Why the design of continuous beam is always more economical than beam with supports?

Assuming that I have a beam that spans across a few supports. I can design the beam as a continuous beam (a beam that is loaded and has more than two supports), or individual beams that each encompass between two supports.

From what I know, the design of continuous beam is always more economical than beam with supports. Why this is the case? I understand that this is because the forces and moment in continuous beam are less than in beams with multiple supports. But I am looking for a deeper, more physical explanation on why the forces are smaller in continuous beam.

I see another answer has given input on the strength side of things, where the economics isn't as clear, so I will add that one of the crucial aspects that usually make continuous beams economical is deflection.

For a given load the deflection of a continuous beam is smaller (usually much smaller) than what it would be for a simply supported beam of the same section and span due to the addition of end restraint(s) and their influence on the curvature distribution. For a problem governed by deflection limits in the service state (and many if not most are) this means that a continuous beam can use a smaller section for the same deflection compared with a simply supported beam. Smaller section means less material and less expense.

• Hi, would you like to elaborate on how the deflection of a continuous beam is actually smaller than simply supported beam? May 28 '16 at 2:57
• @Graviton Assume a single span 20' beam split into two beams with 10' span each, the uniform load remains the same, then which one you expected to deflect less, even without consideration of the negative bending moment at the middle support?
– r13
Jun 26 at 14:39

The reality is a bit more complicated, and depends greatly on the materials being used.

Let's take the following structure as our example, where the middle support may or may not be hinged (creating a beam which is either continuous or not):

Since it is symmetric, I can actually simplify it to these two cases. The one on the left assumes the structure was continuous, the one on the right, that it was hinged:

Let's start by observing that the continuous beam is strictly worse than the simply-supported one with regards to shear force. It increases the maximum shear force for no benefit. So, now that we've gotten that out of the way, let's move on.

Here we notice that the maximum bending moment (ignoring sign) is the same in both diagrams. If your section is symmetric around its transversal x-axis, then its resistance to positive and negative bending moments is the same, so this is irrelevant. If your section isn't symmetric, then this question can't be reasonably answered because the "right" answer depends greatly on the actual section being used. Therefore I will now assume that the section is symmetric.

If you're dealing with a steel beam, the continuous beam is best. It'll have to resist the same forces in either case, but this diagram is actually friendlier to a steel beam because it implies in a larger $C_b$, a coefficient which controls lateral-torsional buckling (LTB) given by the equation

$$C_b = \dfrac{12.5M_{max}}{2.5M_{max} + 3M_a + 4M_b + 3M_c}$$

where all moments are in absolute value and $M_a$, $M_b$ and $M_c$ are the bending moments at one-quarter, one-half, and three-quarters of the span, respectively. In the continuous beam all three of these are smaller than in the simply-supported one, implying in a larger $C_b$ and therefore a higher strength against LTB.

If you're dealing with a reinforced concrete structure, however, I'd argue that the simply-supported one may be more efficient. The reason is that LTB usually isn't relevant (reinforced concrete beams tend to be far less slender than equivalent steel beams), but you can control where you put your reinforcement. The maximum necessary reinforcement will be equal in both situations since the maximum bending moment is the same. However, the continuous beam will require you to properly reinforce both faces of the beam, while the simply-supported one will allow you to properly reinforce only the bottom face and just add the minimal reinforcement on the top face.

That being said, the best answer for concrete structures is dependent on the actual case. After all, the simply-supported beam will have to reinforce against the maximum bending moment twice: once for each span, while the continuous beam only suffers that once. However, the continuous beam needs to truly reinforce both faces of the beam, while the simply-supported one can get by with minimal reinforcement on the top face. Depending on the actual situation, the better solution may change. If the positive bending moment on the continuous beam actually ends up being equal (or close to) the minimal reinforcement, then the continuous beam will be better. On the other hand, if this reinforcement is significant, the simply-supported beam may come out on top.

The text above describes cost-optimization on the beams themselves, without taking into consideration the costs of implementation. This is where continuous beams shine. They are trivial to construct. Simply-supported beams, on the other hand, raise many problems: you'll need expansion joints to close the gap between the beams and you'll need twice the joints (bearing pads or what have you), among others issues.

• Continuous beams also shine once you reach plasticity. Provided they have enough ductility and other resistances they can carry additional loads long after the support section yields. The collapse in the continuous beam doesn't occur until the both the support section and the mid section has reached yielding but in a simply supported beam once you reach yielding you're in trouble. So if circumstances allow plastic design you'd have a lot more capacity in a continuous beam than if you're limited to linear design. May 23 '16 at 10:00