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enter image description here

Why is a bridge designed like this?

The depth of the section at pillars is more than the depth at middle.

If I model this as a simply supported beam having load at mid span then the bending moment will be maximized at the middle and the area is also less at the middle. So, this will lead to higher bending stress.

So, why is it designed like that?

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    $\begingroup$ Don't model it like a simply supported beam, because it is not. Without seeing the design drawings are getting a close up inspection of the support conditions at the piers and abutments its hard to say EXACTLY how it should be model, but simply supported is definitely the wrong method. You can look at this bridge as being continuous for live live load. Place a roller connection at each abutment and 1 pier and a pin connection at the other pier. This is a good starting position but may not be the actual configuration since the support conditions are not actually know from the photo. $\endgroup$
    – Forward Ed
    Commented Jun 13, 2022 at 18:47
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    $\begingroup$ Secondly a bridge is not designed for a point load. It is designed for a moving load. Though this is simulated usually by iterating to a series of point loads at different positions. The design has to accommodate the resulting envelope of maximum and minimum forces. $\endgroup$
    – Forward Ed
    Commented Jun 13, 2022 at 18:50
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    $\begingroup$ "If I model this as a simply supported beam having load at mid span then bending moment will be maximum at middle and since the area is also lesser at middle section." Hold a bag of potatoes with your arm straight out. Which part of your arm gets tired first? Remember that the bridge doesn't just hold up cars. It holds up itself. $\endgroup$
    – DKNguyen
    Commented Jun 13, 2022 at 20:19
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    $\begingroup$ I thought it was that with an arch, the material compresses and concrete/stone is good at that. With a beam, the mid-point separates the material and concrete/stone is inferior for that. $\endgroup$
    – chux
    Commented Jun 14, 2022 at 15:04
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    $\begingroup$ The reason for choice of material to build the bridge can vary substantially: ie, economics, availability, political, trade skill set, environmental, regional precedents, area engineer preference, policy, etc. $\endgroup$
    – Forward Ed
    Commented Jun 15, 2022 at 17:56

5 Answers 5

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If I model this as a simply supported beam having load at mid span [...]

I suspect that this is where your analysis went awry.

First off, you should always model bridges with distributed loads, not a single concentrated load at midspan. The most significant load on a bridge will almost always be its own self-weight; load-trains are heavy but, well, so are bridges.

Secondly, I assume you're thinking of the bridge like this:

enter image description here

Indeed, we can see here that the bending moment is greater at midspan.

However, that's not the bridge we're looking at, it's missing the cantilevers! So in fact we get:

enter image description here

Now, I chose a midspan-to-cantilever ratio which exactly cancels out the bending moment at midspan. It's entirely possible that the real bridge has a positive bending moment at midspan, but it'll certainly be much smaller than the negative moment at the supports.

(the cantilevers might actually be supported at the ends; that would reduce the negative moment at the central supports and therefore increase the positive moment at midspan, but it'd still be much lower than if it were a pure simply-supported beam)

Obviously, the moment envelope from the load-train will have a positive component at midspan, but it won't be anything the thinner cross-section can't handle.

All diagrams created with Ftool, a free, educational 2D frame analysis tool.

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    $\begingroup$ What prg are you using to generate those images? $\endgroup$
    – Forward Ed
    Commented Jun 14, 2022 at 1:59
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    $\begingroup$ @MechaTrex You state lesser material as if it were just one thing. It compounds. Less material means less material to hold up that material and less material to hold up that material, etc. So less material everywhere. $\endgroup$
    – DKNguyen
    Commented Jun 14, 2022 at 20:36
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    $\begingroup$ @ForwardEd: Ftool, a free frame analysis tool. Disclosure: I worked in the development of versions 3.01 and 4.0 a few years back, but have used it since year 2 of uni. $\endgroup$
    – Wasabi
    Commented Jun 14, 2022 at 23:21
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    $\begingroup$ And of course less landfill once the bridge is replaced. $\endgroup$
    – arne
    Commented Jun 15, 2022 at 8:08
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    $\begingroup$ @Peter-ReinstateMonica I wouldn't be as emphatic as that. At worst, it's only highly conservative. Modeling it as a straight beam ignores the fact that the arch transforms a lot of the load into simple compression, which would substantially reduce the requisite reinforcement and cross-section. So modeling the arch as an arch is certainly more correct, but not doing so isn't exactly criminally wrong, just inefficient. $\endgroup$
    – Wasabi
    Commented Jun 17, 2022 at 17:33
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Since this bridge is crossing over a waterway, besides aesthetics, the arch-shaped bridge provides several advantages:

  • Less restrictive over the height of marine traffic due to more headroom in the mid-span.

  • More dead weight is concentrated on the piers which makes the piers more stable.

  • Regarding your analysis, you have ignored the effect of the varying depth of the girders, and, most importantly - the "Arch Action". Due to the very large rigidity at the piers, we can assume the arched middle span is fixed on both ends, for which, the moment due to a concentrated load in the midspan is $\frac{3PL}{64}$, much less than the moment for a straight fixed end beam $\frac{PL}{8}$. Note, for the arch with pin ends, there is no moment, but thrust, throughout the span. (The moment comparison tends to give the arch another advantage - longer clear span.)

Note that The first two reasons usually are the controlling factors in the selection of types of bridges over the waterway.

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  • $\begingroup$ So these connected arches (fixed at piers) need to take negligible bending moments only, by design? $\endgroup$
    – Narasimham
    Commented Jun 14, 2022 at 18:06
  • $\begingroup$ @Narasimham The two arms are called a propped cantilever, which is fixed at the pier, simply supported on the abutment. The moment is much less than a true cantilever. $\endgroup$
    – r13
    Commented Jun 14, 2022 at 18:51
  • $\begingroup$ I mean arch action at middle segment of full bridge between pier fixity at shear web depth $d_1$. $\endgroup$
    – Narasimham
    Commented Jun 14, 2022 at 19:02
  • $\begingroup$ @Narasimham Yes, At piers, M=PL/32. In the midspan M = 3PL/64 (I made a mistake in the answer previously). Note, that the moments are based on an arch with a uniform section. $\endgroup$
    – r13
    Commented Jun 14, 2022 at 20:34
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    $\begingroup$ "the arch-shaped bridge provides several advantages: Less restrictive over the height of marine traffic due to more headroom in the mid-span" → And, for the same reason: more difficult to get clogged in a flood. $\endgroup$
    – walen
    Commented Jun 15, 2022 at 7:44
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Spreading the upward reaction load from the support pillars is one reason, the function of an arch translating vertical load to horizontal thrust is another. But there are already good answers saying that.

There's another answer that isn't about structure : A lighter construction would be three arches - or one and two half-arches - of thickness d2, but there's been an assumption that traffic would probably prefer a level surface along the top than passing over three bumps (or two troughs).

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    $\begingroup$ Looks like a railway bridge, so yeah, the trains would probably mind three bumps. :) $\endgroup$
    – pipe
    Commented Jun 14, 2022 at 12:49
  • $\begingroup$ @pipe - could be. I'll tweak for non-specificity... $\endgroup$
    – ItWasLikeThatWhenIGotHere
    Commented Jun 14, 2022 at 17:29
  • $\begingroup$ The structure is not an arch in terms of structural behaviour and cannot be analysed as such. $\endgroup$
    – PM-14
    Commented Jul 15, 2022 at 10:30
  • $\begingroup$ @PhilM - I thought the arch answer (and now its associated comments) gave some useful information in addition to the selected answer concerning cantilever bending moments. I mentioned both here because on its own "to keep the top straight", though a valid design criterion, would have looked silly. If only there had been some way I could have indicated that my answer wasn't about structure... $\endgroup$
    – ItWasLikeThatWhenIGotHere
    Commented Jul 15, 2022 at 22:26
  • $\begingroup$ Thanks for your clarification - your sentence "the function of an arch translating vertical load to horizontal thrust is another" sounds like it is suggesting that this structure works in that way - but it doesn't and cannot. Possibly though you were not referring to the structure in the photo but to actual arch bridges! $\endgroup$
    – PM-14
    Commented Jul 15, 2022 at 22:44
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Simply put, it is because the mass of the spans has to be supported and each pillar has to support 1/2 the mid span plus some of the end section.

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  • $\begingroup$ This is partially along the right lines. The vertical shear forces in the spans will be much greater closer to the piers than at midspan, and the increasing section depth close to the piers provides greater shear capacity. The high shear forces closer to the piers correspond to the high hogging moments that will also be developed in those portions of the spans. $\endgroup$
    – PM-14
    Commented Jul 15, 2022 at 10:29
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The bridge in the photo appears to be a post-tensioned concrete box girder balanced - cantilever. This is a 'continuous' structural form, meaning the spans are not simply supported, but are continuous over the tops of the piers. As pointed out in the digrams in one of the other answers, this will lead to large 'hogging' type moments (of opposite sign to the 'sagging' moments at mid-spans) over the piers.

Depending on how the cantilevers are balanced, the bending moments under dead loads may be designed to be effectively zero at the midspans (but not always designed this way). Under different live load conditions both hogging and sagging type moments may be experienced at the midspan, again depending on the design.

The bridge spans may have a shallow arched shape, but it is certainly not an arch in terms of structural action and there is effectively no 'arching action' - contrary to what some of the other answers have implied. To model or analyse it as an arch would be seriously incorrect.

To develop arching action the supports would require exceptional rigidity against longitudinal movement (even more so in this case since the 'arch' is very shallow). As it is, the leaf piers are relatively flexible and do not have anything like the necessary rigidity to develop arching action. Additionally, box cantilever decks of this form are usually supported on guided bearings at all but one of the piers/abutments. These bearings permit free longitudinal movement to release thermal and creep (and arching) effects.

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