Two horizontal $5$ N forces are applied to pin B of the assembly shown. Each of the three pins at $A,B,$ and C has the same diameter d $.$

A$.$ Solve this problem analytically.

B$.$ Write a Excel formulation to calculate for values of d from $0\mathrm{.}50$ to $1\mathrm{.}50$ mm $,$ using $0\mathrm{.}05$ mm $.$ increments,

$(1)$ the maximum value of the average normal stress in member AB $,$

$(2)$ the average normal stress in member BC$,$

$(3)$ the average shearing stress in pin A $,$

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

$(5)$ the average bearing stress at $A$ in member $AB,$

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

$(7)$ the average bearing stress at B in member BC $.$

C$.$ Check your program by comparing the values obtained for $d=0\mathrm{.}8mm$

D$.$ Use this program to find the permissible values of the diameter $d$ of the pins, knowing that the allowable values of the normal, shearing, and bearing stresses for the steel used, are respectively, $22$MPa,$13$MPa, and $36$ MPa $.$

E$.$ Solve part D assuming that a new design is being investigated, in which the thickness and width of the two members are changed, respectively, form $0\mathrm{.}5$ to $0\mathrm{.}3$ mm and from $1\mathrm{.}8$ to $2\mathrm{.}4$ mm $.$