-

~ented by Chen (4) (See Fig.l).

G.L.

Wetted zone

one

Fig. 1. Short pile in expansive soil

'P he uncertain environment in the short pile case

includes three categories

a Uncertainties about soil properties.

b Uncertainties about model parameters.

c - Uncertainties about design values.

For such complex situations the distribution-based

reliability analysis becomes difficult and extremely

invaluable. The First Order Second Moment (FOSM) approach

that has recently been presented (S) becomes a more con-

venient approach.

This paper presents a progress report on an ongoing

research programme in the University of Khartoum dealing

with reliability analysis of foundations on expansive soils.

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pr.o- 2. DETERMINISTIC MODEL

on

Fig. 1 illus`rates the model proposed by Chen (4)

for the short pile in expansive soils. Forces acting on

viour the pile are Fd which is the design load transmitted from

tical columns, Fu is uplirt force exerted by soil in the wetter.

sbil- zone whose depth is dw Fr is the resisting force activa-

3ist- ted by skin friction in the non-wetted zone of depth (D-dw1

pre- wheFe D is the total Length of the pile.

Chen also proposed to compute Fu and Fr as

is

Fu = 2.~. r.dw.f. Sp. ............(1)

and

where .

r is the radius of pile

f is the swelling pressure coefficient (=0.15;

Sp is the swelling pressure

~ is the cohesion coefficient (=0.4;

C is the cohesion.

The soil properties in this case are the swelling

pressure, Sp, and cohesior, C. These may be determined

in the laboratory from representative soil samples. `Phe

model parameters are f,~ and dw. While values for these

parameters ere re proposed by Chen (4), experimental :^ves-

tigations are needed to improve our understanding ^: the-

ir magnitudes. The design values of Fd given by the str-

uctural engineer and r and D proposed by the geoLechnical

engineer are all incorporated in this model.

3. PROBABILISTIC ANALYSIS

In the probabilistic approach a structure whose

strength is R and which is sub;ected to load L is consid-

ered. Both R and L• are conside-ed as uncertain; quantities

and hence may be represented by random variables. The sa-

fety margin, SM, is

s M-3

iand the probability of failure, Ff, is then tr

d~

Ir, a previous study (6) the soil properties Sp and ~~

C were represented by normal variates and all other model ~ st

parameters and design values were considered to be deter.- h~

ministic. This led to the estimation of probability of f-

ailure for such cases. The soil properties and model and

design values used in that analysis are all given in Table tt~

i. a

e~

Table 1. Soil properties, model and design values used Ae

in the probabilistic analysis WE

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Property Mean value, ~ Standard Deviation,6re

------ c tC 152 KN/m2 43 KN/m~

Sp 213.58 KN/mZ 164 KN/m2

r 0.3 m 0.0

C 2.4 m 0.0 a

dw 1.5 m 0.0

f 0.15 0.0 ~~

a 0.4nc

0.0

Fd 50 KN 0.0Z

In this report the FOSM approach will be outlined

first and the results of such analysis are given next.

4. FOSM APPROACH

For a particular failure model, a limit state

function is expressed as

Z = 9(X1 XZ X3 .... Xn) - (5)

The probability of failure Pf is

Pf = F ~ Z ~ 0~ = P ~g(X1,XZ,...Xn)S 0~ (6;

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~ As X1, Xz, ... Xn are all random variables,

then ~ is also a random variable. Generally there is no

s~;fficien~ information about the tails of the probability

distribution functions of X1, Xz, ... Xn. Hence the

ind value Pf is computed based on estimates of the mean and

le 1 ,~standard deviation of Z (5). This is why this approach

.er- i

has been referred to as the Second Moment approach.

f-

ind When there are several variables X1, X2, ., Xn,

able then the failure boundary becomes a surface. Nevertheless

~ a planar failure boundary may be assumed and Taylor's

expansion can be used to approximate the failure surface.

As only the first order terms of the Taylor expansion

were used, the method was called the First Order Second

Moment approach.

6 Smith (5) presented a method to determine the

reliability of structures. This approach was used in the

I current investigation and the results were discussed next.

5. RESULTS OF THE ANALYSIS

The FOSM method was used to analyse the reliability

of the short pile in expansive soils. The limit state fu-

nction was expressed as .

Z= g(X X ,Xn)= 27CX tX X X X X X X X X~+X ----(711' 2' " 1 2 3 8 2 3 4 4 5 6 7

Where

X1 = r

X2 = C

x3 = d

X4 = dw

X5 = f

X6 = SP

X~ = Fd

X$ = D

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Using reduced variables and substituting

y _ Xi - i'i'i (8)i

~ i

where f'' and U are the mean value and standard dev-

iation respectively, we have

Xi = ( ~i yi + µi ) ( 9 )

Hence we can express Z in terms of y

i.e. Z = h(Y1, Yz, y3,....,Yn) (10)

Using Taylor's expansion and approximating the exp-

ression for Z as a linear expansion at the point yi*

on the failure boundary such that at yi*,Z is equal

to zero we get .

nZ =~ (yi - yi*) h (yi*) (111

i=1

and hence

~'`'Z = - ~ yi* h~ (yi*) (12)i=1

andn ~

~ Z = ~ ai h (yi*) (13)i=1

As the mean of a standardized variable is zero and

its standard deviation is one, ai is a sensitivity

factor given as .

ai = h' ~Yi* )

~ Z(14)

The distance from the mean of Z,~Z to the

failure boundary i.e. the point at which Z = 0, can

be expressed in terms of CZ the standard deviation

of Z, and equal p ~Z ~ is known as the reliability

index and is a measure of the safety of the system

l•

tt

tl

t

i

rr

tt

f

f

L

i

t

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tJ

(5). Hence ~ can be expressed as

P= ~°'-z

i.e. - ~ yi* h~ (yi* )i=Z

n ,ai h (yi*

- ~=1

ne °~i h (yi} p =-.~ yi.*h (yi")

i.=1 i=1

or

Hence

n~ d.i h (yi*)p + yi* h (yi*) 0i=1

n

L J

The solution of which is ;5):

cls~

~ From the above it is seen that the distance :rim

~ the origin to y * 's a measure of the reliability inc=x.

~ It can be obtained from the expression (5)M

q ~~~n 2

(" r = Y1 ~ 17'E Y l—i

An iterative procedure for determining ~3, pro-

posed by Fiessler (1980}, can be used with reduced va-

riables and give the value of 13 A suitable procedu-

re is as follows (5)

1. Determine an expression for g(x;.

~• Evolve are expression for h(y},

3. Determine expressions for all first derivative

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4. Set yi = 0 and 9= 0

5. Evaluate all hi' values

6. Evaluate h(y)

7. Evaluate standard deviation of Z from

_ ~Z i_ 1 (hi')2

8. Evaluate new values for y from

y -hi L~ ~ C —~1 ~Z 7

9. Evaluate

n

cvi~ z

i=1

10. Repeat steps (5) to (9) until values converge.

Provided that the variables involved in a cert-

ain problem have probability distributions that are

close to normal and provided that. the linear approxim-

ation of the failure surface is realistic, then an ex-

act value for the probability of failure, Pf, can be

obtained from the expression (5).

It can safely be assumed that, for almost all

engineering problems, the linear approximation of the

failure surface will be adequate and that if there

are several variables of roughly equal weight, the

resulting Z function tends to be normal, ever, when the

separate variables are noY. themselves normal (5).

Analysis was carried out in three stages:

(i) Only Sp and C were considered as random values.

When both Sp and C were random and using means

z

t

P

0

r

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J

ir

i

1

Y

t

•

t

a

f

•

0

r

•

R

and standard deviations equal to those used by

Ali and Abbadi (6), results obtained are given

in Table 2.

It is clear that in this situation FOSM analy-

sis was identical to distribution - based ana-

lysis.

Table 2. Results obtained using probabilistic anal-

-_ ysis and FOSM Approach — —

Method Probability of Remarks

failure Pf

Probabilistic 7% actual value

method obtained is 6.85'=

rounded to 7=

FOSM method 6.8~

To investigate the effect of uncertainty of each

random variable on Pf, then the coefficient ~f varia-

tion, CV, which is the ratio of standard deviation to

the mean was used. Cohesion, was considered as a rand-

om variable with mean and standard deviation as used

in the probabilistic analysis and shown in Table 1 and

the coefficient of. variation, CV, of swelling pressure

was varied while all other parameters were assumed co-

nstant Fig. 2 shows results of this analysis.

It is clear that changes in CV of swelling pre-

ssure bring about significant changes in Pf values .

Furthermore it is noted that the CV of Sp based on av-

ailable experimental data is equal to 0.77 which is

quite large. Hence it is imperative that research ef-

forts be made to increase our understanding of this b-

ehaviour and to reduce uncertainties in this material

property.

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Pf •/

Sp

C

0.1 0.2 0,3 0.4 05 0.6 0,7 0,8 0.9

Fig.2. CV Versus Pf for Sp and C'

The same procedure was carried out for the coh-

esion regarding the swelling pressure as random variable

with mean and standard deviation as indicated in Table1 anc the results obtained are given in Fiy. c. Our 1uncertainty abo~.:t cohesion is about 0.3 based on expe-rimental results. Reduction in CV leads to smaller ,changes in Pf values.

iii) Both C and Sp were considered as random proper-

ties and each of the individual model parameters

d, f and dw were represented by random variables

of varying CV values. The remaining model parame-

ters and design values were assumed constant. ,

Fig. 3 shows the effects of each of the individu-

al parameters, dw f andoC. As the uncertainty

of each model parameter increases, Pf incre- ~ases. However, it is quite clear that dw is the

most significant model parameter. Hence resea-

rch effort should target the dw parameter as the

first priority followed by f.

(iii) Design values Fd D and r were also individ- le

ually considered as random variables while the ~ de

C and Sp values are random and all other param- sh

eters constants. Fiq. 4 shows the results obt- ~ uc

ained.

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{

Pf'/

11

1~

1

1

li

r

E

e

f

0 0.1 0.2 0.3 0 4

F~y.3. CV Versus Pf for dw t a:id ~

P t'

0.1 Q2 0.3 0.4

cv

CV

0.5

Fig. 4. CV Versus Pf for D, F~ and r

It is clear that increased uncertainty at ~..r pile

length is a major concern followed by uncertain*~~s in

sdesign column load and pile radius. Significant efforts

should be made to ensure uniform pile length and co red-

~ uce uncertainties about column loads and pile- rah= :._.

n

0

u

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. DISCUSSION

The FOSM analysis presented previously gave an

insight about the effects of uncertainties in different.

soil properties, model parameters and design values .

This gave clear indications about research needs and

required engineering activities in the construction st-

age.

While the basic assumption of statistical indepe-

ndence between the different random variables was made,

this needs revision. The correlation structure between

different variables has to be investigated and incorpo-

rated in the FOSM approach.

7. CONCLUSIONS

A reliability-based analysis for the foundations

on expansive soils was presented and the case of the sh-

ort pile was detailed to show the potential usefulness

of this methodology.

For the same level of data available it is found

that the results obtained from the FOSM analysis were

identical to those obtained using the distribution -bas-

ed analysis.

It is concluded that as the level of uncertainty

of each parameter increases, then Pf increases. And it

is clear that the uncertainty of the dw parameter is the

most significant.

Chen (4) gave a value of. 1.5 m for the depth dw, )

it Ls clear that this parameter depends largely on the

mineralogical composition of the expansive soil and on

the environmental conditions, especially the amount and

duration of rainfall, therefore, research effort is nee-

ded to reduce uncertainties in this property and alsc to

improve our understanding of ail other sources of uncer- l

tainties involved. 1

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8. REFERENCES

1. Gromko, G.J.

" Review of expansive soils."

Journal of the geotechnical engineering division,

ASCE, Vol. 100, No. GT6, PP 667 - 687,June,1974.

2. O'Neill, M.W. and Ghazzaly, O.I

" Swelling Potential related to building perfor-

mance."

Journal of the geotechnical engineering division,

ASCE, Vo1.103, No. GT 12, PP 1363 - ].379, Decem-

ber, 1977.

3. Abbadi, S.M.

" Reliability of Foundations on expansive soils."

M.Sc. thesis, Faculty of Eng. and Arch., University

of Khartoum, Sudan, 19P5.

4. Chen F.H.

" Foundations on expansive soils."

Elsevier, Amsterdam, The Netherland, 1975.

5. Smith, G.N.

" Probability and statistics in civil engineering"

Collins Professional and Technical Books, ~or~~r., 1986.

6. Ali, E.M. and Abbadi, S.M.

" Statistical Modelling of Sudanese expansive =lay

properr.'ies." Accepted for Publication in ~~.3~danese

Eng. Soc., 1986.

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INTERNATIONAL CONFERENCEON

STRUCTURAL FAILURE(ICSF 87 -SINGAPORE)

VOLUME 2

30 -31 MARCH 1987

ORGANISED BY

SINGAPORE CONCRETE INSTITUTENANYANG ?ECHNOLOGICAL INSTITUTE