# Virtual Work - Determining Relationships in deflections

I'm having problems understanding the worked answer for this problem:

Here's part of the worked answer:

I don't understand why $\delta_C = \dfrac{4}{9} \delta_A$ . Can you please explain that part of it?

This is just similar triangles. The corresponding sides of the two triangles are in proportion. You basically equate each triangle's base-to-height ratio:

$\dfrac{\text{Base of Triangle 1}}{\text{Height of Triangle 1}} = \dfrac{\text{Base of Triangle 2}}{\text{Height of Triangle 2}}$

$\dfrac{\text{AB}}{\delta_A} = \dfrac{\text{BC}}{\delta_C}$

$\delta_C = \dfrac{\text{BC}}{\text{AB}} \delta_A$

$\delta_C = \dfrac{0.04}{0.09} \delta_A$

$\delta_C = \dfrac{4}{9} \delta_A$

Think back to your geometry days. In this case the two triangles in question are similar because the three pairs of corresponding angles are equal (AAA). Therefore the corresponding sides are all in proportion.

The same can be done for the other two triangles drawn in the solution.

Hope that helps.

Between ABC there is a single member. Hence the angle between AB and BC is always 90 degrees. Hence if AB rotates by theta, then BC also rotates by theta. As the length AB is 90mm, and the length AC is 40mm, then (by similar triangles):

$\delta_C = \dfrac{40}{90} \delta_A$