2. Reference solutions

For the solution without reinforcement, and considering the symmetries of the problem, the displacement can be written in the form:

\(U(r,z)=g(r){u}_{r}+h(z){u}_{z}\)

With:

The boundary conditions then make it possible to explain the constants:

with \({R}_{2}\) the radius at the interface between the concrete and the circumferential reinforcement sheet.

One gives here the solution with \({P}_{2}\) not zero because that will be useful later.

In the case where there is a circumferential frame on the external face of the cylinder, this frame exerts pressure. It is therefore necessary to determine this pressure (a function of the displacement), then apply the previous results.

The « boilermaker formula » (presence of reinforcements) is applied:

To relate the stress to the displacement solution, we go through deformation (the deformation of the reinforcements being the deformation of the cylinder):

By applying the previous results with \({P}_{2}=P\), we obtain the desired solution.