Understanding the concepts on Degrees of Freedom

1A)

Mass matrix m =

Modified mass matrix 

Model property  Ln         

For the first mode 

n=1

Modified mass matrix M1 =

m1=3.861m

Model property  L1 =

L1=4.121m

Model participation factor 

         Γ1=L1/M1

             =4.121m/3.861m

             =1.067

Effective modal mass 

                 M1* =(4.121m)^2/3.861m

                 M1*=4.398m

For the second mode 

n=2

Modified mass matrix M2 =

m2=3.861m

Model property  L2 =

L2=-1.293m

Model participation factor 

         Γ2=L2/M2

             =-1.293m/3.861m

             =-0.335

Effective modal mass 

                 M2* =(-1.293m)^2/3.861m

                 M2*=0.436m

For the third mode 

n=3

Modified mass matrix M3 =

m3=3.861m

Model property  L3 =

L3=0.6834m

Model participation factor 

         Γ3=L3/M3

             =-1.293m/3.861m

             =0.177

Effective modal mass 

                 M3* =(-1.293m)^2/3.861m

                 M3*=0.121m

For the fourth mode 

n=4

Modified mass matrix M4 =

m4=3.861m

Model property  L4 =

L4=-0.3783m

Model participation factor 

         Γ4=L4/M4

             =-0.3783m/3.861m

             =-0.098

Effective modal mass 

                 M4* =(-0.3783m)^2/3.861m

                 M4*=0.038m

For the fifth mode 

n=5

Modified mass matrix M5 =

m5=3.861m

Model property  L5 =

L5=0.1737m

Model participation factor 

         Γ5=L5/M5

             =0.1737m/3.861m

             =0.045

Effective modal mass 

                 M5* =(0.1737m)^2/3.861m

                 M5*=0.008m

Modal mass participation ratio= Mn*/ Mj

Mj=Total mass of building=m+m+m+m+m=5m

Modal mass participation ratio for first node (n=1)= M1*/ Mj         

                                                                              =4.398m/5m =.88

Modal mass participation ratio for second node (n=2)= M2*/ Mj         

                                                                              =0.436m/5m =.09

Modal mass participation ratio for third node (n=3)= M3*/ Mj         

                                                                              =0.121m/5m =.02

Modal mass participation ratio for fourth node (n=4)= M4*/ Mj         

                                                                              =0.038m/5m =.01

Modal mass participation ratio for fifth node (n=5)= M5*/ Mj         

                                                                              =0.008m/5m =.0002

Adding first two modal mass participation ratios we get, 0.88 + 0.09 = 0.97  i.e 96%

To achieve 90% first two modes are enough

Base Shear

Vbn = Mn* An

For first node 

From graph

A1=.27g

Vb1 = M1* A1

       =4.398m/.27g

       =1.187mg

        

For second node 

From graph

A2=.75g

Vb2 = M2* A2

       =0.436m/.75g

       =0.327mg

        

For third node 

From graph

A3=1.05g

Vb3 = M3* A3

       =0.121m/1.05g

       =0.126mg

For fourth node 

From graph

A4=1.02g

Vb4 = M4* A4

       =0.038m/1.02g

       =0.039mg

For fifth node 

From graph

A5=1g

Vb5 = M5* A5

       =0.008m/1g

       =0.008mg

                        

Average Base shear  Vb =square root of(Vb1^2+Vb2^2+Vb3^2+Vb4^2+Vb5^2)

                                        =square root of(1.187^2+0.327^2+0.126^2+0.039^2+0.008^2)

                                        =1.15mg