Enter the load magnitude (or load intensity) and the resulting response (reaction/effect) into the calculator to determine the corresponding influence value. For a point load, the influence line ordinate at that load position is IL = response ÷ load. For a uniform load over the full length, the calculator uses the average influence ordinate ILavg = response ÷ (w·L).

Influence Line Calculator

Pick beam type, enter the span, and see where a moving unit load causes the largest effect.

Simply Supported
Cantilever

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.

\begin{aligned}
E(x) &= P \cdot IL(x) \\
IL(x) &= \frac{E(x)}{P} \\
IL_{\text{avg}} &= \frac{E}{W\,L}\quad(\text{uniform load }W\text{ over length }L)
\end{aligned}

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 x₀, or a moment at x₀)
  • 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 outline a common way to calculate an influence line (especially for statically determinate structures):

  1. Choose the response quantity you want (e.g., reaction at a support, shear at x₀, moment at x₀, or deflection at a point).
  2. Select a series of load positions along the structure.
  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. Repeat for multiple unit-load positions to obtain a set of response values versus load position.
  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 x₀).

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.