At a critical point in a component, the state of stress is given as σ_xx = 100 MPa, σ_yy = 220 MPa, σxy = σ_yx = 80 MPa and all other stress components are zero. The yield strength of the material is 468 MPa. The factor of safety on the basis of maximum shear stress theory is _______ (round off to one decimal place).

The correct answer is 1.8.

Step 1: Identify the given stress components
The state of stress at the critical point of the component is given by the following non-zero components:

${\sigma }_{\mathrm{xx}}=100MPa$

${\sigma }_{\mathrm{yy}}=220MPa$

${\sigma }_{\mathrm{xy}}={\sigma }_{\mathrm{yx}}=80MPa$

The yield strength of the material is:

$S_y=468MPa$

Step 2: Calculate the principal stresses in the plane
The in-plane principal stresses (${\sigma }_1$ and ${\sigma }_2$) are determined using the principal stress formula:

${\sigma }_{1,2}=\frac{{\sigma }_{\mathrm{xx}}+{\sigma }_{\mathrm{yy}}}{2}\pm \sqrt{{\left(\frac{{\sigma }_{\mathrm{xx}}-{\sigma }_{\mathrm{yy}}}{2}\right)}^2+{\sigma }_{\mathrm{xy}}^2}$

Substituting the given values:

${\sigma }_{1,2}=\frac{100+220}{2}\pm \sqrt{{\left(\frac{100-220}{2}\right)}^2+{80}^2}$

${\sigma }_{1,2}=160\pm \sqrt{{\left(-60\right)}^2+{80}^2}$

${\sigma }_{1,2}=160\pm \sqrt{3600+6400}$

${\sigma }_{1,2}=160\pm \sqrt{10000}$

${\sigma }_{1,2}=160\pm 100$

Therefore, the two non-zero principal stresses are:

${\sigma }_1=260MPa$

${\sigma }_2=60MPa$

Step 3: Identify the absolute maximum and minimum principal stresses
Since all other stress components are zero, the third principal stress is:

${\sigma }_3=0MPa$

Comparing the three principal stresses, we find:

${\sigma }_{\max }=260MPa$

${\sigma }_{\min }=0MPa$

Step 4: Calculate the Factor of Safety using Maximum Shear Stress Theory
According to the Maximum Shear Stress (Tresca) Theory, the yield condition is given by:

${\tau }_{\max }=\frac{S_y}{2\cdot \text{FOS}}$

The absolute maximum shear stress is:

${\tau }_{\max }=\frac{{\sigma }_{\max }-{\sigma }_{\min }}{2}=\frac{260-0}{2}=130MPa$

Hence, the factor of safety (FOS) is calculated as:

$\text{FOS}=\frac{S_y}{{\sigma }_{\max }-{\sigma }_{\min }}=\frac{468}{260-0}=1.8$