A W 360 X 79 steel beam is fixed at A. The beam has a length of 2.5 m and is subjected to a linearly varying distributed load with maximum intensity q₀ = 500 N/m on segment AB and a uniformly distributed load of intensity q₀ on segment BC. Calculate the state of plane stress at point D located 220mm below the top of the beam and 0.3 m to the left of point B. Find the principal normal stresses and the maximum shear stress at D. Include the weight of the beam. See Table F-1(b), Appendix F, for beam properties.

Table F-1(b)

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Properties of Wide-Flange Sections (W Shapes)-SI Units (Abridged List) Mass Web Flange per Thick- Thick- Designation Meter Area Depth ness Width ness 1 kg mm² mm mm mm mm W 760 x 314 314 40100 785 W 760 x 196 196 25100 770 W 610 x 241 W 610 x 140 W 460 x 177 W 460 x 106 W 410 x 149 W 410 x 114 W 410 X 85 W 410 x 46.1 W 360 X 179 W 360 x 122 W 360 x 79 W 360 X 39 W 310 X 129 W 310 X 74 W 310 X 52 W 310 X 21 241 140 177 106 30800 635 17.9 330 17900 617 13.1 230 22600 483 16.6 287 13400 470 12.6 194 149 114 85.0 10800 46.1 5890 19000 432 14.9 14600 419 11.6 19.7 384 15.6 267 179 22800 122 15500 79.0 10100 39.0 4960 129 16500 74.0 9420 52.0 6650 21.0 2680 264 25.0 262 19.3 417 10.9 181 18.2 404 6.99 140 11.2 353 9.40 353 6.48 318 310 318 302 33.5 25.4 31.0 22.2 368 15.0 373 23.9 363 13.0 257 21.7 205 128 26.9 20.6 16.8 10,7 13.1 307 20.6 9.40 205 16.3 7.62 167 13.2 5.08 101 5.72 S r x 106 mm¹ x 10³ mm³ mm 10900 328 6230 310 4290 2400 2150 1120 912 487 620 462 316 156 574 367 225 102 308 163 119 Axis 1-1 36.9 6780 3640 264 251 3790 201 2080 191 2870 2200 1510 773 180 178 171 163 3110 158 2020 154 1270 150 578 144 137 132 1930 1050 747 133 244 117 I x 106mm 315 81.6 184 45.4 105 25.1 77.4 57.4 17.9 5.16 206 61.6 24.0 3.71 100 23.4 10.2 Axis 2-2 0.982 S r x 10³ mm³ mm 1640 610 1120 393 736 259 585 441 198 73.6 1110 480 234 58.2 651 228 122 19.5 88.6 57.2 77.5 50.3 68.3 43.2 63.8 62.7 40.6 29.7 95.0 63.0 48.8 27.4 78.0 49.8 39.1 19.1