When using the moment-area method to find the deflection of a beam, how does the location of the tangent reference point specifically affect the direct calculation of the relative rotation between two points?

When using the moment-area method, the first moment-area theorem states that the relative rotation (or change in slope) between the tangents at any two points on the elastic curve of a beam is equal to the area under the M/EI diagram between those two points. Here, M is the bending moment and EI is the flexural rigidity of the beam. The tangent reference point is one of the two chosen points on the elastic curve whose tangent serves as the baseline or zero-angle reference for measuring the rotation of the tangent at the other point. Its location specifically affects the direct calculation of relative rotation in the following ways:

Firstly, the chosen tangent reference point directly defines one of the integration limits for calculating the area under the M/EI diagram. If point A is designated as the tangent reference point and we wish to find the relative rotation between A and point B, then the area under the M/EI diagram is calculated from point A to point B. This calculation directly yields the angular displacement of the tangent at point B *with respect tothe tangent at point A, often denoted as θ_B/A.

Secondly, changing the location of the tangent reference point reverses the perspective of the calculated relative rotation. If point B is chosen as the tangent reference point, and we are still considering the same two points A and B, then the area under the M/EI diagram is calculated from point B to point A. This calculation directly yields the angular displacement of the tangent at point A *with respect tothe tangent at point B, denoted as θ_A/B. This value, θ_A/B, will have the same magnitude as θ_B/A but the opposite sign, because θ_A/B = -θ_B/A. For example, if the tangent at B rotates 0.01 radians clockwise relative to the tangent at A, then the tangent at A rotates 0.01 radians counter-clockwise relative to the tangent at B.

Therefore, the location of the tangent reference point establishes which tangent is considered the base from which the relative angle is measured, thus dictating both the direction of the M/EI area integration and the sign convention of the resulting relative rotation. It does not alter the absolute magnitude of the relative rotation between the two points, but it precisely defines the start point for the angular measurement and the interpretation of its sign.