# Accounting for the Spatial Variability of Seismic Motion in the Pushover Analysis of Regular and Irregular RC Buildings in the New Italian Building Code

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Pushover Analysis: Brief Overview about Conventional and Non-Conventional Methods

_{b}) vs. roof displacement (δ

_{R}), monitored on a control node (CN). To define the safety level of the building under investigation, the capacity curve is compared with the seismic demand according to established approaches, such as the N2 method [8] as applied in [9], without forgetting the absolute validity of the capacity spectrum method, as implemented in [10] and in [11] for irregular buildings.

_{n}, Equation (4):

## 3. New Provisions about Conventional Pushover Analyses in the Italian Building Code

- It is necessary to consider alternative control nodes, such as the corner points of the last storey, especially when the structural system presents a significant coupling among translations and rotations;
- It is possible to use a load profile proportional to the distribution of the storey shears as computed by an elastic response spectrum analysis (RSA). This load pattern can be used in any case; it can also be used if the participating mass M[%] in the direction considered is lower than 75%. Furthermore, the use of this profile is mandatory if the fundamental period of the structure is greater than 1.3 times the corner period (T
_{c}); - The effects of seismic motion in the 3 directions must be combined according to the same formula already present for RSA, which takes into account the 3 spatial components of an earthquake. In particular, the principal component is considered with a unitary coefficient, while the others are reduced to 30%. The analysis must be repeated by alternating the reduction coefficients among the components. Assuming that the direction X is the principal direction, the combination rule is written as follows, according to Equation (5):$${E}_{X}+0.3{E}_{Y}+0.3{E}_{Z}$$
_{X}, E_{Y}and E_{Z}(usually neglected for typical building structures) are, respectively, the responses in the two main horizontal directions and the vertical direction of the building. The safety assessment shall consider the heaviest effects obtained.

## 4. Application of Directional Pushover on 3D RC Archetypes: General Information

_{REG}) is extremely regular in both main directions (M[%] > 85%). Starting from this model, the other ones were derived by varying some of the structural elements in order to obtain a variation in the M[%] in the Y direction from 75% to 45%, with a progressive step of 5% (the models have been labeled as B

_{75}, B

_{70}, B

_{65}, B

_{60}, B

_{55}, B

_{50}, B

_{45}). More details about the numerical models are given in Section 4.1.

_{X+03Y}), as shown in Equation (6):

_{Y+03X}), Equation (7):

#### 4.1. Description of the Case Studies and Numerical Modelling

_{REG}is a RC building designed according to NTC2018 in “Low” ductility class. From a geometrical point of view, it has a regular shape in-plan, with dimensions 9.00 m in the X direction and 10.00 m in the Y direction, 2 equal bays in both main directions and 2 floors with an inter-storey height of 3.00 m, as shown in Figure 2. No staircases are considered, and floor slabs are disposed as a checkboard. The materials considered are a class C28/35 concrete (cylindrical compressive strength of 28 N/mm

^{2}) and a class B450C steel for reinforcements (tensile yielding stress of 450 N/mm

^{2}). Vertical service loads are evaluated in the case of residential use. Concerning the seismic action, the response spectrum has been computed for the municipality of Bisceglie (latitude = 16.5052°, longitude = 41.243°), Puglia Region, Southern Italy, which is a low-medium seismicity site. According to a probability of exceedance of 10% in 50 years and an A soil category, for a nominal life of 50 years and a second usage class, the spectral parameters are: a

_{g}= 0.134g, F

_{0}= 2.495; and T

_{c}

^{*}= 0.377s (according to NTC18 or Eurocode 8 symbology; where g is the constant of gravity acceleration). The vertical component of the seismic action has been neglected, while a behaviour factor q of 3.9 is accounted.

_{REG}, the other archetype buildings have been generated by varying the M[%] in the Y direction. To this scope, the Y-dimension of column 6 has been progressively increased, obtaining the M[%] values described in Section 4. For all the models, the steel reinforcements of columns 6 have been designed according to the minimum prescriptions provided by NTC2018. Results are summarized in Table 1.

_{y}and θ

_{u}) were computed according to the formulation provided in the Annex of NTC2018, and the acceptance criteria were defined in terms of deformation. Confined concrete has been considered for RC sections and reduction coefficients, used for taking into account the missing efficiency of ribbed steel rebars and adequate overlap and anchor lengths, have been avoided. Limit chord rotation values have been defined in accordance with the limit-states of immediate occupancy (IO—deformation equal to θ

_{y}), life safety (LS—deformation equal to ¾ θ

_{u}) and near collapse (NC—deformation equal to θ

_{u}). As previously mentioned, shear failures have been not considered. Finally, it is worth mentioning that for columns 6, at both storeys, the values of the moments and rotations in the X direction have been kept constant in all models (although strictly speaking the constitutive laws should vary), in order to avoid modifications in the structural behaviour along the X direction. Figure 3 shows the constitutive law of column 6 for all models and both directions.

#### 4.2. Eigenvalue and Pushover Analyses

_{X}, T

_{Y}and T

_{θ}(considering the two main translational components X and Y and the rotational one, indicated with “θ”) and the corresponding participating masses, M[%]

_{X}, M[%]

_{Y}and M[%]

_{θ}, respectively, are reported in Table 2.

_{X}and M[%]

_{X}remain unchanged in all numerical models, while T

_{Y}, T

_{θ}, M[%]

_{Y}and M[%]

_{θ}progressively decrease, as shown in the table. It is worth observing that for B

_{REG}and B

_{75}the first period is the one in the Y direction (the second one is in X direction), while for the other models the reduction of T

_{Y}causes a modal shifting, and T

_{X}becomes the first mode.

_{REG}model in both directions, for all load profiles and we choose CN as the centre of masses of the last storey (Figure 4).

## 5. Discussion of Results

_{u}(according to NTC18) by the first element, indifferent as to whether the first element is a beam or column.

#### 5.1. Effects of the Control Node Selection

_{R,NORM}), for the different load profiles and models.

- For all load profiles, excluding F, the capacity curve peaks are invariant from the B
_{REG}to B_{60}models. After those models, there is an increasing divergence; - Assuming T, U, M and S load profiles in the Y direction, the peaks are always close to the reference curve, presenting the greatest difference for B
_{45}, quantifiable to about 10%; - Assuming the S
_{0.3X+Y}load profile, monitored in the Y direction, the peaks are more further from the reference curve, if compared with the effect shown for the above load profiles. In particular, for B_{45}the difference can be quantified to be about 20%; - Assuming the F load profile in the Y direction, the peaks are very far from the reference curve, achieving a difference of 50% for the B
_{55}model; - Assuming the S
_{X}_{+0.3Y}load profile, monitored in the X direction, the peaks are very close to the reference curve (and to the other curves obtained by mono-directional load profiles), presenting the greatest difference for B_{45}and quantifiable to about 5%;

_{NORM}(normalized demand/capacity ratio with respect to the value relative to the reference curve), as the load profiles and models vary. It can be observed that:

- Except for the S
_{X+0.3Y}load profile, the D/C ratios for the different models do not always follow an increasing trend and in some cases show some reversal points (e.g., the T load profile and the B_{50}model). This is a consequence of the N2 method, the results of which are an effect of the balance between the stiffness of the single degree of freedom (SDOF) bi-linear curve and the seismic demand; - For the load profiles “T”, “U”, “M”, “S” and “S
_{0.3X+Y}” in the Y direction, the D/C ratios are far from the reference curve for all models, with variable differences. For B_{45}, the differences are about 15% for control node 7 and about 20 to 25% for control node 9; - Assuming the load profile “F” in the Y direction, the differences in terms of the D/C ratio seem to have no observable systematic pattern: in some cases they are extremely conservative and in other cases they are in line with the other profiles;
- For the load profile “S
_{X+0.3Y}” monitored in the X direction, the D/C ratios are very close to the reference curve (and to the other curves obtained with mono-directional load profiles), with greatest difference for the B_{45}model, namely about 5 to 7%.

#### 5.2. Use of the Load Profile Proportional to Storey Shears

_{b}, δ

_{R}and the D/C ratio; (ii) the shapes of the load profiles are similar for all models. Figure 7 shows, for all models, the shapes of the “S” and “T” load profiles (normalized to 1) and the related pushover curves.

_{b}are slightly higher than in the case of the load profile “T”.

_{45}, after which they become larger. No difference can be observed, even here, between the load profiles “S” and “T”.

#### 5.3. Application of Bi-Directional Pushover Analysis

_{0.3X+Y}profile is compared with the “T” and “M” profiles, which are those that showed the widest return range of variation in the pushover response (see Figure 4, post-elastic branch of the pushover curves).

_{0.3X+Y}” profile for the various models.

_{0.3X−Y}” gives a slightly different value of the base shear V

_{b}, with displacement higher than that of the “T” profile. With regard to load profile “M”, several differences can be observed, in particular an increasing difference in terms of V

_{b}.

_{0.3X+Y}” profile are always between those obtained by “T” (higher values) and “M” (lower values). If a conservative approach is taken, “T” would seem to be the most appropriate profile. The results for the “S

_{0.3X+Y}” profile could be more accurate, but this should be further investigated and validated by non-linear dynamic analyses and by extending the building sample.

_{X}–M

_{Y}interaction domains are reported, respectively, in Figure 10, Figure 11, Figure 12 and Figure 13.

_{X+0.3Y}” and “S

_{0.3X+Y}” load profiles) and varying the numerical models. It is worth noting that, for column 6, both yielding (Y

_{SURF}) and ultimate (U

_{SURF}) moment surfaces are variable in the Y direction. Of course, there would also be a variation in the X direction, which has been neglected since it is not as significant as the Y variation. Figure 11 reports only the two surfaces corresponding to B

_{REG}and B

_{45}.

- With regard to column 5, which is coincident with the centre of the masses, the mono-directional pushover analyses in the Y direction always follow the same direction, whereas the bi-directional analyses reach the yield surface Y
_{SURF}with a different slope and in the post-elastic field the paths are parallel to the dominant component of the load profiles (both for the X and Y directions); - In column 6, which is the column that is progressively modified in the different models, the mono-directional pushover analyses in the Y direction follow this same direction. For the bi-directional analyses, in the elastic branch a slope variation can be observed depending on the variation of the column section. In the Y-direction, the variation is more evident than in the X-direction, with a post-elastic behaviour disturbed by numerical convergence problems;
- With regard to columns 7 and 9, which are the columns with the greater displacements, both mono- and bi-directional pushover analyses have different slopes in the elastic field, due to the increasing torsion. In general, in the post-elastic field, the path tends to re-align to the direction of the dominant component of the load profiles, but due to the presence of numerical convergence problems, this happens with no recognizable rule. Probably, in these two columns, the results obtained are actually governed by the high variability of the axial load.

## 6. Conclusions and Future Developments

- The use of different control nodes in the analyses, varying models and load profiles, has provided a large set of capacity curves. It is interesting to look at the variations obtained in view of the selection of the most conservative choice in terms of safety level, even if this is in contrast with the spirit of a non-linear analysis, which is supposed to be accurate rather than cautious. In this sense, anyway, the more convenient result is given by the control node that is the farthest from the centres of mass and stiffness;
- The use of the “S” load profile deriving from an RSA analysis has not given significant differences from a classic “T” load profile in any numerical model. This evidence cannot be generalized, but seems to suggest that possibly the “S” load profile could be very useful in the case of buildings presenting irregularities in-elevation, which deserves further investigation;
- The use of bi-directional pushover analyses has been tested in order to fulfil the new indication of the Italian law about the spatial combination of effects. It seems to provide some benefits, since it offers the possibility to investigate how structural elements behave under the spatial effects of ground motion. On the other hand, with regard to the classical pushover approach, it does not provide a conservative approach, since the response is between the extreme load profiles. This solution could be more precise, but this should be validated with more accurate spatial analyses.

## Author Contributions

## Funding

## Conflicts of Interest

## List of Symbols and Abbreviations

δ_{R} | Roof Displacement |

δ_{R,NORM} | Normalized Roof Displacement |

θ_{y} | Yielding Moment |

θ_{u} | Ultimate Moment |

λ | Amplitude Factor of Load Pattern |

Φ | Steel Rebar diameter |

$\overline{{\Phi}_{1}}$ | Eigenvector of the First Vibration Mode |

$\overline{{\Phi}_{n}}$ | Eigenvector of the n^{th} Vibration Mode in [17] |

Ψ | Time-Independent Shape Vector |

3D | Three-Dimensional |

a_{g} | Peak Ground Acceleration on type A [6] |

A | Soil Category of type A |

B_{REG} | Reference Model |

B_{75} | Model with 75% Participating Mass in Y Direction |

B_{70} | Model with 70% Participating Mass in Y Direction |

B_{65} | Model with 65% Participating Mass in Y Direction |

B_{60} | Model with 60% Participating Mass in Y Direction |

B_{55} | Model with 55% Participating Mass in Y Direction |

B_{50} | Model with 50% Participating Mass in Y Direction |

B_{45} | Model with 45% Participating Mass in Y Direction |

B450C | Steel Reinforcement Class |

BPA | Bidirectional Pushover Analysis |

C28/35 | Concrete Class |

CN | Control Node |

CQC | Complete Quadratic Combination |

D/C | Demand/Capacity Ratio |

D/C_{NORM} | Normalized Demand/Capacity Ratio |

DAP | Displacement Adaptive Pushover |

E_{X} | Response in X Direction |

E_{Y} | Response in Y Direction |

E_{Z} | Response in Z Direction |

$\overline{F}$ | Load Profile |

F | Unimodal Load Profile |

F_{0} | Maximum Amplification Factor [6] |

FAP | Force Adaptive Pushover |

GPA | Generalized Pushover Analysis |

g | Gravity Acceleration |

$\overline{I}$ | Identity Vector |

IO | Immediate Occupancy |

LS | Life Safety |

[M] | Mass Matrix |

M[%] | Participating Mass |

M[%]_{X} | Participating Mass for the Main Translational Period in X direction |

M[%]_{Y} | Participating Mass for the Main Translational Period in Y direction |

M[%]_{θ} | Participating Mass for the Main Rotational Period |

M | Mass Proportional Load Profile |

M_{X} | Moment around X Direction |

M_{Y} | Moment around Y Direction |

MDOF | Multi Degree of Freedom |

MMC | Method of Modal Combination |

MPA | Modal Pushover Analysis |

NC | Near Collapse |

NTC18 | Italian Building Code |

q | Behaviour Factor |

RC | Reinforced Concrete |

RSA | Response Spectrum Analysis |

S | Storey Shears Load Profile |

S_{X}_{+0.3Y} | Storey Shears Load Profile (applied in both Directions and scaled to 0.3 in Y Direction) |

S_{Y}_{+0.3X} | Storey Shears Load Profile (applied in both Directions and scaled to 0.3 in X Direction) |

SDOF | Multi Degree of Freedom |

SRSS | Square Root of Sum of Square |

t | Time Step |

T | Inverse Triangular Load Profile |

T_{c} | Corner Period |

T_{c}^{*} | Parameter to define the Corner Period [6] |

T_{X} | Main Translational Period in X Direction |

T_{y} | Main Translational Period in Y Direction |

T_{θ} | Main Rotational Period |

U | Uniform Load Profile |

U_{SURF} | Ultimate Surface |

$\overline{{u}_{max}}$ | Maximum Displacements Vector in [17] |

V_{b} | Base shear |

Y_{SURF} | Yielding Surface |

z_{n} | Participation Factors in [17] |

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**Figure 1.**New indications of the Italian Building Code NTC2018 about the pushover analysis: (

**a**) use of more CNs, (

**b**) use of the storey shears load profile, (

**c**) spatial combination of load profiles.

**Figure 2.**Structural configuration of the in-plan of RC archetype (span lengths in m), 3D numerical model and information on the structural elements (beam and column dimensions in cm).

**Figure 4.**Pushover curves obtained for the reference model (B

_{REG}), for all load profiles (unimodal—F; inverse triangular—T; uniform—U; proportional to masses—M; proportional to the storey shears—S) and both main directions.

**Figure 5.**Pushover curves obtained with the “M” and “U” load profiles in the Y direction, for all the models, by monitoring node 5 (continuous lines), node 7 (dashed lines) and node 9 (dash-dot lines).

**Figure 6.**Variation of the pushover curves peaks as the control node varies for all models and all load profiles (unimodal—F; inverse triangular—T; uniform—U; proportional to masses—M; proportional to the storey shears—S); D/C variation as the control node varies for all models and all load profiles.

**Figure 7.**Load profiles and pushover curves in the Y direction, using “S” and “T” load profiles, for all models.

**Figure 8.**Moment–rotation responses in the Y direction for columns 5, 6, 7 and 9, for all models and for “S” and “T” load profiles.

**Figure 9.**Pushover curves in Y direction, using “S

_{0.3X+Y}”, “M” and “T” load profiles for all the models investigated.

**Figure 10.**Interaction M

_{X}–M

_{Y}domains (yielding and ultimate) for column 5, with the paths followed by T, M, S

_{X+0.3Y}and S

_{0.3X+Y}, for all numerical models.

**Figure 11.**Interaction M

_{X}–M

_{Y}domains (yielding and ultimate) for column 6, with the paths followed by “T”, “M”, “S

_{X+0.3Y}” and “S

_{0.3X+Y}”, for the B

_{REG}and B

_{45}models.

**Figure 12.**Interaction M

_{X}–M

_{Y}domains (yielding and ultimate) for column 7, with the paths followed by “T”, “M”, “S

_{X+0.3Y}” and “S

_{0.3X+Y}”, for all numerical models.

**Figure 13.**Interaction M

_{X}–M

_{Y}domains (yielding and ultimate) for column 9, with the paths followed by “T”, “M”, “S

_{X+0.3Y}” and “S

_{0.3X+Y}”, for all numerical models.

Model | X Dimension (cm) | Y Dimension (cm) | Steel Reinforcement (X/Y) |
---|---|---|---|

B_{REG} | 30.00 | 50.00 | (4 + 4 Φ 16)/(4 + 4 Φ 16) |

B_{75} | 93.31 | (4 + 4 Φ 16)/(5 + 5 Φ 16) | |

B_{70} | 105.89 | (4 + 4 Φ 16)/(6 + 6 Φ 16) | |

B_{65} | 117.57 | (4 + 4 Φ 16)/(7 + 7 Φ 16) | |

B_{60} | 131.04 | (4 + 4 Φ 16)/(8 + 8 Φ 16) | |

B_{55} | 148.69 | (4 + 4 Φ 16)/(10 + 10 Φ 16) | |

B_{50} | 170.99 | (4 + 4 Φ 16)/(12 + 12 Φ 16) | |

B_{45} | 232.08 | (4 + 4 Φ 16)/(16 + 16 Φ 16) |

Model | T_{X} (s) | T_{Y} (s) | T_{R} (s) | M[%]_{X} | M[%]_{Y} | M[%]_{R} |
---|---|---|---|---|---|---|

B_{REG} | 0.3197 | 0.3558 | 0.2740 | 86.495 | 87.263 | 87.670 |

B_{75} | 0.3245 | 0.2507 | 75.061 | 75.861 | ||

B_{70} | 0.3194 | 0.2411 | 69.981 | 69.892 | ||

B_{65} | 0.3159 | 0.2314 | 65.032 | 64.632 | ||

B_{60} | 0.3128 | 0.2197 | 60.093 | 59.424 | ||

B_{55} | 0.3103 | 0.2045 | 55.056 | 54.212 | ||

B_{50} | 0.3077 | 0.1868 | 50.066 | 49.832 | ||

B_{45} | 0.3049 | 0.1507 | 44.986 | 44.706 |

D/C | B_{REG} | B_{75} | B_{70} | B_{65} | B_{60} | B_{55} | B_{50} | B_{45} |
---|---|---|---|---|---|---|---|---|

T_{Y} | 0.176 | 0.136 | 0.125 | 0.116 | 0.115 | 0.121 | 0.129 | 0.079 |

U_{Y} | 0.153 | 0.12 | 0.108 | 0.096 | 0.101 | 0.105 | 0.112 | 0.068 |

M_{Y} | 0.144 | 0.112 | 0.103 | 0.096 | 0.096 | 0.102 | 0.107 | 0.066 |

F_{Y} | 0.169 | 0.141 | 0.14 | 0.144 | 0.154 | 0.216 | 0.213 | 0.289 |

S_{Y} | 0.171 | 0.133 | 0.123 | 0.114 | 0.112 | 0.118 | 0.126 | 0.076 |

S_{X}_{+0.3Y} | 0.126 | 0.128 | 0.127 | 0.128 | 0.128 | 0.136 | 0.157 | 0.168 |

S_{Y}_{+0.3X} | 0.163 | 0.128 | 0.118 | 0.108 | 0.107 | 0.111 | 0.113 | 0.069 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ruggieri, S.; Uva, G.
Accounting for the Spatial Variability of Seismic Motion in the Pushover Analysis of Regular and Irregular RC Buildings in the New Italian Building Code. *Buildings* **2020**, *10*, 177.
https://doi.org/10.3390/buildings10100177

**AMA Style**

Ruggieri S, Uva G.
Accounting for the Spatial Variability of Seismic Motion in the Pushover Analysis of Regular and Irregular RC Buildings in the New Italian Building Code. *Buildings*. 2020; 10(10):177.
https://doi.org/10.3390/buildings10100177

**Chicago/Turabian Style**

Ruggieri, Sergio, and Giuseppina Uva.
2020. "Accounting for the Spatial Variability of Seismic Motion in the Pushover Analysis of Regular and Irregular RC Buildings in the New Italian Building Code" *Buildings* 10, no. 10: 177.
https://doi.org/10.3390/buildings10100177