Question $2$

$[50/100$ marks$]$

For the two$-$storey reinforced concrete moment resisting frame given in Fig.Q$1,$ using the Type $1$ elastic design spectrum for the horizontal component of the PGA according to EC$8,$ you need to calculate:

a$)$ The maximum modal displacements

marks$]$

b$)$ The maximum displacement of the third floor, using any of the modal combination rules that you know $($give a brief explanation$).$

marks$]$

c$)$ The maximum modal force vectors.

marks$]$

d$)$ The maximum base shear $V_{b,\text{m}\text{a}\text{x}}$ acting on the structure.

marks$]$

e$)$ Check if in the first mode of vibration $P-$ effects need to be taken into account at the ground floor level.

marks$]$

The Mass and Stiffness matrices as well as the eigenvectors and the natural periods of vibration are given below:

$]|$

$[0\mathrm{.}0,1\mathrm{.}00|]|$

$[-1\mathrm{.}0,1\mathrm{.}0|]|$

$[1\mathrm{.}5,-1\mathrm{.}0|$

$T_1=0\mathrm{.}204s,T_2=0\mathrm{.}083s$

Basic Data:

The supports are fully fixed

The beam to column joints are consider to be rigid $($No rotations$).$

The columns are axially rigid.

The moment of inertia of the beams is infinite $I_b=.$

PAPER CODE: CEN$308/22-23/$S$2$ Final Examination

Page $3$ of $8$

Assume that the mass of each storey is lumped at the center of mass of the floors.

Assume soil type B

The reference peak ground acceleration is $0\mathrm{.}245g$

The structure's importance category is III

Assume medium ductility class and regularity both in plan and in elevation

$-g=10\frac{m}{s^2}$

$L=4\mathrm{.}5m$

$I_c=2\mathrm{.}13**{10}^{-3}{m}^4$