What is the Optimal Shape for a (trussed) Arch?

Arch bridges date back to Antiquity. Steel trussed walkable arch (such as the one shown in the picture above) can be attractive because they can be prefabricated and thus speed up construction time on site. The deck can be cambered to either allow vertical clearance below and compensate for deflection under its own weight. However the maximum slope of the walkable arch is set by accessibility slope guidelines and needs to be shallow. Because of this shallowness, the arch is prone to in-plane snap-through buckling. This means that the arch can assume an inverted equilibrium position. Since the bridge is also lightweight, it natural vibration can coincide with the pedestrian-induced vibration as was experienced by the visitors to the Millenium Bridge on the day of its opening. When that happens, resonance occurs which can lead to severe structural damage.

Screen Shot 2016-07-06 at 10.57.21 AM
Illustration of arch snap-through buckling (Left) and resonance (right)

So what happens when we try to optimize the buckling or dynamic behavior of the walkable trussed arch bridge by allowing the nodes of the truss top chord to displace? The resulting truss forms, optimized in 2D (nodes only allowed to move in x,y plane) and 3D (nodes allowed to move in all 3 directions) are given in the table below.

Screen Shot 2016-07-06 at 10.57.28 AM
Optimization of the form of a trussed arch bridge by allowing the top chord nodes to move in the y and x direction (2D)

The resulting truss shapes all adhere to the slope guidelines and show a wide variety of forms including non-standard top chord topologies, global bow string topologies, tapered deck profiles and bowtie profiles in plan. When these optimized forms are evaluated for other structural criteria such as maximum axial member load and global deflection, all forms outperform the initial (in optimization called “ground”)  form.

Screen Shot 2016-07-06 at 10.57.38 AM
Walkable trussed arch bridge 2D and 3D optimized for buckling (first 3 rows), dynamic behavior (4th row) and stiffness (5th row)

When we start optimizing the ground form for different boundary conditions, a whole new realm of forms is revealed to us.  This research demonstrates that the right form can substantially improve the buckling and dynamic behavior of a walkable trussed arch. Much to our surprise, we found that the optimized forms also brought along substantial improved behavior for other less critical structural design criteria. However the take-home lesson is that even for a simple structural system like a walkable trussed arch, there is a whole wealth of superiorly behaving unexplored forms waiting to be built.   With these new forms, the vocabulary of the structural designer just got richer.

This post is based upon the work presented doctoral thesis ‘Stability and serviceability optimization of footbridges’ of my Ph.D. student Allison Halpern ’14.

Screen Shot 2016-07-06 at 10.57.45 AM
A whole wealth of walkable trussed arch forms optimized under varying support conditions.

Author: Sigrid Adriaenssens

Editor: Jacob Essig


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