Rational nonlinear analysis of framed structures and curved beams considering joint equilibrium in deformed state

Based on the continuum mechanics principles, a rigorous formulation is presented for the linearized stiffness equation of three-dimensional beam elements with account taken of the joint moment equilibrium in the deformed configuration C 2 . By sticking to the Bernoulli–Euler hypothesis of plane sections and elasticity definitions for stress resultants, the bending moments and torque of the element are shown to be quasi- and semi-tangential, respectively, in the updated Lagrangian formulation. Further, by invoking the moment equilibrium conditions for structural nodes at C 2 , the induced moment matrix that first appears to be antisymmetric on the element level turns out to be symmetric upon assembly of all elements on the structural level. The joint equilibrium conditions at C 2 , as represented by the induced moment matrix, are central not only to the out-of-plane buckling analysis of angled frames, but also to the simulation of curved beams by the straight-beam elements. Examples on the buckling of angled frames and curved beams are provided to support the theory presented.