3446
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
The transformation must be done using mean q values
ci
The next step is to calculate V[E
Si
]. Observe that, as only
one empirical correlation was adopted, V[α] =0 and then,
V
2
[E
Si
] becomes automatically null.
.
4 CONCLUSIONS
Following, the settlement mean and variance contribution of
each sublayer has to be evaluated. Table 1 shows the main
calculation steps and results for the given example, where: q
ci
and E
Si
are given in MPa and predicted settlements results (ρ,
σ[ρ]) are given in mm. Variances are given in square units.
It has been proposed and briefly discussed three simplified
methodologies for probabilistic analysis of settlements of
footings in sands, which adopts the soil stratification to compute
the only considered random variable (deformability modulus).
Table 1. Evaluation of CPT results, uncertainties in E
Si
, and application
of the SOSM method.
Despite the presented limitations adopted on methodologies
proposal, it can be assumed as a first approximation for
evaluating the uncertainties (especially in deformability
modulus) at the SLS analysis of a foundation. The association
between probabilistic analysis and settlement predictions can
become an interesting tool for geotechnical engineering in the
knowing of soil variability and related uncertainties.
Sublayer q
ci
V[q
ci
] E
Si
V [E
Si
]
ρ
i
V[ρ
i
] % in V[ρ]
1
10,0 10,3 20,1 48,2 0,252 0,007
0,2
2
9,6
9,9 19,2 46,0 0,791 0,077
2,0
3
9,7
9,9 19,4 46,0 1,308 0,207
5,4
4
9,2
9,4 18,4 44,1 1,937 0,481
12,5
5
8,9
9,3 17,9 43,3 2,576 0,884
22,9
6
9,4
9,6 18,8 44,7 2,607 0,845
21,9
7
9,7
9,8 19,3 45,7 2,358 0,672
17,4
8
11,9 12,1 23,8 56,4 1,737 0,297
7,7
9
13,3 13,5 26,6 63,0 1,414 0,176
4,6
10
15,4 15,5 30,8 72,2 1,104 0,092
2,4
11
18,1 18,0 36,2 83,8 0,839 0,045
1,2
12
21,5 21,6 42,9 100,8 0,628 0,022
0,6
13
24,2 24,7 48,4 115,3 0,489 0,012
0,3
14
24,8 25,8 49,7 120,5 0,413 0,008
0,2
15
21,8 22,6 43,7 105,5 0,400 0,009
0,2
16
20,4 21,0 40,7 97,9 0,352 0,007
0,2
17
19,2 19,8 38,5 92,3 0,291 0,005
0,1
18
16,1 16,3 32,3 76,1 0,250 0,005
0,1
19
15,9 16,0 31,7 74,5 0,153 0,002
0,0
20
15,9 16,0 31,8 74,5 0,051 0,000
0,0
Sum
-
-
-
-
19,95 3,86
100,0
σ[ρ]
-
-
-
-
-
1,96
-
COV (%)
-
-
-
-
-
9,84
-
Therefore, any attempt to quantify the sources of
uncertainties and its effects in geotechnical analysis, through
probabilistic models, may become an important tool for helping
engineers to make better and consistent design decisions.
5 REFERENCES
Aoki, N.; Cintra, J. C. A. and Menegotto, M. L. 2002.
Segurança e
confiabilidade de fundações profundas
. 8th Congresso Nacional de
Geotecnia, vol. 2, p. 797-806, Lisboa.
Baecher, G. B. and Christian, J. T. 2003. Reliability and statistics in
geotechnical engineering. John Wiley and Sons, Chichester,
England.
Berardi, R. and Lancellotta, R. 1991. Stiffness of granular soils from
field performance. Gèotechnique 41, No. 1, p. 149-157.
Bredja, J. J. et al. 2000. Distribution and variability of surface soil
properties at a regional scale. Soil Science Society of America
Journal, 64, p. 974-982.
Burland, J. B. and Burbidge, M. C. 1985. Settlement of foundations on
sand and gravel. Proceedings of Institution of Civil Engineers, Part
1, 78, Dec., p. 1325-1381.
Campanella, R. G.; Wickremesingue, D. S. and Robertson, P. K. 1987.
Statistical treatment of cone penetrometer test data. Department of
Civil Engineering, University of British Columbia, Vancouver
B.C., Canada, p. 1010-1019.
The mean and variance of the predicted settlement are then
the sum of the increments of each sub-layer, as suggested by the
sum at the bottom of the table 1. So, the predicted settlement
can now be represented by the form:
Fenton, G. A. and Griffiths, D. V. 2002. Probabilistic foundation
settlement on spatially random soil. ASCE Journal of Geotechnical
and Geoenvironmental Engineering, 128(5), p. 381-390.
Gimenes, E. A. and Hachich, W. 1992.
Aspectos quantitativos em
análises de risco geotécnico
. Solos e Rocha, São Paulo, 15, (1), p.
3-9.
2 20 ) (
mm
(19)
For the complete characterization of the solicitation curve
(predicted settlement) lognormal distribution was used. Figure 4
shows the results for the probability of the predicted settlement
to exceed different values of limiting settlements in a range
between 10 to 50 mm. For example, the probability of the
predicted settlement to exceed 25 mm is about 1,1%. For
exceeding values of over 40 mm, P [ρ≥40mm] ≈0.
Goldsworthy, J. S. 2006. Quantifying the risk of geotechnical site
investigations. PhD. The University of Adelaide, Australia,
January.
Griffiths, D. V.; Fenton, G. A. and Tveten, D. E. 2002. Probabilistic
geotechnical analisys. How difficult does it need to be?, Proc. of the
Int. Conf. on Probabilistics in Geotechnics: Technical and
Economic Estimation, R. Pottler, H. Klapperich and H. Schweiger
(eds.), Graz, Austria, United Engineering Foundation, New York,
September.
Negulescu, C. and Foerster, E. 2010. Parametric studies and quantitative
assessment of the vulnerability of a RC frame building exposed to
differential settlements. Natural Hazards and Earth System
Sciences. Sci., 10, p. 1781-1792.
Schmertmann, J. H. 1970. Static cone to compute static settlement over
sand. Journal of the Soil Mechanics and Foundations Division,
ASCE, vol.96, n° SM.3, p. 1011-1043.
Schmertmann J. H.; Hartman, J. P. and Brown, P. R. 1978. Improved
strain influence factor diagrams. Journal of the Geotechnical
Division, ASCE, 104(8), p. 1131-1135.
Sivakugan, N. and Johnson, K. 2004. Settlement predictions in granular
soils: a probabilistic approach. Gèotechnique, LIV (07): p. 499-502.
Figure 4. Probability of the predicted settlement to exceed different
values of limiting settlement.
The analysis of the sources of uncertainties indicates that
about 80% of the settlement variance is influenced by the
uncertainties due to inherent soil variability and measurement
test errors. It is important to emphasize that the uncertainties
due to transformation model was not evaluated in the example.
