# Why are arched bridges stronger than flat bridges?

To keep the question brief: in bridge design, why is the arch structure favoured compared to a simple flat one?

In other words, how does the curved platform alter the force decomposition of the load on the bridge, such that it can uphold larger loads? I imagine that intuitively the load is no longer applied in a fully normal manner (orthogonal) onto the bridge, but I cannot convince myself.

• I guess a simple explanation is that bars hold larger loads under compression than under extension.
– user130529
Commented Jan 2, 2017 at 15:31
• Could you add an illustration? There are many kinds of bridges - also flat ones... Commented Jan 2, 2017 at 15:36
• If you analyze the bending stresses, in the arc design, the bending stresses are much lower. Commented Jan 2, 2017 at 15:38
• I encourage you to look for the "Bridge Construction Set" game. It simulates bridges and your task as player is to construct ever more complicated ones. Sure, the simulation will not be 100% accurate, but it is semi-realistic, and will give you a lot of appreciation if you watch real life bridges (and other, similar, constructions) later!
– AnoE
Commented Jan 2, 2017 at 22:21
• I note that several people have voted to close this as off-topic as engineering, but I don't think that close reason applies. Asking about the reason arc bridges are stronger than flat ones seems on topic here. Commented Jan 3, 2017 at 20:40

Fracture happens under tension - that is, when you pull on something hard enough, it rips. The key to understanding the arc design hinges on understanding that it lowers the maximum tensile force.

Take a simple beam, support it at the ends, and hang something off the center:

Tension at the bottom, and compression at the top, are needed to balance the torque created by the vertical forces of the supports, and the load in the middle. Obviously, the further apart the supports are, or the greater the load, the greater the tension. When that tension reaches a critical value the beam will fail.

Now if we shape the bridge into an arc, we get this:

The additional lateral forces on the arc cause compression in the beam, this reduces the net tension at the bottom and makes the beam better able to support the load. You can make things even better by spreading the load more evenly, designing the shape of the arc to better optimize the load distribution, etc - but the diagram should give you a sense of the underlying principle.

## Update

The lateral forces are perhaps most easily understood by looking at a V shaped structure: you know intuitively that such a structure would collapse unless you provide some torque at the apex to keep the legs together, or provide sufficient friction at the base of the legs to keep them together. You can also see that the force needed near the hinge (which is provided by the red "tension" stress in my upper diagram) would need to be much greater than the force provided by friction at the bottom (lateral forces from the support on the arch).

• Thanks a lot Floris. If you don't mind, I have a couple of questions after reading your answer, mainly to better understand the arguments you've provided: i) in the flat case, why do we have torques when the endpoints of the bridge are fixed? Is this only of relevance when the load is off center? ii) why do we have tension only at the bottom? Last question: iii) where do the additional lateral forces come from in the arc case? Many thanks in advance for your explanations, and also very nice diagrams. Commented Jan 2, 2017 at 17:02
• i) torques are caused whenever there is a force that is applied offset from a fixed suspension point. The suspension point here are the endpoints of the bridge. ii) Think about what happens when you bend something into an arc. The part on the inside of the arc has to get shorter relative to it's initial length and the outside has to get longer. Same action is happening here. iii) They come from the normal force of whatever the bridge is supported on. The shape of an arc naturally distributes some of it's weight laterally. Commented Jan 2, 2017 at 21:33
• Why the horizontal forces? Static (Newtonian) analysis only proves that the horizontal force in the left pier and the one on the right pier are equal and opposite (in this symmetric case). It can't prove that they are zero and in fact they are not, unless the bridge is infinitely stiff. Forces like that are called "indeterminate". You need to look at the bridge deformation and stiffness to find out what those forces are. The details depend on the material's properties and bridge construction. If you assume uniform material, it's not that hard. This is a freshman engineering problem. Commented Jan 3, 2017 at 3:47
• For a simplified version of the problem, imagine a miniature bridge made of rubber with a hefty load, sitting on a table top. It's not hard to visualize that the bridge will bend down quite a bit and the legs will try to separate. Except in a real arch, the legs are "tied down" encased in the foundations, so they can't move. The foundation is pushing sideways on the legs to keep them in place; that's where the horizontal forces come from. You can also see that the force amount necessarily depends on how much the bridge "wants to bend" (it's stiffness) and static analysis can't determine that. Commented Jan 3, 2017 at 4:03

In general "arc" structures like trusses etc. have a large height and less self weight. The more height of a cross-section means a greater value of the 'second moment of area' and thus less stress on the beams. So this technique allows the engineers to design structures that bridge long distances between supports (pillars, columns etc) such as this kind of bridge:

Now if you mean by 'arc' why the deck of the bridge is curved, the main reason is that with this technique the tensile stresses of the beam-deck get decreased, and all materials fail under tension before compression. Also the vertical deflection of the deck is minimized:

Another case is the stone bridges with arches which are well known from antiquity:

The following picture demonstrates how the loads are distributed to the ground:

It's interesting if you search about the use of the 'keystone', which is used in this kind of bridges:

Finally you can take a look at a relevant previous answer of mine.

• A good answer, but it contains an inaccuracy: "all materials fail under tension before compression". What about string? More concretely, a suspension bridge is supported by cables under tension. The only parts under compression are the towers. Commented Jan 3, 2017 at 22:29
• Yes indeed ,in fact not all but most materials fail under tension. I don't understand what you mean about cables, they 'work' only under tension.
– user98038
Commented Jan 3, 2017 at 22:33