The change in slope between any two points on the elastic curve equals the area of the M/EI (moment) diagram between these two points.

${\displaystyle \theta _{A/B}={\int _{A}}^{B}\left({\frac {M}{EI}}\right)dx}$ 

where,

• ${\displaystyle M}$  = moment
• ${\displaystyle EI}$  = flexural rigidity
• ${\displaystyle \theta _{A/B}}$  = change in slope between points A and B
• ${\displaystyle A,B}$  = points on the elastic curve^[1]

The vertical deviation of a point A on an elastic curve with respect to the tangent which is extended from another point B equals the moment of the area under the M/EI diagram between those two points (A and B). This moment is computed about point A where the deviation from B to A is to be determined.

${\displaystyle t_{A/B}={\int _{A}}^{B}{\frac {M}{EI}}x\;dx}$ 

where,

• ${\displaystyle M}$  = moment
• ${\displaystyle EI}$  = flexural rigidity
• ${\displaystyle t_{A/B}}$  = deviation of tangent at point A with respect to the tangent at point B
• ${\displaystyle A,B}$  = points on the elastic curve^[2]

The deviation at any point on the elastic curve is positive if the point lies above the tangent, negative if the point is below the tangent; we measured it from left tangent, if θ is counterclockwise direction, the change in slope is positive, negative if θ is clockwise direction.^[3]

The following procedure provides a method that may be used to determine the displacement and slope at a point on the elastic curve of a beam using the moment-area theorem.

• Determine the reaction forces of a structure and draw the M/EI diagram of the structure.
• If there are only concentrated loads on the structure, the problem will be easy to draw M/EI diagram which will results a series of triangular shapes.
• If there are mixed with distributed loads and concentrated, the moment diagram (M/EI) will results parabolic curves, cubic, etc.
• Then, assume and draw the deflection shape of the structure by looking at M/EI diagram.
• Find the rotations, change of slopes and deflections of the structure by using the geometric mathematics.

• Area-Moment Method. (n.d.)