Speculation continues to mount regarding the cause of Florida International University’s deadly Pedestrian Bridge Collapse. After a day has passed with no knowledge of the origins of the bridge’s collapse, this article addresses a principal cause of the FIU Bridge Collapse, and why a 19th century German structural engineering principle known as Müller-Breslau Matters Most (as depicted in the above cover diagram of this piece constructed with the help of my fine students at Texas Tech University, Mr. Virgilio A. Gomez (@virgilioAgomez), mechanical engineering masters degree candidate under my guidance inside the Edward Whitacre College of Engineering, and Ms. Alexandria Reeves (https://www.linkedin.com/in/alexandria-reeves-b30760118), undergraduate junior business major inside the Jerry Rawls College of Business).

Video Credit: @OliverMcGee on #FIUBridge Collapse, #FoxNews Your World with Neil Cavuto (3-16-18)

Five dozen kids currently enrolled in my Aircraft Structures, and Aircraft Jet Engines and Rocket Propulsion classes at Texas Tech University, including receiving my Elementary Structural Analysis and Intermediate Structural Analysis principles are entering into a brief introduction to Herr Müller-Breslau.

These students can now understand why Müller-Breslau‘s proven structural engineering principle is as relevant for them to learn today in the wake of the tragic FIU Bridge Collapse in Miami, Florida, as it was for me decades ago. I had my similar training at The Ohio State University Department of Civil Engineering’s elementary and intermediate structural analysis classes and highway bridge analysis and design class of The Late Professor Charles B. Smith. This led me to the privilege to teach similar structural analysis classes at Ohio State, Georgia Tech, MIT, Howard University, and Texas Tech.

But first, let’s take a closer look at the backstory here, specifically at what occurred to cause this catastrophic bridge collapse?

A newly installed bridge heralded days earlier, as an innovation of prefabricated construction engineering suddenly collapsed on Southwest Eighth Street, an eight-lane high-traffic highway near Florida International University’s (FIU) campus in Miami on Thursday, March 15, 2018, at about 1:30pm Eastern, Miami-Dade Fire Rescue Division Chief Paul Estopinan said in a press conference Thursday afternoon.

Multiple automobiles were trapped underneath the collapsed pedestrian bridge. Firefighters, who are now in “search and rescue mode,” are utilizing trained canines, search cameras and sensitive listening devices, throughout the night and into the morning. They have pulled out at least six deceased people from the rubble, as of Friday, March 16, 2018, and ten other people were taken to nearby Kendall Regional Medical Center, Miami-Dade Fire Chief Dave Downey said at a Thursday evening press conference and Associated Press.

This is how the FIU Bridge of integrated truss and post-tension prestressed concrete construction appeared in this video days before collapsing.

This surveillance video shows the moment the bridge collapsed and local commentary reveals that the bridge lacked necessary mid-span support as suggested in the cover diagram of this piece. In addition, the local commentary questions why the traffic along Southwest Eighth Street was not halted as construction engineers performed their stress tests on Thursday.

President Trump tweeted: “Continuing to monitor the heartbreaking bridge collapse at FIU – so tragic. Many brave First Responders rushed in to save lives. Thank you for your courage. Praying this evening for all who are affected.”

FIU officials said in a statement: “We are shocked and saddened about the tragic events unfolding at the FIU-Sweetwater pedestrian bridge.”

“This bridge was going to provide a safe transportation for pedestrians to cross between the university and the City of Sweetwater,” said Orlando Lopez, mayor of Sweetwater.

FIU, one of the 10 largest American universities with nearly 54,000 students enrolled, has been rocked by this tragic transportation infrastructure collapse.

The 174-foot long pedestrian bridge was assembled on-site days earlier on Saturday from prefabricated post-tension prestressed concrete spans along the closed sidewalk of the highway. After which, the assembled prefabricated continuous prestressed concrete beam that was hinging on one support was swung 90 degrees across the highway, and was then hinged on the other support on the other side of the Southwest Eighth Street highway.

“The $14.2 million dollar bridge had been partially assembled by the side of the highway, in order to not obstruct the flow of traffic on the seven-lane highway during construction, and was slated to open in 2019,” according to the Miami Herald. But its “innovative installation,” which “saw workers move the walkway into place before the main support tower had been installed, was risky,” University of California, Berkeley engineering professor Robert Bea told the Associated Press.

As reported in Time: “The bridge was also unusually heavy, employing concrete elements, such as trusses and a concrete roof, rather than lighter weight steel,” according to Ralph Verrastro, an engineer and expert in accelerated construction projects.

Munilla Construction, a family-own firm that worked on the bridge, called the accident a “catastrophic collapse” and promised to conduct “a full investigation to determine exactly what went wrong.”

“Munilla Construction has also been fined more than $50,000 for 11 safety violations over the past five years,” according to Occupational Safety Health Administration records, Time reports.

Two construction workers were on the pedestrian bridge when it collapsed, Miami-Dade Fire Rescue confirmed, and “were believed to be conducting a stress test on the unfinished bridge,” reports the Miami Herald. “Over tightening steel cables that run through the bridge slab sections can lead the structure to “camber,” or buckle,” experts told the Miami Herald.

Construction engineers were performing post-tension moment stress tests across the 8-span continuous prestressed concrete bridge, when it suddenly collapsed onto the 8-lane Southwest Eighth Street highway, akin to what we recently observed with the Amtrak 501 derailment outside of Seattle, Washington!

The prefabricated concrete bridge down at #FIU is called “virtual construction” of structural engineering akin to “virtual manufacturing” used in aircraft designs at #Boeing & #Airbus. Prefabricated Civil Engineering Systems are the future of construction & installation of America’s Infrastructure.

https://twitter.com/olivermcgee/status/974393367449792512

Who is Müller-Breslau?

Let me introduce you to a German structural designer and classical structural engineering pioneer.

Heinrich Franz Bernhard Müller (born May 13, 1851 in Wroclaw, Poland and died April 24, 1925 in Grunewald, Germany, “known as Müller-Breslau from around 1875 to distinguish him from other people with similar names”) was a German civil engineer. He made early advances in the structural analysis of continuous beams and rigid frames used in modern pedestrian and highway bridges and tall buildings.

Essentially, Müller-Breslau establishes a longstanding principle utilized by structural engineers to sketch qualitative influences of continuous bridge supporting forces, spanwise forces, transverse shear stresses, and transverse bending moment stresses, as a basis for pedestrian and highway bridge design and analysis, including experimental stress testing of bridges.

Why Müller-Breslau Matters in the FIU Bridge Collapse.

U.S. Senator Marco Rubio (R-FL) tweeted on Thursday: “The cables that suspend the #Miami bridge had loosened and the engineering firm ordered that they be tightened. They were being tightened, when it collapsed today.”

This FIU Bridge Collapse, I add, is a result of a missing essential center safety support tower mechanism located at midspan across the enormously long 174-foot span of the integrated truss post-tension prestressed concrete continuous beam construction, as shown in the cover diagram of this article.

Thus, the bridge’s failure collapse mechanism occurred around the center (as shown as a red (failure deflected) dashed line in the cover diagram of this piece), as a result of positive moment distribution stress failure (which could have been mitigated by the missing negative moment distribution stress over the missing midspan support) of the bridge under its own dead-load weight of 950-tons or 5.5 tons per linear foot of bridge span length of uniformly-distributed deadweight loading.

Essentially, as shown in the above cover diagram, the post-tension prestressed concrete pedestrian bridge is supposedly fundamentally designed to deflect as a “smile” under positive bending moment stress between end-span supports.

And, it is supposedly designed to deflect as a “frown” under negative bending moment stress over middle-span supports – which was apparently missing during Thursday’s stress tests. This essentially caused the FIU bridge to collapse through a huge 140% over traverse bending deflection (as discussed below) during its prefabricated construction and installation.

Ultimately, the FIU pedestrian bridge’s designed live-loading was to withstand a Category 5 Hurricane over a hundred years!

Müller-Breslau‘s principle demands that a middle support tower mechanism is essential to prevent the FIU pedestrian bridge collapse mechanism, as indicated by the red (failure deflected) dashed line in the cover diagram. The green (true deflected) dashed line in the cover diagram is the properly midspan tower supported equilibrium shape of the FIU pedestrian bridge, carrying its 5.5 tons per linear foot uniformly-distributed deadweight loading. This is shown atop the idealized depiction of the integrated truss post-tension prestressed concrete continuous span bridge.

Under the FIU pedestrian bridge deadweight loading, including ideally the properly constructed midspan support tower mechanism, Müller-Breslau’s principle says the bridge’s shear stress distribution is actually proportional to a linear function of the spanswise coordinate (x) shown: V(x)=wL((5/8)-(x/L)) with a midspan support tower maximum shear stress proportional to (5wL/8).

The bridge’s bending moment stress distribution is actually proportional to a quadratic function of the spanwise coordinate (x): M(x)=wL(Td+(x/L)Tr)(L/8), wherein as first introduced by Müller-Breslau, Td is a sensitivity of the bridge’s bending moment stress undergoing a linearly distributed transverse shear stress, and wherein Tr is a sensitivity of the bridge’s transverse bending moment stress undergoing a constant transverse shear stress distribution. This altogether leads to a midspan support tower negative transverse bending moment stress proportional to (wL)(L/8).

Finally, the bridge’s properly midspan supported transverse deflection must actually be according to code: Ely(x)=(x/L)(2Wr+Wd)(wL**4)/48, (where L**4 symbolizes now and hereafter the bridge span length, L, raised to the fourth exponent power). And wherein, as first originated by Müller-Breslau, Wd is the bridge’s transverse bending moment stress undergoing a linearly distributed transverse shear stress, and wherein Wr is the bridge’s transverse bending moment stress undergoing a constant transverse shear stress distribution.

This altogether leads to a maximum transverse deflection according to code at Ely(at x=50 feet from the midspan support equal to (27/5000)(wL**4), or about 0.0054(wL**4), wherein E is the elastic modulus of the bridge’s concrete material and I is the bridge’s moment of inertia or second moment of cross-sectional area transverse to the bridge’s span-wise coordinate (x).

Under the FIU pedestrian bridge deadweight loading without the midspan tower support, as it happened during Thursday’s collapse, Müller-Breslau’s principle says the bridge’s transverse shear stress distribution is actually proportional to a linear function of the span-wise coordinate (x) with V(x)=Tr(wL/2) vanishing at the bridge’s midspan.

The bridge’s transverse bending moment stress distribution is actually proportional to a quadratic function of the spanwise coordinate (x) with M(x)=(wL/2)L, having a midspan non-supported positive transverse bending moment stress maximum proportional to (wL)(L/8).

Finally, as the bridge’s midspan is non-supported, its transverse bending deflection is Ely(x)=(x/L)(Wd-(x/L)Wr(x/L))(wL**4)/24, having a maximum transverse bending deflection of Ely(at x=87 feet, at the non-supported midspan) equal to (5/384)(wL**4) or about 0.013(wL**4).

In conclusion, Müller-Breslau’s principle says the FIU pedestrian bridge collapsed mathematically under an absolute value of (1-(0.013/0.0054))(100%)=140% error in its failure collapse transverse bending deflection mode (shown as a red dashed line in the cover diagram) underneath the bridge’s own deadweight. This is measured relative to its ideally proper midspan tower supported transverse bending deflection mode (shown as a green dashed line in the cover diagram) underneath the bridge’s own deadweight.

“My thoughts and prayers are with the victims of this tragedy and their families. Incidents such as this, while thankfully rare, remind us of the complexity of structural systems and the great responsibility that structural engineers and contractors have to public safety,” says Indianapolis-based practicing structural engineer, Michael I. Owings, P.E., S.E. “Many eyes in this industry will be focused on the investigation over the coming weeks to ascertain the cause of the FIU bridge collapse, prevent future loss of life and restore the public’s trust.”

I wholeheartedly agree with my very dear friend and former masters degree graduate student at Ohio State, Mr. Owings.

We are reminded that the FIU Bridge Collapse is still an ongoing investigation. We all hope to have some definitive answers very soon, that we enable us to better understand structural engineering and infrastructure safety and security threats, whether accidental or resulting from unintentional consequences, so as to avoid this kind of catastrophic and tragic extreme event in the future.

It’s just a matter of time, as people will be thinking about this bridge collapse continuously before we all really know completely all the truths behind this extreme infrastructure event and its aftermath of human recovery.

Danke Herr Müller-Breslau

Our most compelling interest in pedestrian and historic bridge safety and security in the age of America’s crumbling infrastructure remains an ongoing and essential contemporaneous priority in discussing advanced structural engineering technology and education, as well as, considering the public’s understanding of science, engineering and technology. And, most of all, we must facilitate the diverse cultural participation in structural engineering safety and security by a global workforce of experts, working through the aftermath investigation of a pedestrian bridge collapse on the 10th largest university campus in America at Florida International University.

Besides all the talk of prefabricated concrete bridge construction, midspan tower support mechanisms, innovative bridge installation procedures, German structural engineering, post-tension prestressed concrete materials, and so forth, what investigators also have on their side are basic scientific and engineering principles.

Bridges don’t just collapse, and they don’t just fall onto busy highways. They go up and they don’t fall down.

Like everything else in this world, bridges are bound by fundamental rules of science and engineering — things like transverse shear stresses, transverse bending moment stresses, bridge deadweight to live-load ratios and, not the least, simple gravity of Sir Issac Newton.

We all owe a great debt of gratitude to Herr Heinrich Franz Bernhard Müller-Breslau for his pioneering innovations in structural engineering analysis and design, and for his fundamental Müller-Breslau principle. This is now aiding America’s ongoing efforts in managing our stressed infrastructure and rebuilding and retrofitting it in preparation for the next generation and the generation after – On Getting to 2076 – America’s Tercentennial!