This research studies the large-displacement response of a fractal-architecture mechanism with circular-axis flexible hinges by formulating an efficient and accurate nonlinear finite element model. Two three-dimensional line elements are proposed whose nodal degrees-of-freedom include the three spatial Tait–Bryan angles. The nonlinear finite element is generated using the minimum potential energy condition for the entire deformed structure in a non-incremental approach. The error does not depend on the number of load steps since one step is sufficient to achieve the final, deformed state. The method is applied to predict the nonlinear, large, out-of-plane displacement of the fractal-hinge compliant mechanism. The model predictions are validated by finite element code simulation and experimental testing. The nonlinear finite element force-displacement data coincide with the linear compliance model predictions of Lobontiu et al. (2019, “Stiffness Design of Circular-Axis Hinge, Self-similar Mechanism With Large Out-of-Plane Motion,” ASME J. Mech. Des., 141(9), p. 092302) for approximately one-fourth lower portion of the load range and display the expected hardening-spring features for the load range remainder.