Understanding the concepts on Degrees of Freedom
Answer
(1) To find the parameters for each modes
Let the mass matrix (m) =`[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]]`
The fundamental period of vibration (T1 = 2 seconds) and its associated mode shape
`phi_1= ((0.334),(0.641),(0.895),(1.078),(1.173))`
(i) Modified Mass matrix
`M_1 =phi_n^T.m.phi_n`
= `(0.334,0.641,0.895,1.078,1.173).[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]].((0.334),(0.641),(0.895),(1.078),(1.173))``
`M_1=3.861m`
`(ii) Modal Property (L_n)`
`L_1=phi_n^T*m*L`
Let Assume L = 1
`[ 0.334,0.641,0.895,1.078,1.173] ((m,0,0,0,0),(0,m,0,0,0),(0,0,m,0,0),(0,0,0,m,0),(0,0,0,0,m))*((1),(1),(1),(1),(1))`
`L_1 = 4.121 m`
(iii) Modal Participation Factor
`gamma_1 = L_1/M_1 = 4.121m/3.861m=7.982`
(iv) Effective modal mass
`M_1^ast=L_n^2/M_1=(4.121m)^2/(3.861m)=4.398m`
(v) Effective modal mass partification factor
`m_(j1)= M_1^**/m_j`
`m_j=` Total mass of the building = 5m
Therefore `m_(j1)=(4.398m)/(5m) = 0.88`
Similarily for the the various modes
The fundamental period of vibration (T2 = 0.685 seconds) and its associated mode shape
`phi_2= ((-0.895),(-1.173),(-0.641),(0.334),(1.078))`
(i) Modified Mass matrix
`M_2 =phi_n^T.m.phi_n`
= `(-0.895,-1.173,-0.641,0.334,1.078).[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]].((-0.895),(-1.173),(-0.641),(0.334),(1.078))``
`M_2=3.861m`
`(ii) Modal Property (L_n)`
`L_1=phi_n^T*m*L`
Let Assume L = 1
`[ -0.895,-1.173,-0.641,0.334,1.078] ((m,0,0,0,0),(0,m,0,0,0),(0,0,m,0,0),(0,0,0,m,0),(0,0,0,0,m))*((1),(1),(1),(1),(1))`
`L_2 = -1.297 m`
(iii) Modal Participation Factor
`gamma_2 = L_2/M_2 = -1.297m/3.861m=-0.335`
(iv) Effective modal mass
`M_2^ast=L_n^2/M_1=(-1.297m)^2/(3.861m)=0.436m`
(v) Effective modal mass partification factor
`m_(j2)= M_2^**/m_j`
`m_j=` Total mass of the building = 5m
Therefore `m_(j2)=(0.436m)/(5m) = 0.087`
Similarly by doing for each we get
The fundamental period of vibration (T3 = 0.434 seconds)
`M_3=3.861m`
`L_3=0.685m`
`gamma_3=0.177`
`M_3^**=0.121m`
`m_(j3)=0.024`
The fundamental period of vibration (T4 = 0.338 seconds)
`M_4=3.861m`
`L_4=-0.38m`
`gamma_4=-0.0984`
`M_4^**=0.037m`
`m_(j4)=7.4**10^(_3)`
The fundamental period of vibration (T5 = 0.2966 seconds)
`M_5=3.861m`
`L_5=0.173m`
`gamma_5=0.044`
`M_5^**=7.75**10^-3m`
`m_(j5)=1.55**10^(_3)`
(2) To find the base shear of each modes
From graph the pseudo spectral accelaration shall be evaluated as
For `T_1= 2 sec, therefore A_(n1)= 0.27g`
Therefore the Base shear at mode 1
`V_(B1)= M_1^** *A_(n1)= 4.398m **0.27 g= 1.189mg`
Similarily for the other base shear with respect to the value time period
For `T_2=0.685 sec`
`V_(B2)= M_2^** *A_(n2)= 0.436m **0.75 g= 0.327mg`
For `T_3=0.434 sec`
`V_(B3)= M_3^** *A_(n3)= 0.121m **1 g= 0.121mg`
For `T_4=0.338 sec`
`V_(B4)= M_4^** *A_(n4)= 0.037m **1 g= 0.037mg`
For `T_5=0.2966 sec`
`V_(B5)= M_5^** *A_(n5)= 7.75**10^(-3)m **1 g= 7.75**10^(-3)mg`
(3) How many modes are to be considered such that a modal mass participating ratio of 90% is obtained?
The summation of modal mass participation ratio we get the percentage
therefore by adding the first two modes 0.88+0.087= 0.967 which is above than the 90% which we required
Therefore first two modes itself is enough