$2\mathrm{.}11$ EXAMPLE $-$ FORCED VIBRATION $-$ STEADY STATE

A horizontal dynamic force $P(t)=50sin28t,(kN)$ is applied to joint $A$ of the

portal frame shown below:

Given:

Size of the columns: $500mm500mm$

Damping ratio: $5\%$

The girder supports a mass of $20$ tones

Modulus of elasticity $=30$GPa

The beams $($floor system$)$ is rigid

$($infinitely stiff$).$

The axial deformation in columns can be neglected and assume that the frame

remains in the elastic region.

Stiffness of the columns $=\frac{12EI}{L^3}$

$u_{sS}(t)=\frac{(u_{st}{)}_{o}sin(t-)}{(1-{}^2{)}^2+(2{)}^2},$;$,=ta{n}^{-1}(\frac{2}{1-{}^2})$;$=\frac{}{{}_n}$

$R_d=\frac{1}{(1-{}^2{)}^2+(2{)}^2}$

Required:

Compute the lateral stiffness of each column

Compute the lateral stiffness of the frame

Compute the natural angular frequency

Compute the natural cyclic frequency

Compute the natural period

Compute the static deflection

Compute the frequency ratio $()-1$

Find the dynamic response factor $(R_d)$

Find the steady state amplitude of the frame vibration

Develop the steady state equation of motion

Find the steady state displacement response at time $t=1\mathrm{.}5$ seconds.

Plot the steady state response of the frame in the first $2$ seconds

Compute the maximum dynamic base shear of the frame