Module 17 · Lesson 17.4
Influence lines for trusses and indirect loading
When the load reaches the girder through cross beams, and how truss members respond to it.
Why this matters
Bridge decks do not sit on the main girders directly. The load lands on the deck, travels to cross beams, and only then reaches the girders at discrete points. That changes the influence line in a specific and useful way. And truss members need influence lines of their own, which the method of sections supplies almost immediately.
By the end of this lesson you should be able to
- Explain why indirect loading makes an influence line piecewise linear
- Construct the influence line for a girder loaded through cross beams
- Use the method of sections to get the influence line for a truss chord
- Distinguish the influence lines for chord and diagonal members
What you should already know
- Influence lines for reaction, shear and moment (this module)
- The Müller-Breslau principle (this module)
- Method of sections for trusses (Module 4)
So far the load has been assumed to sit directly on the member being analysed. In a real bridge it does not. The deck spans between cross beams, the cross beams sit on the main girders at panel points, and the girder therefore only ever receives load at those points.
Put a unit load on the deck between two panel points, a distance a from the left one in a panel of length p. The deck spans as a simple beam between the two cross beams, so it delivers:
- (p − a)/p of the load to the left panel point
- a/p to the right one
Both of those are linear in a. The effect on the girder is the sum of each share times the influence-line ordinate at its own panel point — and a linear combination of two fixed numbers, with linear weights, is itself linear.
So the influence line for indirect loading is straight between panel points, joining the ordinates of the direct influence line at those points.
Predict first
A moment influence line for a girder peaks at a point one third of the way along a panel. What does indirect loading through cross beams do to that peak?
Truss members need influence lines too, and the method of sections gives them directly.
Cut the truss through the panel containing the member of interest, and take moments about the joint where the other two cut members meet.
For a chord member. The other two cut members meet at a joint on the opposite chord. Taking moments about it, the chord force is the bending moment at that joint divided by the truss depth:
Fchord = M/d
So the influence line for a chord force is the moment influence line at that joint, divided by d. It is a triangle, peaking under the joint, with ordinates in units of 1/length.
For a diagonal member. The diagonal is the only cut member with a vertical component, so resolving vertically gives:
Fdiagonal = V/sin θ
The influence line for a diagonal is the shear influence line for that panel, divided by sin θ. It therefore has the characteristic shape of a shear influence line — two branches of opposite sign — which is why diagonals near mid-span can go into either tension or compression depending on where the load stands, and why counter-bracing is sometimes needed there.
Worked example
Influence lines for a panelled truss girder
Given
- Truss girder spanning 24.0 m in panels of 4.00 m, depth 3.00 m
- Bottom chord member in the panel from 8.00 to 12.0 m
- Load delivered through cross beams at the panel points only
Find
The influence line for that chord force, its peak ordinate, and the force under a 100 kN load at the panel point.
Practice
A truss spans 24.0 m with a depth of 3.00 m. For a bottom chord member whose section is taken about a joint at 8.00 m, what is the peak ordinate of the chord force influence line, in 1/m?
Practice
A girder has an influence line whose ordinates at panel points 3.00 m and 6.00 m are 2.00 and 2.00 respectively. With indirect loading through cross beams, what is the ordinate at 4.50 m?
Practice
A truss diagonal makes an angle of 36.87° with the horizontal, so sin θ = 0.600. If the shear influence line ordinate for its panel is 0.450, what is the ordinate of the diagonal force influence line?
Summary
- Indirect loading delivers the load only at panel points, in linear proportions
- The influence line is therefore straight between panel points
- Ordinates at panel points are unchanged; interior peaks are cut off
- A shear influence line's jump becomes a ramp across the panel containing the section
- Chord force influence line = moment influence line at the joint ÷ truss depth
- Diagonal force influence line = shear influence line for the panel ÷ sin θ
- Diagonal influence lines have two signs, so mid-span diagonals may reverse
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint