Which of the following is the correct expression for slope \(\left(\frac{d y}{d x}\right)\) at any distance' X' in a cantilever beam shown in the figure according to the double integration method?

Where, EI is the flexural rigidity of beam section.

Explanation:

Consider a section X at a distance x from the fixed end A. The bending moment at this section is given by

M_x = - W (L - x)

But BM at any section is also given by

\(M = EI{d^2y \over dx^2}\)

Equation both the equations we get

\(M = EI{d^2y \over dx^2} = -W(L-x) = - WL +Wx\)

Integrating, this equation we get

\(EI {dy \over dx} = - WLx +{ {Wx^2}\over 2} + C_1\ \)

Integrating again we get

\(EI y = {- WLx^2\over 2} +{ {Wx^3}{}\over {2\times 3}} + C_1 \times x +C_2 \)

Where C₁ and C₂ are constant of integration, and their values are obtained from boundary condition, which are 1) at x = 0, y = 0 2) at x = 0 dy/dx = 0

1) by substituting x = 0, y = 0 in the above equation we get

0 = 0 + 0 + 0 + C₂ 

∴ C₂ = 0

2) By substituting x = 0 and dy/dx =0, we get

0 = 0 + 0 + C₁

∴ C₁ = 0

Substituting C₁ and C₂ we get

\(EI{dy \over dx} = -WLx + W{x^2 \over 2} =-W(Lx - {x^2\over 2})\)

\(\frac{d y}{d x}=-\frac{W}{2 E I}(2 l x-x^2) \)