**Errors Revealed in Bažant's Calculations Further Bury the Official Story as ASCE and NIST Continue to Stonewall Calls for a New WTC Investigation**

**Errors Revealed in Bažant's Calculations Further Bury the Official Story as ASCE and NIST Continue to Stonewall Calls for a New WTC Investigation**

**In other words, it's wrong.**

**PROFESSOR BAŽANT’S INCORRECT CONCEPT OF THE DESTRUCTION OF THE WTC TWIN TOWERS ON 9/11 AND ERRORS IN HIS MATHEMATICAL ANALYSIS INVALIDATE HIS THEORY AND THE NIST REPORT **

**March 4, 2024**

*Why did the World Trade Center Collapse? – Simple Analysis*and was co-authored with one of his graduate students, Yong Zhou. This technical paper put forward a theory, which came to be known as the Crush Down Crush Up (CDCU) theory, that attempted to explain the mechanics of how the Twin Towers at the World Trade Center were destroyed on 9/11. The JEM published his paper in January 2002 after some additions were made. Since then, the CDCU theory has been adopted by the National Institute of Standards and Technology (NIST) as the official U.S. government explanation for the complete collapse of the Twin Towers.

^{1}that Bažant and various co-authors produced between 2001 and 2022 to describe and defend his theory, and, as such, is the foundation that establishes the NIST narrative of the event and provides a mathematical analysis that attempts to provide proof of its validity.

*Why did the World Trade Center Collapse? – Simple Analysis*

^{2}

- The planes crashed into the towers severing some of the steel columns and damaging others in the zone of the aircraft impacts.
- The debris from the planes tore the fireproofing off some of the steel floor trusses in the impact zones.
- Fires caused by the explosion of the jet fuel erupted in the impact zones.
- The floor trusses were weakened by the fires and sagged, pulling in the perimeter columns.

- After burning for the next hour or so, the heat from the fires weakened the structure and the weight of the building above the fire and impact damaged zone overcame the ability of the damaged zone to bear the weight of the upper part of the building. The upper part of the building then fell down through the damaged zone and collided with the lower part of the building.
- The force of the collision was sufficient to overcome the ability of the columns in the lower part of the building to withstand the impact.
- The impact caused the lower part of the building to give way.
- The destruction of the building continued all the way from the impact zone to the ground, driven by the falling weight of the upper part of the building, the mass of the damaged zone and the accumulating mass of the destroyed lower part of the building.
- When the upper part of the building hit the debris below it on the ground, it was also destroyed.

**Pd/Po = 1 + The Square Root of the quantity [1 + (2Ch/mg)] ****≈ 31**

- The equation guarantees that Pd was at least twice Po, no matter what the values of the stiffness, mass of the upper part of the building, and height of one story were. This is counterintuitive. How can his equation guarantee this minimum ratio, no matter what the dimensions and properties of the building itself were?

- While Bažant’s analysis requires that the upper part of the building fell through one collapsed story prior to the elastic phase of the impact, he uses the fall through this full story height (h) as the elastic deformation. The correct distance for the elastic deformation is only a small fraction of the height of one story.

- The factor of “2” was apparently inserted under the radical to represent the two separate springs of the upper and lower parts of the building acting together when the upper part hit the lower part, which would be incorrect, since springs acting in series produce an equivalent stiffness, where 1/Ceq = 1/Cu + 1/CL, [not Cu + CL]. The insertion of the factor “2” in the equation is incorrect.

- Bažant also underestimated the design load capacity of the lower part of the building, Po, as mg. However, mg is only the static weight of the previously supported structure above the impact. The columns in the building had safety factors included in their design load capacities: for the perimeter columns it was 5 to 1 and for the core columns it was 3 to 1. Therefore, since the perimeter and core columns essentially shared the static load, Bažant should have used the average of these safety factors, (5 + 3)/2 = 4 for the design load capacity Po = 4mg.

- There is no basis for the inclusion of both factors “1” inside and outside the radical, nor is there a basis for taking the square root of a portion of the quantity, as the ratio is simply the elastically generated force (C times the elastic deformation) divided by the design load capacity, (Po = 4mg).

- Therefore, the correct form of Equation (1) is:

**Pd/Po = (Ceq) X (elastic deformation)/4mg**

**The correction of Bažant’s concept and the errors in equation (1)**

**Bažant’s first error**

**was his conceptualization of the process of destruction.**He posited that the upper part of the building acted as a solid body impacting the lower part of the building, with the lower part of the building acting as a spring during the initial elastic impact. He illustrated this concept in Figure 2(a) of his paper [reproduced below], in which the upper part of the building with mass m falls through a distance of one story, h, and impacts the lower part of the building acting as a spring with stiffness C. There are two problems with this concept.

- In reality, since for every action there is an equal and opposite reaction, in an elastic reaction the upper part of the building is also acting as a spring under the force of the collision between the two bodies. Therefore, the correct concept of the interaction must include two springs acting in series, each with their own stiffness, whereas Bažant only theorizes one spring – that of the lower part of the building. Correction of his error can be done by calculating the equivalent stiffness of the two springs acting in series, and this concept is also compared to Bažant’s conceptualization below. Bažant’s error affects his mathematical model of the event.

**BAŽANT’S FIGURE 2(a)**

**WHAT BAŽANT’S FIGURE 2(a) SHOULD HAVE BEEN**

- As noted above, the elastic interaction of the two parts of the building could not involve the descent of the upper portion for a full story, h, of the building. The actual distance could only be the distance due to the compression of the two parts of the building, Pd/Ceq, calculated below, which only amounts to a small fraction of the distance of a full story that Bažant used.

**Bažant’s second**error was his estimation of the stiffness, C, of the lower part of the building (which is re-labeled CL in the calculations below to correct his equation). Without giving us the details of his calculation for the stiffness of the lower part of the building, he asserted it has an estimated value of 71 GN/m (Giga Newtons per meter). Upon checking this calculation, which Szamboti and MacQueen performed in their paper THE MISSING JOLT

^{3}, the actual estimated value of CL was found to be 7.1 GN/m. Thus, Bažant overestimated the stiffness of the lower part of the building by 10X.

**The third error**was his claim that the downward elastic displacement from the initial equilibrium position to the point of maximum deflection of the lower part of the building was (h + Pd/C). According to Bažant, this would have been a distance of (3.7m + Pd/C). In actuality, in an elastic collision, the deflection would only be Pd/Ceq, and Ceq must be calculated as the equivalent stiffness of the upper and lower springs acting in series, as explained above. Using Szamboti and MacQueen’s method, Pd/Ceq = 7.21 in. = 0.183m.

^{4}According to Bažant, the elastic deflection would have been a distance of (3.7m + 0.183m) = 3.883m, but this is incorrect. The height of one story of the building should not have been added to the elastic deformation, Pd/Ceq, because any potential overload causing the complete failure of the columns would only be due to the elastic interaction. Thus, Bažant has overestimated the elastic deformation of the lower part of the building by a factor of 3.883/0.183 = 21.2X.

**His fourth error**was his estimation of the mass of the upper part of the North Tower as 58 X 10⁶ kg. However, the actual in-service load of the upper part of the building (per NIST it was 12 stories in WTC 1) was 33 X 10⁶ kg.

^{5}

**His fifth error**was representing the value of Po = mg as the design load capacity of the lower part of the building. As explained above, mg, the static load, should be increased by the average factor of safety, 4, in order to represent the true design load capacity.

##### Correcting these errors in Equation (1),

**First**, finding the equivalent stiffness, Ceq, for the two springs acting in series:

^{th}floor)

^{th}floor = 149.24 in X 12 stories = 1,791 in.

**Second**, correcting the distance of the elastic deformation by replacing h with the elastic deformation:

**Third**, correcting the actual working load of the upper part of WTC 1: m = 33 X 10⁶ kg

**And fourth**, correcting the capacity of the lower part of the building to withstand the actual design load capacity with factors of safety incorporated,

**Therefore, using Bažant’s corrected Equation (1):**

**The CDCU theory does not provide a load amplification factor of 31. It actually doesn’t even provide an overload, as the ratio is less than one, and is not large enough to overcome the design load capacity of the lower part of the building to resist collapse.**

**Collapse would not be initiated and could not occur.**

**Bažant’s theoretical proof that the elastic response of the lower part of the building cannot withstand the impact of the upper portion of the building is not valid.**

**Bažant’s theory, when corrected, does not explain the destruction of the Twin Towers.**

**The remainder of Bažant’s first paper is an analysis of the inelastic energy dissipation caused by the collapse, which is now irrelevant, since the design load capacity of the lower part of the building could not have been overcome by the impact.**

**The subsequent four papers by Bažant are in defense of his first paper, which has been shown to be incorrect. There is no need to review his subsequent defenses of his theory, as they failed to address his fundamental misconception of the process and the errors noted in his initial analysis. In fact, those efforts specifically reinforced the analysis put forward in the first paper.**

**For example, in the second paper (JEM March 2007), the following statement is made:**

*“Up to the moment of collapse trigger, the foregoing scenario (impact, removal of fireproofing, heating of members, collapse of floor trusses) was identified by meticulous, exhaustive, and very realistic computer simulations of unprecedented detail, conducted by S. Shyam Sunder’s team at NIST. The subsequent progressive collapse was not simulated at NIST because its inevitability, once triggered by impact after column buckling, had already been proven by Bažant and Zhou’s (2002A) comparison of kinetic energy to energy absorption capability. The elastically calculated stresses caused by impact of the upper part of the tower onto the lower part were found to be 31 times greater than the design stresses…. (…. Eq. 1, rather than 2, is decisive).”*(Emphasis added)**In conclusion, the NIST Report purporting to explain the complete collapse of the Twin Towers on 9/11 fails, since it relies on Bažant’s CDCU theory to accurately portray the event, and that theory has been shown to be invalid.**

^{6 }which was published by The International Journal of Protective Structures in June 2013. There the authors showed that if the correct values were used for available impact energy and energy absorption capacity that a collapse would not have propagated and that Bažant’s ratio of these two values was wildly inaccurate.

**NOTES:**

*1.Mechanics of Progressive Collapse: Learning from World Trade Center and Building Demolitions*, Zdeněk P. Bažant and Mathieu Verdure, Journal of Engineering Mechanics, March 2007; *What Did and Did Not Cause Collapse of World Trade Center Twin Towers in New York?*, Zdeněk P. Bažant, Jia-Liang Le, Frank R. Greening, and David B. Jenson, Journal of Engineering Mechanics, October 2008; *Why the Observed Motion History of World Trade Center Towers is Smooth*, Zdeněk P. Bažant and Jia-Liang Le, Journal of Engineering Mechanics, January 2011; *Spontaneous Collapse Mechanism of World Trade Center Twin Towers and Progressive Collapse in General*, Zdeněk P. Bažant and Jia-Liang Le, Journal of Structural Engineering, April 8, 2022.

*2. NIST (National Institute of Standards and Technology). 2005. Federal Building and Fire Safety Investigation of the World Trade Center Disaster: Final Report on the Collapse of the World Trade Center Towers*. NIST NCSTAR 1-6 Structural Fire Response and Probable Collapse Sequence of the World Trade Center Towers (authored by John L. Gross and Therese P. McAllister), paragraph 9.4.4, p. 323, wherein they explicitly state (referring to Bažant’s theory),* “NIST agrees with the assessment of the tower’s required structural capacity to absorb the released energy of the upper building section as it began to fall as an approximate lower bound.”*

**NIST also references other studies: Weidlinger Associates, Inc. with Hughes Associates and ArupFire (2002 & 2003); University of Maryland and Israel Institute of Technology (2002); University of Edinburgh (2003 & 2005); and ARUP (2005). However, none of these other reports posited the collapse mechanism of the lower part of the buildings – only Bažant’s paper put forward a theory purporting to explain why the lower part of the buildings failed. Furthermore, in other parts of the NIST Report, references are made to the failure of the lower part of the building for the reasons given in the Bažant theory, e.g., NIST NCSTAR 1, Abstract, p. iii,**

*“Global collapse occurred as potential energy of the falling upper structure exceeded the strain energy capacity of the deforming structural members.”*

*3. The Missing Jolt*, Szamboti & MacQueen, Journal of 9/11 Studies, January 2009, Appendix C.

*4. The Missing Jolt*, Szamboti & MacQueen, Journal of 9/11 Studies, January 2009, Appendix D.

*5. Analysis of the Mass and Potential Energy of World Trade Center Tower 1,* Gregory Urich, Journal of 9/11 Studies, December 2007.

*6. Some Misunderstandings Related to WTC Collapse Analysis,* Gregory Szuladzinski, Anthony Szamboti, Richard Johns, Journal of Protective Structures, June 2013.