Determine the force and moment reactions at the support A of the built-in beam which is subjected to the sine-wave load distribution. The force reaction RA is positive if upward, negative if downward. The moment reaction MA is positive if counterclockwise, negative if clockwise.

Explanation:

The beam is subjected to the sine-wave load distribution as shown in the figure.

As the beam is in equilibrium condition, so net force and moment in any direction are zero.

Assuming the length, , of the beam is along the x-axis and the loading direction is along the y-axis.

The load density, w, per unit length, at a distance of x from the point A, for the sine-wave load is

,

where is constant (maximum load density)

is positive if upward, so w is negative as it is acting in the downward direction.

A small force, dF, in the downward direction, due to load on a small element dx at a distance of x from the point A is

in the downward direction

The moment, dM about point A, due to small force, dF, is

As the moment in the clockwise direction is negative, so

At equilibrium state, net force along the y-direction will be zero, i.e

From equation (i)

The center of F is at the centroid of the sine-curve in the downward direction.

Putting this value in the equation (iii), we have

Again, at the equilibrium state, net force along the y-direction will be zero, i.e

 [from (ii)]

Where is the x-coordinate of the centroid.

Due to symmetry,

So,

[ using (iv)]

Hence, the reaction force and the moment at point A are

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explanation:

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