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We present formulations for dealing with the singularities encountered at an apex of a yield surface of generalized plastic hinges located at both ends of a 3D beam-column element. At an apex of a yield surface in a 3D stress space, the number of active yield conditions may be more than that of the dimension of the stress space. In such a case, direct application of Koiter's generalized plastic flow law leads to indeterminacy of the tangent stiffness matrix. To overcome the difficulty, we propose a simple and systematic formulation for deriving the tangent stiffness matrix, wherein application of Koiter's rule is effectively bypassed if necessary. This formulation is exact in the sense that it does not introduce any additional approximations like smoothing or cut off of a yield surface around an apex. The validity of the present formulation is demonstrated in a moment resisting frame example, where the results obtained by the present approach are compared with those by the 2D analysis.

TANGENT STIFFNESS MATRIX OF 3D BEAM-COLUMN ELEMENT AT APEX OF YIELD SURFACE FOR GENERALIZED PLASTIC HINGES