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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.

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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 $x_o$, $\ M= wx*x/2=wx^2/2$

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