# The Strength of Materials

The Strength of Materials

``````I realize that this question is somewhat oversimplified, but perhaps it is still amenable to an answer.
Let’s say you have a simple bridge over a stream. The bridge is made of wood.
The bridge will not last forever. But x number of automobiles will be able to cross the bridge, one at a time, before the bridge will collapse.
Now imagine that y number of automobiles is placed on the bridge all at once on top of one another.. This is a much earlier the point at which the bridge will collapse.
There is a relationship of y to x. Obviously, y is a much, much lower number than is x.
Now imagine the ratio for other materials that compose the bridge.
QUESTION: Is there any more or less consistent relationship between y and x for different materials or is this entirely a function of the specific material?

NOTE: Perhaps I should speak of the weight o f the automobiles, rather
than the number of automobiles, but the question remains unchanged.``````

No. That’s not how it works.

The bridge can support a maximum y[sub]max[/sub] number of cars at the same time, based on the total weight of the cars, the strength of the materials and the geometry of the structure.

The lifespan of the bridge is more complex and defined by a number of factors:
The durability of the material - as wood rots or metal rusts, the effective cross section of the structural members is reduced, reducing the load it can safely support. i.e. a 2x8 that is 50% eaten by termites is now effectively a 2x4.

Fatigue - Think of it as bending a paperclip. You can do it a couple times with no problem. Keep bending it and it will eventually snap. Same thing with a bridge, building, airplane or anything else subjected to cyclical loads. Each vehicle bends the bridge a bit. The changing of the seasons bends the bridge a bit. Every gust of wind bends the bridge a bit. This phenomenon is more pronounced with steel structures, but wood ones also have steel fasteners that are subject to the same phenomenon.

Calculating x[sub]max[/sub] is more complex. Back in my structural engineering days, my school had a big lab where they would take giant steel girders and have a big machine flex them over and over, millions of times over days and weeks to see how long they would last.

Some materials have a fatigue limit, below which they can withstand an indefinite number of stress cycles without failure; others do not. See the graph comparing the properties of steel & aluminum here:

I would expect that, for most bridge materials, the bridge’s lifespan is mostly independent of the number of cars that have crossed it (beyond it being “the next car after it’s weakened too much”). The actual weakening, I’d expect, would be due to rot or rust or acid rain or whatever, not to repeated stress.

Hydraulic failure (that is water over the bridge) has been the leading cause of bridge failures. About 65% of the US bridge failures from 1966 to 2005 have hydraulic failure as the cause(followed by collision and overload).

Page 41 of 58

However , as many have pointed out above - fatigue is a big cause of concern for structural engineers since it is hard to predict requiring dynamic (frequency response ) analysis. The cause of the vibrations can be anything from wind to a sudden application of loads.

Corrosion, especially stress corrosion cracking is a big concern too.

Rot is the reason that old wooden bridges were built with covers, to keep sun, rain, and snow off of the functional deck.

Are we to presume that no maintenance is being done during the life of the bridge?

It is not an either/or; while corrision and age-related degradation certainly reduce both the absolute strength and its fatigue limit, the number and weight of vehicles will also have an impact upon the lifespan of the bridge. For instance, as the members and connections of a steel truss bridge corrode, they become weaker and more cracks form. If the bridge is lightly used, the cracks don’t propagate very much, and the lifespan can be virtually indefinite until it literally rusts through or thermal cycles and water ice expansion cause a failure or for the connections to be loose enough that the bridge no longer has adequate integrity. If the bridge is heavily used, however, cracks can propagate quickly to something that seriously degrades the carrying capacity of the bridge, and it can fail suddenly and catastrophically, hence the need for periodic inspections of bridges and other civil structures.

There is also the potential for latent defects in members and connections, or design errors that were not captured in a review process and that do not result in failure upon initial proof testing but which substantially reduce the resilience of the structure to fatigue and near design limit stresses resulting in sudden and unexpected failure.

Wood is a fantastic material from a strength-to-weight ratio, but like all composites (tree wood is a composite of cellulose fiber and lignan matrix) the failure modes are nonlinear (that is, it does not have a need modulus of elasticity and a clearly designed yield point below which it undergoes only pretty linear and non-destructive deformation) and it can develop flaws internally that are difficult to inspect but can seriously reduce its strength, particularly in buckling or interlaminar shear. Wood is also hygroscopic (absorbs water) unless heavily treated by the injection of a non-porous resin compound or otherwise sealed. It also has enormous flexibility but that can also lead to undesirable modal responses in complex structures that can feel unsafe even if the structure itself is well within design loads. And while wood does not corrode per se, it will degrade with exposure to water and sun, and the metallic fasteners generally used to connect it can rust away to nothing as mariners can attest to. All of these reasons are why wood—which is otherwise a great construction material—is rarely used for modern civil structures like bridges or many-story buildings.

Stranger

Stress corrosion cracking (SCC) is mostly a concern with high strength fasteners, threaded rod, or machined/heat treated members under constant or high cyclic tension, espeically when made of materials with low elongation such as high carbon alloy steels and aerospace-grade aluminum (6000 and 7000 series in particular). SCC can potentiallly occur in any area that is exposed and under some degree of tension but for most common civil structural materials like ASTM A36 and A572, they are so “gummy” and have such a large range of elastic-plastic deformation that failures are generally observable long before the propagate to failure as long as good design practices are followed. Overstressing a member with too much preload, or failing to account for coefficient of thermal expansion (CTE) loads on top of nominal fatigue are the most common way that preloaded threaded fastener joints fail in civil structures. Corrosion, again, will exacerbate this, but good material selection and the appropriate design load knockdowns should account for loss of strength from corrosion.

Stranger

Agreed. I was specifically referring to the cables on suspension bridges.

The cables themselves are generally made of an alloy that while high in carbon (for strength) is resistant to SCC, and is then either galvanized or coated in grease or a paraffin-based wax, or in the case of main bridge cables may be encased in a multi-layer acrylic sheath as protection against water intrusion. Cables can see a lot of cycles over their lifespan (tens of thosands of cycles a day, so in the tns of millions to billions) so they are designed for “infinite life” by selecting very large design margins and keeping the change in stress state small (and hopefully preventing stress reversal, because fully unloading a cable can cause very high and unstable dynamic loads). It is extremely rare for a suspension bridge to collapse due to cable failure; failures are usually either in the connections, or more rarely in the deck or footings.

Stranger