## 1 Introduction  
### Background  
In recent modern multistory building construction, the implementation of horizontal and vertical irregularities is getting popular due to different building functions and architectural needs. Vertical irregularity often regarded as a setback affects discontinuities in mass, stiffness, and structural strength distributions [1; 2]. Under seismic excitations, this irregularity has a considerable effect on structure responses [3]. The existence of the setback may cause torsional response [4] and stress and damage concentration on the irregular part [5]. Studies on the seismic response of buildings with setbacks had been conducted by several researchers. [PERSON] and [PERSON] [6] assessed the effect of simultaneous action of two orthogonal ground motion components on the performance of structure with different setback ratios and found that the effect of ground motion components is significant for the seismic performance of setback structures. [PERSON] et al. [7] investigated the modal response history analysis of setback buildings and instead of using the conventional SDOF modal system, the 2 DOF modal system was employed in modal response history analysis of building with specific vertical irregularities, to characterize the distinctly different modal response of tower and base structures. It was found that the 2 DOF approach adequately captured the characteristic of drastic change in the peak interstortower and base. [PERSON] and [PERSON] [8] proposed the equations to minimize the rotational response due to the mass eccentricity of setback buildings by changing the location of lateral load resisting elements. The parameters in the proposed equations involved the first mode frequency and the displacement ratio between the tower and the base structure. The proposed equations were also applied for minimizing the rotational response of medium-rise buildings with irregular plans under seismic excitations [9]. [PERSON] and [PERSON] [10] developed equations to determine the Park-Ang damage index [11; 12] of setback buildings using three independent parameters, namely overall drift (which is the ratio between the maximum roof displacement to the total height of the frame), the natural period, and the irregularity indices of the building. The adopted irregularity indices were originally introduced by [PERSON] et al. [13], involving the geometry of setback buildings, such as the number of stories and bays and the height and the width of setback buildings. [PERSON] et al. [14; 15] also predicted the Park-Ang damage index of setback structures using three primary parameters: the ratio of modal participation factor between irregular and regular buildings, the beam to column stiffness ratio, and the displacement ductility. [PERSON] et al. [16] assessed the Park-Ang damage index of low to mid-rise buildings using parameters generated from dynamic analyses, such as the maximum interstory drift, the peak roof displacement, and the maximum joint rotation of the members. [PERSON] et al. [17]. also predicted the Park-Ang damage index by multi-variable regression using the artificial neural network. In these studies, the Park-Ang damage index was computed through nonlinear dynamic analysis for both tower and base sections, and the damage distribution could be retrieved after. This procedure could take time and resources for the dynamic analysis. The objective of this paper is to propose the formula to determine the damage distribution of setback buildings based on the geometric measure of the degree of irregularity without conducting dynamic analysis. First, to verify the accuracy of the analytical method, the experimental tests for two types of setback buildings are analyzed. The experimental tests are a towered structure of fifteenth-scale 9-story 3-bay RC frame with a setback at level 3 by [PERSON] [18] and a quarter model of 6-story 2-bay setback RC frame by [PERSON] and [PERSON] [19].