Q$1.$ Overlapped K$-$Connection with Rectangular HSS

A planar roof truss contains the welded HSS $45$ overlapped K$-$connection shown in Figure bellow. Note

that the chord moment is necessary for equilibrium because of the noding eccentricity. The connection is

a balanced K$-$connection because the vertical component of the compression branch member force is

equilibrated $($within $20\%)$ by the vertical component of the tension branch member force. The through

branch is the wider and thicker branch member. For fabrication the compression $($through$)$ branch is

tacked initially to the chord, the diagonal $($overlapping$)$ branch is then tacked into place, and finally the

whole connection is welded together. The loads shown consist of live load and dead load in the ratio of

$3$:$1.$ Determine the adequacy of the connection under the given loads. Assume the welds are strong

enough to develop the yield strength of the connected branch wall at all locations around the branch.

All the HSS are H$500$ Grade B$.$

Please use bellow equations to calculate the $O_v$ :

$O_v=(\frac{q}{p})100\%$

$p=$ projected length of the overlapping branch on the chord

$q=$ overlap length measured along the connecting face of the chord beneath the two branches, from

geometry

$q=(\frac{H_{bj}}{2\text{S}\text{i}\text{n}{}_{bj}}+\frac{H_{bi}}{2\text{S}\text{i}\text{n}{}_{bi}})-(\frac{e+\frac{H}{2}}{\frac{\text{S}\text{i}\text{n}{}_{bj}\text{S}\text{i}\text{n}{}_{bi}}{\text{S}\text{i}\text{n}({}_{bj}+{}_{bi})}})$

Please assume reasonably any missing information and justify your assumptions.