Influence lines
Influence lines of a continuous beam for every support reaction and for the moment and the shear at a point you choose, and the largest and smallest effect of a train of point loads and of a uniform load placed where they govern. Use it for crane, vehicle and other moving loads on beams and girders, for pattern loading, and to see how a load anywhere on the beam reaches a support.
Inputs of the influence lines #
- Spans and supports: 1 to 6 spans, each with the support at its right end: pin or roller, fixed, or none (a free end, or a beam continuous over a removed support). The beam is prismatic; influence lines of forces do not depend on EI, so no stiffness is entered.
- Point: where the moment and the shear are wanted, from the left end. The shear is taken just right or just left of it: at a support the two differ by the reaction.
- Load train: point loads at fixed distances from the first load (up to 16), run across the beam both ways or one way. Loads are as entered: apply load factors and impact yourself.
- Uniform load: a lane load placed only where the influence line is positive (for the largest effect) or negative (for the smallest).
- Add the train and the uniform load: both at once, each at its own worst placement; otherwise the larger of the two governs.
The influence lines have no auto-design: there is nothing to size.
Checks performed #
| Result | Basis |
|---|---|
| Influence line of each reaction, of the moment and of the shear at the point | Müller-Breslau principle; ordinates by a unit load at every node of one stiffness analysis (Maxwell-Betti) |
| Largest and smallest ordinate, positive and negative areas | The lines as computed |
| Train: largest and smallest effect with the position of the first load | Every position with a load on a node, both sides of the shear jump, both ways |
| Uniform load: w ∫IL⁺ and w ∫IL⁻ | Loading only the parts of the beam of one sign |
Reactions are positive upward, moments positive sagging, shears positive with the left part pushed up.
Limitations of the influence lines #
- Prismatic beams on rigid supports; no internal hinges, springs or varying EI (use the beam with varying stiffness for a haunched beam under fixed loads).
- Between nodes the lines are interpolated linearly on 48 elements per span; the ordinates at the nodes are exact.
- Loads are vertical and as entered; no impact, factors, dynamic effects or deflection influence lines. The member is not designed.
Verification examples #
Verification status: 17 values compared in 5 examples (2 published, 3 hand calculations from the cited equations): 17 within tolerance, 0 recorded deviations. Thin: only 2 published examples. Code basis: the provisions it uses were read in the target edition.
| Example | Source | Values |
|---|---|---|
| Propped cantilever, 20 ft | Udoeyo, Structural Analysis (2020), Example 13.1 | By at 5, 10, 15 ft, Ay at 5 ft, MA at 10 ft |
| Two spans of 3 m | Udoeyo Example 13.2 | By, Ay and Cy at the quarter points |
| Two loads on a 40 ft span | Hand calculation | Largest midspan moment, end reaction, shear just right of midspan both signs |
| Uniform load on two 20 ft spans | Hand calculation, three-moment equation | Support moment, interior reaction, moment at 0.4 l both signs |
Udoeyo's Table 13.1 prints its ordinates to two decimals; the exact values of its own equations are used, with the printed ones in the value names. Its moment at C is not used: those printed values do not satisfy the statics of its own reactions.