[en] In most of the calculations of piping systems using simplified methods, the flexibility of elbows is accounted for by means of the well-known flexibility factors. These factors are determined for elbows, considered as a part of an axi-symmetrical torus assuming an uniform behaviour along the bend. In reality, adjacent straight pipes or even flanges tend to restrict the deformation of the elbow cross-section, thus lowering the flexibility factor and stress intensification factors. The aim of this paper is to present analytical solutions for the propagation of ovalization and the variation of the flexibility factor along pipe bends terminated by straight pipes or flanges, under in-plane bending, assuming an elastic material behaviour. The solutions are compared to existing and experimental results