Falling Bridges and the Slide Rule

The is a great article in the Babbage blog on the Economist.  The intersection of bridge failure, indeterminate structural analysis/design, and the slide rule.  The present and future are always more important to the public and political class.  But to an engineer, the past is also important - - our future always had a past.  From historical rainfall data to calculations in 1965, history may be for historians, put history also belongs to engineers and engineering.

"BY ALL accounts, the four-lane bridge over the River Skagit in Washington state, along Interstate 5 between Seattle and the Canadian border, was in pretty good shape for a 58-year-old structure. It had been inspected for cracks as recently as last November, and was not even on the state’s list of bridges judged “structurally deficient”. Yet when, on May 23rd, a cross-member of the bridge’s superstructure was damaged by a passing flatbed truck carrying an oversize load, the whole span promptly collapsed. The truck made it across, but two cars on the bridge plunged into the river below. Thankfully, no one was killed.

The Skagit Valley Bridge, which carried some 71,000 vehicles a day, was part of the main route between California and Canada. Diversions are expected to last for several weeks while a temporary fix is prepared. A more lasting repair will not be ready until autumn. Beyond that, questions are now being raised about what, in the immediate future, can be done about the thousands of other bridges in America that are technically similar.

Much of the country's original 42,000-mile (67,500km) Interstate Highway System was built in a rush during the Cold War, from 1956 onwards—ostensibly so people could be evacuated in case of a nuclear attack, and for the armed forces to move around the country swiftly. The system was championed by President Dwight Eisenhower, who had seen at first hand how effective Germany's autobahns (unlike its more vulnerable railways) had been at moving men and materiel during the second world war.

Today, America has nearly 4m miles of public roads, including now over 47,000 miles of Interstate highways and 115,000 miles of freeways and other roads built to similar standards. Of the 607,000 bridges in the country, the Interstates alone account for more than 55,000—two-thirds of which were built during the 1950s and 1960s.

Their age is one concern. In some cases, however, their design is an even bigger problem. Most bridges built during the early years of Interstate construction used trusses that had just enough structural members to maintain their rigidity. In other words, they had no redundancy. Knock out one tensioned member, and the rigid structure would become a floppy mechanism. Engineers call such a design “fracture-critical”.

At its simplest, bridge-building is all about stringing triangles of structural cross-members together to form of a truss that transfers the various loads acting on it to the ground. Imagine a pair of triangles in the form of a square in the vertical plane—ie, four structural members connecting the four corners, and a fifth joining one pair of diagonally opposite corners. For simplicity, assume there are pin-joints at each corner. Then, no matter how much any corner is pushed, the frame will (more or less) maintain its square shape. But remove, say, the one diagonal member, and the whole frame instantly collapses. It is no longer a structure capable of carrying loads, but a mechanism that cannot even support its own weight.

As a fledgling aeronautical engineer, Babbage learned to distinguish between static structures and dynamic mechanisms by means of a simple rule. Doubling the number of joints and subtracting three gave the number of load-carrying members needed to prevent a structure reverting to a mechanism and collapsing. For instance, the above square frame with four corner joints requires five members (two times four minus three equals five). A truss with 14 joints needs 25 members, and so on.

The photograph of the bridge over the Skagit River (above) shows that the trusses on either side of the roadbed each have 14 joints and 25 load-bearing members—the minimum required for rigidity. The vertical trusses are connected at the top by a horizontal lattice that itself has little or no redundancy. The bridge itself is a “through-truss” design, meaning traffic travels through the structure rather than over it, increasing the chance of contact with passing vehicles.

Unfortunately, apart from being a fracture-critical structure and a through-truss design, the bridge over the Skagit River is what is known as “functionally obsolete”. This does not necessarily mean it is unsafe, but simply that it was built to standards that are not used today. Lane widths, shoulder widths and vertical clearances are inadequate by current standards. In the event, the oversize load that caused the span to collapse caught a cross-member of the bridge’s lower-than-normal superstructure.

More than 84,000 bridges in America are classified as functionally obsolete. Another 66,000 are deemed structurally deficient, meaning they need to be repaired or replaced. In the meantime, they can remain open to traffic provided certain restrictions are applied—such as reduced weight limits or speed restrictions. Together, these two problematic classes account for a quarter of all bridges in the country. Of these, some 18,000 are fracture-critical designs.

Which raises the question of why they were built in the first place. The short answer is that they are cost-effective. Embodying only as many members as absolutely necessary makes them extremely efficient. And by saving material, they are among the quickest and cheapest to build. Also, despite their lack of redundancy, they have had a reasonable record for safety—though the I-35W bridge in Minneapolis that suddenly collapsed in 2007, killing 13 people and injuring 145 others, was a fracture-critical design.

Decades ago, engineers favoured such bridge designs for another reason, too. Having no redundant components, calculating the forces carried by the various members, the deflections they cause, the shear stresses involved and the bending moments around their joints could be be done by applying simple Newtonian mechanics. Adding reinforcements to the structure (ie, increasing the redundancy) made the design "statically indeterminate" and the calculations horribly complicated.

Or so it did until the late 1960s, when digital computers and numerical techniques, such as finite-element analysis, began to take the grunt-work out of such calculations. Few fracture-critical bridges have been built since. Nor have any through-truss designs been adopted for road use, though they are still used occasionally for railways. Besides, finite-element analysis—devised originally for calculating the stresses in aircraft fuselages and wings—has allowed exquisite thin-shell structures to be built in concrete to carry loads over previously unimagined distances. Bridge-building today is as much architecture as it is structural engineering.

In his state-of-the-union address earlier this year, President Obama proposed a “Fix-It-First” programme to put people to work as soon as possible on America’s most urgently needed repairs. He specifically mentioned the “nearly 70,000” structurally deficient bridges across the country. But the $50 billion needed for the whole Fix-It-First endeavour is proving hard to come by.

Not counting the country's functionally obsolete bridges, the putative cost of the current backlog of work waiting to be done on the National Highway System’s bridge rehabilitation and replacement programme is more than $32 billion (in 2004 dollars). It would take over $5 billion a year for 20 years to catch up. Doing so at a pace President Obama had in mind would require at least $20 billion a year.

The trouble is that the Highway Trust Fund, which receives money from fuel taxes to pay for repairing bridges and other infrastructure, is expected to go broke next year. Revenue from fuel taxes has declined steadily as Americans have switched to more efficient cars, and also started driving less as a result of persistently high unemployment and the anaemic economy. Meanwhile, the chance of raising federal taxes on petrol and diesel, which have not seen an increase since 1993, is effectively zero—given the present gridlock in Congress.

Babbage hesitates even to suggest it, but he believes the only answer is for individual states and local authorities to introduce tolls for bridges and other pieces of infrastructure that have fallen into disrepair. In principle, given that road and fuel taxes are inevitable, motorists should not have to pay twice for the right to use public highways and crossings. But if tolls offer the only means for rehabilitating the country's crumbling and troublesome infrastructure, then reluctantly so be it."