Influence lines for beams and trusses

So far every load has had a fixed position. Bridges, crane runways, parking structures and the access road over the operations center's culvert carry loads that move. For a moving load the question changes. Instead of asking what the shear and moment are everywhere for a given load position, we ask: for a particular quantity at a particular point, such as the moment at midspan or the reaction at a support, where should the load be placed to make that quantity as large as possible, and how large does it get? The tool that answers this is the influence line.

What an influence line is

An influence line is a graph of the value of one response quantity (a reaction, a shear or moment at one section, or a member force) as a unit load moves across the structure. The horizontal axis is the position of the unit load; the vertical axis is the value of the quantity when the unit load is at that position. Do not confuse it with a shear or moment diagram. A moment diagram shows the moment at every section for one fixed loading; an influence line shows the moment at one fixed section for every position of the load.

Because the structure is linear, the effect of any real load follows from the influence line by scaling. A point load P at a position where the influence line ordinate is \(\eta\) produces a value \(P\eta\). A uniform load w over a stretch of the structure produces w times the area under the influence line over that stretch. And a group of loads produces the sum of their individual effects. Influence lines therefore turn the search for the worst loading into a geometry problem.

Influence lines by equilibrium

The direct way to construct an influence line is to place a unit load at a general position x, write equilibrium, and express the quantity as a function of x. For a simply supported beam AB of span L:

• Reaction at A: with the unit load at x from A, \(R_A = (L - x)/L\). The influence line is a straight line from 1 at A to 0 at B. Similarly \(R_B = x/L\), from 0 at A to 1 at B.
• Shear at a section C, a distance a from A (with \(b = L - a\)): when the load is to the right of C, the shear at C equals \(R_A = (L - x)/L\); when it is to the left, the shear equals \(R_A - 1 = -x/L\). The influence line is two parallel lines with a jump of 1 at C, from \(-a/L\) just left of C to \(+b/L\) just right of it.
• Moment at C: when the load is to the right of C, \(M_C = R_A\,a = a(L - x)/L\); to the left, \(M_C = R_B\,b = bx/L\). The influence line is a triangle with its peak at C, of height \(ab/L\).

These shapes tell you at once where to put loads. For maximum moment at C, load the whole span, and put the heaviest point load at C. For maximum positive shear at C, load only the part to the right of C, because loads to the left produce negative shear there. For the maximum reaction at A, load the whole span with the heaviest point load at A.

Overhangs and internal hinges

For a beam with an overhang, the influence lines simply extend beyond the support as straight lines. The influence line for the reaction at A on a beam ABC with the overhang BC continues past B below the axis, because a load on the overhang lifts A. The moment influence line at a section in the main span also extends past B with negative ordinates: a load on the overhang causes hogging moment at the section. For a section on the overhang itself, the moment influence line is zero for loads on the main span and grows linearly toward the tip, as for a cantilever. Internal hinges make influence lines kink at the hinge, and the segment beyond a hinge pivots about the next support. The quickest way to draw all of these is the Müller-Breslau principle.

The Müller-Breslau principle

In 1886 Heinrich Müller-Breslau showed that the influence line for any force quantity has the same shape as the deflected shape of the structure when that quantity is released and given a unit displacement in its own direction. To draw the influence line for a reaction, remove the support and push that point up by one unit; the structure's new shape is the influence line. For shear at a section, cut the beam there so that it can slide but not rotate, and give it a unit relative sliding displacement with the two pieces staying parallel. For moment, insert a hinge at the section and give it a unit relative rotation.

For a determinate structure, releasing one restraint turns it into a mechanism, so the displaced shape is made of straight segments and is easy to draw. For an indeterminate structure, releasing one restraint leaves a structure that still has to bend, so the influence line is curved. The principle still gives its shape qualitatively, which is enough to decide which spans to load for the worst effect. This is the basis of pattern loading in continuous beams and frames: for maximum positive moment in a span, load that span and alternate spans; for maximum negative moment over a support, load the two adjacent spans and then every other span beyond.

Girders with floor systems

Many bridges and some building floors carry the deck on stringers that span between floor beams, and the floor beams deliver the load to the main girders or trusses only at panel points. A load between panel points reaches the girder as two point loads at the adjacent panel points, shared in proportion to its position, like the reactions of a simply supported stringer. The influence line for any quantity in the girder is therefore a straight line between panel points, even for sections between them. The influence line for shear in a panel, called the panel shear, is the same for every section within the panel, because no load is applied to the girder between its panel points.

Influence lines for trusses

Truss bridges carry their deck at the panel points of the top or bottom chord, so truss influence lines are also straight between panel points. To construct the influence line for a member force, place the unit load at each panel point in turn and find the member force by the method of sections, then join the values with straight lines.

For a parallel-chord truss, the method of sections links the member forces to beam quantities. A chord member's force equals the moment at the opposite panel point divided by the depth, so its influence line is the moment influence line for that panel point, scaled by \(1/d\). A diagonal's vertical component equals the panel shear, so its influence line is the panel shear influence line scaled by the ratio of the diagonal's length to the depth.

Influence lines by computer

Because an influence line is just the response to a unit load at many positions, a program can build it by brute force: move a unit load in small steps and record the quantity at each step. Sofia's version does this for the moment and shear at a section of a simple span, then integrates the ordinates to find the effect of a uniform load on the most unfavorable stretch, which reproduces the hand results.

L, a = 60.0, 24.0                        # span and section position (ft)
n = 2000
xs = [i * L / n for i in range(n + 1)]

def response(x, quantity):
    RA = (L - x) / L                     # unit load at x
    if quantity == "M":
        return RA * a - (a - x if x < a else 0.0)
    if quantity == "V":
        return RA - (1.0 if x < a else 0.0)

for q in ("M", "V"):
    eta = [response(x, q) for x in xs]
    dx = L / n
    pos = sum(max(e, 0.0) for e in eta) * dx
    neg = sum(min(e, 0.0) for e in eta) * dx
    peak = max(eta, key=abs)
    print(f"{q}: peak ordinate {peak:6.3f}, positive area {pos:7.2f}, negative area {neg:7.2f}")

w, P = 0.64, 25.0
Mmax = w * sum(max(response(x, "M"), 0) for x in xs) * L / n + P * max(response(x, "M") for x in xs)
Vmax = w * sum(max(response(x, "V"), 0) for x in xs) * L / n + P * max(response(x, "V") for x in xs)
print(f"Max moment at C = {Mmax:.0f} kip*ft, max positive shear at C = {Vmax:.1f} kips")

M: peak ordinate 14.400, positive area  432.00, negative area    0.00
V: peak ordinate  0.600, positive area   10.81, negative area   -4.79
Max moment at C = 636 kip*ft, max positive shear at C = 21.9 kips

The numerical areas agree with the triangle areas to within the step size. For an indeterminate structure, the same loop works if the response function calls a frame analysis program, which is how bridge design software builds influence surfaces for decks.