A SHELL FOR
Emmanuelle Sainte Suzan ibrahim
Broken Shell From baren shores we will set forth to rock in a sea of luminescent reflections nestled within an open shell our bodies cupped inside its silky, shallow depths here, we can extend to reach beyond ourselves and the confines of our feeble bodies in search of the harmony hidden in contradiction finding in balance a strange and transient beauty
40 horizontal layers of laminated Birch Veneer Plywood
Analysis 1 : Weight of Chair and Reactions WEIGHT OF CHAIR Using the mass of a single 1/4” sheet of 5’ x 5’ birch plywood, the density of plywood is calculated. 5’x5’x0.25” Birch veneer plywood = 9.5 kg 5’x5’ = 2 322 576 mm2 = 9.5 kg / 2 322 576 mm2 = 4.1x10-6 kg/mm2 By multiplying the sum of the surface area of each layer of the chair by density of plywood, the mass of the chair is determined. Total Layer Surface Area = 2877015.108 mm2 Density of plywood
= 4.1x10-6 kg/mm2
Mass of Chair = 4.1x10-6 kg/mm2 x 2877015.108 mm2 = 11.8 kg To convert the mass of the chair into its weight, the mass was multiplied by its acceleration towards the earth caused by gravity. Weight of Chair =mxg = 11.8 kg x 9.8 m/s2 =115.7N
REACTION OF CHAIR As only the bottom-most layer, Layer 40, touches the ground, the reaction will be distributed accross this layer’s surface. The shape of this surface is such that in the x-axis, the centre of gravity of the chair falls on the layer’s axis of symmetry, whereas in the y-axis, the centre of gravity falls eccentrically to the axis of symmetry. Because of the symmetry in the x-axis, we can assume that the reaction of the chair against the floor will be uniformly distributed in this direction. However, because of the eccentricity of the load in the y-axis, the reaction of the chair will not be uniformly distributed in this direction. The result is a reaction that takes the shape of a wedge. In order to calculate this reaction the following simplifying assumption was made: 1 - The reaction will occur in a rectangular area over the face of the layer The width of this rectangle is roughly that of the width of the layer, such that the centre of gravity is at the halfway point between edges AD and BC. The length of this rectangle is such that the center of gravity is at the 1/3 point between the edges AB and CD. [The short edge furthest to the centroid (CD) represents the line where the reaction goes to zero, and therefore any area past this edge can be assumed to lift of the ground and not contribute to the reaction.]
The value RAB is assigned to the maximum reaction between the floor and the chair - exerted at the AB edge of the rectangle. Because of the distribution of the load, the maximum reaction between the floor and the chair is twice what it would be if the reaction were uniformly distributed, (x). Therefore, RAB = 2x or x = RAB/2
x is found by dividing the weight of the chair (W) by the area of the reaction rectangle (A). To find RAB, x is simply multiplied by 2.
Area of Reaction Rectangle =lxk = 178 mm x 110 mm = 19 580 mm2 x
= W/A = 115.7 N / 19 580 mm2 = 0.0059 N/mm2 RAB = 2x = 2 (0.0059 N/mm2) = 0.0118 N/mm2
The wedge-like shape of the reaction means that the reaction goes to zero at edge CD. Therefore, RCD
= 0 N/mm2
REACTION OF CHAIR + PERSON The reaction of the chair against the floor was calculated after adding the weight of a 1000N person sitting on the chair. This was done by calculating the reaction the chair itself, and of the person sitting on the chair separately, and adding these two reactions at the end. In order to calculate this reaction the following simplifying assumption was made: 1 - The centroid of the distributed weight of the person on the chair corresponds with the centroid of bottom layer, and therefore of the simplified reaction rectangle.
PERSON The reaction of the person on the reaction rectangle (RAB2) is found by dividing the weight of the person (W) by the area of the rectangle (A). W A
= 1000N = 19 580 mm2
RAB2 = W/A = 1000 N / 19 580 mm2 = 0.0511 N/mm2 Because the person’s weight is at the center, the load of the person is uniformly distributed accross the reactions, which means that RAB2 is equal to RCD2. RCD2 = RAB2 = 0.0511 N/mm2 CHAIR The reaction of the chair on the floor is the same as what was previously calculated. RAB RCD
= 0.0118 N/mm2 = 0 N/mm2
TOTAL REACTION OF CHAIR + PERSON The individual reactions for the chair and the person are then added up. Reactions at ABtot = RAB + RAB2 = 0.0118 N/mm2 + 0.0511 N/mm2 = 0.0629 N/mm2 Reactions at CDtot = RCD + RCD2 = 0 N/mm2 + 0.0511 N/mm2 = 0.0511 N/mm2
Sub-shape
Area (mm2)
1
14388.71585 1 2 3
2
26736.23412 1 2 3
2b 3
3985.305983 25202.71953 1 2 3 4 5 6 7 8 9 0 11
3b
15579.84594 1 2
4
54526.46517 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
5 6
52465.87516 41940.79152 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
7 8 9
68303.91646 67898.81496 76122.30574 1 2 3 4 5 6 7
Sub-shape Area 16206 1781 5953 8472 28694 3136 12260 13299 3984 27001 9926 2224 866 624 529 514 1120 3390 4061 2077 1671 16479 9361 7118 57920 6949 3675 1068 584 584 584 280 280 584 2587 21014 1517 1450 1450 1777 3781 3874 1528 1907 2443 46062 4687 2496 1722 1074 584 584 584 584 584 584 584 6902 950 950 958 1053 1231 3488 4311 1327 6164 1511 1043 1022 1083
82928 6951 12202 6478 1820 1758 1542 1671
Mass (kg) 0.059008124 0 0 0 0.109645296 0 0 0 0.01634374 0.103356353
0.063892948
0.223613034
0.215162554 0.171999186
0.280114361 0.27845304 0.312177576
Weight (N) 0.578672016 0 0 0 1.075253043 0 0 0 0.160277336 1.013579577 0 0 0 0 0 0 0 0 0 0 0 0.626575781 0 0 2.192894757 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.110023861 1.686735818 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.746983502 2.730691506 3.061416224 0 0 0 0 0 0 0
xi
yi
zi
Aixi 237
155.8 169.84 200.57
295 245 173
139 160 216 530.4
327 256 157 166
149 100 127 146 157 174 211 271 288 256 234
301 226 165 145 122 107 84 98 153 208 257
537 500
194 129
137 80 97 118 143 156 182 207 232 274 400 289 260 231 202 501 584 590 568 521
372 285 203 174 149 124 106 94 81 93 134 192 230 277 314 213 232 182 137 102
128 66 56 63 87 104 121 148 174 202 232 361 490 527 564 597 622 630 557 470 367 274 244 215 193
437 380 299 233 194 167 141 117 98 79 62 52 63 72 91 120 165 244 277 245 218 263 308 354 398
231
225
219
213 207
201 195 189 192 126 43 61 87 133 180
574 484 330 229 176 127 89
2987754.979 277438.7311 1010987.444 1699328.804 7383207.131 435888.432 1961572.608 2872495.807 2113250.284 13908006.73 1478921.18 222405.39 109995.0048 91159.1442 83081.4327 89479.2042 236311.56 918621.0576 1169507.03 531607.68 390922.038 0 5026856.463 3559139.55 19876565.23 952042.0083 294034.2088 103614.1347 68963.93334 83574.93617 91172.65764 50970.51188 57971.95582 135590.1062 708882.4678 8405679.168 438460.9047 376909.5577 334869.6455 359011.2986 1894414.02 2262703.605 901706.8188 1082955.865 1273037.428 0 15526023.44 599940.1728 164762.781 96448.61 67670.38551 50846.28984 60781.77176 70717.25368 86497.13673 101692.5797 118056.9028 135590.1062 2491533.014 465324.9865 500461.7712 540104.5895 628353.8959 765583.1516 2197150.226 2401462.315 623652.0593 2262196.429 414038.9351 254377.3831 219833.5587 208947.1319 0 0 61139142.67 1334633.933 1537506.287 278562.85 111016.2969 152952.2521 205019.8189 300751.2585
Aiyi 3449443.201 525317.238 1458383.913 1465742.05 6913217.224 1025435.376 3138516.173 2087878.897 661386.7782 8486423.169 2987619.296 502636.1814 142906.896 90534.7665 64560.0942 55024.5681 94076.64 332195.0688 621300.6099 431931.24 429345.999 0 1816033.806 918258.0039 11347769.9 2585106.767 1047496.869 216841.952 101692.5797 87081.57685 72470.57402 29686.1223 26325.42921 47339.64916 240606.0931 2815902.521 291295.826 333419.9933 401553.6441 558067.0682 805409.5536 898882.2539 278153.6288 261205.904 249231.8957 0 10392621.91 2048233.246 948634.1937 514966.6856 250273.0131 113381.3819 97601.49888 82406.05594 68379.49323 57275.13108 46170.76893 36235.28701 358891.182 59827.49827 68374.28373 87144.53484 126302.2906 203088.7782 850959.7699 1194264.024 325095.2224 1343757.007 397416.934 321099.3197 361958.5105 430885.7954 0 0 23895576.47 3989999.362 5905976.531 2137807.919 416766.0981 309420.6479 195770.8045 148704.7889
Aizi 3410125.657 0 0 0 6176070.081 0 0 0 0 5670611.895 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11941295.87 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11175231.41 8681743.844 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13729087.21 13240268.92 14387115.79 0 0 0 0 0 0 0
x-bar
y-bar
z-bar
184.363072
212.8521078
237
257.30563
240.9264271
231
515.086516
314.2968097
225
343.173019
195.9215999
219
337.070751
225.6243454
213 207
737.260115
288.1501882
201 195 189
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 10 11 12
1671 1042 9574 1584 1549 1844 1699 3323 8524 2257 2257 8413 2257 2257 2257 86514.45626 99103.59992 110972.0908
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 13 14 15
115652.9335 114538.8315 109409.3173 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
16 17 18
92243.07993 92177.44761 84366.76197 1 2 3 4 5 6 7 8 9 10
120152 6368 4835 3036 3036 9085 2340 2340 2340 2340 2340 2340 9524 2340 2340 4666 6636 6614 2340 2340 7087 6023 8408 21439
116394 13944 2340 2340 2340 2340 14031 2340 2340 2340 2340 2340 9469 2340 2340 2340 2340 2340 4730 2340 2340 6142 5568 18055 2340 2340 2340
91460 12427 2256 2256 2256 14388 2786 2786 2786 12835 2786
0.354795785 0.406423863 0.455096544
0.47429268 0.469723748 0.44868761
0.378288871 0.378019713 0.345988091
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.479358086 3.985656579 4.462972525 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4.651222312 4.606416395 4.400122353 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.709746555 3.707107015 3.392984111 0 0 0 0 0 0 0 0 0 0
229 274 4001 533 578 618 648 665 610 510 463 365 264 232 209
61 49 32 58 93 131 186 252 322 304 279 256 314 368 420 183 177 171
23 145 93 61 27 53 72 101 141 192 244 394 544 595 653 684 625 524 473 367 263 285 411
656 607 551 500 372 249 199 152 103 77 50 33 62 93 158 277 381 367 343 319 399 517 602 165 159 153
359 204 153 101 64 39 40 67 105 150 202 254 393 534 587 630 672 685 698 683 630 566 490 379 554 516
678 656 626 586 537 373 208 159 111 68 46 39 55 84 132 181 229 313 395 435 489 454 414 557 605 654 147 141 135
347 152 98 63 51 101 152 206 404 594
633 603 554 504 313 132 87 42 10 63
382622.4344 285388.2072 38303805.86 844295.1953 895483.1695 1139716.671 1100696.083 2209494.42 5199677.957 1150926.894 1044861.082 3070744.827 595773.9216 523558.9008 471654.3546 0 0 39300889.08 146469.1255 701012.4292 282303.8273 185167.0265 245292.865 124005.1456 168459.8205 236311.6926 329900.4818 449226.188 570891.6139 3752569.748 1272807.533 1392133.239 3046634.07 4538993.781 4133469.621 1226013.138 1106687.432 2600967.3 1584040.565 2396275.99 8811256.447 0 0 42817119.78 5006052.708 477302.8247 357977.1185 236311.6926 149742.0627 547199.8565 93588.78916 156761.2218 245670.5715 350957.9593 472623.3852 2405192.497 919509.8535 1249410.335 1373415.481 1474023.429 1572291.658 3239865.786 1633124.371 1598028.575 3869491.975 3151526.375 8846797.365 886753.7773 1296204.73 1207295.38 0 0 32973920.09 4312002.734 342976.6057 221129.6536 142154.7773 733793.8752 281356.7114 423427.922 573856.2628 5185480.352 1654711.748
101921.2598 51036.5772 306353.8584 91874.52406 144082.9321 241590.4271 315940.5424 837282.096 2744748.036 686042.6976 629624.7126 2153727.878 708609.8916 830472.7392 947822.148 0 0 44974182.57 4177554.189 2934583.066 1672574.289 1517762.512 3379590.585 582590.2125 465604.2261 355637.3988 240991.1321 180158.4191 116985.9864 314301.5271 145062.6232 217593.9348 737164.1394 1838159.762 2519763.081 858677.1405 802523.867 2260786.291 2403164.203 4346928.726 12906025.26 0 0 44339904.75 9454327.956 1534856.142 1464664.55 1371075.761 1256429.494 5233475.55 486661.7036 372015.4369 259708.8899 159100.9416 107627.1075 369301.2102 128684.5851 196536.4572 308843.0042 423489.2709 535795.8179 1480405.826 924189.2929 1017778.082 3003462.818 2527902.781 7474641.039 1303223.889 1415530.436 1530176.703 0 0 30003846.01 7865987.696 1360624.297 1250059.471 1137238.219 4503480.058 367713.7218 242356.7712 116999.8206 128353.4741 175499.7308
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15832145.5 17541337.19 18976227.52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19082734.02 18211674.21 16739625.55 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13559732.75 12997020.11 11389512.87 0 0 0 0 0 0 0 0 0 0
327.093119
374.3107599
183 177 171
367.864655
380.9477108
165 159 153
360.527956
328.0539667
147 141 135
11 12 13 14 15 16 17 19 20 21
2786 10249 2786 2786 2786 2786 9721 83521.54194 83868.58899 83470.41294
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 22 23 24
82903.45114 81975.92004 80628.93474 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
25 26 27 28 29
73491.98939 72041.1695 75621.61586 73993.97757 72095.94392 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
89898.2559 9127.674 2784.9672 2784.9672 2784.9672 11593.4526 2784.9672 2784.9672 2784.9672 2784.9672 9583.1244 2784.9672 2784.9672 2784.9672 9351.666 2784.9672 2784.9672 2784.9672 2784.9672 2784.9672 8467.8309
86130.5238 6325.9074 2274.4521 2274.4521 2274.4521 2274.4521 7663.3263 2274.4521 2274.4521 2274.4521 2274.4521 2274.4521 10926.1431 2274.4521 2274.4521 2274.4521 2274.4521 7258.2741 2274.4521 2274.4521 2274.4521 2274.4521 2274.4521 2274.4521 2274.4521 8467.8309
76619.2635 11205.1998 2274.4521 2274.4521 2274.4521 6935.3523 2274.4521 2274.4521 2274.4521 2274.4521 2274.4521 7124.8122 2274.4521 2274.4521 2274.4521 2274.4521 8769.2868 2274.4521 2274.4521 2274.4521 8467.8309
0.342521844 0.343945083 0.342312163
0.339987053 0.336183248 0.330659261
0.301390648 0.295440836 0.310124247 0.303449302 0.295665466
0 0 0 0 0 0 0 3.358991837 3.372949053 3.356935578 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.334134035 3.29683145 3.242659646 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.955632603 2.897284876 3.041279944 2.975821097 2.899487742 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
643 676 650 622 586 540 381
113 277 440 489 540 590 408 129 123 117
331 167 105 72 63 105 135 188 250 405 560 620 650 686 662 632 600 560 500 383
655 613 569 519 337 155 105 55 26 13 35 86 136 285 436 486 535 585 615 412 111 105 99
309 206 155 105 87 77 104 126 162 205 254 407 558 607 653 682 687 672 646 609 569 511 463 413 378
660 619 587 547 496 351 205 153 107 70 44 36 57 95 146 196 313 429 478 526 565 600 621 640 410 93 87 81 75 69
358 191 145 118 111 131 160 192 243 292 410 529 578 620 653 650 617 566 519 378
603 560 510 460 343 223 174 124 88 65 65 79 116 166 215 357 498 525 547 407
1791211.539 6928031.495 1810711.509 1732711.628 1632426.068 1504283.407 3703653.802 0 0 33460754.66 3021260.094 465089.5224 292421.556 200517.6384 730387.5138 292421.556 375970.572 523573.8336 696241.8 3881165.382 1559581.632 1726679.664 1810228.68 6415242.876 1843648.286 1760099.27 1670980.32 1559581.632 1392483.6 3243179.235 0 0 32890154.64 1954705.387 468537.1326 352540.0755 238817.4705 197877.3327 590076.1251 236543.0184 286580.9646 368461.2402 466262.6805 577710.8334 4446940.242 1269144.272 1380592.425 1485217.221 1551176.332 4986434.307 1528431.811 1469296.057 1385141.329 1294163.245 1162245.023 1053071.322 939348.7173 3200840.08 0 0 0 0 29235642.1 4011461.528 434420.3511 329795.5545 268385.3478 769824.1053 297953.2251 363912.336 436694.8032 552691.8603 664140.0132 2921173.002 1203185.161 1314633.314 1410160.302 1485217.221 5700036.42 1403336.946 1287339.889 1180440.64 3200840.08
314785.2315 2838853.142 1225712.406 1362212.197 1504283.407 1643568.908 3966117.457 0 0 29966469.2 5978626.47 1707184.894 1584646.337 1445397.977 3906993.526 431669.916 292421.556 153173.196 72409.1472 124580.6172 97473.852 239507.1792 378755.5392 2665224.81 1214245.699 1353494.059 1489957.452 1629205.812 1712754.828 3488746.331 0 0 29334758.56 4175098.884 1407885.85 1335103.383 1244125.299 1128128.242 2689827.531 466262.6805 347991.1713 243366.3747 159211.647 100075.8924 393341.1516 129643.7697 216072.9495 332070.0066 445792.6116 2271839.793 975739.9509 1087188.104 1196361.805 1285065.437 1364671.26 1412434.754 1455649.344 3471810.669 0 0 0 0 26069583.31 6756735.479 1273693.176 1159970.571 1046247.966 2378825.839 507202.8183 395754.6654 282032.0604 200151.7848 147839.3865 463112.793 179681.7159 263836.4436 377559.0486 489007.2015 3130635.388 1132677.146 1194087.353 1244125.299 3446407.176
0 0 0 0 0 0 0 10774278.91 10315836.45 9766038.314 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9202283.077 8607471.605 7982264.54 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6834755.013 6267581.747 6125350.885 5549548.318 4974620.13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
372.206939
333.3376037
129 123 117
381.864096
340.5849316
111 105 99
340.2484195
93 87 81 75 69
381.570388
30 31 32
70078.27615 67833.81642 65490.17507 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
33 34 35
62915.08147 60221.72679 57201.12314 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
36 37
54158.79152 50211.24316 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
38
81976.41532
39
55522.79746
40
35662.60586
70354.9539 7117.3458 2274.4521 2274.4521 7061.3478 2274.4521 2274.4521 2274.4521 2274.4521 9065.1429 2274.4521 2274.4521 2274.4521 9075.4092 2274.4521 2274.4521 2274.4521 2274.4521 8467.8309
61220.7468 5989.9194 2274.4521 2274.4521 6658.1622 2274.4521 2274.4521 2274.4521 7899.4512 2274.4521 2274.4521 2274.4521 7186.41 2274.4521 2274.4521 2274.4521 8467.8309 54816.4422 4218.516 2274.4521 2274.4521 5357.142 2274.4521 2274.4521 2274.4521 7413.2019 2274.4521 2274.4521 6615.2304 2274.4521 2274.4521 2274.4521 8467.8309
0.28739101 0.278186481 0.268575208
0.258014749 0.246969302 0.234581806
0.222105204 0.205916308
0.336185279 87818.8635 0.227698992 59967.3249 0.146252347 38365.1631
Total
2877015.108
11.79863896
2.818343053 2.728077455 2.633823063 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.53026034 2.421941501 2.300461668 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.178107999 2.019349164 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.296851369 0 2.232964324 0 1.434245575 0 115.7051227
63 57 51 341 232 183 144 172 196 233 282 408 537 578 616 638 599 549 499 450 383
598 541 494 366 241 193 143 104 98 121 174 220 346 471 501 535 551 412 45 39 33
354 253 210 189 216 259 297 408 518 561 590 604 558 508 459 383
543 496 450 342 236 179 133 133 141 191 242 357 467 490 509 412
374 295 246 215 244 279 320 416 538 563 569 545 494 445 383
511 495 448 362 265 218 166 146 192 242 343 440 473 494 408
27 21
15 9 412
341 3
0 0 27834559.07 2427014.918 527672.8872 416224.7343 1016834.083 391205.7612 445792.6116 529947.3393 641395.4922 3698578.303 1221380.778 1314633.314 1401062.494 5790111.07 1362396.808 1248674.203 1134951.598 1023503.445 3243179.235 0 0 24259119.44 2120431.468 575436.3813 477634.941 1258392.656 491281.6536 589083.0939 675512.2737 3222976.09 1178166.188 1275967.628 1341926.739 4340591.64 1269144.272 1155421.667 1043973.514 3243179.235 0 21847948.22 1577724.984 670963.3695 559515.2166 1151785.53 554966.3124 634572.1359 727824.672 3083891.99 1223655.23 1280516.532 3764066.098 1239576.395 1123579.337 1012131.185 3243179.235 0 0 24706537.86 24706537.86 0 0 0
0 0 24112973.06 4256172.788 1230478.586 1123579.337 2584453.295 548142.9561 438969.2553 325246.6503 236543.0184 888384.0042 275208.7041 395754.6654 500379.462 3140091.583 1071266.939 1139500.502 1216831.874 1253223.107 3488746.331 0 0 20672453.14 3252526.234 1128128.242 1023503.445 2277091.472 536770.6956 407126.9259 302502.1293 1050627.01 320697.7461 434420.3511 550417.4082 2565548.37 1062169.131 1114481.529 1157696.119 3488746.331 0 18709367.65 2155661.676 1125853.79 1018954.541 1939285.404 602729.8065 495830.5578 377559.0486 1082327.477 436694.8032 550417.4082 2269024.027 1000758.924 1075815.843 1123579.337 3454875.007 0 0 20448857.79 20448857.79 0 0 0
4414931.397 3866527.536 3339998.928 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2831178.666 2348647.345 1887637.064 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1462287.371 1054436.106 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1229646.23 0 499705.1771 0 106987.8176 0 0
395.630407
342.7331229
63 57 51
396.25651
337.6707116
45 39 33
398.565601
341.3094119
27 21
444.980062
368.2966047
15 9 3 387.551177
310.3290451
126.3158
Analysis 2 : Overturning and Friction FRICTION Using the coefficient of static friction (μ) for wood against a wooden surface, and the normal force (FN), the necessary applied force for the chair to slide from a static position is determined. μ[wood against wood] = 0.3 FN = 115.7 N FFriction = FN x μ = 115.7 N x 0.3 = 34.71 N OVERTURNING The edge of the bottom layer of the chair, or the footprint perimeter forms a pivot point at which the chair can overturn. By comparing the distance of this pivot point from the chair’s center of gravity (dPivot) and the chair’s weight (WChair), with the distance of an applied force from the pivot point (dApp), it is possible to determine the magnitude of force necessary to overturn the chair (FApp). WChair x dPivot FApp
= FApp x dApp = (WChair x dPivot)/ dApp = (115.7 N x 136 mm) / 240 mm = 15 735.2 Nmm / 240 mm = 65.56 N
The minimum force necessary to overturn the chair is greater than the force necessary for it to overcome static friction, therefore, the chair would slide instead of overturning.
Overturning footprint Perimeter
PERSON If a person were to sit on one edge of the chair such that their center of gravity fell outside of the footprint perimeter of the chair, they could disturb the stability of the chair and cause it to tip over. The distance at which this would occur (dPerson)can be determined by comparing the person’s weight (WPerson) with the weight of the chair (WChair), and the distance between the center of gravity of the chair and the pivot point (dPivot). Assuming a person of 1000N, WPerson x dPerson = WChair x dPivot dPerson = (WChair x dPivot) / WPerson = (115.7 N x 136 mm) / 1000N = 15 735.2 Nmm / 1000 N = 15.73 mm Therefore a distance of 15.73 mm between the pivot point and a person’s center of gravity would be sufficient to make the chair tip over.
Analysis 3 : Frame Analysis RACKING
SPREADSHEET FORMULAE
Because of the solidity and geometry of the chair, it cannot rack. It will instead rock, provided that the applied force is smaller than the force necessary to overcome friction (see Analysis 2). If the force is greater than FFriction, the chair will slide accross the floor.
Active Area Percentage
= (Overlap Area / Total Area) x 100%
Mass of Shape
= Area of Layer x Density of Plywood = Area x 4.1x10-6 kg/mm2
By taking the sum of the moments around the center of gravity of the chair, it is possible to determine whether the chair will rock or slide, as a result of the minimum necessary force being applied laterally at the highest point of the chair with a magnitude great enough to set the chair in motion. FN
= 115.7 N
Total height of chair
= 240 mm
dFriction = z-bar = 114 mm dt
= Total height - dFriction = 240 mm - 114 mm = 126 mm
ΣMCOG = FFriction (dFriction) + FApp (dt) FApp = (34.71 N)(114 mm) / 126 mm = 31.4 N The minimum force required to set the chair in motion is smaller than the force necessary to overcome friction, and therefore the chair would rock.
Total Mass Above = Σm1 + m2 + m3... [where mn = mass of layer] Total Weight Above =Mxg = M x 9.8 m/s2 [were M= Σmn] Bearing Stress = Total weight above / Area of Layer
Shape 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Total Area (mm^2) 14389 26736 25203 54526 52466 41941 68304 67899 76122 86514 99104 110972 115653 114539 109409 92243 92177 84367 83522 83869 83470 82903 81976 80629 78744 72041 75622 73994 72096 70078 67834 65490 62915 60222 57201 54159 50211 81976 55523 35663
Overlap Area (mm^2) 0 14389 16067 24716 29811 22655 50703 50803 51840 60454 70184 79278 85656 85170 91032 84923 85962 79059 76943 67846 78869 76820 75636 72073 69595 67093 64813 62444 59639 56749 53793 51102 48287 45011 41742 38237 34743 19186 55523 35640
Active Area Percentage 0 54 64 45 57 54 74 75 68 70 71 71 74 74 83 92 93 94 92 81 94 93 92 89 88 93 86 84 83 81 79 78 77 75 73 71 69 23 100 100
Mass of Shape (kg) 0.059008124 0.109645296 0.103356353 0.223613034 0.215162554 0.171999186 0.280114361 0.27845304 0.312177576 0.354795785 0.406423863 0.455096544 0.47429268 0.469723748 0.44868761 0.378288871 0.378019713 0.345988091 0.342521844 0.343945083 0.342312163 0.339987053 0.336183248 0.330659261 0.322930085 0.295440836 0.310124247 0.303449302 0.295665466 0.28739101 0.278186481 0.268575208 0.258014749 0.246969302 0.234581806 0.222105204 0.205916308 0.336185279 0.227698992 0.146252347
Total Mass Above (kg) Total Weight Above (N) 0 0.059008124 0.16865342 0.272009773 0.495622806 0.71078536 0.882784546 1.162898908 1.441351948 1.753529524 2.108325309 2.514749172 2.969845716 3.444138396 3.913862145 4.362549755 4.740838626 5.118858338 5.464846429 5.807368273 6.151313356 6.49362552 6.833612573 7.169795821 7.500455082 7.823385167 8.118826003 8.42895025 8.732399552 9.028065018 9.315456029 9.59364251 9.862217718 10.12023247 10.36720177 10.60178357 10.82388878 11.02980509 11.36599037 11.59368936
0 0.578672016 1.653925059 2.667504636 4.860399393 6.970423254 8.657159071 11.40414257 14.13483408 17.1962503 20.67560839 24.66126497 29.12423749 33.77545981 38.3818762 42.78199855 46.49174511 50.19885212 53.59183623 56.95082807 60.32377712 63.6807127 67.01484674 70.31167819 73.55433783 76.72120015 79.61848503 82.65976497 85.63558607 88.53507381 91.35341686 94.08149432 96.71531738 99.24557772 101.6675192 103.9679809 106.1460889 108.1654381 111.4622894 113.6952537
Bearing Stress (N/mm^2) 0 4.02171E-05 0.000102939 0.000107928 0.000163041 0.000307677 0.000170742 0.00022448 0.000272665 0.00028445 0.000294591 0.000311074 0.000340015 0.000396565 0.000421629 0.000503776 0.000540843 0.000634954 0.000696509 0.000839408 0.000764864 0.000828956 0.000886021 0.000975557 0.001056885 0.001143503 0.001228439 0.001323745 0.001435902 0.00156013 0.001698237 0.001841038 0.002002921 0.002204905 0.002435621 0.002719024 0.003055187 0.005637793 0.002007505 0.003190131
0 N/mm2
4.02x10^-5 N/mm2
1.03x10^-4 N/mm2
1.08x10^-4 N/mm2
1.08x10^-4 N/mm2
3.07x10^-4 N/mm2
1.71x10^-4 N/mm2
2.24x10^-4 N/mm2
2.72x10^-4 N/mm2
2.84x10^-4 N/mm2
void bearing area footprint of layer above outline of shape
2.93x10^-4 N/mm2
3.11x10^-4 N/mm2
3.4x10^-4 N/mm2
3.97x10^-4 N/mm2
4.22x10^-4 N/mm2
5.04x10^-4 N/mm2
5.41x10^-4 N/mm2
6.35x10^-4 N/mm2
6.97x10^-4 N/mm2
8.39x10^-4 N/mm2
7.64x10^-4 N/mm2
8.29x10^-4 N/mm2
8.86x10^-4 N/mm2
9.76x10^-4 N/mm2
void bearing area footprint of layer above outline of shape
1.06x10^-3 N/mm2
1.14x10^-3 N/mm2
1.23x10^-3 N/mm2
1.32x10^-3 N/mm2
1.44x10^-3 N/mm2
1.56x10^-3 N/mm2
1.70x10^-3 N/mm2
1.84x10^-3 N/mm2
2.00x10^-3 N/mm2
2.20x10^-3 N/mm2
2.44x10^-3 N/mm2
2.72x10^-3 N/mm2
3.06x10^-3 N/mm2
5.64x10^-3 N/mm2
2.01x10^-3 N/mm2
3.19x10^-3 N/mm2
void bearing area footprint of layer above outline of shape
Analysis 4: Columns & Beams COLUMN In order to calculate the forces in this column, we made the following simplifying assumptions: 1 - the column is carrying 1/3 of the self-weight of the chair 2 - the centre of gravity of the system created by the weight of the person on the chair and the weight of the chair on the column passes through the centre of the column. 3- the column is approximated into a circle with a diameter of 100 mm BUCKLING The critical buckling stress for the column correlates to the modulus of elasticity of the material (E), and its slenderness ratio (l/r). It is given by the formula (P/A)cr = (π2E) / (l/r)2 In order to find the slenderness ratio, it is necessary to determine the radius of gyration of the column (r), given by the moment of inertia (I), and the cross-sectional area (A). Radius of column (R) = 50 mm
ICircle = (πR ) / 4 = (π x (50 mm)4) / 4 = 4 908 379 mm4 4
ACircle = πR2 = π (50 mm)2 = 7 854 mm2 r = √(I/A) = √(4 908 379 mm4/ 7 854 mm2) = 25 mm
The radius of gyration is then inserted into the formula for critical buckling stress. E
= 8 273 MPa [from plywood properties table]
l
= number of layers x layer thickness = 23 x 6 mm = 138 mm
l/r
= 138 mm / 25 mm = 5.52
(P/A)cr = (π2E) / (l/r)2 = (π2 x 8 273 MPa) / (5.52)2 = 14 792 MPa
From the critical buckling stress, the critical load for buckling can be determined. (P/A)cr = 14 792 MPa Pcr = 14 792 MPa x 7 854 mm2 = 116 x 106 N Because the critical buckling load is so much bigger than the weight of a person (1000N), there is no need to worry about buckling.
CRUSHING The load at which the column is in danger of crushing is determined by the crushing stress of wood (FC), which can be found in a list of the properties of plywood, and by the cross-sectional area of the column (A). FC
= 2.34 MPa [from plywood properties table]
A
= 7 854 mm2
P
= A x FC = 7 854 mm2 x 2.34 MPa = 18 378 N
Because the critical crushing load is so much bigger than the weight of a person (1000N), there is n need to worry about crushing. AXIAL DEFORMATION Axial deformation (δ) in the column can be measured as a function of a critical load (P), the column’s length (l), cross-sectional area (A), and modulus of elasticity (E). P
= 33 674 N
l
= 138 mm
A
= 7 854 mm2
E
= 8 273 MPa
δ
= Pl/AE = (33 674 N x 138 mm) / (7 854 mm2 x 8 273 MPa) = (4 647 012 Nmm) / (64 976 142 N) = 0.0715 mm
Because of the small result, no need to worry about axial deformation.
Section A
Section B
BEAM 1 In this cross-section of the chair, the top of the chair can essentially be considered two beams that are each supported by a side of the chair and by the column. In this section, we will assume a person with an average mass of 60kg, or 588 N in weight, and this weight is distributed accross 350mm. BEAM 1 - SECTION A
Placing the weight of a person on this beam such that its centroid falls at the half-way point of the beam’s length would place this centroid outside the footprint perimeter of the chair, causing it to tip. The resulting slope of the chair on the floor can be determined using a ratio of the sides of the triangle formed by the chair and the floor. Ratio 92 mm : 224 mm 4 : 10 Slope a2 + b2 = c2 42 +102 = c2 √(116) = c 10.8 = c It is then necessary to project the weight of a person onto the sloped member, which can be done by using the method of similar triangles to find the projected load’s components parallel and perpendicular to the slope. (FPERP / 10) = (588 N / 10.8) FPERP = 54.4 N x 10 = 544 N
With this value, it is now possible to find the magnitude of the reactions using the sum of the perpendicular values. ΣV = 0, therefore R1 + R2 = 544 N
uniformly distributed load, therefore R1 = R2 R2 + R2 = 544 N 2R2 = 544 N R2 = (544 N / 2) = 272 N
The beam can be checked for deflection, by using the formula for deflection caused by a uniformly distributed load found in the beam tables. Δmax = 5wl4 / 384 EI w
=W/l = 544 N / 280 mm = 1.9 N/mm
l
= 0.280 m
EI
= 440 000 Nmm2 [from plywood properties table]
Δmax = 5wl4 / 384 EI = (5 x 1.9 N/mm x (0.280 m)4 ) / (384 x 440 000 mm2) = 0.05839232 Nmm3 / 168 960 000 Nmm2) = 3.36 x 10-10 mm Comparing this deflection to the permissible deflection due to live load of L/240, it is evident that this beam is more than adequate in deflection. L/240 = 280 mm / 240 = 1.16 mm
1.16 mm >> 3.36 x 10-10 mm
The area under the shear diagram corresponds to the value of the maximum moment. ATriangle = (272 N x 140 mm) / 2 = 38 080 Nmm / 2 = 19 040 Nmm MMax = ATriangle = 19 040 Nmm
BEAM 1 - SECTION B Placing the weight of a person on this beam such that its centroid falls at the half-way point of the beam’s length would place this centroid outside the footprint perimeter of the chair, causing it to tip. The resulting slope of the chair on the floor can be determined using a ratio of the sides of the triangle formed by the chair and the floor. Ratio 40 mm : 136 mm 2 : 7 Slope a2 + b2 = c2 22 + 72 = c2 √(53) = c 7.3 =c It is then necessary to project the weight of a person onto the sloped member, which can be done by using the method of similar triangles to find the projected load’s components parallel and perpendicular to the slope. (FPERP / 7) = (588 N / 7.3) FPERP = 80. 5 N x 7 = 563 N
With this value, it is now possible to find the magnitude of the reactions using the sum of the perpendicular values. ΣV = 0, therefore R1 + R2 = 563 N
uniformly distributed load, therefore R1 = R2 R2 + R2 = 563 N 2R2 = 563 N R2 = (563 N / 2) = 282 N
The area under the shear diagram corresponds to the value of the maximum moment. ATrapezoid = (282 N x 33 mm) + (282 N x 175 mm / 2 ) = 9 306 Nmm + 24 675 Nmm = 33 981 Nmm MMax = ATrapezoid = 33 981 Nmm
BEAM 1 The bending and shear stresses that occur in the beam relate directly to its cross-section. In order to calculate the forces in this column, we made the following simplifying assumptions: 1 - The bending and shear stresses are calculated here for a unit thickness of 10 mm 2 - The average section of the beam is 3 layers thick BENDING STRESS The maximum bending stress in the beam is found using the maximum moment occuring in the beam (MMax), and the distance from the neutral axis to the extreme fiber (c), over the moment of inertia (I). FB
= MMaxc / I
MMax1 = 19 040 Nmm MMax2 = 33 981 Nmm c
= h / 2 [by symmetry, the neutral axis is located at middepth] = (6mm x 3 layers)/ 2 = 9 mm
IRectangle = bh3 / 12 = 10 mm (18 mm)3 / 2 = 4 860 mm4 With the variables established, the maximum bending stresses in each section of the beam can be calculated.
FB1
= MMax1c / I = (19 040 Nmm x 9 mm) / 4 860 mm4 = 171 360 Nmm2 / 4 860 mm4 = 35.2 MPa
FB2
= MMax2c / I = (33 981 Nmm x 9 mm) / 4 860 mm4 = 305 829 Nmm2 / 4 860 mm4 = 62.9 MPa
SHEAR STRESS The shear distribution along the cross section of the beam can be found by taking the shear stress at different heights, using values from the shear diagram (V), and the statical moment of area (Q), over the moment of intertia (I) and the breadth of the cross section (b). FV
= VQ / Ib
V
= 272 N
Q1
= 9 mm (10 mm)(4.5 mm) = 405 mm3
Q2
= 4.5 mm (10 mm)(6.25 mm) = 281.25 mm3
I
= 4 860 mm4 [from bending stress calculation]
b
= 10 mm
[from shear stress diagram]
With the variables established, the maximum shear stresses in each section of the beam can be calculated. @ level 1 FV1 = (272 N x 405 mm3) / (4 860 mm4 x 10 mm) = 110 160 Nmm3 / 48 600 mm5 = 2.26 MPa @ level 2 FV2 = (272 N x 281.25 mm3) / (4 860 mm4 x 10 mm) = 76 500 Nmm3 / 48 600 mm5 = 1.57 MPa
Shear Distribution Diagram
BEAM 2 Beam 2 Section A is omitted, due to its similarities with the same section of Beam 1. BEAM 2 - SECTION B Ratio 120 mm : 260 mm 6 : 13 Slope a2 + b2 = c2 62 +132 = c2 √(205) = c 14.3 = c It is then necessary to project the weight of a person onto the sloped member, which can be done by using the method of similar triangles to find the projected load’s components parallel and perpendicular to the slope. (FPERP / 13) = (588 N / 14.3) FPERP = 41.1 N x 13 = 535 N
With this value, it is now possible to find the magnitude of the reactions using the sum of the perpendicular values. ΣV = 0, therefore R1 + R2 = 535 N Taking the ΣMR1,
clockwise
= counterclockwise
535 N (231 mm) = R2 (406 mm) 123 585 Nmm /(406 mm) = R2 R2 = 304 N R1 = 535 N - 304 N =231 N
The area under the shear diagram corresponds to the value of the maximum moment. MMax = ARectangle + ATriangle ARectangle =bxh = 231 N + 56 mm = 12 913 Nmm ATriangle
= (b x h) / 2
bTriangle = a - x [ x / 304 N = 350 mm - 199 mm [ x =151 mm ATriangle
= 350 mm / 535 N ] = 199 mm ]
= (b x h) / 2 = (151 mm x 231 N) / 2 = 34 881 Nmm
MMax = 12 913 Nmm + 34 881 Nmm = 47 794 Nmm The beam can be checked for deflection, by using the formula for deflection found in the beam tables. First, all the variables are determined. w
=W/l = 535 N / 350 mm = 1.5 N/mm
a
= 350 mm
b
= 56 mm
x
= 199 mm
EI
= 440 000 Nmm2 [from plywood properties table]
l
= 406 mm
With these variables established, it is possible to use the equation from the beam tables. Δmax = (wx / 24EIl) (a2 (2b)2 - 2ax2 (2b) + lx3)
= [(1.5 N/mm x 199 mm) / (24 x 440 000 Nmm2 x 406 mm)] x [(350mm)2 (2 x 56 mm)2 - 2(350 mm)(199 mm)2 x (2 x 56 mm) + 406 mm (199 mm)3] = (298.5 N / 4 287 360 000 Nmm3 ) (1 536 640 000 mm4 - 3 104 718 400 mm4 + 3 199 523 194 mm4)
= 6.9 x 10-8 1/mm3 x 1 631 444 794 mm4 = 112.6 mm
Comparing this deflection to the permissible deflection due to live load of L/240, it is found that this beam insufficient in deflection. L/240 = 406 mm / 240 = 1.69 mm
1.16 mm << 112.6 mm
BENDING STRESS The maximum bending stress in the beam is found using the maximum moment occuring in the beam (MMax), and the distance from the neutral axis to the extreme fiber (c), over the moment of inertia (I). MMax = 47 794 Nmm c
= (6mm x 3 layers)/ 2 = 9 mm
IRectangle = bh3 / 12 = 10 mm (18 mm)3 / 2 = 4 860 mm4
With the variables established, the maximum bending stresses in each section of the beam can be calculated. FB
= MMax1c / I = (47 794 Nmm x 9 mm) / 4 860 mm4 = 430 146 Nmm2 / 4 860 mm4 = 88.5 MPa
SHEAR STRESS The shear distribution along the cross section of the beam can be found by taking the shear stress at different heights, using values from the shear diagram (V), and the statical moment of area (Q), over the moment of intertia (I) and the breadth of the cross section (b). FV
= VQ / Ib
V
= 304 N
Q1
= 9 mm (10 mm)(4.5 mm) = 405 mm3
Q2
= 4.5 mm (10 mm)(6.25 mm) = 281.25 mm3
I
= 4 860 mm4 [from bending stress calculation]
b
= 10 mm
[from shear stress diagram]
With the variables established, the maximum shear stresses in each section of the beam can be calculated. @ level 1 FV1 = (304 N x 405 mm3) / (4 860 mm4 x 10 mm) = 123 120 Nmm3 / 48 600 mm5 = 2.53 MPa
@ level 2 FV2 = (304 N x 281.25 mm3) / (4 860 mm4 x 10 mm) = 85 500 Nmm3 / 48 600 mm5 = 1.75 MPa
Shear Distribution Diagram
A Column Bearing B Outer Ring Bearing C ALL Bearing on the Column Trib. Area Outline
Analysis 5: Load Take Down Three layers, from the top, middle and bottom of the chair respectively, are examined to see how the self-weight of the chair and the weight of the person affects each layer. In order to calculate how loads are carried through the structure, we made the following simplifying assumption: 1 - Each layer falls into one or more out of three possible zones of loading. Zone A - area directly bearing on column; any area that falls within the columnâ&#x20AC;&#x2122;s tributary area also carries most of the personâ&#x20AC;&#x2122;s seated weight Zone B - outer- ring; the area unsupported by the column, bearing directly onto the layers below Zone C - outer ring seating area; area that will be occupied by person, where their load would be directly transfered to the layers below LAYER 15 Zone A The area falling into Zone A of layers from 1 to 14 is calculated to determine the weight of the chair bearing on the column. Ex Zone ALayer2
= 1105 mm2
Density of plywood
= 4.1x10-6 kg/mm2
[from AutoCAD file]
mass = 1105 mm2 x 4.1x10-6 kg/mm2 = 0.00452 kg weight = 0.00452 kg x 9.8 m/s2 = 0.04 N
This is repeated from layer 1 to 14. The total combined weight in Zone A for these layers is 9.27 N. This load is assumed to be bearing on layer 15 in the area within the tributary area of the column or Zone A. The area of this zone is determined from the AutoCAD file, and used to determine the bearing stress on the layer. Zone ALayer15 = 24 966 mm2 PC
[from AutoCAD file]
= 9.27 N
FC
=P/A = 9.27 N / 24 966 mm2 = 3.7 x 10-4 MPa = 370 Pa To get the effect on Zone A of this layer caused by a personâ&#x20AC;&#x2122;s weight on the chair, the bearing stress in the overall seating area, (Zone A + Zone C) of this layer must be determined. Zone ACLayer15 = 42 342 mm2 PC
[from AutoCAD file]
= 588 N
FC
=P/A = 588 N / 42 342 mm2 = 0.0139 MPa = 13.9 kPa Adding up these two values for bearing stress, we get the total bearing stress in Zone A of layer 15. FC total = FCa + FCac = 0.37 kPa + 13.9 kPa = 14.2 kPa
Comparing this to the allowable bearing stress for plywood, the member is found more than adequate in compression. FC allowable
= 2.34 MPa = 2 340 kPa
[from plywood properties table]
2 340 kPa >> 14.2 kPa Zone B The area falling outside the tributary area of the column from layers from 1 to 14 is calculated to determine the weight of the chair bearing on the sides of the chair in layer 15. Zone BCLayer15 = 75 668 mm2
[from AutoCAD file]
PC = 29.1 N
[area x density of zone c layers 1-14]
FC
=P/A = 29.1 N / 75 668 mm2 = 3.8 x 10-4 MPa = 380 Pa Zone C
The bearing in Zone C is found by adding the bearing caused by the weight of the person, to that caused by the weight of the chair left unsupported by the column. FC total = FCac + FCbc = 13.9 kPa + 0.38 kPa = 14.3 kPa
Comparing this to the allowable bearing stress for plywood, the member is found more than adequate in compression. 2 340 kPa >> 14.3 kPa
LAYER 26 Zone A
Zone B
The weight of the layers of the column from layer 15 - 26 is added to the self-weight of the chair in Zone A calculated in the previous section. Combined with the weight of the person distributed within Zone A, the bearing on this layer of the column is determined.
The weight of layers 15 to 25 are added to the previously determined self-weight to determine the total bearing in Zone B onto the outer ring of layer 26.
Zone ALayer 15 = 24 966 mm AColumn = 9 130 mm2 2
P(layers 16-25)
[from AutoCAD file]
= (9 130 mm2 x 4.1x10-6 kg/mm2 x 9.8 m/s2 ) 10 layers = 0.37 N x 10 layers = 3.7 N
Zone BCLayer15-25= 828 268 mm2 P(layers 15-25)
= 828 268 mm2 x 4.1x10-6 kg/mm2 x 9.8 m/s2 = 33.2 N PZone B = 33.2 N + 29.1 N = 62.4 N
P(layers 15)
Zone BCLayer26 = 68 054 mm2
P(layers 1-14)
= 9.27 N
FC
Pperson
= 588 N (Zone A15/ Zone AC15) = 588 N (24 966 mm2 / 42 342 mm2 ) = 588 N (0.5896) = 346 N
= 24 966 mm2 x 4.1x10-6 kg/mm2 x 9.8 m/s2 = 1.0 N
PZone A = 3.7 N + 1.0 N + 9.27N + 346 N = 360 N FC =P/A = 360 N / 9 130 mm2 = 0.0394 MPa = 39.4 kPa Comparing this to the allowable bearing stress for plywood, the member is found more than adequate in compression. 2 340 kPa >> 39.4 kPa
[from AutoCAD file]
=P/A = 62.4 N / 68 054 mm2 = 9.16 x 10-4 MPa = 916 Pa
Zone C
LAYER 40
The bearing in Zone C is found by adding the bearing caused by the weight of the person, to that caused by the weight of the chair left unsupported by the column.
In this layer, the loads have been redistributed along the two solid layers above, and the assumption has been made that bearing is more or less even. Therefore, there is only one active zone on this layer.
Zone CLayer26 = 14 784 mm2 Pperson = 588 N - 346 N = 242 N
[from AutoCAD file]
FC total = FC + FCbc = (P / A ) + 0.916 kPa = (242 N / 14 784 mm2 ) + 0.916 kPa = 16.4 kPa + 0.916 kPa = 17.3 kPa
The total weight of the chair above is added to the weight of the person to determine bearing at this layer. Players 1-39 = 113.7 N
Pperson = 588 N
[from analysis 3 spreadsheet]
Ptotal = 113.7 N + 588 N = 701 N ALayer40 = 35 663 mm2 FC total = P / A = 701 N / 35 663 mm2 = 0.0197 MPa = 19.7 kPa Comparing this to the allowable bearing stress for plywood, the member is found more than adequate in compression. 2 340 kPa >> 19.7 kPa
Comparing this to the allowable bearing stress for plywood, the member is found more than adequate in compression. 2 340 kPa >> 17.3 kPa
ARCH 365 Structural Design Workshop University of Waterloo School of Architecture