English is currently available.
Civil Engineering DISCUSSION
In my first structures course we learned that a beam under uniform load has its largest bending moment, w × L² / 8, at mid-span. Yet many multi-span road bridges have girders that are clearly deepest over the piers and thinnest in the middle of each span, the opposite of what that result suggests.
Is this an architectural choice, or does the bending moment really peak at the supports in these bridges? If so, why is it different from the beam in my textbook?
The w × L² / 8 result belongs to a simply supported beam, whose ends are free to rotate. A girder that runs continuously over a pier cannot rotate freely there, because the adjacent span holds it. That restraint creates a hogging moment over the support, with tension in the top, and reduces the sagging moment in the span.
Take the limiting case of a span fully fixed at both ends under uniform load: the moment is w × L² / 12 at each support and only w × L² / 24 at mid-span. With w = 20 kN/m and L = 30 m the simply supported moment would be 2250 kN·m at mid-span; the fixed span has 1500 kN·m at the supports and 750 kN·m in the middle. The support moment is the larger one by a factor of two.
A two-span continuous girder shows the same thing with realistic supports. For two equal spans under uniform load, the hogging moment over the central pier is w × L² / 8, and the largest sagging moment is 9 × w × L² / 128, located 0.375 L from each end support. With the figures above that is 2250 kN·m over the pier against 1266 kN·m in the spans.
Shear peaks at the same place. The central pier carries 1.25 × w × L = 750 kN, against 0.375 × w × L = 225 kN at each end support. So the section over the pier has the largest moment and the largest shear, and that is where depth is needed.
Varying the depth is also efficient in a way that goes beyond matching the moment diagram. For long spans most of the load is the girder's own weight. Removing depth, and therefore weight, from mid-span takes load away from exactly the position where it causes the most bending. In addition, a continuous structure distributes moment according to stiffness: making the pier regions deeper and stiffer draws still more moment to the supports and relieves the spans further.
The construction method points the same way. Many of these bridges are built as balanced cantilevers, segment by segment outward from each pier. During construction each arm is a cantilever with moment w × a² / 2 at the pier and zero at the tip, so a depth that tapers away from the pier follows the demand closely.
The reversal of moment changes where the material goes inside the section, not just how deep it is. Over the piers the top is in tension and the bottom in compression, so a concrete bridge has its prestressing tendons or main reinforcement near the top there, and box girders have a thickened bottom slab near the piers to carry the compression. Towards mid-span the tendons drop to the bottom.
The same rule applies to an ordinary building frame: continuous beams and slabs need top steel over columns and interior supports. Putting all the reinforcement at the bottom because that is where it goes in a simply supported beam is a classic early mistake.
Continuity is a choice with a price. Its advantages are smaller peak moments, much smaller deflections (the mid-span deflection of a fixed-ended span is one fifth of that of a simply supported one) and fewer expansion joints and bearings to maintain. Against that, a continuous girder is statically indeterminate, so uneven settlement of one pier or a temperature difference between top and bottom induces bending moments that must be designed for.
A chain of simply supported spans is indifferent to settlement and simple to build from precast beams, which is why both forms are still in use. When you see a constant-depth deck on many short spans it is often that second type.
