Principal of virtual work

[Dell]

using the principal of virtual work, find the reactions of the following structure

as far as i know, to solve using virtual work i take the statically determinate structure and release one of the support reactions making it indeterminate (1st degree),

after picking my brain with the geometry i came to the assumption that when the right side drops δ, the left side rises δ, (when i release the "y" reaction on the left side), this gives me

Ay*δ=P*δ
Ay=P (↑)

when i release the middle "y" reaction, i get

By*δ/2=P*δ
By=2P (↓)

since i can use statics to solve these i know that they are correct,

but for the "x" reactions i know (from statics) that Ax=Bx=0
but using virtual work, when i release one of the "x" reactions, let's say the left one,
since the support A can move(↔) since the lengths of the bars don't change, when the support moves to the right, the point at P, must drop, leaving me with

Ax*Δ=P*δ
Ax=P*δ/Δ

havent bothered trying to find the relation of δ-Δ since i know that Ax is meant to be 0.

a) what is the best way to prove that when the left side rises δ, the right side drops δ,
b) what am i doing wrong with my calculations for Ax, Bx?

 

[Valkarie]

a) The best way to prove that when the left side rises δ, the right side drops δ, is to use the principle of virtual work. This states that the total work done by the forces on a system remains unchanged if the system undergoes a virtual displacement. Thus, if the left side rises δ and the right side drops δ, the total work done by the forces must remain the same. This can be verified by calculating the work done by the forces on both sides before and after the displacement. If the work done is equal, then the displacement is valid. b) For the reactions Ax and Bx, you are correct in using virtual work to calculate the change in the forces due to the displacement. However, the result you are getting (Ax=P*δ/Δ) is incorrect since the displacement of the support A is not related to the force P. The correct result should be Ax=0.