PSI - Issue 18

Bilal L. Khan et al. / Procedia Structural Integrity 18 (2019) 108–118 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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generally exist and are unmeasurable vector signals. The matrices A, B, C, and D for a structure are shown below. They are used as input parameters for state space analysis. More details on state space representation can be found in literature (Van Overschee & De Moor, 1994; Van Overschee & De Moor, 2012). A = [ − − ∗ − − ∗ ] (4) B = [ [ − ] [ ] ] (5) C = [ − − ∗ − − ∗ ] (6) D = [ [ − ] [ ] ] (7) In this paper, the response of three different base isolated systems differentiated on the basis of their effective damping (Low Damping Rubber bearing: LDRB 3% damping, High Damping Rubber bearing: HDRB 13% Damping, Lead Core Rubber Bearing: LCRB 25% damping) is analyzed. As the values of mass matrix, stiffness matrix and damping ratio matrix are known, we can calculate the response of structure under earthquake excitation by using equations (viii) and (ix) below. Sys = ss(A,B,C,D) (8) Y = lsim(Sys,u,t) (9) Using equation (ix), the acceleration, displacement and velocity at each storey are calculated and the results like interstorey drift, transmissibility ratio and peak displacement are compared. Following procedure is adopted for designing of base isolated structure. In this procedure time period of the structure is increased. Time period of base isolated assumed should be minimum 3 times the time period of fixed base structure (T iso / T fix ) min = 3 (Clemente & Buffarini, 2010). The time period of the fixed base structure calculated through state space approach is 1.1 seconds. Therefore, to de sign base isolation for this structure, the time period of the base isolated structure is taken as 3.5 seconds. The detailed steps are as follows: • First, calculate the natural time period and natural frequency of fixed base structure. In our case the time period is 1.1 seconds as mentioned above and natural frequency is 0.9 Hz. • Now set a value of time period of base isolated structure. The value should be at least three times greater than the time period of fixed base structure(Clemente & Buffarini, 2010). Select a time period of base isolated structure and calculate angular frequency of base isolation (Manarbek, 2013). • Use modal analysis and calculate ̃ , which is the equivalent modal stiffness and m ̃ , which is the equivalent modal mass. They are given by the following expressions (Manarbek, 2013): ̃ = (10) ̃ = (11) • Having established the numerical values for the modal mass and stiffness, we can select the appropriate stiffness coefficient so that the time period of the isolated structure would be more than 3 times greater than time period of fixed base. Calculate Γ using following formula: Γ = (12) • Γ is also given by following formula from which we calculate stiffness of base isolation, as we know the stiffness of the structure (Manarbek, 2013). 3.2. Design of Base Isolated System