Structural Engineering & Statics
where the load goes
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Influence line — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Influence line for the bending moment at x = 3. The bending moment at one fixed station, plotted against the position of a unit load walking across the span. The beam was re-solved at 301 load positions. The worst position is x = 3.00, giving 2.100.

    The worst place to stand

    A bridge is not designed for a load. It is designed for a load that moves, and for every station along it there is a different position of that load that does the most damage.

    part 1 · internal-forces
  2. The worst position is not the obvious one. Three axles totalling 320 units, marched across a span of 20 in steps of 0.02. The envelope is the largest moment each station ever sees; its peak is 1160.2 at 9.88 along the span, which is 0.12 off midspan and occurs under the axle nearest the resultant rather than under the heaviest one. Barré's construction, which places midspan halfway between that axle and the resultant, independently gives 1160.3 at 9.88. The dashed curve is the envelope the same total weight would produce as one load rather than three: its peak is 1600.0, which is 38% more — spreading a load out is worth something.

    The train that is worse than its heaviest axle

    An influence line says where to stand one load. A vehicle is several loads at fixed spacings, and the worst arrangement never puts the heaviest one at the peak.

    part 2 · internal-forces
  3. Whole spans miss the shear inside a span. The shear along three spans of 8.0, 8.0, 8.0 m, under 20 kN/m of dead load everywhere and 15 kN/m imposed: the largest and least when the imposed load stands wherever it raises or lowers the shear at each section (solid), and when it may only cover whole spans (dashed). At the supports the two agree. At the middle of the second span the exact envelope reaches 23.8 kN either way and the whole spans 10.0 kN; at a tenth of the way along it, 122.4 against 122.0.

    The shear whole spans cannot find

    A continuous beam is designed by loading its spans in patterns, whole spans at a time, and for the bending moment that is very nearly exact: the moment's influence line keeps one sign over each span, so a span is either worth loading or not. The shear's influence line changes sign at the section itself, inside the span, and no pattern of whole spans can load one side of a section and not the other. At the middle of the inner span of three equal spans the imposed shear is 0.198 qL; whole spans find 0.083 qL, and the straight line to qL/8 that is safe on a simple span finds 0.125.

    part 3 · internal-forces

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