I'd like to calculate the bending moment of a cantilever, fixed at its base, and submitted to a certain stress on a specific spot, but I can't find the proper definition of this bending moment (first time I'm encountering this name, and google didn't not help me so far).
Let us assume that the cantilever is fixed on a wall at the RHS end, and use that end as reference for measuring the distance, $x$, along the cantilever away from the wall. Let us also assume that the force $F$ is applied at a point $a$ units away from the wall (obviously the $a\le L$, where $L$ is the length of the canrilever). The bending moment at some point P which is distance $x$ from the wall, the bending moment is the algebraic sum of the moments of all loads lying to the right of the point P, and taken with respect to the point P itself. So we can write the forllowing:
for $x\le a$ the bending moment $M(x) =-F((L-x))$
for $x\gt a$ the bending moment $M(x)=0$ as there are no loads beyond the point $x=a.$
When you combine this with the equation for the deflection, v, you get
where $E$ is the Young modulus of the material, and $I$ is the second moment of area of the cantilever (also known as moment of inertia, calculated in a similar way). You need to solve this equation using youe boundary conditions, to get the diflection at various values of $x$. Hope this helps.