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Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

Determine the slope at A and the maximum deflection of the simply supported beam. EI is constant.

Expert Solution

Step 1

The reaction force will not develop in the beam as there is no external force is applied. Taking a section at x distance from A then the equation of bending moment is,

$M_{x} = - M_{0}$

The equation of double integration is,

$E I \frac{d^{2} y}{d x^{2}} = M_{x} E I \frac{d^{2} y}{d x^{2}} = M_{0} E I \frac{d y}{d x} = - M_{0} x + C_{1} . . . . . . (1) E I y = - M_{0} \frac{x^{2}}{2} + C_{1} x + C_{2} . . . . . . . (2)$

Step 2

Applying the boundary condition at x = 0, y = 0.

Substitute in equation (2),

$\begin{array}{rcl} E I y & = & - M_{0} \frac{x^{2}}{2} + C_{1} x + C_{2} \\ 0 & = & 0 + 0 + C_{2} \\ C_{2} & = & 0 \end{array}$

Applying the boundary condition at x = L, y = 0.

Substitute in equation (2),

$\begin{array}{rcl} E I y & = & - M_{0} \frac{x^{2}}{2} + C_{1} x + C_{2} \\ 0 & = & - M_{0} \frac{L^{2}}{2} + C_{1} L + 0 \\ C_{1} & = & \frac{M_{0} L}{2} \end{array}$

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