# Inelastic Dynamic Eccentricities in Pushover Analysis Procedure of Multi-Story RC Buildings

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Non-Linear Model

#### 2.2. Definition of the “Capable Near Collapse Principal System” and of the “Capable Near Collapse Torsional Radii” of Multi-Story Buildings

#### 2.3. Definition of Inelastic Dynamic Eccentricities for the Safe Prediction of the Ductility Demands at the Stiff and Flexible Sides (in-Plan Irregularity)

#### 2.4. Handling of Accidental Eccentricities. Definition of Inelastic Design Eccentricities

#### 2.5. Consideration of the Higher-Mode Effects

#### 2.6. Target Displacement of the Proposed Pushover Analysis at NC and Capacity Curves

#### 2.7. Evaluation of the Seismic Demand at the NC State Due to the Spatial Seismic Action

#### 2.8. Significant Damage (SD) and Damage Limitation (DL) Performance Levels: Verification Using the Proposed Procedure

- (a)
- For the verification of the building at the DL state, it is suggested to provide each structural member of the nonlinear model with the effective bending stiffness equal to $0.25{\rm E}{I}_{\mathrm{g}}\le {\rm E}{I}_{\mathrm{eff},\mathrm{DL}}=2{\rm E}{I}_{\mathrm{sec}}\le 0.5{\rm E}{I}_{\mathrm{g}}$, where ${I}_{\mathrm{g}}$ is the is the moment of inertia of the geometric section. The target displacement, at the point of application of the lateral loading on the top floor of the building, is determined from response spectrum analysis or Ν-LRHA (or even LRHA) or with the use of the informational Annex B of EN 1998-1 for the DL earthquake or from the reduced by 75% values determined from the alternative estimation of the NC target displacement in Section 2.6.
- (b)
- For the verification of the building at the SD state, it is suggested to provide each structural member of the nonlinear model with the effective bending stiffness equal to the average of the corresponding values used for the DL and NC states, i.e., the average of ${\rm E}{I}_{\mathrm{eff},\mathrm{DL}}$ and ${\rm E}{I}_{\mathrm{sec}}$. The target displacement, at the point of application of the lateral loading on the top floor of the building, is determined from Ν-LRHA or with the use of informational Annex B of EN 1998-1 for the SD earthquake or from the reduced by 30% values determined from the alternative estimation of the NC target displacement in Section 2.6.

## 3. Numerical Example of a Six-Story Building

#### 3.1. Building Description

^{2}, respectively. The Center of Mass (CM) of each floor coincides with its geometrical center (CG) and all CM lie on the same vertical axis. The concrete grade is C30/37 of mean compressive strength equal to 38 MPa while the reinforcement steel grade is B500c of mean strength 550 MPa. All columns have a square section of dimension 0.55 m from the base of the building to the third floor and 0.5 m from the third floor to the top. The beams are considered having T section with flange 1.5/0.17 m and web 0.3/0.6 m for stories 1 to 4 and 0.3/0.55 m for stories 5–6. The beams have a similar section in all frames except in one of the inclined sides which has T beams with a wider web of dimension 0.40 m. The two walls are of orthogonal shape of dimensions 1.2/0.3 and 1.5/0.3 m with the smaller one (along the x-axis) having a boundary barbell 0.45/0.45 m. The total height of the building is 21.5 m, the height of the first floor is 4 m, while the height of the other five stories is 3.5 m.

#### 3.2. Design of the Six-Story Building

_{1}= 1), with effective peak ground acceleration α

_{g}= 0.16 g, soil category D and total behavior factor q = 4. The linear model of the building was analyzed by performing modal response spectrum analysis. In the design process, all the structural elements of the linear model of the building have been provided with their effective flexural and shear stiffness that is equal to one-half of their respective uncracked (geometric) stiffness. The building is classified into the structural type of dual buildings, equivalent to wall buildings, along both the x, y-axes, according to EN1998-1. Additionally, the building is characterized as torsionally non-sensitive since both the torsional radii ratios, ${r}_{\mathrm{I}\mathrm{I},\mathrm{des}}/{r}_{\mathrm{m}}=1.03$ and ${r}_{\mathrm{I},\mathrm{des}}/{r}_{\mathrm{m}}=1.10$, are greater than 1. The translational uncoupled periods of the building are 1.06 sec along the II

_{des}axis and 0.99 sec along the I

_{des}axis. The horizontal ideal principal axes I

_{des}and II

_{des}of the building are rotated by −17.5° relative to the x, y-axes and the double static eccentricity ${e}_{\mathrm{I};\mathrm{IIdes}}$ (distance between CR

_{des}and CM in the floor-diaphragm closest to the 0.8H

_{tot}level from the base) is about equal to $0.10{L}_{\mathrm{I};\mathrm{IIdes}}$ along both the horizontal ideal principal axes. The designed building has appropriate longitudinal/confinement steel reinforcement details that provide an overall high ductile behavior, spreading the ductility demands to the end-sections of all beams and to the base end-sections of all columns/walls (beam-sway mechanism). The numbering of the structural members in the mathematical model of the building, the section properties and the reinforcement details are presented in Figure A1 and in Table A1, Table A2 and Table A3 of Appendix A.

#### 3.3. Non-Linear Model

_{3}and P-M

_{2}-M

_{3}hinges are inserted at the end-sections of beams and columns/walls, respectively, with constitutive laws according to Mander et al. [40] for the unconfined/confined concrete and according to Park [41] for the steel reinforcement. The plastic hinge length ${L}_{\mathrm{pl}}$ (Equation (4)) divided by the ${\gamma}_{\mathrm{el}}$ factor is provided to the analysis software to convert the M-φ curves to Μ-θ ones. The “Capable Near Collapse Principal System, $II{I}_{\mathrm{sec}}\left({\mathrm{CR}}_{\mathrm{sec}}\right),{I}_{\mathrm{sec}},I{I}_{\mathrm{sec}}$” of the six-story RC building is determined according to Section 2.2 at the floor-diaphragm closest to the level $0.8{H}_{\mathrm{tot}}=17.20\mathrm{m}$ from the building base, i.e., at the fifth floor with height equal to 18 m measured from the building base. The in-plan position of the “inelastic” center of stiffness ${\mathrm{CR}}_{\mathrm{sec}}$ and the orientation of the horizontal ideal “inelastic” principal axes ${I}_{\mathrm{sec}},I{I}_{\mathrm{sec}}$ of the building are shown in Figure 6. The latter are turned relative to x, y axes by 23.82° clockwise. The inelastic static eccentricities are equal to ${e}_{\mathrm{R},\mathrm{Isec}}=1.974\mathrm{m}$ and ${e}_{\mathrm{R},\mathrm{IIsec}}=$ 1.524 m and their normalized values are equal to ${e}_{\mathrm{R},\mathrm{Isec}}/{L}_{\mathrm{Isec}}=0.12$ and ${e}_{\mathrm{R},\mathrm{IIsec}}/{L}_{\mathrm{IIsec}}=0.11$, where ${L}_{\mathrm{Isec}}$ and ${L}_{\mathrm{IIsec}}$ are the maximum plan dimensions along the axes ${I}_{\mathrm{sec}}$ and $I{I}_{\mathrm{sec}}$, respectively. The (mean) normalized “inelastic” torsional radii are equal to r

_{I,sec}/r

_{m}= 1.104 and r

_{I,sec}/r

_{m}= 1.033, where r

_{m}= 6.928 m is the radius of gyration of the floor-mass at the fifth floor.

^{2}, which is by 67% higher from the nominal value (12,000 tn‧m

^{2}). Therefore, this case refers to a more torsionally sensitive building where the radius of gyration of the floor-mass becomes equal to ${r}_{\mathrm{m}}=8.944\mathrm{m}$ and the smaller of the two torsional radii ratios, ${r}_{\mathrm{I},\mathrm{sec}}/{r}_{\mathrm{m}}=0.855$ and ${r}_{\mathrm{I},\mathrm{sec}}/{r}_{\mathrm{m}}=0.80,$ is well below the limit value 1.10 of Equation (9). The first three uncoupled modes of the “case 2” nonlinear model have periods equal to 2.19, 1.87, and 1.75 s, where the first one is torsional around z-axis while the second and third ones are translational along the $I{I}_{\mathrm{sec}}$ and ${I}_{\mathrm{sec}}$ axis, respectively. Accidental eccentricity will not be considered in this numerical example.

#### 3.4. Calculation of Inelastic Dynamic Eccentricities

#### 3.5. Application of the Floor Lateral Forces in Plan and in Elevation

#### 3.6. Target Displacement of the Eight Separate Pushover Analysis (Per Pattern) by N-LRHA

#### 3.7. Verification Procedure

_{sec}(CR

_{sec}), I

_{sec}, II

_{sec}” of the multi-story RC building. The appropriate floor enforced displacements are given in this procedure by graphs or tables following an extended parametric analysis of asymmetric ductile multi-story RC buildings, which is mentioned in Section 2.3, Section 2.6 and Section 2.8. The translational components of the enforced displacements are determined by proposed values of the floor angular deformations at the in-plan position of the vertical ideal principal axis $II{I}_{\mathrm{sec}}$. These values are adjusted better to the examined building by performing two sets of temporary pushover analyses. In these analyses, the floor lateral static forces act at the in-plan position of the vertical $II{I}_{\mathrm{sec}}$ axis and along the horizontal ${I}_{\mathrm{sec}}$ and $I{I}_{\mathrm{sec}}$ axes following the two patterns of Figure 5 and with target displacement given by Table 1. The sixteen (16) simultaneous combinations of the floor enforced displacements are given by tables: 8 combinations ${\psi}_{\mathrm{Isec},\mathrm{i}}\pm 0.3{\psi}_{\mathrm{I}\mathrm{Isec},\mathrm{i}}\pm {\psi}_{\mathrm{R},\mathrm{I}\mathrm{I}\mathrm{I}\mathrm{sec},\mathrm{i}}$ and 8 combinations $0.3{\psi}_{\mathrm{Isec},\mathrm{i}}\pm {\psi}_{\mathrm{I}\mathrm{Isec},\mathrm{i}}\pm {\psi}_{\mathrm{R},\mathrm{I}\mathrm{I}\mathrm{I}\mathrm{sec},\mathrm{i}}$, where ${\psi}_{\mathrm{Isec},\mathrm{i}}$ and ${\psi}_{\mathrm{I}\mathrm{Isec},\mathrm{i}}$ are the translational components and ${\psi}_{\mathrm{R},\mathrm{I}\mathrm{I}\mathrm{I}\mathrm{sec},\mathrm{i}}$ is the rotational one inside each i-floor. The first 8 combinations maximize the displacements along the ${I}_{\mathrm{sec}}$ axis, while the second 8 combinations maximize the displacements along the $I{I}_{\mathrm{sec}}$ axis.

#### 3.8. Analysis Results

## 4. Conclusions

- (a)
- The N2 (EN 1998-1) pushover procedure seriously underestimates the floor displacements and the floor angular deformations at the stiff sides. This was recorded for both the examined cases of torsional sensitivity, where the first one refers to a torsionally non-sensitive building according to EN 1998-1. Additionally, it underestimates the seismic demand at the higher floors throughout the building. Therefore, inelastic dynamic eccentricities and appropriate loading patterns must be used in the framework of pushover analysis.
- (b)
- The extended N2 pushover procedure corrects the unsafe estimates of the N2 procedure and provides in general conservative estimates of the floor angular deformations throughout the building.
- (c)
- The “corrective eccentricities” pushover procedure, with the use of the two modal patterns of the floor lateral static forces proposed herein, also corrects the unsafe results of the N2 procedure at the higher floors but it still provides in general unsafe estimates of the floor angular deformations at the stiff sides of the building due to the small value of the corrective eccentricity.
- (d)
- (e)
- The “inelastic dynamic eccentricities” pushover procedure on multi-story RC buildings provides in general safe results for the floor displacements and the floor angular deformations at the stiff and flexible sides as well as for those at the higher floors, at the NC state. Wherever unconservative values are shown, they are just below the seismic demand. Additionally, the conservatism of the proposed procedure is not higher than in other examined pushover procedures. The maximum story shears are a little underestimated by the proposed pushover procedure. The plastic chord rotations of the end-sections of beams and columns/walls are in general predicted with safety. Τhe developed plastic mechanism of the building at the NC state is also conservatively assessed by the proposed procedure. Additionally, the proposed procedure provides conservative estimates of the seismic demand for the verification at the SD state, close enough to the predictions of Ν-LRHA.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Structural Element | Story | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

All Columns | 55/55 | 50/50 | ||||

Wall W1 | 120/30/45/45 | |||||

Wall W2 | 30/150 | |||||

Beams 1–6 and 9–12 | Τ 30/60/150/17 | Τ 30/55/150/17 | ||||

Beams 7–8 | Τ 40/60/160/17 | Τ 40/55/160/17 |

**Table A2.**Longitudinal and transverse confinement reinforcement (hoops) details of columns/walls of the six-story building.

Columns, Walls | Story | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

C 1 | 16Ø20 hoops, 5 ties Ø8/84 | 12Ø20 hoops, 4 ties Ø10/80 | 12Ø20 hoops, 4 ties Ø8/84 | 4Ø20+8Ø14 hoops, 4 ties Ø8/84 | ||

C 5 | 16Ø20 hoops, 5 ties Ø8/84 | 12Ø20 hoops, 4 ties Ø10/80 | ||||

C 2–4, 6–9 | 12Ø20 hoops, 4 ties Ø10/80 | |||||

W 1 | 10Ø20 + 12Ø20(Column) + 6Ø10 hoops, 4 ties Ø10/80 | (2Ø20 + 8Ø14) + (4Ø20+8Ø14) (Column) + 6Ø10 hoops, 4 ties Ø8/100 + 4Ø8/84 | ||||

W 2 | 2 × (10Ø 20) + 12Ø10 hoops, 4 ties Ø10/80 | 2 × (2Ø20 + 8Ø16) + 12Ø10 hoops, 4 ties Ø8/100 |

**Table A3.**Longitudinal reinforcement details of beams of the six-story building. All beams have steel bars of 16mm diameter, and the number of bars is reported at the upper and lower fibers of the end-sections, at the start (s) and at the end (e) of beams (e.g., s 8-7 means 8Ø16 at the upper fiber and 7Ø16 at the lower fiber of the start section of the beam).

Beam | Story | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |||||||

s | e | s | e | s | e | s | e | s | e | s | e | |

Number of Longitudinal Steel Bars D = 16 mm in the Upper and Lower Fibers at the Start and End Sections of Beams | ||||||||||||

Β 1 | 8-7 | 9-7 | 8-7 | 9-7 | 7-6 | 8-6 | 6-5 | 7-5 | 5-4 | 5-4 | 4-4 | 4-4 |

2 | 9-7 | 10-8 | 9-7 | 10-8 | 8-6 | 9-7 | 7-5 | 8-6 | 5-4 | 6-5 | 4-4 | 4-4 |

3 | 10-8 | 9-7 | 10-8 | 9-7 | 9-7 | 8-6 | 8-6 | 6-5 | 6-5 | 5-4 | 4-4 | 4-4 |

4 | 9-7 | 9-8 | 9-7 | 9-8 | 8-6 | 7-6 | 6-5 | 6-5 | 5-4 | 5-4 | 4-4 | 4-4 |

5 | 8-6 | 8-6 | 8-6 | 7-5 | 7-5 | 6-5 | 6-4 | 5-4 | 5-4 | 4-4 | 4-4 | 4-4 |

6 | 8-6 | 8-6 | 7-5 | 8-6 | 6-5 | 7-5 | 5-4 | 6-4 | 4-4 | 5-4 | 4-4 | 4-4 |

7 | 11-9 | 10-7 | 11-8 | 10-7 | 10-7 | 9-7 | 8-6 | 7-5 | 6-4 | 6-4 | 5-5 | 5-5 |

8 | 10-7 | 11-9 | 10-7 | 11-8 | 9-7 | 10-7 | 7-5 | 8-6 | 6-4 | 6-4 | 5-5 | 5-5 |

9 | 5-5 | 7-5 | 5-5 | 7-5 | 5-5 | 6-5 | 4-4 | 5-4 | 4-4 | 4-4 | 4-4 | 4-4 |

10 | 7-5 | 7-6 | 7-5 | 7-7 | 6-5 | 6-5 | 5-4 | 5-4 | 4-4 | 4-4 | 4-4 | 4-4 |

11 | 8-6 | 9-7 | 8-6 | 9-7 | 7-6 | 8-6 | 6-5 | 7-5 | 4-4 | 4-4 | 4-4 | 4-4 |

12 | 9-7 | 8-6 | 9-7 | 8-6 | 8-6 | 7-6 | 7-5 | 6-5 | 4-4 | 4-4 | 4-4 | 4-4 |

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**Figure 1.**Three elastic analyses (

**a**–

**c**) for the determination of (

**d**) “Capable Near Collapse Principal System ${\mathrm{CR}}_{\mathrm{sec}}\left(II{I}_{\mathrm{sec}}\right),{I}_{\mathrm{sec}},I{I}_{\mathrm{sec}}$” and two more elastic analyses (

**e**,

**f**) for the determination of “Capable Near Collapse torsional radii ${r}_{\mathrm{I},\mathrm{sec}},{r}_{\mathrm{II},\mathrm{sec}}$ ”, in the non-linear model ($E{I}_{\mathrm{sec}})$ of the multi-story RC building.

**Figure 2.**Application of the lateral static loads on each i-floor at two different positions determined by the inelastic dynamic eccentricities ${e}_{\mathrm{flex}}$ and ${e}_{\mathrm{stiff}}$ along each horizontal principal direction. Positions 3, 1 and 4, 2 are used, respectively, to safely estimate the ductility demands at the flexible and stiff sides of the multi-story building along the ${I}_{\mathrm{sec}}$ and $I{I}_{\mathrm{sec}}$ axes, when the accidental eccentricity is not considered in analysis.

**Figure 3.**Normalized inelastic dynamic eccentricities, left: ${e}_{\mathrm{stif}\text{}\mathrm{I};\mathrm{IIsec}}/{r}_{\mathrm{m}}$ for the stiff side, right: ${e}_{\mathrm{flex}\text{}\mathrm{I};\mathrm{IIsec}}/{r}_{\mathrm{m}}$ for the flexible side.

**Figure 4.**Application of the lateral static loads on each i-floor at two different positions along each horizontal principal direction ${I}_{\mathrm{sec}}$ and $I{I}_{\mathrm{sec}}$ (1, 2 and 3, 4) determined by the inelastic design eccentricities ${e}_{1}$, ${e}_{2}$ and ${e}_{3}$, ${e}_{4}$, respectively, when the accidental eccentricity is also considered in analysis.

**Figure 5.**Vertical loading principal planes defined by the inelastic dynamic or design eccentricities along the ${I}_{\mathrm{sec}}$ or $I{I}_{sec}$ axes. Two patterns of floor lateral static forces are used in the proposed pushover procedure: an uncoupled translational modal one for each direction ${I}_{\mathrm{sec}}$ and $I{I}_{sec}$ and a modal one with a reduced base shear but with an additional top force.

**Figure 6.**Plan and elevation view of the six-story RC building. “Capable Near Collapse Principal System, $II{I}_{\mathrm{sec}}\left({\mathrm{CR}}_{\mathrm{sec}}\right),{I}_{\mathrm{sec}},I{I}_{\mathrm{sec}}$” defined in the nonlinear model (${\rm E}{I}_{\mathrm{sec}}$ ) of the building.

**Figure 7.**Eight (8) pushover analyses per pattern. The floor lateral static forces are applied at the positions determined by the inelastic dynamic eccentricities ${e}_{\mathrm{flex}}$ and ${e}_{\mathrm{stiff}}$.

**Figure 8.**Two different in elevation loading patterns: (

**a**) proportional to the fundamental uncoupled translational modes along ${I}_{\mathrm{sec}}$ and $I{I}_{sec}$ axis for a base shear equal to 1 kN, (

**b**) similar to the first but for 80% of the unit base shear and the remainder 20% is applied as an additional lateral force at the building top.

**Figure 9.**(

**a**) Three pairs of unit-normalized artificial accelerograms (a

_{g}∙S = 1.00∙g, t

_{d}= 25 s, strong motion duration 19 s), (

**b**) Elastic acceleration spectra of the 5 accelerograms and their mean acceleration spectrum relative to the design spectrum of EN 1998-1 (damping 0.05, a

_{g}∙S = 1∙g and soil D).

**Figure 10.**Earthquake spatial action of concurrently acting floor enforced-displacements in the framework of 16 pushover analyses: (

**a**) 8 combinations ${\psi}_{\mathrm{Isec},\mathrm{i}}\pm 0.3{\psi}_{\mathrm{I}\mathrm{Isec},\mathrm{i}}\pm {\psi}_{\mathrm{R},\mathrm{I}\mathrm{I}\mathrm{I}\mathrm{sec},\mathrm{i}}$, to maximize the displacement along ${I}_{\mathrm{sec}}$ axis, (

**b**) 8 combinations $0.3{\psi}_{\mathrm{Isec},\mathrm{i}}\pm {\psi}_{\mathrm{I}\mathrm{Isec},\mathrm{i}}\pm {\psi}_{\mathrm{R},\mathrm{I}\mathrm{I}\mathrm{I}\mathrm{sec},\mathrm{i}}$, to maximize the displacement along $I{I}_{\mathrm{sec}}$ axis.

**Figure 11.**Plan inelastic displacement profiles of case 1 building at NC: (

**a**) ${u}_{\mathrm{I}\mathrm{I}\mathrm{sec}}$ along the $I{I}_{\mathrm{sec}}$ axis, (

**b**) ${u}_{\mathrm{I}\mathrm{sec}}$ along the ${I}_{sec}$ axis. Proposed pushover procedure vs. N-LRHA.

**Figure 12.**Plan inelastic displacement profiles of case 2 building at NC: (

**a**) ${u}_{\mathrm{I}\mathrm{I}\mathrm{sec}}$ along the $I{I}_{\mathrm{sec}}$ axis, (

**b**) ${u}_{\mathrm{I}\mathrm{sec}}$ along the ${I}_{sec}$ axis. Proposed pushover procedure vs. N-LRHA.

**Figure 13.**Floor angular deformations (rad) of case 1 building at NC: (

**a**) ${\gamma}_{\mathrm{I}\mathrm{I}\mathrm{sec}}$ along the $I{I}_{\mathrm{sec}}$ axis, (

**b**)${\gamma}_{\mathrm{I}\mathrm{sec}}$ along the ${I}_{sec}$ axis. Proposed pushover procedure vs. N-LRHA.

**Figure 14.**Floor angular deformations (rad) of case 2 building at NC: (

**a**) ${\gamma}_{\mathrm{I}\mathrm{I}\mathrm{sec}}$ along the $I{I}_{\mathrm{sec}}$ axis, (

**b**)${\gamma}_{\mathrm{I}\mathrm{sec}}$ along the ${I}_{sec}$ axis. Proposed pushover procedure vs. N-LRHA.

**Figure 15.**Comparison of floor angular deformations at NC resulted from the examined pushover procedures and N-LRHA for case 1: (

**a**) ${\gamma}_{\mathrm{I}\mathrm{I}\mathrm{sec}}$ along the $I{I}_{\mathrm{sec}}$ axis, (

**b**)${\gamma}_{\mathrm{I}\mathrm{sec}}$ along the ${I}_{sec}$ axis.

**Figure 16.**Comparison of floor angular deformations at NC resulted from the examined pushover procedures and N-LRHA for case 2: (

**a**) ${\gamma}_{\mathrm{I}\mathrm{I}\mathrm{sec}}$ along the $I{I}_{\mathrm{sec}}$ axis, (

**b**)${\gamma}_{\mathrm{I}\mathrm{sec}}$ along the ${I}_{sec}$ axis.

**Figure 17.**Maximum story shears resulted from the “inelastic dynamic eccentricity” pushover procedure on the case 2 building: (

**a**) ${V}_{\mathrm{I}\mathrm{sec}}$ along the ${I}_{\mathrm{sec}}$ axis, (

**b**)${V}_{\mathrm{I}\mathrm{Isec}}$ along the $I{I}_{sec}$ axis. Comparison with the seismic demand from N-LRHA.

**Figure 18.**Mean plastic chord rotations ${\theta}_{\mathrm{pl},3}$ (rad) of the beams’ end-sections at: (

**a**) the flexible sides and (

**b**) the stiff sides of the case 2 building resulted by the “inelastic dynamic eccentricities” pushover procedure at the NC state.

**Figure 19.**Capacity curves of the case 2 building resulted from: (

**a**) N2 pushover procedure (EN 1998-1), (

**b**) “inelastic dynamic eccentricities” pushover procedure. Load numbering is shown in Figure 7.

**Figure 20.**Plan inelastic displacement profiles of case 2 building at SD: (

**a**) ${u}_{\mathrm{I}\mathrm{I}\mathrm{eff}}$ along the $I{I}_{\mathrm{eff}}$ axis, (

**b**) ${u}_{\mathrm{I}\mathrm{eff}}$ along the ${I}_{\mathrm{eff}}$ axis. Proposed pushover procedure vs. N-LRHA.

**Figure 21.**Floor angular deformations (rad) of case 2 building at SD: (

**a**) ${\gamma}_{\mathrm{I}\mathrm{I}\mathrm{eff}}$ along the $I{I}_{\mathrm{eff}}$ axis, (

**b**)${\gamma}_{\mathrm{I}\mathrm{eff}}$ along the ${I}_{\mathrm{eff}}$ axis. Proposed pushover procedure vs. N-LRHA.

**Table 1.**Seismic target angular deformation ${\gamma}_{\mathrm{t},\mathrm{top}}$ of the building at the NC state (mean values), on the top of the vertical axis $II{I}_{\mathrm{sec}}$ and along the horizontal axes ${I}_{\mathrm{sec}}$ and $I{I}_{\mathrm{sec}}$. Values for dual systems (frames and walls) or coupled (via beams) wall systems, as well as for different number of stories, can be found by linear interpolation.

Number of Stories | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Pure frame buildings without walls | 0.0300 | 0.0295 | 0.0235 | 0.0205 | 0.0195 |

Pure wall buildings without frames | 0.0280 | 0.0290 | 0.0260 | 0.0240 | 0.0230 |

**Table 2.**Mean values of the ratio ${\rm E}{I}_{\mathrm{sec}}/{\rm E}{I}_{\mathrm{g}}$ of the secant stiffness at yield to the geometric stiffness in each story, separately for the columns, walls, and beams. The local axes of structural elements are denoted by 2 and 3, where 3 is normal to their strong direction.

Mean Values of 9 Columns | Mean Values of 2 Walls | Mean Values of 12 Beams | |||
---|---|---|---|---|---|

Story | $\mathit{{\rm E}}{\mathit{I}}_{3,\mathbf{s}\mathbf{e}\mathbf{c}}/\mathit{{\rm E}}{\mathit{I}}_{3,\mathbf{g}}$ | $\mathit{{\rm E}}{\mathit{I}}_{2,\mathbf{s}\mathbf{e}\mathbf{c}}/\mathit{{\rm E}}{\mathit{I}}_{2,\mathbf{g}}$ | $\mathit{{\rm E}}{\mathit{I}}_{3,\mathbf{s}\mathbf{e}\mathbf{c}}/\mathit{{\rm E}}{\mathit{I}}_{3,\mathbf{g}}$ | $\mathit{{\rm E}}{\mathit{I}}_{2,\mathbf{s}\mathbf{e}\mathbf{c}}/\mathit{{\rm E}}{\mathit{I}}_{2,\mathbf{g}}$ | $\mathit{{\rm E}}{\mathit{I}}_{3,\mathbf{sec}}/\mathit{{\rm E}}{\mathit{I}}_{3,\mathbf{g}}$ |

1 | 0.15 | 0.17 | 0.31 | 0.27 | 0.128 |

2 | 0.13 | 0.15 | 0.29 | 0.25 | 0.129 |

3 | 0.14 | 0.14 | 0.20 | 0.20 | 0.117 |

4 | 0.15 | 0.15 | 0.16 | 0.19 | 0.104 |

5 | 0.13 | 0.13 | 0.12 | 0.17 | 0.088 |

6 | 0.10 | 0.10 | 0.08 | 0.14 | 0.082 |

Target Displacement | By Table 1 | Ν-LRHA (0.39 g) | Inf. Annex Β ΕΝ 1998-1 | |||||
---|---|---|---|---|---|---|---|---|

(m) | $\mathit{I}\mathit{I}{\mathit{I}}_{\mathit{s}\mathit{e}\mathit{c}}\text{}\mathbf{Axis}\text{}\mathbf{Top}$ | Case 1 | Case 2 | Case 1 | Case 2 | $\mathit{I}\mathit{I}{\mathit{I}}_{\mathit{s}\mathit{e}\mathit{c}}\text{}\mathbf{Axis}\text{}\mathbf{Top}$ | ||

$\mathit{I}\mathit{I}{\mathit{I}}_{\mathit{s}\mathit{e}\mathit{c}}\text{}\mathbf{Axis}\text{}\mathbf{Top}$ | Point 3/1 | Point 4/2 | Point 3/1 | Point 4/2 | ||||

${u}_{\mathrm{t}.\mathrm{Isec}}$ | 0.52 | 0.53 | 0.54 | 0.54 | 0.53 | 0.54 | 0.57 | 0.462 |

${u}_{\mathrm{t},\mathrm{IIsec}}$ | 0.53 | 0.55 | 0.56 | 0.53 | 0.55 | 0.55 | 0.50 |

**Table 4.**Error (%) on the seismic floor angular deformations ${\gamma}_{\mathrm{I}\mathrm{I}\mathrm{sec}}$ at the stiff and flexible sides along the $I{I}_{\mathrm{sec}}$ axis resulted from the examined pushover procedures for case 1 building.

Pushover Procedures | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Inel. Dynamic Ecc. | Enforced Displ. | Corrective Ecc. | EXT N2 | N2 | ||||||

Story | Flexible | Stiff | Flexible | Stiff | Flexible | Stiff | Flexible | Stiff | Flexible | Stiff |

1 | 8 | 27 | 2 | 54 | −3 | −16 | 5 | 8 | 1 | −28 |

2 | 19 | 39 | −8 | 28 | 15 | −1 | 22 | 26 | 19 | −16 |

3 | 15 | 50 | −7 | 35 | 12 | 10 | 18 | 37 | 15 | −7 |

4 | 4 | 40 | −14 | 27 | −3 | 4 | −5 | 22 | −7 | −17 |

5 | −6 | 38 | −10 | 28 | −11 | 1 | −4 | 26 | −28 | −34 |

6 | −13 | 46 | −3 | 40 | −17 | 2 | 9 | 50 | −43 | −45 |

**Table 5.**Error (%) on the seismic floor angular deformations ${\gamma}_{\mathrm{I}\mathrm{sec}}$ at the stiff and flexible sides along the ${I}_{\mathrm{sec}}$ axis resulted from the examined pushover procedures for case 1 building.

Pushover Procedures | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Inel. Dynamic Ecc. | Enforced Displ. | Corrective Ecc. | EXT N2 | N2 | ||||||

Story | Flexible | Stiff | Flexible | Stiff | Flexible | Stiff | Flexible | Stiff | Flexible | Stiff |

1 | 28 | 15 | 39 | 31 | 25 | −31 | 28 | −2 | 28 | −32 |

2 | 25 | 20 | 1 | 14 | 22 | −24 | 25 | 5 | 24 | −28 |

3 | 13 | 24 | −6 | 17 | 13 | −24 | 16 | 4 | 15 | −29 |

4 | 10 | 19 | −6 | 13 | 8 | −30 | 6 | −10 | 5 | −38 |

5 | 8 | 15 | −5 | 9 | 6 | −35 | 5 | −12 | −15 | −50 |

6 | −15 | 39 | −6 | 26 | −2 | −32 | 7 | 9 | −37 | −56 |

**Table 6.**Error (%) on the seismic floor angular deformations ${\gamma}_{\mathrm{I}\mathrm{I}\mathrm{sec}}$ at the stiff and flexible sides along the $I{I}_{\mathrm{sec}}$ axis resulted from the examined pushover procedures for case 2 building.

Pushover Procedures | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Inel. Dynamic Ecc. | Enforced Displ. | Corrective Ecc. | EXT N2 | N2 | ||||||

Story | Flexible | Stiff | Flexible | Stiff | Flexible | Stiff | Flexible | Stiff | Flexible | Stiff |

1 | 11 | 23 | 19 | 34 | 10 | −14 | 12 | 20 | 12 | −28 |

2 | 34 | 29 | 9 | 9 | 33 | −4 | 35 | 32 | 35 | −21 |

3 | 20 | 27 | 3 | 6 | 22 | −4 | 23 | 30 | 23 | −22 |

4 | 3 | 16 | −8 | −2 | 2 | −11 | −3 | 14 | −3 | −31 |

5 | −5 | 22 | −3 | 5 | −6 | −8 | −2 | 22 | −23 | −43 |

6 | 0 | 30 | 16 | 14 | −2 | −7 | 22 | 38 | −32 | −53 |

**Table 7.**Error (%) on the seismic floor angular deformations ${\gamma}_{\mathrm{I}\mathrm{sec}}$ at the stiff and flexible sides along the ${I}_{\mathrm{sec}}$ axis resulted from the examined pushover procedures for case 2 building.

Pushover Procedures | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Inel. Dynamic Ecc. | Enforced Displ. | Corrective Ecc. | EXT N2 | N2 | ||||||

Story | Flexible | Stiff | Flexible | Stiff | Flexible | Stiff | Flexible | Stiff | Flexible | Stiff |

1 | 34 | 12 | 37 | 21 | 37 | −32 | 39 | 14 | 39 | −35 |

2 | 32 | 14 | 2 | 2 | 34 | −26 | 35 | 20 | 35 | −32 |

3 | 17 | 20 | −5 | 7 | 24 | −24 | 26 | 21 | 26 | −31 |

4 | 9 | 20 | −12 | 8 | 11 | −27 | 7 | 11 | 7 | −37 |

5 | 9 | 19 | −9 | 5 | 11 | −31 | 9 | 10 | −12 | −49 |

6 | 11 | 28 | −4 | 23 | 13 | −25 | 20 | 36 | −30 | −55 |

**Table 8.**Error (%) on the mean seismic plastic chord rotations of the beams’ end-sections at the flexible and stiff sides of the case 2 building resulted by the “inelastic dynamic eccentricities” pushover procedure at the NC state.

Story | Flexible Sides (%) | Stiff Sides (%) |
---|---|---|

1 | 22 | 33 |

2 | 26 | 34 |

3 | 3 | 20 |

4 | −6 | 6 |

5 | −2 | 1 |

6 | 29 | 4 |

**Table 9.**Error (%) on the mean seismic plastic chord rotations ${\theta}_{\mathrm{pl},3}$ and ${\theta}_{\mathrm{pl},2}$ of the base sections of columns and walls of the case 2 building resulted by the “inelastic dynamic eccentricities” pushover procedure at the NC state.

Base Sections | ${\mathit{\theta}}_{\mathbf{p}\mathbf{l},2}\text{}(\%)$ | ${\mathit{\theta}}_{\mathbf{p}\mathbf{l},3}\text{}(\%)$ |
---|---|---|

Columns | 23 | −7 |

Walls | −15 | 22 |

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## Share and Cite

**MDPI and ACS Style**

Bakalis, A.; Makarios, T.; Athanatopoulou, A.
Inelastic Dynamic Eccentricities in Pushover Analysis Procedure of Multi-Story RC Buildings. *Buildings* **2021**, *11*, 195.
https://doi.org/10.3390/buildings11050195

**AMA Style**

Bakalis A, Makarios T, Athanatopoulou A.
Inelastic Dynamic Eccentricities in Pushover Analysis Procedure of Multi-Story RC Buildings. *Buildings*. 2021; 11(5):195.
https://doi.org/10.3390/buildings11050195

**Chicago/Turabian Style**

Bakalis, Athanasios, Triantafyllos Makarios, and Asimina Athanatopoulou.
2021. "Inelastic Dynamic Eccentricities in Pushover Analysis Procedure of Multi-Story RC Buildings" *Buildings* 11, no. 5: 195.
https://doi.org/10.3390/buildings11050195