2. Benchmark solution

2.1. Calculation method used for the reference solution

The problem is solved analytically.

The result of the efforts (equal to the total weight) is equal to:

• concrete weight: \(\mathrm{Pb}=V{\rho }_{b}g\)

• cable weight: \(\mathrm{Pc}=AL{\rho }_{c}g\)

in the direction in which gravity is applied.

The structure is isostatic. Prestressing forces are self-balanced.

Let \({S}_{b}\) be the area of the concrete in a plane perpendicular to the cable \({S}_{b}=(2\times \mathrm{0,6}){m}^{2}\), \({E}_{a}\) and \({E}_{b}\) the modules of steel and concrete, \({N}_{a}\) the tension in the cable and \({\sigma }_{b}\) and the stress in the concrete after tension.

The balance of the concrete and cable assembly is written: \({N}_{a}+{\sigma }_{b}{S}_{b}\mathrm{=}0\) therefore \({\sigma }_{b}\mathrm{=}\mathrm{-}\frac{{N}_{a}}{{S}_{b}}\)

Since the macro command CALC_PRECONT is used, and since there is no friction or losses in the cable, the tension in the cable is equal to the initial tension, unlike the case where RELA_CINE_BP is used, which suffers prestress losses due to the shortening of the concrete (see test SSNP108, [V6.03.108])

The deformation of concrete is: \({\varepsilon }_{b}=\frac{{\sigma }_{b}}{{E}_{b}}\)

2.2. Benchmark results

• Result of efforts: \(R=132N\)

• Stress in concrete: \({\sigma }_{b}\mathrm{=}\mathrm{-}\mathrm{1,66666667}{10}^{5}\mathit{Pa}\)

• Normal force in steel: \({N}_{a}\mathrm{=}2{10}^{5}\mathit{Pa}\)

• Deformation in concrete: \({\varepsilon }_{b}=-\mathrm{5,555555555}{10}^{\text{-}6}\)

2.3. Uncertainty about the solution

It is an analytical solution.

The solution gives the average stress in concrete. When there are several elements (models \(B\) and \(C\)) it is necessary to average the values of the cells.