Stability of beams under uniform loading q combined with end moments Ma and Mb

Abstract

: Calculation of the critical moments M cr of the beams under various combinations of the uniform loading q and the end moments. Solution of the differential equations is presented in the graphical form. Two diagrams enable to obtain the factors C 1 and C 2 , which depend on the parameters and . Approximate formula for calculation of the critical moments depending on three factors C 1 , C 2 , C 3 and three dimensionless parameters , , was derived in (2000a). The values of the factors C 1 , C 2 , C 3 for basic loadings of the beams with the various combinations of the boundary conditions in vertical and horizontal bending and torsion for the monosymmetric I-sections were published in (2000b). They were later accepted for Eurocode (2007) and many national annexes. The presented results are valid for the beams simply supported with double symmetric I-sections. They may be used also for the continuous beams being slightly on the safe side. Numerical examples show that our results equal to the exact results of commercial computer programs. The procedure gives quickly practically exact results and therefore it is efficient tool for designers