Features of the analytical solution of the problem of displacement cantilever steel beams with variable flange depth

Abstract

The article is devoted to the problem of generalization of the influence of the variability of stiffness of elastic steel elements on thedisplacement and angles of rotation in the Cartesian coordinate system when placing the origin ofthe coordinates in the center of gravity of the largest cross section. Cantilever elastic I-beam steelbeam with variable flange depth is considered.The obtained formulas are shows the influenceof the variability of the I-beam cross-section on themoment of inertia in two main axis. This made itpossible to write the differential equation of beambending as a linear equation with variable coefficients.The solution of the differential equation makesit possible to obtain analytical formulae for determining the displacements and angles of rotation ofthe cross section of cantilever I-beams with variable flange depth. To confirm the obtained analytical expressions in the transition to the definition ofdeflections and angles of rotation of the I-beamswith a constant cross-section, the Lopital-Bernoullirule was used. A variant of the formula for theconsequences of the second remarkable (special)boundary in the disclosure of uncertainty is obtained.This makes it possible to prove by an analyticalapproach the coincidence of the obtained solutionswith the solutions for constant cross-section Ibeams. Numerical studies also confirmed the obtained result. This approach can be applied to Ibeams with variable flange depth under differentsupport conditions.The obtained displacement formulas make itpossible to check the stiffness of cantilever steel Ibeams with a linear change in their stiffness of thebeams according to the deflection limits. The obtained results can also be used for research of Ibeams with variable flange depth under differentsupport conditions. The obtained displacementformulas make it possible to check the stiffness ofcantilever steel I-beams with a linear change intheir stiffness of the beams according to the deflection limits. The obtained results can also be usedfor research of I-beams at linear change of stiffness, including a change of the modulus of deformation of steel at limited plastic work deformations on different sections of a beam.