A rigid beam BCD$,$ shown in Figure $2,$ is supported by a pin at C and $2$ identical cables $($length $=$

$L,$ diameter $=d),$ at $B$ and $D.$ The cable at $B$ is perpendicular to the beam, whereas the cable at $D$

is inclined $$ from the vertical direction. The beam is loaded with a uniform vertical distributed

load with intensity q $,$ which makes the beam rotate such that point D moves down $.$ It is also

known that the deformation of the cable ED leads to a stress in that cable of ODE, whereas the stress

in cable $AB({}_{AB})$ can be calculated as ${}_{AB}=\frac{{}_{DE}}{{}_{DE}}{}_{AB}.$

a$)$ If $={}_1,$ find the strains and changes in length in cables DE and AB using approximate

methods $($i$.$e$.$ using small$-$angle approximation$).$

b$)$ If $={}_1,$ find the value of $q$ needed to produce the strains and changes in length in part a$).$

c$)$ If $={}_2,$ find the strains and changes in length in cables DE and AB using approximate

methods $($i$.$e$.$ using small$-$angle approximation$).$

d$)$ If $={}_2,$ find the strains and changes in length in cables DE and AB using exact methods

Note: With exact methods, point D will not move directly downwards. Take $$ to be the

change between $D\text{'}$s final and initial positions.

Figure $2$: Rigid beam with cables and distributed load

Take $a=1\mathrm{.}4m,L=3m,d=7mm,$ and $=30.$

Take ${}_{DE}=1\mathrm{.}4$GPa,${}_1=21cm,{}_2=42cm.$