Subject : Structural Analysis 1

Topic : General Principles

Difficulty : High

"In a Linear elastic structure, the displacement of point B of a structure due to unit load acting at point A is equal to the displacement of point A when the unit load is acting at point B."

consider a simply supported beam shown in the figure.

The deflection at b due to load p applied at a.

$\therefore M_{a}=p(M_{a})$ where, $M_{a}$=bending moment due to unit load at 'a'

According to the dummy unit load method, to find the deflection at b apply unit load at b.

Let $m_{b}$ is the bending moment at any section due to unit load at b.

The deflection at b due to load p at 'a'

$q_{ba}=\int_{}^{}\frac{p(m_{a})(m_{b})}{EI}dx \space\space\space\space\space\space --(1)$

The deflection at a due to load p at 'b'

$q_{ab}=\int_{}^{}\frac{p(m_{b})(m_{a})}{EI}dx \space\space\space\space\space\space --(2)$

$\therefore$ From equation (1) and (2), we get

$q_{ab}=q_{ba} \space\space\space\space\space\space --(3)$

The equation (3) doesn't confine to vertical deflection.

Now, consider a general case of a linear elastic system as shown.

Reciprocity of deflections - linear elastic system

$(P)\partial_{BA}=(M)\alpha_{AB} \space\space\space\space\space\space --(4)$

$(n)\partial_{BA}=(n)\alpha_{AB} \space\space\space\space\space\space --(5)$

$\therefore$ Deflection at B due to moment (M=n) at a is equal to the rotation at A due to load (p=n) at B.