Brief Introduction to Rocking Structures
Rocking structures avoid structural damage by shifting the burden of energy dissipation to non-critical, replaceable structural elements, and by preventing weak story failure (see figure 1.2). Damage that would result in severe injury, and also damage that would prevent future serviceability of the structure, can be addressed by allowing structures to move, relative to their foundations. By enabling structures to be serviceable after a seismic event, rocking systems are a highly sustainable approach to structural design in earthquake-prone regions.
For newly constructed buildings, there are many ways to prevent seismic damage. For example the entire frame can be allowed to rock, with energy dissipation being performed by replaceable fuses (figure 1.1).4,p3 However clearly for existing structures, this is not possible, and we must find other ways to apply the benefits of rocking structures.
The concept of rocking shear walls, as differentiated from fixed-base shear walls, was introduced by Ajrab et al in 2004. The work they presented was built on studies which originated in 1963, when Housner investigated the free vibration of rigid rocking blocks.1,p1
Mander and Cheng (1997, cited in Ajrab et al1,p1) defined an approach to rocking, structural flexibility, and prestressing, as damage avoidance design. The desired performance objective for the DAD philosophy for a maximum assumed earthquake (MAE) is that the structure remains elastic at all times during ground shaking. For example elastic rotations might be defined as rotations of less than 1%, 0.6°. Usually much less is preferred, for example the design criteria for the Tokyo Institute of Technology G3 building retrofit project was a peak story drift angle of 1/250 radian (0.4%, 0.23° ).13,p6 For a maximum considered earthquake (MCE), which is defined in FEMA document 27310,p32, the structure may yield with limited damage (for example defined as plastic rotations of less than 0.5%) to the conventional reinforced concrete framing elements.1,p4 Rocking walls are one method that can achieve these objectives, and which are particularly relevant to retrofit applications.
In this report, we first review some of the current literature available on rocking walls, and then introduce models and simulations which are intended to show quantitatively how rigid a rocking wall would need to be for a given general application, and also to lay the groundwork for further study.
2.1. Rocking Wall Design
A number of central issues emerge in rocking wall design. First, adequate resistance to lateral loads must be maintained. The structure has lost one moment bearing element at the base of each wall, and this load bearing capacity must be maintained by the moment capacity of shear wall-frame connections, by the weight of the rocking shear wall itself, and by any additional mechanisms, such as bracing, or post-tensioned tendons running through the wall (figure 2.1). Pekcan et al (2000, cited in Ajrab et al1,p1) suggested that such tendons be draped to match the shape of the moment diagram induced under the assumed inertial loading. Note that the original rocking wall concept presented is that of a flat base. If the rocking wall is pinned at the base, as for example in the retrofit at the Tokyo Institute of Technology by Wada et al2, then the weight of the wall itself offers no moment resistance.
Figure 2.1. A non-hinged
rocking wall on rigid foundation, with supplemental supportive
(adapted from Ajrab et al.1,p2)
Secondly, the rocking wall-supported structure must offer sufficient damping. Mander et al (1998, cited in Ajrab et al1) conclude that a tendon-supported rocking structure provides only limited damping, for example 1-2% of critical. Percassi (2000, cited in Ajrab et al1) showed that the addition of damping devices to tendons could enhance the damping offered by the system.
The third central issue in rocking wall design, is the degree to which the structure is serviceable following a seismic event. Kishiki & Wada9,p1 discuss how following the Northridge and Kobe earthquakes, many buildings became structurally unviable, leading to a termination of social and industrial activities, and consequently severe economic loss. A flat-based rocking wall as illustrated in figure 2.1 may suffer from toe crushing, and if rigid wall-frame connections are used, these may also be significantly damaged. Such severe damage invariably requires that the building be reconstructed, since further seismic performance cannot be predicted.2,p1 A step towards a truly mechanical building is shown by Wada et al2,p7, where an open pin design, fabricated from cast iron, is used at the base of the rocking wall to prevent severe damage to the wall toes during rocking (figure 2.2).
2.2. Lateral Load Bearing Capacity
For a flat-based rocking wall on a rigid surface, and which is otherwise unsupported, it can be seen from statics that the lateral load that can be supported is
Vmax = Ww(b - h)/Heff
where Ww is the weight of the wall, Heff is the point of application of the load, b is half the width of the wall, h is half the height of the wall, and q is the angle through which the wall has moved, as in figure 2.1.
Further load-bearing capacity is provided by the moment capacity of floor-wall connections, and frame bracing, and these contributions may be derived trivially. Example formulae providing the loading-bearing capacity offered by floor-wall connections and supportive tendons are derived in Ajrab et al1,p2
Of course in the case of a retrofit application of rocking walls, rather than in the design of a new building, it will usually be reasonable to assume that the existing structure has sufficient lateral load bearing capacity, except for seismic loading.
2.3. Damping of Rocking Wall Systems
The four sources of damping in the structure are inherent damping, which is typically taken to be 5% for concrete structures, radiation damping due to the impact of a flat-based rocking wall with the ground, hysteretic damping due to plastic behavior within the frame, and supplemental damping such as dampers attached to the system. Formulae to illustrate radiation damping and hysteretic damping are given by Ajrab et al1,p3
Various types of damping devices have been proposed, such as the damping/fuse elements in series with tendons proposed by Ajrab et al1,p2, and externally-mounted mild steel dampers proposed by Marriott et al3,p2. Damping devices may be installed with the intent that they be replaced after a seismic event.2,p2
2.4. Design Motivation
The introduction of rocking wall system to a frame building may be motivated by showing the benefit of a global failure mode with low rotations as opposed to a local failure mode with high rotations. Consider for example the simple frame with the three failure modes shown in figure 2.3.
Figure 2.3. A simple
frame and the three failure modes possible for such a frame
(adapted from Wada et al.2,p3)
Referring to figure 2.3, It can be readily seen that as the number of plastic hinges increases, so does the ability of the structure to dissipate energy. The first two failure modes shown have low energy dissipation, and so the rotations induced will be large, with high potential for loss of life and severe structural damage. However the final failure mode uses all possible plastic hinges, and thus has the maximum energy dissipation per unit rotation. Thus in this mode the rotations will be smaller, and the probability of saving life and further structural serviceability is maximized.2,p2 Hence the focus for seismic retrofit need not be strengthening the individual members which would deform excessively under seismic loading, but rather the control of the global behaviour of the structure to prevent damage from weak modes.2,p6
Hence if we can design a rocking wall that is stiff enough to resist the partial failure modes C1 and C2, the frame will tend to fail in the preferable global failure mode (figure 2.4).2,p4 A rocking wall also suppresses higher mode vibrations2,p10. Hence we might consider the problem of exactly how strong or stiff a rocking wall must be to resist the partial failure modes.
Figure 2.4. A rocking wall can encourage high energy dissipation and thus low rotations (Wada et al.2,p4)
1) Ajrab et al (2004) Rocking WallFrame
Structures with Supplemental Tendon Systems.
Available at: http://dx.doi.org/10.1061/(ASCE)0733-9445(2004)130:6(895) (Accessed: 18 November 2010)
2) Wada et al (2009) Seismic Retrofit Using Rocking
Walls and Steel Dampers.
Available at: http://www.quzhe.net/Resource/Struct/2009_ATC-SEI_G3Bld.pdf (Accessed: 18 November 2010)
3) Marriott et al (2008) Dynamic Testing of Precast,
Post-Tensioned Rocking Wall Systems with Alternative Dissipating
18 November 2010)
4) G. Deierlein (2010) Presentation: Damage Resistant
Braced Frames with Controlled Rocking and Energy Dissipating
Available at: http://peer.berkeley.edu/events/2010/semm_seminar_deierlein.html (Accessed: 28 November 2010)
5) Van der Merwe, Johann Eduard (2009) Rocking shear wall
foundations in regions of moderate seismicity: Master of Science
thesis in Civil Engineering. Engineering University of
Available at: https://scholar.sun.ac.za/handle/10019.1/1957 (Accessed: 4 December 2010)
6) Browne et al (2006) The Analysis of Reinforced
Concrete Rocking Wall Behaviour.
Available at: http://db.nzsee.org.nz/2006/Paper15.pdf (Accessed: 18 November 2010)
7) Ma et al (2006) An Alternative Mathematical Model
for a Controlled Rocking System.
Available at: http://db.nzsee.org.nz/2006/Paper16.pdf (Accessed: 18 November 2010)
8) American Society of Civil Engineers (2006) Minimum Design Loads for Buildings and Other Structures, ASCE/SEI 7-05.
9) Kishiki & Wada. (2009) New Dynamic Testing Method on
Available at: http://peer.berkeley.edu/events/2009/icaese3/cd/files/pdf/KISHIKI_WADA_21.pdf (Accessed: 18 November 2010)
10) Federal Emergency Management Agency (FEMA) (1997) NEHRP
Guidelines for the Seismic Rehabilitation of Buildings.
Available at: http://www.wbdg.org/ccb/FEMA/ARCHIVES/fema273.pdf (Accessed: 20 November 2010)
11) Nippon Steel (2005) Development of U-shaped Steel
Damper for Seismic Isolation System, Technical Report No. 92.
Available at: http://www.nsc.co.jp/en/tech/report/pdf/n9211.pdf (Accessed: 20 November 2010)
12) Akira Wada (2010) Talk on the rocking wall retrofit of building G3. Massachusetts Institute of Technology, Cambridge, Massachusetts, 22 Oct 2010
13) Wada (2010) Strength, Deformability, Integrity and
Strong Columns in Seismic Design of Multi-Story Structures.
Available at: http://www.bridge.t.u-tokyo.ac.jp/apss/lectures/Prof.Wada_July26_APSS2010.pdf (Accessed: 25 November 2010)