Stability and non-linear second-order elastic analyses of beam and framed structures with semi-rigid connections using the cross method

The main objective of this publication is to present an extended version of the Moment Distribution Method (MDM) for the stability and non-linear second-order analysis of indeterminate beams and framed structures made of beam-columns of symmetrical cross-section including the combined effects of shear and bending deformations, axial loads, and semi-rigid connections. The proposed method along each member has the following advantages: (1) it can be utilized in the first- and second-order analyses (including buckling analysis) of indeterminate beams and framed structures made of beam-columns with rigid, semi-rigid, and simple end connections; (2) the effects of semi-rigid connections are condensed into the bending stiffness and fixed-end moments without introducing additional degrees of freedom and equations of equilibrium; and (3) it is accurate, powerful, practical, versatile, and an excellent teaching tool. Analytical studies indicate that shear deformations, semi-rigid connections, and axial loads increase the lateral deflections and affect the internal moments and reactions of continuous beams and framed structures. These effects must be taken into account particularly in slender structures and when they are made of beam or columns with relatively low effective shear areas (like laced columns, columns with batten plates or with perforated cover plates, and columns with open webs) or with low shear stiffness (like short columns made of laminated composites with low shear modulus G when compared to their elastic modulus E) making the shear stiffness GAs of the same order of magnitude as EI/L2. These effects become even more significant when the external supports are not perfectly clamped. Three comprehensive examples are included that show the effectiveness of the proposed method.