(Solved):   As shown in fig 1 . a thin-walled cylindrical pressure vessel \( \left(\sigma_{3} \approx ...

 

As shown in fig 1 . a thin-walled cylindrical pressure vessel \( \left(\sigma_{3} \approx 0\right) \) made of aluminum is subjected to an internal pressure \( p \). \( 113.1 .1111- \) v a1104uy1111411va1 Knowing that: \[ \begin{array}{l} p r^{2} \pi-\sigma_{1} 2 r \pi t=0 \\ \sigma_{2} 2 t d-p 2 r d=0 \end{array} \] 1- Write the expression of stress, in each direction ( \( \sigma_{1}, \sigma_{2} \), and \( \sigma_{3} \) directions), find \( \lambda \), with, \( \sigma_{1}=\lambda \sigma_{2}(10 \) points) 2- Write the expression of effective stress \( \bar{\sigma} \) (von Mises) as \( \bar{\sigma}=\mathrm{f}\left(\sigma_{2}\right) \) (10 points) 3- Write the expression of elastic and plastic strain \( \varepsilon_{2 e} \) and \( \varepsilon_{2 p} \) (10 points). 4- Since we have plane stress, write the equation for the strain \( \varepsilon_{2}(\Delta \mathrm{r} / \mathrm{r}) \) in the direction of \( \sigma_{2} \) as a function of \( \sigma_{2} \) and material constants E, H, and n from the uniaxial curve, consider the material follows the Ramberg-Osgood uniaxial stress-strain curve (10 points). 5- Calculate the strain \( \varepsilon_{2} \). Given: \( l=5 \mathrm{~m}, r=1 \mathrm{~m}, t=10 \mathrm{~mm} \), \( v=0.3 \) \( P=1 M P a \)., \( H=690, E=69 \mathrm{Gpa} \) and \( n=0.14 \). (10 points).