# Basics to Determine and Evaluate Influence Lines

### Technical Article

#### General

The basis for internal forces are cross-section designs which are, in most cases, determined for loads with known and fixed application point. However, when designing craneways or street bridges, movable live loads occur as well. The application points of these loads are variable in time.

To design a supporting structure, the maximum and minimum internal forces as well as the support reactions must be known. To find the governing load positions, influence lines can be used.

#### Determination of Influence Lines

As for the non-moving load, the basis is the definition of a section for which the internal forces should be determined. The loading is applied with a unit load whose load position is variabel. Figure 01 shows the comparison of the evaluation of the internal force M_{y} for a non-moving load F_{z} with x = 1.25m and the influence line ηM_{y }for a moving unit load F_{z} = 1 and the evaluation position x = 1.25 m.

To determine the influence line, a single load is "rolled" over the supporting structure and for each load position, the influcence on the section to evaluate is recorded as ordinate η. The ordinates can be positive and negative. If the ordinate has the value η = 0, it is also called load separation point. At this location, the influcence line cuts the baseline. A load applied at this location has thus no influence on the examined internal force. In Figure 01, this is the case as soon as the load is directly above the support.

If all end points of the determined ordinates are connected, it results in the influence line.

Figure 01 - Internal Force My and Influence Line for My for x = 1.25 m

#### Evaluation of Influence Lines

In Figure 02, the structural acting load has been applied at x = 3.50 m in the upper two-span beam. The influence line in the lower two-span beam is still evaluated for x = 1.25 m. The term "influence" should be clarified here. At location x = 1.25 m, it results in a moment M_{y} = -0.12 kNm provided that the load Fz = 1.0 kN is at x = 3.50 m.

If the influence line ηM_{y }is considered now, an ordinate η = -0.12 is located at x = 3.50 m.

Summary: Any load F_{z }which is located at x = 3.50 m, generates a moment M_{y} = -0,12 ⋅ F_{z }at x = 1.25 m. (The value -0.12 is rounded. The pure value amounts to -0.11507.)

Figure 02 - Internal Force My for P = 1 with x = 3.50 m and Influence Line My for x = 1.25 m

Another simple example shows in Figure 03 the evaluation of the influence line for the support force in the left support for a train load resulting from two single loads. The unit load F_{z} = 1 generates a force of 1.0 kN in the support. The ordinate η is thus 1.0. Any load P generates thus the force P ⋅ η in the support. If several loads are available, each single load P_{i }generates a force P_{i} ⋅ η_{i }in the support.

The support force results in:

$\mathrm A\;=\;\sum_{\mathrm i\;=\;1}^{\mathrm i\;=\;\mathrm n}{\mathrm P}_\mathrm i\;{\mathrm\eta}_\mathrm i$

Example from Figure 03:

$\mathrm A\;=\;10\;\mathrm{kN}\;\cdot\;1.0\;+\;20\;\cdot\;0.6345\;=\;22.69\;\mathrm{kN}$

To determine a maximum support load, the train load has to be placed in the positive area of the influence line. If the minimum should be determined, the train load has to be placed accordingly in the negative area.

General rule: To find an extreme value, the bigger load is positioned directly over the maximum value of the influence line. The graphical display of the influence line via RF-INFLUENCE is useful here.

Figure 03 - Support Reactions for Set of Loads and Influence Lines for the Left Support

#### Summary

Influence lines are used to find extreme values of support forces and internal forces of movable loads.

The shape of the influence line allows conclusions to be drawn about the governing loading to find a limit value at the evaluated section for the corresponding internal force.

The influence line is displayed over the whole length of the structural component to evaluate, but all indicated ordinates η describe the influence on a structural value at only one location of the beam.

RF-INFLUENCE is the tool to determine influence lines of members and influence surfaces of surfaces for internal forces, support forces and deformations.

#### Keywords

influence line extreme value

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