The relevant cross-sectional details of a compound beam comprising a symmetric $${\rm I}$$-section and a channel section (with welded connections), proposed for a steel gantry girder, are given below (all dimensions are in $$mm$$)

$$(a)$$ Determine the depth of the centroidal axis $$\overline y $$ and the second moment of area, $${{\rm I}_{xx}}$$ and $${{\rm I}_{yy\,\,eff}}$$ of the compound section. For computing $${{\rm I}_{yy\,\,eff}}$$ include the full contribution of the channel section, but only the top flange of the $${\rm I}$$-section.

$$(b)$$ Determine the maximum compressive stress that develops at a top corner location on account of a vertical bending moment of $$550.0$$ $$kN$$-$$m,$$ combined with a horizontal bending moment of $$15.0$$ $$kN$$-$$m.$$

Two wheels, placed at a distance of $$2.5$$ $$m$$ apart, with a load of $$200\,\,kN$$ on each of them are moving on a simply supported girder ($${\rm I}$$-section) of span $$6.0$$ $$m.$$ The top and bottom flanges of the $${\rm I}$$-section are of $$200 \times 20\,\,mm$$ and the size of web plate is $$800 \times 6\,\,mm$$. If the allowable and average stresses are $$110\,MPa,$$ $$165$$ $$MPa$$ and $$100$$ $$MPa$$ respectively, check the adequacy of the section against bending and shear stress (self-weight of the girder, may be neglected)