Introduction

The seismic demands placed upon reinforced concrete RC walls can be visualized as a set of equivalent lateral loads with not necessarily equal magnitude at every floor level. The trivial case of this is a uniform distribution. To evaluate the maximum elastic (or nominal yielding) displacement of a cantilever wall at the roof level, the so called inverted triangle distribution is typically used. This corresponds to a linear variation of the lateral load with a maximum at the roof (top) level, and equal to zero at the base *(e.g.*^{Wallace and Moehle, 1992}). Alternatively, it is often assumed that all the equivalent lateral load is concentrated at the roof level, as in direct displacement based design procedures (^{Priestley and Kowalsky, 1998}; ^{Paulay, 2002}; ^{Priestley et al., 2007}), and previous design guides (^{Park and Paulay, 1975}, ^{Paulay and Priestley, 1993}).

The load patterns mentioned in the previous paragraph are aimed at imposing a deflected shape representative of the first mode of vibration of the wall. Nevertheless, it is acknowledged that the actual distribution of equivalent lateral forces along the height of the wall varies in time, and depends on the relative predominance of the modes of vibration and their periods, not only the first one. The consequence is a possible overestimation of the maximum elastic roof displacement of cantilever walls, generically named *δ _{te},* when calculated per the aforementioned assumptions.

In the following, a simple method built upon the analogy with a cantilever beam loaded with a concentrated vertical force at a given distance from the support is presented. The deflection at the top of a cantilever wall within the elastic range, generically named *δ _{t},* is calculated as the result of the action of a lumped lateral load V, placed at a height

*h*from the critical section, such that it produces the same reactions at the base of the wall, compared to a distributed load pattern. Firstly,

*h*is calculated for the first mode of vibration, such that

*h*=

*h*

_{1}. Subsequently, the formulation incorporates a reduction of

*h*

_{1}via

*ω*a dynamic amplification factor (

_{ν},^{Paulay and Priestley, 1992};

^{Priestley et al., 2007};

^{Rutenberg, 2013}), to account for dynamic effects in the calculation of

*δ*, as initially proposed by

_{t}^{Paulay and Priestley (1992)}for estimating the shear demands placed upon RC walls during earthquakes, following capacity design principles. Finally, by imposing yielding conditions at the critical section of the wall, such that

It is shown that the proposed formulae provide a more conservative approach for estimating *δ _{te}*, compared to others proposed in the literature (

*e.g*.

^{Wallace and Moehle, 1992};

^{Priestley et al., 2007};

^{Massone et al., 2015}).

Equivalent lateral load location, dynamic effects and shear demands

In the approach introduced in this article, the lateral load pattern corresponding to the equivalent lateral seismic actions along the height of the wall, is represented by an equivalent concentrated lateral load V. Per equilibrium, this force is equal to the shear at the base of the wall, and is located at *h* = M/V, where M is the overturning moment at the base of the wall (the critical section in this case) produced by V. Figure 1 presents two scenarios for this location: (1) V = V_{1} and *h* = *h*_{1}, the lateral force and its location associated to the first mode; and (2) V = V* _{v}* and

*h*=

*h*, the lateral force and its location which account for the dynamic effects produced by the higher modes.

_{v}
Figure 1a shows the equilibrium scenario of a slender cantilever wall subjected to an inverted triangle load distribution, associated to the first mode of vibration. In this case, the position of the equivalent load V1, is *h*1 = 2/3*h _{w}*, by definition. In the limit at the onset of the nominal yielding,

*i.e.*when M

_{1}= M

*and the curvature ϕ = ϕ*

_{y}*at the base of the wall, the base shear and equivalent lateral load is well determined by (1), such that:*

_{у}The resulting *δ _{t}* associated to this scenario is named

*h*assuming EI constant (

_{w},^{Wallace and Moehle, 1992}) and is given by (2):

Figure 1b shows a different situation, where the load distribution is not fully determined, but it imposes a deflected shape representative of higher-modes of vibration upon the wall, such that the equivalent lateral force is V* _{v}*. If in this case the elastic limit at the base of the wall is also imposed, M

*= M*

_{v}*, with M*

_{y}*the bending moment at the base of the wall due to the higher-mode load pattern. As depicted in Figure 1,*

_{v}*h*is smaller than

_{v}*h*

_{1}, such that

*h*=

_{v}*h*

_{1}/

*ω*, with

_{v}*ω*> 1.0, the dynamic amplification factor.

_{ν}One consequence of the above is that the base shear V* _{v}*, depends not only on M

*and*

_{y}*h*but also on

_{w},*ω*, such that:

_{v}This is the principle behind the capacity-based procedure for shear design originally proposed by ^{Paulay and Priestley in 1992}. Nevertheless, as this method focuses on an ultimate limit state, the over-strength of the resisting moment at the base of the wall must be included, *e.g.* via *Ω*_{o} = λM* _{n}*/M

*where the probable resisting moment at the base of the wall is defined as the nominal resisting moment M*

_{y},*times a factor*

_{n}*λ*> 1.0 that accounts for hardening, and an increased yielding stress of the reinforcing steel with respect to the nominal value. In this situation, the capacity-based shear demand is given by (4):

a well-established formula since 1992 (^{Paulay and Priestley, 1992}; ^{SNZ 3101, 2006}). It is important to recall that the first-mode reference force V_{1} is equal to the base shear resulting from the application of the lateral load distribution shown in Figure 1a, or, alternatively, of the equivalent lateral loads prescribed by codes, as required by ^{SNZ 3101 (2006)}, for example. In any case, V_{1} should not be the shear obtained with a modal spectral analysis and a certain modal combination, because it already includes the effect of the higher modes, in a different way. Hence, this effect would be doubled when including the factor ω* _{v}* in the procedure.

A second implication of *h _{v}* <

*h*

_{1}, as explained later, is that

*ad hoc*intuitive lateral load distribution representative of that shown in Figure 1b (such as the inverted triangle for the first mode), which would allow obtaining

*(e.g.*in

^{Priestley et al., 2007}).

In the following, an analytical expression for calculating
*ω _{ν}* is developed.

Lateral roof displacement including dynamic effects

Consider a cantilever beam of length L and constant stiffness EI, loaded with a vertical force F located at a distance *a* from the fixed end, as shown in Figure 2. As can be determined using energy methods, described in most structural analysis textbooks, the vertical displacement Δ at the free end of the cantilever beam shown in Figure 2, considering flexural deformations only, is given by (5):

Similarly, referring to Figure 1b, the lateral displacement at the top of a slender cantilever wall in the elastic range,
* _{v}*, acting at the height

*h*is:

_{v}where, EI is the flexural stiffness of the cross-section of the wall (moment-curvature stiffness). Note that the right hand side of (5) is valid only if EI is constant along the height of the wall. In this case, as a first approximation, whose limitations are included at the end of this article, EI is taken constant along *h _{w}*, and equal to that of the critical cross-section of the wall, as explained below.

With reference to Figure 1b, by equilibrium, M* _{v}* = V

*Up to the yielding point, in the elastic range, this moment is linearly proportional to the curvature of the crossed-section at the base of the wall ϕ, such that M*

_{v}h_{v}.*= EIϕ. Equating these two expressions and rearranging, it follows that:*

_{v}Replacing (7) into (6), leads to (8):

In the elastic limit or yielding point, the curvature at the base of the wall is ϕ = ϕ* _{y}*, the nominal yielding curvature. This curvature is further defined (

^{Paulay, 2002}) as in (9) and (10):

where, M’* _{y}*, ϕ’

*and*

_{y}*β*, are the bending moment, curvature, and neutral axis depth (as a fraction of

*l*), respectively, at first yield; and

_{w}*ε*the yielding strain of the reinforcing steel.

_{γ}Combining (10) and (9), an expression for *η* takes the following form:

It has been shown (^{Priestley and Kowalsky, 1998}; ^{Paulay, 2002}; ^{Priestley, 2003}) that *η* is approximately constant, and can be taken as *η* = 2.0 for rectangular reinforced concrete walls, for example, within a plus minus 15% error (^{Priestley, 2003}). Nevertheless, it is easy to calculate this number on a case by case basis using a sectional analysis and (11).

Imposing ϕ = ϕ* _{y}* in (8), such that

*, (8) becomes (12):*

_{y}Defining α = *h*_{1}/*h _{w},* the normalized height of the equivalent first-mode lateral load pattern, and recalling that

*h*=

_{v}*h*

_{1}/

*ω*(12) can be rewritten as:

_{ν},Dividing (13) into *h _{w},* and rearranging, it becomes:

Further, defining *A _{r}* =

*h*/

_{w}*l*, the aspect ratio of the wall; and

_{w}
(15) is a novel expression that defines the maximum elastic roof displacement capacity of a slender cantilever wall, accounting for dynamic effects, in dimensionless terms. Obviously, when *ω _{v}* = 1.0, (15) reduces to the case where

*h*=

_{v}*h*

_{1}, and

To construct an expression for comparison with other nondimensionless formulas with the form of (2), (14) is firstly divided by *h _{w}*, such that:

Noting that the third factor of the right-hand side of (17) is equal to ϕ* _{у}* as shown in (10), it follows that:

Defining

The parameter γ defined in (19) serves for comparison with (2), where *γ* = 11/40, as well as with other limits proposed by other researchers, as shown later on.

Selection of ω_{ν}

The variable ω_{ν} depends on the height of the wall, or, indirectly, on the number of storeys of the building, as proposed by ^{Paulay and Priestley (1992)}. There are several other expressions for *ω _{ν}.* For a comprehensive review of the literature on this subject, the reader is referred to

^{Rutenberg (2013)}.

In the formulation of ^{Paulay and Priestley (1992)} considered herein, *ω _{ν}* varies linearly from 1.0 to 1.8 for one and six storeys, and it is limited to 1.8 for buildings of six storeys or more. Nevertheless, based on numerical work done by

^{Quintana Gallo (2008)}, it is suggested that the upper limit of

*ω*2.0 be considered. This value is used in the formulation of the simplified expressions presented at the end of this article. On the other hand, the factor

_{ν}=*ω*is explicitly included in the general formulation for

_{ν}*ad hoc*amplification factors if desired.

Example for discussion

As an example for discussion, consider the case of a rectangular cantilever wall with the following properties: *h _{w}* = 25 m, and

*l*= 5 m, such that A

_{w}*= 5. Take*

_{r}*η*= 2.0 for a rectangular wall (

^{Priestley and Kowalsky, 1998};

^{Paulay, 2002};

^{Priestley et al., 2007}), and consider a steel with

*ε*= 0.002 = 0.2%, as in the Chilean practice. Assume the action of an inverted triangle lateral load distribution (see Figure 1a), such that α = 2/3. For now, neglect the dynamic effects,

_{γ}*i.e.*take

*ω*= 1.0.

_{ν}Using the novel expression proposed in (19), *γ* = 7/27 ≈ 0.26. From the classical expression presented in (2), *γ* = 11/40 = 0.275. Hence, expression (19) very closely approximates the elastic displacement obtained with (2) (^{Wallace and Moehle 1992}). Now consider α = 1, such that the equivalent lateral load is located at the roof level. In this case, (19) gives *γ* = 1/3 ≈ 0.33, the value used in direct displacement based design (^{Priestley et al., 1998}; ^{Paulay, 2002}) to estimate the yielding roof displacement of a wall, as initially proposed by ^{Park and Paulay (1975)} for cantilever beams.

Now include the dynamic effects. Note that for an interstorey height, *h _{s} =* 3 m, typical of the New Zealand construction practice, the equivalent number of storeys of the wall of the example is

*n*= 25/3 ≈ 8.3 > 6. Note that in Chile

*h*is typically equal to 2.6 m, such that

_{s}*n*in this case, would be larger than its New Zealand counterpart. Hence, the upper limit of

*ω*applies in both cases, and is herein conservatively taken as

_{ν}*ω*= 2.0, as mentioned before. Replacing this and the other data into (19), γ= 4/27 ≈ 0.15, which is significantly smaller than the previously examined values. This approximation is also more conservative than that proposed by

_{ν}^{Massone et al. (2015)}, for example, who suggest using

*γ*= 0.22, based on the results of dynamic analyses.

Calculating *dr _{te}* for the wall of the example with (16), without consideration of dynamic effects, i.e. for the first mode only:

*h*. On the other hand, considering

_{w}*ω*= 2.0 and (15):

_{ν}*δ*

_{te}.The estimation of *δ _{te}* is important in the design and detailing of confinement boundary elements of RC walls, within a plastic-hinge model approach, currently required by the Chilean RC code provisions (DS60, 2011). The reason is that an overestimation of

*δ*leads to a smaller required plastic roof displacement

_{te}*δ*(and equivalently smaller plastic rotations at the base of the wall), for achieving the same ultimate roof lateral displacement

_{tp}*δ*. As a result, smaller horizontal extensions of the boundary confinement elements would be required. Therefore, the approach introduced in this article might serve as a more conservative, yet rational, tool for design.

_{tu}Complete and simplified proposed expressions

To be considered within the Chilean code requirements, the following expressions are suggested for computing
*ν* and *t* are dropped, such that

with

and

where, *F*_{1,k} and *h*_{1,k} are the magnitude and height of the lateral force associated to the storey *k (k* = 1 to *n*), obtained with a code-prescribed equivalent lateral force analysis, *e.g.* that required by the Chilean standard NCh433 (^{INN, 2009}).

Alternatively, assuming an inverted triangle load pattern, α = 2/3. Replacing this value in (20) yields:

In both formulations, the parameter *η* can be calculated using (11), or can be taken as:

*η* = 2.0 for rectangular and asymmetric (flanged) walls with the flange in tension,

*η* = 1.5 for asymmetric (flanged) walls with the flange in compression.

Further simplification of (23) by taking *ω _{ν}* = 2.0, it reduces to (24):

Note that (24) is appropriate for a single degree of freedom (SDOF) system, where *α* = 1/3 and *ω _{ν}* = 1.0, by definition. Replacing these values into (20) also leads to (24). Hence, for SDOF systems, (24) should be the equation to refer to.

As a rule of thumb, for rectangular walls, (24) can be additionally simplified taking *η* = 2.0, and *ε _{y}* = 0.2%, as in the Chilean practice, such that:

For walls with flanged cross-sections with the flange acting in compression, the right hand side of (25) should be multiplied by 3/4, as in that case *η* = 1.5 instead of 2.0 (^{Priestley et al., 2007}; ^{Quintana Gallo, 2008}, ^{2014}).

Finally, note that if one neglects the dynamic effects *(i.e. ω _{ν}* = 1.0), as in a pushover analysis, and uses an inverted triangle load pattern such that

*α*= 2/3, as in (20), using

*η*= 2.0 and

*ε*= 0.2% leads to:

_{y}Again, for flanged walls with the flange in compression, the right-hand side of (26) should be multiplied by 3/4. By comparison of (25) with (26), it is found that when neglecting the dynamic effects, *δ _{e}* is overestimated by approximately 70%, under all the assumptions considered in the simplified versions of (20).

Limitations of the approach and further research

The assumptions made for constructing the formulae introduced in this paper are discussed to open opportunities for its evaluation and rational criticism in the sense of ^{Popper (1963)} (see also ^{Miller (1994)} and Verdugo (1995)).

Firstly, it was assumed that the cross-section stiffness EI is constant along the height of the wall: this is not true when the wall is placed within a building, in particular, as the axial load decreases with the height, and normally so does the amount of longitudinal reinforcing steel. This results in a decreased M* _{n}* in the upper floors, and consequently a reduced flexural stiffness EI, due to both effects. The implication is that, for the same externally imposed lateral load pattern, the curvature of the wall will be larger along its height when the strength variation is included compared to when is not. However, this would traduce into greater values of

*δ*compared to those calculated with the aforementioned assumption. Hence, the approximation leads to conservative results.

_{te}Secondly, the value of *ω _{ν}* considered for developing the simplified formulas, might not be appropriate for all cases, and should be understood as a ‘current’ upper bound, which could well be increased in the future, depending on the evidence.

Lastly, any connection of the wall with the surrounding structure is neglected. Therefore, at least the coupling effects of the floor slabs and/or beams, which can be more pronounced for walls ending in the façade of a building, are neglected. This, in turn, means neglecting the variation of the axial load imposed to the wall by coupling with the rest of the structure via these members.

As future research to cover some of the aspects outlined above, and critically evaluating the proposed formulae, numerical simulations of a building with rectangular walls of different aspect ratios A* _{r}*, modelled with macro and fibre elements, are currently under preparation. Additionally, collaborative efforts with researchers working on the same topic, are expected to provide a more comprehensive evaluation of the approach, when including the results of nonlinear dynamic analyses of buildings with asymmetric (flanged) walls.

Summary

This article provides a simple formulation for calculating the maximum elastic (yielding) roof displacement of a slender cantilever RC wall, accounting for dynamic effects. This displacement is calculated as a function of the equivalent lateral force resulting from a certain lateral load distribution, and its equivalent height, measured from the critical section of the wall. The equivalent height is firstly calculated for a load distribution associated to the first mode of vibration *(e.g.* an inversed triangle), and is subsequently reduced to account for a load pattern representative of a higher-mode response. The ratio between both heights corresponds to the dynamic amplification factor *(ω _{ν}* > 1.0) used in capacity-based design for shear actions, as proposed in the past by other researchers. An expression for the maximum lateral roof elastic drift ratio of a slender cantilever wall is formulated, including dimensionless numbers only, using a well-established expression for the yielding curvature of RC members, and assuming that the nominal yielding point at the base of the wall is reached when the equivalent load acts at the reduced height. Developing a common parameter for comparison, and using a simple numerical example, it is shown that the proposed novel formula predicts smaller maximum elastic deflections compared to expressions previously presented in the literature. Finally, different versions of the proposed formulae, with various levels of simplification, are presented, aiming at its consideration for its use within the Chilean RC code, after a thorough critical evaluation with nonlinear analyses.