3.1 Displacements

A parametric study was conducted to understand the variations and continuity or discontinuity of beams to find out the best performing system under lateral loading, considering 8, 10, 12, 14, and 16 storied buildings. 16 storied structures with continuity and discontinuity in beams are taken to understand the top story displacement. Table 5 depicts the maximum displacements of multi-story structures in both X and Y directions considering the combined loading conditions.

It was observed that, with any story levels or height within the design limit, the frames act in a non-linear way in terms of displacement. Moreover, it can also be stated that the Model- A1 and Model-A3 show minimum and maximum displacement, respectively. Model-A2 exhibits the best performance and it almost performs closely like Model-A1 which has beam discontinuity at every level. However, the performance of the buildings varies with the continuity or discontinuity of beams, and displacements increase with the increment of building heights.

Displacement in structures usually arises due to lateral loadings such as earthquakes and winds. At first, earthquake and wind loads were applied to the O-shaped model considering both X and Y axes. The results obtained from the analysis were quite similar for both axes. Ultimately, the ENVD combo loading system was applied to simplify the process of obtaining the maximum displacement from both axes for earthquake and wind loads. Moreover, the displacement gets significantly higher as it goes to the higher floors as depicted in Fig. 5. In Fig. 5, Deqx, Deqy, Dwx, Dwy denoted the displacement regarding earthquake and wind load in both X and Y axes; respectively and Dc represented the displacement combining both quake and wind loads considering X-axis values as the magnitude of the X-axis was higher than Y-axis. The overall displacement changes rectilinearly and follows a linear regression equation which is developed and shown in the Figure.

Fig. 5 Displacement of O-shaped building under earthquake and wind loadings

3.2. Bending Moment

The bending moment of multistoried buildings with beam discontinuity was investigated by taking a representative section of beam and column named BEAM-5EF and COLUMN-6D. This two sections of beam and column is selected randomly to represent the bending moment significance. Through, Table 6, Fig. 6, and Fig. 7 an attempt was conducted to explore the bending within the beam and column section considering discontinuous beams systems. Combine lateral loadings were applied to Model-A1, A2, and A3 accordingly to examine the bending moment variation in BEAM-5EF and COLUMN-6D at story level-3 as presented in Table 6.

The maximum bending moment was located just right after the discontinuous beams in Model-A. In the case of Model-A1, where beams continue up to the story level has to resist fewer moments than the other models. Fig. 6 reveals that the BEAM-5EF was affected by the discontinuity of the beam in that system. When lateral load acts on the frame system, the load path was switched when an element is missing in the line of force and simultaneously increased the bending moment of the beam.

Bending moment variation in COLUMN-6D was evaluated at story-3 for all three models as shown in Table 6. Due to discontinuity, an extra potential moment was developed in the models A2 and A3. Each model exhibits significant deviation in bending moments from each other. It is mandatory to take necessary steps to resist extra developed moments for discontinuity else catastrophic damages can be combatted due to earthquake or heavy wind loading. A maximum significant result was found at level 3 as presented in Fig 7.

Both positive and negative moments were developed in beams and columns of the O-shaped structure. The maximum positive and negative moments for the beam at B60 and B78 were achieved as 3274.22, 6242.37, and 3707.64, 6459.07 (10-3 kN-m) respectively. In the case of columns, C20 and C13 maximum positive and negative moments obtained from analysis were 5561.46, 4513.8 and 3618.86, 4014.91 (10-3 kN-m), respectively. It can be seen that the O-shaped frame structure has to resist the least amount of both positive and negative moments. To understand the bending moments in both column and beam, three models such as C6, C13, and C36, and one beam B60 were considered. Here “B” and “C” denotes the beam and column and by number is mentioned the respective beam and column number. Fig. 8 provides to locate the bending moment in the concentrated columns and beams. Fig. 9 represents the values of the bending moment of a single column, considering the maximum bending moment at the base, and observed that maximum moments were generated on the top floor only. Moreover, this bending moment exhibits a linear trend by following a linear relapse equation as presented in Fig. 9. In Fig. 9, “M” is denoted as the bending moment.

Fig. 6 Increased bending moment in BEAM- 5EF

Fig. 7 Bending moment in COLUMN-6D

Fig. 8 Bending moment locations for specific columns (C13) and beams (B60)

Fig. 9 Bending moment and axial force

3.3 Axial Force on Base Column

Though axial forces are very important for low-rising building eventually it is neglected. Axial forces can produce more overturning moments and a massive level of compression and tensions than horizontal motions. Moreover, vertical motions may create a negative impact on columns along with horizontal motions. The axial force on every floor of the O-shaped building indicates its susceptibility to overturning. A single column is taken for the assessment of the O-shaped structure which reveals the highest axial forces as the results are presented in Fig. 9. In Fig. 9, “AF” is denoted as the axial force. The P-M interaction curve indicates the capacity for P and M that reinforced concrete can resist and an interaction diagram displays the combinations of the acceptable moment and axial capacities of a structural member. The equivalency between an eccentrically applied load and an axial load–moment combination. The P-M curve for axial on base column is depicted in the Fig. 10 and indicated none of points exceeded the controlling zone.

Moreover, COLUMN-4E was selected to examine the maximum axial forces at the base level for all of the three multistory models –A1, A2, and A3. The axial force is produced under the action of combined applied lateral loads and maximum magnitudes at story-16 for all three models as shown in Table 7 (see columns 2-4). There is a significant change in axial force for the column analyzed in different multistory frame systems as shown in Fig. 11.

Fig. 10 P-M interaction diagram of base column

Fig. 11 Axial force on base COLUMN-4E

3.4 Torsion on Beam

Regarding the evaluation of torsion on the beam for the models A1, A2, and A3, BEAM E-56 at story-3 was considered for the discussion. Table 7 (see columns 5-7) shows the torsional magnitude obtained from the analysis. There is a major change in the magnitude of torsion in different multistory frame system models established for the study. The beam analyzed, being located next to the discontinued beam provides an enhanced torsional value as shown in Fig. 12.

It is noticed from the analysis that; the discontinuity of beams generates a huge amount of torsional stress in beams that come in the way of force. Moreover, comparing the 3 models it can be concluded that, without even continuing the beams in upper stories the Model-A2 performs better than the other two models in terms of torsional stress.

Fig. 12 Torsion on Beam-E56

3.5 Story Drift

Fig. 13 provides the details values of story drifts of each floor level regarding the O-shaped model. As it can be seen from the figure, the story drifting is quite closer for both axes. To simplify the analysis, maximum values were considered and X-axis presents a higher magnitude than the Y. To ease the assessment ENVD combined loading was imposed on the model considering the values of the X-axis. From both axes, maximum drifts were observed on the first floor and insignificant drifts were noticed from story 8 to the top floor. From Fig. 13, it can be seen that the story drifting was manifesting a specific linear pattern ensuing in a linear regression analysis. In Fig. 13, SDeqx, SDeqy, SDwx, and SDwy represents the story drift regarding earthquake and wind both for X and Y-axis and SDc denotes the combined story drift by considering both earthquake and wind effect by using the X-axis value as the X-axis provided the maximum magnitudes.

Fig. 13 Story Drifts for O-shaped

3.6 Comparison with Previous Research

In this section, current research is compared with three previous studies conducted by ^{Chaudhary and Mahajan (2021)}, ^{Sazzad and Azad (2015)}, and ^{Mahato and Kumar (2019)}, respectively. ^{Chaudhary and Mahajan (2021)} numerically analyzed several different shaped high rise buildings including O-shaped structures of 12 and 16 storied building systems considering a heavy mass utilizing ETABS. ^{Sazzad and Azad (2015)} carried out a computer-aided analysis to evaluate the performances of different irregular-shaped buildings with an O-shaped frame system (7-storied). Again, ^{Mahato and Kumar (2019)} evaluated the structural performance of G+18 buildings for O-shaped structures using ETABS. All the three authors analyzed several models for different shaped structures, for validation only O-shaped structures are considered. Details of these three previous studies are presented in the Table 8. In Fig. 13 and 14, SDc represents the obtained combined story drift from this current study considering the effect of both earthquake and wind load only for X-axis as this axis provided maximum value.

Fig. 14, presents the displacement validation of the current research with previous research. In the case of MO-1 and MO-1’, the displacement pattern almost matches the current study, by following a linear trend. When compared to MO-2, authors showed the displacement and drifting results both for earthquake and wind for both axes, where the values of both axes were exactly similar, thus only X-axis values were picked for the comparison and expressed MO-2 (W) and MO-2 (EQ); correspondingly. However, the results patterns were significantly matches with the present study. In comparison with MO-3, the result doesn’t exactly coincide with the current study, as there may be two reasons such as- number storied they examine and code provision discrepancies between this study and current one. It can be also observed from the figure that, magnitudes don’t match at all with the current and the previous studies due to have discrepancies in the code provision used for the analysis.

Fig. 15, provides a good representation regarding the comparison of story drifting ratios between the current study and three previous studies. It can be seen from the figure that, MO-2 (W) and MO-2 (EQ) exhibits exactly same pattern as the current study. There were some differences in magnitudes between MO-2 and current study as different code provision was considered. In case of MO-3, the drifting pattern doesn’t coincide with the present study due to have difference in the story number analyzed and variation in the code provision used.

Fig. 14 Comparison of displacement with three models

Fig. 15 Comparison of story drift with three models

Buildings with Beam Discontinuity:

∙ Based on the beam discontinuity some features are noticeable like beam discontinuity from the base and mid-height exhibits a large amount of displacement and internal torsions show interesting results.

∙ Taller buildings reveal fewer deflections than the buildings that have no discontinuous beams in both axes. Moreover, axial force produces less amount in these types of systems, though it has great significance in the case of tall buildings. Theoretically, the model with discontinuity from the upper heights is safer and more economical than the model with no beam discontinuity due to its lightweight. It was observed that Model-A2 performs well in terms of safety, cost-effectiveness, and efficiency.

O-shaped Building:

∙ In the case of an O-shaped structure, the amount of displacement production is proportional to the building height while applying earthquake and wind loads.

∙ The moment was distributed properly and moment fluctuations from one point to others were negligible. It can be seen that maximum moments were generated at the top floors only and maximum bending moments at the base.

∙ In the case of story drifting, maximum drifts were noticed on the first floor, and from the story level-8 to the top floors, drifting was insignificant.

∙ O-shaped building structures are less vulnerable to overturning due to their less resistivity to lateral load and show the least amount of axial force arising in their column and making this shape relatively safe. However, its resistance toward lateral load can be greatly improved by focusing on and strengthening a few critical points.

∙ Results obtained from the analysis were validated by previous research and the results are quite similar to the present work.

Future Scope:

∙ Present research work was carried out only for 10-storied buildings, leaving scope to analyze the multi-storied building with varying heights.

∙ As O-shape buildings were selected for the present study, the performance of different irregularly shaped building such as H, U, T, L, M-shape, plus shape, etc. shall be adopted for further study.

∙ A new research study can be conducted considering the bracing systems such as cross bracing, zig-zag bracing, V type, etc. in the structural frame system.

∙ Another study shall be carried out by changing the seismic zone of seismic analysis.

∙ Different design software like SAP2000, if used, can also generate new research work as it may produce different results. A good comparative study can be conducted in this case.