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AIM:- To obtain Ln, Mn and Γn for each of the five modes associated wih the model, calculate the effective modal mass M*n and the modal mass participating ratios for the five modes and also obtain the base shear as per SRSS rule. INTRODUCTION- All the solutions are going to be explained with step by step procedure.…
Sandeep Ghosh
updated on 02 Jul 2022
AIM:- To obtain Ln, Mn and Γn for each of the five modes associated wih the model, calculate the effective modal mass M*n and the modal mass participating ratios for the five modes and also obtain the base shear as per SRSS rule.
INTRODUCTION- All the solutions are going to be explained with step by step procedure.
SOLUTION:-
1.1 Calculation of Ln, Mn and Γn for each modes, effective modal mass and modal mass participating ratios
a) For first mode
ΦT1 = [ 0.334 0.641 0.895 1.078 1.173]
Mass matrix , m = [ [ m , 0 , 0, 0, 0 ] , [ 0 , m , 0 , 0, 0], [0,0,m,0,0], [ 0, 0, 0,0,m ,0 ], [ 0 , 0, 0, 0, m] ]
M1 = ΦT1 * m * Φ1 = 3.861 m
L1 = ΦT1 * m * I = [ 0.334, 0.641, 0.895, 1.078, 1.173 ] * m * [ [1], [1], [1], [1], [1] ] = 4. 121 m
Γ1 = L1 / M1 = 4.121 m / 3.861 m = 1.067
M*1 = L21/ M1 = 4.397 m
Modal mass participation ratios for first mode is given by = (4.397 m/ 5 m) * 100 = 87.94%
b) For Second mode
Mass matrix , m = [ [ m , 0 , 0, 0, 0 ] , [ 0 , m , 0 , 0, 0], [0,0,m,0,0], [ 0, 0, 0,0,m ,0 ], [ 0 , 0, 0, 0, m] ]
M2 = ΦT2 * m * Φ2 = 3.861 m
L2 = ΦT2 * m * I = - 1.3 m
Γ2 = L2 / M2 = -1.3 m/ 3.861 m = -0.336 m
M*2 = L22/ M2 = (1.3m)^2/3.861 m = 0.437 m
Modal mass participation ratios for second mode is given by = (0.437 m/ 5 m) * 100 = 8.74 %
c) For third mode
Mass matrix , m = [ [ m , 0 , 0, 0, 0 ] , [ 0 , m , 0 , 0, 0], [0,0,m,0,0], [ 0, 0, 0,0,m ,0 ], [ 0 , 0, 0, 0, m] ]
M3 = ΦT3 * m * Φ3 = 3.861 m
L3 = ΦT3 * m * I = [ 1.173, 0,334 , -1.078, -0.641, 0.895 ] * m * [ [1], [1], [1], [1], [1] ] = 0.683 m
Γ3 = L3 / M3 = 0.683 m/ 3.861 m = 0.177 m
M* 3 = L23/ M3 = (0.683m)^2/3.861 m = 0.12 m
Modal mass participation ratios for third mode is given by = (0.12 m/ 5 m) * 100 = 2.4 %
d) For fourth mode
Mass matrix , m = [ [ m , 0 , 0, 0, 0 ] , [ 0 , m , 0 , 0, 0], [0,0,m,0,0], [ 0, 0, 0,0,m ,0 ], [ 0 , 0, 0, 0, m] ]
M4 = ΦT4 * m * Φ4 = 3.864 m
L4 = ΦT4 * m * I = [-1.078, 0.895, 0.334, -1.173, 0.641] * m * [ [1], [1], [1], [1], [1] ] = -0.381 m
Γ4 = L4 / M4 = -0.381 m/ 3.864 m = -0.098 m
M*4 = L24/ M4 = (0.381m)^2/3.861 m = 0.037 m
Modal mass participation ratios for fourth mode is given by = (0.037 m/ 5 m) * 100 = 0.74 %
e) For fifth mode
Mass matrix , m = [ [ m , 0 , 0, 0, 0 ] , [ 0 , m , 0 , 0, 0], [0,0,m,0,0], [ 0, 0, 0,0,m ,0 ], [ 0 , 0, 0, 0, m] ]
M5 = ΦT5 * m * Φ5 = 3.861 m
L5 = ΦT5 * m * I = [0.641, -1.078, 1.173, -0.895, 0.334] * m * [ [1], [1], [1], [1], [1] ] = 0.175 m
Γ5 = L5 / M5 = 0.175 m/ 3.864 m = 0.045 m
M*5 = L25/ M5 = (0.175 m)^2/3.861 m = 7.931 * 10^-3 m
Modal mass participation ratios for fifth mode is given by = ( 7.931 * 10^-3 m/ 5 m) * 100 = 0.15 %
1.2 Calculation of base shear forces
Vb1 = M*1 * A1 = 4.397 m * 0.27g ( T1 = 2s) = 1.187 m*g =2.658 kips
Vb2 = M*2 * A2 = 0.437 m * 0.7g ( T2 = 0.6852 s) = 0.3059 m*g =0.685 kips
Vb3 = M*3* A3 = 0.12 m * 1.039 g ( T3 = 0.4346 s) = 0.279 kips
Vb4 = M*4* A4 = 0.037 m * 1.039 g ( T4 = 0.3383 s) = 0.085 kips
Vb5 = M*5* A5 = 7.931 * 10^-3 m * 0.76 g ( T5 = 0.2966 s) = 0.011 kips
1.3 Combined base shear force as per SRSS rule
Vb = Sq rt ( 2.658^2 + 0.685^2+ 0.279^2 + 0.085 ^2 + 0.011^2 ) = 3 kips
1.4 Modal mass participating ratios
Modal mass ratio is obtained in 2 model = (8.74+ 87.94 ) % = 96.68%
RESULT:- a) Ln, Mn and Γn for each of the five modes associated wih the model have been illustrated properly.
b) Effective modal mass have been illustrated properly
c) modal mass participating ratios for the five modes are also been obtained the base shear as per SRSS rule.
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