Understanding the concepts on Degrees of Freedom

Aim

Is to Understanding the concepts on Degrees of Freedom by solving the problem sum given as the challenge.

In questoion given values are each storey has mass ‘m’ and the 5 periods of vibration T_n and their corresponding mode shapes, φ_n.

Ans 1

fundamental period of vibration (T₁ = 2 seconds)

_{all the steps are to followed is same so its mention below so i skip some of the steps.}

_{second mode of vibration (T2 = 0.6852 seconds)}

_{now calculating for Ln as formula mention in ques.}

_{Now calculating for Γn as formula mention in ques.}

calculate the effective modal mass M_n* and the modal mass participating ratio for the 5 modes.

Note; percentage figures are rounded up with one unit place.

Calculating the base shear for each of the 5 modes of vibration and the combined base shear as per SRSS rule.

Estimating how many modes are to be considered such that a modal mass participating ratio of 90% is obtained.

_Result

We have calculated the effective modal mass Mn for all the modes considering each levels.

Then we have obtain Ln for the same as the formula mention in the question.

By obtaining so we calculate the modal mass participation ratio then we go through the base shear calculation for each modes as per SRSS rule.

Then we considered such that a modal mass participating ratio of 90% is obtained which is shown in calculation.