Determine the slope of the beam shown in Fig. a at point B. EI is constant. MEIPBBPLEI(b)(a)-tan A,Ов/АATOBBtan B(c)

Consider the given figure. The moment at the section x-x is given by, Mx=-WL-x=-WL+Wx=Wx-WL The…

Answered: M EI P B B PL EI (b) (a) -tan A ,Ов/А A…

Materials Science And Engineering Properties

1st Edition

ISBN:9781111988609

Author:Charles Gilmore

Publisher:Charles Gilmore

Chapter8: Engineering Materials

Section: Chapter Questions

Problem 85CQ

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Question

Determine the slope of the beam shown in Fig. a at point B. EI is constant.

M
EI
P
B
B
PL
EI
(b)
(a)
-tan A
,Ов/А
A
TOB
B
tan B
(c)

Expert Solution

Step 1

Consider the given figure.

The moment at the section x-x is given by,

$\begin{array}{rcl} M_{x} & = & - W (L - x) \\ = & - W L + W x \\ = & W x - W L \end{array}$

The differential equation of the elastic curve of the beam is given by,

$E I \frac{d^{2} y}{d x^{2}} = M$

Substitute the value in the above expression.

$E I \frac{d^{2} y}{d x^{2}} = W x - W L$

Integrate the above expression.

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

Integrate the above expression again.

$E I y = \frac{{W x}^{3}}{6} - W L \frac{x^{2}}{2} + C_{1} x + C_{2} . . . (2)$

Step 2

Boundary conditions.

At point $x = 0$ , the slope will be zero.

$\frac{d y}{d x} = 0$

Substitute the value in equation (1).

$\begin{array}{rcl} E I (0) & = & \frac{{W (0)}^{2}}{2} - W L (0) + C_{1} \\ C_{1} & = & 0 \end{array}$

At point $x = 0$ , the deflection will be zero.

$y = 0$

Substitute the value in equation (2).

$\begin{array}{rcl} E I (0) & = & \frac{{W (0)}^{3}}{6} - W L \frac{{(0)}^{2}}{2} + C_{1} (0) + C_{2} \\ C_{2} & = & 0 \end{array}$

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M EI P B B PL EI (b) (a) -tan A ,Ов/А A TOB B tan B (c)