3473
Technical Committee CFMS /
Comité technique CFMS
greater than that measured). However, the model exhibited
significant variability, with
COV
= 85 percent. The accuracy in
the selected approach decreases significantly with increases in
magnitude of displacement. For example, at displacements of
25 and 50 mm, Eqn. (6) and Figure 1 produced mean biases and
COV
s of 0.46 and 88 percent, and 0.17 and 54 percent,
respectively. The
COV
at 50 mm is somewhat smaller due to
the reduction in the number of bearing pressure-displacement
data pairs at larger displacements available in the database.
5 DEVELOPMENT OF PROPOSED MODEL
The evaluation of the elasticity-based approach presented above
indicated a need for more accurate immediate settlement
calculations. An accurate model should account for the non-
linear response of footings loaded rapidly on clays. An
approach that incorporates common triaxial strength test data
within an elastic stress field is described below.
5.1
Selected constitutive response
The Duncan-Chang hyperbolic model (Duncan and Chang,
1970; Duncan et al. 1980) is a non-linear soil constitutive model
that expresses the development of the principal stress difference
as a function of axial strain, initial Young’s modulus, and
effective confining pressure. The stress path that develops
below the center of a footing is similar to an undrained triaxial
compression stress path (Stuedlein and Holtz, 2010). The failure
criterion can be defined as the point at which half of the
principal stress difference exceeds the available shear strength:
u
ult
s
2
'
'
3
1
(8)
where
(σ’
1
- σ’
3
)
ult
is the principal stress difference at failure.
The original hyperbolic model developed by Kondner (1963) is
given as:
1
3
1
3
'
'
1
'
'
in
ult
E
(9)
where
σ’
1
and
σ’
3
can represent the vertical and horizontal
stresses below the center of a footing, respectively,
ε
is the axial
strain and
E
in
is the initial undrained Young’s modulus, which
remains constant during undrained loading (Duncan, et al.,
1980). Note that
E
in
represents the initial tangent Young’s
modulus and is typically measured at small strains; the range in
strain associated with
E
u
as reported by Duncan and Buchignani
(1987) is not known.
5.2
Calculation of footing displacements
The distribution of vertical, horizontal, and shear stress beneath
the center of the footing was generated for each footing in the
load test database using elasticity theory assuming undrained
conditions (
ν
s
= 0.5). During loading, the change in vertical and
horizontal stresses,
Δσ
1
and
Δσ
3
, can be modeled as the change
in vertical and radial stresses,
Δσ
v
and
Δσ
r
, respectively, by
assuming that square footings can be treated as equivalent
circles (Davis and Poulos, 1972).
Substitution of Equation (8) into Equation (9) and
rearranging for axial strain produces an expression for
displacement based on the integration of strains over the
assumed depth of influence. This study considered an effective
depth of 2
B
eq
for the integration of strains. The displacement
resulting from an applied load,
δ
i
, can be calculated using:
2
0
2
eq
Z j
Z j
Z j
Z j
B
vr
i
vr
in
u
d Z
E
s
j
(10)
where
Δσ
vr
is the principal stress difference and
ΔZ
j
= an
increment of depth. Pertinent soil parameters (
s
u
,
OCR
, and
PI
)
were averaged over a depth of
B
eq
below the footing where the
majority of the large strains develop.
Due to the asymptotic nature of the constitutive model
adopted, unreasonable displacements are computed when the
applied shear stress approaches
s
u
within a given
ΔZ
j
. To
mitigate this effect, the shear stresses were limited to 99 percent
of the available
s
u
(i.e.,
Δσ
vr
/2 < 0.99
s
u
). Although, excessive
displacements result at higher loads, the calibrated hyperbolic
model may be used to estimate the non-linear pre-failure
displacements without performing a time-consuming numerical
study.
5.3
Displacement prediction using the non-linear model
Bearing pressure-displacement curves were calculated using the
Duncan-Chang model and elastic stress fields. The
E
u
was
estimated using Figure 1 and Equation (5). The observed and
predicted
q-δ
curves were compared statistically with the bias.
Bearing pressure-displacement points corresponding to
Δσ
vr
/2 ≥
0.99
s
u
were omitted.
On average, the non-linear approach produced a slight
under-prediction of displacements for a given bearing pressure,
with a mean
= 1.13 for each bearing pressure-displacement
curve in the database, but exhibited significant variability (
COV
= 105 percent). The relatively large
COV
is the result of the
inherent variability in soil strength, transformation model error
in the calculation of undrained modulus, and model error. The
tendency for the selected non-linear constitutive model and
elastic stress-field to under-predict the displacement at a given
bearing pressure resulted from excessive strains calculated as
the mobilized shear stresses approached the undrained shear
strength.
6 BACK-CALCULATION OF INITIAL MODULUS
Another application of a non-linear constitutive model within an
elastic stress field is the estimation of the initial Young’s
modulus of the soil. Equation (10) can be rearranged for initial
Young’s modulus and its value back-calculated using least
squares regression on the observed bearing pressure-
displacement curve:
2
0
1
2
eq
Z j
j
j
Z j
B
vr
in
j
vr Z
Z
u
E
d Z
s
(11)
where
E
in
is the initial undrained Young’s modulus averaged
over a depth
B
eq
. Again, data-pairs corresponding to shear
stresses approaching the ultimate stress difference were omitted.
The back-calculated initial Young’s modulus depends on
the shape of the predicted bearing pressure-displacement curve.
In some cases the predicted curvature of the bearing pressure-
displacement curve was not in agreement with the observed
curvature and in these instances the fitting procedure was
modified to estimate the initial portion of the bearing pressure
displacement curve. This was done to focus on the initial
stiffness characteristics (Strahler 2012).
6.1
Young’s modulus comparison
The calculated undrained Young’s modulus and back-calculated
initial Young’s modulus,
E
in
, were compared using the bias and
its distribution. In general, the
E
u
calculated using the Duncan
and Buchignani (1987) correlation under-predicts the back-
calculated initial modulus (mean
= 3.05) and exhibits a
significant amount of variability (
COV
= 99%). This level of
under-prediction is not surprising, given that the Duncan-Chang
model uses an initial undrained Young’s modulus that is
typically based on the first 0.1 to 0.25% of axial strain or less.
The type or strain level of the Young’s modulus referenced by
