Effect of soil–structure interaction and spatial variability of ground motion on seismic risk assessment of bridges - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 593

where p u is the ultimate bearing capacity at depth H , y is the lateral defl ec-
tion and k is the initial modulus of subgrade reaction which is both depth-
and diameter-dependent despite the fact that in many cases (Pender, 1993)
the modulus of the subgrade reaction is assumed to be independent of
diameter.
The particular expression is also applied both for dynamic or non-linear
problems after a simple transformation to a bi-linear relationship by assum-
ing a specifi c threshold deformation D y for entering the inelastic range
(typically equal to 2.5 cm for cohesionless soils) and a second branch stiff-
ness reduced to a quarter of the initial soil stiffness (Kappos and Sextos,
2001). As an alternative to the above procedure, the static stiffness extrapo-
lated from the complex dynamic stiffness matrix is also used in practice as
discussed in the following section.
In addition to the springs attached along the pile shaft, a horizontal
inelastic soil spring can be used at the top of the pile to represent the
strength and stiffness provided by passive soil resistance against the pile
cap, while a vertical, uniaxial, inelastic spring is commonly used at the pile
tip to account for downward and upward capacity of the supporting soil
(Pender, 1993).
For the case of non-uniform, liquefaction susceptible soil profi les, the
lateral subgrade reaction of piles and the maximum reaction force of the
laterally spreading soils have to be appropriately reduced at the corre-
sponding locations along the pile length. This reduction factor lies in the
range of 0.1-0.2 (Finn, 2005) or 0.05-0.2 (Elgamal et al. , 2006; Suzuki et al. ,
2006). As a rule of thumb, the lateral stiffness provided along the liquefi ed
soil layers is reduced to the level of 10-20% depending on the estimated
shear deformation.
Based on the above discussion, from a static response point of view, it
can be concluded that the use of lateral soil resistance-defl ection curves is
a convenient approach for estimating the dynamic characteristics of the
bridge-structure system (Kappos and Sextos, 2001). Nevertheless, despite
the wide application of the P - y approach for the assessment of the struc-
tural response in the design practice, there are certain limitations that have
to be stressed:

• Uncertainty of estimating the parameters involved when load tests are

not available (especially of defi ning p u and k ), is disproportionally high
compared to the simplicity of the approach. It is notable that although
eq. 22.1 is adopted by both the Multidisciplinary Center for Earthquake
Engineering Research and American Technology Council (MCEER/
ATC, 2003) as well as the California Department of Transportation
(CALTRANS, 2006) guidelines, the proposed sets of the required sub-
grade moduli differ on average by a factor of 4 (Finn, 2005).