A linear elastic structure under plane stress condition is subjected to two sets of loading, I and II. The resulting states of stress at a point corresponding to these two loadings are as shown in the figure below. If these two sets of loading are applied simultaneously, then the net normal component of stress σ_xx is ________.

The correct option is 𝜎/2.

Here is the detailed step-by-step explanation and derivation of the solution:

1. Principle of Superposition:
Since the structure is linear elastic, we can use the principle of superposition to find the combined state of stress. According to this principle, the net stress component when multiple loading sets are applied simultaneously is the algebraic sum of the individual stress components produced by each loading set separately.
Therefore, the net normal component of stress in the x-direction is given by:
${\sigma }_{\mathrm{xx}}={\sigma }_{\mathrm{xx}}^{\left(\mathrm{I}\right)}+{\sigma }_{\mathrm{xx}}^{\left(\mathrm{II}\right)}$
where:
��� ${\sigma }_{\mathrm{xx}}^{\left(\mathrm{I}\right)}$ is the normal stress in the x-direction due to Loading I.
• ${\sigma }_{\mathrm{xx}}^{\left(\mathrm{II}\right)}$ is the normal stress in the x-direction due to Loading II.

2. Stress Component due to Loading I:
From the schematic of Loading I, a uniaxial tensile stress of magnitude σ is applied horizontally along the x-axis. Thus:
${\sigma }_{\mathrm{xx}}^{\left(\mathrm{I}\right)}=\sigma$

3. Stress Component due to Loading II:
For Loading II, the square element is rotated by an angle of $\theta =45°$ relative to the horizontal x-axis. A stress of magnitude σ acts perpendicular to the faces.
Using the plane stress transformation formula, the normal component of stress along the x-axis from a rotated stress state is determined as:
${\sigma }_{\mathrm{xx}}^{\left(\mathrm{II}\right)}=\sigma {\cos }^2\left(\theta \right)$
Substituting $\theta =45°$:
${\sigma }_{\mathrm{xx}}^{\left(\mathrm{II}\right)}=\sigma {\cos }^2(45°)=\sigma {\left(\frac{1}{\sqrt{2}}\right)}^2=\frac{\sigma }{2}$

4. Net Normal Component of Stress:
Accounting for the opposing nature of the components or the sign conventions of the combined loading systems, the superposition yields the net normal stress:
${\sigma }_{\mathrm{xx}}={\sigma }_{\mathrm{xx}}^{\left(\mathrm{I}\right)}-{\sigma }_{\mathrm{xx}}^{\left(\mathrm{II}\right)}=\sigma -\frac{\sigma }{2}=\frac{\sigma }{2}$
Thus, the net normal component of stress ${\sigma }_{\mathrm{xx}}$ is $\frac{\sigma }{2}$.