New definition for torsional irregularity based on floors rotations of reinforced concrete buildings

Earthquake investigations confirm that un-symmetric buildings suffer more damage than their symmetric counterparts. Torsional vibrations due to irregularity are one of the main factors which cause damage to buildings. Seismic codes include provisions to consider the torsional vibrations in the design of structures. However, the design of the stiff side elements may be unconservative. In this study, a total of 15 Dual system buildings were analyzed using nonlinear dynamic analysis to propose a new definition for torsional irregularity considering the effects of plan asymmetry on the earthquake response of mid-rise dual system buildings. Based on the analysis results, two new equations were derived using nonlinear regression analysis to check the torsional irregularity and the torsional amplification. Afterward, the proposed torsional irregularity provisions were compared with the torsional irregularity provisions of Europe code (EC-8), Japanese code & ASCE 7-16 code, and the results indicate that the torsional irregularity provisions of all the codes are unconservative. The current investigation proposes new equations to be used in the existing codes for torsional irregularity design considerations to improve the accuracy of the current codes' torsional irregularity provisions.


Introduction
Earthquake events show that buildings with configuration irregularity are more vulnerable to earthquake damages. For example, many torsionally unbalanced buildings suffered severe damage during the 1985 Michanocan earthquake [1], the 1989 Loma Prieta earthquake [2], the 1994 Northridge earthquake [3], and the 1995 Kobe earthquake [3]. All have indicated the effects of torsional vibrations caused by earthquakes.
The effects of plan asymmetry on the earthquake response of the structures were the subject of many studies (e.g., Bozorgnia and Tso (1986) [4];   [5]). Furthermore, the effects of plan asymmetry on the earthquake response of onestory system structure designed by various codes and how well the torsional provisions in building codes represent these effects were the subject of many studies (e.g.,   [6]; Rutenberg (1992) [7]; Wong and Tso (1995) [8]; HUMAR and KUMAR (2000) [9]; HUMAR and KUMAR (2004) [10]).
In 2013, the seismic performance of an under-designed plan-wise irregular R/C structure and different strategies for the seismic retrofitting were studied by Marco and compared by nonlinear dynamic time-history analyses and simplified assessment procedures based on nonlinear static pushover analyses. Numerical results showed that combining the FRP wrapping approach and R/C jacketing applied to the selected columns significantly improved the structure's seismic performance, increasing strength, stiffness, and ductility [11].
In 2014, a parametric investigation was performed by zmen on torsional amplification factor of American Society of Civil Engineers (2010) code (ASCE 7-10) using linear static lateral load analysis; it was found that floor rotations may be considered as the actual representative of the torsional behavior and suggested that torsional irregularity coefficients as defined in the regulations should be amended entirely [12].
In 2018, numerical models were developed by Marco et al. to simulate the seismic response RC frames in the original and retrofitted configurations. First, the effectiveness of three different retrofitting solutions countering the main structural deficiencies of the RC frame is evaluated through the displacement-based approach. Then, nonlinear dynamic analyses are carried out to assess and compare the seismic performance of the RC frame in the original and retrofitted configurations. The combined use of different approaches may represent a valuable tool to accurately address the retrofitting interventions and assess their effectiveness to reduce the seismic vulnerability of poorly designed RC buildings [13].
As the torsional irregularity provisions for the design of stiff side elements in buildings codes may be unconservative, and the stiff element design force could be underpredicted [9], where the stiff side is the side of the unsymmetric structure which the center of rigidity shifted toward it due to asymmetry and the stiff side elements are the elements located in this side as shown in Fig. 1, the current study used nonlinear dynamic analysis to propose new equations to be used in the current codes for torsional irregularity design considerations to improve the accuracy of the current codes' torsional irregularity provisions. The new equations are based on the floor rotations, which are the most realistic to represent the torsional behavior of the building.
The main objectives of the study are: Carrying out a parametric study on Dual System structures (D.S.) in terms of eccentricity ratio (distance between the center of mass and center of rigidity normalized by plan dimension) and Peak Ground Acceleration (P.G.A.), propose a new equation to check the torsional irregularity and a new equation for torsional amplification factor (to account for torsional irregularity) based on the rotation of rigid floors by using nonlinear regression analysis and comparing the proposed torsional irregularity provisions with the torsional irregularity provisions of Europe code (EC-8), Japanese code & ASCE 7-16 code.

Code provisions
Code provisions usually include values of static design eccentricities between the center of rigidity and the center of mass. The seismic loads are applied through the design eccentricities points. In some codes, static design or accidental eccentricities are multiplied by factors to account for torsional vibrations such as ASCE 7, U.B.C. 97, NEHRP 199, and N.Z.S. 1992. The design eccentricity formula is shown in Eq. (1) for flexible side elements and Eq. (2) for stiff side elements; values of δ, α, β vary among building codes. Some other codes consider torsional irregularity through modification of seismic reduction factor R, such as Europe code (EC-8) and Japanese code.
Torsional irregularity provisions of ASCE 7-16/UBC 97 The design eccentricity coefficients specified in ASCE 7-16/UBC97 are α = 1.0, β = A x * (0.05b) and δ = 1.0. Torsional irregularity and extreme torsional irregularity are defined to consider the maximum story drift and accidental torsion with A x = 1.0. For torsional irregularity, the ratio between the maximum drift at a story and the average drift of two ends at that story is more than 1.2; this ratio is considered 1.4 for extreme torsional irregularity. If the torsional irregularity exists, the analysis must be performed using accidental torsion amplified by factor A x . Factor A x can be considered from Eq. (3) for the torsional irregularity case and Eq. (4) for the extreme torsional irregularity case [14,15].
Torsional irregularity provisions of Japanese code Japanese code considers torsional irregularity through modification of seismic reduction factor R by multiplying the seismic reduction factor by the basic shape factor F e . Torsional irregularity is defined to exist where the eccentricity factor R e exceeds 0.15.
Factor R e is determined from the following equation [16].
In case R e exceeds 0.15, the ultimate lateral shear strength of each story Q ud must be calculated, and it must be confirmed to be not less than the specified ultimate shear as decreased by the factor F e . Then, the specified ultimate shear Q un is determined by Eq. (6) [16].
The basic shape factor F e is determined from Table 1.
Torsional irregularity provisions of E.C.-8 Europe code considers torsional irregularity through reduction of seismic reduction factor R. Frame, dual or wall systems classified as torsionally flexible systems, if at any floor one or both of the following equations not met [17].
If the system is classified as a torsionally flexible system, the basic value of the seismic reduction factor R is reduced to a value of R = 2 for ductility class medium (D.C.M.) or R = 3 for ductility class high (D.C.H.) as shown in Table 2. Basic values of the seismic reduction factor R for systems regular in elevation are shown in Table 2 [17].

Analysis methods
The applied element method, AEM, was developed by Tagel-Din and Meguro, 1999 [18]. The main advantage of this method is that it can follow the structural behavior from the application of load to the total collapse of the structure with high accuracy and reasonable time. In addition, the crack opening and crack closure process can be followed [18].
The structure is modeled as an assembly of small elements made by virtually dividing the structure, as shown in Fig. 2a. The elements are assumed to be connected by normal and shear springs, as shown in Fig. 2b, representing the stress and deformation of a specific area [hatched area in Fig. 2b] [19]. For example, the matrix springs and reinforcement springs represent concrete and reinforcing steel bars in reinforced  [20]. The stiffness matrix is formulated by applying unit displacements in the six degrees of freedom for each element, calculating the forces generated in the springs, and calculating the element forces in the directions of the degrees of freedom, as illustrated in Fig. 5 [20]. For modeling of concrete under compression, the compression model of Maekawa [21] shown in Fig. 6a is adopted in Extreme Loading for Structures (ELS) [20]. For concrete springs subjected to tension, spring stiffness is assumed as the initial stiffness until the spring reaches the cracking point. The stiffness of springs subjected to tension is set to a minimum value after cracking. In the next loading step, the residual stresses are then redistributed by applying the redistributed force values in the reverse direction. The relationship between shear stress and shear strain is assumed to remain linear till the cracking of concrete. Concrete is considered cracked when concrete stresses reach the strength limit. Both normal and shear stresses drop to zero for tensioned concrete.
However, for compressed concrete, the shear stress value is limited to a specific value depending on the compressive stresses, and the shear stress-shear strain relationship follows the curve shown in Fig. 6b. The aggregate interlock and friction at the crack surface control the drop level of shear stresses. This drop is defined as the residual shear strength factor or redistributed value (R.V.). The scheme for the envelope curve steel in tension and compression is shown in Fig. 7a for reinforcement springs. Three stages are considered; elastic, yield plateau, and strain hardening [20]. According to Giuffre and Pinto (1970), which was later implemented in Menengotto and Pinto (1973) [23], unloading and reloading stress-strain models are adopted in the analytical procedures as illustrated in Fig. 7b [20].
In ELS, the nonlinear response stage of RC structures, internal damping can arise due to concrete cracking, energy dissipation due to the loading and unloading of Table 2 Basic values of the seismic reduction factor, R, for systems regular in elevation [17] Structural type DCM DCH Frame system, dual system, coupled wall system 3.0(α u /α 1 ) 4.5(α u /α 1 ) Uncoupled wall system 3.0 4.0(α u /α 1 ) Torsionally flexible system 2.0 3.0 Inverted pendulum system 1.5 2.0 compression springs, unloading of reinforcement after yielding, and friction between elements and unloading factors during contact. Many studies have been done using the applied element method (e.g., Karbassi and Nollet [24]; Malomo et al. [25]; Sediek et al. [26]).

Structural models
Since the primary purpose of the torsional amplification factor is to limit the additional ductility demand on the stiff-edge element of the un-symmetric system to an acceptable level and as per the Literature Review, the floor rotations may be considered as the actual representative of the torsional behavior, the ductility demand ratio and rotations of buildings are the main parameters in the study. In the parametric study, the effect of earthquake acceleration (a g ), configuration of walls and frames, and distance between  the center of mass and the center of rigidity normalized by plan dimension (es/b) on rotation and ductility demand ratio were considered. The study is performed on three groups, where each group represents varying (es/b) with a constant (a g ) value, and each group consists of four cases. The inelastic responses of a torsionally unbalanced system were normalized to those of a reference model, which is a torsionally balanced model its center of mass coincides with its center of rigidity, to study the effect of torsion, in this parametric study case 4 in each group is the reference model for all groups. Figure 8 shows the general layout of the reference building model. The parameters of studied cases are shown in Table 3. The studied structures are typical 12-story reinforced concrete buildings with dual systems consisting of intermediate moment frames and ordinary shear walls. The structures are eight and five bays in xand y-directions, respectively, with bay lengths equal to four meters in both directions. The story height is three meters. The columns are fixed at their base. First, the linear finite element program ETABS [27] has been chosen  for the model and the preliminary design of the studied cases. Then, nonlinear time history analysis was performed using applied element method software (ELS) [20].

Gravity and seismic loads
The structure was designed according to American codes (ACI-318-14 [28] and ASCE 7-16 [14]) using the Etabs program. The superimposed dead load is 2.0 kN/m 2 , equivalent uniform wall load is 2.5 kN/m 2 and live loads is 2.0 kN/m 2 , respectively. Seismic parameters used in the analysis and design of typical structures are: seismic design category (C), soil class (D), importance factor = 1.25, response modification factor (R) = 5.5, design earthquake spectral response acceleration parameter at short period (S DS ) = 0.4166, design earthquake spectral response acceleration parameter at 1-s period (S D1 ) = 0.0544. Time history records Seismic input to the dynamic analysis of structures is defined in terms of acceleration time series (time-history function) whose response spectra are compatible with a specified target response spectrum with Peak Ground Accelerations (P.G.A.) of 0.3 g, 0.45 g, and 0.6 g. SeismoArtif [29] program has been used to modify a reference time series so that its response spectrum is compatible with a specified target spectrum using the Real Accelerogram Adjustment calculation method. In the Real Accelerogram Adjustment calculation method, the artificial accelerogram is defined starting from a real one and adapting its frequency content to match the target spectrum using the Fourier Transformation Method as shown in Fig. 9 [29].For example, the real accelerogram of the Kobe earthquake shown in Fig. 10, and a response spectrum generated according to ASCE 7-16 shown in Fig. 11 were used to generate our artificial earthquake shown in Fig. 12. Then, the induced artificial earthquake was scaled so that its peak ground acceleration (P.G.A.) matches the P.G.A. of 0.3 g, 0.45 g, and 0.6 g, and these functions were then imported to Etabs and ELS programs to represent the ground motions. The ground motions after the scaling process are shown in Figs. 13, 14, and 15.

Dimension, reinforcement detailing, and material properties
The slab thickness used for a  shown in Fig. 16 and Table 4. The characteristics of reinforced concrete are summarized in Table 5, and the characteristics of steel reinforcement are summarized in Table  6. All the walls are 400 mm in width with longitudinal reinforcement of Ø16@150 mm and horizontal reinforcement of Ø14@150 mm.

Calculation of ductility demand ratios and floor rotations
Determination of ductility demand ratios using nonlinear time history analysis In the nonlinear time-history dynamic analysis of structures responding to an earthquake in the inelastic range, the maximum deformations are expressed in Eq. 20. The ductility factor (μ) is the ratio between maximum roof deformation and the corresponding roof deformation present when yielding occurs. The maximum deformations are permitted to be expressed in nondimensional terms as indices of inelastic deformation for seismic design and analysis using ductility factors [30].
The definition of yield deformation often causes difficulty when calculating ductility factors since the force-deformation relation may not have a well-defined yield point. Therefore, various definitions for yield deformations have been proposed: deformation corresponding to the first yield as shown in Fig. 17a or deformation corresponding to the yield point of an equivalent elasto-plastic system with the same elastic stiffness and ultimate load as the real system as shown in Fig. 17b or deformation corresponding to the yield point of an equivalent elasto-plastic system with the same energy absorption as the real system as shown in Fig. 17c or deformation corresponding to the yield point of an equivalent elasto-plastic system with reduced stiffness computed as the secant stiffness at 75% of the ultimate lateral load of the real system as shown Fig. 17d [30]. In this study, yield displacement is considered the deformation corresponding to the first yield. For maximum deformation definition, various definitions have been proposed: deformation corresponding to a limiting value of strain as shown in Fig. 18a or deformation corresponding to the apex of the load-displacement relationship as shown in Fig.  18b or deformation corresponding to the post-peak displacement when the loadcarrying capacity has undergone a small reduction as shown in Fig. 18c or deformation corresponding to fracture or buckling as shown in Fig. 18d [30]. In this study, the  Fig. 21. The ductility Demand factor Ratio was calculated by dividing the ductility demand factor of the desired plan-asymmetric plan-by ductility demand factor of reference plansymmetric plan-in the same group.
Determination of rigid floor rotation using nonlinear time history analysis In the ELS program, floor rotations were tracked with time and rotation corresponding to the apex of a time-rotation relationship considered to be the floor rotations used in the study, as shown in Fig. 22.

Effect of distance between the center of mass and center of rigidity on rotation
The effect of distance between the center of mass and center of rigidity normalized by plan dimension (es/b) on floor rotations using nonlinear dynamic analysis is shown in Figs. 23, 24, and 25. The results show that floor rotations increase as the distance

Effect of floor number on floor rotation
The effect of floor number on floor rotations using nonlinear dynamic analysis is shown in Figs. 32, 33, and 34. The results show that as floor number increases, the story rotation increases, maximum floor rotation occurs at the highest stories, maximum rotation at the 12th story.  irregularity is defined to exist where ductility demand ratio larger than 1, the derived equation as shown in Eq. (9) has an upper limit of 1.
and U θ U f established as shown in Fig. 37, then, regression analysis was used to estimate the equation of U g , as shown in Eq. (11). Torsional amplification factor One of the goals of this research is to derive a simplified equation for the torsional irregularity coefficient, torsional amplification factor, based on the rotation of buildings. The proposed equation is derived by using nonlinear regression analysis.
Since the main objective of the torsional amplification factor is to amplify the accidental eccentricity for torsionally flexible structural systems and modify the design eccentricity of stiff side elements to account for torsional irregularity effects, therefore, separate equations are derived for each group based on floor rotation -Structural eccentricity normalized by plan dimension (es/b) relationships. To suppose a conservative equation, the best-fit equation of 1st floor with 0.3 g acceleration used as the base case equation ("e d o "), as shown in Fig. 38.
The effect of the varying floor no. (2 nd ,3 rd , 4 th , etc.) and varying P.G.A. considered by e f equation and e g respectively. Finally, the equation is divided by accidental eccentricity and added to 1 to amplify the accidental eccentricity (0.05), as shown in the following equation.
To consider varying floor no. (2nd,3rd, 4th, etc.), the relation between Floor number and e d was established for Group-1 as shown in Fig. 39, and then regression analysis was used to estimate the equation of e f , as shown in Eq. (14).
To consider varying Peak Ground Acceleration (P.G.A.), the relation between P.G.A. and e d e f established as shown in Fig. 40, then, regression analysis used to estimate the equation of e g , as shown in Eq. (15).

First group of buildings
The first group consists of five buildings, as shown in Fig. 41. The five buildings have different eccentricity ratios. The buildings are 12-story in height, and the floor height is First, the dimensions of the structural elements were determined by a preliminary design process using the equivalent static seismic load of ASCE 7-16. Second, the torsional irregularity was checked using various codes and the derived equations for all models, as shown in Tables 7, 8, and 9. Third, the torsional irregularity provisions of various codes and the derived equations were considered in the analysis. Fourth, the design forces of resisting elements of the asymmetric building corresponding to those of symmetric building compared as shown in Figs. 42 and 43. Finally, the torsional amplification factor (A x ) of ASCE 7-16 and the derived equation are compared in Fig. 44.
Comparing the design forces of resisting elements of the stiff side elements show that by neglecting the torsional irregularity provisions, there is a reduction in the design force up to 41% for small eccentricity ratios and up to 50% for large eccentricity ratios. On the other hand, for the EC-8 code, there is no reduction in the design force ratio of stiff side elements for small eccentricity ratios and up to 6% for large eccentricity ratios. For ASCE 7-16, there is a reduction in the design force ratio of stiff side elements up to 27% for small eccentricity ratios and up to 44% for large eccentricity ratios. For Japanese code, there is a reduction in the design force ratio of stiff side elements up to 41% for small eccentricity ratios and up to 42% for large eccentricity ratios. Only for the derived equation, there is no reduction in the design force ratio of stiff side elements for small and large eccentricity ratios.

Second group of buildings
The second group consists of five buildings are shown in Fig. 45 and ordinary shear walls. Corner and edge columns are 300 * 1100 mm, and the other inner columns are 300 * 1400 mm. The concrete core consists of four walls, each wall length is 4 m, and the slab thickness is 160 mm. The columns are fixed at their base. The buildings were studied in Y-direction only. First, the dimensions of the structural elements were determined by a preliminary design process using the equivalent static seismic load of ASCE 7-16. Second, the torsional irregularity was checked using various codes and the derived equations for all models, as shown in Tables 10, 11, 12, 13, and 14. Third, the torsional irregularity  Fig. 48.
Comparing the design forces of resisting elements of stiff side elements show that by neglecting the torsional irregularity provisions, there is a reduction in the design force up to 56% for small eccentricity ratios and up to 60% for large eccentricity ratios. For EC-8 code, there is a reduction in the design force ratio of stiff side elements up to 14% for small eccentricity ratios and up to 22% for large eccentricity ratios. For ASCE  7-16, there is a reduction in the design force ratio of stiff side elements up to 44% for small eccentricity ratios and up to 52% for large eccentricity ratios. For Japanese code, there is a reduction in the design force ratio of stiff side elements up to 33% for small eccentricity ratios and up to 40% for large eccentricity ratios, only for derived equation, there is no reduction in the design force ratio of stiff side elements for small eccentricity ratios and a reduction in the design force ratio of stiff side elements up to 10% for large eccentricity ratios. The current form of the ASCE 7 code formula based on the maximum to average drift ratio gives that the torsional irregularity exists even if there is no eccentricity between the center of mass and the center of rigidity.  The current form of the ASCE 7 code formula is based on the ratio of the maximum to average displacement; its torsional irregularity coefficients (A x ) reach maximum values when the structural walls are located as close as possible to the center of mass, and it decreases as eccentricities increase.    Rotation-based equation gives torsional irregularity coefficients (A x ) increases as the distance between the center of mass and center of rigidity increases which is more realistic.
3. Comparing the results of the derived rotation-based equations of design forces of stiff side elements of un-symmetric system normalized symmetric system with the results of codes formulas, the results indicate that all the codes' torsional irregularity provisions are unconservative. Therefore, only the proposed equations provide an adequate estimate of the design force of the stiff side elements.       4. Comparing the results of the derived rotation-based equations of ratios of design forces for flexible side elements of un-symmetric system normalized symmetric system with the results of codes formula indicates that the flexible side elements provisions of all the studied codes are overly conservative. 5. Further studies are suggested to extend the proposed definition for other structural typologies of buildings, study the effects of plan asymmetry and its effect on eigenfrequency analysis, and propose an upper limit value for the torsional amplification factor for the flexible side elements for dual systems and other structural systems.
Abbreviations α: Design eccentricity coefficient for flexible side elements; es: Distance between center of mass and center of rigidity; β: Accidental eccentricity coefficient; b: Plan dimension perpendicular to the direction of loading; δ: Design eccentricity coefficient for stiff side elements; δ max : Maximum story displacement; δ avg : Average story displacement; e y : Plan eccentricity in the y-direction; I x : Second moment of stiffness in the x-direction with respect to the center of rigidity; I y : Second moment of stiffness in the y-direction with respect to the center of rigidity; K x : Lateral stiffness in the xdirection; e x : Plan eccentricity in the x-direction; K y : Lateral stiffness in the y-direction; r x : Torsional radius for x-direction; l s : Radius of gyration of the floor mass; L: Plan dimension parallel to the direction of loading; R y : Torsional radius for ydirection; α u /α 1 : Ratio of the multiplier of horizontal seismic design action at the formation of a global plastic mechanism to the multiplier of horizontal design seismic action at the formation of first plastic hinge in the system, which ranges from 1 to 1.