Which of the following relations is true ?

Question

With full explanation and steps....

Answer 1

Correct option is (4) 2G(1 + v) = Y

Relation b/w Y and G

\(\sigma_A=\tau sin2\theta\)

\(\sigma=\tau\times sin 2\times45° \)

\(\sigma = \tau\times sin 90° \)

\(\sigma=\tau\)

Principle stress along diagonal = \(\tau\)

e_BD = \(\frac{\tau}E+\frac{\mu.\tau}{E}\) 

Triangle strain along BD = \(\frac{\tau}E+\frac{\mu.\tau}E\) 

 = \(\frac{\tau}E(1+\mu)\)------(i)

Shear strain = θ = \(\frac{DD'}{AD}\) 

Triangle strain along BD = \(\frac{BD'-BD}{BD}\) 

 = \(\frac{BE+D'E-BE}{BD}\) 

 = \(\frac{DE'}{BD}\) 

\(\therefore\) Δ DD'E D'E = DD' x cos45°

 = DD' x 1/√2

\(\therefore\) BD = \(\sqrt{AB^2+AD^2}\) 

 = \(\sqrt{(AD)^2+(AD)^2}\) 

BD = √2AD

Triangle strain along BD = \(\cfrac{DD'\times\frac1{\sqrt2}}{\sqrt2 AD}\) 

 = \(\frac12\frac{DD'}{AD}\) = \(\frac12\theta\)------(ii)

equation (i) and (ii)

\(\frac{\tau}{e}(1+\mu)=\frac12\theta\)

\(\theta=\frac{\tau}c\)

\(\frac{\tau}E(1+\mu)=\frac12\frac\tau{c}\) 

2c (1 + \(\mu\)) = E

E replace Y

\(\mu\) replace v

c replace G

2G(1 + v) = Y