Mar 162018

FIU Bridge Collapse: Why Müller-Breslau Matters

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 (, 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.

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, Portland 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!

(Visited 148,108 times, 1 visits today, 76,721,997 overall visits across all posts)

Thank you so much for your time in reading this article. Will you please share it across your Facebook, Twitter, Google and LinkedIn social media? I do await your comments on this article.

  22 Responses to “FIU Bridge Collapse: Why Müller-Breslau Matters”

  1. so interesting. thanks for sharing your expertise in this manner. I understand that they had made plans for building a center support. We are all wondering why that wasn’t at least partially built prior to moving bridge into place and then finished once positioned. prayers for all in this tragedy.

    • Great question JED, which is why my students and I initiated this public conversation. Thank you for joining in with your fine question.

      • This is not even remotely relevant to the collapse of this bridge. Dr McGee and Mr Gomez both have it wrong. Please take another look at the drawings for this bridge and you will see two important facts. If you fail to acknowledge them now before it is too late, then you may lose your credibility perhaps permanently. I am trying to help you here, but you do what you want. You guys have jumped to a conclusion and are now embarrassing yourselves with your wild over-confidence.

        1. The bridge section that fell was only HALF of the final bridge. The center support that you are so sure of yourself about is at the end of this first section across the road from FIU. There is a whole section left to be built over the canal, so yes at the end it will appear to have a center support, but in reality it will be two bridges.

        2. The future “cable stays” are not stays at all and are cosmetic pipes mostly for show. The design details show a flat plate welded onto the end of 16″ pipe with a few anchors at the tower and on top of the bridge. Hardly a suitable connection for a cable or even a pipe stay, which would be some kind of heavy duty pinned connection.

        We do not know what happened here and it will come out, but I’ll stake my reputation on the fact that this bridge is not “missing” a center support.

        • “… so yes at the end it will appear to have a center support” quoting you Bryan, as your statement here is the educational purpose of this article.

          This article is not about definite rights or wrongs at all. How can it be written with all the facts still to be determined and the investigation still to be completed.

          The article is for educational purposes only sir.

          Continuous beam (bridge) analysis and design, which I have taught for decades, and which has been outlined in the original 19th century German writings of Mueller-Breslau, of which I own the only existing copies of to-date, and of which I specifically referenced alongside my wonderful students, particularly Mr. Gomez and Ms Reeves, stipulates every statement written inside this educational piece for my students to learn and observe.

          I will defend this higher education purpose for future engineers and engineering education to the floor sir.

          Sometimes education is presenting fundamental questions to discuss with the public and not just about answering those public questions with definitive right or wrong answers that tends to definitely shut down all further questions and answers and further discussions inside the public forum, and most specifically, inside the higher education classroom.

          We have presented this piece for further Socratic discussions inside my spring semester classroom at Texas Tech University in which I am currently teaching Aircraft Structures, and Aircraft Jet Engines and Rocket Propulsion. In that delightful end, the higher purpose of this piece is immediate and is aimed specifically for future minds and capabilities of future civil, mechanical and aerospace engineers – of which I am each of these. I am only serving as a humble teacher here, and as an actively engaged science, technology, engineering and mathematics role model here – and not as a National Transportation Safety Board (NTSB) Member – of which I have not been called upon to serve as humbly speaking.

          Thank you for your engagement of this article and my website.


  2. I knew nothing about engineering until I read this article. It is very enlightening and well written. Now I have to question, if this information has been around for so many years, why did the companies involved in building it not follow the rules? Poorly trained engineers or just greedy for the money? Sad that people had to die for whatever the reason.

    • Andy, that remains to be seen as the investigation continues, who really knows until then. Me and my students were attempting to do an exercise to share with others to join in our conversation, and thank you for joining in. Above all else, this about encouraging and teaching kids to become fine engineers.

  3. Speaking as a structural bridge engineer, this article reads as technical mumbo-jumbo. Lost beneath the factoids and anecdotes is the simple fact that engineers today are entirely capable of designing a bridge able to span the full distance of this bridge without a central support.

    Until we know better whether there was an engineering design issue, a construction sequencing issue, or a material failure at FIU, the assertions here are speculative at best (and not particularly coherent at that).

    • Respectfully, I disagree with you.

    • Speaking as a Mechanical Engineer with a Master’s degree on the vibrations of three-dimensional structures, there is no such thing as a bridge without central support. Even suspension bridges, which do not have a physical central support tower, have central supports in the form of suspension cables. Or even arch bridges, which do not have a central support tower, provide central support by redistributing the dead weight loads horizontally. If you look at the FIU article below, you can see that the proposed design incorporated a type of suspension support. Before rotating the bridge 90 degrees onto the highway, they should have installed a temporary central support until the suspension cables were fully installed. Your statement of being a structural engineer is coming into question. I wouldn’t want you designing a doggy house.

    • I am a retired structural engineer (35 years or work). ianebersole is absolutely 100% correct. Dr. McGee should be commended for showing his students the possible advantage of continuous span construction over simple span construction. But in the case of the FIU bridge the simple “truss” span could easily be designed to accomade the bridge dead load and construction live load until the mast and cable stays were completed. I suspect something went wrong when jacking additional tension in the post tensioned tendons.

      • I wholeheartedly agree with you Tom and thank you for getting the point of our educational piece for my students. They are wonderfully energetic kids and they deserve our engagement from the experts like you in structural engineering, kind sir. We so appreciate your reading our piece and your online engagement inside our structural engineering classroom. Our five dozen kids now know you herein and are extremely pleased to meet you, Tom! I am pleased to have you join our social media classroom here – as this website is an online media forum for my Aircraft Structures and Aircraft Jet Engines and Rocket Propulsion classes I am teaching this spring semester at Texas Tech University. Our kids look forward to reading more about your experiences, and what you may further want to offer here, as further investigation continues on the FIU Bridge Collapse.


  4. You’re wrong. The bridge did not break in the middle. The video of the actual collapse shows that the bridge first broke at the bottom of the last angled roof support on the right end of the bridge in the last picture. All of the angled roof supports were of similar angle except that the angle of the support where the bridge broke was more vertical than the others.

  5. I was a registered designer of industrial systems(Wisconsin).I agree with Mr McGee’s analysis.
    There is no doubt this collapse could have been easily avoided if QUALIFIED engineers had reviewed the methods planned for this installation.

    • On behalf of our fine young engineers at Texas Tech (who’s basketball athletes have just made the NCAA Sweet 16 in Basketball), Thank you so much, Peter.

      My above replies to others gives the purpose(s) of offering this piece to a wanting public wanting to discuss this publicly tragic accident, as they naturally would. We are humbly only one such article among so many others in which similar discussions and learning are taking place. Warmly, Oliver

  6. The original design shows a central support tower with cable stays with the angled web members being in Tension. (You can see the substantial attachment points on the roof.) The section was installed as a “simple span” Without the cable suspension. Wouldn’t that put the top and bottom “flanges” (roof and deck) in excessive and possible opposite stress forces? Seems it would be more difficult/ costly to design a structure both ways?
    I also disagree with the narrator in the collapse video who stated the bridge was a “bad design” when it appears the more likely cause to be an Incorrect installation sequence? I’m sure this case will be extensively studied in Civil Engineering classes from now on. So sad at the loss of innocent lives. Details Matter!

    • We so appreciate your thoughtful comments, your fine questions raised and your suggestive contributions to our discussion here, J.W. Jones! Thank you, Oliver

  7. No plans? Unprofessional. Confused analysis. Incorrect “true deflection” curve. Given that it was designed as a cable stayed bridge and I suspect the stresses were high in the interim and a center support is an easy way out. Without plans I would not draw any conclusions. I love Mueller but he was all about deflection and your curve is discontinuous for a continuous beam. Brian Blum PE, MSME MSCE former aircraft stress analyst.

    • “a center support is an easy way out”

      Not an easy way out Brian, the safest way! Construction and installation and retrofitting America’s Crumbling Infrastructure must always be about safety first at all cost always, sir.

      “your curve is discontinuous for a continuous beam”

      I beg to differ here Brian, and so does the original 19th Century German works of Mueller-Breslau, which I own inside my library, sir.

      Continuous beam (bridge) analysis and design, which I have taught for decades, and which has been outlined in the original 19th century German writings of Mueller-Breslau, of which I own the only existing copies of to-date, and of which I specifically referenced alongside my wonderful students, particularly Mr. Gomez and Ms Reeves, stipulates every statement written inside this educational piece for my students to learn and observe.

  8. I This question is directed to Dr. McGee-
    The bridge was to be supported by cables linked to a vertical”post” located at “curb” side . The length of the section supported over the street is approximately 2x the length towards the curb.
    Would the support post eventually fail due to greater stresses on the street side?

Leave a Reply