57
Honour Lectures /
Conférences honorifiques
are indeed allowed to slide during the design earthquake. The
time is therefore ripe for soil–foundation–structure interaction
(SFSI) to also move from imposing “safe” limits on forces and
moments acting on the foundation (aiming at avoiding pseudo-
static “failure”) to performance–based design in which all
possible conventional “failure” mechanisms are allowed to
develop, to the extent that maximum and permanent
displacements and rotations are kept within acceptable limits.
3 THE CONCEPT OF “ROCKING ISOLATION”
IN FOUNDATION DESIGN
The paper addresses the case of structure-foundation systems
oscillating mainly in a rotational mode (rocking).
Subjected to strong seismic shaking, structures tend to
experience large inertial forces. For tall-slender structures these
forces will lead to overturning moments onto the foundation
that may be disproportionally large compared to the vertical
load. As a result, a shallow foundation may experience
detachment (uplifting) of one edge from the supporting soil.
This in turn will lead to increased normal stresses under the
opposite edge of the foundation. Development of a bearing
capacity failure mechanism is quite possible if such a
concentration leads to sufficiently large stresses. But, in
contrast to a static situation, even
then
failure may not occur.
Thanks to the
cyclic
and
kinematic
nature of earthquake
induced vibrations : (i) the inertial forces do not act
“forever”
in the same direction to cause failure (as would be the case with
static load), but being cyclic, very soon reverse and thereby
relieve the distressed soil; and (ii) the developing inertial forces
are not externally applied predetermined loads, but are
themselves reduced once the soil-foundation system reaches its
(limited) ultimate resistance
the foundation system acts like a
fuse. As a result, the system experiences nonlinear-inelastic
rocking oscillations, which may or may not result in excessive
settlement and rotation. But failure is almost unlikely.
In the last 10 years a number of research efforts have
explored the consequences of substantial foundation rocking on
the response of the supported structure, theoretically and
experimentally : Kutter et al 2003, Gajan et al 2005, Harden et
al 2006, Kawashima et al 2007, Apostolou et al 2007, Paolucci
et al 2008, Chatzigogos & Pecker 2010, Deng et al 2012. The
results of these studies confirmed the idea that strongly-
nonlinear rocking oscillations under seismic excitation can be of
benefit to the structure.
Taking the whole idea one small step farther, it is proposed
that the design of a shallow foundation should actively “invite”
the creation of two simultaneous “failure” mechanisms:
substantial foundation uplifting and ultimate bearing-capacity
sliding. This would be accomplished by substantially under-
designing the foundation
e.g., by reducing its width and
length to, say, one-half of the values required with current
design criteria. This can be thought of as a reversal of the
“capacity” design: “plastic hinging” will take place in the
foundation-soil system and not at the column(s) of the structure.
Fig. 1 elucidates the main idea of Rocking Isolation. The
benefits of designing the foundation to work at and beyond its
conventional limits will become evident in the sequel. To this
end, three examples will elucidate the dynamics of “Rocking
Isolation” in comparison with the dynamics of the conventional
design :
(a) a bridge pier, free to rotate at its top
(b) a two-storey two-bay asymmetric frame (MRF)
(c) a three-storey retrofitted frame
−
shearwall structure.
In each case, the two alternatives ( the conventional and the
rocking-isolated system) are subjected to numerous acceleration
time histories the overall intensity of which is either within or
well beyond the design earthquake levels.
4 ROTATIONAL MONOTONIC RESPONSE
OF SHALLOW FOUNDATIONS
Much of the research in earlier years on dynamic rocking of
foundations and dynamic soil
−
structure interaction had focused
on linear response. Elastic stiffness and damping as functions of
frequency have been developed and utilised to describe the
dynamic action of the foundation system. The various US
seismic codes in the last 30
+
years have promulgated linear
approximations to deal with seismic soil
−
structure interaction.
The behavior of “Rocking Foundations” significantly
deviates from linear visco-elasticity: uplifting introduces strong
geometric nonlinearity and even damping due to impact ; soil
yielding and plastic deformation generate hysteresis, implying
significant frequency-independent damping, while when
bearing-capacity slippage mechanisms develop a limiting
plateau restricts the passage of high accelerations from the
ground into the superstructure.
In monotonic loading, a most crucial parameter controlling
the moment
−
rotation, M
−
θ
, relation of a specific foundation is
the factor of safety against vertical static bearing capacity
failure :
F
s
= N
uo
/N
(1)
where
N
uo
is the ultimate load under purely vertical loading and
N the acting vertical load. Fig. 2 offers typical results for a
homogeneous (
G
and
s
u
) soil for three
F
s
values : a very high
one (20), a low one (2), and an extremely low one (1.25). M is
normalized by N
uo
B, where B is the width of the footing in the
direction of loading. This leads to curves which, for the
homogeneous profile considered, depend solely on the so-called
“rigidity index”,
G/ s
u
, and the shape of the footing.
Also shown in Fig. 2 are the snapshots of the deformed soil
and the contours of plastic strain as they develop when the
maximum moment is reached
apparently at different angles
of rotation. The following are worthy of note in the figure:
•
The foundation with
F
s
= 20
(which can be interpreted either
as a very-lightly loaded foundation or as a “normally”-loaded
foundation on very stiff soil) despite its largest initial elastic
rocking stiffness fails at the smallest value of applied
moment:
M
u
≈
0.025 N
uo
B
(2a)
Indeed if
F
s
→ ∞
, i.e. there is no vertical load onto the
foundation,
M
u
would vanish, due to the tensionless nature of
the soil
−
footing interface.
•
As expected from the literature (Meyerhof 1963, Georgiadis
and Butterfield 1988, Salençon and Pecker 1995,
Α
llotey and
Naggar 2003, Apostolou and Gazetas 2005, Gajan and Kutter
2008, Chatzigogos et al. 2009, Gouvernec 2009, Gajan and
Kutter 2008) the largest maximum moment is attained by the
F
s
= 2
footing :
M
u
≈
0.13 N
uo
B
(2b)
but its elastic initial rocking stiffness is smaller than for the
F
s
= 20
foundation. Evidently, the extensive plastic deformations
upon the application of the vertical (heavy) load soften the
soil so that a small applied moment meets less resistance
hence lower stiffness. However,
F
s
= 2
achieves the largest
ultimate
M
u
as it leads to an optimum combination of uplifting
and bearing-capacity mobilization.
•
A more severely loaded foundation, however, with the (rather
unrealistic)
F
s
= 1.25
will only enjoy an even smaller initial
stiffness and a smaller ultimate moment than the
F
s
= 2
foundation. Notice that in this case no uplifting accompanies
the plasticification of the soil.
The failure envelope (also called interaction diagram) in N-
M space is given in Fig. 3 for the specific example. It was
