Cardiovascular diseases remain the primary cause of mortality worldwide. Therefore, the efficacy of cardiac catheterisation for diagnosing heart conditions is of paramount significance. This procedure involves inserting a long, slender, and flexible tube through a blood vessel to reach the heart. Navigation through blood vessels is highly dependent on the specific patient and intervention, often requiring the reinsertion of multiple catheters with different tip shapes, prolonging the procedure time and elevating the risk of embolism. To mitigate reinsertion, and therewith ease the procedure, reduce operating time and risk, there is a need for a versatile compliant tip that can smoothly morph between predefined shapes. To this end, we will develop a gradient-based optimisation framework for the design of single-wire actuated shape-morphing flexible tubular structures. We employ geometrically and material nonlinear isogeometric analysis techniques to predict the behaviour of initially-curved Euler–Bernoulli beams. The placement of the actuation wire relative to the neutral axis of the beam introduces a highly nonlinear design- and state-dependent loading. These nonlinearities present challenges in the analysis process yet open up a new world of design freedom. In addition to studying the influence of the initial shape of the neutral axis, the cross-sectional shape and material properties, this work investigates the influence of variable wire placement along the length of the beam on a selection of morphing performance metrics.