The equivalent building should comprise same number of floors and diaphragms. The columns of each floor should have specific dimensions, thus for each floor relative lateral and torsional stiffnesses are used.
Each diaphragm is replaced by an equivalent one consisting of 4 fixed-ended columns placed symmetrically with respect to its centre of stiffness.
The equivalent diaphragm i is assumed to behave as an one-storey structure of fixed-ended columns. The four relative translations of the centre of stiffness along X, Y axes δXXoΖ,i, δXYoΖ,i , δYXoΖ,i, δYYoΖ,i as well as the rotation angle θXZMZ,I are defined as functions of the actual building displacements as follow.
δXXoΖ,i= δXXo,i-δXXo,i-1 , δXYoΖ,i=δXYo,i-δXYo,i-1 ,
δYXoΖ,i= δYXo,i-δYXo,i-1, δYYoΖ,i=δYYo,i-δYYo,i-1,
θXZΜZ,i = θXΖΜ,i –θXZΜ,i-1.
Therefore, the data of the equivalent diaphragm are: rotation angle az,i=2δXYoΖ,i/( δXXoΖ,i- δYYoΖ,i),
torsional stiffness KθZ,i=H·cY/ θXZΜZ,i , lateral stiffnesses KxxZ,i= H/(δXXoZ,i+δXYoZ,i·tanaZ,i) και KyyΖ,ι=H/(δYYoZ,i-δXYoZ,i·tanaZ,i) (equations C.9.2 and C.9.3 of §C.9).
Also rxZ,i=√KθZ,i/KyyZ,i, ryZ,i=√KθZ,i/KxxZ,i
The location of the equivalent diaphragms is determined by translating the centre of stiffness to point 0.0, 0.0. In this way the equivalent building is created and for its diaphragms i the following relations apply:
δXXo_equal,i=Σ(δXXoΖ,k) όπου k=i έως 1 à δXXo_equal,i=Σ(δXXo,i-δXXo,i-1)= (δXXo,i-δXXo,i-1)+ (δXXo,i-1-δXXo,i-2)+…+ (δXXo,1-0.0)=δXXo,i à δXXo_equal,i=δXXo,i
Using the same logic δXYo_equal,i=δXYo,i , δYYo_equal,i=δYYo,i και θXZ_equal,i= θXZΜ,i
Therefore all diaphragm quantities Kxx_equal,i , Kyy_equal,i , Kθ_equal,i are equal to those of the actual building.
The seismic assessment of the actual building can be performed in an easy, direct and descriptive way by means of the equivalent building.