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Continuum robots, which are composed of multiple concentric, precurved elastic tubes, can provide dexterity at diameters equivalent to standard surgical needles. Recent mechanics-based models of these “active cannulas” are able to accurately describe the curve of the robot in free space, given the preformed tube curves and the linear and angular positions of the tube bases. However, in practical applications, where the active cannula must interact with its environment or apply controlled forces, a model that accounts for deformation under external loading is required. In this paper, we apply geometrically exact rod theory to produce a forward kinematic model that accurately describes large deflections due to a general collection of externally applied point and/or distributed wrench loads. This model accommodates arbitrarily many tubes, with each having a general preshaped curve. It also describes the independent torsional deformation of the individual tubes. Experimental results are provided for both point and distributed loads. Average tip error under load was 2.91 mm (1.5% - 3% of total robot length), which is similar to the accuracy of existing free-space models.