In this investigation, a three dimensional gradient deficient beam element (BEAM9) using the absolute nodal coordinate formulation (ANCF) is introduced. This element has nine coordinates per node, this includes the position vector and the two gradient vectors rx and ry. Like most of the ANCF elements, this element has constant mass matrix and zero centrifugal and Coriolis inertia forces. The plane strain elastic force model and the elastic line approach are two elastic force models presented in this paper in order to simulate the element internal resistance. Both models support resistance to the general bending and twist moments. The possibilities of employing nonlinear material models will be discussed in future work. Furthermore, the proposed element has the advantage of easy integration over general cross section area that is not easy to perform using the fully parameterized ANCF beam element (BEAM12). Comparing to the ANCF cable element (BEAM6), the proposed element can resist general bending and twist loads. Moreover, shear deformations in the xy plane due to shear force and in the yz plane due to twist moment are considered with the gradient deficient beam element proposed in this work. However, no shear deformations are considered with the ANCF cable element. Comparing to the fully parameterized ANCF beam element, the gradient deficient beam element (BEAM9) avoids some locking issues, shows better computational efficiency and offers better convergency characteristics. Numerical examples are presented in order to validate the proposed gradient deficient beam element and to compare with other ANCF beam elements.

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