Influence lines

A bridge girder does not carry a fixed load; trucks and trains move across it. An influence line answers a narrower question than a bending moment diagram: how does one particular quantity, say the reaction at A or the moment at mid-span, change as a single unit load moves along the structure? Its ordinate at each position is the value of that quantity when the unit load stands there.

For determinate structures influence lines are made of straight segments. The influence line for a simply supported beam's reaction at A falls linearly from 1 at A to 0 at B. For the moment at a section at distance a from A and b from B, it is a triangle with peak a b / L under the section. For shear, it jumps by one at the section. The Müller-Breslau principle gives the shape directly: release the restraint for the quantity and give it a unit displacement; the deflected shape is the influence line. For indeterminate structures the lines are curved.

To use an influence line, multiply each point load by the ordinate beneath it and add, or multiply a uniform load by the area under the line where it acts. To find the worst position of a moving load train, place loads near the peak and test; for a moment at a section, the maximum occurs when the average load on each side of the section is equal. Bridge codes specify standard vehicle trains for this purpose.