59
Honour Lectures /
Conférences honorifiques
7 SEISMIC RESPONSE OF BRIDGE PIER ON SHALLOW
FOUNDATION
The concept of “Rocking Isolation” is illustrated in Fig. 5 by
comparing the response of a 12 m tall bridge pier carrying a
deck of four lanes of traffic for a span of about 35 m
typical
of elevated highways around the world.
The bridge chosen for analysis is similar to the Hanshin
Expressway Fukae bridge, which collapsed spectacularly in the
Kobe 1995 earthquake. The example bridge is designed in
accordance to (EC8 2000) for a design acceleration
A
= 0.30 g,
considering a (ductility-based) behavior factor
q
= 2. With an
elastic (fixed-base) vibration period
T
= 0.48 sec the resulting
design bending moment
M
COL
≈
45 MNm.
The pier is founded through a square foundation of width
B
on an idealized homogeneous 25 m deep stiff clay layer, of
undrained shear strength
s
u
= 150 kPa (representative soil
conditions for which a surface foundation would be a realistic
solution). Two different foundation widths are considered to
represent the two alternative design approaches. A large square
foundation,
B =
11 m, is designed in compliance with
conventional capacity design, applying an overstrength factor
γ
Rd
= 1.4 to ensure that the plastic “hinge” will develop in the
superstructure (base of pier). Taking account of maximum
allowable uplift (eccentricity
e
=
M
/
N
<
B
/3, where
N
is the
vertical load), the resulting safety factors for static and seismic
loading are
F
S
= 5.6 and
F
E
= 2.0, respectively. A smaller,
under-designed,
B
= 7 m foundation is considered in the spirit
of the new design philosophy. Its static safety factor
F
S
= 2.8,
but it is designed applying an “
understrength
” factor
1/1.4
≈
0.7
for seismic loading. Thus, the resulting safety factor for seismic
loading is lower than 1.0 (
F
E
≈
0.7).
The seismic performance of the two alternatives is
investigated through nonlinear FE dynamic time history
analysis. An ensemble of 29 real accelerograms is used as
seismic excitation of the soil–foundation–structure system. In
all cases, the seismic excitation is applied at the bedrock level.
Details about the numerical models and the requisite
constitutive relations can be seen in Anastasopoulos et al, 2010,
2011.
Results are shown here only for a severe seismic shaking,
exceeding the design limits: the Takatori accelerogram of the
1995 M
JMA
7.2 Kobe earthquake. With a direct economic loss of
more than $100 billion, the Kobe earthquake needs no
introduction. Constituting the greatest earthquake disaster in
Japan since the 1923 M
s
= 8 Kanto earthquake, it is simply
considered as one of the most devastating earthquakes of
modern times. Of special interest is the damage inflicted to the
bridges of Hanshin Expressway, which ranged from collapse to
severe damage. The aforementioned bridge chosen for our
analysis is very similar to the Fukae section of Hanshin
Expressway, 630 m of which collapsed during the earthquake of
1995. It is therefore logical to consider this as a reasonably
realistic example of an “
above the limits
” earthquake. In
particular, the Takatori record constitutes one of the worst
seismic motions ever recorded : PGA = 0.70 g, PGV = 169
cm/s, bearing the “mark” of forward rupture directivity and of
soil amplification.
Fig. 5 compares the response of the two alternatives, in terms
of deformed mesh at the end of shaking with superimposed the
plastic strains. In the conventionally designed system there is
very little inelastic action in the soil; the red regions of large
plastic deformation are seen only under the severely “battered”
edges of the rocking foundation
but without extending below
the foundation. “Plastic hinging” forms at the base of the pier,
leading to a rather intense accumulation of curvature
(deformation scale factor = 2).The P
−
δ
effect of the mass will
further aggravate the plastic deformation of the column, leading
to collapse.
In stark contrast, with the new design scheme the “plastic
hinge” takes the form of mobilization of the bearing capacity
failure mechanisms in the underlying soil, leaving the
superstructure totally intact. Notice that the red regions of large
plastic shearing are of great extent, covering both half-widths of
the foundation and indicating alternating mobilization of the
bearing capacity failure mechanisms, left and right.
The above observations are further confirmed by the time
history of deck drift shown in Fig. 5(c). The two components of
drift, are shown, one due to footing rotation in blue and one due
to structural distortion in green. Their sum is shown in red.
Evidently, the conventional design experiences essentially only
structural distortion which leads to uncontrollable drifting
collapse. In marked contrast, the system designed according to
the new philosophy easily survives. It experiences substantial
maximum
deck drift (about 40 cm), almost exclusively due to
foundation rotation. Nevertheless, the
residual
foundation
rotation leads to a tolerable 7 cm deck horizontal displacement
at the end of shaking.
Fig. 5(d) further elucidates the action of the foundation-soil
system. The M-
θ
relationship shows for the 11m
2
foundation a
nearly linear viscoelastic response, well below its ultimate
capacity and apparently with no uplifting. On the contrary, the
7m
2
(under-designed) foundation responds well past its ultimate
moment capacity, reaching a maximum
θ
≈
30 mrad
, generating
hysteretic energy dissipation, but returning almost to its original
position, i.e. with a negligible residual rotation.
However, energy dissipation is attained at a cost : increased
foundation settlement. While the practically elastic response of
the conventional (
over-designed
) foundation leads to a minor 4
cm settlement, the
under-designed
foundation experiences an
increased accumulated 15 cm settlement. Although such
settlement is certainly not negligible, it can be considered as a
small price to pay to avoid collapse under such a severe ground
shaking.
Perhaps not entirely fortuitously, the residual rotation in this
particular case turned out to be insignificant. The recentering
capability of the design certainly played some role in it.
8 SEISMIC RESPONSE OF TWO
−
STOREY TWO BAY
ASYMMETRIC FRAME
The frame of Fig. 6 was structural designed according to EC8
for an effective ground acceleration A = 0.36 g and ductility-
dependent “behavior” factor q = 3.9. The soil remains the stiff
clay of the previous example. Two alternative foundation
schemes are shown in the figure .
The conventionally
over-designed
footings can mobilize a
maximum moment resistance
M
u
from the underlying soil,
larger than the bending moment capacity of the corresponding
column
M
COL .
. For static vertical loads, a factor of safety
F
S
≥
3
is required against bearing capacity failure. For seismic load
combinations, a factor of safety
F
E
= 1 is acceptable. In the
latter case, a maximum allowable eccentricity criterion is also
enforced:
e
=
M/N
≤
B/3
. For the investigated soil–structure
system this eccentricity criterion was found to be the controlling
one, leading to minimum required footing widths
B
= 2.7 m, 2.5
m and 2.4 m for the left, middle, and right footing, respectively.
Bearing capacities and safety factors are computed according to
the provisions of EC8, which are basically similar to those
typically used in foundation design practice around the world.
The
under-sized
footings of the rocking isolation scheme, are
“weaker” than the superstructure, guiding the plastic hinge to or
below the soil–footing interface, instead of at the base of the
columns. The small width of the footings promotes full
mobilization of foundation moment capacity with substantial
uplifting. The eccentricity criterion is completely relaxed, while
F
E
< 1 is allowed. The static
F
S
≥
3 remains a requirement as a
measure against uncertainties regarding soil strength. Moreover,
it turns out that
F
S
≥
4 might be desirable in order to promote
uplifting–dominated response, and thereby limit seismic
settlements [Kutter et al. 2003, Faccioli et al. 2001,Pecker &
