Subscribe $4.99/month

Un-lock Verified Step-by-Step Experts Answers.

Find the Source, Textbook, Solution Manual that you are looking for in 1 click.

Our Website is free to use.
To help us grow, you can support our team with a Small Tip.

All the data tables that you may search for.

Need Help? We got you covered.

For Arabic Users, find a teacher/tutor in your City or country in the Middle East.

Products

Find the Source, Textbook, Solution Manual that you are looking for in 1 click.

For Arabic Users, find a teacher/tutor in your City or country in the Middle East.

Need Help? We got you covered.

Chapter 10

Q. 10.18

Figure 10.33 shows a compound beam loaded at its free end. If the flexural rigidity is constant throughout the beam, calculate the total strain energy stored. Using this strain energy, find the deflection at point E.

10.33

Step-by-Step

Verified Solution

Let us draw the free-body diagram of the beams as shown in Figure 10.34.

From Figure 10.34(b) above

\sum M_{ C }=0 \Rightarrow R_{ D }=2 P(\uparrow)

Therefore,

\sum F_y=0 \Rightarrow R_{ C }=P(\downarrow)

Again from Figure 10.34(a)

\sum M_{ A }=0 \Rightarrow R_{ B }=2 R_{ C }=2 P(\downarrow)

and            \sum F_y=0 \Rightarrow R_{ A }=P(\uparrow)

So, both beams are symmetrically and identically loaded. Thus, the total strain energy of the system is

U_{\text {bending }}=2\left[\left\lgroup \frac{1}{2 E I} \right\rgroup \int_0^a M_x^2 d x\right]_{ AC }+2\left[\left\lgroup \frac{1}{2 E I} \right\rgroup \int_0^a M_x^2 d x\right]_{ CE }

=\left\lgroup \frac{2}{E I} \right\rgroup \int_0^a M_x^2 d x

=\frac{2}{E I} \int_0^a P^2 x^2 d x=\frac{2}{3} \frac{P^2 a^2}{E I}

The total strain energy of the system is 2 P^2 a^3 / 3 E I

To calculate the deflection at point E, we can apply Castigliano’s second theorem [refer Eq. (10.66)].

\frac{\partial U}{\partial Q_i}=\Delta_i ; \quad 1 \leq i \leq n             (10.66)
Therefore,

\Delta_{ E }=\frac{\partial U}{\partial P}=\frac{4 P a^3}{3 E I}

Thus, vertical deflection at E is

\Delta_{ E }=\frac{4 P a^3}{3 E I}(\downarrow)

10.34