Calculate left-reaction, section-shear and section-moment influence ordinates for an ideal simply supported beam. Single loads and fixed two-axle groups are evaluated only while all entered loads are on the span; critical positions and one-sided shear limits are checked directly.

Downward point load on a simply supported span.

Ideal simply supported beam, static downward point loads and linear superposition. Shear is RA minus loads left of the section; a load exactly at the cut uses the right-of-section ordinate. Extrema include stated one-sided limits. No dynamic amplification or structural capacity check.

Influence Line Formula

An influence line gives the value of a chosen structural response (reaction, shear, moment, deflection, etc.) at a point due to a unit moving load. For a single point load, the response equals the load magnitude times the influence line ordinate at the load position. For a distributed load, the response equals the load intensity integrated against the influence line over the loaded length.

E(x) & = P × IL(x) \ IL(x) & = (E(x)) / (P) \ ILavg & = (E) / (W L) (uniform load W over length L)

Variables:

  • IL(x) is the influence line ordinate for the selected response at load position x (often unitless for reactions/shear; has units of length for moment influence lines)
  • E is the resulting response (e.g., a support reaction, a shear at xâ‚€, or a moment at xâ‚€)
  • P is a point load magnitude
  • W is a uniform load intensity (force per length)
  • L is the loaded length used with the uniform load (often the full span length)

For a point load, divide the response by the point load magnitude to get the influence line ordinate at that load position (IL = E/P). For a uniform load of constant intensity applied over the full length L, the response is E = W·L·ILavg, where ILavg is the average influence ordinate over that loaded length.

What is an Influence Line?

An influence line is a graphical representation used in structural engineering to illustrate how a moving load on a structure (such as a bridge or a beam) affects a particular response quantity, such as a support reaction, internal shear, internal moment, or deflection at a specified point. The x-axis represents the position of the moving load along the structure, and the y-axis represents the value of the response quantity produced by a unit load at that position. Influence lines help engineers identify where a moving load causes the maximum (or minimum) effect for design and analysis.

How to Calculate an Influence Line

The following steps describe influence-line construction in general. This widget evaluates the simply supported reaction, shear and moment relations for one or two downward point loads; distributed loads and deflection are reference context only.

  1. Choose the response quantity you want (e.g., reaction at a support, shear at xâ‚€, moment at xâ‚€, or deflection at a point).
  2. Identify load positions at supports and at the section of interest, plus positions where each axle crosses those points.
  3. Place a unit load at one position and compute the response quantity at the point of interest (using statics for determinate structures, or appropriate structural analysis methods for indeterminate structures).
  4. For this determinate beam, the response is linear between critical load positions. Evaluate those positions and both admissible sides of a shear discontinuity instead of relying on a sampled sweep.
  5. Plot the computed response values against load position and connect the points to form the influence line (noting any discontinuities, such as the jump in the shear influence line at xâ‚€).

Example Problem:

Simply supported beam length L = 10 m. A point load P = 25 kN is located at x = 6 m from support A. The support reactions are:

RA = P·(L − x)/L = 25·(10 − 6)/10 = 10 kN

RB = P·x/L = 25·6/10 = 15 kN

The influence line ordinates at that load position are ILRA = RA/P = 10/25 = 0.4 and ILRB = RB/P = 15/25 = 0.6.