Calculation of Stresses:

Letting the assumption that their longitudinal strain is constant, we may write,

σl/ E - v  (σ_h / E )+ v( σ_r/ E) = constant

Since long longitudinal strain is constant, constant. Therefore, grouping all the constant terms to the right hand side, we might write,

σ_h  - σ_r  = some constant, say 2a

Substituting for σ_h,

-σ _r - r (d σ_r /  dr) - σ_r = 2a

 or,

- r (d σ_r/ dr)  = 2σ _r + 2a = 2 (σ_r + a)

i.e.

 d σr/ dr - 2 (σr  + a)/             r

i.e.

d σ_r/ (σ_r  + a) =- 2 dr/r

Integrating,     ln (σ_r + a) = - 2 ln r + constant

Taking the constant as ln b,

ln (σ_r + a) = ln b - 2 ln r

or,  ln (σr + a) = ln b - ln r ²

 i.e. ln (σ_r + a) = ln  (b /r₂)

Thus, σ_r + a =  b /r²

or,

σ_r =  (b/r²  ) - a

But, we know that σ_h - σ_r = 2a, thus σ_h = σ_r + 2a.

Thus,

σ_h =   b/r²  + a

These expressions for the radial stress and hoop stress are called as Lames expressions. The constants a and b shall be found from the known external pressure & internal pressure.