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 $$15.0$$ $$kN$$-$$m.$$

The bending moment (in $$kN$$-$$m$$ units) at the mid-span location $$X$$ in the beam with overhangs shown below is equal

The degree of static indeterminacy, $${N_s}$$ and the degree of kinematic indeterminacy, $${N_k}$$ for the plane frame shown below, assuming axial deformations to be negligible, are given by:

Identify, from the following, the correct value of the bending moment $${M_A}\,\,$$ (in $$kNm$$ units) at the fixed end $$A$$ in the statically determinate beam shown below (with internal hinges at $$B$$ and $$D),$$ when a uniformly distributed load of $$10$$ $$kN/m$$ is placed on the spans. (Hint: Sketching the influence line for $${M_A}\,\,$$ or applying the Principle of Virtual Displacements makes the solution easy).