Stiffness matrix for a symmetric structure

[Amaelle]

Good day All
While trying to solve the following exercice, I was stucked by a couple of issues

for the first part in which we have to find the simplest configuration ( symmetry)
according to my basic understanding Symmetry must be :

• geometry
• load
• support

here I don t have the third condition ( the roller in node 1 ) break the rule so i don't think we can talk about symmetry? Please correct me if I m wrong.

But despite that I have decided to follow up with the symmetry approach and I have chosen
this symmetry ( the red line represent the axe of symmetry)

and I ended having the following matrix

I have projected the reaction of the roller on node 1 on the axis u1 and u2
(R*cos 45 and R*sin 45°)
I've putted the boundary conditions
but while trying to slove the matrix to find U1
I ve ended up with two unkowns U1 and R
any help would be highly appreciated

Thanks a million!

 

[Valkarie]

Your understanding of symmetry is correct. The roller support at node 1 does break the symmetry and so you cannot use the symmetric approach to solve this problem. To solve the problem, you will need to consider the reaction force at node 1 as an unknown variable and solve the equations of equilibrium and boundary conditions with the 6 unknowns (three forces in each direction). This will give you a system of 6 equations with 6 unknowns which can be solved to find the reactions.