A code was written to optimize the geometry of a of a good conductor to maximize heat transfer by minimizing compliance. The method of homogenization was used, whereby a hypothetical laminate with infinitely small laminate widths is assumed to exist over the whole region, in which the laminate is always oriented with its strong axis in the direction of the temperature gradient. The percentage content of the good conductor is optimized for all elements in the region, using LaGrange multipliers to insure total area of the good conductor did not exceed a given percentage of the region.
The formulation variation is used to determine the change in the objective function in in response to a perturbation of content change. Then, a formula for a perturbation in the content was found that will always (assuming step is sufficiently small) reduce the compliance. This perturbation is calculated for each successive iteration, and the percentage content of material A is optimized. Finally, penalization is used to discretize the results so that any given point is either made of good conductor or bad conductor.
Code was written in FreeFEM, an open-source finite element analysis solver based on C++, to perform iterations, calculate perturbations, and penalize. Code was based on examples by G. Allaire available here. The code is generalized, and can be applied with different regions, different heat loads, different materials, and isotropic and anisotropic conductors.
Example: Optimizing the shape of material B (good conductor, limited to 20% of area), and material A (bad conductor, 80% of area), with a parabolic source of heat imposed. The problem is shown in figure 1. The optimized percentage content of material A in the laminate at each point is shown below in figure 2.
Penalization was then utilized to discretize the region into either material A or material B. Penalization was integrated into the optimization iteration, with a function below, where θ is the percentage content of material A at each point, and r is a number in the range [.95, .99], in which a lower number gives faster penalization.
As expected, root-like structures developed, which pulls heat from the source to the sink, shown in figure 3.
A final temperature gradient was calculated, shown below in figure 4. A valley-ridge pattern of the temperature is visible at the heat source. This effect occurs where individual roots cause little valleys in the temperature at their point of contact with the heat source.
The UC San Diego Steel Bridge Team designs, analyzes, optimizes, and fabricates a model steel bridge. The team optimizes for high stiffness, low weight, and fast constructability.
Extensive testing was performed on the model bridge for the 2022 UC San Diego Steel Bridge Team. The testing was performed to insure sufficient strength of members and connections, to accurately predict local load-deformation behavior of connections and global load-deformation of the whole bridge, and to perform a vibration system identification of the bridge.
1. Vibration Testing
Vibration testing was performed with a home-made setup of accelerometers and Arduino micro-controllers in an attempt to identify system properties. First natural freq. of 7.9 Hz aligns fairly closely with predicted 8.8 Hz from analytical dynamic FEA model. Significant variation observed in higher modes.
Notes on instrumentation:
3-axis accelerometers stick to bridge with magnets (right pic)
Arduino nano micro controllers (left pic) communicate with up to 3 accelerometers
Computer with C++ and MATLAB to coordinate microcontrollers and compile/postprocess data.
Karl Johnson, a UCSD electrical engineering student, coded and integrated the accelerometer system, teaching UCSD steel Bridge members how to sauder and wire circuit boards in the process, and how to communicate with Arduino microcontrollers with laptops. Open source code is published here: GitHub Accelerometer Code
Smartphones can also be utilized to record vibration data. Saul and teammates created a document with MATLAB code for how to remotely control your phone as an accelerometer through the MATLAB app, and graph acceleration data. The document is here: Acceleration from MATLAB phone app
2. Destructive Pullout-Testing
The destructive connection testing was used as part of the connection design process. A robust set of design, analysis, prototyping, and testing techniques were used, including:
Hand calculations based on AISC § J3 to check for bearing capacity exceedance, net section failure, and pull out failure
Finite Element Analysis models using “worst case” loading scenarios for connections with complex geometries (left picture).
Destructive prototype testing (center, right) to verify results of hand calculations, Finite Element Analysis, to determine weld strength and loading behavior. Force-deformation results were then applied to “weak members” in a SAP 2000 to accurately estimate bridge deflection, both from local connection deformation and global elastic deformation.
3. Cyclic Loading Tests
We performed 3 full-scale loading and unloading tests prior to competitions to (1)insure sufficient strength and (2)measure experimental deflection and compare to analytical linear-elastic FEA model.
The right figure shows loading curves of the bridge. The pink test was performed after stiffness-increasing updates were made in preparation for nationals.
The bridge does not fully rebound to initial position, indicating irreversible “play” or other inelastic deformation in the connections. This “play” was quantified through the destructive pull-out testing, allowing the team to accurately model the inelastic behavior of the bridge in SAP 2000. Connection are being built for the 2023 steel bridge with heat-treated chromoly, in order to reduce local yielding of connections and thus reduce inelastic behavior of the bridge. Carefully consideration is being used to balance the goal of local stiffness with the need for ductility.
I am deriving a mathematical model – and writing optimization code in Python, C++, and AMPL – to be able to optimize trusses (which could be a skyscraper, bridge, crane, etc…) to support a given set of loads, or multiple sets of loads, while (1) minimizing weight, (2) maximizing stiffness, (3) minimizing construction difficulty, or (4) for any multi-objective function that combines these factors. So far, I have been able to optimize for the any of the three factors on their own – albeit only with smaller problems for construction difficulty, due to the discrete variables inherent in evaluating construction difficulty – as well as with for combinations of minimizing weight and maximizing stiffness.
The code is being published under an open-source license on my GitHub. A full document with detailed mathematical derivations and optimization routines for many truss optimization examples is available, also on my GitHub here: TrussOptimizationDoc. Below are some fun examples of optimized trusses.
Minimizing Number of Members and Adding Member Length Constraint
Volume Minimization with Joint Cost to Simplify Constructability
Code adapted from “A Python script for adaptive layout optimization of trusses”, L. He, M. Gilbert, X. Song, Struct. Multidisc. Optim., 2019.
Optimizing a Balance Between Low Weight and High Stiffness
This #seismicsaturday is riveting! We feature a 1904 railroad bridge across the Susquehanna River near Cooperstown, NY, which your Seismic Saturday correspondent spotted while canoeing.
In late 19th century and the early 20th century, when Cooperstown and surrounding areas were manufacturing hubs, the bridge was part of a bustling train route that brought raw goods up to the factories of Cooperstown, and manufactured goods back down to New York City.
The bridge looks like a modified warren truss (fig. 2). It looks to have several Warren Trusses overlain on top of one another. In contrast to the warren truss in figure 2, the Susquehanna Bridge’s design is redundant, because if one of the diagonal elements fails, the load can travel through the members of other adjacent diagonals to the base at either side. The corner joint of the bridge, shown in figure 3, is crucial in that it connects on the left with two diagonal elements and one vertical element, each of which travels downward to support a diagonal coming in from the left side.
The foundation of the bridge is made out of flat stones (fig. 4). This may be a foundation from a pre-1904 bridge, when flat river stones were commonly used as foundations (for example underneath the original white house). Stones were popular because they could be gathered from nearby. In this case, the stones were probably gathered from the adjacent riverbed. In modern bridges, river stones have been replaced with steel reinforced concrete.
See all those bumps (fig. 5, 6, 7)? Those are rivets. Riveted connections are fascinating. They are done with a cylinder with a smooth head, which is inserted into a punched hole. The cylindrical side is then smashed down to create a pin connection (see process figure 8). This type of riveting was used up till the mid 1900s, and was used to build the Golden Gate and Brooklyn bridges.
In civil structures like bridges, rivets have now been largely replaced by bolts, which are easier to install, don’t require an on-site furnace, and do better in earthquakes due to their ductility. A concern of rivets is that because they cool down so fast after taken out of the furnace, they are essentially quenched, and thus very brittle. However, riveting is still commonly used in aircraft (pic 9), and even is used on SpaceX’s 2021 Starship Rocket (pic 10). It is riveting to see technology of the past being used in structures of the future!
Note: The UC San Diego Steel Bridge Team did a 2-day Bridge Design-athon Challenge based on designing a replacement for this bridge. It includes all sorts of fun constraints, including the presence of an endangered fish species in an inlet stream. Here is a link: Bridge Designathon: Susquehanna River Bridge
High wind has caused deadly consequences for gondolas. This #seismicsaturday we tell the story of the Squaw Valley Tram Car failure.
On April 15, 1978, a blizzard with gale force winds caused the tram car (pictured above), which was heading down the mountain, to unlatch from the first of it’s support “track” cables. Track cables weight of the cable car (which rolls on wheels along them) while a propulsion cable pulls the car along. After detaching from the track cable, the gondola plummeted 75 feet, and then shot back up when the other track cable tightened. As the car bounced back up, the first track cable swung down, slicing through the roof of the cabin and pinning 12 people to the floor. Four of those people tragically died. The rescue operation of the other 40 people on the cable car would continue for 10 hours into the night amidst a blizzard of 60 mph winds (See Frohlich 2008). Figure 1 and 2 show the effects of the cable slicing through the car.
Why did the first cable derail? There is no clear answer – one passenger reported “swinging” and “twisting” of the car in the wind just before the accident, but the clamp should have been been able to handle this movement. Dr. Karl Bittner, a leading expert in cable cars, was called in to investigate the cause of one cable unlatching, but couldn’t figure out the specific cause. Following a State Investigation, the district attorney and Cal-OSHA officially declared the accident ‘an act of god’ meaning that it was due to unknown causes and could not have been prevented with “reasonable” foresight (Renda 2015).
It is likely that something went wrong in the clamping mechanism, causing one of the two cables to detach. The Squaw Valley Gondola was a multi-cable detachable grip system (see fig. 4). In this system, one cable is for propulsion, while the other cables are called “track” cables and are rolled on with wheels to provide stability and take vertical weight. The Gondola can detach from the cable mechanism so that, upon boarding of the gondola in the station, the car can moves slowly. These detachable systems are essential in maintaining a high capacity of modern gondola systems, as the propulsion cable doesn’t have to stop or slow down every time a gondola car enters the station to pick off or drop off passengers. The detachable clamp is designed to only detach in the station when subjected to a special un-clamping mechanism. A benefit of multi-cable systems is they have better stability than their single cable counterparts (where the single cable is the propulsion cable) due to their added stabilizing track cables, and therefore can reach higher speeds (See “Bicable Detachable Gondolas”).
Although the specific cause of the track cable derail was never determined, engineers tried to learn from what they think may have been the cause of failure to build the next cable car better. Dr. Bittner, in his inconclusive investigation, recommended that the new Squaw Valley cable cars system be built with improvements including deeper grooves on the cable hanger, a longer cable guard, and cable clamps on the support towers. These improvements were implemented when the new Squaw cable car system went into effect the next winter season in December of 1978 (Frohlich 2008). Squaw Valley also now conducts daily inspections of the brakes, the counterweights, and other machinery. A new Squaw Valley cable car is shown in fig. 4, with the glistening lake Tahoe in the background. Modern cable car hangers, like the Leitner 3S Carriage, feature innovative lateral damping system to improve wind resistance, a modern system of shock absorption, and an automatic shutdown which immediately notifies the operator when defects in the rolling mechanism are detected.
Note: in my research for this post, I was unable to find any technical report detailing the old clamping mechanism of the Squaw Valley Gondola, or any other technical report on the failure. If you find one, can you please share it below?I wonder if anyone has taken another look at the mystery as to what caused the initial cable derailment since Dr. Bitner’s inconclusive report in 1978 and OSHA’s “Act of God” determination.
What did bridges look like before steel and reinforced concrete? This #seismicsatueday we feature the Smolen-Gulf Bridge in Ashtabula, Ohio. Measuring 613 ft, the bridge is the longest wooden truss covered bridge in the US.
Popular in the 1800s, a “covered bridge” is a wooden truss bridge with a truss roof. The roof protects the bridge deck from snow, rain, and sun, extending the life of the structure. The Smolen-Gulf bridge was built in 2008 and due to its roof covering, shown in fig. 2, engineers estimate it to have a design life of over 100 years.
The bridge was built in homage to Ashtabula County’s many old covered bridges. The county is known for its covered bridges, and hosts the anual Covered Bridge Festival. It is a dream of this bridge blogger to attend some day!
The Smolen-Gulf bridge is designed as a Pratt Truss (see fig. 3). The top and bottom chords are made by sistering together planks of wood. For examble the bottom, shown in figure 4, uses four 2X planks of southern yellow pine connected together with a steel plate. The diagonals of the bridge are made with three steel rods that stretch diagonally across the frame (fig. 4). These rods are in tension, and slot in between the pieces of wood in the bottom chord, as shown in figure 4. In the 1800s, wood planks would have been used in place of these rods.
The bridge is built in 4 sections, with small gaps in between. These gaps of give the sections space to expand and contract in the extreme temperature variations of Ohio (fig. 6), which can reach 100 degrees F in the Summer and plunge below 0 degrees F in the winter.
The wooden deck is directly paved with asphalt. After 15 years of wear, some asphalt appears to be worn down (fig. 6). It looks to be overdue for a repaving. The pavement may have been initially applied in a thin layer to reduce the weight added onto the structure, meaning that it wore down faster.
The Smolen-Gulf bridge structure hybridizes 20th-century bridge technology, like steel rods and modern reinforced concrete abutments, with 19th-century covered wooden bridge building techniques. The bridge is a wonderful break from conventional modern highway bridges, and is a fantastic homage to the many historic covered bridges of Ashtabula County.
How do you build an earthquake-resistant structure that looks like coral? This Seismic Saturday, we travel to Santiago, Chile, to feature the Cruz del Sur Building.
The first thing that the eye catches is the structure’s surprising form, starting thin and expanding outward up the building (figure 2). A thick central shaft, which takes both the base shear as well as the overturning moment, runs from the ground foundation all the way to the roof. Diagonal columns jut out from the central shaft to support the edges of the concrete slab, as well as the vertical columns which travel up the building.
This unconventional design causes an interesting load path under gravity load. The diagonal columns are in compression(shown in red), and the first floor slab is under tension (shown in blue) (figure 3). Concrete is very weak in tension; thus massive cables are routed through the first floor slab to carry the tension. The ends of the cables are capped with steel cylinders to allow for later access (figure 4). A structural detail of the slab is shown in figure 5, in which the the steel post-tension cable system can be seen with three slots for cables, as well as an angled cap to press inward (and slightly upward) against the concrete. Axes (or “eje” en Español) for the vertical column, the diagonal column, and the slab meet at a central point.
For any structure in Santiago, Chile, the million dollar question is, of course, what happens in an earthquake? Most of Chile borders the the Nazca Plate, which is being thrust underneath the south-American plate at a rate of ~80 cm/year, creating a subduction zone. Huge earthquakes happen every decade up and down the coast of Chile. For instance, in the last 12 years, Chile has been jolted by the 2015 Coquimbo Earthquake (8.3 Mw), the 2014 Iquique Earthquake (8.2 Mw), and the 2010 Maule Earthquake (known also as the Concepción Earthquake) (8.8 Mw). Santiago can experience lenghy earthquakes which cause peak ground accelerations in excess of .3 g (Hussain et. al. 2020). Imagine falling sideways 30% of the acceleration in which you fall downwards; that is how fast the ground accelerates. Since acceleration and motion is amplified for higher floors, the top floor of a structure, under these ground accelerations, can easily be jerked back and forth with an acceleration in excess of 1 g.
For structures with thin bases, like the Cruz del Sur Building, withstanding these extreme earthquake forces is challenging. To understand why, think of a person on a metro. When the train accelerates away from a station (and no handhold is in reach), people naturally stand with their legs wide apart, in line with the train’s acceleration. Anyone with their feet together would fall over. The same problem exists for the Cruz Del Sur structure with its slim base – the base has to resist the urge to overturn with “its feet close together” and thus a small lever arm. This overturning scenario which manifests as tensile and compressive stresses induced by the earthquake forces as well as base shear, is shown in figure 6. The solution found by the structural engineers to prevent the overturning is to extend the shaft deep into the ground and design a mat slab beneath. When the building tries to turn, the soil pushes back on the mat slab (green arrows) and on the shaft (pink arrows) to prevent it from doing turning. To deal with the stresses induced by the overturning, which can cause tensile failure or compressive failure as shown in fig. 6, the walls on either side the central shaft incredibly thick (~2 m deep). These walls are loaded with both vertical rebar to take tension, and confining rebar to prevent a spalling compressive failure. Essentially, if the building were a person on the metro, it would have its feet close together, but its feet would be latched to the ground and its legs would be bigger than Dwayne “The Rock” Johnson’s thuder-things.
One of my favorite aspects of the building is that it reminds me of a coral reef (figure 7). The base starts out thin, and then the building expands as it goes upward. It is interesting to think that coral reefs face a similar constraint to buildings in a city environment – there is a high demand for ground-level space, but upon building upward, there is more space available to expand. For coral, only a small anchor point is needed to flourish upward and then outward (and to be able to host the largest number of photosynthetic algae). For the Cruz del Sur building, a smaller anchor point allows for a public Plaza underneath, while expansion as the building travels upwards allows for more rentable office space at higher (and more desirable) elevations.
A fascinating effect, whereby light reflects off of the surface of a small pool onto the column of the structure, reminded me of the moving light patterns on the ocean floor caused by reflection through the surface (figure 8).
Also notable in the structure is the use of circular windows, both in the floor slab above the lobby, and in the 2 meter-thick shaft wall. For me, the circles add to the oceanic aesthetic, as they resemble the windows in a ship’s hull. It is interesting to add that the circular windows are located within the reinforcing wall and slabs, in areas that would be subject to high flexural, shear, and axial demand. The circular form may do a better job at preventing stress concentrations that occur in the corners of square windows. (This is why airplanes have oval windows: /https://aerospaceengineeringblog.com/dehavilland-comet-crash/). Of course, this would also depend on the steel rebar design around the circular windows.
Also of note in the structure is the strategic hiding of circular vents behind the diagonal column groups, to keep the the visible surface of the building cleaner. Additionally, the ~60 cm overhang of the slab above each window which shades the window while the sun is at its highest angle during the Summer. The structure maintains the modern window-facade look from afar, while being more energy efficient than glass-walled office buildings.
The Cruz del Sur’s unconventional shape, seismically resistant technology, and bio-inspired form, make it fascinating from the perspective of both engineers and architects. As I walk down Apoquindo (the Wall Street banking hub of Chile nicknamed San-hattan), I’m drawn towards the structure by its overhanging inverted form, and its audacity to be supported by a thin shaft in one of the Earthquake capitals of the world.
Shrouded by eucalyptus in Maple Canyon is one of San Diego’s most impressive bridges. This #seismicsaturday we feature the 1st Avenue Bridge.
The 463-foot bridge was built in 1931, and is nicknamed “The People’s Bridge” as it was funded by San Diego’s first public infrastructure tax. The bridge was actually assembled first in Ohio, before pieces were loaded on trains and brought to San Diego. The main span is a massive steel arch (fig. 1). The arch is connected to concrete abutments on either side with slanted pin connections (fig. 2). The pin allows the beams and columns to rotate slightly, allowing the bridge to flex in earthquakes and when heavy trucks drive across without putting repetitive stress on the support.
A 13-million dollar seismic retrofit was completed in 2010. If you look closely, you can spot the differences between old and new:
Above the pinned connections, sleeves have been added around the bridge columns (fig. 3). The sleeves help secure the main columns of the bridge to the steel embedment into the foundation. One can spot these sleeves by noticing that they are secured by bolts with hexagonal nuts, as opposed to the original components which are are all connected by rivets with circular heads.
To tie the bridge to the side abutments, large threaded rods have been added (fig. 4). The rods are anchored into the concrete abutments on one side and bolted to the deck on the other, keeping the bridge tied to it’s abutment in an earthquake, and preventing the deck from being thrown-off and separating from the concrete abutment. These rods are are analogous to tension ties (fig. 5) used when building wooden decks beside houses.
Old and New Nuts Reveal Confining Sleeve
Now is where things get super cool. Look closely at the nuts in fig. 6 and 7. Do you notice a difference?
The first is slightly conical on top while the second is flat. The difference likely means that the bolts were installed at different times, with the second bolt probably installed during the 2010 retrofit. The second bolt looks to be part of a layered metallic sleeve (fig. 8), that encases the the concrete of the original foundation to prevent shearing during an earthquake and deterioration over time.
The People’s Bridge is a wonderful example of how old, corroding structures can be retrofitted to be safe and beautiful once again. It must have taken a lot of creativity on the part of the retrofit bridge engineers, T.Y. Lin International, to add modern reinforcement while retaining the bridge’s rustic appearance.
What is the oldest functioning bridge in San Diego? This #seismicsaturday we feature the Quince Street Bridge.
The Quince St. Bridge was built in 1905 to connect residents of the then fast-growing Bankers Hill neighborhood to the trolley line in 4th avenue. The bridge is 263 ft long and 60 ft tall. It cost just 850 dollars in 1905 to build. The bridge has a wooden truss structure.
The bridge was originally made out of Redwood. Redwood contains a natural chemical called tannin, which makes the wood rot, bug, and fire resistant and gives the wood it’s beautiful red color. The tannin in the redwood allowed the bridge to last 82 years outdoor under the elements until 1987. Other notable San Diego structures built with old growth redwood include the Hotel del Coronado, old Alpine City Hall, the Del Mar Library, and the incredible Goat Canyon Trestle Bridge, as well as many of the city’s old victorians-style homes.
In 1987, the Quince Street Bridge was suddenly closed after a city inspection found it to be “infested with termites, full of rotting wood, and generally unsafe” (McDonell 1987). A consultant hired by the city determined that the bridge should be torn down. Local residents were were up in arms about the possibility of their romantic redwood bridge being torn down and they mobilized to have the bridge saves, convincing the City to designate it as a historic landmark. In 1990, the bridge underwent a major refurbishment, in which much of the redwood was replaced by pressure-treated pine.
One can still see some of the original redwood columns on the bridge along with the original bolts from 1905 (fig. 3). The fact that these columns are still standing is a testament to the incredible durability of redwood.
The pressure-treated pine from the 1990 rebuild can be identified by the staple-sized indents (fig. 4). These indents are made during the treatment process so that chemicals, including chromium (bactericide), copper (fungicide), and arsenic (insecticide), can penetrate into the wood. These chemicals make pressure treated pine last much longer outdoors than untreated pine.
The bridge is not in pristine condition. The dirt around one of the foundations is quite eroded (fig. 5), and some of the steel strips that sister beams together are coming off (fig. 6).
The Quince Street Bridge is a beautiful and historic redwood structure. Thanks to the efforts of dedicated community citizens in the 1980s, it has been saved for generations to come.
McDonell, Patrick. “Despite Emotional Attachments, Future Bleak for Quince Street Bridge.” Los Angeles Times. 28 November 1987.
How do you build a 200 ft long wooden bridge over a ravine just 10 miles from the San Andreas fault? This #seismicsaturday we feature a beautiful glu-laminated wooden pedestrian bridge at UC Santa Cruz.
The bridge deck is supported by three thick glu-laminated beams (fig. 2). These “glu-lam” beams are made by gluing many strips of wood together, and can be used to create continuous beams that are to large to be cut from one tree.
The bridge is supported by two large columns which are braced diagonally to provide lateral strength in side to side shaking from an earthquake. One mystery strucuture is circled in pink. Anyone know what this is?
UC Santa Cruz is less than 10 miles from major fault lines (fig. 3, USGS).
A structure that is this close to a fault experiences not just sideways shaking, but also violent up-and-down acceleration. Research conducted following the 1994 Northridge Earthquake showed that vertical acceleration is typically around 2/3 of horizontal acceleration for structures near an earthquake epicenter (Borzogonia etc. al. 1996). We can think of the effect of this vertical acceleration with two scenarios. First, when a vertical acceleration of “1 g” works with gravity, the weight of the structure can double. Alternatively, in tbe second scenario, when the “1 g”works against gravity, a structure can, for that instant, essentially become weightless. The momentary weightlessness reduces the friction between a bridge (or a house) and its foundation and can, when combined with side to side shaking, throw beams off columns and columns off foundations. The UC Santa Cruz Bridge is designed ingeniously to resist these upward forces, as well as normal downward, and lateral (sideways) loads. Check out how the different components in tbe bridge work together to resist these three distinct loads in figure 4.
Diagonal pieces of steel are hidden underneath the deck (fig. 5). Called “bridging,” these diagonals tie the deck beams (or stringers) to the beams, preventing them from toppling over sideways in an earthquake.
Sometimes, earthquake-resting mechanisms (like huge “in your face” steel diagonal bracing) can take away from a structure’s aesthetic. However, the primarily wooden and minimally steel components blend in with the UC Santa Cruz redwood forest, making the bridge both aesthetically pleasing and earthquake safe.
US Geological Survey. “Earthquake Probabilities in the San Francisco Bay Region: 2000 to 2030. A Summary of Findings By the Working Group on California Earthquake Probabilities.” USGS Open-File Report 99-517. 1999.