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Unlike traditional rigid linked robots, soft robotic manipulators can bend into a wide variety of complex shapes due to control inputs and gravitational loading. This paper presents a new approach for modeling soft robotic manipulators that incorporates the effect of material nonlinearities and distributed weight and payload. The model is geometrically exact for the large curvature, shear, torsion, and extension that often occur in these manipulators. The model is based on the geometrically exact Cosserat rod theory and a fiber reinforced model of the air muscle actuators. The model is validated experimentally on the OctArm V manipulator, showing less than 5% average error for a wide range of actuation pressures and base orientations as compared to almost 50% average error for the constant-curvature model previously used by researchers. Workspace plots generated from the model show the significant effects of self-weight on OctArm V.