الفهرس 
Static constitutive models
Damping and maximum damping ratiosOnly 14 pages are availabe for public view
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Abstract
Liquefaction phenomenon causes devastating damages to structures and
facilitates and most importantly risks to peoples’ lives. Seed and Idriss (1971)
proposed a simple empirical method to assess the liquefaction hazard. The
method is called the “Simplified Procedure”. The procedure is based on
calculations of the cyclic shear stresses initiating liquefaction and the soil
strength needed to resist them. If the factor of safety against liquefaction
(FSliq) is less than 1.0, the soil is considered completely liquefied, and hence
the soil shear strength drops to zero. A suitable mitigation method shall be
applied to improve the soil condition. If (FSliq) is greater than 2, the
generation of excess pore pressure is relatively small, and the allowable
seismic bearing capacity can be calculated by increasing the static value by
33%. For the intermediate case where (FSliq) ranges between 1.0 and 2.0, Day
(2010) assumed a punching shear failure mechanism, and modified the
general bearing capacity equation to include the effect of the excess pore
pressure. The punching shear mechanism assumption leads to a significant
underestimation of the seismic bearing capacity. Accordingly, the design of
footings tends to be conservative. The main objective of the whole research
study is to check the validity of this conservative assumption via numerical
modeling. The soil under study is partially liquefiable saturated sand, which is
found to correspond to medium dense condition with SPT blow count ranging
127
from 10 to 30 for the peak ground accelerations in the range of interest. An
extensive parametric study is conducted considering different footing widths,
foundation depths, soil density conditions, input motion intensities and
durations to study the failure mechanism and the values of the ultimate
seismic bearing capacity. The conducted analyses are two-dimensional plane
strain analyses. Hence, only strip footings are investigated. The results are
interpreted in terms of the stress-settlement curve for each case, and the
relative impacts of each of the studied parameters on the seismic bearing
capacity.
The nonlinear dynamic model is adopted to simulate the soil constitutive
behavior under seismic conditions. The stress-strain relationship in the
nonlinear model is approximated by a hyperbolic curve. The curve is defined
by two parameters; the slope at zero strain (the small strain or the maximum
shear modulus) and the asymptotes at large strains (the maximum shear
strength). The nonlinear dynamic model accounts for the variation of the
excess pore pressure at each time step during the earthquake. Hence, the
volumetric changes that occur after the end of the seismic shaking can be
accurately predicted.
The soil profile under study consists of 10.0-11.5 m of medium dense
saturated sand, followed by a 15.0m-thick dense sand layer overlying a
15.0m-thick very dense sand. The input motion used in the parametric study is
an artificial earthquake time history based on the standard response spectrum
of the Uniform Building Code (1994).
Four footing widths are investigated: 1.0 m, 3.0 m, 5.0 m and 7.0 m. The
effect of the foundation depth is investigated by comparing the seismic
bearing capacity of surface footings with that of footings founded at the depth
of 1.5 m below the ground surface (in addition for foundation depth at 2.5 m for some cases). The groundwater table is always taken at the foundation
level. The base analyses are conducted by applying a peak acceleration
response of 0.15g at the foundation level, and earthquake duration of 20
seconds. The ultimate seismic bearing capacity is determined graphically
from the stress-settlement curve. Each point on the curve is determined by
conducting numerical analyses in three steps. The first step is to establish the
initial static conditions due to the effective overburden stresses and the
applied footing stress. The second step is to apply the seismic excitation at the
lower boundary of the model in order to calculate the excess pore pressures.
The third step re-distributes the unbalanced stresses from the seismic analysis
due to the excess pore pressures. Hence, the post-earthquake settlement
corresponding to each applied stress is calculated.
5.2. Conclusions
The study results show that each of the footing width, foundation depth and
soil relative density has a significant impact on the ultimate seismic bearing
capacity of partially liquefiable sand. While the effects of the footing width
and relative density are logic and expected, the positive impact of the
foundation depth on the seismic bearing capacity requires the need for a
significant enhancement to Day (2010) equation. It is concluded that a footing
resting on partially liquefiable medium dense saturated sand will not be
subjected to a punching shear failure as assumed by Day (2010). Study results
show that both the q-term and -term should be included in the seismic
bearing capacity equation with reduced values of the seismic bearing capacity
factors N
q and N. The new developed values of the seismic bearing capacity
factors are presented in the form of charts as a function of the internal friction angle. The use of the recommended values of Nq, N and pore pressure ratio in
the ultimate seismic bearing capacity equation yields about 0.8-1.2 the values
deduced from the stress-settlement curve.
The study presents, also, useful design charts that can be used to estimate
the allowable seismic bearing capacity, based on the specified allowable postearthquake settlement. In order to satisfy both the ultimate and serviceability
limit states, the conducted research provides a design framework to enable the
design engineer to determine the allowable seismic bearing capacity of
shallow foundations resting on partially liquefiable saturated sands. The
proposed framework is summarized in the following steps:
a. The ultimate seismic bearing capacity is calculated based on the
suggested equation and using the recommended seismic bearing
capacity factors and pore pressure ratio.
b. The effect of the peak acceleration response is introduced through a
developed equation based on the numerical analyses conducted in this
research study.
c. A first estimate of the allowable seismic bearing capacity is calculated
by considering a factor of safety of 5.0, as suggested by Day (2010).
d. A second estimate of the allowable seismic bearing capacity is
determined from the design charts developed in this research study,
based on the allowable post-earthquake settlement by the design code.
e. The final design seismic bearing capacity is the minimum value
obtained from (c) and (d).
The outcome of this research study provides more insight into the
mechanism of the seismic bearing capacity of partially liquefiable sand and
the different factors influencing its value. Given the poor coverage of this
important design issue in the literature, the current research study closes an important gap and provides the design engineer with a useful and optimized
design guideline for the problem.
The factors influencing the ultimate seismic bearing capacity can be
categorized into factors related to the footing characteristics, the soil strength,
the soil deposit thickness and the earthquake characteristics.
The proposed seismic bearing capacity equation for partially liquefiable
sand accounts primarily for the effects of the footing characteristics and soil
strength. The soil deposit thickness and earthquake characteristics are
partially considered in the value of the pore pressure ratio. The effects of
these two important factors are further investigated in this research study, and
their impact on the seismic bearing capacity is quantified.
The main conclusions of the research study can be summarized in the
following main points:
1. The mode of failure of shallow foundations resting on partially
liquefiable sand is not a punching shear mode. This is evidenced by the
noticeable effect of the foundation depth on the ultimate seismic
bearing capacity.
2. The seismic bearing capacity factors Nq and N should not be equal to
the static values. Representative values are derived in this research
study, based on the internal friction angle of the sand.
3. The exact factor of safety against seismic bearing capacity failure
should be determined after considering the tolerable post-earthquake
settlement.
4. The study recommends a design framework to obtain the allowable
seismic bearing capacity of partially liquefiable sand.