$20$ kN load is applied as shown to the horizontal member ABC $.$ Member ABC has a $10mm50mm$ uniform rectangular cross$-$section and is supported by four vertical links, each of $8mm36mm$ uniform rectangular cross$-$section. Each of the four pins at A$,$B$,$C and $D$ has the same diameter d$.$

A$.$ Solve this problem analytically.

B$.$ Write a Excel formulation to calculate for values of d from $10$ to $30$ mm $,$ using $1$ mm increments,

$(1)$ maximum value of the avarage normal stress in the links connecting to pins B and D$,$

$(2)$ the average normal stress in the links connecting to pins C and E $,$

$(3)$ the average shearing stress in pin B

$(4)$ the average shearing stress in pin $C$

$(5)$ the average bearing stress at B in member ABC

$(6)$ the average bearing stress at C in member ABC

C$.$ Check your formulation by comparing the values obtained for $d=16mm$

D$.$ Use this formulation to find the permissible values of the diameter d of the pins.

${}_{\text{a}\text{l}\text{l}\text{o}\text{w}\text{,}\text{n}\text{o}\text{r}\text{m}\text{a}\text{l}}=150$MPa

${}_{\text{a}\text{l}\text{l}\text{o}\text{w}\text{,}\text{s}\text{h}\text{e}\text{a}\text{r}}=90$MPa

${}_{\text{a}\text{l}\text{l}\text{o}\text{w}\text{,}\text{b}\text{e}\text{a}\text{r}\text{i}\text{n}\text{g}}=230$MPa

E$.$ Solve part D $,$ assuming that the thickness of member ABC has been reduced from $10$ to $8$ mm $.$