MOMENT AREA METHOD OF STRUCTURAL ANALYSIS:
Theorem – I
The product of 1/EI is the difference in slope between tangents drawn to the elastic curve at any two points A and B.
Between those two places (1/EI) and the region of the moment diagrams.
Theorem – II
The product of the departure of any point B from a tangent drawn to the elastic curve at any other location A, in a direction perpendicular to the beam’s original position, is equal to
the area moment between points A and B of that segment of the moment diagram.
This method is useful for finding slope and deflection at a specified position.
One of the most successful ways for determining bending displacement in beams and frames is the moment-area approach.
The slope and or deflections at certain positions along the axis of the beam or frame are computed using the area of the bending moment diagrams in this method.
For beams made of homogenous and isotropic material, with symmetrical cross-sections and lengths much greater than their cross-section dimensions, the second assumption is realistically valid (10 times or more is a common rule of thumb).
In other words, if the beam deforms appreciably in any way other than symmetric bending, the assumption of normal and plane cross sections is violated.
Short beams, sandwich type cross-sections, narrow cross-sections, and open unsymmetrical cross-sections are examples of such instances.
Two theorems, commonly known as’moment area theorems’ or ‘Mohr’s theorems,’ underpin the moment area approach. The first is concerned with the change in slope between any two places on the beam, while the second is concerned with the deflection at a specific position.
After the following graphic, which will be used as a reference, the two theorems will be stated. It shows a simple beam that has been deflected by some random loading, as well as the bending moment diagram that goes with it.