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Origin of Bending equation

CircleThe fundamental equation used to analyze beams is equation where y is the displacement shown in the figure, and x is the displacement in the figure, measured from the orange clamp at the bottom. M is the bending moment (torque) experienced by the beam at the location x. E is the Young's modulus, and I is called the "moment of area," or (incorrectly, but commonly enough that everyone understands) the "moment of inertia."

moment of inertia

where the integral is over the cross sectional area of the beam. (If dA were an element of mass this would be a moment of inertia.)

To glimpse the origin of this formula, imagine a rectangular beam bent (gently) into a circle of radius R. In the figure, c is the distance from the beam center (strictly the centroid) to the outer edge of the beam. A point on the outer edge is marked with a dot, a distance R+c from the center.

In xy coordinates, starting at the origin, a point which lies a distance S from the origin, has risen a distance y above its original horizontal position. The location, x, of this point is shown. It is approximately true that x=S. We will take the limit of small x to improve this approximation.

For small x, the angle theta can be related to the derivative of y with respect to x for the outer edge of the beam.

tan theta

We eliminate R from this expression by considering the strain induced in the outer edge of the beam when it is flexed into this curved shape. In bending, the inner edge of the beam is compressed, the outer edge is stretched, and the centroid is unchanged in length. The strain in the outer edge is:Strain

so that: Epsilon

This in turn is related to Young's modulus, E, and the tensile stress s in the outer edge:

Stress Strain

Using the result thatmcbyiat the outer edge, we have


Now enforce the requirement that s be small. In that case, s is the same as x, and in particular


Taking the derivative of both sides of the dy/dx equation, we can eliminate s by forcing it to be small in the limit for the derivative:


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