I need help to check a calculation I did. I want to know if it is possible to use this method or if I am using an assumption which is wrong. Let me explain the problem, a beam with length $l$ is fastened in one end. A force $F$ an a moment $M_v$ is applied at the end of the beam, see figure below. The beam has a circular cross-section. Due to the force, the end of the beam will deform a length $\delta$. Only the deflection is known and the geometric parameters, such as the length and diameter.
Using Euler-Bernoulli beam theory the deflection can be expressed as:
$$\delta = \frac{Fl^3}{6EI} \tag{1}$$
Where $E$ is the Young's Modulus of the material and $I$ the inertia, which is $I=\frac{\pi d^4}{64}$ for a circular scross section. Here $d$ is the diameter of the beam.
Inserting the inertia in (1) and rearranging it as an expression of $F$ gives:
$$F = \frac{3 \delta \pi d^4 E}{32l^3} \tag{2}$$
This can be inserted in the general formula for maximum bending stress in a cross-section
$$\sigma_{max}= \frac{Fl}{\frac{\pi d^3}{32}} = \frac{32Fl}{\pi d^3} \tag{3}$$
Here the bending resistance for a circular-cross section has allready been inserted in the formula and the bending moment has been replaced for the maximum moment which is $Fl$.
This is the part which I am not so sure on, I use the force from (2) and insert it in (3) to get the maximum stress. Please let me know if this is possible or if I am making an error.
Furthermore, the shear stress can be calculated from $\tau = \dfrac{M_v}{W_v}$ where $W_v = \dfrac{\pi d^3}{16}$, which is the torsion resistance in the material. I then proceed to use the von Mises yield criterion to get an estimate of the maximum stress in the material.
$$\sigma_{von\ Mises} = \sqrt{\sigma^2+3\tau^2}$$
As I asked before, I am mainly interested if this is a possible way to proceed with solving this problem or if I am using some methods/assumptions which are wrong.