58
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
obtained with the same numerical (FE) analysis as the curves
and snapshots of Fig. 2, and can be expressed analytically as a
function of the static factor of safety (
F
S
) as
=
1 −
(3)
The specific plot is in terms of
N/N
uo
which is
1/F
s
which
ranges between 0 and 1. Notice that heavily and lightly loaded
foundations with
1/F
s
symmetrically located about the
1/F
S
=
0.5
value where the
M
u
is the largest, have the same moment
capacity : yet their behavior especially in cyclic loading is quite
different as will be shown subsequently.
5 MONOTONIC RESPONSE ACCOUNTING FOR P
−
δ
EFFECTS
An increasingly popular concept in structural earthquake
engineering is the so-called
“pushover”
analysis. It refers to the
nonlinear lateral force-displacement relationship of a particular
structure subjected to monotonically increasing loading up to
failure. The development (theoretical or experimental) of such
pushover relationships has served as a key in simplified
dynamic response analyses that estimate seismic deformation
demands and their ultimate capacity. We apply the pushover
idea to a shallow foundation supporting an elevated mass, which
represents a tall slender structure with
h/B = 2
(or “slenderness”
ratio
h/b = 4, where b = B/2
). This mass is subjected to a
progressively increasing horizontal displacement until failure by
overturning. Since our interest at this stage is only in the
behavior of the foundation, the structural column is considered
absolutely rigid. The results are shown in Fig:4(a) and (b) for
two
F
s
values : 5 and 2.
The difference in the
M-
θ
response curves from those of
Fig. 2 stems from the so-called P-
δ
effect. As the induced lateral
displacement of the mass becomes substantial its weight induces
an additional aggravating moment,
mgu = mg
θ
h
, where
θ
is the
angle of foundation rotation. Whereas before the ultimate
moment
M
u
is reached the angles of rotation are small and this
aggravation is negligible, its role becomes increasingly
significant at larger rotation and eventually becomes crucial in
driving the system to collapse. Thus, the (rotation controlled)
M-
θ
curve decreases with
θ
until the system topples at an angle
θ
c
. This critical angle for a rigid structure on a rigid base
(F
S
=
∞
)
is simply :
,
∞
=
(4)
where b = the foundation halfwidth. For very slender systems
the approximation
,
∞
≈
(4a)
is worth remembering.
As the static vertical safety factor (
F
S
) diminishes, the
rotation angle (
θ
c
) at the state of imminent collapse (“critical”
overturning rotation) also slowly decreases. Indeed, for rocking
on compliant soil,
θ
c
is always lower than it is on a rigid base
(given with
Eq. 4
). For stiff elastic soil (or with a very large
static vertical safety factor)
θ
c
is imperceptibly smaller than that
given by Eq. 4, because the soil deforms slightly, only below
the (right) edge of the footing, and hence only insignificantly
alters the geometry of the system at the point of overturning. As
the soil becomes softer, soil inelasticity starts playing a role in
further reducing
θ
c
. However, such a reduction is small as long
as the factor of safety (
F
S
) remains high (say, in excess of 3).
Such behaviour changes drastically with a very small
F
S
: then
the soil responds in strongly inelastic fashion, a symmetric
bearing-capacity failure mechanism under the vertical load N is
almost fully developed, replacing uplifting as the prevailing
mechanism leading to collapse
θ
c
tends to zero.
The following relationship has been developed from FE
results by Kourkoulis et al, 2012, for the overturning angle
θ
c
=
θ
c
(F
s
) :
,
∞
≈ 1 −
+
1 −
(5)
6 CYCLIC RESPONSE ACCOUNTING FOR P
−
δ
EFFECTS
Slow cyclic analytical results are shown for the two
aforementioned systems having static factors of safety (
F
S
= 5
and 2). The displacement imposed on the mass center increased
gradually; the last cycle persisted until about 4 or 5 times the
angle
θ
u
of the maximum resisting moment. As can be seen in
the moment
−
rotation diagrams, the loops of the cyclic analyses
for the safety factor
F
S
= 5
are well enveloped by the monotonic
pushover curves in Figure 7(a). In fact, the monotonic and
maximum cyclic curves are indistinguishable. This can be
explained by the fact that the plastic deformations that take
place under the edges of the foundation during the deformation-
controlled cyclic loading are too small to affect to any
appreciable degree of response of the system when the
deformation alters direction. As a consequence, the residual
rotation almost vanishes after a complete set of cycles
―
an
important (and desirable) characteristic. The system largely
rebounds, helped by the restoring role of the weight. A key
factor of such behaviour is the rather small extent of soil
plastification, thanks to the light vertical load on the foundation.
The cyclic response for the
F
S
= 2 system is also essentially
enveloped by the monotonic pushover curves. However, there
appears to be a slight overstrength of the cyclic “envelope”
above the monotonic curve. For an explanation see
Panagiotidou et al, 2012.
But the largest difference between monotonic and cyclic, on
one hand, and
F
S
= 2
and 5, on the other, is in the developing
settlement. Indeed, monotonic loading leads to monotonically-
upward movement (“heave”) of the center of the
F
S
= 5
foundation, and slight monotonically-downward movement
(“settlement”) of the
F
S
= 2
foundation. Cyclic loading with F
S
= 5 produces vertical movement of the footing which follows
closely its monotonic upheaval.
But the
F
S
= 5
foundation experiences a progressively
accumulating settlement
much larger that its monotonic
settlement would have hinted at. The hysteresis loops are now
wider. Residual rotation may appear upon a full cycle of
loading, as inelastic deformations in the soil are now
substantial.
The above behavior is qualitatively similar to the results of
centrifuge experiments conducted at the University of
California at Davis on sand and clay (e.g., Kutter et al. 2003,
Gajan et al. 2005) large-scale tests conducted at the European
Joint Research Centre, (Negro et al. 2000, Faccioli et al. 1998),
and 1-g Shaking Table tests in our laboratory at the National
Technical University of Athens on sand (Anastasopoulos et al
2011, 2013, Drosos et al 2012).
In conclusion, the cyclic moment
−
rotation behavior of
foundations on clay and sand exhibits to varying degrees three
important characteristics with increasing number of cycles :
•
no “strength” degradation (experimentally verified).
•
sufficient energy dissipation
large for small F
S
values,
smaller but still appreciable for large ones. (Loss of energy
due to impact will further enhance damping in the latter
category, when dynamic response comes into play.)
•
relatively low residual drift especially for large FS values
implying a
re-centering
capability of the rocking
foundation.
These positive attributes not only help in explaining the
favorable behavior of “Rocking Foundation”, but also enhance
the reliability of the geotechnical design.
