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Moving load

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ExampleReferenceRecordsLargest deviation
Influence line for the support reactionClosed form10%
Influence line for the mid-span momentClosed form20%
Single travelling load, moment envelopeClosed form10%
AASHTO HS20 truck, moment envelopeClosed form10%
AASHTO HS20 truck, absolute maximum momentClosed form10.301%
Partial strip loadClosed form10%
Tapered bridge girder under the HS20 load trainClosed form11.19%
Two cranes, independent positions with a minimum approach distanceClosed form10%
Bridge crane, both rails at the same stationClosed form10%
Envelope from the influence line (one crane, two cranes, a crane kept on the rail)Closed form10%
Braking force on a route rotated in planClosed form10%

Problem. A moving load is a loading that advances step by step along a route. The influence line, the moment envelope of a single travelling load, the envelope of a truck train and the direction of the braking force on a route that turns in plan are tested.

Reference document. The reference is statics. Barré's rule gives the position of the absolute maximum moment.

The moment under travelling point loads is computed from this relation. (⋅)+(\cdot)_+ keeps positive values only.

M(x)=RA x−∑P (x−a)+M(x) = R_A\,x - \sum P\,(x-a)_+

The influence lines of the support reaction and the mid-span moment are given by these relations.

RA(x)=1−xL,M(x)=P x (L−x)LR_A(x) = 1 - \dfrac{x}{L},\qquad M(x) = \dfrac{P\,x\,(L-x)}{L}

Numerical model. The route is defined as a chain of elements, given in mixed order and reversed direction. The beam is simply supported at both ends.

Influence line for the support reaction

Parameters. L = 20 · sampling point = 41

Closed form. The support reaction from statics

RA(x)=1−x/LR_A(x) = 1 - x/L
QuantityReferenceComputedDeviation
Largest absolute deviation02.16×10−132.16 \times 10^{-13}0%

Note. The drawing shows the step at the middle of the route. The influence line is made of the reaction values at every step.

Influence line for the mid-span moment

Parameters. L = 20

Closed form. Triangular influence line, peaking at mid-span with the value L/4L/4

QuantityReferenceComputedDeviation
Peak value550%
Location of the peak10100%

Note. The drawing shows the unit load at mid-span.

Single travelling load, moment envelope

Parameters. P = 150 · L = 20 · section = 20

Closed form. The closed-form moment relation

M(x)=Px(L−x)/LM(x) = Px(L-x)/L
QuantityReferenceComputedDeviation
Largest relative deviation across all sections02.328×10−132.328 \times 10^{-13}0%

Note. The drawing shows the load at mid-span, where it produces the peak of the envelope.

AASHTO HS20 truck, moment envelope

Parameters. L = 20 · steps = 0.25 · position = 115

Closed form. An independent static calculation, enveloped over the same steps

M(x)=RAx−∑P (x−a)+M(x) = R_A x - \sum P\,(x-a)_+
QuantityReferenceComputedDeviation
Largest relative deviation across all sections04.58×10−134.58 \times 10^{-13}0%

Note. The AASHTO LRFD design truck has axle loads of 35.6, 142.3 and 142.3 kN at 4.27 m spacing. The drawing shows the truck where it produces the peak of the envelope.

AASHTO HS20 truck, absolute maximum moment

Parameters. L = 20 · element count = 20 · steps = 0.1

Closed form. By Barré's rule mid-span bisects the distance between the resultant and the critical axle. The value itself follows from equilibrium.

QuantityReferenceComputedDeviation
Maximum moment along the span1,229.291,225.590.301%

Note. The envelope is read at nodal sections, so it stays slightly below Barré's rule. The drawing shows the truck where it produces the largest moment.

Partial strip load

Parameters. w = -9 · start = 4.3 · end = 13.7

Closed form. The total reaction is the distributed load times the loaded length

QuantityReferenceComputedDeviation
Total support reaction84.684.60%

Tapered bridge girder under the HS20 load train

Parameters. L = 20 · element count = 10

Closed form. In a symmetric beam the largest mid-span deflection occurs when the resultant sits at mid-span

QuantityReferenceComputedDeviation
Location where the largest deflection occurs12.8471131.19%

Note. The drawing shows the train where it produces the largest mid-span deflection.

Two cranes, independent positions with a minimum approach distance

Parameters. combinations = 703 · minimum approach = 2 · bogie = 3.5

Closed form. An independent static calculation, enveloped over the same position combinations

QuantityReferenceComputedDeviation
Largest relative deviation across all sections04.014×10−134.014 \times 10^{-13}0%

Note. The drawing shows the two cranes where they produce the largest moment.

Bridge crane, both rails at the same station

Parameters. steps = 48 · wheel A = 90 · wheel B = 32 · wheel spacing = 3.5

Closed form. An independent static calculation, each rail with its own wheel loads at the same positions

QuantityReferenceComputedDeviation
Largest relative deviation over all steps and sections of both rails07.38×10−137.38 \times 10^{-13}0%

Note. The trolley is next to rail A, so the wheels on rail A are heavy and those on rail B light. Both rails are at the same station at every step, and the vertical reactions add up to the wheel loads on the girders at every step.

Envelope from the influence line (one crane, two cranes, a crane kept on the rail)

Parameters. station spacing = 0.5 · position pairs = 703 · minimum approach = 2

Closed form. The stepped solution on the same grid of positions

QuantityReferenceComputedDeviation
Largest relative deviation from the stepped solution04.955×10−134.955 \times 10^{-13}0%

Note. The envelope is built from the influence line of a unit load, and the worst placing of two cranes is searched on that line. The model solved again at the position found gives the same value.

Braking force on a route rotated in plan

Parameters. rotation = 37 · braking force = 40

Closed form. The braking force acts along the tangent to the route

QuantityReferenceComputedDeviation
Direction of the resultant horizontal reaction, in degrees37370%

Note. The vertical wheel load and the braking force act on the same axle, and the direction follows from the route geometry.