This work is focused on the modeling of rod-like elastic bodies with thin-walled curved cross-sections such as tape springs, which develop localized folds due to a flattening of the cross-section. Starting from a complete nonlinear elastic shell model, a rod model with highly deformable cross-section is derived for large displacements, large rotations and dynamics, by introducing an elastica kinematics to describe the in-plane changes of the cross-section shape. This model is able to handle the formation of localized folds which can move along the rod line, merge or split, allowing simulation of complex scenarios of coiling, folding and deployment.