60
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
Pender 2000, Kawashima et al. 2007, Chatzigogos et al. 2009;
Panagiotidou et al. 2012]. Applying the methodology which has
been outlined in Gelagoti et al. 2012, the footings were designed
to be adequately small to promote uplifting, but large enough to
limit the settlements. Aiming to minimize differential
settlements stemming from asymmetry, the three footings were
dimensioned in such a manner so as to have the same
F
S
. Based
on the above criteria, the resulting footing widths for the
rocking–isolated design alternative are
B
= 1.1 m, 1.8 m, and
1.3 m, for the left, middle, and right footing, respectively:
indeed, substantially smaller than those of the code-based
design. Footing dimensions and static factors of safety against
vertical loading of the two designs are summarized in Table 1.
Table 1. Footing dimensions and corresponding factors of safety
(computed following the provisions of EC8) against vertical loading for
the seismic load combination (
G
+ 0.3
Q
) for the two design alternatives
of Fig. 6.
Conventional Design
Rocking Isolation
Footing
B (m)
F
S
Footing
B (m)
F
S
Left
2.7
32.6 Left
1.1
5.4
Middle
2.5
10.6 Middle
1.8
5.4
Right
2.4
18.1 Right
1.3
5.4
The performance of the two design alternatives is compared
in Fig. 6. The deformed mesh with superimposed plastic strain
contours of the two alternatives is portrayed on top
(Fig. 6a).
With the relentless seismic shaking of the Takatori motion, the
conventionally designed frame collapses under its gravity load
(due to excessive drift of the structure, the moments produced
by
P–
δ
effects cannot be sustained by the columns, leading to
loss of stability and total collapse). As expected, plastic hinges
firstly develop in the beams and subsequently at the base of the
three columns, while soil under the footings remains practically
elastic. The collapse is also evidenced by the substantial
exceedance of the available curvature ductility of the columns
(
Fig. 6b). Conversely, the rocking–isolated frame withstands the
shaking, with plastic hinging taking place only in the beams,
leaving the columns almost unscathed (moment-curvature
response: elastic). Instead, plastic hinging now develops within
the underlying soil in the form of extended soil plastification
(indicated by the red regions under the foundation. The time
histories of inter-storey drift further elucidate the
aforementioned behavior of the two design alternatives
(
Fig.
6d).
Thanks to the larger bending moment capacity of the column
than of the footing, damage is guided “below ground” and at the
soil–foundation interface in the form of detachment and
uplifting
evidenced in
Fig. 6d
by the zero residual rotation,
unveiling the re-centering capability of the under-designed
foundation scheme.
The price to pay: large accumulated settlements. Moreover,
despite the fact that the three footings have been dimensioned to
have the same static factor of safety
F
S
(in an attempt to
minimize differential settlements exacerbated from asymmetry),
the central footing settles more than the two side footings,
leading to a differential settlement of the order of 3 cm. The
difference in the settlement stems of course from their
differences in width. As previously discussed, the central
footing was made larger (
B
= 1.8 m, compared to 1.1 m and 1.3
m of the two side footings) in order to maintain the same
F
S
.
Since the latter is common for the three footings, if the loading
is more-or-less the same, their response should be similar.
However, such equivalence refers to dimensionless quantities,
not absolute values [see Kourkoulis et al., 2012b]. In other
words, while the three footings sustain almost the same
dimensionless settlement
w/B
, which is roughly equal to 0.025
(
≈
3 cm/1.2 m) for the two side footings and 0.033 (
≈
6 cm/1.8
m) for the central one, the latter is substantially larger in width
and hence its settlement is larger in absolute terms. Naturally,
the three footings are not subjected to exactly the same loading,
something which further complicates the response. Such
differential settlements may inflict additional distress in the
superstructure, and are therefore worthy of further investigation.
9 THREE
−
STOREY FRAME RETROFITTED WITH
SHEAR
−
WALL
The results presented now are not from numerical analysis as
the previous one, but from Shaking Table experiments. They
refer to a 3-storey two-bay frame which was designed according
to the pre-1970 seismic regulations, for a base shear coefficient
of 0.06. Because of the small value of this coefficient and the
otherwise inadequate design, the frame has columns of cross-
section 25 x 25 cm
2
and beams 25 x 50 cm
2
resulting in a strong
beam
−
weak column system. Naturally, it fails by first “soft-
story” type of collapse when excited by motions corresponding
to today’s codes with effective ground accelerations of the order
of 0.30g and more. To upgrade the frame, a strong and stiff
Shear Wall 1.5 m x 0.3 m in cross-section is constructed
replacing the middle column, as shown in Fig. 7.
The 1:10
−
scale model is supported on dense fine
−
grained D
r
≈
80% sand. The original footings of all three columns were 1.5
m square. For the retrofitted frame the two columns retained
their original 1.5 x 1.5m
2
footings. The foundation of the Shear
Wall (SW) is of special geotechnical interest : due to its
disproportionately large lateral stiffness the SW tends to attract
most of the seismically induced shear force and hence to
transmit onto the foundation a large overturning moment. By
contrast, its vertical load is relatively small. To meet the
eccentricity limit e = M/N < B/3, a large foundation 6.0m x 0.80
m is thus necessary. Hence, the conventional solution of Fig. 8.
Of course the resulting vertical bearing-capacity factor of safety
is unavoidably large, F
S
≅
10, and the seismic apparent factor of
safety against moment bearing-capacity is also far more than
adequate :
F
E
= 2.
The decision to reduce the footing width to merely B = 3.5 m
is not only economically favorable, but in the harsh reality of
old buildings it may often be the only feasible decision in view
of the usual space limitations due to pipes, small basements,
walls, etc, present in the base. We will see if it is also favorable
technically in resisting a strong seismic shaking.
To be practical, in the above sense, no change is made to the
column footings. (1.5 m square).
We subject all three structures [ i.e., “a” the original frame,
“b” the retrofitted with a SW founded on conventionally-
conservative footing, and “c” the retrofitted with the
underdesigned SW footing] to a number of strong ground
excitations. Frame “a” easily fails as sketched in
Fig. 8,
where
the physical collapse was artificially prevented by an external
protective barrier in the Shaking Table experiment. The
conventionally retrofitted SW-frame “b” could withstand most
excitations. But with some of the strongest motions it developed
substantial plastification at its base and led to residual top drift
of an unacceptable 8%.
The unconventionally–founded system “c” behaved much
better with residual top drift of merely 2%.
Figure 8 sketches the deformation pattern of the three
systems while Fig. 7 plots the time histories of
structural
−
distortion and foundation
−
rotation induced top drift
ratio. It is seen that not only is the total drift of the Rocking-
Isolated system only 2% but at least half of it is solely due to
foundation rotation, rather than damage to the SW.
The penalty to pay is the increased settlement (1.5 cm rather
0.8 cm) which nevertheless in this particular case would be
acceptable for most applications.
