9. Conclusion

The finite elbow elements that we describe here can be used for elasticity or plasticity pipe calculations. Pipes can be subjected to various combined loads - internal pressure, plane and anti-plane flexures, twisting, extension.

For the moment, the element produced is a linear element of the beam type, straight or curved, with three nodes, in small rotations and deformations, with local elasto-plastic behavior under plane stresses. It allows for ovalization, warping and swelling to be taken into account. It combines the properties of shells and beams. The beam kinematics for the elbow axis is increased by shell kinematics, of the Love-Kirchhoff type without transverse shear, for the description of the behavior of the transverse sections. This last kinematics is discretized in \(M\) Fourier modes, including the number \(M\), whose number, which the literature encourages us to choose equal to 6 [bib8], [bib13], must both be sufficient to obtain good results in plasticity and not too big to limit the calculation time. In elasticity, for relatively thick pipes (the ratio of thickness to radius of the cross section greater than 0.1), we can be satisfied with \(M\mathrm{=}2\) or \(M\mathrm{=}3\).