Stiffness Matrix Issue: Get Help Now!

[Amaelle]

Good day All,
while trying to solve this exercice

I was puzzeld by the solution approach
indeed, they use the symmetry of the structure, they have made a cut on the hinge where the force F is applied (the force F has been divided by 2 for the symmetry reason), and ONLY replace it with a HORIZONTAL REACTION!
according to my basic understanding, when we do a cut in a hinge we replace it with two reactions HORIZONTAL and VERTICAL!

Any help would be highly appreciated !
Thanks

 

[Orodruin]

A vertical reaction would break the symmetry.

 

[Amaelle]

thanks for your prompt answer, but could you elaborate more?

 

[Orodruin]

The symmetry is a mirror symmetry between the left and right half. If you mirror the free-body diagrams of the left and right parts, they should transform into each other. However, if you introduce a vertical component of the reaction force in the up direction on the left part, then the 3rd law pair on the right part is in the down direction. Mirroring the diagrams therefore does not give you back the same diagrams as you will get a left part with the force in the down direction and a right part with the force in the up direction. Thus, any reaction force in the vertical direction breaks the symmetry.

 

[Amaelle]

The symmetry is a mirror symmetry between the left and right half. If you mirror the free-body diagrams of the left and right parts, they should transform into each other. However, if you introduce a vertical component of the reaction force in the up direction on the left part, then the 3rd law pair on the right part is in the down direction. Mirroring the diagrams therefore does not give you back the same diagrams as you will get a left part with the force in the down direction and a right part with the force in the up direction. Thus, any reaction force in the vertical direction breaks the symmetry.

Thanks a million!