Innovations in rocking wallframe systemstheory and development
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Abstract
The need to improve the seismic performance of buildings has brought about innovative systems such as rocking wallmoment frame (RWMF) combinations. The behavior of RWMFs can best be visualized by the momentframe (MF) restraining the wall in place, and the rigid rocking wall (RRW) providing additional damping and imposing uniform drift along the height of the frame. A novel method of analysis followed by the development of a new lateral resisting system is introduced. The proposed concepts lead to an efficient structural configuration with provisions for selfcentering, reparability, performance control, damage tolerance and collapse prevention. Exact, unique, closed form formulae have been provided to assess the collapse prevention and selfcentering capabilities of the system. The objective is to provide an informative account of RWMF behavior for preliminary design as well as educational purposes. All formulae have been verified by independent computer analysis. Parametric examples have been provided to verify the validity of the proposed solutions.
Keywords
Link beams Rocking wallframes Uniform drift Collapse prevention Recentering ReparabilityIntroduction
 1.
The relative stiffnesses of the structure and the rocking system (MacRae et al. 2004);
 2.
The relative strengths of the two systems (Grigorian and Grigorian 2016).
 3.
Local seismicity and structural archetype (FEMA 2009).
 4.
Interactions of the two systems at common interfaces (Garlock et al. 2007; Dowden and Bruneau 2011).
 5.
The response of the freestanding MF on its own (Grigorian and Grigorian 2015).
 6.
The fact that RWMFs cited in this article have all passed tests of experimentation as well as time–history analysis (Zibaei and Mokari 2014).
These conditions have been employed to propose a new building archetype that incorporates posttensioned RRWs, link beams (LBs), buckling restrained braces (BRBs), and a grade beam restrained moment frame (GBRMF). The most compelling utility of the proposed solution is that it allows RWMFs to be designed as damage tolerant systems or in extreme scenarios as collapse prevented structures. All symbols are defined as they first appear in the text.
Conceptual design philosophies

The RRW acts like a vertical simply supported beam rather than a fixed base cantilever.

The axial stiffnesses of the BRBs, activated LBs and the RRW can be replaced by equivalent rotational stiffnesses.

The RRW suppresses all higher modes of vibration to lower levels than associated with axial deformations of the frame.

A RRW may be considered as rigid if its maximum elastic drift does not exceed 10% of the allowable drift ratio.

The RWMF remains a single degree of freedom (SDOF) system throughout the loading history of the structure.

Drift concentration is either nil or insignificant during all loading stages of the structure.

Gap opening can happen between any two planes perpendicular to the axis of a member.

The analysis can be immensely simplified, without loss of generality, by considering the lumped stiffnesses, equilibrium and drift compatibility of adjoining subframes, instead of those of the constituent elements of the system.
These ideas have all been utilized to develop the proposed building system and the corresponding analytical model.
Designled system development

Increasing energy dissipation, through steel tendon stretching (Christopoulos et al. 2002),

providing wall supported and other types of structural dampers and devices, such as LBs, BRBs, friction devices, etc,

reducing global stiffness in order to increase natural periods of vibrations (Chancellor et al. 2014),

making demand capacity ratios of as many members as close to unity as possible,

controlling mode shapes by allowing the first mode of vibrations to suppress all higher modes,

enforcing sway type collapse mechanism to prevent soft story failure (Wada et al. 1992),

reducing drift concentration, thereby improving structural performance (Hamburger et al. 2007),

reducing damage by using specific boundary support conditions such as those in GBRMFs,

designing relatively inexpensive PT cables to remain elastic during earthquakes; and

increasing repairability, by limiting damage to beams and replaceable items only.
Potential design advantages

The new system is ideally suited for reducing drift and preventing soft story failures in new and existing buildings,

the new system lends itself well to selfcentering, collapse prevention and damage reduction strategies,

the proposed system attracts substantially less residual stresses and deformations due seismic effects,

no major anchor bolt, base plate and footing damage can occur due to seismic moments,

gap openings dissipate seismic energy and provide opportunities for selfcentering and collapse prevention,

RRWs can be used as elements of structural control for pre and postearthquake conditions,

the drift profile is not sensitive to minor changes in wall stiffness,

RWMFs have longer natural periods of vibration than their fixed base counterparts and attract smaller seismic forces,

RRWs tend to rotate as rigid bodies without significant in or out ofplane deformations,

RRWs also provide effective protection against nearfault effects at all performance levels,

the displacement profile remains a function of the same single variable for all loading conditions,

the structure is a SDOF system, and as such lends itself well to equivalent energy studies,

the limit state drift ratios are smaller than those of identical frames with fixed and pinned boundary support conditions,

the magnitude and distribution of Pdelta moments are more favorable than in geometrically similar free standing MFs,

the restraining effects of BRBs can be expressed as notional equivalent overturning moments; and

RRWs tend to induce points of inflection at mid spans of all beam and columns,

and the earthquake resisting MFs are designed in accordance with the requirements of the prevailing codes of practice.
Principle design features

In GBRMFs plastic hinges are forced to form at the ends of the grade beams (not at column supports). The gravity system and the earthquakeresisting MFs are designed in accordance with the requirements of the prevailing codes of practice.

Conventional PT gap opening systems utilize flat bearing ends. Fully bearing gapopening devices tend to expand the frame beyond its original span length (Garlock et al. 2007); Dowden and Bruneau 2011). The uncontrolled span expansion induces drift concentration, additional column moments and tends to damage floor level diaphragms. In order to alleviate these effects, a truncated version of the same LB (Fig. 2a, c) has been introduced. The proposed LB consists of a pin ended, steel wide flange beam that contains the PT cables. In order to avoid contact between the column and the truncated ends of the LB, the width of the initial gap should be larger than \(\bar{\phi }d/2.\)

The disposition of Posttensioned (PT) tendons along the wall and the LBs should be in strict conformance with engineering principles. The length, layout, crosssectional areas, guide plates, the PT forces, etc., should be assessed in terms of the required drift angle, selfcentering and collapse prevention requirements. The cable layouts presented in Fig. 2a, b have been devised to reduce loss of stretching due to simultaneous gap opening and closing at both ends of the LB.
 Unsupplemented RRWs neither increase the strength nor the stiffness of the MFs. Seismic shear is transferred to the RWMF through direct shear, PT tendons, LBs, BRBs, as well as shear connectors between the slab and the RRW (Fig. 3a, b). The physical separation between the slab and the wall and the LBs prevents the slab and the wall from being damaged during earthquakes. Figure 3a allows horizontal shear transfer without inhibiting the vertical movement of the wall at the junction. The detail also provides out of plane stability at all floor levels. Figure 3d shows a typical base detail.
BRBs act as axially hysteretic elements. The use of BRBs increases the ductility, strength and stiffness of the RWMF. They can be highly instrumental in implementing drift control, damage reduction and collapse prevention strategies (AISC/SEAOC 2001). The challenge, therefore, is to select the brace force T _{ B.r,i }, at rth stage loading, for ith level, subframe, in such a way as to reduce the effects of the total external overturning moments including the Pdelta and out of straightness, ϕ′ effects (Fig. 1l) to more manageable levels. This is achieved by defining an equivalent moment of resistance M _{ B.r } and equivalent rotational stiffness K _{ B.r } for the braced frame of Fig. 1b. An innovative short cut method for relating T _{ B.r,i } to the global drift ratio ϕ _{ r } has been proposed for the specific purposes of the current article.
Theoretical development
Elastoplastic response

instead of modeling the axial restraining effects of the LBs, BRBs and the RRW tendons, equivalent rotational schemes have been utilized to capture the restoring effects of these devices against external forces.
 The additional stiffness of a horizontal subframe containing a BRB can be expressed as that of an equivalent pinjointed subframe with modified properties (Fig. 5) and that,

The redistribution of moments through formation of plastic hinges at beam ends (Fig. 4b) due to sway type failure forces the points of inflection towards mid spans.
Subframe response due to story level shear
The use of these observations leads to accurate solutions with insignificant margins of error. This can be attributed to the imposition of uniform drift by the RRW and the spread of plasticity over the entire structure. The solution becomes exact at incipient collapse. Since the drift angles ϕ _{ r,i } = ϕ _{ r } and initial imperfection or out of straightness, ϕ′ (Fig. 1l) are the same for all subframes i at rth stage loading, then the drift increment equation of any subframe, as in Fig. 1e in terms of story level racking, M _{0,r,i } = V _{ r,i } h _{ i } and Pdelta moments, M _{ PΔ,r,i } = (ϕ _{ r } + ϕ′)h _{ i } ∑ _{ i } ^{ m } ∑ _{ j=0} ^{ n } P _{ i,j } and subframe stiffness K _{ r,i } can be expressed (Grigorian and Grigorian 2012) as follows:
The denominator of Eq. (2) represents the rotational stiffness of the unsupplemented RWMF under lateral and Pdelta effects. It represents a closed form solution that can estimate lateral displacements and member forces of RWMFs throughout the entire linear and nonlinear static ranges of loading. In Eq. (2) shear, panel zone, hinge offset, axial load and other secondary effects have been discarded in favor of simplicity. Equation (2) is highly versatile in that it contains all plausible performance levels that affect structural response, safety and property protection. The contributions of supplementary LBs, BRBs and the posttensioned RRW are discussed in the few sections.
Development of subframe and LB load–displacement relationship
Development of subframe and BRB load–displacement relationship
Consider the displacements of the imaginary braced frame of Fig. 1, composed of the end column at j = n and the RRW as its vertical chords, and LBs and BRBs as its horizontal and diagonal elements, respectively. The RRW imposes a straight drift profile on the MF and the braced frame. As a result, each subframe, such as that shown in Fig. 5b, displaces an amount Δ_{ r,i } = ϕ _{ r } h _{ i }, with respect to its lower chord. The challenge here is to relate the brace force T _{ B.r,i } to the drift ratio ϕ _{ r }.
Development of MF and RRW displacement relationship
Equation (17) is the most generalized characteristic load–displacement equation of the RWMF, where K _{ r } ^{*} represents the global stiffness of the system at any loading stage, r. K _{ r } ^{*} contains a continuum of ten distinct and several intermediate levels of response: r = 0 (at rest), r = E (elastic MF, before first yield), r = Y (MF at first yield), r = C (MF at incipient collapse), r = B _{ E } (at BRB elastic level), r = B _{ Y } (BRB at first yield), r = L _{ E } (at LB elastic level), r = L _{ Y } (LB at first yield), r = C _{ E } (wall cables at elastic level), r = C _{ Y } (wall cables at first yield) or r = W (at wall failure) Intermediate levels can be defined in terms of fractions of stages of r, e.g., r = 0.6 yield (60% first yield) or r = 0.3, device (30% device failure), etc.
Effects of initial imperfections and Pdelta moments
Nonlinear static analyses
Once again, if a state of damage tolerant design is specified, then the last term in Eq. (19) would have to be replaced with its elastic counterpart, \(2\sum\nolimits_{j = 1}^{n} {\sum\nolimits_{i = 0}^{m} {\bar{M}_{i,j} } }\). Equation (19) can also be used to establish the load factor needed to assure collapse prevention and selfcentering, in which case it would be appropriate to assume; \(M_{C}^{P} > 2\sum\nolimits_{i = 0}^{m} {M_{D,i}^{P} } > M_{B}^{P} > 2\sum\nolimits_{j = 1}^{n} {\sum\nolimits_{i = 1}^{m} {\bar{M}_{i,j}^{P} } }\). Equation (19) allows the following four distinct plastic failure scenarios to be envisaged.
RWMF with no supplementary devices, M_{C}^{P} = M_{D}^{P} = M_{B}^{P} = 0
Note that the RRW, being a mechanism, cannot improve the ultimate carrying capacity of a device free MF. Substitution of K _{ F } = K _{ C } = K _{ D } = K _{ B } = 0, into Eq. (17) will lead to the corresponding drift ratio described by Eq. (10) above.
RWMF with no wallmounted supplementary devices, M _{ D } ^{ P } = M _{ B } ^{ P } = 0 and M _{ C } ^{ P } ≠ 0.
The corresponding drift ratio can now be estimated by inserting K _{ F } = K _{ D } = K _{ B } = 0 in Eq. (17). In conclusion, a properly designed rocking wall can actually prevent plastic collapse of the entire system.
Fully supplemented RWMF with no wall tendons M _{ C } ^{ P } = 0, M _{ D } ^{ P } ≠ 0 and M _{ B } ^{ P } ≠ 0.
Following the arguments leading to Eq. (17), the drift ratio can now be computed by replacing \(K_{i} h_{i}^{2} (1 + \bar{K}_{D,i} )\) with αK _{ D,i } and inserting K _{ C } = 0 in Eq. (17). The post failure static stability of the MF suggests that a welldesigned array of LBs and BRBs can either on their own or in conjunction with a properly designed, pretensioned RRW help prevent total collapse and recenter the system after a major seismic event. Naturally, collapse prevention can be achieved if a preassigned drift ratio can be sustained while safely resisting Ω(M _{0} ^{ P } + M _{ PΔ} ^{ P } ), where Ω is the over strength factor defined by the codes.
Determination of wall strength, K_{D} = K_{B} = K_{C} = 0
With M _{0} ^{ P } known, the value of M _{ w } ^{ P } can be extracted from Eq. (23). The use of Eq. (23) is demonstrated in “Appendix 1”.
RWMF design strategies
While Eq. (28) contains a large number of solutions, two extreme but important scenarios come to mind, S _{ i } = 0_{ i } and S _{ i } = F _{ i }. The two limiting cases describe the use of RWMF combinations as either counterproductive, or highly efficient. These limiting cases are known as MFs of uniform sections or uniform shear (MFUS) and MFs of uniform response (MFUR), respectively. The attributes of both cases are briefly discussed in the next two sections.
Case 1, S_{r,i} = 0, attributes of MFUR
MFUR are highly optimized systems that can be used to assess the efficiencies of geometrically identical structures under similar loading and boundary conditions. Since ϕ _{ i } = ϕ and the frame obeys the rules of proportionality, it can no longer benefit fully from the stiffness of the wall. The obvious conclusion drawn here is that it would be counterproductive to use RRWs in conjunction with MFUR. The attributes of a typical MFUR are briefly illustrated in “Appendix 2”.
Case 2 S_{i} = F_{i}, attributes of MFUS
The free body diagrams of the subframes and the rocking wall of the subject RWMF are shown in Fig. 7c through f. The resulting frame as depicted in Fig. 7c is known as an MFUS. While the use of free standing MFUS may appear counterintuitive, their combination with properly designed RRWs can lead to the development of highly efficient RWMFs. An even more counterintuitive but highly efficient condition arises when h _{ i } = h. For equal or nearly equal story heights, K _{ u,i } = K _{ u }, i.e., I _{ i,j } = I, implying that all horizontal members can be the same. Similarly, since J _{ i,j } = J, all columns can also be the same. Example 3 (“Appendix 3”) provides a tractable comparison between the performances of an idealized MFUS and a seemingly inefficient MFUS as part of a simple RWMF without supplementary devices.
Determination of wall stiffness
The nature of the interactive forces S _{ i } and Q _{ m } suggests that the wall tends to bend as an upright simply supported beam with a rigid body tilt ϕ. Hence, the conclusion that the stiffer the wall, the better the expected performance of the RWMF. The reactive forces reach their maxima, as the wall becomes stiffer. However, if the rigidity of the wall were to be large but finite and h _{ i } = h, then the following design data in the form of maximum wall drift or end slope ψ _{max} may be found useful for preliminary estimation of wall stiffnesses under commonly occurring distributions of lateral forces, e.g.,
For practical design purposes the stiffness of the wall can be related to a fraction of the prescribed uniform drift say 5%ϕ or ψ _{max} = ɛϕ; however, convergence is rapid and results in highly workable initial values.
Collapse prevention and self centering
ASCE (2007) guidelines for the rehabilitation of existing buildings define specific performance levels for immediate occupancy, life safety and collapse prevention, where collapse prevention is defined as “the post earthquake damage state in which the building is on the verge of partial or total collapse”. The current section focuses briefly on collapse prevention employing RWMF technologies for both new aswellas existing structures. Seismic collapse is usually triggered by structural instability or the Pdelta phenomenon, preceded by the formation of partial or complete ductile failure mechanisms. Plastic failure modes such as those shown in Fig. 6d and f undergo large lateral displacements that in turn lead to catastrophic collapse. While gravity forces, as active components of the Pdelta effect, are constant quantities, lateral displacements can be controlled even reversed by means of RWMF capabilities suggested by Eqs. (17) or (18), provided that residual effects are small (MacRae and Kawashima 1997), the wall remains elastic and suppresses soft story failure.
The preventive mechanism
Subscripts C and T refer to concrete and tendon, respectively. Here, the wall height, H, and the cable length, L _{ T, } are not necessarily the same. Theoretically speaking, T _{ W } should be sufficiently large to recenter the structure; otherwise, residual deformations under seismic loading can significantly affect the recentering capacity of the system.
Conclusions
A relatively new seismic structural system that combines BRBs, LBs and RRWs with GBRMFs has been introduced. The seismic behavior of the proposed RWMF can be characterized by the combined responses of the ductile MF and the supplementary devices. In addition to BRBs, both vertical as well as horizontal gap opening devices have been provided to ensure collapse prevention and active recentering. PT provides restoring forces at the ends of the LBs and the RRW that tend to prevent catastrophic collapse and force the frame and the wall to return to their preearthquake positions. The proposed mathematical model lends itself well to SDOF treatment. Several theoretically exact formulae for the preliminary design of regular RWMFs have been presented. The proposed concepts lead to minimum weight solutions. A new gap opening LB that does not induce unwanted moments in the columns and causes no damage to the diaphragms has also been introduced. It has been shown that the magnitude of the LB gap opening is a function of the link beam offsets from the centerlines of its supporting walls or columns. In the interim two new classes of moment frames, MFUR and MFUS have also been introduced. It has been argued that the use of RRWs in conjunction with MFUR is counterproductive; in contrast the MFUS–RRW combination can lead to highly efficient earthquakeresisting buildings. While the accuracy of the proposed formulae has been verified by independent computer analysis, results may differ due to shear and axial strains, shrinkage and tendon relaxation, etc. The proposed structural scheme is still in its infancy and needs the test of time before being recognized as a viable earthquakeresisting system.
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