# Displacement-Based Seismic Assessment of the Likelihood of Failure of Reinforced Concrete Wall Buildings

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Probabilistic Displacement Based Assessment

_{cap}, together with new empirical relationships between intensity and displacement (ductility) demands, to compute the median spectral acceleration capacity, S

_{a,cap}, as per Figure 1e.

_{i}, is then estimated. This can be done using the equivalent SDOF mass, m

_{e}, yield displacement, Δ

_{y}, and yield strength, V

_{y}, as per Equation (1):

_{R}, for a limit state, as per Figure 1f. Finally, the annual probability of exceeding the limit state, P

_{LS}, can be computed via Equation (2), proposed by Vamvatsikos [16].

_{a}, are exceeded at a site, assuming the second-order expression proposed by Vamvatsikos [16] and shown as Equation (3), and p is given by Equation (4).

_{D}, and capacity, β

_{c}, as per Equation (5).

_{0}, k

_{1,}and k

_{2}) could be provided as part of national seismic hazard information. Furthermore, assessment guidelines could provide information on typical values of dispersion in demand and capacity, or directly provide the value of β

_{Tot}(which will typically be between 0.4 and 0.6).

#### 1.2. Empirical Relationships between Seismic Intensity and Displacement Ductility Demand

_{a}is the spectral acceleration intensity and a and b are scaling coefficients.

_{a}, and displacement ductility demand, µ.

## 2. Assessing the Likely Failure Mechanism for a RC Wall Building

#### 2.1. Flexural Plastic Hinging or a Wall Shear-Mechanism

_{n}, is first calculated. To do this, one should use traditional RC section analysis approaches, but with expected (as opposed to characteristic) yield strength values for the reinforcing steel and concrete. Priestley et al. [11] suggest that in-lieu of more accurate data on the materials, one could take the expected strength of reinforcing steel and concrete as being 1.1 and 1.3 times the characteristic values, respectively.

_{i}is the mass of level i, h

_{i}is the height above the base hinge to level i and Δ

_{i}is the lateral displacement of level i of the wall system at the limit state of interest. The summations are done considering all levels in the building.

_{y},

_{i}, as [11]:

_{i}is the height of level i above ground, H is the total height of the wall, and ${\varphi}_{y}$ is the nominal yield curvature (${\varphi}_{y}$) of the wall base, which can be found either from moment-curvature section analyses or via Equation (11) (from [11]):

_{w}, in the case of RC walls, ε

_{y}is the yield strain of the longitudinal reinforcement and $e$ is an empirical constant that varies according to the section shape. The value for $e$ is 2.0 for rectangular sections [11] and 1.4 for U-shaped and I-shaped sections bending parallel to the web (see [29]).

_{i}, can then be found via Equation (12):

_{i}can be taken as the yield displacement profile for the wall system (via Equation (10)).

_{i,E}is the wall shear expected for level i considering the base flexural strength (i.e., the shear profile shown in Figure 2) multiplied by the plastic hinge overstrength, φ

_{o}, and ${\omega}_{v}$ is a dynamic magnification factor (to account for higher modes) given by:

#### 2.2. Flexural Plastic Hinging or Foundation Overturning?

_{OT}, associated with flexural hinging at the base of the wall can be found as:

_{b}is the peak base shear force, M

_{wall,Ov}is the overstrength flexural resistance of the base plastic hinge and h

_{f}is the foundation height (see Figure 2). Note that when computing the peak overturning demand allowance for higher mode effects on shear demands can be made as per the previous section, but this should have limited effect on the overturning demands.

#### 2.3. Influence of Uncertainties on the Expected Mechanism?

## 3. Identification of Limit State Deformation Capacity

_{cap}(also referred to as ∆

_{sys}) in Figure 1, will be a function of the deformation capacity of the structural and non-structural elements as well as the displaced shape of the structure. The displaced shape will in turn depend on the expected mechanism. The following sub-sections illustrate how the displaced shape can be computed considering different failure mechanisms.

#### 3.1. Displacement Capacity for the Case of Flexural Hinging at Wall Base

_{p,min}is the minimum allowable plastic hinge rotation at the base of the wall, considering structural and non-structural deformation limits. This can be found by taking the minimum of θ

_{p,NS}and θ

_{p,S}from below:

#### 3.2. Displacement Capacity for the Case of a Wall Shear Mechanism

_{i,ls}, can be approximated as:

_{b}is the base shear capacity associated with the shear mechanism, ${\u2206}_{y,i}$ is the displacement profile of the wall that would be expected if flexural yielding were to occur and the terms in the denominator represent the base shear force at flexural yielding amplified to account for higher mode effects and overstrength. This expression should only be applied for cases where flexural cracking of the wall is expected prior to shear failure. If flexural cracking is not expected, the displacements will be considerably lower and could be computed using elastic analysis with gross (uncracked) wall section properties.

#### 3.3. Substitute Structure Characteristics

## 4. Quantification of the Limit State Median Intensity Capacity and Annual Probability of Failure

_{ay}, and ductility factor ($\mu $) are implemented into Equation (28) to estimate the median spectral acceleration capacity expected:

_{cap}reduced such that the limit is satisfied.

## 5. Quantifying the Annual Probability of Exceeding the Assessment Limit State

_{f,ik}is the probability of ith element failure because of kth accelerograms at mth mechanism. The first term of Equation (31) illustrates the probability of system failure by virtue of activation of m failure mechanisms in general, which has been simplified into the second term as per the assumptions above.

## 6. Gauging the Accuracy of Probabilistic Displacement-Based Assessment

_{c}A

_{g}) is 2.3%, 3.5%, 5.3%, and the longitudinal reinforcement ratio is 0.85%, 1.1%, and 1.6% for the 4-, 8-, and 12-story walls, respectively. However, in order to investigate possible mixed shear-flexure mechanisms, the transversal reinforcements spacing is taken as 170 mm for the 4-story, and 200 mm for the 8- and 12-story buildings; and the transversal rebar diameter is 10 mm for all case study buildings. Furthermore, the characteristic concrete strength is 30 MPa, and the reinforcement’s effective yield and ultimate strength is taken to be 500 MPa and 650 MPa, respectively. The median seismic mass is taken as 602 t for the 4- and 8-story, and 722 t for the 12-story building at each level (noting that variations in mass are considered as part of the probabilistic seismic assessment). The median base shear at yield of the reinforced concrete walls is 3438 kN, 3664 kN, and 13,834 kN for the 4-, 8-, and 12-story buildings, respectively. These base shear resistances correspond to design strength coefficients (lateral yield strength divided by weight) that are again typical of existing buildings in New Zealand, ranging from 0.10 to 0.19.

#### 6.1. Rigorous Probabilistic Assessment of the Multi Story RC Wall Buildings

_{eff}= 0.84 m

^{4}, A = 1.8 m

^{2}, A

_{v}= 0.38 m

^{2}for the 4-story, I

_{eff}= 2.24 m

^{4}, A = 2.4 m

^{2}, A

_{v}= 0.42 m

^{2}for the 8-story, and I

_{eff}= 4.12 m

^{4}, A = 2.4 m

^{2}, A

_{v}= 0.64 m

^{2}, for the12-story walls. The wall flexural nonlinear behavior was modeled by employing Giberson beam elements with the Takeda hysteresis rule indicating alpha and beta equal to 0.5 and 0.0 [11], respectively. The foundations were assumed rigid for all case study buildings. The tangent-stiffness proportional Rayleigh damping model was used with 5% damping specified at the fundamental mode and the other one with more than 90% mass contribution. The fundamental mode was found to be 1.0 s for the 4-story building and 2.0 s for the 8- and 12-story case study buildings (adopting cracked section properties). The ground motions employed for MSA were selected to be hazard consistent by [38] for a conditioning period of 1.0 s for the 4-story case study and 2.0 s for the 8- and 12-story case studies for nine different intensity levels (stripes) at a soil-type C site in Wellington [38], following the generalized conditional intensity measure procedure detailed in [39].

_{y}, the wall effective (i.e., cracked) section second moment of inertia, I

_{eff}, the wall yield curvature, ${\varphi}_{y}$, and the ultimate curvature capacity of the walls, ${\varphi}_{ult}$, as well as the seismic mass (with dispersion of 0.1) and the modeling damping coefficient (with dispersion of 0.6). These variables were selected noting that O’Reilly et al. [40] and Gokkaya et al. [41] had noted they tended to have the most significant impact on the seismic assessment results of existing RC buildings. The sampling procedure followed by [40] was used herein. As such, the available random function in MATLAB (2017b) [42] was utilized to generate 250 samples from each structural parameter distribution, as illustrated in Figure 6. Hence, the randomly generated sample structures were modeled in Ruaumoko [37] and exposed to 40 ground motions selected at each stripe leading to 10,000 nonlinear analyses.

#### 6.2. Simplified Probabilistic Assessment of the Multi Story RC Wall Buildings

## 7. Discussion

_{a}= 0.40 g for the 8-story building and a capacity of S

_{a}= 0.45 g for the 12 story building), one would anticipate that the seismic risk for the 8-story building is higher than that of the 12-story building. However, from Table 8, we see that the 12-story building actually has a higher annual probability of exceeding the shear limit state than the 8-story building has of exceeding the 2% drift limit (the annual probability of exceeding the limit state is 0.00130 for the 8-story building and is 0.00177 for the 12-story building) and, thus, should be more of a priority for seismic retrofit. This highlights the fact that computing the seismic risk could indeed impact decision making in practice.

^{−3}, 1.50 × 10

^{−3}, respectively. However, it changes to 2.95 × 10

^{−3}, 3.35 × 10

^{−3}for curvature and shear failure, respectively. As such, the system performance failure assuming all failure mechanisms are independent and may occur simultaneously can be computed as follows.

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

_{b}) are assumed equal to (approximately) 170 and 24 mm, respectively. Hence, the curvature capacity is computed as follows.

_{0}= 8.54 × 10

^{−4}, k

_{1}= 148.95 × 10

^{−2}, and k

_{2}= 5.78 × 10

^{−2}, as illustrated in Figure A2. These parameters are implemented in Equations (2)–(5) to compute the annual probability of exceeding the limit state (as presented in Table 8) associated with the given mechanism. The required hand calculation is presented below for illustration.

**Figure A2.**Wellington 1.0 s spectral acceleration hazard curve and fit curve as per recommendation by Vamvatsikos [16].

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**Figure 1.**Overview of probabilistic displacement-based assessment approach, Adapted with permission from [12]. Copyright 2021 Journal of Earthquake Engineering.

**Figure 2.**Equivalent first mode lateral displacement, shear, and moment diagrams for a cantilever RC wall building.

**Figure 4.**Influence of second-order P-delta effect on the lateral force-displacement response of a SDOF system (Adopted with permission from [13]. Copyright 2013 IUSS press; F is the total lateral force equivalent to base shear before considering geometric nonlinearity).

**Figure 5.**Case study buildings: (

**a**) 4-story; (

**b**) 8-story; (

**c**) 12-story reinforced concrete wall buildings’ layout.

**Figure 6.**Probabilistic modeling approach (

**a**) overview of variables included in the moment-curvature capacity curve and (

**b**) sampling process for each capacity parameter.

**Figure 7.**Multi-stripe analysis results for the 4-story case study building; (

**a**) inter-story drift ratio demands obtained at each intensity level; (

**b**) probability of limit state exceedance and associated fragility curve.

**Figure 8.**Multi-stripe analysis results for the 8-story case study building; (

**a**) inter-story drift ratio demands obtained at each intensity level; (

**b**) probability of limit state exceedance and associated fragility curve.

**Figure 9.**Multi-stripe analysis results for the 12-story case study building; (

**a**) inter-story drift ratio demands obtained at each in-tensity level; (

**b**) probability of limit state exceedance and associated fragility curve.

**Table 1.**Median b values obtained from regression, as a function of period and hysteretic model, for use in Equation (7).

Period | Bilinear | Takeda | Flag (λ = 5.67) | SINA | ||||
---|---|---|---|---|---|---|---|---|

T(s) | $\widehat{\mathit{b}}$ | $\mathit{\beta}$ | $\widehat{\mathit{b}}$ | $\mathit{\beta}$ | $\widehat{\mathit{b}}$ | $\mathit{\beta}$ | $\widehat{\mathit{b}}$ | $\mathit{\beta}$ |

0. | 3.10 | 0.08 | 5.55 | 0.27 | 5.65 | 0.21 | 5.89 | 0.22 |

0.2 | 1.54 | 0.02 | 2.16 | 0.08 | 2.87 | 0.04 | 4.05 | 0.21 |

0.3 | 1.30 | 0.04 | 1.72 | 0.12 | 2.05 | 0.20 | 2.45 | 0.24 |

0.4 | 1.24 | 0.05 | 1.49 | 0.08 | 1.86 | 0.18 | 2.63 | 0.26 |

0.5 | 1.18 | 0.05 | 1.44 | 0.09 | 1.73 | 0.15 | 2.04 | 0.24 |

0.6 | 1.14 | 0.07 | 1.36 | 0.09 | 1.62 | 0.11 | 1.79 | 0.16 |

0.8 | 1.09 | 0.05 | 1.29 | 0.07 | 1.58 | 0.07 | 1.71 | 0.16 |

1.0 | 1.10 | 0.08 | 1.23 | 0.06 | 1.51 | 0.07 | 1.57 | 0.12 |

1.5 | 1.12 | 0.10 | 1.20 | 0.12 | 1.37 | 0.06 | 1.31 | 0.10 |

2.0 | 1.10 | 0.07 | 1.23 | 0.07 | 1.47 | 0.10 | 1.30 | 0.07 |

2.5 | 1.17 | 0.10 | 1.24 | 0.07 | 1.45 | 0.09 | 1.26 | 0.09 |

3.0 | 1.24 | 0.14 | 1.28 | 0.11 | 1.48 | 0.14 | 1.33 | 0.12 |

Period | Bilinear | Takeda | Flag (λ = 5.67) | SINA | ||||
---|---|---|---|---|---|---|---|---|

T(s) | b | β | b | β | b | β | b | β |

0.2 ≤ T < 0.6 | 1.28 | 0.12 | 1.57 | 0.24 | 1.86 | 0.34 | 2.50 | 0.42 |

0.6 ≤ T ≤ 3.5 | 1.12 | 0.10 | 1.26 | 0.12 | 1.49 | 0.12 | 1.38 | 0.17 |

Case Study * | M_{y} (kN·m) | β | I_{eff} (m^{4}) | β | ${\mathit{\varphi}}_{\mathit{y}}\left({\mathbf{m}}^{-1}\right)$ | β | ${\mathit{\varphi}}_{\mathit{u}\mathit{l}\mathit{t}}\left({\mathbf{m}}^{-1}\right)$ | β |
---|---|---|---|---|---|---|---|---|

4-story | 20,820 | 0.3 | 0.84 | 0.3 | 8.33 × 10^{−4} | 0.3 | 3.5 × 10^{−3} | 0.3 |

8-story | 41,677 | 0.3 | 2.24 | 0.3 | 6.25 × 10^{−5} | 0.3 | 2.62 × 10^{−3} | 0.3 |

12-story | 76,558 | 0.3 | 4.12 | 0.3 | 6.25 × 10^{−5} | 0.3 | 2.62 × 10^{−3} | 0.3 |

**Table 4.**Median intensity and dispersion for different failure limit states obtained via rigorous MSA and Mont Carlo simulations.

Limit State | 4-Story | 8-Story | 12-Story | |||
---|---|---|---|---|---|---|

${\widehat{\mathit{S}}}_{\mathit{a}}\left[\mathbf{g}\right]$ | $\mathit{\beta}$ | ${\widehat{\mathit{S}}}_{\mathit{a}}\left[\mathbf{g}\right]$ | $\mathit{\beta}$ | ${\widehat{\mathit{S}}}_{\mathit{a}}\left[\mathbf{g}\right]$ | $\mathit{\beta}$ | |

IDR = 0.01 | 0.39 | 0.50 | 0.18 | 0.40 | 0.22 | 0.60 |

IDR = 0.02 | 0.80 | 0.45 | 0.40 | 0.50 | 0.58 | 0.70 |

Shear failure | 0.52 | 0.70 | 0.28 | 0.70 | 0.45 | 0.95 |

Curvature failure | 0.49 | 0.55 | 0.33 | 0.58 | 0.50 | 0.45 |

**Table 5.**Annual probability of limit state exceedance obtained for the three case study buildings through rigorous analysis.

Mechanism | APOE ^{*} 4-Story | APOE 8-Story | APOE 12-Story |
---|---|---|---|

IDR1 = 0.01 | 38 × 10^{−4} | 35 × 10^{−4} | 31 × 10^{−4} |

IDR2 = 0.02 | 15 × 10^{−4} | 13 × 10^{−4} | 8 × 10^{−4} |

Shear | 32 × 10^{−4} | 25 × 10^{−4} | 18 × 10^{−4} |

Curvature | 31 × 10^{−4} | 18 × 10^{−4} | 9 × 10^{−4} |

**Table 6.**Median intensity associated with different failure mechanisms found through the simplified approach and multi-stripe analysis (MSA).

Limit States | 4-Story, S_{a}(T_{1}) [g] | 8-Story, S_{a}(T_{1}) [g] | 12-Story, S_{a}(T_{1}) [g] | |||
---|---|---|---|---|---|---|

Simplified | MSA | Simplified | MSA | Simplified | MSA | |

IDR = 0.01 | 0.39 | 0.39 | 0.20 | 0.18 | 0.30 | 0.22 |

IDR = 0.02 | 0.78 | 0.80 | 0.39 | 0.40 | 0.57 | 0.58 |

Shear | 0.58 | 0.52 | 0.34 | 0.28 | 0.50 | 0.45 |

Curvature | 0.43 | 0.49 | 0.34 | 0.33 | 0.46 | 0.50 |

**Table 7.**Annual probability of exceeding different limit states for the case study buildings obtained through rigorous assessment and via the simplified approach with assumed values of total dispersion.

Limit States | APOE * 4-Story | APOE 8-Story | APOE 12-Story | |||
---|---|---|---|---|---|---|

Simplified | Rigorous | Simplified | Rigorous | Simplified | Rigorous | |

β_{Assumed} | β_{Rigorous} | β_{Assumed} | β_{Rigorous} | β_{Assumed} | β_{Rigorous} | |

IDR1 | 3.94 × 10^{−3} | 3.80 × 10^{−3} | 3.23 × 10^{−3} | 3.50 × 10^{−3} | 2.00 × 10^{−3} | 3.10 × 10^{−3} |

IDR2 | 1.50 × 10^{−3} | 1.50 × 10^{−3} | 1.34 × 10^{−3} | 1.30 × 10^{−3} | 7.90 × 10^{−4} | 8.40 × 10^{−4} |

Shear | 3.13 × 10^{−3} | 3.20 × 10^{−3} | 2.74 × 10^{−3} | 2.50 × 10^{−3} | 1.29 × 10^{−3} | 1.77 × 10^{−3} |

Curvature | 3.60 × 10^{−3} | 3.10 × 10^{−3} | 1.75 × 10^{−3} | 1.80 × 10^{−3} | 1.11 × 10^{−3} | 9.00 × 10^{−4} |

**Table 8.**Annual probability of exceeding limit states for the case study buildings obtained through rigorous multi-stripe analyses and via the simplified approach using rigorous (numerically found) values of total dispersion.

Limit States | APOE * 4-Story | APOE 8-Story | APOE 12-Story | |||
---|---|---|---|---|---|---|

Simplified | Rigorous | Simplified | Rigorous | Simplified | Rigorous | |

β_{Rigorous} | β_{Rigorous} | β_{Rigorous} | β_{Rigorous} | β_{Rigorous} | β_{Rigorous} | |

IDR1 | 4.10 × 10^{−3} | 3.80 × 10^{−3} | 3.13 × 10^{−3} | 3.50 × 10^{−3} | 2.16 × 10^{−3} | 3.10 × 10^{−3} |

IDR2 | 1.50 × 10^{−3} | 1.50 × 10^{−3} | 1.39 × 10^{−3} | 1.30 × 10^{−3} | 1.02 × 10^{−3} | 8.40 × 10^{−4} |

Shear | 2.95 × 10^{−3} | 3.20 × 10^{−3} | 2.60 × 10^{−3} | 2.50 × 10^{−3} | 1.68 × 10^{−3} | 1.77 × 10^{−3} |

Curvature | 3.80 × 10^{−3} | 3.10 × 10^{−3} | 1.87 × 10^{−3} | 1.80 × 10^{−3} | 1.06 × 10^{−3} | 9.00 × 10^{−4} |

**Table 9.**Annual probability of exceedance limit state accounting for three failure mechanisms before and after the retrofitting process.

Building | LS1 | LS2 | LS3 | LS4 | Simultaneous Failures and Retrofit | |||
---|---|---|---|---|---|---|---|---|

Retrofit None | Retrofit LS[1] | Retrofit LS(1,2) | Retrofit LS(1,2,3) | |||||

4-story | 4.10 × 10^{−3} | 1.5 × 10^{−3} | 2.95 × 10^{−3} | 3.35 × 10^{−3} | 11.8 × 10^{−3} | 7.78 × 10^{−3} | 6.29 × 10^{−3} | 3.35 × 10^{−3} |

8-story | 3.13 × 10^{−3} | 1.39 × 10^{−3} | 2.60 × 10^{−3} | 1.87 × 10^{−3} | 8.97 × 10^{−3} | 5.85 × 10^{−3} | 4.47 × 10^{−3} | 1.87 × 10^{−3} |

12-story | 2.16 × 10^{−3} | 1.02 × 10^{−3} | 1.68 × 10^{−3} | 1.06 × 10^{−3} | 5.91 × 10^{−3} | 3.76 × 10^{−3} | 2.75 × 10^{−3} | 1.06 × 10^{−3} |

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**MDPI and ACS Style**

Orumiyehei, A.; Sullivan, T.J.
Displacement-Based Seismic Assessment of the Likelihood of Failure of Reinforced Concrete Wall Buildings. *Buildings* **2021**, *11*, 295.
https://doi.org/10.3390/buildings11070295

**AMA Style**

Orumiyehei A, Sullivan TJ.
Displacement-Based Seismic Assessment of the Likelihood of Failure of Reinforced Concrete Wall Buildings. *Buildings*. 2021; 11(7):295.
https://doi.org/10.3390/buildings11070295

**Chicago/Turabian Style**

Orumiyehei, Amirhossein, and Timothy J. Sullivan.
2021. "Displacement-Based Seismic Assessment of the Likelihood of Failure of Reinforced Concrete Wall Buildings" *Buildings* 11, no. 7: 295.
https://doi.org/10.3390/buildings11070295