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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 point A of the beam. EI is constant.

Expert Solution

Step 1

Draw the free body diagram and calculate the reactions:

Consider the equilibrium of vertical forces:

$R_{B} - R_{C} = 0 R_{B} = R_{C}$

Consider the moment equilibrium about point C:

$(R_{B}) (a) - M_{o} = 0 R_{B} = \frac{M_{o}}{a}$

$R_{C} = R_{B} = \frac{M_{o}}{a}$

Step 2

Write the bending moment equation at section x-x at distance x from the point D:

$M_{x} = - (R_{C}) (x - a) + R_{B} (x - 2 a)$

From double integration method:

$E I \frac{\partial^{2} y}{\partial x^{2}} = - M_{x} E I \frac{\partial^{2} y}{\partial x^{2}} = R_{C} (x - a) - R_{B} (x - 2 a) i n t e g r a t e t h e a b o v e e q u a t i o n : E I \frac{\partial y}{\partial x} = \frac{R_{C}}{2} {(x - a)}^{2} - \frac{R_{B}}{2} {(x - 2 a)}^{2} + C_{1} . . . . . (1) i n t e g r a t e a g a i n : E I y = \frac{R_{C}}{6} {(x - a)}^{3} - \frac{R_{B}}{6} {(x - 2 a)}^{3} + C_{1} x + C_{2} . . . . . (2)$

Use following boundary conditions in equation (2) to calculate the value of constants:

At x = a, y = 0 and at x = 2a, y = 0.

Which gives:

$C_{1} = 0 a n d C_{2} = \frac{- M_{o} a^{2}}{6}$

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