Abstract In the present study, flattening deformation of rectangular tubes (thickness: t, width of the bottom wall: C 1, width of the sidewall: C 2) in bending was investigated using the finite element method. It is found that the flattening ratio μ 2 can be expressed by a sole function of nondimensional curvature ${\kappa }_g{C}_1^2/t$ (κ g: the bending curvature of the tube's central axis) independent of tube thickness in elastic bending; and in plastic bending, the relation between flattening ratio μ 2 and ${\kappa }_g{C}_1^2/t$ is influenced by the tube thickness and the material parameters of yield stress and strain hardening, however, these effects are very small. A mechanical model is proposed to analyze the cross-sectional flattening of a rectangular tube in bending. In the analysis of flattening in plastic bending, it is essential to determine moduli E q and E p, the coefficients relating the stress and strain in the axial direction and relating the bending moment and curvature in circumferential direction, respectively. Based on the investigation carried out on the results of FEM, it seems that a simple approximation using a combination of E t and E s gives a solution with a reasonable accuracy. Moreover, when the tube is short, the cross-sectional flattening is also blocked in the middle part. The ratio of decrease of flattening ratio $A_{\mu }$ can be approximately expressed as a function of nondimensional length ratio ξ ($\xi =L/\left[C_1\right(3+C_2/C_1\left)\right]$, L: tube length) alone. Also, based on the fact that the ratio between the flattening ratios of short and long tubes in plastic bending is almost the same as that in elastic bending, a formula for evaluation of the flattening ratio μ 2 in plastic bending of a tube with finite length is proposed.

Highlights • In the present study, flattening deformation of rectangular tubes in bending was investigated using the finite element method. It is found that the flattening ratio can be expressed by a sole function of nondimensional curvature independent of tube thickness in elastic bending; and in plastic bending, the relation between flattening ratio and the nondimensional curvature is influenced by the tube thickness and the material parameters of yield stress and strain hardening, however, these effects are very small. • A mechanical model is proposed to analyze the cross-sectional flattening of a rectangular tube in bending. In the analysis of flattening in plastic bending, it is essential to determine moduli E q and E p, the coefficients relating the stress and strain in the axial direction and relating the bending moment and curvature in circumferential direction, respectively. Based on the investigation carried out on the results of FEM, it seems that a simple approximation using a combination of E t and E s gives a solution with a reasonable accuracy, where E t is given by J 2 flow theory and E s is given by J 2 deformation theory. • Moreover, when the tube is short, the cross-sectional flattening is also blocked in the middle part. The ratio of decrease of flattening ratio can be approximately expressed as a function of nondimensional length ratio alone. Also, based on the fact that the ratio between the flattening ratios of short and long tubes in plastic bending is almost the same as that in elastic bending, a formula for evaluation of the flattening ratio in plastic bending of a tube with finite length is proposed.