Understanding the concepts on Degrees of Freedom

AIM:

To compute `L_(n), M_(n) and T_(n)` for each of the five modes from the given equations

To calculate the effective modal mass `M_(n)^*` and the modal mass participating ratio for the 5 mode

To obtain the base shear for each of 5 modes of vibration and the combined base shear as per SRSS rule

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

The fundamental period of vibration (T₁ = 2 seconds) and its associated mode shape:

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

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

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

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

• L_n , M_n and Γ_n for each of the five modes are to be computed from Equations below.

• 

Taking first equation of Mn into consideration.

`M_(n)= phi_(n)^Tmphi_(n)`

`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]]`

SOLUTION

`M_(1)= phi_(1)^Tmphi_(1)`

`M_(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]]` `((0.334),(0.641),(0.895),(1.078),(1.173))`

`M_(1)`= 3.614m

Similarly

`M_(2)`= [-0.895 -1.173 -0.641 0.334 1.078]

`M_(2)` =`[[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

`M_(3)`= [1.173 0.334 -1.078 -0.641 0.895 ]

`[[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]]``((1.173),(0.334),(-1.078),(-0.641),(0.895))`

`M_(3)`= 3.861m

`M_(4)`= [-1.078 0.895 0.334 -1.173 0.641]

`[[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]]``((-1.078),(0.895),(0.334),(-1.173),(0.641))`

`M_(4)`= 3.861m

= [0.641 -1.078 1.173 -0.895 0.334 ]

`[[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]]``((0.641),(-1.078),(1.173),(-0.895),(0.334))`

`M_(5)`= 3.861m

`Ln= phi_(n)^Tml`

`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]]`

I = `((1),(1),(1),(1),(1))`

SOLUTION

`L_(1)= phi_(1)^Tml`

`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)`= 4.121m

Similarly,

`L_(2)`= -1.29 m

`L_(3)`= 0.68 m

`L_(4)`= 0.381 m

`L_(5)`=0.175 m

`T_(n)=L_(n)/M_(n)`

SOLUTION

`T_(1)=L_(1)/M_(1)`

4.121/3.8614

`T_(1)`= 1.067

Similarly:

`T_(2)`= -0.336

`T_(3)`= 0.177

`T_(4)`= 0.009

`T_(5)`= 0.045

`M_(n)^*=L_(n)^2/M_(n)`

Participating ratio= `M_(N)^*/M_(i)`

`M_(1)^*`= 4.398 m

Participating ratio= 0.8796

`M_(2)^*= 0.435m`

Participating ratio= 0.087

`M_(3)^*= 0.121m`

Participating ratio= 0.0242

`M_(4)^*= 0.037m`

Participating ratio= 7.4 x 10^-3

`M_(5)^*= 0.008m`

Participating ratio= 1.6 x 10^-3

• 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.

`T_(1)=2`

`A_(1)=0.27`g

`T_(2)=0.6852`

`A_(2)=0.76`g

`T_(3)=0.43446`

`A_(3)=1.03`g

`T_(4)=0.3383`

`A_(4)=1.03`g

`T_(5)=0.2966`

`A_(5)=1.03`g

`V_(bn)=M_(n)^*A_(n)`

SOLUTION:

`V_(b1)=M_(1)^*A_(1)`

`V_(b1)=1.1874m`

Similarly,

`V_(b2)=0.3306m`

`V_(b3)=0.12463m`

`V_(b4)=0.3811m`

`V_(b5)=`8.24 x 10^-3m

As per SRSS method:

=`V_(b)= root`(1.187)^2 + (0.3306)^2 + (0.12463)^2 + (0.3811)^2 + (0.00824)^2`

`V_(b)= 1.457m`

• Modal effective mass:

Modal mass participating ratio= `M_(n)^*/M_(j)`

`M_(n)^*=`effective modal mass`

`M_(j)`= Total mass = M+M+M+M+M=5M

Mode 1

`M_(1)^*/M_(j)`

= 4.39m/5m= 0.88

=88%

Mode 2:

`M_(2)^*/M_(j)`

0.43m/5m= 0.086

=8.6%

Mode 3

`M_(3)^*/M_(j)`

0.12m/5m= 0.024

= 2.4%

Mode 4

`M_(4)^*/M_(j)`

0.03m/5m=0.006

0.6%

Mode 5

`M_(5)^*/M_(j)`

0.007m/5m=0.001

=0.1%

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

THE TWO MODES ARE TO BE CONSIDERED SUCH THAT A MODEL MASS PARTICIPATING RATIO = 90% IS OBTAINED

`M_(1)^*/ M_(j) + M_(2)^*/ M_(j)`

= 88% + 8.6% = 96.6%

First two modes are enough for us to obtain 90% modal mass participating ratio

RESULT:

Calculated `L_(n), M_(n) and T_(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 considered such that a modal mass participating ratio of 90% is obtained