25.6 : モーメント面積定理

The moment-area theorem provides geometric properties of an elastic curve for determining deflection and slope at any point on the beam supporting a building floor.

When plotted along the beam's length, the ratio M/EI creates a diagram similar to the bending moment diagram.

The moment-area theorem derives the slope and tangential deviation equation from the beam diagram. The first moment-area theorem equates the angle between tangents at points P and Q to the area between these points under the M/EI diagram.

The second moment-area theorem relates the tangential deviation of point P from Q to the first moment of the same area concerning the vertical axis through P.

This theorem is also expressible in terms of the product of the area and the distance from the centroid of the area to the vertical axis through P.

The tangential deviation is the vertical distance from Q to the tangent at P. It's calculated by multiplying the area under the M/EI diagram by the distance from its centroid to the vertical axis through Q.