Task:

Consider the two$-$storey shear frame shown in the figure.

The following characteristics are provided:

Mass matrix $m=m([2,0],[0,1])$

Stiffness matrix: $k=\frac{24EI}{h^3}([3,-1],[-1,1])$

The natural frequencies: ${}_1=3\mathrm{.}464\sqrt[\phantom{2}]{\frac{EI}{m{h}^3}},{}_2=6\mathrm{.}928\sqrt[\phantom{2}]{\frac{EI}{m{h}^3}}$

The natural mode shapes: ${}_1=\{[1],[2]\},{}_2=\{[1],[-1]\}$

The peak response of the two$-$storey shear frame to a horizontal ground motion characterised by

the design spectrum shown in the figure which is scaled to $0\mathrm{.}5$ g peak ground acceleration.

$0T{T}_B,A=g*(1+\frac{T}{T_B}1\mathrm{.}5)$

$T_{B}T{T}_C,A=g*2\mathrm{.}5$

$T_{C}T{T}_D,A=g*25\frac{T_C}{T}$

$T_B=0\mathrm{.}15s_1$

$T_C=0\mathrm{.}40s$

$T_D=3\mathrm{.}0s$

Assume, $m={10}^{5}kg,E=210{10}^8\frac{N}{m^2},I=4\mathrm{.}2{10}^{-4}{m}^4,h=4m$

You are required to:

a$)$ Determine the influence vector.

b$)$ Find the peak pseudo accelerations $A_{sp,1}$ and $A_{sp,2}.$

c$)$ Determine the peak modal accelerations $A_1$ and $A_2.$

d$)$ Calculate the peak lateral displacements of the frame, $u_{1,\text{m}\text{a}\text{x}}$ and $u_{2,\text{m}\text{a}\text{x}}-$

e$)$ Find the shear forces at both floors of the frame using the SRSS combination rule.

f$)$ Determine the bending moments at both floors of the frame using the SRSS combination

rule.