1                     Abstract

The mass and potential energy of one of the Twin Towers is calculated based on available data. The mass for each floor is established based on floor types, documented design loads, and estimated in-service live loads. The calculated mass of 288,100 metric tons (317,500 short tons) is found to correspond with two other comparable structures in terms of mass per unit floor area, NIST’s SAP2000 model, and the reported amount of recovered debris. The calculated mass refutes the popular notion that the building weighed 500,000 tons.

2                     Introduction

In the aftermath of the World Trade Center Disaster, the Federal Emergency Management Agency (FEMA) and the National Institute of Standards and Technology (NIST) conducted investigations of considerable scope regarding building performance and the collapse of the World Trade Center Towers. The FEMA report described a pancake-type progressive floor collapse scenario causing the removal of lateral support on several floors leading to buckling in unshored columns which were weakened by fire and partially damaged by aircraft impact.³ The FEMA report was not rigorous and the conclusions regarding collapse initiation and progressive collapse can only be considered to be an educated guess by the investigators. The more rigorous NIST reports described aircraft impact damage and collapse initiation based on forensic evidence and computer simulations. Unlike scientific articles, the NIST investigation reports do not provide enough information to be able to reproduce the models or any results derived from the models.

Analyses from independent researchers regarding aircraft impact damage and collapse scenarios have appeared during and after the official investigations. Earlier analyses were severely limited by a lack of information and were overly simplified. Later analyses have been more substantial, but as seen in Bazant et al. (2007)¹⁰, the mass and potential energy are probably grossly overestimated.

Due to these limitations, many have questioned both the government’s specific account of collapse initiation and the general theory of gravity-driven, progressive collapse. These questions can only be answered by better modeling and a truly scientific approach. To be valid, further analyses and models must be based on the correct mass and mass distribution throughout the building.

The purpose of this paper is to establish a substantiated mass, mass distribution and potential energy in World Trade Center Tower 1 (north tower) within a reasonable margin of error. Here, the NIST “Federal Building and Fire Safety Investigation of the World Trade Center Disaster” (called NCSTAR) documents provide a wealth of information regarding the structural design, dimensions, building materials, contemporaneous building codes and an approach to modeling.

2.1                  Popular numbers

Many references can be found with different values for the mass of and the amount of potential energy stored in the WTC twin towers. A number of references are shown in Table 1 below. None of these references provide any data or calculation method on which the mass and potential energy are based.

Table 1: Different values for mass and potential energy

* The Bazant et al. number is calculated here based on the following:

“Near the top, the specific mass (mass per unit height) µ = 1.02 × 10⁶ kg/m. In view of proportionality to the cross section area of columns, µ = 1.05 × 10⁶ kg/m at the impact level (81st floor) of South Tower. Generally, we assume that µ(z) = k₀e^k2z + k₁ (where k₀, k₁, k₂ = constants), with a smooth transition at the 81st floor to a linear variation all the way down (precise data on µ(z) are unavailable). The condition that ∫₀ ^H µ(z)dz be equal to the total mass of tower (known to be almost 500,000 tons) gives µ = 1.46 × 10⁶ kg/m at the base.” ¹⁰

Since µ(z) is unknown we can approximate the value for floors 82-110 using a linear variation from the value at floor 81 to the value at floor 110 (29 floors) and the proportion of the height for those floors. The height of WTC1 from the base to the roof is 437.69 m. The total number of floors is 116. µ(z)avg_81-110  = 1.035 × 10⁶ kg/m.  µ(z)avg_B6-81 = 1.2475 × 10⁶ kg/m.

Mass_82-110 = µ(z)avg_81-110 x (29/116) x h = 113.3 × 10⁶ kg

Mass_B6-81 = µ(z)avg_B6-81 x (87/116) x h = 409.5 × 10⁶ kg

The total mass is then 522.8 × 10⁶ kg or, converting to short tons, 576,000 tons. Bazant et al. most likely assumed metric tons for the popular 500,000 ton number but that doesn’t explain why we get 522.8 × 10⁶ kg. The maximum error of using the linear approximation instead of the exponential equation is less than 2 × 10⁶ kg. If Bazant et al. used the nominal height of the building (414.63 m from the concourse level to the roof) the result would be 493.9 × 10⁶ kg which corresponds better to the statement “known to be almost 500,000 tons” assuming metric tons.

2.2                  Original Design

A number of original design documents are provided in NCSTAR1-1 and NCSTAR1-1A. NIST NCSTAR1-1A (p. 5)⁹ presents definitions from the original design as follows:

1.      “Floor inside of core”. That part of the floor bounded by the outside faces of columns 501, 508, 1001 and 1008.

2.      “Floor outside of core”. That part of the floor between the outside walls and the “Floor inside of core”.

3.      “Code live load”. The load specified in the New York Building Code for a given occupancy.

4.      “Live load for floor design”. The actual live load used for the design of the parts of the floor which load may not be less than the “Code live load”, and may be reduced for tributary areas as defined in “Live load reduction”.

5.      “Live load for column design”. The code live load, reduced as defined in “Live load reduction” for columns.

6.      “Construction dead load”. The weight of the bare structure (i.e. the slab and beam) used in design of unshored composite beams.

7.      “Construction live load”. The allowance for the weight of any equipment and/or forms which is not permanent and does not form part of the total load summation.

8.      “Superimposed dead load”. The weight of ceilings, floor finish, walls or partitions of known location, mechanical and electrical equipment and similar items not included in the “Superimposed live load” or “Construction dead load”.

9.      “Dead load”. The sum of items 6 and 8 above.

10.  “Superimposed live load”. The weight of the design live load, based on occupancy, plus the weight of partitions if their location is subject to change.

Values for construction dead load (CDL), superimposed dead load (SDL) and superimposed live load (SLL) are also given in the design documents presented in NCSTAR1-1A for some of the different types of floors within the building, inside and outside the core. CDLs include steel used in floors such and beams, trusses, deck and concrete reinforcement.

2.3                  Amount of Steel

NIST gives the total the weight of structural steel in the two WTC towers as 200,000 tons.¹¹ NIST describes steel contracts in NCSTAR1-3 (p.16), and the values are shown in Table 2 below.³ These contracts do not include trusses outside the core, steel deck, concrete reinforcements or grillages.

Table 2: Weight of steel from supplier contracts

2.4                  NIST Reference Models

In the NCSTAR1-2 series, NIST presents the methods used for developing the reference structural models the WTC towers. These models were used to assess the towers’ ability to withstand gravity and wind loads and to establish the reserve capacity in the structures to withstand unanticipated events. According to NIST:

“The reference models included the following: Two global models of the primary structural components and systems for each of the two towers (and) two models, one of a typical truss-framed floor (tenant floor) and one of a typical beam-framed floor (mechanical floor), within the impact and fire regions. All reference models were linearly elastic and three-dimensional, and were developed using the Computers and Structures, Inc. SAP2000 software. SAP2000 is a commercial finite-element software package that is customarily used for the analysis and design of structures.” ⁷

The databases for the reference models were developed based on original structural drawings. The databases were reviewed and checked against the original drawing books. According to NIST:

“The original structural drawings of the WTC Towers were issued in two main formats: (1) Large-size drawing sheets containing plan and elevation information, and (2) Smaller book-sized drawings containing details and tabulated information of cross-sectional dimensions and material properties. The larger-sized drawings referred to the structural drawing books in their notes, section and details. The structural databases, developed in Microsoft Excel file format, were generated from these drawing books and included modifications made after construction. The databases were generated for use in the development of reference global models of the towers.” ⁷

None of the original structural drawings were released by NIST. However, the larger drawing sheets for WTC-1 (north tower) were leaked subsequently to the general public and are generally available.¹⁷ The smaller drawing books still have not been made public.

2.5                  NIST’s “Tower and Aircraft Impact Models”

NIST describes the “Tower and Aircraft Impact Models” in NIST NCSTAR 1-2. These models were developed using the LS-DYNA 2003 software package.

“The WTC models for the impact analysis required considerably greater sophistication and detail than was required for the reference models described in Chapter 2. The reference models provided a basis for the more detailed models required for the impact simulations. The impact models of the towers, which utilized the structural databases described in Chapter 2 (see also NIST NCSTAR 1-2A), included the following refinements…” ^{7 (p. 93)}

The loading of the structure for the analysis was determined by NIST as follows:

“The densities of the tower components (workstations and gypsum walls) were scaled by the appropriate ratios to obtain the desired distribution of live loads in the core and truss floor areas. The densities of all the remaining tower structural components were scaled proportionately to obtain the desired superimposed dead loads. These additional loads were important for obtaining an accurate mass distribution in the towers and inertial effects in the impact response. The in-service live load used was assumed to be 25% of the design live load on the floors inside and outside the core. The in-service live load was selected based on a survey of live loads in office buildings (Culver 1976) and on engineering judgment.” ^{7 (p. 106)}

NIST NCSTAR 1-2B (p. 53) gives an SDL (36.2 psf) which is in fact applied to the structural components (columns).¹³ The SDL mass being applied to columns, is not a problem when calculating the mass. However, the impact analysis must be significantly affected by reducing the probability of debris coming into contact with core contents.  The effect is that impacting debris has a free shot at core structural members and is more likely to pass all the way through the core. It is unclear if the partitions are included in the SDL, SLL or both.

NIST NCSTAR 1-2B (p. 53) gives a summary of superimposed dead loads and live loads and floor areas to which they are applied.¹³ The values are shown in table 3 below.

Table 3: Summary of superimposed dead loads and live loads

3                     Method

The mass for the building is calculated on a floor by floor basis based on information in the NIST reports and the architectural drawings. In some cases there are deviations from NIST values and motivations for alternative values are described. In cases where there is not enough information in the NIST reports, dimensions or materials are used from similar areas of the building. As described in the introduction above, the design documentation for WTC1 has the structural loads divided into construction dead loads, superimposed dead loads, and superimposed live loads. These divisions are also used here.

3.1                  Floor Areas

According to NIST, the floor areas inside the core and outside the core are 8,694 sq ft and 31,257 sq ft respectively (see Table 3). However, the architectural drawings give the distance between the center of the external columns on one side to the center of the external columns on the other side as 207’-8”.¹⁷ NIST gives the width of the external column flanges as 13.5” and the spandrel thickness as 5/8”. Together these are roughly 14” contributing approximately 7” on each side to the 207’-8” dimension. Thus the overall floor dimensions must be 206’-6” x 206’-6” with a gross floor area of 42,642 sq ft. The outer dimensions of the core were 137’ x 85’ giving a gross core area of 11,745 sq ft. Thus the floor area outside the core is 30,897 sq ft. It may be that NIST subtracted the areas taken up by core columns, elevator shafts and utility shafts in the core area, which would account for the difference of roughly 25%. Generally in this analysis, the floor areas used inside the core and outside the core are 11,745 sq ft and 30,897 sq ft respectively.

For the purposes of establishing CDLs in the core, the floor areas inside the core were adjusted to account for empty space due to elevator and utility shafts. The actual floor areas were approximated by sampling a number of representative floors using the architectural drawings.¹⁷ Two sizes of elevators predominated and the other shafts were split into three groups: small, medium and large. The areas for the shafts in each group were established by taking the dimensions of all shafts on floors 11-16 from the architectural drawings (core plans), grouping them, and taking the average size for each group.¹⁷ Elevators and shafts were then counted on the representative floors and grouped by size. Elevators and shafts on average take up 41% of the core floor area. The sampled floors, number of elevators and shafts, area with no floor, and the percentage of empty space in the core are shown in Table 4. See Diagram 1 for examples of elevators and shafts.

Table 4: Elevators and shafts on representative floors

Diagram 1: Example of architectural drawings - core plan floors 11-16. (Colored areas with number designations are examples of the groups described above.)

3.2                  Floor Types

A table of floors and diagrams of 15 different floor framing types are given in NIST NCSTAR 1-2A, Appendix G (p192-196).¹⁴ The table shows which type of floor framing was used for each particular floor. The diagrams show how the different types of framing (i.e. truss or beam) were used in different floor types. The approximate percentages of truss versus beam areas can easily be deduced from the diagrams. Unfortunately, there is no indication of concrete types or thicknesses. For the purposes of this analysis, floors are divided up into normal, mechanical, special and sublevel floors.

3.2.1                   Normal Floors

All floors are considered normal unless they are mechanical floors, sublevels, or special floors as described below. The floor numbers for normal floors are 10-40, 44-74 and 78-106. These floors, which are predominantly offices and related areas, comprise eleven floor types (type 1-11) that are predominantly truss framed. Some of these types have sections of beam framed floor and two types have heavier angles or have reinforced trusses. All of these floor types are treated as type 1 (100% truss framed) to simplify the calculation of mass. The total error induced by this simplification is less than 1/1000 and can be calculated as follows:

Err% = A_Avg% x S% x (B-T)/B% = 0.073%

Where A_Avg is the average proportion of beam area (1.45%), S = the floor frame steel proportion of the total mass of the building (approx. 10%), T is the truss design construction dead load (10 psf) ⁹ and B is the design construction dead load (20 psf) ⁹ for beam framed floors. See table 5 for calculation of A_Avg.

Table 5: Floor types, count and beam framed area for calculation of average beam area.

* Note: Floor 106 is type 10 which has reinforced trusses. The reinforced trusses are assumed to be heavier than normal trusses and lighter than beam frames. Thus the floor is given as being 50% beam framed to account for the extra weight for the purpose of calculating the error due to simplification.

3.2.2                   Mechanical floors

The mechanical floors are 7-9, 41-43, 75-77, and 108-110, which are all beam framed floors. In each group of three floors, the upper and lower floors are type 12 and the middle floor is type 13 (mechanical mezzanine). The mechanical mezzanines were 50% open (no floor) outside the core so the floor area is 15,448.5 sq ft.

3.2.3                   Sublevels

Sublevel floors were beam framed floors, designated B1-B6, and are type 14. As seen in Table 2, 6000 tons of steel were used for slab support below grade. There is a minor discrepancy between the NIST documentation and the architectural drawings. In the architectural drawings, the floor below floor 1 is called the “Service Level” and the five floors below are named B1-B5.

3.2.4                   Special floors

Special floors include the Concourse level (floor 1), Plaza level or mezzanine (floor 2), and the roof, which are beam framed floors. Floors 3-6 had no floors outside the core. The Concourse level which was a high pedestrian traffic area is type 14 and probably had stronger than normal floors. The Plaza level was type 15 and was partially open. Floor 107 was the restaurant “Windows on the World” and had beam-framed floors.

3.3                  Gravity Loads

3.3.1                   Foundation

The mass of the foundation provides no load on structural components other than itself and contributes a negligible amount to potential energy. The mass of the foundation is nonetheless approximated based on the film footage from the Port Authority of New York and New Jersey.¹ Dimensions are established by comparison to objects of known size, i.e. humans. The total mass of the foundation is shown in Table 7.

The foundation for the core columns was comprised of steel reinforced concrete footers and steel grillages built up out of I-beams. One steel grillage is made up of 17 I-beams, each with approximate dimensions 0.75m x 0.2m x 2m and a plate thickness of around 0.03m. Each grillage also had a base plate for the core column with average approximate dimensions 1m x 1m x 0.1m. It is assumed that there is one grillage per core column. Using a density of 7.784 metric tons per cubic meter for the density of A36 steel, the total mass for the grillages is approximately 484 metric tons. Each grillage was placed on a concrete footer with approximate dimensions 2.5m x 2.5m x 2m. Using a density of 2.4 metric tons per cubic meter, the total mass for the concrete footers was approximately 1410 metric tons.

The foundation for the external columns was comprised of a continuous, steel reinforced, concrete footer and base plates ranging from 7 to 9 sq ft (approx. 0.74 m2).⁶ The thickness of the base plate is unknown but a thickness of 3 cm is assumed. Using a total number of 80 exterior columns (transition to 238 columns at 7th floor), the total mass of the base plates is approximately 14 metric tons. The concrete footer for the external columns had a perimeter of 252 m. The other dimensions of the footer are unknown but are approximated using 2 m for depth and 2 m for width. The total mass for the concrete footer was approximately 2420 metric tons.

Table 6: Mass of the foundation

3.3.2                   Amount of Core Column Steel

As described in the introduction, the steel contracts included 6,500 tons for core box columns below the 9^th floor, 15,500 tons for core box columns above the 9^th floor and 12,950 short tons for rolled columns and beams. The amount of steel attributed to rolled columns (wide flange shapes) is calculated in Appendix 2 as 3,268 short tons. Thus the total core column steel is 25,268 short tons.

3.3.3                   Variation of Core Column Steel

Core columns dimensions have been extracted from NIST’s SAP2000 model, which was released based on a Freedom of Information Act (FOIA) request. These dimensions are currently available on the internet.¹⁵ It can be seen in this data that the variation of core columns steel is non-linear in the areas from floor B6 to floor 7 and from floor 107 to the roof. There are also non-linear variations at the mechanical floors where the columns were somewhat heavier, but these are ignored. The variation of core column steel mass is shown in Table 7, which is based on calculations of core column steel per floor for selected floors (see Table 19 in Appendix 3).

Table 7: Variation of Core Column Steel

When the core column steel mass is varied in this manner, the total core column steel becomes 24,576 tons with 5,801 tons below floor 9. This amount of core column steel below floor 9 should be 6,500 tons according to the steel contr