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Multistory Building Frames and Shear Walls Founded on “Rocking” Spread Footings

  • G. GazetasEmail author
  • D. Dais
  • F. Gelagoti
  • R. Kourkoulis
Chapter
  • 963 Downloads
Part of the Geotechnical, Geological and Earthquake Engineering book series (GGEE, volume 46)

Abstract

The seismic performance of a two-story 2D frame and a five-story 3D frame–shear-wall structure founded on spread (isolated) footings is investigated. In addition to footings conventionally designed in accordance with “capacity-design” principles, substantially under-designed 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 full-scale bridge pier-foundation 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 under-designed foundations against very strong seismic shaking. Equally-supportive conclusions have been drawn by numerous experimental campaigns with small-scale shaking-table tests under both 1-g and centrifuge conditions.

Most of the theoretical and experimental studies on rocking foundations refer to a single footing supporting a simple inverted-pendulum type structure like a single-column bridge pier. A more limited number of studies have dealt with simple frame structures, as well as with frame–with–shear-wall 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 under-designed “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 multi-story and multi-spam 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 two-story 2-span moment-resisting frame; and (b) a three-dimensional five-story building frame with no and with four shear walls.

13.2 Two-Story Frame on two Types of Footings

The frame of Fig. 13.1 was structurally designed according to EC8 for an effective ground acceleration A = 0.36 g and ductility-dependent “behavior” factor q = 3.9. The soil is stiff clay with Su = 150 kPa and Go = 105 MPa. Two alternative foundation schemes are examined (Anastasopoulos et al. 2014):
  1. (a)

    Conventionally over-designed 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 FS ≥ 3 is required against bearing capacity failure. For seismic load combinations a factor of safety FE = 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.

     
  2. (b)
    Under-sized 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 FE < 1 is allowed. The static FS ≥ 3 remains a requirement as a measure against uncertainties regarding soil strength. Moreover, it turns out that FS ≥ 4 might be desirable in order to promote uplifting–dominated response, and thereby limit seismic settlements and increase re-centering. 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 FS. 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 code-based 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)

    FS

    Footing

    B (m)

    FS

    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

     
Fig. 13.1

The 2-story RC plane frame: geometry and reinforcement

The performance of the two design alternatives is compared in Fig. 13.2. The deformed mesh with superimposed plastic strain contours of the two alternatives is portrayed on the figure. 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. 13.2b). 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 plasticization (indicated by the red regions under the foundation.
Fig. 13.2

Comparison of performance of two alternatives to Takatori motion. (a) Deformed mesh with plastic strain contours; (b) column moment-curvature response

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 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 FS (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 more-or-less 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 Five-Story Existing Frame: Seismic Petrofit with Shear Walls

13.3.1 Existing Building

A five-story reinforced-concrete frame (shown in Fig. 13.3) consists of 12 columns connected with beams and carrying 5 slabs. It has been designed according to the norms and practices of the 1970s, with a base-shear coefficient of 0.06. The soil is a stiff clay with uniform with depth Su ≈ 150 kPa and the footings (squares or rectangles) were designed to have a static factor of safety against bearing-capacity, Fs > 3 and a seismic one FE > 2; The resulting dimensions are given in Tables 13.1 and 13.2.
Fig. 13.3

The 5-story RC framed building: plan view and elevation

Table 13.2

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

The frame–foundation–soil system is modelled with 3D finite elements using ABAQUS (Fig. 13.4a). We subjected this frame to the motion recorded in Lefkada 2003 earthquake, which according to our current understanding roughly corresponds to the assumed design base-shear coefficient. The structure fails. Fig. 13.4b shows the computed time histories at the top and the first floor. The two curves are almost identical and reveal failure of the “soft first story” type, with the upper structure moving as a block. Therefore, there is a need for seismic retrofit to upgrade the structure.
Fig. 13.4

(a) The 3D finite element discretisation of the system; (b) displacement time history of first floor (red dashed line) and roof (black bold line) during the Lefkada 2003 earthquake

13.3.2 Upgrading with Shear Walls on Conventional Foundations

For a simple retrofitting scheme, we construct four shear (structural) walls, two in each direction, in place of the columns K2, K4, K9 and K11 (as shown in Fig. 13.5). The structural design of each wall was based on the current greek codes (EAK and EKΩΣ) and resulted in a wall with lw = 1.7 m, bw = 0.4 m, and hw = 9 m. The latter is the height of the building. The (ultimate) moment capacity of each wall, MRD was computed equal to 2 MNm (in its long direction). The design effective ground acceleration is A = 0.24 g and the ductility-depended factor q = 3. Details can be found in the thesis od Dais (2015).
Fig. 13.5

Plan of 1st floor of the retrofitted building showing the location and size of the shear walls (top); first mode deformation of the structure showing the conventional footings (bottom)

The footing of each shear wall obeys the standard capacity-design rules: Fs ≥ 3, FE ≥ 1, e = M/N ≤ L/3, and loading increased by an over-strength factor aCD ≈ 1.3. The latter ensures that the foundation system’s maximum moment resistance Mu exceeds the structural moment capacity, MRD ≈ 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 closely-spaced 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.

So after some trials we select for the structural walls footings:
$$ \mathrm{L}=3.2\;\mathrm{m}\;\mathrm{and}\;\mathrm{B}=1.8\;\mathrm{m} $$
which are indeed much smaller than those imposed by capacity design, L = 6 m and B = 2.2 m. The small footing nevertheless complies with the static requirement since Fs = 6. It violates only the aforementioned criterion of (seismic) eccentricity:
$$ \mathrm{e}={\mathrm{M}}_{\mathrm{RD}}/\mathrm{N}\approx 2000/820\approx 2.4\;\mathrm{m} $$
which by far exceeds not only L/3 ≈ 1.1 m but also L/2 = 1.6 m! Clearly, a pseudo-static way of thinking would not have allowed the resultant force to fall outside the foundation as this comparison implies.

13.3.4 Comparison of the Seismic Performance of the Two Alternatives

The retrofitted five-story building whose structural walls are supported with (a) the conventional (L = 6 m, B = 2.2 m) and (b) the unconventional (L = 3.2 m, B = 1.8 m) foundation is subjected to two ground motions:
  • 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 design-level 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.

The comparison for the Lefkada excitation is given in Figs. 13.6 and 13.7, referring to the response: of the whole structure, of one of the shear structural walls, and of one representative column. Specifically, Fig. 13.6a, b compares the moment-curvature relations and the shear force time histories at the base of wall T11, from which it is evident that the wall of the “rocking isolation” design responds more favorably. Indeed Fig. 13.6c, d shows that this unconventional solution results in smaller roof displacement and small drift ratio of the first floor.
Fig. 13.6

Comparison of the response of the retrofitted building on conventional and unconventional foundations induced by the 2003 Lefkada motion: (a) moment-curvature relation of shear wall T11; (b) shear force time history at the base of shear wall T11; (c) drift ratio time history of the 1st floor; and (d) roof displacement time history

Fig. 13.7

Response of column K12 to 2003 Lefkada: (a) time history of normalised axial force; (b) moment-curvature relations at the base of the column

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.

With the stronger San Salvador CIG excitation, Fig. 13.8 shows the moment-curvature relations for two walls (T4 and T9), as well as the time history of roof displacement. No doubt the walls of the unconventional system respond more favorably. They remain in the linear range with maximum moment limited to about 1 MNm. In addition, the top displacement time-history of the unconventional system is consistedly smaller. Evidently, the differences between the two systems are larger with the stronger excitation. And in spite of the rocking of the foundation the shear walls act like the backbone in humans preventing the ribs from slipping out: the floor slabs are held from experiencing significant drift, and hence the un-improved columns do not suffer much.
Fig. 13.8

Comparison of the response of the retrofitted building on conventional and unconventional foundations induced by the 1986 San Salvador motion: (a) moment-curvature relation of shear wall T9; (b) moment-curvature relation of shear wall T4; (c) time history of roof displacement

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 Over-Conservatism 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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • G. Gazetas
    • 1
    Email author
  • D. Dais
    • 1
  • F. Gelagoti
    • 1
  • R. Kourkoulis
    • 1
  1. 1.National Technical University of AthensAthensGreece

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