Beam with two supports + spring

[Jakob]

Homework Statement

I have beam supported in three points (one of the support is spring) .
What i know:
P_R = 100kN/m
spring constant Z = 10kN/m
I have to calculate bending moments and shear forces in points A, B and Z.

Homework Equations

[/B]
ΣM = 0
ΣF = 0
Spring force : F_z=-zx
This is not enoug to solve the problem. I have two equations and three unknowns (V_A, V_B, and x)

The Attempt at a Solution

[/B]
I was thinking about finding deflection of the beam in point Z however the shape of the beam between points Z and A is quite complicated. Maybe there is some simpler method and I can't see?

I will be very gratefull for help/hint with that!

 

[haruspex]

I was thinking about finding deflection of the beam in point Z

That is the right approach, but of course it is a bit more complicated here because the deflection and the force at Z depend on each other.
You will need to step through the usual procedure for figuring out the shape of the beam, but treating the force at Z as an unknown.