A structural member under loading has a uniform state of plane stress which in usual notations is given by σ_x = 3P, σ_y = -2P and τ_xy = √2 P, where P > 0. The yield strength of the material is 350 MPa. If the member is designed using the maximum distortion energy theory, then the value of P at which yielding starts (according to the maximum distortion energy theory) is

The correct option is 70 MPa.

To determine the value of $P$ at which yielding starts according to the maximum distortion energy theory (also known as the Von Mises yield criterion), we begin by identifying the given state of plane stress:
$${\sigma }_x=3P$$
$${\sigma }_y=-2P$$
$${\tau }_{\mathrm{xy}}=\sqrt{2}P$$
The yield strength of the material under uniaxial tension is:
$${\sigma }_y=350\text{ MPa}$$

According to the Maximum Distortion Energy Theory, yielding occurs when the equivalent Von Mises stress (${\sigma }_v$) reaches the yield strength of the material. For a state of plane stress, the Von Mises stress is defined as:
$${\sigma }_v=\sqrt{{\sigma }_x^2+{\sigma }_y^2-{\sigma }_x{\sigma }_y+3{\tau }_{\mathrm{xy}}^2}$$

Substitute the given stress values in terms of $P$ into the equation:
$${\sigma }_v=\sqrt{{\left(3P\right)}^2+{(-2P)}^2-\left(3P\right)(-2P)+3{\left(\sqrt{2}P\right)}^2}$$

Simplify each term inside the square root:
$${\sigma }_v=\sqrt{9P^2+4P^2+6P^2+3\left(2P^2\right)}$$
$${\sigma }_v=\sqrt{9P^2+4P^2+6P^2+6P^2}$$
$${\sigma }_v=\sqrt{25P^2}$$
Since $P>0$, this simplifies to:
$${\sigma }_v=5P$$

To find the value of $P$ at the start of yielding, we equate the Von Mises stress to the material's yield strength:
$${\sigma }_v=S_{\mathrm{yt}}$$
$$5P=350\text{ MPa}$$
$$P=\frac{350}{5}=70\text{ MPa}$$

Thus, yielding will initiate when the parameter $P$ reaches 70 MPa.