Understanding the concepts on Degrees of Freedom

Question 1:

A five-story structure with 5 translational DOF is shown here. Each story 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.

SOLUTION:

As given in the question,

CALCULATING Mn: (Modal Mass)

The expression for computing value of Mn is as follows:

Now, by substituting the relevant values in the above formula we get the following results for each mode shape.

Using the same approach executed above, the following values for Mn related to every mode shape were obtained.

M2 = 3.861M

M3 = 3.85M

M4 = 3.86M

M5 = 3.86M

CALCULATING Ln:

As we know that the formula for obtaining Ln is given as:

Substituting the relevant values for each modal shape, we get:

Now, following the same approach executed above for the remaining modes, we get the following values for the Ln corresponding to the respective modal shape.

Ln2 = -1.297

Ln3 = 0.687

Ln4 = -0.381

Ln5 = 0.175

CALCULATING Mn*: (Effective Modal Mass)

The formula for calculating Mn* goes as follows,

Substituting the relevant values for mode 1, we get:

M1* = Ln1^2/M1 = (4.121M)^2/3.86M = 4.39

Similarly,

M2* = 0.43

M3* = 0.12

M4* = 0.03

M5* = 0.008

CALCULATING THE MODAL PARTICIPATION RATIO:

The modal participation ratio = Mn*/Mj

here, Mj = total lumped mass present in the structure which is equal to 5 in the given five-story frame.

Therefore, the following values were obtained for the respective modal shapes,

The ratio for mode 1 = 4.39 / 5 = 0.88

Similarly,

For mode 2 = 0.086

For mode 3 = 0.024

For mode 4 = 0.006

For mode 5 = 0.001

CALCULATING MODAL PARTICIPATION FACTOR: Γ_n

The formula goes as follows for calculating Γ_n,

Substituting the relevant values for the first mode shape, we get:

Γ1 = L1/M1 = 4.121M/3.86M = 1.067

Similarly,

Γ2 = -0.335

Γ3 = 0.177

Γ4 = -0.098

Γ5 = 0.045

CALCULATING THE PSEUDO ACCELERATION: An

The relative value for An can be obtained from the graphical representation given in the question,

From the above graph, we can compute the following values for the pseudo acceleration for each mode deflection.

A1 = 0.3

A2 = 0.76

A3 = 1.03

A4 = 1.03

A5 = 1.03

CALCULATING THE Vbn:

The formula is,

Substituting the relevant values for each mode, we get the following results:

Vb1 = (L1^2/M1)A1 = M1*.A1 = 4.39 x 0.3 = 1.317

Similarly,

Vb2 = 0.327

Vb3 = 0.124

Vb4 = 0.061

Vb5 = 0.008

OBTAINING THE TOTAL RELATIVE BASE SHEAR FOR THE ENTIRE STRUCTURE:

As we have got five different base shear values relating to each modal deflection. So, we need to combine it into a single relevant applicable base shear force. And the same is done by applying the SRSS rule.

Therefore, by applying the SRSS rule,

= 1.364

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

As indicated by the modal participation ratio calculated in one of the previous sections, the modal participation ratio for the first mode is 0.88 which means it accounts for 88% of the total deflection taking place.

Similarly,

The modal participation ratio for the second mode is 0.086 = 8.6%

Thus, when we combine both these values, we get:

88 + 8.6 = 96.6 > 90%

Hence, we only need to consider the first two modes in order to obtain the modal participating ratio of more than 90%.