Question

In a two dimensional stress analysis, the state of stress at a point is shown in the figure
If $\sigma =120$ Mpa and $\tau =70$ MPa, then ${\sigma }_x$ and ${\sigma }_y$ are respectively

• A. 26.7 MPa and 172 MPa
• B. 54 MPa and 128 MPa
• C. 67.5 MPa and 213.3 MPa
• D. 16 MPa and 138 MPa

Answer 1

The correct option is C 67.5 MPa and 213.3 MPa

AB = 4
BC = 3
AC = 5
$cos\theta =\frac{4}{5}$

$sin\theta =\frac{3}{5}$

Normal stress , $\sigma =120$ MPa
Shear stress , $\tau =70$ MPa
$\Sigma F_x=0$
[$-{\sigma }_x\times Area+\sigma cos\theta \times Area-\tau sin\theta .Area$]=0

$-{\sigma }_x.(AB\times 1)+120\times \frac{4}{5}\times (AC\times 1)$
$-70\times \frac{3}{5}\times AC\times 1=0$
$-{\sigma }_x.4+96\times 5-42\times 5=0$
$\Rightarrow {\sigma }_x=67.5MPa$

$\Sigma F_y=0$
$[-{\sigma }_y\times BC+\sigma sin\theta \times AC+\tau cos\theta \times AC]=0$

$-{\sigma }_y.3+120\times \frac{3}{5}\times 5+70\times \frac{4}{5}\times 5=0$
$\Rightarrow {\sigma }_y=213.3MPa$