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From the picture above, in order to calculate the moment of load acting on the beam we take moment $M=\frac{W}{L}×L×\frac{L}{2}$.

Why can't we calculate the moment as $M=\frac{W}{L}×L×L$?

Why do we need to take the distance from center of gravity to a point as the distance?

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A point force produces a moment with respect to a point equal to the product of the force by the distance. To compute the moment produced by a system of forces, we can compute the moment for each force and sum the individual moments, or we can just sum the forces and multiply them by the average distance, therefore getting the same moment. Finding the average distance is the same as finding the centre of gravity of the system of forces, since the distance to the center of gravity is the same as the average distance to the forces in the system.

If I'm understanding you, in your formula you are using the distance to the most distant point with distributed load, but most of the load is acting at a shorter distance and therefore producing less moment that the value you are accounting for. If we use the distance to the centre of gravity, some loaded points would be closes to reference points (producing less moment) and some will be further (producing more moment) but the global result will be right.

However, beware that the moment you gave in your question is valid for a cantilever beam with uniform load, and that's not what is in your drawing shows. Your drawing shows a simple supported beam, that is, supported at both ends. The maximum moment in a simple supported beam is $M=\frac{W}{L}\times\frac{L^2}{8}$.

And just a last note: beware that the definition of moment I've used is valid when the direction of the force is perpendicular to the distance to the point, as usually happens when computing moments in beams. If they are not perpendicular, definition of moment should be adjusted.

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  • $\begingroup$ Why is the maximum moment M=W/L × L2/8? Can you explain further about this? Thank you $\endgroup$
    – Wendy lly
    Jul 25, 2016 at 1:42
  • $\begingroup$ In short: Copute reactions (W/LLL/2 at each end), slice the beam at center and take moments of load and reactions in one side relative to the slicing point. Moment of reaction will be (W/LLL/2)*L/2 and moment of load -(W/LLL/2)*L/4 (please notice opposite signs). By summing and multiplying you get W/L*L^2/8. There are a lot of tutorials about moments in a simply supported beam, for instance bendingmomentdiagram.com/tutorials/… or youtube.com/watch?v=EPWBGqTfjwY $\endgroup$
    – Pere
    Jul 25, 2016 at 14:27

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