Problem $2.$ For the give beam section, determine the following:

a$)$ The value of $b$ that places the plastic neutral axis $(y_p)$ at the mid$-$height of the section. $(5$ points$)$

b$)$ The maximum value of $b$ that keeps the plastic neutral axis $(y_p)$ within the top flange of the I$-$section. $(5$ points$)$

c$)$ The minimum value of $b$ that keeps the plastic neutral axis $(y_p)$ within the bottom flange of the I$-$section. $(5$ points$)$

d$)$ The value of $b$ such that the top elastic section modulus is twice the bottom elastic section modulus. $(5$ points$)$

e$)$ The value of $b$ where $y_p\frac{?}{b}=ar(y)($the centroidal axis$).$ In this case, compute the elastic and plastic section moduli, yield moment, plastic moment, and shape factor. Use ASTM A$572$ steel $(15$ points$)$

$($Note: it is recommended to derive the mathematical relationships between $b$ and $y_p,$ as well as $b$ and $\frac{?}{\text{b}\text{a}\text{r}}(y)).$

b

$($Using SAP $2000)$

Problem $3.$ The frame given in the figure below has fixed supports a A and D$.$ Connection at B is rigid while C is a pin connection. All frame members are HSS $1410\frac{1}{2}.$ Use American ASTM A$500$ Grade B steel with modulus of elasticity equals to $210$ GPa $.$ The loads applied to the frame are $P_{D1}=50KN,P_{D2}=40KN,P_{L1}=30KN,P_{L2}=60KN,W_D=20K\frac{N}{m},W_L=5K\frac{N}{m}.$

a$)$ Use the default load combinations generated by SAP $2000,$ draw the axial, shear, and bending moment diagrams for the frame using dead load only and $1\mathrm{.}2D+1\mathrm{.}6L$ combination. $(6$ points$)$

b$)$ Find the value and loacation of the maximum axial force, shear, and moment. $(6$ points$)$

c$)$ Compute the lateral displacment of joint $B$ from all load combinations and service loads $($organize them in a table$).(6$ points$)$

d$)$ Why meshing is important in SAP $2000?(5$ points$)$