Multistory Building Frames and Shear Walls Founded on “Rocking” Spread Footings
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Abstract
The seismic performance of a twostory 2D frame and a fivestory 3D frame–shearwall structure founded on spread (isolated) footings is investigated. In addition to footings conventionally designed in accordance with “capacitydesign” principles, substantially underdesigned footings are also used. Such unconventional (“rocking”) footings may undergo severe cyclic uplifting while inducing large plastic deformations in the supporting soil during seismic shaking. It is shown that thanks to precisely such behaviour they help the structure survive with little damage, while experiencing controllable foundation deformations in the event of a really catastrophic seismic excitation. Potential exceptions are also mentioned along with methods of improvement.
13.1 Introduction: Isolation Via Rocking Foundation
In the last 15 years, “Rocking Foundations” have been found to be not only an economic but, in many cases, a technically superior seismic solution to conventionally designed foundations. Their superiority stems from the fact that they constrain the transmitted to the superstructure accelerations (thanks to the cutoff provided by their reduced moment capacity), and that they lead to increased natural period and hysteretic damping (Pecker 1998; Gajan et al. 2005; Kawashima et al. 2007; Anastasopoulos et al. 2010; Deng et al. 2012; Makris 2014; Gazetas 2015; Kutter et al. 2016). A nearly fullscale bridge pierfoundation seismic experiment, conducted by the teams of Professors Panagiotou and Kutter (Antonellis et al. 2016) on the UC San Diego shaking table, demonstrated the outstanding performance of highly underdesigned foundations against very strong seismic shaking. Equallysupportive conclusions have been drawn by numerous experimental campaigns with smallscale shakingtable tests under both 1g and centrifuge conditions.
Most of the theoretical and experimental studies on rocking foundations refer to a single footing supporting a simple invertedpendulum type structure like a singlecolumn bridge pier. A more limited number of studies have dealt with simple frame structures, as well as with frame–with–shearwall structures (Gelagoti et al. 2012; Kourkoulis et al. 2012; Anastasopoulos et al. 2014, 2015; Antonaki 2013; Dais 2015), with also encouraging results for the beneficial role of underdesigned “rocking” foundations in protecting the superstructure. The only drawback is the possible remaining settlement and rotation of the foundation. And whereas footing settlement of a statically determinate structure may not be a major problem, for the highly indeterminate multistory and multispam frames the consequences may be difficult to absorb in design. Hence the need to investigate the feasibility and usefulness of “rocking” spread foundations in such structures.
Two such systems are examined here: (a) a plane twostory 2span momentresisting frame; and (b) a threedimensional fivestory building frame with no and with four shear walls.
13.2 TwoStory Frame on two Types of Footings
 (a)
Conventionally overdesigned footings that can mobilize a maximum moment resistance M_{ u } from the underlying soil larger than the bending moment capacity of the corresponding column M_{ RD } . 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. The maximum allowable seismic 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. Notice that the left corner footing is required to be the largest because of its smallest axial load, an hence a tendency for larger eccentricity. 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.
 (b)Undersized footings of the rocking isolation scheme whose geotechnical capacity is smaller than the structural capacity of the columns, “guiding” the plastic hinge at 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 and increase recentering. 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–isolation 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 codebased design. Footing dimensions and static factors of safety against vertical loading of the two designs are summarized in Table 13.1.Table 13.1
Footing dimensions and corresponding factors of safety (computed with the provisions of EC8) against vertical loading, for the two design alternatives of Fig. 13.1
Conventional design
Rocking isolation
Footing
B (m)
F_{S}
Footing
B (m)
F_{S}
Left
2.7
32.0
Left
1.1
5.5
Middle
2.5
11.0
Middle
1.8
5.5
Right
2.4
18.0
Right
1.3
5.5
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. 13.2b by the zero residual rotation, unveiling the recentering capability of the underdesigned 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 FS. Since the latter is common for the three footings, if the loading is moreorless the same, their response should be similar. However, such equivalence refers to dimensionless quantities, not absolute values. 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. Pertinent amelioration measures are discussed later.
13.3 FiveStory Existing Frame: Seismic Petrofit with Shear Walls
13.3.1 Existing Building
Dimensions of half the footings (advantage of symmetry)
Direction  K1  K2  K3  K4  K5  K6 

X  1.8  2.0  1.8  1.4  2.4  1.4 
Y  1.8  1.4  1.8  2.0  2.4  2.0 
13.3.2 Upgrading with Shear Walls on Conventional Foundations
The footing of each shear wall obeys the standard capacitydesign rules: F_{s} ≥ 3, F_{E} ≥ 1, e = M/N ≤ L/3, and loading increased by an overstrength factor a_{CD} ≈ 1.3. The latter ensures that the foundation system’s maximum moment resistance M_{u} exceeds the structural moment capacity, M_{RD} ≈ 2 MNm. As a result of the small axial load and the (disproportionately) high overturning moment transmitted onto the footing by the wall, the required footing plan dimensions are L = 6 m and B = 2.2 m. This is the “conventional” foundation.
13.3.3 Unconventional (Rocking) Foundation
It is highly desirable in practice to be able to reduce these huge footing dimensions. Not so much for the (appreciable) savings in concrete, as for the frequent lack of space between closelyspaced footings in an actual building. All kinds of utilities may exist passing through this space. Hence, it is interesting to investigate the feasibility of solution with a rocking foundation.
To this end, we decided to reduce only the dimensions of the new structural wall while leaving the spread footings of the columns intact. At first, one might expect that such an action would “shed load” from the walls to the columns, as their overall stiffness has increased relative to the stiffness of the walls. And, hence, the columns may suffer from disproportionally high moments. Yet, when retrofit is attempted, this is by far the most desirable and easy solution, even though not technically optimal.
13.3.4 Comparison of the Seismic Performance of the Two Alternatives

the Lefkada 2003 earthquake record in Lefkada.

the San Salvador 1986 earthquake record at CIG.
The first is a moderately strong motion with PGA = 0.42 g; its response spectral value at T ≈ 0.75 s (the natural period of the retrofitted structure) only slightly exceeds the (EAK) design spectral value, while its spectral values at larger periods (T > 0.8 s) drop below those of the (EAK) design spectrum. It is therefore a designlevel excitation.
The San Salvador motion is fairly strong, exceeding the (EAK) design spectrum for all periods, and being some 50% to 100% larger than the Lefkada spectrum at the periods of interest T > 0.75 s. Hence it is a higher than design excitation.
On the other hand, the columns pay a very small penalty despite their increased share of the load. Indeed as seen in Fig. 13.7, column K12, the most severely stressed, experiences an increased ductility demand that is easily within the acceptable range. The axial load carried by the column also slightly increases.
13.3.5 Comments and Limitations
The analysis presented above, the specific building, and its proposed retrofit are only an example aimed to show the potential benefits of rocking foundations, even when upgrading existing buildings. The solution investigated is by no means optimal. But it does reinforce the conclusion reached in many studies, experimental and theoretical, that being overly conservative in foundation design does not lead to increased seismic safety of the structure they support. Recall the wisdom of the seminal 1977 article by the late Professor Ralf Peck on “The Pitfalls of OverConservatism in Foundation Design.”
One of the limitations of the “rocking isolation” for multicolumn buildings on spread footings is that the settlements and rotations of the individual footings will induce differential displacements between the columns of the structural system, and thereby cause damage. This is indeed a potential that must be investigated during analysis and its consequences must be accounted for in the design of the framing system. One solution may be the use of tie beams. In many cases their use is compulsory. But if their construction, as usual, fixes them on the top of the footing, rocking will be severely hindered and the “isolation” it provides will practically vanish. Continuous tie beams hinged at the base of the columns have been proposed by Anastasopoulos et al. (2014) which allow the beneficial rotation while they minimize differential settlement and permanent rotation of the footings. However, implementing such ideas in practice requires detailed thorough analysis with realistic modelling of the hinged connections – not a trivial task for engineering practice.
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