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In order to calculate the temperature reached by the steel and subsequent reduction in yield strength and capacity / demand ratio, we must examine the WTC construction in some detail. NIST have digitised the original drawings into databases, and have identified and corrected some errors. Although the complete specifications have not been made public, the information available is more than adequate for a validity test of fire collapse theories. For a high estimate of the gravity load presented by the upper floors, we make some correction for their lower content of steel and / or concrete. After compiling inventories (which will inevitably have omissions) for steel and concrete per Tower, the total weight of this material is deducted from the total building weight to obtain the mean remaining mass per floor. Then the corresponding concrete and steel content of a typical floor above the fire zone is added. A better, lower estimate would also allow for some reduction of the superimposed dead load and service live load with height. There was much uncertainty here, due to a dearth of detail on the distribution of this remaining weight. Fortunately, NIST's 2004 publications have commendably helped to fill in much of the missing variables. Appendix D of June 2004 provided information on dead and live loads on the relevant 1WTC floors, enabling a lower, more accurate evalualation of the gravity load. This was followed in October 2004 by publication of the column demand / capacity ratios (before and after aircraft impact). The gravity load of the upper floors was confirmed at the lower estimate, with two independent methods providing a match within 2.5%. Four floors (41, 42, 75, and 76) housing heavy mechanical equipment were double-height storeys. This accounts for the extra few inches when one divides 1,368 feet by 110 floors, rather than 114. A normal floor was 12 feet in height. Thus, in calculating average weights per floor it is sometimes appropriate to allow for the extra "floors". For items spanning the mechanical floors we shall divide by an extra 4 floors in the case of columns, and an extra 2 floors for "columns plus beams". It is clearly preferable to deal with total weights rather than per floor averages where possible. The total weight of each Tower is widely quoted as 500,000 tons (tons taken to be short US tons unless otherwise stated). This would include the seven basement levels, but not the underground Plaza complex or ancillary buildings outside each Tower's footprint of slightly under an acre. It was said that the attacks left 1.2 million tons of steel, concrete, and glass on the ground. This would also include 7WTC and structure damage to buildings such as St Nicholas Greek Orthodox Church. Some reports claim 1.5 million tons for "the WTC" or "The Towers"; presumably this would include the whole complex above and below ground. The total debris removed by July 2002 was said to be over 1.6 million tons, including north of Vesey Street where 7WTC had stood. The air conditioning equipment alone weighed 49,000 tons, with 60,000 tons of cooling capacity. Much of this would not be included in the 500,000 tons per Tower, as it was contained under the central Plaza. The 4th basement level contained the 2.5 acre refrigeration plant, with intake and outflow pipes running to the Hudson river 1,500 feet away. But some 100,000 supply and return air-conditioning outlets, and 24,000 induction units, were installed within the Towers. The total weight of steel within each Tower is generally quoted at 86,000 to 100,000 tons. NIST published an incomplete, though useful, inventory in an interim report on structural steel specifications (appendix E, Table E-10, source Feld 1971), showing the various steel contracts for the WTC construction. The total - excluding items such as grillages, floor trusses, and steel decking - came to 158,200 tons or 79,100 tons per Tower as below: 55,800 Exterior columns and spandrels, 9th to 107th floor 25,900 Rolled columns and beams above 9th floor, in cores; also exterior wall steel above 107th floor 6,800 Perimeter bifurcation columns (trees) 4th to 9th floor 13,600 Perimeter box cols. below the bifurcation cols. to 4th floor 13,000 Core box columns below the 9th floor 31,100 Core box columns above 9th floor and built-up beams 12,000 Support for slabs below grade ------- 158,200 TOTAL (The "141,170" total listed by NIST appears to be an error. And it seems reasonable to count all of Levinson's 12,000 tons of below-grade 14WF sections as being within the Towers' footprints, rather than partly used for the sub-Plaza area. The Attachment 1 annex lists the 12,000 tons and Plaza separately.) Calculation of the weight of steel decking is quite straightforward, although the corrugations lead to an error bound of a few hundred tons per Tower. The core area 137' x 87' is 11,919 ft^2 or 1,107 m^2 out of 208' x 208' which is 43,264 ft^2 or 4,019 m^2, making it 27.5% of the total floor area. However, 50% of the core area was typically taken up by services such as elevator shafts and stairwells (from NIST appendix E, fig. E-7). The sky lobbies were on the 44th and 78th floors. If we assume 50% as the fraction of the core area lost to shafts over the middle stories, the upper floors above the floor 78 sky lobby gained core space by losing 11 or 12 express elevator shafts, and the lower floors up to about the 44th lost core space to a similar number of extra shafts. So when we come to consider the fire zone floors of 1WTC, which were all clearly in the upper section, they would have lost about 40% of core floor space. (One source quotes 13% as the proportion of the total area occupied by elevator shafts. This equates to 47% of the core area, and stairwells would add a little to this. If 56 elevator shafts take up 522 m^2, then 11 or 12 shafts account for some 107 m^2, which is about 10% of the core area.) Floors 9 to 106, excluding four floors housing heavy mechanical equipment (41, 42, 75, and 76) and the floors above (43 and 77), incorporated 4 inch thick lightweight concrete poured on 22-gauge, 1.5" fluted non-composite steel decking with composite floor trusses outside the 137' x 87' core area. Extension of the truss diagonals above the top chord provided a shear connection and composite behaviour with the concrete. Within the core, these regular floors featured 5" thick normal-weight concrete slabs on 1.5" fluted steel deck, supported by rolled steel structural shapes acting compositely with the slabs. The mechanical floors and floors 43 and 77 employed rolled steel structural shape framing throughout, typically wide flange "W-shapes" (shaped like an 'H'). Normal-weight concrete was poured onto 1.5" fluted steel deck, acting compositely with the steel beams. On the four mechanical floors slab thickness was 5.75"; on floors 43 and 77 the concrete was 8" thick within the core and 7.75" thick outside. Floors 107 to 110 were also used for mechanical services, although apparently were not double-height storeys. Details of the flooring was not provided by FEMA. NIST (Appendix D) has tables of dead and live loads which indicate a slab thickness (normal-weight) ranging from 5.5 to 8 inches. 22-gauge steel is 0.0299 inches thick. According to the drawing (FEMA Chapter 2, Fig. 2-9) which is not totally to scale, each flute has the steel plate diverting diagonally by going up 3/2" and across 1/2", and then down 3/2" and across 1/2", rather than simply continuing horizontally for 1". Each diagonal is SQR[(1/2)^2 + (3/2)^2] = 1.581". So the total additional length along the axis perpendicular to the double trusses is 2 * (1.581" - 1/2") = 2.162" per flute. Assuming 17 flutes between each double truss, i.e. every 6' 8", there is an extra 17 * 2.162 per 80 length units or 45.9%. Most diagrams and description of steel decking imply that the corrugations only add about 10% to 25% to the area or volume. The average floor had 4019 - 1107 / 2 = 3465 m^2 of decking. Taking the density to be 7860 kg/m^3 and allowing a compromise figure of 30% extra for the corrugations, a single floor contained 7860 * 1.3 * (0.0299 / 39.37) * 3465 / 907.2 = 29.64 tons of steel decking. Details of the lower floors are rather sparse. If we allow for 102 floors (from 9 to 110), these collectively contained 3023 tons of decking, which raises the NIST incomplete total from 79,100 to 82,123 tons of steel. Core columns were attached to two layers of steel grillage foundations. Each grillage was fixed atop a massive reinforced concrete footing that was socketed into the bedrock. These concrete slabs were 2.l metres in height, say 7 feet, and a diagram (Multi-Storey Buildings in Steel, Godfrey G B 1985) shows the length and breadth of a slab to be about three times its height. The length of the lower grillage layer would be about the length of the concrete slab, half of this length supported the orthogonal beams of the upper layer, and the core column rested on half the length of those beams. Thus, the load was spread across four times the area. However, the column spacing would not have allowed all footings to be 21 x 21 feet. Some core columns were only about 10 feet apart, although the typical separation was 20 feet. With the beams of the lower grillage layer aligned to enable the greatest length, this would allow for 16 to 20 feet, say an average of 18 feet. The upper layer of orthogonal beams could be from 10 to 20 feet in length, say an average of 15 feet. The height of each layer appears to have been about the same as the concrete at 7 feet. If the boundary between the core and perimeter load areas is taken to be four lines at the midpoint of the non-core area, the core load area is about 172' by 148'. The perimeter load area consists of two rectangles 30' by 208' which include the corners, and two 18' by 148'. The relative sizes of these areas provide some indication of the respective loading. Thus, the perimeter load is 17,808 / 208^2 = 41.2% of the total, with the core being 25,456 / 208^2 = 58.8% of the total. With a total load of 500,000 tons per Tower, the demand on the core is 294,000 tons, and the perimeter 206.000 tons. With 47 core columns, the demand per column at the grillages is 6255 tons. There are 236 perimeter columns, so the demand per column is much lower at 873 tons. Suppose the grillages were specified as 42 ksi steel, with the design providing a demand / capacity ratio of 1:3. For the core, this would require a cross-section of 6255 * 2000 * 3 / 42000 = 894 square inches for the load bearing section. If there are 9 lower beams which are 18 feet long and support the load across half their length, the web thickness of each beam would be 894 / (9 * 12 * 18 * 0.5) = 0.92 inches. The total cross-section including the non-loaded half of the length is twice the 894 ins^2 at 1788 ins^2. So multiplying this by the height (7 feet) obtains the volume of the web (vertical section) of lower layer beams: 1788 * 84 = 150,192 ins^3. Viewed from above or below, the flanges (horizontal sections) - top and bottom of all nine beams - are collectively about 18' by 15'. So assuming the same plate thickness, the volume for two sets of flanges is 2 * 0.92 * 144 * 18 * 15 = 71,539 ins^3 to give a total lower layer volume of 221,731 ins^3. If the upper layer beams are 15 feet long and support the load across half their length, the total cross-section (viewed from above in line with the axial load) is twice the required 894 ins^2 at 1,788 ins^2. Assuming 6 beams, each would be 894 / (6 * 12 * 15 * 0.5) = 1.65 inches thick. The web volume is the height times the cross-section equalling 1788 * 84 = 150,192 ins^3 as before. Viewed from above, the flanges are about 15' by 9'. If the plate thickness is the same 1.65 inches, the volume for both top and bottom flanges of all 6 beams is 2 * 1.65 * 144 * 15 * 9 = 64,152 ins^3 to give a total upper layer volume of 214,344 ins^2. The volume of both layers is 436,075 ins^3 = 7.146 m^3, which assuming 7860 kg/m^3 obtains a total of 61.91 tons of steel per core column grillages, or 2910 tons for all 47 core columns. There are 236 perimeter columns to take less than half of the load, so the mass of grillages per column would be considerably less than 61.91 tons. The total mass of steel required for all perimeter grillages would reflect the 41.2% of the total load collectively supported by exterior columns. If the total steel used for all core column grillages was 2910 tons, and the core columns bore 58.8% of the total load, the total steel for core and perimeter grillages would have weighed in the region of 2910 / 0.588 = 4949 tons. So the accumulated total of steel increases from 82,123 to 87,072 tons per Tower. Given the uncertainty arising from the grillages and decking, it would not be unreasonable if the sometime quoted 86,000 tons included both of the latter but excluded the trusses. Now for the trusses. We divide the non-core office area into four rectangles. The larger two - the long span sections - A and B, are adjacent to each long 137' core edge and also include the corners, hence each are about 60' by 208'. C and D - the short span sections - are adjacent to the short 87' core edge, hence are each about 35' by 87'. Details of the trusses are taken from FEMA (Chapter 2) which provides the framing plan, and from NIST (Appendix E Interim Report on Structural Steel Specifications) which provides some additional detail of a truss cross-section. Areas A and B each had: 4 transverse trusses in the long 208' direction. 5 intermediate support angles in the long 208' direction. 28 double trusses in the short 60' direction. 1 intermediate support angle in the short 60' direction. 1 double truss adjacent to core in the long 208' direction. Areas C and D each had: 2 transverse trusses in the long 87' direction. 4 intermediate support angles in the long 87' direction. 12 double trusses in the short 35' direction. 1 double truss adjacent to core in the long 87' direction. The total length of double trusses was: 2 * (28 * 60 + 1 * 208 + 12 * 35 + 1 * 87) = 4790 feet. The total length of transverse trusses was: 2 * (4 * 208 + 2 * 87) = 2012 feet. The total length of intermediate support angles was: 2 * (5 * 208 + 1 * 60 + 4 * 87) = 2896 feet. Throughout most of the length of a truss, the cross-section comprised two chords and a diagonal rod. The rod diameter varied but was typically 1.09" in diameter, making its cross-sectional area pi * 0.545 ^ 2 = 0.933 ins^2. The distance from peak to peak ("wavelength") was 40". The height difference between peaks and troughs ("amplitude") was about 31", slightly more than the separation of top and bottom chords, as the diagonals extended slightly above the top chord to act as shear studs for the slabs. Over each 20 inch "half-cycle", the length of the diagonal was SQR(20 ^ 2 + 31 ^ 2) = 36.9", or twice this every 40". So the length was increased by a factor of 36.9 / 20 = 1.845 times. It is more convenient to factor this into the cross-section rather than adjust the lengths. So for the rod cross-section we will allow 1.845 * 0.933 = 1.72 ins^2. The bottom chord was C-shaped channel, 3/8" plate, with a 3" web and two 2" flanges. Thus, its cross-section was: 3 * 0.375 + (2 - 0.375) * 0.375 * 2 = 2.34 ins^2. The top chord was 1/4" plate, with a 2" web and 1.5" flanges, making its cross-section 2 * 0.25 + (1.5 - 0.25) * 0.25 * 2 = 1.13 ins^2. The total cross-section is therefore 1.72 + 2.34 + 1.13 = 5.19 ins^2. This would have represented a single or transverse truss. Information on the "intermediate support angle" was not provided, but this would have been less substantial than a single truss. We shall assume 5.19 ins^2 for the transverse trusses, double i.e. 10.38 ins^2 for the double trusses, and 3 ins^2 for the intermediate support angles. The volume at the perimeter would have increased due to, e.g., the gusset plates. But, on the other hand, the bottom chords stopped short of the core columns. We shall not attempt any further adjustments. The double trusses amount to 4790 * 12 * 10.38 = 596,642 ins^3. The transverse trusses amount to 2012 * 12 * 5.19 = 125,307 ins^3. The intermediate angles amount to 2896 * 12 * 3 = 104,256 ins^3. This makes the total per floor volume of trusses 826,205 cubic inches or 13.54 cubic metres (which is nearly 1/1000th of the actual floor volume). Assuming 7860 kg/m^3, the mass is 106,424 kg = 117.3 tons per floor. Most of the floors from 9 to 110 utilised trusses; there were six floors featuring normal-weight concrete and rolled steel structural shapes throughout. So 96 floors x 117.3 tons per floor obtains 11,261 tons of steel used in the trusses. The accumulated total increases from 87,072 to 98,333 tons of steel per Tower. Excluding trusses, the total with decking and grillages could have been 86,000 tons, but the trusses would surely take the total over 86,000 tons. From the contracts list and our calculations above, we have a steel inventory (tons per Tower) as follows: 27,900 Exterior columns and spandrels, 9th to 107th floor 12,950 Rolled columns and beams above 9th floor, in cores; also exterior wall steel above 107th floor 3,400 Perimeter bifurcation columns (trees) 4th to 9th floor 6,800 Perimeter box cols. below the bifurcation cols. to 4th floor 6,500 Core box columns below the 9th floor 15,550 Core box columns above 9th floor and built-up beams 6,000 Support for slabs below grade 3,023 Steel decking 4,949 Grillages 11,261 Floor trusses ------ 98,333 TOTAL Now for the steel content of the upper floors. We shall concentrate on 1WTC, where the fire zone was mainly floors 94 to 99. Video evidence showed the worst fires to be in the upper part of the damaged area, around floor 98. The core box columns stop at floor 95, so we'll ignore the "core box columns above 9th floor". The rolled columns above 9th floor would have tapered towards the top; on the other hand they actually start to make an appearance at various transition points between box and wide flange columns. We will include these, the exterior columns, decking and trusses as part of the upper floors. The core columns and beams total 12,950 tons over 101 floors (10 to 110). Since this includes four double-height storeys - the middle mechanical floors - we should divide by 101 floors for beams and 105 for columns. With the column:beam ratio an unknown, we divide by 103 floors to obtain 125.7 tons per floor. The total length of the beams was 1960 feet per floor, calculated from FEMA's diagram. The total length of core columns per floor was 47 * 12 feet = 564 feet. Following calculations to determine the most likely specification of columns around floor 98, it was found that the mean core column cross-section was about 46 ins^2. Details of these calculations precede the spreadsheet data below on predicted final column temperatures under various conditions. This mean size placed the core columns at 44.3 tons per floor at this level, The beams were therefore 125.7 less 44.3 which is 81.4 tons per floor. With total beam length at 1960 feet, the mean weight was around 83 pounds per linear foot. The density is taken as 490.7 lb/ft^3, and so the mean beam cross-section was 83 / 490.7 ft^2 = 144 * 83 / 490.7 ins^2 = 24.4 ins^2. (The 12,950 tons included a small proportion for wall steel above floor 107, so those beam figures are slightly high. We find below that the exterior columns and spandrels were about 60 tons per floor at the top, so this exterior wall steel might have amounted to some 200 tons which is only 1.5% of the 12,950 tons.) According to NIST, the gauge of steel used for the exterior columns varied considerably with height. The perimeter column flanges for the lower stories were often more than 2" thick, whereas the steel could be only 1/4" thick in the upper storey columns. The thickness depended on the calculated gravity load and wind load. At the 9th floor lower limit (before the "trees" took over"), the gravity load of 101 floors (or more if allowing for the mechanical floors) compares with a load of only 14 floors at floor 96 for example. In addition, the ratio is more than 7.2:1 due to the higher storeys having less steel, although the steel was only some 1/5 of the total building weight. The wind load, however, was certainly greater at the higher storeys, which placed an eventual lower limit on column gauge. Each exterior column comprised two 13.5 inch flanges, an outer web that was 14 inches less twice the flange thickness, and a 15.75 inch inner web. Per floor, we shall take the height as 12 feet. However, the spandrel plates were an integral part of the inner web, so for the inner web height per floor we deduct the 52 inch spandrel height to leave 92 inches. Per column, each spandrel plate covered the 40 inches between columns. The web thickness was about half that of the flanges, with the spandrels being a heavier gauge (though lower yield strength) than the webs, say, about three- quarters of the flange thickness. Let f be the exterior column flange thickness in inches and let p be the spandrel plate thickness in inches. Then each column per floor (1/236 of a floor) comprises two f" x 13.5" x 144" flanges, a 0.5f" x (14 - 2f)" x 144" outer web, a 0.5f" x 15.75" x 92" inner web, and s" x 40" x 52" of spandrel plate. Allowing 0.000142 tons/ins^3, the mass in tons per floor is: Mass = -4.824 * f ^ 2 + 188.3 * f + 69.68 * s which simplifies to: Mass = -4.824 * f ^ 2 + 240.6 * f if the spandrels are 3/4 of the flange thickness. Since an "average" floor contained 27,900 / 103 tons (99 floors from 9 to 107 plus four double-height mechanical floors) for perimeter columns and spandrel plates, i.e. 270.9 tons, the mean flange thickness is 1.15 inches. This is consistent with a range from 1/4 inch to "more than two inches". The gage of flanges versus storeys might have gone something along the lines of eight floors at 1/4" (107 to 100) up to a gravity load of 12 storeys which allows for one extra, then eight at 0.42" (99 to 92) with the load up to 20 storeys so 1/4 * 20/12 = 0.42, then 7 more sets of eight at 0.58", 0.75", 0.92", 1.08", 1.25", 1.42", and 1.58", then 3 sets of nine floors at 1.77", 1.96", and 2.15" places the mean flange thickness at 1.2". (This version hasn't allowed for any double-height storeys.) The most severely burning floor on 1WTC was probably around floor 98, and gages here were unlikely to be lighter than 1/4". But to be fair to fire collapse theories, for our model we will suppose that the flanges were only 1/4", with 3/16" spandrels and 1/8" webs. So from Mass = -4.824 * f ^ 2 + 240.6 * f the exterior columns and spandrels total about 59.8 tons per floor in the 1WTC fire zone, which is about 211 tons less than an average floor. With the upper section saving core space due to a lower requirement for express elevator shafts, the core floor area was some 60% of core space. The total floor space is 4019 - 40% of core space = 4019 - 1107 * 0.4 = 3576 m^2. So the steel decking in the upper third of floors is (given density of 7860 kg/m^3 and allowing an extra 30% for the corrugations) 7860 * 1.3 * (0.0299 / 39.37) * 3576 / 907.2 = 30.6 tons per floor. The steel inventory for the upper floors (tons per Tower) is now: Core columns and beams: 12950 / 103 (floors 10 to 110, plus 2) = 125.7 Exterior columns and spandrels: (1/4" flanges, 1/8" webs) = 59.8 Steel decking: 30.6 Trusses: 117.3 TOTAL steel: approx 333 tons per floor at around the 98th floor ...or 40% of the average floor's content of 98,333 / 117 = 840 tons. The WTC construction used 425,000 cubic yards of concrete. It is likely that a good proportion of this was outside the Towers' footprints: used at or below ground level, or on ancillary buildings. As an upper limit, let us suppose for the moment that the entire volume was applied to the Towers. Most of the concrete used in the Towers was the lightweight variety, but some was normal-weight. The density of lightweight concrete extends from about 160 to 1920 kg/m^3, but structural lightweight would be towards the upper end. We will assume 1750 kg/m^3 for the lightweight and the typical average 2300 kg/m^3 for normal-weight. If one-quarter by volume was normal-weight, then the mean density is 1750 + 0.25 * (2300 - 1750) = 1887.5 kg/m^3. 425,000 yards^3 is very nearly 325,000 m^3 or 162,500 m^3 per Tower, which gives a total weight of 1887.5 * 162,500 / 907.2 equalling 338,094 tons per Tower. 5,737,500 ft^3 / 208^2 ft^2 per Tower would be a total height of 132.6 feet per Tower, 13.6 inches per floor, or nearly 10% of the entire volume! For a more reasonable evaluation, let's start with the 91 typical office floors which had 4" lightweight poured outside the core, and 5" normal-weight poured over an average of half of the core area. A single floor had 4019 - 1107 m^2 of the 4" giving (4 / 39.37) * 2912 * 1750 / 907.2 = 570.7 tons, and 1107 / 2 m^2 of the 5" giving (5 / 39.37) * 553.5 * 2300 / 907.2 = 178.2 tons. Thus, a typical regular floor had 748.9 tons of concrete floor slab. The upper office floors above the floor 78 sky lobby had slightly more concrete than the middle office floors. With an extra 10% of core space gained from a saving in elevator shafts, the additional 5" concrete amounted to (5 / 39.37) * 0.1 * 1107 * 2300 / 907.2 = 35.6 tons, so 748.9 plus 35.6 obtains 784.5 tons of concrete per floor at the fire zone. With the sky lobbies being on the 44th and 78th floors, both heavy floors (43 and 77) above the mechanical floors had the same express elevators and core space as the 34 or so floors below each of them. Floor 77, which was still within the middle circulation zone, would have retained about 50% of its 1107 m^2 core floor area. This had 8" thick, normal-weight concrete, with the 2912 m^2 non-core area filled with 7.75" thick normal-weight concrete. The total weight is (8 / 39.37) * 0.5 * 1107 * 2300 / 907.2 = 285.1 tons plus (7.75 / 39.37) * 2912 * 2300 / 907.2 = 1453.3 tons for a total of 1738.4 tons. Floor 43, in the lowest zone, had about 60% of core space taken up by service shafts. So 40% of 1107 m^2 had 8" normal-weight concrete, and the 2912 m^2 outside the core had 7.75 normal-weight. The 8" weight totalled (8 / 39.37) * 0.4 * 1107 * 2300 / 907.2 = 228.1 tons, and the 7.75" was (7.75 / 39.37) * 2912 * 2300 / 907.2 = 1453.3 tons to give a total of 1681.4 tons. The mechanical floors 41, 42, 75 and 76 featured 5.75 inch normal-weight concrete supported by structural steel shapes rather than trusses, outside as well as within the core. The former two were in the lower zone, with 40% of core space retained. 75 and 76 were in the middle zone with 50% of core area remaining. The weight of concrete per floor was respectively: (5.75 / 39.37) * (0.4 * 1107 + 2912) * 2300 / 907.2 = 1242.2 tons (5.75 / 39.37) * (0.5 * 1107 + 2912) * 2300 / 907.2 = 1283.2 tons The upper mechanical floors 107 to 110 had 7802 tons in total, including 962 tons on the roof, according to NIST's table of dead loads, Appendix D. So 6840 / 4 is 1710 tons per floor, similar to floors 43 and 77. The design documents, provided by LERA, specified 100 pcf for lightweight concrete, and 150 pcf for normal-weight. 100 pcf is about 1600 kg/m^3, but NIST found samples to have a density of 110 pcf, and used that in their models. Our assumed 1750 kg/m^3 is close to 110 pcf. We shall stick to 2300 kg/m^3 as a generally accepted value for normal-weight concrete, within a few per cent margin of error. The mechanical floors housed air conditioning and other technical services, along with the first nine levels and floors within the substructure. The sub-grade floors were constructed of reinforced concrete slabs, with structural steel columns support. From the above it is evident that the average regular office floor had about 749 tons of concrete, with the heavier floors averaging between an extra 500 to 1,000 tons. According to NIST Appendix B, the 5.75" slab mechanical floors had a topping slab of 2" maximum. Together with the fact that the density may have been over 2300 kg/m^3, and the lower floors were probably at least 8", the average heavy floor was probably nearer 1,000 than 500 tons heavier than the office floors. If we suppose there were 110 - (9 + 10) = 91 regular floors at 749 tons, and another 26 including 7 sub-grade at an average 749 + 900 tons, this is 68159 + 42874, i.e. 111,033 tons of concrete per Tower, well short of our upper estimate of 338,094 tons per Tower. The steel grillage foundations were fixed on top of massive reinforced concrete footings, 7 feet deep. Spacing within the core was not quite sufficient to allow 47 of these slabs with a footprint of 21 x 21 feet. If we suppose that the entire core area plus an extra 10 feet was covered in concrete to a depth of 7 feet, the total volume is 147 * 97 * 7 = 99,813 ft^3 = 2826 m^3 = 7166 tons of normal-weight concrete. As calculated previously, each perimeter column at the grillages had to support around 873 tons. The rock at a depth of 22.5 metres had a permissible bearing pressure of 39 kg/cm^2. (Within the core area, the demand came to some 23 kg/cm^2.) 873 tons / 39 kg indicates a minimum area of 20,307 cm^2 or 21.86 ft^2 per perimeter column. With 3' 4" intervals between columns, a width of 10 feet would provide a reasonable reserve capacity. If we say approximately 4 * 208 * 10 * 7 ft^3 = 58240 ft^3 = 1649 m^3, this is another 4181 tons of concrete. The total per Tower weight of concrete now stands at 111033 + 7166 + 4181 = 122,380 tons. The perimeter footings would just about qualify as concrete within a Tower's footprint, and part of the 500,000 tons per Tower. As for the remaining concrete, there was a 3 feet thick by 70 to 80 feet deep reinforced diaphragm wall along the site perimeter; there was the Plaza substructure housing a shopping mall, parking for 6,000 cars, and a subway station, with reinforced concrete flat slabs providing lateral support for the perimeter diaphragm, there were the ancillary buildings 3WTC to 7WTC, and any other concrete at or below ground level beyond each Tower's perimeter footings. One report (Ahmed Ghoniem) claimed that concrete fireproofing was used for the core columns. However, FEMA reports that floors 40 and up of 1WTC, and all of 2WTC, used spray-applied (asbestos-free) mineral fibre insulation for fire protection of structural elements. The material used on the steel floor trusses included cement-type binders as well as inorganic fibres. The concrete inventory (tons per Tower) now stands at: 68,159 Regular office floors, 91 off at 749 tons (average) 42,874 Heavy floors, 26 off at 1649 tons (average) 7,166 Core foundations 4,181 Perimeter foundations ------- 122,380 TOTAL Storeys within and above the fire zone were regular office floors up to floor 106. As calculated previously, upper office floors above the floor 78 sky lobby had slightly more concrete due to the extra 10% of core space. An extra 36 tons compared to an average office floor amounted to some 785 tons of concrete per floor. Floors 107 to 110 had 2,000, 1,380, 1,380, and 2,080 tons respectively, and the roof had 962 tons of concrete, according to NIST Appendix D, table of dead loads. Let's suppose failure of members occurred at floor 98, and we are interested in the gravity load for floors above this point. We'll take it as 13 floors, so nine floors from 98 to 106 at 785 tons plus the top four floors and the roof totals 14,867 tons of concrete, an average of 1,144 tons per floor. So if the average concrete content was 122,380 / 117 = 1,046 tons per floor, this set of 13 floors was actually some 100 tons per floor over the average due to a greater preponderance of heavy floors and a roof. Their structural steel content was low by about 500 tons per floor, due to tapering of perimeter columns and phasing out of core box columns. We have probably omitted some steel, but will take account of NIST's load tables for the more accurate, lower estimate. The total weight per Tower (500,000 tons) - steel inventory total (98,333 tons) - concrete inventory total (122,380 tons) = 279,287 tons of remaining mass per Tower. Let us suppose half of this remaining load was doubled up for each of the four double storey mechanical floors. So allowing for approximately two extra floors, 279,287 tons divided into 119 floors is 2,347 tons per floor, excluding inventoried steel and concrete. The 13 upper floors averaged 1,144 tons of concrete and 333 tons of steel. So the total weight for floors 98 to 110 and the roof is 2,347 plus 1,144 plus 333 equalling 3,824 tons per floor, i.e. 49,712 tons in total. The inventoried steel and concrete is the dead load, and the remaining 2,347 tons per floor consists of the superimposed dead load (SDL) and the service live load (SLL). NIST's model takes the area as 60.9 m by 60.9 m, which is very nearly 40,000 ft^2, for calculating the load per unit area. Where a load is deemed to be distributed over the entire floor, we shall use 40,000 ft^2 as the area, which places the per floor load in tons at twenty times the mean load in psf. The remaining mass excluded from the inventories, at 2,347 tons per floor from 98 through to the roof, is 117 psf, as our high estimate for the SDL and SLL. For our low estimate, we consult NIST's Appendix D. For a typical office floor up to 106 (96 was taken as the example floor), the superimposed dead load outside the core is quoted as 2 psf for each of mechanical and electrical, ceiling, and floor covering, and an average of 2.8 psf for fireproofing. No figure is given for tenant partitions, and the total is shown as the sum of those four items. Floor 107 has 12 psf for partitions but was a heavy mechanical floor; FEMA assumed 10 psf for partitions in 3WTC which was a hotel. We will count the non-core SDL as 8.8 psf for the tenant floors. Within the core, SDL in these floors varied from 29 to 49 psf. We shall take the average 39 psf. NIST uses 30,647 ft^2 for the non-core area and 9,274 ft^2 for the core. So for the non-core, 8.8 * 30647 / 2000 = 135 tons, and for the core, 39 * 9274 / 2000 = 181 tons, totalling 316 tons per floor. Service live loads were assumed to be a quarter of the design live loads. Outside the core, the SLL averaged 16.7 psf. Over the core, the SLL varied from 10 to 25 psf. We shall assume 17.5 psf. So for the non-core, 16.7 * 30647 / 2000 = 256 tons, and for the core, 17.5 * 9274 / 2000 = 81 tons, totalling 337 tons. So the SDL plus the SLL for a typical light office floor (96) was 316 plus 337 tons = 653 tons. We have already calculated the dead load as 785 tons of concrete and 333 tons of steel. Thus, the total weight was 1,771 tons per floor. Floor 107 had a SDL consisting of partitions (12 psf or 240 tons), ceiling (40 tons), mechanical and electrical (40 tons), fireproofing (40 tons), and flooring (1,140 tons), for a total SDL of 1,500 tons. The SLL was 500 tons. The dead load figures are provided for floors 107 and up, so we shall assume these values. There were concrete slabs (2,000 tons), reinforcing steel (40 tons), steel decking (40 tons), and structural steel (260 tons), for a total dead load of 2,340 tons. Hence, the total load of floor 107 was 1,500 plus 500 plus 2,340 equalling 4,340 tons. Floor 108 had a SDL consisting of ceiling (200 tons), mechanical and electrical (60 tons), fireproofing (100 tons), and flooring (620 tons), for a total SDL of 980 tons. The SLL was 376 tons. The dead load consisted of concrete slabs (1,380 tons), reinforcing steel (60 tons), steel decking (40 tons), and structural steel (400 tons), for a total dead load of 1,880 tons. Hence, the total load of floor 108 was 980 plus 376 plus 1,880 equalling 3,236 tons. Floor 109 had a SDL consisting of ceiling (200 tons), mechanical and electrical (60 tons), fireproofing (100 tons), and flooring (620 tons), for a total SDL of 980 tons as for floor 108. The SLL was 750 tons. The dead load consisted of concrete slabs (1,380 tons), reinforcing steel (60 tons), steel decking (40 tons), and structural steel (400 tons), for a total dead load of 1,880 tons, the same as floor 108. Hence, the total load of floor 109 was 980 plus 750 plus 1,880 equalling 3,610 tons. Floor 110 had a SDL consisting of mechanical and electrical (1,000 tons), and fireproofing (100 tons), for a total SDL of 1,100 tons. The SLL was 376 tons. The dead load consisted of concrete slabs (2,080 tons), reinforcing steel (60 tons), steel decking (40 tons), and structural steel (400 tons), for a total dead load of 2,580 tons. Hence, the total load of floor 110 was 1,100 plus 376 plus 2,580 equalling 4,056 tons. The roof featured a 362 ton antenna, which would be included amongst the following loads. The SDL of the roof consisted of mechanical and electrical (1,000 tons), fireproofing (100 tons), and flooring (100 tons), for a total SDL of 1,200 tons. The SLL was 750 tons. The dead load consisted of concrete slabs (962 tons), reinforcing steel (40 tons), steel decking (40 tons), and structural steel, for which no figure is given but a footnote states that it is included in NIST's model. If we assume 400 tons, the same as each of the top three floors, the total dead load equals 1,442 tons. Hence, the total load of the roof was 1,200 plus 750 plus 1,442 equalling 3,392 tons. So for floors 98 through to 106, the gravity load is 9 times 1,771 equalling 15,939 tons. Floors 107 through to 110 and the roof were 4,340, 3,236, 3,610, 4,056 and 3,392 tons respectively. Therefore the total gravity load of this upper section was 34,573 tons, or an average 2,659 tons per floor, for our more accurate, lower estimate. Of course, with the mechanical floors having more than double the weight of the tenant floors, the per floor weight will vary considerably with the floors section's boundaries. The higher, rough estimate based on the oft-quoted 500,000 tons per Tower was 49,712 tons, or an average 3,824 tons per floor, for the same section of floor 98 to roof. The service live load, at 25% of the design live load, was taken as 337 tons for floors 98 to 106, and then 500, 376, 750, 376, and 750 tons for floor 107 up to the roof. With a total SLL of 5,785 tons for this section, the difference between the DLL and SLL is three times the SLL placing it at 17,355 tons. If we add this to the 34,573 tons estimate, 51,928 tons exceeds the high estimate. So unrealistically high assumptions regarding the live load can more than account for the differing estimates. The 500,000 tons figure probably estimates the live load as nearly 100% of the design maximum. We shall take NIST's 25% of the design live load as a realistic estimate of the service live load, making 34,573 tons the preferred estimate for the total weight of this section of the building. The next check involves determining the weight of this upper section from NIST's October 2004 publication of the column demand / capacity ratios, and from details provided of the column members. This helps to confirm the specifications of the core columns. We begin by considering the perimeter columns. From NIST Appendix E (Fig. E-3), each of these columns (per floor) comprises two f" x 13.5" x 144" flanges, webs of approximately 0.5f" x (14 - 2f)" x 144" and 0.5f" x 15.75" x 92", and s" x 40" x 52" of spandrel plate part of which is an integral part of the inner web, where f is the flange thickness and s the spandrel plate thickness. The spandrel plate was a heavier gage than the webs, but a lower yield strength. We shall assume the gage difference is balanced by the difference in yield strength, with the axial load capacity determined by the flanges and webs. At the fire zone, around floor 98, the flange thickness f was only about 1/4". We shall assume the web was half the flange thickness. So the cross-section of each column was 2 * 0.25 * 13.5 + 0.125 * (13.5 + 15.75) = 10.4 square inches. Appendix E (Table E-4) lists the number of perimeter columns of various specified grades within the fire zone of floors 92 to 100 (1WTC). They range from one of only 46 ksi to sixteen at 100 ksi. Totalling all columns, we find that the average perimeter column yield strength is 64.2 ksi. There are 59 perimeter columns each side, making a total of 236. Hence, the total cross-section of all columns is 236 times 10.4 = 2,454 square inches. Multiplying by the yield strength, the axial load capacity of all exterior columns is 2454 * 64200 / 2000 = 78,773 tons. Concrete comprised a significant proportion of the load, and had a fairly uniform distribution throughout the horizontal plane. This also applied to flooring on the mechanical floors. As FEMA states, the various trusses combined to allow the floor system to act as a grillage, with the load distributed between columns. In the absence of a detailed plan of distribution of mass across each floor, we shall assume a uniform load. As calculated previously when determining the mass of the grillages, the core load area is about 172' by 148', which places the perimeter column load at 17,808 / 208^2, or 41.2% of the total. According to NIST's October 2004 publication of column demand / capacity ratios (floors 93 to 98, 1WTC), the perimeter columns' demand varied from 0.17 to 0.2 of capacity, before the aircraft impact. Let us assume 0.185 as the average demand / capacity ratio. Hence, the demand on the columns was 0.185 times 78,773 tons equalling 14,573 tons. If the perimeter columns handled 41.2% of the load, the total gravity load was 14,573 / 0.412 = 35,371 tons. This is within 2.5% of our 34,573 tons evaluation above, calculated for floors 98 through to the roof. The 34,573 tons was a more detailed calculation, with the 35,371 tons subject to interpretation of the exterior column specifications and the lower bound of the upper section. So we shall take 34,573 tons as the actual gravity load of floor 98 and up. The core columns' load area of 25,456 ft^2 is 58.8% of the total. For the demand on the core columns, we assume 34,573 tons times 0.588 equalling 20,329 tons. So, at floor 98, the demand was 20,329 / 47 = 433 tons on average per core column, and 14,244 / 236 = 60.4 tons per perimeter column. We shall select the likely specifications for each core column and sum the radiant heat contributions from each unit of floor space assuming maximum possible energy density for the fires. After determining the heat absorbed, the final temperatures are calculated, along with effect of the fires on column demand / capacity ratio for several scenarios with various proportions of the fire resistive coating removed. If the trusses retained sufficient strength and floors did not collapse, then the upper section could be treated as a solid object. Provided the total column capacity could meet the demand and sufficient columns remained to prevent toppling of this upper section, then we would only be interested in the total column capacity. However, there are various collapse sequences which do not enable sufficient redistribution of load to neighbouring columns. And some scenarios involving increases in the slenderness ratio could lead to instability and potentially serious reductions in column capacity. So we shall treat columns individually. The (x, y) co-ordinates for each core column were determined from NIST Appendix E Figure E-7. It was found that predicted final member temperatures, when working in metres and summing the heat contributions from each square metre of floor space, were within a few degrees K of the equivalent summing of square feet. With a tenfold difference in computation time, it was decided to work in metres. Viewed with the north facade at the top of the page, column 1001 at (11, 18) was closest to the SW corner, and the NE corner was rounded to (63.5, 63.5). The boundaries of each column's load area was deemed to be midway between the column and its adjacent member. For example, the core columns with the least demand were 704 and 705. 705 was at (34, 33). The x bounds were 32 and 35.5 and the y bounds 32 to 36.5 to give a total load area of 15.75 m^2. The corner columns had the greatest demand. Column 1001 load area spanned an x range of 6 to 14, and a y range of 9 to 21 to give a total of 96 m^2. Load areas were divided by 63.5^2, with the resulting proportion multiplied by the 34,573 tons of floor 98 to the roof.