- Open Access

# An Estimate of the Yield Displacement of Coupled Walls for Seismic Design

- Enrique Hernández-Montes
^{1}Email author and - Mark Aschheim
^{2}

**11**:188

https://doi.org/10.1007/s40069-017-0188-5

© The Author(s) 2017

**Received:**26 August 2016**Accepted:**31 January 2017**Published:**29 May 2017

## Abstract

A formula to estimate the yield displacement observed in the pushover analysis of coupled wall lateral force-resisting systems is presented. The estimate is based on the results of an analytical study of coupled walls ranging from 8 to 20 stories in height, with varied amounts of reinforcement in the reinforced concrete coupling beams and walls, subjected to first-mode pushover analysis. An example illustrates the application of these estimates to the performance-based seismic design of coupled walls.

## Keywords

- coupled walls
- shear walls
- earthquake engineering

## 1 Introduction

Because changes in lateral strength are achieved by changing the amount of material used, rather than the inherent strengths of the steel and concrete materials, the changes in lateral strength are associated with changes in lateral stiffness. The yield displacement observed in a nonlinear static (pushover) analysis is nearly invariant with changes in lateral strength. This is easily explained for individual structural elements (Priestley et al. 1995; Hernández-Montes and Aschheim 2003), and is also observed in a more generalized way for entire buildings (Paulay 2002; Tjhin et al. 2007).

Estimates of yield curvature provided in Priestley et al. (2007).

Circular concrete column | 2.25ε |

Rectangular concrete column | 2.10ε |

Rectangular concrete wall | 2.00ε |

Symmetrical steel section | 2.10ε |

Flanged concrete beam | 1.70ε |

_{CB}, quantified as:

_{CB,b}is the base moment resistance associated with a couple resulting from the shears carried by the coupling beams and M

_{OTM}is the overturning moment at the base induced by the applied lateral loads. The yield displacement of the wall is then estimated by considering curvatures over the height of a wall, recognizing that the influence of coupling beam resistance on the bending moments within the walls. The degree of coupling beam, β

_{CB}, affects the location of zero moment within the wall, and is determined from empirical studies.

## 2 Behavioral Assumptions for Coupled Walls

Coupled walls can be considered to be an extension of the strong-column weak-beam philosophy of seismic design, applied to shear walls. This philosophy seeks to ensure that primary elements critical to structural integrity maintain gravity load resistance throughout the seismic action, while yielding develops within the beams. Yielding at the base of the columns or walls is accepted as an unavoidable part of the mechanism that develops during inelastic response, although yielding of the beams is the preferred way to confer ductility to the lateral force-resisting system.

An estimate of the yield displacement observed in a first-mode nonlinear static (pushover) analysis is useful for seismic design. In such an analysis, lateral forces are applied over the height of the building in proportion to the amplitude of the first mode, ϕ_{1,i} and mass, m_{i}, at each floor (i). Recognizing that the first mode shape may vary with β_{CB}, thus affecting the lateral force distribution and moments (and curvatures) over the heights of the walls, analytical studies were conducted to calibrate a simple expression for the yield displacement of a cantilever wall.

*Ø*

_{y}, the displacement at the top of the wall is given (by integrating the curvature twice) as:

*Ø*

_{y}) may be represented parametrically as

_{y}is the yield strain of the reinforcing steel and κ is a coefficient to be deduced for coupled walls that accounts for the complicated mechanical behavior of coupled walls undergoing response in a first-mode pushover analysis. The depth of the wall, D

_{cw}, in Eq. 3 is the distance between the center of gravity of the primary longitudinal reinforcement at one boundary of the coupled wall and the extreme concrete fiber of the remote edge (i.e., the overall section height of the cross section of the entire coupled wall less the cover to the centroid of the boundary longitudinal reinforcement), as illustrated in Fig. 3.

## 3 Numerical Study of Coupled Walls

_{yk}= 500 MPa and expected yield strength of 575 MPa) and the concrete is C-30 (having characteristic strength f

_{ck}= 30 MPa and expected compressive strength of 39 MPa) for all the walls.

The design of coupling beams is normally governed by shear strength limits. Thus, the analytical study considered two design shear levels, 500 and 1000 kN, representing unit shear stresses on the gross section of 0.33 and 0.65 √f_{ck} (MPa units) and 3.95 and 7.91 \( \sqrt {{\text{f}}_{\text{c}}^{\prime} } \) (psi units), and corresponding bending moment strengths at the wall face of 250 and 500 kN m (184 and 369 k-ft), respectively, following EC-2 (2002).

_{c}, between the limits of 0.002A

_{c}and 0.04A

_{c}). An additional case was considered, in which the longitudinal reinforcement within the end zone is increased by 50%. Thus, two levels of longitudinal reinforcement at the base of the wall were considered, for the 8-, 12-, and 20-story buildings.

In the uniform coupled wall cases considered herein, where cross sectional dimensions of all components and reinforcement within the coupling beams is held uniform, the proportion of overturning moment resisted at the base of the wall system resisted by the coupling action (represented by β_{CB}) increases as the number of stories increase. As will be shown later in the paper, yielding at the base of the wall may be postponed to much later in the lateral (pushover) analysis. As the number of stories increases, the reinforcement provided at the base of the wall has less influence on the lateral strength of the coupled wall because the resistance provided by the coupled beams may be substantial.

_{ye}= 575 MPa and f

_{ce}= 39 MPa rather than at the nominal characteristic strengths. The models of each coupled wall were subjected to nonlinear static (pushover) analysis using lateral forces applied to the coupled wall in proportional to the first mode forces. The applied force to story i is F

_{i}, defined by EC-8 (§ 4.3.3.2.3) (2004).

_{b}is the shear at the base of the wall, and s

_{i}and s

_{j}are the displacements of masses m

_{i}, m

_{j}, respectively, in the fundamental mode. The pushover curve is obtained using a displacement-controlled analysis, in which the roof displacement is gradually increased with F

_{b}adjusted to provide equilibrium at the nodes. The results are typically displayed as a “capacity curve,” which plots F

_{b}on the ordinate and roof displacement on the abscissa.

It can be observed that solid lines and dashed lines get closer as the number of stories of the coupled walls increases, as shown in Figs. 6, 7, and 8. For the case of 20-story wall and V_{cb} = 500 kN, both lines coincide. This is congruent with the fact already commented that as the number of stories increases, the reinforcement provided at the base of the wall has less influence on the lateral strength of the coupled wall.

_{y}as a function of ε

_{y}H

^{2}/3D

_{cw}for the three sets of coupled walls. The empirical results are approximately linear; the slope of the curve provides an estimate of κ equal to 0.52 for the coupled walls. Therefore, an estimate of the displacement at the top of a coupled wall at yield in a first mode pushover analysis is given by:

## 4 Example of Seismic Design Based on an Estimated Yield Displacement

In this section the preliminary design of a coupled wall is developed to illustrate how the estimated yield displacement can be used in seismic design. First, the design is based on application of the equal displacement rule to an elastic design spectrum. Then, to illustrate an alternative approach, Yield Point Spectra are used.

The coupled wall is part of the perimeter of a 12-story RC frame structure (Fig. 4). The height of the first story is 4.5 m while the overlying stories are 3.4 m high, resulting in a total height of 41.9 m. The coupled wall consists of two rectangular section walls having a plan length of 4.5 m and thickness of 0.4 m. B500 steel reinforcement (f_{yk} = 500 MPa) and C-30 concrete (f_{ck} = 30 MPa) are used. The material safety factors used in the design are γ_{s} = 1.0 and γ_{c} = 1.0.

_{u,drift}) given by EC-8 (§4.4.3.2) to

The seismic action is calculated based on EC-8 (2004). The horizontal seismic action is represented by the elastic response spectrum Type 1 (Ms >5.5, EC-8 §3.2.2.2 where Ms is the surface-wave magnitude). The type of soil is B (EC-8 Table 3.1), and according EC-8 Table 3.2: *T*
_{
B
} = 0.15 s, *T*
_{
C
} = 0.5 s, *T*
_{
D
} = 2.0 s, and *S* = 1.2. *T*
_{
B
} and *T*
_{
C
} are the lower and the upper limit of the period of the constant spectral acceleration branch, respectively. *T*
_{
D
} is the value defining the beginning of the constant displacement response range of the spectrum and *S* is the soil factor.

The reference peak ground acceleration is a_{gR} = 0.3 g, where g is the acceleration of the gravity. The building is classified as importance class II, meaning γ_{I} = 1.0 [EC-8 Table 4.3 and §4.2.5(5), where γ_{I} is the importance factor]. Thus the peak ground acceleration a_{g} = γ_{I}a_{gR} = 0.3 g. Damping of 5% is considered by imposing η = 1.0.

### 4.1 Design Based on an Elastic Spectrum

_{d,displ}= S

_{d}(T/2π)

^{2}] are shown in Fig. 10.

_{1}, should be approximately 1.46 for a coupled wall building 12 stories in height (NEHRP 2009), the associated peak displacement of an “equivalent” SDOF system (NEHRP 2009) is

_{C}, we can use the spectral displacement plot of Fig. 10 to determine the period of a long-period system whose spectral displacement is 0.192 m. This period is 1.72 s; the corresponding pseudo-spectral acceleration (S

_{d}) value (Fig. 10) is 2.57 m/s

^{2}.

Considering similar triangles, the required S_{d} value for the yielding SDOF oscillator is given by (2.57 m/s^{2})(0.078/1.46/0.192) = 0.72 m/s^{2}.

The tributary mass per story is 234,000 kg; the total reactive weight is (234,000 kg)(12)(9.81 m/s^{2})(1 kN/1000 N) = 27,546 kN. The first-mode effective mass coefficient *α*
_{1} is approximately 0.79 for a coupled wall building of this height (NEHRP 2009). Therefore, the required base shear strength at yield is estimated to be 0.79(0.72 m/s^{2})(27,546 kN)/(9.81 m/s^{2}) = 1597 kN.

### 4.2 Design Using Yield Point Spectra

_{d}and abscissa yield displacement D

_{y}, determined parametrically as a function of T. The spectral design acceleration S

_{d}(T) is given by EC-8 (§3.2.2.5), while D

_{y}(T) is given by:

As before, the roof displacement at yield is estimated according to Eq. 6 as

Because we have estimated D_{y} be 0.078, the q factor for this limit is q ≈ μ = D_{u,drift}/D_{y} = 7.95.

_{1}= 1.46. Therefore, we enter the Yield Point Spectra with an estimated yield displacement of

_{d}= 0.72 m/s

^{2}. The associated period of vibration, applicable to both the SDOF system and the first mode of the MDOF system, is

The result is identical to that obtained in Sect. 4.1. However, the Yield Point Spectra format may be appreciated as more direct, and applies more generally, including portions of the spectrum where short period displacement amplification is present.

As previously calculated, the required base shear strength at yield is V_{b} = 1597 kN. The horizontal seismic forces can then be calculated according to EC-8 §4.3.3.2.3(3).

_{OTM}, at the base of the coupled wall, due to the horizontal seismic forces indicated in Table 2 is

Horizontal forces.

Story | F |
---|---|

12 | 240 |

11 | 221 |

10 | 201 |

9 | 182 |

8 | 162 |

7 | 143 |

6 | 123 |

5 | 104 |

4 | 84 |

3 | 65 |

2 | 45 |

1 | 26 |

0 | 0 |

_{CB}(Priestley et al. 2007) is chosen. β

_{CB}should be established for design between 0.25 and 0.75 (Priestley et al. 2007). In this example, we chose β

_{CB}to be equal to 0.4.

_{CB,b}is the total moment of the coupling beams at the base. Assuming that the shear carried by all coupling beams is identical (V

_{i}), with the coupling beams having dimensions L

_{w}= 4.5 m and L

_{CB}= 1.0 m, then V

_{i}is equal to 564.1 kN

_{CW,b}, is

_{CW,b}, acts with an axial force in tension of 3150 kN. The cross section of Fig. 15, designed following EC-2 prescriptions, contains 10ϕ25 bars within each boundary.

_{i}= 564.1 kN, along with a flexural moment of 282.1 kN m at the face of the wall. The resulting reinforcement is indicated in Fig. 16.

The coupled wall was modeled using SeismoStruct (2016), with concrete modeled assuming f_{ce} = 1.3·f_{ck} = 39 MPa (per Priestley et al. 2007) and steel modeled assuming f_{ye} = 1.15·f_{yk} = 575 MPa. The resulting period is 1.7 s. Because this period is slightly less than 1.72 s, we are confident the spectral displacement will be acceptable. An eigenvalue analysis of the preliminary design determined Γ_{1} = 1.47, which is very close to the value of 1.46 assumed at the start of the design process. The eigenvalue analysis determined α_{1} = 0.62, which is less than the value of 0.79 assumed when establishing the design base shear. Relative to the values assumed in preliminary design, the reduction in period will cause the spectral displacement to be slightly smaller, while the increase in Γ_{1} will cause a slight increase in the roof displacement relative to the spectral displacement, representing a minor combined effect. Design for the higher value of α_{1} confers greater lateral strength than is needed (which is also reflected in greater stiffness that results in a slightly lower period). While the ductility and interstory drift demands should be acceptable, a minor refinement using the values of Γ_{1} and α_{1} determined for the initial design can be done, if such precision is needed.

In comparison, the EC-8 estimate of period for a coupled wall of this height is given by 0.05H^{3/4} = 0.823 s. Similarly, the ASCE-7 (§12.8.2.1) estimate is 0.804 s. These two estimates of period, relied upon in conventional code-based seismic design approaches, are suggested without regard to lateral strength, stiffness, or mass, and thus are seen to be less precise than that determined based on seismic performance objectives and an estimate of the yield displacement. Because the code period estimates are less than half of the computed first mode period, peak displacements would be substantially underestimated using the code period estimates; any updating to recognize the eigenvalues would necessitate a series of iterative design refinements. The approaches herein (Sects. 4.1 and 4.2) led to an excellent preliminary design in a single step. Past behaviors of walls under earthquake motions (Wallace 2012; Kim et al. 2016) force us to consider improvements in the design.

In the case of walls with non-uniform coupling beams, walls having different geometry, or non-uniform mass distributions or story heights, the yield displacement will deviate from the estimates developed herein. However, the stability of the yield displacement will apply to these systems as well, i.e., the yield displacement observed in a first mode pushover analysis will remain approximately constant for proportional changes in strength. Thus, the estimate of yield displacement and modal parameters used in the initial design can be updated using values computed in the analysis of the first design.

## 5 Conclusions

New expressions to estimate the yield displacement of coupled wall systems in a nonlinear static (pushover) analysis are presented herein. The expressions were calibrated to uniform coupled walls having a range 8–20 stories, for wall cross-sections of 10 × 0.4 m, with coupling beams of 1 × 0.7 × 0.4 m and story heights of 3.4 m. The expressions are stated in terms of parameters that are known or may be estimated early in the design process. A design example using an “equivalent” single-degree-of freedom system in conjunction with Yield Point Spectra was provided to illustrate the application of these estimates to the design of a RC coupled wall. The design example and method more generally demonstrates that the fundamental period of vibration is a consequence of the lateral strength (and stiffness) provided to satisfy the seismic performance objectives, and is estimated with poor fidelity by current code formulae for the so-termed “approximate period,” T_{a}. The accuracy of the yield displacement estimate allowed the preliminary design to be achieved in a single step, whereas the use of conventional code estimates of fundamental period of vibration is likely to require a series of design iterations in order to obtain a preliminary design that achieves the desired seismic performance objectives.

## Declarations

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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