The Evaluation of the Seismic Performance of Unsymmetric-Plan Tall Buildings Using Modal Spectral Time History and Multi-Mode Pushover Analysis ^†

Abstract

In recent seismic events that occurred worldwide and in Peru, it has been observed that irregular structures in plan present greater structural damage compared to regular structures. Investigations carried out after seismic events indicate that irregular plan structures collapse due to erroneous structural conception and poor seismic analysis. Likewise, the Peruvian earthquake-resistant standard does not establish a permissible limit for the degree of irregularity under analysis, instead qualitatively assessing the structural irregularity. The objective of this article was to study the effect of plan irregularities using innovative methodologies on the structural response of tall 10-story reinforced concrete buildings. In this sense, seventeen (17) structural models are proposed that reflect different irregular configurations in plan: 06 structures Type L, 05 structures Type I, 05 structures Type I, and one regular building. These buildings are numerically modeled using ETABS software V.18.0 through modal analysis, Modal Spectral and Linear Time History (MSLTH), and Multi-Mode Pushover (MPA). For the MSLTH, seven (07) pairs of representative Peruvian earthquakes were analyzed. The results of the modal analysis evaluated in the first two vibration modes demonstrated that Type L irregular structures change their behavior from translational to torsional when the structures present an irregularity greater than 57%. Type I and O structures present translational behavior. Furthermore, the results of the Modal Spectral and MSLTH analysis demonstrate that Type L structures present greater displacements and drifts in both directions. The shear force and the overturning moment for Types L, I, and O decrease as the irregularity in plan increases. Finally, the results of the MPA for irregular Type L structures demonstrated that the lateral stiffness of the structures decreases as the irregularity in plan is critical, increasing the possibility of the formation of plastic mechanisms in the structural elements.

1. Introduction

Earthquakes are one of the most unpredictable and devastating natural phenomena that affect the economy of a country and result in the loss of human life due to the collapse of civil structures [1]. Recent research confirms that irregular buildings present greater structural damage compared to regular buildings [2]. This situation reflects the lack of knowledge of some professionals about the factors that influence the seismic response of a structure [3]. To design a safe building, seismic performance must be adequately understood; however, currently, the structural behavior of irregular buildings in the event of an earthquake is still unknown [1]. The Peruvian territory, due to its geographical location at the interaction of the Nazca and South American plates, is located in one of the most active seismic zones in the world with 80% of the earthquakes that have occurred. The last major earthquake in Peru occurred on 15 August 2007 with a magnitude of 7.0 ML and 7.9 Mw [4]. The damage and losses caused by this seismic event left 596 people dead and an economic loss of a total of S/. 3977.8 million [5]. After this seismic event, it was observed that the structures collapsed and presented greater structural damage due to structural irregularity [6]. On the other hand, structural irregularities in buildings begin in the initial part of the architectural design phase [7]. Modern buildings are widely being designed as irregular structures, which entails performing detailed structural analysis to ensure correct structural performance in the face of seismic excitation [1]. Taking into account that buildings of the same structural system, region, and danger, the damage to the structural elements is not equal or homogeneous in the event of an earthquake [8]. The fundamental condition to ensure the correct seismic performance of buildings is to maintain a certain regularity of the structural system in both plan and height [9]. Seismic-resistant standards recommend designing and building structures with symmetrical geometries since this type of structure presents well-defined force transmission responses between structural elements [3]. However, the earthquake-resistant standards of South American countries such as Ecuador, Colombia, and Peru do not establish permissible limits for the degree of irregularity under analysis, so it is assessed qualitatively and not quantitatively as appropriate [10]. Therefore, the main objective of this research was to study the effect of plan irregularities on the structural response of asymmetrical buildings in plan.

2. Irregularity

Irregular structures present important discontinuities in the configuration or in their structural system resistant to lateral forces [11]. Likewise, irregular buildings lack symmetry and present discontinuity in geometry, mass, and the structural elements that resist the acting loads [12]. The main earthquake-resistant regulations in the world have classified structural irregularity mainly as irregularities in plan and height. On the other hand, the E.030 standard of Peru classifies plan irregularity as Torsional Irregularity, Incoming Corners, Diaphragm Discontinuity, and Non-Parallel Systems [13].

3. Modeling in Tall Building

In this study, seventeen (17) tall buildings with different irregularity conditions and a regular typology pattern model structure are analyzed. The details of the structural and other elements are shown in

Table 1. Design parameters.

Description	Value
Number of stories	10
Story height	3 m
Slab size	4 × 4 m
Beam	0.35 × 0.50 m2
Column	0.60 × 0.60 m2
Slab thickness	0.20 m
Compressive strength of concrete (f’c)	210 kg/cm2
Modulus of elasticity of concrete (E)	217,370 kg/cm2
Concrete poisson ratio (u)	0.15
Steel (fy)	4200 kg/cm2
Live load- slab	0.2 ton/m2
Live load- roof	0.1 ton/m2

and

Table 2. Structural models analyzed.

Structure	Model	% Irregularity
1st Structure	Base	0.00%
2nd Structure	L-1	14.29%
3rd Structure	L-2	28.57%
4th Structure	L-3	42.86%
5th Structure	L-4	57.14%
6th Structure	L-5	71.43%
7th Structure	L-6	85.71%
8th Structure	I-1	4.08%
9th Structure	I-2	12.24%
10th Structure	I-3	24.49%
11th Structure	I-4	40.82%
12th Structure	I-5	61.22%
13th Structure	O-1	2.04%
14th Structure	O-2	6.12%
15th Structure	O-3	18.37%
16th Structure	O-4	30.61%
17th Structure	O-5	51.02%

. Likewise, the structural models are represented in Figure 1 and Figure 2.

4. Method of Analysis

4.1. Modal Spectral (MS)

Spectral modal analysis is a linear dynamic method that estimates the contribution of each vibration mode to obtain the maximum response of a structure [14]. A response spectrum is defined as the maximum response of all possible oscillators of a single degree of freedom (SDOF) system, which can be described by their natural frequency, damping coefficient, and natural period [15]. For this investigation, a spectrum defined according to the following parameters based on the E.030 Standard [13] shown in

Table 3. Seismic parameters.

Parameter	Factor	Description
Z	0.45	Zone 4
U	1.00	Common use
S	1.00	Very rigid soil
R	8.00	Portico C.A.

was considered.

With the parameters shown in Table 3, the spectrum of pseudo-accelerations is obtained for a return period of 475 years with a 10% probability of being exceeded in 50 years based on the E030 Standard [13] as shown in Figure 3.

4.2. Linear Time History (LTH)

Time history analysis provides a linear and nonlinear evaluation of the dynamic structural response to a seismic load as a function of time [14]. Likewise, this analysis involves a step-by-step evaluation of the structural response of a building using real seismic records [3]. In this research, an LTH analysis is performed. Seven (07) pairs of seismic acceleration records available on the Centro Peruano Japonés de Investigaciones Sísmicas y Mitigación de Desastres (CISMID) web portal were selected, which are presented in

Table 4. Seismic acceleration records.

Station	Date	Magnitude	Depth	Acceleration (cm/s2)
				EO	NS
Parque Reserva (Lima)	17-Oct-66	8.1 Mw	24.00 km	180.56	268.24
Zarate (Lima)	05-Jan-74	6.1 mb	91.70 km	138.94	156.30
Parque Reserva (Lima)	03-Oct-74	6.6 mb	13.00 km	194.21	180.09
Vizcarra Vargas (Tacna)	23-Jun-01	6.9 mb	33.00 km	295.15	220.00
Univ. San Agustin (Arequipa)	07-Jul-01	6.5 mb	33.00 km	123.21	120.52
Univ. San Luis Gonzaga (Ica)	15-Aug-07	7.0 ML	40.00 km	272.82	333.66
Univ. Basadre (Tacna)	05-May-10	6.5 ML	36.00 km	154.00	190.00

.

4.3. Multimodal Pushover (MPA)

Seismic evaluation of a structure will require consideration of its nonlinear response [16]. It is important to know the capacity of a building that depends mainly on the resistance and maximum deformation of its individual structural components [3]. To determine this important parameter, a nonlinear static pushover analysis is conducted. In this research, a method called Multimodal Pushover Analysis (MPA) is selected, which has demonstrated high-precision results [17,18]. The seismic demands for the structures analyzed in the present study are not dominated by the fundamental period of vibration, which is why the MPA is chosen, as it, considers the influence of the higher modes on the seismic response [19].

5. Result and Discussion

5.1. Modal Participation Mass Factor

Analyzing the results shown in

Table 5. Modal participation mass percentage—Structure Type L.

Model	Irregularity (%)	Mode	UX	UY	RZ	Observation
L-1	14.29	1	41.68	41.68	0.00	Translational
		2	41.70	41.70	0.00	Translational
L-2	28.57	1	41.65	41.65	0.00	Translational
		2	41.64	41.64	0.08	Translational
L-3	42.86	1	41.61	41.61	0.00	Translational
		2	41.07	41.07	1.12	Translational
L-4	57.14	1	37.84	37.84	7.32	Rotational
		2	41.56	41.56	0.00	Translational
L-5	71.43	1	28.64	28.64	26.28	Rotational
		2	41.99	41.99	0.00	Translational
L-6	85.71	1	19.58	19.58	43.26	Rotational
		2	42.12	42.12	0.00	Translational

, it is observed that structures with incoming corner irregularities (Type L) change their behavior from translational to torsional when the structures present an irregularity greater than 57%, different from what the E.030 Standard indicates [13], which is 15%.

Examining the results shown in

Table 6. Modal participation mass percentage—Structure Type I.

Model	Irregularity (%)	Mode	UX	UY	RZ	Observation
I-1	4.08	1	0.00	83.21	0.00	Translational
		2	83.39	0.00	0.00	Translational
I-2	12.24	1	0.00	83.14	0.00	Translational
		2	83.35	0.00	0.00	Translational
I-3	24.49	1	0.00	82.82	0.00	Translational
		2	83.30	0.00	0.00	Translational
I-4	40.82	1	0.00	82.63	0.00	Translational
		2	83.18	0.00	0.00	Translational
I-5	61.22	1	0.00	83.87	0.00	Translational
		2	85.09	0.00	0.00	Translational

and

Table 7. Modal participation mass percentage—Structure Type O.

Model	Irregularity (%)	Mode	UX	UY	RZ	Observation
O-1	2.04	1	4.73	78.66	0.00	Translational
		2	78.66	4.73	0.00	Translational
O-2	6.12	1	83.23	0.00	0.00	Translational
		2	0.00	83.40	0.00	Translational
O-3	18.37	1	2.90	80.27	0.00	Translational
		2	80.27	2.90	0.00	Translational
O-4	30.61	1	82.89	0.00	0.00	Translational
		2	0.00	83.16	0.00	Translational
O-5	51.02	1	5.58	79.09	0.00	Translational
		2	79.09	5.58	0.00	Translational

, it is observed that structures with diaphragm discontinuity irregularities (Type I and Type O) present translational behavior in all cases. According to the E.030 Standard [13], structures with openings greater than 50% present torsion; however, in the present case, it is observed that the structural behavior of irregular buildings is independent on the percentage of the opening.

5.2. Lateral Displacement

Lateral displacements are important to analyze since excessive lateral movements can affect the main structural elements (beams and columns) and non-structural elements. Figure 4b represents the maximum lateral displacement for the structures under analysis in the X direction. It is observed that the maximum lateral displacement occurs for the L-6 type structure. According to the modal spectral analysis, the maximum lateral displacement for the L-6 structure varies by 113.59% compared to the regular structure. On the other hand, for Type I structures, it is observed that the values of lateral displacements decrease as the irregularity in plan increases by a percentage of less than 10% for the I-5 structure with respect to the regular structure. Likewise, for the type O structure, the displacement decreases by 6.95% for the O-5 structure with respect to the value of the structure without irregularity.

Figure 4b represents the maximum lateral displacement in the Y direction. For L-type structures, it is observed that the maximum lateral displacements increase as the plan irregularity is critical. On the other hand, the values of the lateral displacements for type I structures increase as the irregularity in plan is critical in a percentage of less than 5% for the I-5 structure with respect to the RB structure. However, for the type O structure, the maximum displacements decrease as the irregularity in the plan is critical by a percentage less than 5%.

5.3. Story Drift

Story drifts are important to evaluate, as are lateral displacements, because without proper control they can directly affect the main structural elements and adjacent buildings. Figure 5a,b represent the maximum story drifts, which occur on floor three (03) for all the structures under analysis. It is observed that the values of story drifts for structures L-3, L-4, L-5, and L-6 exceed the permissible value of 0.007 indicated in Standard E.030 [13] for both the X and Y direction.

5.4. Base Shear

This section describes the influence of the geometric shape of buildings on base shear forces. Figure 6a,b represent the forces on the base in the X and Y directions, respectively. In the X and Y directions, it is observed that the maximum shear forces decrease as the plan irregularity is critical. Through the spectral modal analysis (MS), it is determined that the shear forces at the base (X direction) decrease by a percentage of 59.62%, 48.78%, and 39.26% for the critical structures with respect to the regular model.

5.5. Ooverturning Moment

Overturning is a phenomenon that occurs when the structure is tilted against a lateral load on the building [11]. This section analyzes the influence of the geometric shape of buildings that influence overturning moments. Figure 7a,b show that the maximum overturning moments decrease as the irregularity in plan is critical. The spectral modal analysis (MS) demonstrates that the maximum overturning moments (X direction) decrease by a percentage of 59.43%, 48.27%, and 38.96% for the critical structures with respect to the regular model.

5.6. Result MPA

This section presents the results of the MPA for Type L structures. These structures, according to the linear analysis, tend to present maximum displacements and story drifts. Therefore, nonlinear analysis is performed to observe the structural response in the nonlinear range. Likewise, structure L-6 is excluded from this analysis because it does not meet the necessary requirements to carry out the MPA. Figure 8a shows the bilinear representation of the capacity curve based on the equal areas criterion [20,21]. Examining Figure 8a, it is observed that the lateral stiffness of the L-type structures represented by the slope of the straight line decreases as the irregularity in plan is critical.

On the other hand, Figure 8b represents the ductility of the L-type structures obtained as the ratio of the ultimate displacement to the yield displacement. Analyzing Figure 8b, it is observed that the irregularity in plan and the ductility are directly proportional. For the L-5 structure, the ductility increases by 15.58% with respect to the regular structure.

6. Conclusions

• The modal analysis demonstrated that irregular structures in plan present a structural behavior independent of the percentage of irregularity.
• Type L structures evaluated in their first two vibration modes change their behavior from translational to torsional when they present an irregularity greater than 57%. Type I and O structures present translational behavior.
• Type L structures present greater displacement and story drifts than regular structures, varying this value by 114%.
• The maximum lateral displacements and mezzanine drifts for Type I and O structures decrease as the irregularity in the plan is critical by a percentage of less than 10%.
• The values of the mezzanine shear force and overturning moment for the irregular structures Type L, I, and O decrease as the irregularity in plan increases compared to the regular structure.
• The results of the MPA for irregular Type L structures demonstrated that the lateral stiffness of the structures decreases as the irregularity in plan is critical, increasing the possibility of the formation of plastic mechanisms in the structural elements.