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 $1500$ Grade B$.$ Please use bellow equations to calculate the Ov: $0,=(9)*100\%$ 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 H; $2$Sinon, $2$Sini$)$ e $+$H$/2$ Sinon, Sini Sin$($s$)+8)$ Please assume reasonably any missing information and justify your assumptions. $6\mathrm{.}00$ P$-690$ kps P$,-230$ kips $5\mathrm{.}00$ P$.-9.$ kips Po $=23\mathrm{.}0$ kips OO HS$564$ ASTM A$500$ GB HSS$5$x$3$ ASTM A$500$ Gr$.$B $458\mathrm{.}00$ P $-150$ kips P$.-50\mathrm{.}0$ kips $2\mathrm{.}00$ M$-212$ kip$-$in$.$ Mo $-70\mathrm{.}8$ kip$-$in $-256$ kips P$.-853$ kips HSS$8-8\%$ ASTM A$500$ Gr$.$B