# Why is the bending moment equation for a uniformly distributed load (w) across a length x on a structure given by w(x)(x/2)?

So for any structure, simply supported beam, three hinge frame, cantilever etc. where there is a uniformly distributed load. For example, for a simply supported beam with uniform load of w along the length of x. Why is the bending moment equation w(x)(x/2)? I understand that the magnitude of the force is given by w(x), but is the reason for the (x/2) is because the magnitude acts halfway through the load?

Now I know that shear force and bending moment are related, with the shear force being the derivative of the bending moment, so I can just integrate the shear force equation to get w(x)(x/2), but I want to confirm through fundamentals as to why the equation is as it is.

Thanks.

consider an infinitesimally small section of a beam from coordinate $$x_1$$ to $$x_1-dx$$ loaded with a distributed load w. The moment of this load about point $$x=0 \ is \\ M= w*x*dx.$$
Now if we have the load area extended from $$x_0 \ to\ x,$$ we need to integrate the moment over the length of beam loaded with w: $$M= \frac{1}{2}wx^2$$
Or as you noted because the CG of load rectangle w*x is at its middle the moment is the area times the distance of CG to $$x_o$$, $$\ M= wx*x/2=wx^2/2$$