KKMishra's Tutorials: What is an economical span in a bridge?

Today our subject is Civil Engineering and we are going to discuss the topic "What is an economical span in a bridge?"

We know that a bridge has two abutments, one at each end and a number of spans above the river width. The bridge spans are supported on piers which foundations go well below the river bed depending upon various data and design. The number of piers is one less than he number of spans.

In civil engineering when we design a structure, we have to keep two aspects in mind. First, the design should be safe and serviceable. Suppose in the case of a bridge, a safe design means - it should be safe for a considered static and moving loads, it should not develop a crack or should not fall. It should be serviceable, that means even if the structure is safe it should not have such a deflection or sagging so that the movement on it is not smooth or gives fear to the the public for moving on it.

The second aspect of the design is that the structure should be economical. In case of a bridge, if we decrease the number of spans, the span length will increase. Bending moment in the span will increase because it is proportional to the square of the effective length. So we will need heavier section resulting in increased cost of spans. On the other hand, due to the increase in span we will need lesser number of piers. Though pier section will increase somewhat but the total cost of piers will decrease.

And opposite is the case if we increase the number of spans. It will decrease the span length. Span cost will decrease but number of piers will increase. So the total cost of piers will increase.

So the question arises, what is the optimum span length for a bridge so that the cost of a bridge is minimum, that is what is the economical span of a bridge?

A sketch of a bridge

See the above diagram of a bridge. Suppose it has 'n' number of spans and 'n-1' number of piers.

We take the cost of a span as 'S', the cost of a pier as 'P' and the cost of an abutment as 'A'.

Let the ratio of cost of a span to the cost of a pier =R

i.e. S/P =R

→S =PR

Now the cost of the bridge is

Cost = C = 2A + nS + (n -1)P

→ C = 2A + nPR + nP -P

For the cost C to be minimum, dC/dn should be zero

i.e. dC/dn = 0 +PR + P -0 = 0

{Because variable is n.}

→PR + P = 0

→P(R +1) = 0

Now for the left hand side to be zero, either P is zero or (R +1) is zero.

Since P is the cost of a pier, it can not be zero. So definitely (R +1) is zero.

Thus, R + 1 = 0

→R = -1

Since R is a ratio of costs, the negative sign does not apply.

So R = 1

→S/P = 1

→S = P.

That is, for the cost of a bridge to be minimum, the cost of one span of a bridge should be equal to the cost of one pier of the bridge. Such span is called the economical span of a bridge.

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Solutions - "Concepts of Physics" Part-I and Part-II, by H C Verma
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