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

Анотація

The article is devoted to the problem of generalization of the influence of the variability of stiffness of elastic steel elements on the
displacement and angles of rotation in the Cartesian coordinate system when placing the origin of
the coordinates in the center of gravity of the largest cross section. Cantilever elastic I-beam steel
beam with variable flange depth is considered.
The obtained formulas are shows the influence
of the variability of the I-beam cross-section on the
moment of inertia in two main axis. This made it
possible to write the differential equation of beam
bending as a linear equation with variable coefficients.
The solution of the differential equation makes
it possible to obtain analytical formulae for determining the displacements and angles of rotation of
the cross section of cantilever I-beams with variable flange depth. To confirm the obtained analytical expressions in the transition to the definition of
deflections and angles of rotation of the I-beams
with a constant cross-section, the Lopital-Bernoulli
rule was used. A variant of the formula for the
consequences of the second remarkable (special)
boundary in the disclosure of uncertainty is obtained.
This makes it possible to prove by an analytical
approach the coincidence of the obtained solutions
with 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 different
support conditions.
The obtained displacement formulas make it
possible to check the stiffness of cantilever steel Ibeams with a linear change in their stiffness of the
beams according to the deflection limits. The obtained results can also be used for research of Ibeams with variable flange depth under different
support conditions. The obtained displacement
formulas make it possible to check the stiffness of
cantilever steel I-beams with a linear change in
their stiffness of the beams according to the deflection limits. The obtained results can also be used
for 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.