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Wi = the weight of the i th floor, Φik refers to the mode shape of the i th floor due tokth mode.The first example (a single storeyed portal frame) being one storied, the modal masscorresponding to the first mode equals 100% of the total seismic mass, hence the dynamicmethod for computing the inpidualmodes and then the base shear (asmentioned in IS code)becomes more or less similar to the static procedure. In the second example (a two storeythree bay frame), the contribution of first mode towards modal mass is 92.4%. Hence, thecomputation of design base shear using dynamic method is not needed. In the third example(a three storey 2 bay frame), the contribution of firstmode towardsmodalmass is 89%. Beingvery close to the required 90% mark, dynamic method is not adopted in this case either.2.1 IS code based equivalent static evaluation of structureThe equivalent static load approach defines a series of forces acting on a building to repre-sent the effect of earthquake ground motion, typically defined by a seismic design responsespectrum. It assumes that the building responds in its fundamental mode. For this to be true,the building must be low-rise and must not twist significantly when the ground moves. Sincehigher modes are not accounted for, the accuracy of this method may be limited.
To accountfor effects due to “yielding” of the structure, modification factors (like reduction factors) areused. The structures designed by this method could be rather conservative (as can be seen inthe example in later sections).IS1893:2002 uses a horizontal acceleration spectrum factor, Ah, which on multiplicationwith the seismic weight of the structure (Ws ) gives the base shear, Vb (Clause 7.5.3, IS1893:2002).Vb = Ah × Ws (2)Ws is the sum of the seismic weights of all the floors of the building. The seismic weight of afloor is the sum of the factored dead plus factored live loads. Ah can be found for a structureby knowing the following factors (Clause 6.4.2, IS 1893):1. Zone factor (Z): It is a factor to obtain design spectrum taking into account the locationof the building. The country is pided into 4 different zones according to the likelihoodof severe earthquakes to occur (maximum considered earthquake). Z values for differentzones are given below in Table 1. Seismic zone V indicates a zone with high seismic activity where the chances of a majorearthquake occurring is significant. This factor Z is formaximumconsidered earthquake.2. Importance factor (I ): This factor assumes a value of 1 and 1.5, the latter being forstructures of higher importance.3. Response Reduction factor (R): Response reduction factor depends on the allowablesystem ductility and represents the ratio of maximum seismic force on a structure duringa specified ground motion (if it were to remain elastic) to the design seismic force. Thus,actual seismic forces are reduced by a factor R to obtain design forces. In the presentstudy, all the structures have been analysed with an R value of 5.The intent of R is to simplify the structural design process such that only linear elas-tic static analysis is needed for building design. Such a design philosophy implies thatstructural inelastic behaviour is expected. The three factors contributing towards R aresystem overstrength, structural redundancy, and system ductility.However, there is no clear demarcation of the contribution of each factor in R.As a result,a system with low ductility but high overstrength (+ redundancy) and a system with highductility and a low overstrength (+ redundancy) would have the same value of R.Foraportal framed structure, reduction of response due to redundancy becomes insignificant.Assuming a symmetric structure, it is clear that the loadswould be symmetric aswell. So,hinges at different locationswould formsimultaneously and hence structural redundancybecomes almost zero. IS code does not address this. Also, the present R factor is notdependent on the natural period of the structure and does not address the redistributionof forces once non-linearity sets in. Apparently, the reduction of the design forces due toR is compensated by factor of safety imposed on the structure while designing its com-ponents. The most important source of uncertainty in seismic performance evaluation ofstructures is in ground motion itself (for example the structure in Sect. 3.3). Although, ingeneral, greater importance should be given to ductility based reduction than to systemoverstrength, this is beyond the scope of present study.4. Structural Response factor (Sa/g): It is a factor that determines the acceleration responsespectrum of the structure subjected to earthquake ground vibration and depends on thenatural period of vibration and damping of the structure (clause 6.4.5 IS 1893:2002).Once these factors are known Ah can be evaluated as:Ah = ZISa2Rg(3)Equation (2) can then be used to find the total base shear, Vb. As seen in Eq. (3), the zonefactor Z is further pided by 2 for reducing the maximum considered earthquake to designbasis earthquake (Jain and Murthy 2005). This reduction is based on the assumption that thestructure would not be subjected to the maximum considered earthquake during its lifetime.