## 1 Introduction  
A force-based method is adopted in current seismic design codes. It regards the earthquake loads as equivalent lateral inertia force consistent with the fundamental elastic vibration modes of structures [1; 2; 3; 4]. During a whole structural design service life, the occurrence probability of strong earthquakes which exceed the design earthquake intensity cannot be neglected [5; 6]; they may lead to inelastic response, and even more structurally collapse occasionally. Previous experimental and analytical studies showed that buildings which were designed by current codes exceeded the elastic limits and performed inelastic seismic demands under rare earthquakes [7; 8; 9; 10; 11]. Many studies have been carried out to achieve a status of uniform seismic demands. In these studies, pushover analysis was conducted to calculate the inelastic seismic demands [12; 13; 14; 15; 16]. However, pushover analysis cannot consider the higher vibration modes or dynamic characteristics of the earthquake loadings. Consequently, pushover analysis had limitations for the accurate prediction of inelastic seismic demands, especially for a severe damage state [17; 18]. Meanwhile, the cross sections of the structural loading bearing elements were set as the optimization variables, which might lead to unpractical sizes of cross sections [19]. To integrate closely with the seismic design procedure in present design codes, extensive studies have been conducted on proposing a modified lateral load pattern to achieve uniform seismic demands while other procedures remain unchanged. [PERSON] [20] and [PERSON] [21] concluded that the earthquake-induced story shear forces were different from the forces determined by the lateral force pattern in the seismic design codes. New lateral load patterns, which consider the effects of higher vibration modes and the inelastic behavior of structures under earthquake excitations, were proposed. [PERSON] [22] and [PERSON] [23] proposed the theory of uniform deformation. Afterwards, various related investigations [24; 25; 26; 27; 28; 29; 30] were conducted based on this theory to optimize lateral load patterns for frame structures under a series of real and synthetic earthquake records. The input earthquake records were scaled to a specific intensity to match with the design spectra. Meanwhile, the influences of the dynamic characteristics of earthquake ground motion and structural parameters were also discussed. [PERSON] [31; 32] and [PERSON] [33] investigated the influence of the soil-structure interaction on the optimized lateral load pattern for a uniform distribution of seismic demands along the height of buildings. [PERSON] [34] proposed an analytical formula of the lateral load pattern for multi-story buildings to increase the collapse resistant capacity. It showed that the optimized lateral load pattern leads to lower collapse probability than that of the conventional seismic design method with the same construction cost. The advantages of the optimized lateral load pattern were demonstrated, but the shear-beam model used in these studies cannot provide accurate seismic response predictions under severe earthquakes. In addition, most optimized lateral load patterns were impractical because of ignoring the constrained provisions in the design codes. Various studies were conducted on the optimum seismic design by optimizing the cross section of the structural elements considering inelastic performance under severe earthquakes [35; 36; 37]. A heavy computer calculation cost was needed and multiple cross sections were designed, which are unacceptable in practical application. The aforementioned studies have shown the advantages of optimum seismic design methods based on the theory of uniform seismic demands, while limitations still exist.