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 two-span continuous beam shown below is subject to a clockwise rotational slip $${\theta _A} = 0.004$$ radian at the fixed end $$A.$$ Applying the slope-deflection method of analysis, determine the slope $${\theta _B}$$ at $$B.$$ Given that the flexural rigidity $$EI = 25000\,kN$$ - $${m^2}$$ and span $$L=5$$ $$m,$$ determine the end moments (in $$kN$$-$$m$$ units ) in the two spans, and draw the bending moment diagram.

The frame below shows three beam elements $$OA, OB$$ and $$OC$$, with identical length $$L$$ and flexural rigidity $$EI$$, subject to an external moment $$M$$ applied at the rigid joint $$O.$$ The correct set of bending moments $$\left\{ {{M_{OA}},\,{M_{O{\bf{B}}}},{M_{OC}}} \right\}$$