In teaching engineering, I’ve come to realize that it’s essential to help students intuitively understand how it works. A lot of engineering instruction revolves around using formulas to calculate things. For example, students in solid mechanics draw shear and moment diagrams and calculate deflections. But when I taught solid mechanics, I found that students would frequently give completely unreasonable answers. Their deflections might be off by two or three orders of magnitude, yet they didn’t recognize how preposterous their results were.
To understand what’s reasonable and what isn’t, you need to understand how a structure works physically. Sometimes we can use our experience of living in buildings or crossing bridges as a guide. For example, most of us have experienced the vibration of a steel floor system in a building. Considering that experience, we might estimate the amplitude of the vibration at a few millimeters. Much more than that—say, a few inches—and we’d know something was seriously wrong.
Unfortunately, some topics in solid mechanics are beyond our daily experience. Thermal expansion, Poisson’s effect, and elastic deformation in concrete and steel are usually too small to perceive directly. So they’re difficult to understand intuitively. We can learn to calculate them, but we may not think of them as real phenomena with real consequences.
On my first visit to a prestressed concrete plant, I happened to be there when they were making concrete piles in a 600-ft pile line. To tension the steel cable, the jack pulled it about 6 ft, and then another 6 ft. Knowing that prestressing works only in the elastic range, I wasn’t sure at first that I was seeing what I thought I was seeing. The jack had stretched the steel cable just like a rubber band.
Deflected shapes
Students learn in statics class to draw shear and moment diagrams from the structure’s geometry and applied forces. In solid mechanics they learn to calculate the section properties; from that and the elastic modulus they can determine the deflected shape. That is, they’ll sketch the shape of the structural member as it deforms under load.
As a graduate student, I served as a teaching assistant in a beginning solid mechanics class. Nearly all the students were adept at shear and moment diagrams, but hardly any could sketch a deflected shape. In grading their quizzes, I found one that showed no calculations at all, but had a perfect deflected shape. He couldn’t have copied it from anyone—nobody else had it right. Then I noticed in small, faint writing a note that he’d used his plastic ruler to bend to the right shape. That is, his flexible ruler was a model that helped him understand how it works. It was a simple application of the principle of “Think first, then do.”
Segmental construction
Segmental construction is a method for assembling precast concrete elements into bridges. In the video, the elements string together like beads in a necklace, cantilevering out from each pier. Dywidag bars thread through adjacent elements to hold them together temporarily. Once the span is complete, posttensioning cables tie the whole thing together.
Video: Precast concrete segmental bridge construction. Match casting of concrete segments ensures perfect fit on site. Adding an element is like stringing beads: the segment is lifted into place, and Dywidag bars thread through the ducts in the segment. Stressing the Dywidag bars holds the segment in place. Once the span is complete, full-length posttensioning ties everything together.
It’s essential that the engineer analyze the stresses in each element not only for the final load case when everything is in place and the posttensioning is complete. The most severe load case may occur only during construction—for example, in lifting the element from the casting bed, transporting it to the site, or assembling it into the structure.
Modeling to see how it works
In the late 1980s I spent a year as a postdoctoral fellow at Norges tekniske høgskole (now NTNU) in Trondheim, Norway. Part of what made the experience so valuable was that I lived in student housing with Norwegian roommates. Their explanations of everything from Norwegian politics to how to operate the washing machines were extremely helpful.
One day I got a chance to return the favor. As I got home from the lab, my roommate Øystein peeked out from the bathroom. “Put on your rainboots,” he said. Øystein was in the last semester of his engineering diploma studies. His diploma thesis concerned a proposed hybrid bridge-tunnel to span a fjord.
Transportation structures in Norway face challenges from difficult terrain as well as a harsh climate. In this case, a bridge would have to provide enough clearance for ship traffic to pass under it even at high tide, while a tunnel on the seafloor would have forced vehicles to negotiate a long, steep path to get to and from it. The hybrid bridge-tunnel would float under the water surface, allowing ships to pass over it, while providing a reasonable slope for vehicles to negotiate. It would be a catenary in elevation and an arch in plan to withstand the forces due to tidal currents. The construction sequence called for segmental construction from the two abutments, meeting in the middle. Øystein’s assignment was to evaluate two conceptual designs, one in concrete and one in steel.
Making it work
That day he was looking at the stresses the segments would experience during construction. Øystein wasn’t big on calculation, so he decided to model it physically to understand how it works. He obtained a length of surgical tubing to represent the bridge-tunnel segment. Our bathroom sink wasn’t big enough for this purpose, and we didn’t have a bathtub. So he blocked the shower drain and flooded the whole bathroom. With rainboots on, I joined him.
We simulated the construction sequence by forming the tubing into the catenary-arch shape of the bridge-tunnel, gripping one end as the abutment and letting the rest float in the water. Then we simulated adding a new segment to the free end. It soon became clear that the additional weight would induce torsion in the cross section. Neither the concrete nor the steel conceptual design had allowed for torsion. However, thickening and reinforcing the concrete section would enable it to resist torsion. But there was no way to modify the steel section to prevent buckling.
Sadly, our other roommates, who were studying liberal arts subjects, were far less excited than we were at our discovery. They didn’t want to have to wear rainboots in the bathroom. So we drained the floor and went on to other things, but understanding now how the bridge-tunnel would work.