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How do I know when to use an upward or a downward curve when I'm drawing a bending moment diagram for a uniformly-distributed load (UDL) and what's the most reliable way of identifying the maximum bending moment?

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The direction a bending moment diagram is drawn depends on the sign convention being used. The most common sign convention is:

PositiveSignConvention

It is good practice to indicate the sign convention used whenever you draw a BMD.

So if we are interested in the sign of the bending moment for a UDL we can take a cut such as: enter image description here

In which case we can see from the sum of moments at the cut:

$$ M = Rx - w\frac{x^2}{2} $$

And whatever the sign found from this is the sign for the BMD.

For simple structures you can obtain the sign of the bending moment by thinking of the deformed shape. If the deformed shape resembles that shown in the sign convention above then the sign of the BMD will be positive (tension on bottom). So for a simply supported beam with a UDL acting downwards the bending moment would be positive.

Be aware that sometimes a BMD is drawn with the positive moment axis pointing downwards, this is called drawing 'on the tension side' and is common in concrete design as it indicates where reinforcement should be placed.

The fastest, most reliable way of finding the maximum bending moment, especially for simple cases, would be to use established beam design formulas such as: DA 6 - Beam Design Formulas with Shear and Moment Diagrams. If you knew where the location of maximum bending moment was, you could take a cut at that location and calculate it directly as shown above. Otherwise, you will need to resort to integrating the shear force (either directly, or graphically) to obtain the BMD and then locating the maximum that way.

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