Seismic performance and assessment of RC framed structure with geometric irregularities

Over the few decades, irregularly shaped structures have become famous in modern times because of their architectural significance. Symmetrical plan structures often exist, although irregular plan structure is practiced in developing nations. This study investigates the effect of geometric irregularity in reinforced concrete (RC) frame structures under seismic zone IV using the software STAAD Pro. Moreover, the time displacement on the nodes from where the geometric irregularity starts has been analyzed. Three types of 8-storey’s frame models were considered for the dynamic behaviors, i.e., models M1, M2, and M3. Model M3 is geometrically regular in the x–y-plane, and geometric regularity is interrupted in models M1(less disturbed) and M2 (more disturbed) in the identical plane. The results showed that the time displacement is more geometrically irregular structures. The time frequency increases in each mode, which means the higher method has a higher frequency and more in a more geometrically rare form. The irregularity of the structures' geometricity is reflected in the irregularity shape factor (ISF). According to the finding, the rough geometrical shape significantly impacts the dynamic response of the building, and ISF of 0.26 (M2) and 0.24 (M1) greatly varies with frequency, period, velocity, acceleration, and displacements.


Introduction
Past earthquake destruction data have indicated that various building collapses occur due to the structure's vertical irregularities (Quansah & Zhirong, 2017). Hence, regular modifications and enhancements are completed frequently in earthquake-resistant fields (Stefano & Pintucchi, 2008). In multi-story designs, RC structures with vertical geometric asymmetry are commonly considered for their originality (Gerasimidis et al., 2012;Panagiotis et al., 2017;Sarkar et al., 2010). However, each structure's performance diverges, consistent with the assembly of the structural elements that appear in the system. The orientation of the frames depends on the various aspects, i.e., geometry, dimension, and shape (Naveen et al., 2019). The traditional buildings have no considerable discontinuities in the plan. However, Irregular has substantial discontinuities in the program. In engineering practice, geometric asymmetries significantly contribute to the dynamic response in the light of time displacement, frequency, and velocity, including acceleration (Chopra & Goel, 1999). So far, various investigators have examined the influence of earthquake response on buildings having vertical and horizontal asymmetries (Destefano et al., 2005;Sarkar et al., 2010). However, the loading and force are established and intended at the center of the structure's mass (Siva et al., 2019). However, the careful variation of these considerations in the design of buildings enhances the system's effectiveness (Siva et al., 2019;Valmundsson & Nau, 1997). The asymmetries in the structures provide for their aesthetic objectives and effectiveness. The magnitude of variation in behavior varies on the type, intensity, and place of asymmetries appearing in the structure. The earthquake performance of multi-story or high-rise structures with vertical irregularities declines the stiffness by up to 30% and raises the story drift by 20-40% (Valmundsson & Nau, 1997). Also, the static and dynamic assessments of the structures having mass irregularity are ineffective in predicting the response of the buildings (Tremblay & Poncet, 2005). Guevara et al. (1992) aimed at the 1 3 floor plan's impact on the building's seismic performance. The study involves the dynamic analysis of H and L-shaped buildings. It concludes that buildings with H and L-shaped plans must be divided into rectangular blocks separated by seismic joints (Guevara et al., 1992). Wood (1992) examined the impact of seismic response of setback buildings and determined that the occurrence of setbacks did not impact seismic performance. The behavior was like that of a regular structure, having no setbacks. Khoure et al. (2005) studied the response of nine-story steel frames with setback irregularities and observed a higher torsion response at the upper portion of the setbacks (Khoure et al., 2005). The impact of torsional irregularity on structures was determined and revealed that torsion is an irregularity among the mass and stiffness (Gokdemi et al., 2013). Also, the parametric analyses suggested for the six-story building with variables (shear wall positions, surface rotations, and torsional irregularity coefficient) concluded that the torsional irregularity coefficient increases with the floor decrease (Ozmen et al., 2014;Varadharajan et al., 2012). However, L-shaped structures have a higher response than the regular frame (Momen et al., 2016). From time history analysis for traditional multi-story buildings, it has been determined that story drift declines with an increase in story height (Mehta & Rana, 2017). The irregular shape of the structure plan provides a severe high responsibility towards the ground motion and suitability of dynamic analysis compared to static analysis (Kar & Sadhu, 2021). However, the structure's static and dynamic evaluations are ineffective in evaluating the response under the dynamic behavior having mass irregularity (Athanassiadou, 2008;Tremblay & Poncet, 2005). Assorted studies have assessed the performance of structures with stiffness, mass, and vertical geometry variations. However, fewer studies on the effect of the irregularity shape factor (ISF) have a more profound correlation to the behavior of a structure. The irregular structures which constructed in a haphazard manner causing geometric irregularity. Sometimes these geometrical irregularities are also responsible for mass and stiffness irregularities. Thus, the response of geometrically irregular structures was performed among the three models (M1, M2, and M3). Model M1 correspondence to the less periodic system, and M2 parallels more irregular structures than model M3, which is a standard model with a geometrically stable structure. This research aims to evaluate the structure's response, especially displacement on the nodes from where the cut out (geometric irregularity) starts, along with analyzing the structure's response, such as frequency, acceleration, and velocity due to geometric changes.

Methods
In the present work, seismic analyses of multi-story RC frame structures were carried out using the software STAAD Pro. V8i as of its pros, such as its easy interface and capability to analyze design(s) conforming to the Indian Standard (IS:1893 (Part 1) 2002) (IS 1893 (Part-1) 2016). In horizontal load applications such as earthquake and wind loads, geometric irregularity plays a significant role in the response behavior of buildings in terms of displacement, natural frequency, time, stiffness, velocity, and acceleration of the structure (Quansah & Zhirong, 2017). The time history analysis has been selected to check the dynamic response for the models under the earthquake conditions in seismic Zone IV (Mehta & Rana, 2017;Sharma et al., 2021). The time displacement is higher in a geometrically more irregular structure (M2) than in one less irregular (M3). The time frequency increases in each mode, which means the higher method has a higher frequency. The effect of the geometric irregularity of the structures has been reflected in the irregularity shape factor (ISF). A comparison has been made concerning the regular geometrical shaped model M3. Structural models behaving differently have been attributed to the irregularity in geometry created in the structures. The Irregularity Shape Factor reflects the effect of the geometrical shape of the design. In xx, I ggy is the geometrical moment of inertia about XX and YY axes. The Regularity Ratio has been specified as the ratio of I gxy (M1 or M2)/I gxy for (M3) as per the model studies. The flowchart is presented in Fig. 1. Consequently, lateral stiffness in such stories should be increased suitably considering the irregularity shape factor. The outcomes of structural responses among the three models are expressed in the form of displacement, frequency, acceleration, and velocity. The stress-strain of the concrete grade M25 have been shown in Fig. 2.

Modeling
The dynamic behavior has been studied with an 8-story (G + 7) three RC frames structure for the geometrically regular and geometric irregular models in the identical plane. To find the critical areas that require special attention due to necessary time displacement. Also, evaluate the structure's response, especially displacement on the nodes from where the cut-out (geometric irregularity) starts. The analysis (Standard) was done by considering that the connection between columns and beams is completely fixed. The 8-Storey (G + 7) three RC frames models have been selected as Model M1, M2, and M3 dimensions are shown in Table 4. Model M3 is geometrically regular in the XY plane, and geometric regularity has been disturbed in models M1 and M2 in the same plane. The floor-to-floor height (length, height, width ratio) for (M1), (M2), and (M3) models are represented in Table 5. The most disturbed or irregular model is M2, followed by model M1, which is geometrically less disturbed than model M3 (considered a standard model due to geometrical symmetry). The geometrical shape of   Table 3

Design parameter
Various types of loads act on the building under different circumstances, and these loads are as per the standard provision and seismic presented in Tables 1, 2. The properties of the concrete grade M25 have been shown in Table 3. The design characteristics of the concrete have been shown in Tables 4 and 5.

Time history displacement calculation
In all the models, the front frame has been considered an authentic replica of all the structures in its back, with bottom coordinates as x = 0, y = 0; x = 12, y = 0. The acceleration generated under zone-IV, responsible for displacement in the structure, has been tabulated in Table 6.
The data generated from the model M1 have been tabulated in Table 7, showing that the upper values exhibit the displacement along the X-axis in the (+) direction. However, the lower displacement along the nodes indicates displacement in the (−) X-direction. Figure 4a shows that on the 1st floor, the displacement is 11.6 E--3 mm which further increases up to the 8th floor with the displacement of 103 E-3 mm in the upper direction. However, Fig. 4b exhibits the displacement along the lower path, which is 12.6 E-3 mm at the 1st floor level and increases up to the 8th floor displacement is 127 E-3 mm. The displacement increases with the order of increment up floors. This exhibits that the lateral force increases floor-wise in an upward direction. It has been observed that there is a change in displacement one story up and one level down on floor 3. Therefore, floors 2 and 3 are affected by the geometric irregularity created on floor 3.
The data generated from model M2 have been tabulated in Table 8. The displacement increases with the increase of floors. The dynamic response of the lateral force increases floor-wise in the upward direction of the structure. Figure 5a shows that the displacement in the upper path on the 1st floor is 17.5 E-3 mm, which further increases up to the 8th story to the extent of 195 E-3 mm, and Fig. 5b exhibits the displacement in the lower direction on the 1st floor is 20.7 E-3 mm which increases up to 8th story to the extent 233 No. of members 520 520 520 Table 5 Floor-to-floor height for (M1), (M2), and (M3) models 1  3  12  12  3  12  12  3  12  12  2  3  12  12  3  12  12  3  12  12  3  3  12  12  3  12  6  3  12  12  4  3  12  9  3  12  6  3  12  12  5  3  12  9  3  12  6  3  12  12  6  3  12  9  3  12  6  3  12  12  7  3  12  9  3  12  6  3  12  12  8  3  12  9  3  12  6  3 12 12 E-3 mm. The displacement increases with the height of the frame's structure. It has been observed that the change in displacement is one story up and one level down on floor 2. Therefore, floors 1 and 3 are affected by the geometrical irregularity created on floor 2. The data generated from model M3 have been tabulated in Table 9. Figure 6a shows the ground surface floor-0 displacement is 0, which further increases up to the 8th story to the extent of 164 E-3 mm, and Fig. 6b exhibits the displacement in the upper direction at 1st floor to the size of 19.6 E-3 mm which increases up to 8th Floor to the magnitude 195 E-3 mm. The displacement increases with the increase of subsequent base upwards, where each floor attains a height of 3.00 m.   Figure 7a shows the time-frequency mode shape for model M1. However, in Fig. 7b, each mode has variations in frequencies. In the first Mode, the time frequency is 1.024 Hz, which is 4.941 Hz in the 8th mode. The frequency increases with the extent to standard mode M3 subsequently. The increase in the frequency concerning the first to the last mode is 382.51% which means the modal frequency in the previous mode is 4.83 times that first mode. The functional relationship of increment in frequency ratio w.r.t 1st floor for model 1 is shown in Fig. 8a. However, in Fig. 7c, there is a variation in the frequencies from the first node to 8 nodes. The frequency varies from 1.013 to 4.921 H Z . The frequency  increases with every mode, which means the higher mode has a higher frequency than its lower one. The increase in the frequency from the first to the highest is 385.78 percent. That means the highest modal frequency is 4.86 times the first mode frequency. The functional relationship of increment in frequency ratio w.r.t 1st floor in model 2 is shown in Fig. 8b. However, In Fig. 7d, there is variation in the frequencies on each floor, which varies from 0.969 to 4.909 H Z from the 1st floor to the top floor. The frequency increases with the increase in the modes. That means the higher method has a higher frequency than its lower one. The increase in the frequency from the first to the highest is 406.60%. That means the highest modal frequency is 5.07 times the first mode frequency. The functional relationship of increment in frequency ratio w.r.t 1st floor in model 3 is shown in Fig. 8c.  -195 -195 -195 -195 -195 (  Figure 9 shows the relationship of frequency ratio concerning the fundamental frequency for models M1, M2, and M3 w.r.t the ground floor. The time-acceleration increases with the increase in height, which means that the higher floor will have higher acceleration than the lower ones. Figure 9e displays the acceleration at the first-floor level (1st floor) to the extent of 5.58 E-3 m/ s 2 , which further increases on the up bases and reaches 17.9 E-3 m/s 2 on the 8th floor in the upper direction, and Fig. 9f show lower direction time-acceleration, i.e., 3.94 E-3 m/s 2 at (1st floor), which increase 16.2 E-3 m/s 2 at 8th floor in the more downward direction. The time-acceleration increases with the increase in height, which means that the higher floor will have higher acceleration than the lower ones. has different behavior on floors 2 to 4. As discussed, the geometric irregularity has affected a change in velocity one foot up and one floor down. Figure 10e shows the velocity in the upper direction on the 1st floor to 255 E-3 mm/s. At the last base (8th), it is 1130 E-3 mm/s, and Fig. 10(f) shows the lower direction the velocity on the 1st floor is 213 E-3 mm/s, and on the 8th floor, velocity is 1160 E-3 mm/s in the more downward movement. There is variation in time-acceleration at floors/ stories. This is probably due to a change in acceleration from ground to first floor/story. The results reveal much variation in the dynamic response of the structure due to changes in geometrical shape in RC frame models M1, M2, and M3. Each frame has eight floors (8 stories). The fundamental frequency contributes to mass participation of 71.26% in M1, 78.15% in M2, and 83.17% in M3. Consequently, results focus more on floor level one for M1, M2, and M3 RC frames. A comparison has been described between standard model M3 and the other models, M1 and M2. The input data have created 3-D models in STAAD Pro. Software, along with the formation of nodes and elements. The effect of the geometrical shape of the structures has been reflected in the irregularity shape factor ISF. The geometrical moment of inertia has been correlated by the square root of the sum of square (SRSS) method taking into consideration the effect of both geometrical moments of inertia about xx-and yyaxes in the analysis. The geometrical moment of inactivity has been assigned because this is not the actual moment of inertia of the structure. Instead, it is related to the structure's geometry, and so is the name geometry assigned.

Time history displacement
It has been observed from the results that the displacement increases with the increment in the height of the structural frame, and the removal of the floor situated at a higher level has higher displacements, as can be seen in the models M1, M2, and M3. The highest floor (8th floor) has the highest displacement. However, the ground level has zero displacements (see Fig. 11).
Time history displacement has been considered zero on the ground floor as the base is fixed concerning other floors. In model M1, the displacement increases in an upper direction from 11.6 E-3 mm on the first floor to 103 E-3 mm on the 8th floor. It varies from 17.5 to 195 E-3 mm in model M2 and 16.8 to 164 E-3 mm in the case of model M3. In general, it has been observed that the displacement in model M1 is higher than in model M2. Model M2 has a higher displacement in comparison to model M3. The increase in displacement in model M1 is 30.95% (11.6 E-3 mm) and 4.1% (17.5 (cut) has been generated. It also has been observed that there is a change in displacement one story down and one story up in the adjacent stories (M1). This can be observed that the geometric irregularity starts at level 3 in M1, and floor disturbance in displacement has been observed on floors 2 and 4. That indicates that one floor up and one down adjoining the affected floor needs to be paid attention to special structural treatment. This disturbance is due to geometric irregularity. The displacement has also affected the length of 5 modes on floor three. Therefore, a complete story/floor must be paid attention to. It can be restricted that floors 2, 3, and 4 need special attention for structural treatment. Similarly, in the case of M2, the variation in the upper displacement in the outer node varied from 41.2 E-3 mm to 41.8 E-3 mm in story second, where irregularity is present in M2. However, the changes in one floor up and one floor down displacement have been noticed. It also has been observed that the geometric irregularity changes the displacement behavior affecting the entire floor as in the case of M1 on floors 2, 3, and 4 in the case of M2, and affected feet are 1, 2, 3. The displacement in the adjoining floor up and down needs special attention under such circumstances. Appropriate reinforcement treatment may have to be required to treat the concentration of stresses. Figure 9a-f exhibit that the behavior of the function of displacement is almost similar except from floor 2 to 4, which is the area where irregularity in the geometry has been created in M1; however, the function of displacement does change from floor 1 to 3 in case of M2. That means irregularity plays a vital role in the response behavior of the structures.

Time history frequency
It has been observed that the frequency increases with the irregularity generated in the frame structure. This means the models M1 and M2, which have disturbed regularity, have a  Table 10. Time-frequency increases from 1.024 to 4.941 Hz in model M1, from 1.013 to 4.921 Hz in model M2, and from 9.69 to 4.909 Hz in the case of model M3. In general, it has been observed that the frequency in models M1 and M2 is higher than in model M3 except at floor levels 4, 5, 6, where the frequency is lower in M1 than M2. Model M2 has a higher frequency in comparison to model M3. The increase in frequency in model M1 is 5.67% (1.024 H Z ) and 4.54% (1.013 H Z ) in M2 concerning M3 (0.969H Z ) at the first-floor level, and the decrease in frequency in model M1 is 0.132% (4.941 H Z ) and − 0.04% (4.941 H Z ) in M2 concerning M3 (4.909 H Z ) at 8th floor. Percentage increase in frequency M1 concerning M3 from mode-1 to mode-3 is 5.67% to 14.36%, then decreases from mode 4 to 6, and again, it increases from mode-7 to mode-8. The percentage change in frequency M2 concerning M3 from mode-1 to mode-8 is 4.54% to 0.04%. Since mode-3 has shown significant change, it may have some relationship on the 3rd floor level geometrically M1 and M2 on the 2nd floor. Table 10 exhibits the function's behavior of frequency changes at level 3 and level 7 in M1. Therefore, there appears to be some relation between the point of geometric irregularity and frequency modes on floor 3; mode 3 has a significant change in frequency increased to M3. Frequency has another considerable change at mode seven, meaning a band formation in frequency has happened. Therefore, irregularity plays a vital role in the response behavior of the structures. The model M2, which is more irregular, shows more frequency than M1 and has a higher frequency than M3 (regular).

Time history acceleration
The time-acceleration and time-acceleration ratio in seismic zone IV for models M1, M2, and M3 have been presented in Tables 11 and 12.
The acceleration increases along with the floors. It varies from 3.90 to 13.40% in case of M1; 5.84 to 23.2% in M2 and 5.58% to 17.9% in M3 from floor − 1 to floor 8. The acceleration change is almost the same in M1 and M2 concerning M3 after floor − 3, an area of geometric irregularity interference. However, From Fig. 12(a-b), acceleration in model M1 is less s in all floors than in model M2 and model M3. Last floor which is 8th floor in each models has the highest time-acceleration i.e., 13.4 E-3 m/s 2 , 23.2 E-3 m/s 2 , and 17.9 E-3 m/s 2 for M1, M2, M3 respectively. The decrease in Time-acceleration in frame model M1 is  Figure 12c, d exhibits that time-acceleration has tangible effects at level 2, the area of geometric irregularity for M2. Due to geometric irregularity, the time-acceleration behavior differs from floors 2 to 4.

Comparison: time history velocity
The time-velocity of models M1, M2, and M3 has been presented in Table 13. Figure 13    has the highest velocity, which is a most irregular model. The time-velocity is lower in M1 than in M2, which has less geometric irregularity. Therefore, it can be concluded that geometric irregularity affects the time-velocity in the structure, which is different from some of the interference of geometric irregularity.

Irregularty shape factor
An attempt has been made to define an irregularity shape factor (ISF) for a rectangular geometrical structure. The moment of inertia has been made a basis for such a factor. The effect of the geometry of the designs has been reflected in the irregularity shape factor ISF. A comparison has been made concerning the regular geometrical shaped model M3.
The variation in the response behavior has been attributed to the irregularity in geometry created in the structure. The Regularity Ratio (RR) has been defined as the ratio of I gxy (M1 or M2)/I gxy for (M3) as per the model studies in consideration. I gxx , and I gyy are the geometrical moment of inertia about xx and yy axes. The Regularity Ratio (RR) has been defined as the ratio of I gxy (M 1 or M 2 ) / I gxy for (M 3 ) as per the model studies in consideration.

Conclusions
The seismic analysis of geometric irregularity in multistoried RC frames has been carried out. Three types of models, M1 (geometrically less irregular), M2 (geometrically more irregular), and M3 (geometrically regular), have been used under seismic zone IV conditions. It has been concluded from the study that geometric irregularity affects the displacement in the structure. There is more change in dynamic response in more geometrically disturbed models than in the regular ones less disturbed. It has also been found that there is a change in displacement, one story down and one level up from the floor where geometric irregularity has been created. It also has been observed that the cut-out at a base affects the entire floor (horizontally), having changes in displacement at horizontal nodes. Consequently, special attention is required in such stories for structural treatment. One of the primary concerns in geometrical irregularities is the localization of seismic demand, which must be fulfilled per the analysis requirement. The structural geometric irregularity affects the Time-frequency also. It is a more geometrically irregular structure than the regular ones-the time-acceleration increases along with up-floors. The change in time-acceleration is visible at the level of geometric irregularity. The time-velocity increases along up bottoms of the structures. The shift in time-velocity is visible at the level of geometric irregularity on floor 3 for (M1) and floor 2 (M2). The behavior of time-velocity is nonuniform on floors 1 and 3 for M2, and on floors 2 and 4 remains the same for M1. A term defined as Irregularity Shape Factor corresponding to the structure's geometry reflects the design's geometric irregularity. The Irregularity Shape Factor (ISF) for model M1 is 0.24 and 0.26 for the M2 model; from such factor, the degree of irregularity of the structure can be assessed.