1
FOUNDATIONS FOR DYNAMIC LOADS
VIJAY K. PURI SHAMSHER PRAKASH
Professor Professor Emeritus
Civil Engineering Civil Engineering
SIU Carbondale Missouri S&T, Rolla
IL 62901 MO 65409
puri@engr.siu.edu
praskash@mst.edu
ABSTRACT
Design of foundations in earthquake prone areas needs special considerations.
Shallow foundations may experience a reduction in bearing capacity and increase in
settlement and tilt due to seismic loading. The reduction in bearing capacity depends
on the nature and type of soil and ground acceleration parameters. In the case of
piles, the soilpile behavior under earthquake loading is generally nonlinear. The
nonlinearity must be accounted for by defining soil pile stiffness in terms of strain
dependent soil modulus. A comparison of observed and predicted pile behavior under
dynamic loads has attracted the attention of several investigators. The lateral dynamic
pile response of single piles predicted by analytical models often yields higher natural
frequencies and lower resonant amplitudes compared to those determined from field
tests in horizontal vibrations only. This has been found to be due to overestimated
shear modulus and radiation damping of the soil. The authors made an investigation
to determine a simple method to improve the theoretical predictions of piles
embedded in fine soils. Based upon this investigation shear strain dependent
reduction factors are proposed for determining the shear modulus and damping for
pile response calculations.
INTRODUCTION
Structures subjected to earthquakes may be supported on shallow foundations or on
piles. The foundation must be safe both for the usual static loads as well for the
dynamic loads imposed by the earthquakes and therefore the design of either type of
foundation needs special considerations compared to the static case. Shallow
foundations in seismic areas are commonly designed by the equivalent static
approach. The observations during 1985 MichoacanGuerero earthquake (Pecker,
1996) and the Kocaeli 1999 earthquake have shown that initial static pressure and
load eccentricity have a pronounced effect on seismic behavior of foundations. The
soil strength may undergo degradation under seismic loading depending on the type
of soil. Pore pressure buildup and drainage conditions may result in decrease in
strength and an increase in settlement. Most foundation failures due to earthquake
occur due to increased settlement. However, failures due to reduction in bearing
capacity have also been observed during Niigata earthquake (1964) in Japan and
Izmit earthquake (1999) in Turkey (Day, 2002).
Prasad et al (2004) made an experimental investigation of the seismic bearing
capacity of sand. A practical method to account for reduction in bearing capacity
308
2
due to earthquake loading was presented by Richard et al, (1993). The settlement and
tilt of the foundation must also be considered. Gazetas and Anastasopoulos (2008)
have studied the interaction effects of two adjacent buildings, founded on shallow
footings.
For analysis and design of piles and pile groups under seismic loads a simple
approach to account for nonlinear soilpile interaction using strain dependent soil
modulus to define soilpile stiffness and radiation damping can be used and is
discussed in the paper. Some studies are available comparing the predicted and
observed response of piles also. In some of these studies the soil pile stiffness and
damping were arbitrarily modified to match the observed and predicted response of
the soil pile system. A study was conducted under the supervision of the authors in
which the data reported in literature was reanalyzed and reduction factors have been
proposed for the stiffness and radiation damping obtained by using the commonly
approach of Novak and ElSharnouby (1983). Using the proposed reduction factors
the reported test data of Gle (1981) was reanalyzed. The predicted values of natural
frequency and amplitude of vibration at various frequencies showed an excellent
geotechnical comparison with the observed data. The case of shallow foundations is
discussed first followed by several aspects of pile behavior under dynamic loads.
SHALLOW FOUNDATIONS
The problem of static bearing capacity of shallow foundations has been extensively
studied in the past (Terzaghi (1943), Meyerhoff (1951), Vesic (1973) and many
others. Basavanna et al. (1974) obtained analytical solutions for the bearing capacity
under transient loading conditions. The design of foundations subjected to seismic
loading has generally been performed by using psedostatic approach (Prakash;
1981). The foundations are considered as eccentrically loaded and the ultimate
bearing capacity is accordingly estimated. Prakash and Saran (1971) proposed a
method to determine the settlement and tilt of a foundation subjected to vertical load
and moment. The response of a footing to dynamic loads is affected by the (1) nature
and magnitude of dynamic loads, (2) number of pulses and (3) the strain rate response
of soil. To account for the effect of dynamic nature of the load, the bearing capacity
factors are determined by using dynamic angle of internal friction which is taken as
2degrees less than its static value (Das, 1992). Building codes such as International
Building Code (2006) generally permit an increase of 33 % in allowable bearing
capacity when earthquake loads in addition to static loads are used in design of the
foundation. This recommendation is based on the assumption that the allowable
bearing pressure has adequate factor of safety for the static loads and a lower factor
of safety may be accepted for earthquake loads. This recommendation may be
reasonable for dense granular soils, stiff to very stiff clays or hard bedrocks but is not
applicable for friable rock, loose soils susceptible to liquefaction or pore water
pressure increase, sensitive clays or clays likely to undergo plastic flow (Day, 2006).
Richards et al (1993) observed seismic settlements of foundations on partially
saturated dense or compacted soils. These settlements were not associated with
liquefaction or densification and could be easily explained in terms of seismic bearing
3
capacity reduction. They have proposed a simplified approach to estimate the
dynamic bearing capacity q
ue
and seismic settlement S
Eq
of a strip footing
Figure 1. Failure surface in soil for seismic bearing capacity (After Richards et
al, 1993; See also Prakash and Puri, 2008)
Figure 1 shows the assumed failure surfaces. The seismic bearing capacity
(q
uE
) is given by Eq. 1:
q
uE
= cN
cE
+ qN
qE
+ ½ γ BN
γE
(1)
where, γ = Unit weight of soil
D
f
= Depth of the foundation and q= γD
f
N
cE
, N
qE
, and N
γE
= Seismic bearing capacity factors which are functions of φ and
tanψ = k
h
/ (1k
v
)
k
h
and k
v
are the horizontal and vertical coefficients of acceleration due to
earthquake.
For static case, k
h
= k
v
, = 0 and Eq. (1) becomes
q
u
= cN
c
+ qN
q
+ ½ γBN
γ
(2)
in which N
c
, N
q
and N
γ
are the static bearing capacity factors. Figure 2 shows plots of
N
γE
/N
γ
, N
qE
/N
q
and N
cE
/N
c
with tan ψ
and φ.
tanψ = k
h
/ (1k
v
) tan ψ = k
h
/ (1k
v
), tanψ = k
h
/ (1k
v
),
Figure 2. Variation of N
qE
/N
q
, N
γE
/N
γ
and N
cE
/N
c
with φ and tan ψ (After
Richard et al 1993; See also Prakash and Puri, 2008)
SEISMIC SETTLEMENT OF FOUNDATION
Bearing capacityrelated seismic settlement takes place when the ratio k
h
/(1  k
v
)
reaches a critical value (k
h
/1 – k
v
)*. If k
v
= 0, then (k
h
/1  k
v
)* becomes equal to k
h
*
.
Figure 3 shows the variation of k
h
*
(for k
v
= c = 0; granular soil) with the static factor
4
of safety (FS) applied to the ultimate bearing capacity [Eq, (2)], for φ = 10°, 20°, 30°,
and 40° and D
f
/B of 0, 0.25, 0.5 and 1.0.
Figure 3. Critical acceleration
*
h
k
(After Richards et al, 1993; See also Prakash
and Puri, 2008)
The settlement (S
Eq
) of a strip foundation due to an earthquake can be estimated as
4
2
*
( ) 0.1 7 4 t an
k
V
h
S m
E q A E
A g A
α
−
=
(3)
where V = peak velocity for the design earthquake (m/sec),A = acceleration
coefficient for the design earthquake, g = acceleration due to gravity (9.81 m/sec
2
).
tan α
AE
depends on φ and k
h
*, (Richards et al, (1993). In Figure 4, variation of tan
α
AE
with k
h
* for φ
of 15°  40° is shown.
Suppose a typical strip foundation is supported on a sandy soil with B = 2 m, and D
f
= 0.5 m, and γ = 18 KN/m
3
, φ = 34°, and c = 0. The value of k
h
= 0.3 and k
v
= 0 and
the velocity V induced by the design earthquake is 0.4 m/sec. The static ultimate
bearing capacity for this footing for vertical load will be 1,000 KN/m
2
(Eq. 2). The
reduced ultimate bearing capacity for vertical load is calculated as 290 KN/m
2
(Eq 1).
If the footing is designed using a FS = 3 on the static ultimate bearing capacity (i.e.,
for an allowable soil pressure of 333 kN/m
2
), the additional settlement due to
earthquake will be 20.5 mm (Eq 3, Richards et al, 1993). This settlement reduces to
7.0 mm if FS of 4 is used. Besides ensuring that the footing soil system does not
experience a bearing capacity failure or undergo excessive settlement, the
foundations should be tied together using interconnecting beams.
Seismic bearing capacity factors for a strip footing resting on cohesionless soil were
determined by Dormieux and Pecker (1995) using the upper bound theorem of yield.
5
Figure 4. Variation of tan α
AE
with k
h
* and φ (After Richards et al 1993; See
also Prakash and Puri, 2008)
Using the classical Prandtl like mechanism, it was established that the reduction in
bearing capacity was mainly caused by load inclination (Dormieux and
Pecker;1995).Choudhury and Subba Rao (2005) determined seismic bearing capacity
factors for shallow strip footing using the limit equilibrium approach and pseudo
static method of analysis. The reduction in bearing capacity under the combined
effect of vertical and horizontal forces was explained by using smaller failure surface
compared to case when only static vertical loads are applied.
It is thus seen that presently (2009), the pseudostatic approach is being used to
determine bearing capacity and settlement and tilt of the foundations subjected to
seismic loads in nonliquefying soils. Dynamic nature of the load and other factors
which affect the dynamic response are not being accounted for. Also, no guidelines
are available design of footings in liquefying soil.
PILES FOUNDATIONS
Piles may often be the preferred choice of foundations in seismic areas. The seismic
loading induces large displacements or strains in the soil. The shear modulus of the
soil degrades and damping (material) increases with increasing strain. The stiffness of
piles should be determined for these strain effects. The elastic solutions for
determining response of piles subjected to dynamic loads have been presented by
several investigators in the past in several modes of vibrations, (Novak, 1974; Novak
and ElSharnouby, 1983; Novak and Howell, 1977; Poulos, 1971; Prakash and
Puri,1988; and Prakash and Sharma, 1991). Displacement dependent spring and
damping factors for piles for vertical, horizontal and rotational vibrations have been
presented by Munaf and Prakash (2002), Munaf et al. (2003) and Prakash and Puri
(2008). The stiffness of the pile group is estimated from that of the single piles by
6
using group interaction factors. The contribution of the pile cap, if any, is also
included. The response of the single pile or pile groups may then be determined using
principles of structural dynamics. The design of pile foundations subjected to
earthquakes requires a reliable method of calculating the effects of earthquake
shaking and postliquefaction displacements on pile foundations. Keys to good design
include reliable estimates of environmental loads, realistic assessments of pile head
fixity, and a mathematical model which can adequately account for all significant
factors that affect the response of the pilesoilstructure system to ground shaking
and/or lateral spreading in a given situation.
The equivalent spring stiffness and damping of the soilpile system are a function of
Young’s modulus of pile material (E
p
), shear modulus of soil (G
s
), and geometry of
the piles in the group. Shear modulus and hence spring and damping factors are strain
or displacement dependent. There are six independent spring factors for a pilescap
system; ie; k
x
, k
y
,
k
z
, in translation in x, y and z directions, respectively and k
φ
, k
θ
, k
ψ
rotationalsprings about x, y and z directions respectively. There are two rotational
crosscoupled springs; i.e.; k
xφ
and k
yθ
which include 2components of displacement;
i.e., translation and rotation about the appropriate axis. Also there are corresponding
eight damping factors; i.e., c
x
, c
y,
c
z
, c
φ
, c
θ
, c
ψ
, and. c
xφ
and c
yθ
. To develop
displacement dependent relationships for the spring and damping factors, appropriate
relationships between strain and displacement are needed. Also, modulus
degradation with strain needs be built into these relationships. Stiffness and damping
in all the modes; i.e., vertical, horizontal, rocking and torsion and cross coupling in
both the x and y direction have been evaluated (Munaf and Prakash, 2002). The sign
convention is explained in Fig. 5.
Figure 5: Sign Convention
For the case of earthquake loads, the response of piles under horizontal loads are
important and are discussed here.
Sliding and Rocking Stiffness and Damping Factors
Because, the pile is assumed to be cylindrical with a radius r
o,
its stiffness and
damping factors in any horizontal direction are the same. However, in the pile group,
the number of piles in the x and y directions may be different. Therefore the stiffness
and damping factors of a pile group are dependent on the number of piles and their
spacing in each direction, (Figure 6).
7
Sliding (k
x,
c
x
)
1
3
0
x
pp
x
f
r
IE
K
⎥
⎦
⎤
⎢
⎣
⎡
=
(4a)
2
2
0
x
s
pp
x
f
Vr
IE
C
⎥
⎦
⎤
⎢
⎣
⎡
=
(4b)
Rocking (k
φ
, c
φ
) and (k
θ
, c
θ
)
1
2
φθϕ
f
r
IE
KK
o
pp
⎥
⎦
⎤
⎢
⎣
⎡
==
(5a)
2
2
φθφ
f
Vr
IE
CC
so
pp
⎥
⎦
⎤
⎢
⎣
⎡
==
(5b)
Crosscoupling (k
xφ
, c
xφ
) and (k
yθ
,
c
yθ
)
1
2
0
θθϕ
x
pp
yx
f
r
IE
KK
⎥
⎦
⎤
⎢
⎣
⎡
==
(6a)
2
φθφ
x
so
pp
yx
f
Vr
IE
CC
⎥
⎦
⎤
⎢
⎣
⎡
==
(6b)
Where;
I
p
= moment of inertia of single pile about x or y axis
r
o
= pile radius
E
p
= modulus of elasticity of pile material
V
s
= shear wave velocity of soil along the floating pile
f
x1,
f
x2,
f
xφ1,
f
xφ
2
,
f
xφ1,
f
xφ2
are Novak’s coefficient obtained from Table 1 for
homogeneous and parabolic soil profiles, with appropriate interpolation for ν between
0.25 and 0.4.
Group interaction factor
For group effect in lateral directions (Poulos, 1971), obtained a solution for α
L
for
each pile in the horizontal xdirection, considering departure angle β (degrees). α
L
’s
are a function of L, r
o
and flexibility K
R
as defined in figure 7 and departure angle β.
This procedure will also apply for horizontal y direction. The group interaction
factor (∑α
L
) is the summation of α
L
for all the piles. Note that the group
interactionfactor in xdirection and ydirection may be different depending on number
and spacing of piles in each direction.
Group Stiffness and Damping
Figure 6 shows schematically the plan and cross sections of an arbitrary pile group.
This figure will be used to explain the procedure for obtaining the stiffness and
damping for a group of pile for all modes of vibration.
8
Figure 6: Plan and Cross Section of Pile Group
Figure 7: Graphical solution of α
L
(Poulus, 1971)
Sliding and Rocking and Cross Coupled Group Stiffness and Damping Factors
Translation along Xaxis
∑
∑
=
Lx
x
g
x
k
k
α
(7a)
∑
∑
=
Lx
x
g
x
c
c
α
(7b)
9
Table 1. Stiffness and Damping Parameters of Horizontal Response for Pile with
L/R
o
>25 for Homogeneous Soil Profile and L/R
o
>30 for Parabolic Soil Profile
( Novak and ElSharnouby, 1983)
Translation along Y Axis (k
y
g
,
c
y
g
)
∑
∑
=
LA
y
g
y
k
k
α
(8a)
∑
∑
=
Ly
y
g
y
c
c
α
(8b)
Rocking About Y Axis (k
φ
g
,
c
φ
g
)
[ ]
φφφ
α
xccxrz
Lx
g
kzzkxkkk 2
1
22
−++=
∑
(9a)
[ ]
φφφ
α
xccxrz
Lx
g
czzcxccc 2
1
22
−++=
∑
(9b)
Rocking About X Axis (k
θ
g
,
c
θ
g
)
[ ]
θθθ
α
yccyrz
Ly
g
kzzkykkk 2
1
22
−++=
∑
(10a)
10
[ ]
θθθ
α
yccxrz
Ly
g
czzcyccc 2
1
22
−++=
∑
(10b)
CrossCoupling: Translation along X Axis and Rotation about Y Axis. (k
xφ
g
,
c
xφ
g
)
( )
∑
−=
cxx
Lx
g
x
zkkk
φφ
α
1
(
11a)
( )
∑
−=
cxx
Lx
g
x
zccc
φφ
α
1
(
11b)
CrossCoupling: Translation along YAxis and Rotation about X Axis. (k
yθ
g
,
c
yθ
g
)
( )
∑
−=
cyy
Ly
g
y
zkkk
θθ
α
1
(
12a)
( )
∑
−=
cyy
Ly
g
y
zccc
θθ
α
1
(
12b)
COMPARISION OF COMPUTED AND PREDICTED PILE RESPONSE
The present methods for design of pile foundations subjected to dynamic loads are
generally based on the models developed by Novak (1974) and Novak and
El_Sharnouby (1984) and also presented by Prakash and Puri (1988) and Prakash and
Sharma (1990). Several researchers have attempted to make a comparison of the
observed and predicted pile response. Small scale pile tests, centrifuge and full scale
pile tests have been used for this purpose (Gle, 1981; Novak and ElSharnouby, 1984;
Woods, 1984; and Poulos, 2007). Woods (1984) reported results of 55 horizontal
vibration tests on 11 end bearing piles 15  48 m long. The outer diameter of piles was
35.56 cm and the wall thickness varied from 0.47 cm to 0.94 cm. Typical amplitude –
frequency plot for one of the piles in soft clay is shown in Fig. 8. It may be seen from
this plot that the observed natural frequency decreases with an increase in the value of
‘ө’ (increase in ‘ө’ means an increase in dynamic force at the same frequency of
vibrations) indicating nonlinear behavior. Woods (1984) also compared the
observed and computed response of the piles. The stiffness and damping values were
obtained using computer program PILAY which uses continuum model
accommodating soil layers and assumes homogeneous soil in the layer with elastic
behavior. A typical comparison of the pile response so computed with the observed
response is shown in Fig. 9. It may be observed from Fig.9 that the calculated and
computed responses do not match.
Efforts were made to obtain a match between observed and predicted response by
using reduced values of stiffness obtained from PILAY, which did not help much. A
better match could, however, be obtained when a considerably softened or weakened
zone was assumed surrounding the piles (program PILAY 2) simulating disturbance
to soil during pile installation. A loss of contact of the soil with the pile for a short
length close to the ground surface also improved the predicted response.
11
Fig. 8. Response curves; a decrease Fig 9. Typical response curves
in resonant frequency with increasing predicted by PILAY superimposed
amplitudes. (Woods, 1984) on measured pile response.
(Woods, 1984)
Novak and ElSharnouby (1984) performed tests on 102 model pile groups using
steel pipe piles. A typical comparison of the theoretical and experimental horizontal
response is shown in Fig.10. Plot ‘a’ shows the theoretical group response without
interaction effects. Response shown in plot ‘b’ was obtained by applying static
interaction factors to stiffness only. Plot ‘c’ was obtained with arbitrary interaction
factor of 2.85 applied to stiffness only. Plot‘d’ was obtained by using an arbitrary
interaction factor of 2.85 on stiffness and 1.8 on damping respectively. Plot ‘e’ shows
the experimental data. The plot which shows an excellent match with experimental
data was obtained by
arbitrarily
increasing the damping factor by 45%.
Fig. 10. Experiment horizontal response curves and theoretical curves calculated
with static interaction factors. (Novak and ElSharnouby, 1984)
The comparison of theoretical response obtained by using dynamic interaction
factors of Kaynia and Kausel (1982) is shown in plot ‘a’ in Fig.11. Plot ‘b’ in Fig. 11
shows the calculated data based on dynamic analysis of Wass and Hartmann(1981).
12
Fig. 11. Experimental horizontal response curve and theoretical curves.
(a) Calculated with Kaynia and Kausel dynamic interaction factors
(b) Calculated with Wass and Hartmann impedances
(c ) Experimental (Novak and ElSharnouby, 1984).
The experimental data of Novak and ELSharnouby (1984) is shown by plot ‘c’.
Novak and ElSharnouby, (1984) also compared the observed response for vertical
and torsional vibrations with the predicted response.
Jadi (1999) and Prakash and Jadi (2001) reanalyzed the reported pile test data of
Gle (1981) for the lateral dynamic tests on single piles and proposed reduction factors
for the stiffness and radiation damping obtained by using the approach of Novak and
ElSharnouby (1983). The suggested equations for the reduction factors are:
λ
G
= 353500 γ
2
– 0.00775 γ + 0.3244 (16)
λ
c
= 217600 γ
2
– 1905.56 γ + 0.6 (17)
where, λ
G
and λ
c
are the reduction factors for shear modulus and damping and γ is
shear strain at computed peak amplitude without correction factors.
Typical
comparison of the computed pile response using the modified values of stiffness and
damping values and the observed response is shown in figures 1220.
Fig 12 shows prediction and performance of Gle’s pile in Fig 9. Fig 13, 14 and 15
are for the cases where prediction and performance matched well. However in Fig16
18, the prediction and performance does not match well. Fig 19 and 20 shows
prediction and performance from other sites.
COMMENTS ON PREDICTIONS
Novak and El Sharnouby (1984) have attempted to match the observed and the
predicted response by adjusting, arbitrarily, the group stiffness and damping values.
No guidelines were developed to modify these values.
Woods (1984) used PILAY program with modified stiffness to match prediction and
performance.
Jadi (1999) developed rational correction factors to both stiffness and damping to
match the computed and predicted responses. She was successful in her efforts. Her
13
Fig.12 Measured and predicted Fig. 13 Measured and predicted lateral
response of pile of Fig. 11 Dynamic Response of pile GP 137 (ө=2.5
˚
)
(Prakash and Jadi 1991) (Prakash and Jadi, 2001)
Fig. 14 Measured Vs arbitrarily reduced Fig. 15 Measured vs. predicted lateral
Dynamic Response of pile K 167 (ө=5
˚
) Dynamic Response with proposed
(Prakash and Jadi, 2001) reduction factors for pile K 167, (ө=5
˚
)
(Prakash and Jadi, 2001)
Fig.16 Measured and predicted lateral Fig.17 Measured and predicted lateral
Dynamic Response of piles K167 Dynamic Response of pile GP 13 7
(ө=5
˚
)(Prakash and Jadi 2001) (ө=2.5
˚
) (Prakash and Jadi, 2001)
14
Fig.18 Measured and predicted lateral Fig.19 Measured and predicted lateral
Dynamic Response of pile LF 16 Dynamic Response of pile IWES
(ө=10
˚
)(Prakash and Jadi 2001) vibrator λ
G
= 0.31, λ
c
= 0.5
(Prakash and Jadi, 2001)
Fig.20 Measured vs. Reduced predicted lateral Dynamic Response of the 2.4”
Pile tested by Novak and Grigg, 1976, λ
G
= 0.44, λ
c
= 0.34, (Prakash and Jadi,
2001)
approach, however is more scientific and more efforts needs to be devoted to develop
relationships for correction factors for different modes of vibration.
CONCLUSIONS
Considerable attention has been paid to the design of foundations for earthquake
loads, both for shallow foundations and piles. Efforts have been made to understand
the behavior of the foundations under seismic loading. However the shallow
foundations are mostly designed using the equivalent static approach.
For the case of pile foundations, many efforts have been made for comparing the
predicted and observed response of single piles and pile groups under dynamic
loading and arbitrary modifications to stiffness and damping were made. Jadi’s
(1999) method of modifying shear modulus and damping appears reasonable.
15
However this has been tested against a limited data base. More analyses are needed to
make this tool easily usable in practice.
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th
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