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This paper proposes a series of simple mathematical expressions which govern the ultimate behavior of two-dimensional reinforced concrete beam-column joints based on an observation of a failure crack pattern, and its compatible kinematic field, overlooked by the past researchers. The kinematic model is named nine DOF model in this report and combined with non-linear biaxial constitutive relation of concrete and steel to bear simple algebraic expressions for the ultimate moment capacity and moment at balanced failure, applicable to symmetric interior beam-column joints subjected four symmetric moments transmitted among the four ajacent members. The ultimate moment capacity is defined at which tensile yielding of longitudinal bars passing through the joint in the joint causes the expansion of diagonal cracks in the joint as well as local rotation of concrete panel and local crushing of concrete. The balanced failure of beam-column joint is defined at which tensile yielding of longitudinal bars passing through the joint and crushing of concrete start simultaneously. Simple equations for the upper bound value for the amount of reinforcement precluding the joint failure before yielding of longitudinal reinforcement are also brought.
Hitoshi Shiohara (Tue,) studied this question.