Understanding the concepts on Degrees of Freedom

AIM:- Understanding the concepts on Degrees of Freedom

INTRODUCTION:-

• A degree of freedom is the number of possible movements of a structural system. The degrees of freedom can be used to describe displacements and rotations at a nodal point. Thus, each degree of freedom allows for a displacement or a rotation in a certain direction.
• In this project we have study the details conceptual degree of freedom in structural analysis

Question 1:

A five storey structure with 5 translational DOF is shown here. Each storey has mass ‘m’. The eigenvalue problem has been solved and the 5 periods of vibration T_n and their corresponding mode shapes, φ_n has been derived.

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):}}`

The second mode of vibration (T₂ = 0.6852 seconds) and its associated mode shape:

`Phi_2={(-0.895),(-1.173),(-0.641),(0.334),(1.078):}}`

The third mode of vibration (T₃ = 0.4346 seconds) and its associated mode shape:

`Phi_3={(1.173),(0.334),(-1.078),(-0.641),(0.895):}}`

The fourth mode of vibration (T₄ = 0.3383 seconds) and its associated mode shape:

`Phi_4={(-1.078),(0.895),(0.334),(-1.173),(0.641):}}`

The fifth mode of vibration (T₄ = 0.2966 seconds) and its associated mode shape:

`Phi_5={(0.641),(-1.078),(1.173),(-0.895),(0.334):}}`

• L_n , M_n and Γ_n for each of the five modes are to be computed from Equations below.
• Also, calculate the effective modal mass M_n* and the modal mass participating ratio for the 5 modes. The mass matrix will be required to evaluate these parameters.

• Finally, obtain the base shear for each of the 5 modes of vibration and the combined base shear as per SRSS rule. The pseudo acceleration design spectrum to obtain the base shear for each mode is provided below.

• How many modes are to be considered such that a modal mass participating ratio of 90% is obtained?

INTRODUCTION:

• Modal effective mass is a modal dynamic property of a structure associated with the modal characteristics; natural frequencies, mode shapes, generalised masses, and participation factors.
• Modal participation factors - Mass participation ratio specifies the percentage of how much the structural mass of the model is participating for a given direction and mode.modal participation factors are scalars that measure the interaction between the modes and the directional excitation in a given reference frame. Larger values indicate a stronger contribution to the dynamic response.
• Effective mass factors - the effective mass factors associated with each mode represents the amount of system mass participating in that mode in a given excitation direction. This value is given as a percentage of the total system mass. Therefore, a mode with a large effective mass will be a significant contributor to the system’s response in the given excitation direction. A common rule of thumb for linear dynamic analysis is that a mode should be included if it contributes more than 1-2% of the total effective mass.
• Base shear is an estimate of the maximum expected lateral force on the base of the structure due to seismic activity.

PROCEDURE:-

a) Modified mass matrix;

Mn">`M_n= Φ_n^TmΦ_n`ΦnTmΦn">

Mode 1:

`M1=(0.334,0.641,0.895,1.078,1.173)(M) *{(0.334),(0.641),(0.895),(1.078),(1.173):}}``)`

`= M x(0.334^2+0.641^2+0.895^2+1.0782+1.173^2)`(0.3342+0.6412+0.8952+1.0782+1.1732)">

`M1= 3.891 M`

Mode 2:

``M2=(-0.895,-1.173,-0.641,0.334,1.078)(M) *{(-0.895),(-1.173),(-0.641),(0.334),(1.078):}}``

`= M x ((−0.895^2)+(−1.173^2)+(−0.691^2)+(−0.334^2)+(−1.078^2))`(-(0.8952)+(-1.1732)+(-0.6912)+(-0.3342)+(-1.0782))">

`M2= 3.861 M`

Mode 3:

`M3=(1.173,0.334,-1.078,-0.641,0.895)(M) *{(1.173),(0.334),(-1.078),(-0.641),(0.895):}}``

`=M *((1.173^2)+(0.334^2)+(−1.078^2)+(−0.641^2)+(0.895^2))`((1.1732)+(0.3342)+(-1.0782)+(-0.6412)+(0.8952))">

`M3= 3.85 M`

Mode 4:

`M4=(-1.078,0.895,0.334,-1.173,0.641)(M) *{(-1.078),(0.895),(0.334),(-1.173),(0.641):}}``

`=M *((-1.078^2)+(0.895^2)+(0.334^2)+(−1.173^2)+(0.641^2))`

`M4= 3.861 M`

Mode 5:

`M5=(0.641,-1.078,1.173,-0.895,0.334) (M) *{(0.641),(-1.078),(1.173),(-0.895),(0.334):}}``

`= M *((0.641^2)+(−1.078^2)+(1.173^2)+(−0.895^2)+(0.334^2)`

```M5= 3.861 M`

b) Modal property L;

`L_n=ϕ_n^Tml`Ln=ϕnTmi">

Where,l= influence vector(assuming 100% utilization of siesmic forces)=`[[1],[1],[1],[1],[1]]``

Mode 1:

`L1=(0.334,0.641,0.895,1.078,1.173)(M) * [[1],[1],[1],[1],[1]]`(11111)">

`=(0.334+0.641+0.895+1.078+1.173)M`

`L1=4.121 M`

Mode 2:

`L2=(-0.895,-1.173,-0.641,0.334,1.078)(M)* [[1],[1],[1],[1],[1]]``(11111)">

=`(-0.895-1.173-0.641+0.331+1.078)M`

`L2=1.298 M`

Mode 3:

`L3=(1.173,0.334,-1.078,-0.641,0.895)(M)* [[1],[1],[1],[1],[1]]```(11111)">

`=(1.173+0.334-1.078-0.641+0.895)M`

`L3=0.687``

Mode 4:

`L4=(-1.078,0.895,0.334,-1.173,0.641)(M) * [[1],[1],[1],[1],[1]]````(11111)">

`=(-1.078+0.895+0.334-1.173+0.641)M`

L4">`L4=-0.381 M`

Mode 5:

L5">`L5=(0.641,-1.078,1.173,-0.895,0.334)(M) * [[1],[1],[1],[1],[1]]`````(11111)">

`=(0.641-1.078+1.173-0.895+0.334)M`

`L5= 0.175 M`

c) Modal participation factor ;

`Gamma_n=L_n/M_n`∣n=LnMn">

Mode 1:

`Gamma_1=L_1/M_1=(4.121M)/(3.861M)=1.067`

Mode 2:

∣2=L2M2">`Gamma_2=L_2/M_2=(-1.298M)/(3.861M)=-0.33`

Mode 3:

`Gamma_3=L_3/M_3=(0.687M)/(3.85M)=0.178`

Mode 4:

`Gamma_4=L_4/M_4=(-0.381M)/(3.861M)=0.09`

Mode 5:

`Gamma_5=L_5/M_5=(0.175M)/(3.861M)=0.04`

d) Effective modal mass;

Mode 1:

M1⋅=L12M1">`M_1=L_1^2/M_1=(4.121M)^2/(3.861M)=4.39M`

Mode 2:

`M_2=L_2^2/M_2=(-0.298M)^2/(3.861M)=0.43M`

Mode 3:

`M_3=L_3^2/M_3=(0.687M)^2/(3.85M)=0.12M`

Mode 4:

M4⋅=L42M4">`M_4=L_4^2/M_4=(-0.381M)^2/(3.861M)=0.03M`

Mode 5:

M5⋅=L52M5">`M_5=L_5^2/M_5=(0.175M)^2/(3.861M)=0.007M`

e) Modal mass participating ratio

`=M_n/M_j`Mn⋅Mj">

Mj">Mj=Total mass of the building = M+M+M+M+M = 5M

Mode 1:

`M_1/M_j=(4.39M)/(5M)=0.88%`

Mode 2:

M2⋅Mj">`M_2/M_j=(0.43M)/(5M)=0.086%`

Mode 3:

M3⋅Mj">`M_3/M_j=(0.12M)/(5M)=2.4%`

Mode 4:

M4⋅Mj">`M_4/M_j=(0.03M)/(5M)=0.6%`

Mode 5:

M5⋅Mj">`M_5/M_j=(0.007M)/(5M)=0.1%`

• Adding first 2 mode mass participation ratio we get,

= 0.88+0.086= 96.6=96.6%

• To achieve minimum 90% of modal mass participation ratio,first 2 modes are enough

f) WKT, Vbn=Ln2MnAn=Mn⋅An">

Vbn=Ln2MnAn=Mn⋅An">`V_(bn)=L_n^2/(Mn)An=M_nA_n`

here,

T1">`T1= 2sec`, `A1=0.27g`

T2">`T2=0.6352sec`, A2">`A2=0.76g`

T3">`T3=0.4346sec`, A3">`A3=1.03g`

T4">`T4=0.3383sec`, A4">`A4=1.03g`

T5">`T5=0.2966sec`, A5">`A5=0.76g`

g) Base shear Vb

Mode 1:

`V_(b1)=M_1A_1=1.39M *0.27g=2.619kips`

Mode 2:

Vb2=M2⋅A2">`V_(b2)=M_2A_2=0.43M * 0.76g=0.719kips`

Mode 3:

Vb3=M3⋅A3">`V_(b3)=M_3A_3=0.12M * 1.03g=0.272kips`

Mode 4:

Vb4=M4⋅A4">`V_(b4)=M_4A_4V=0.03M * 1.03g = 0.067kips`

Mode 5:

Vb5=M5⋅A5">`V_(b5)=M_5A_5=0.007M * 0.76g = 0.011kips`

h) Average base shear is calculated by the method of square root of sum of square (srss)

`V_b=√V_(b1)^2+V_(b2)^2+V_(b1)^2+V_(b1)^2+V_(b1)^2`

`V_b=2.6132+0.7192+0.2722+0.0672+0.0112"`> `= 3.2 kips`

i) Model mass ration of 90% in obtained by 2 model

`((M_1)/(M_j))+ ((M_2)/(M_j)) =88% +8.6%=96.6%`

RESULT:-

• Calculated L_n , M_n and Γ_n for each of the five modes from the given equations.
• Calculated the effective modal mass M_n* and the modal mass participating ratio for the 5 modes.
• The base shear for each of the 5 modes of vibration are obtained.
• As per SRSS rule the combined base shear is obtained.
• The two modes are to be considered such that a modal mass participating ratio of obove 90% is obtained.