Beam deformation. FEM (solidmechanics)

[Payam30]

Homework Statement

I have this question. I need to know the diformation in P direction. Here is the question

I use the symmetry and finally with a 1/4 modell I get the following:

which is correct according to the instruction. K is the stiffness of the spring. which with symmetry becomes k/2 and also P

now
I have this deformation according to my knowledge

If I apply the formulas for beam deformation which I am allowed to do(in basic level) I got this:

you see that deformation d1 in my third picture will have two componentes and it will not solve the problem. becouse :

you see that i get problem writing f1x and f1z.. Can anybody tell me how to sett the solustion?

f1x is the force in node 1 at x direction and so on.
f2x will be p/2 cos45 and f2z will be -p/2sin45 and M2

Homework Equations

The Attempt at a Solution

What I wrote up there.

 

[nvn]

Payam30: I would say your quarter model and displacements currently look good, except when you quarter the spring, I currently think its stiffness remains k, not 0.5*k. Also, I would probably number the displacements D5 and D7. I would currently say, use the frame member global stiffness matrix, instead of basic. I would probably number your three nodes node 1 for left end of spring, node 2 for lower end of frame member, and node 3 for upper end of frame member. I would probably number your members member 1 for the frame member, and member 2 for the spring.

Obtain the frame member global stiffness matrix for member 1, at a slope angle of alpha = 45 deg. Assemble its partitions into the structure stiffness matrix, uppercase K. The member global stiffness matrix for the spring is just k, which you add to K_7,7.

In the structure stiffness matrix, note that you can cross out columns and rows 1, 2, 3, 4, 6, 8, and 9.

After you do all of this, I think you end up with the equations in the attached file. Check my work, to see if I made any mistake, because I did this very hurriedly.

After that, invert the stiffness matrix in the attached file, or solve the two equations by hand using simultaneous solution, to solve for displacements D5 and D7.

Do you have given numerical values for E, I, A, L, k, and P?