Draws the influence lines of a bridge deck carried as a continuous beam, for the moment, shear, reactions, deflection and rotation at the sections you choose, then moves the axles of a vehicle and a lane load to where each effect is largest and smallest, and gives the envelope along the deck. Every step shows its formula.

What it checks

The system: the deck is a chain of segments between numbered points, one at every support, hinge and change of section. Its degree of indeterminacy is r − (3 + h), with r the reactions, springs included, and h the hinges. In an indeterminate deck the moments depend on the stiffness, and the influence lines are curves: the deflected shapes of the Müller-Breslau principle.
The influence lines, up to six: the bending moment (sagging positive), the shear force just left or just right of a section, the vertical reaction or the reaction moment of a support, the deflection or the rotation of a section, each for 1 kN standing at x.
For each line: y_max and y_min with their positions, and the areas A_+ and A_− of the positive and negative parts.
The axles: Σ Q_k·y(x_k) with the train at its worst position, for the largest and for the smallest value.
The lane load: q_L·A_+ and q_L·A_−, or over the whole deck.
The permanent load: g·(A_+ + A_−), in both the largest and the smallest value.
The result: for each effect, the sum of the largest contributions and the sum of the smallest.
The envelope, optional: the moving load placed at its worst for every section of a regular grid, with the shear on both sides of every interior point, and the absolute extremes M_max, M_min, V_max and V_min along the deck.

What you enter and what you get

You enter up to twelve segments, the modulus E, and for each segment its length and second moment of area, constant or with straight or parabolic haunches. At each point you choose the support (pinned, roller, fixed, free, guided up and down, guided left and right, or elastic with a vertical and a rotational spring) and whether the deck is continuous or hinged. Then how the loads reach the girder, directly or through cross-beams, and the effects you want, each at a section or a support.

For the traffic: up to ten axles at fixed distances, entered from left to right as they stand on the deck, in the order entered only or also reversed; a lane load q_L, wherever it increases the effect or over the whole deck; the uniform permanent load g; and the number of sections per segment for the envelope.

You get the influence lines drawn, the extreme values of every effect with the share of the axles, the lane load and the permanent load, and the envelope of moment and shear.

Assumptions and limits

A straight girder with the same E along its whole length.
In a haunch the width is taken as constant, so I grows with the cube of the depth.
No load factor is applied: enter characteristic or design values. Loads downward are positive.
A lane load is a uniform load of any length that may stand anywhere.
The permanent load is uniform over the whole deck. Other permanent loads are analysed in the continuous-beam module, the longitudinal deck analysis.
An analysis only: no section is verified.

Worked example

A girder of three spans, 30 m, 40 m and 30 m, with E = 35,000 N/mm² and a constant I = 4,500,000·10⁶ mm⁴, pinned at the left end and on rollers elsewhere, loaded directly; five effects (the moments at x = 12 m and 30 m, the shear right of 30 m, the reaction at point 1 and the deflection at 50 m) under two axles of 300 kN 1.2 m apart in both orders, a lane load of 27 kN/m wherever it increases the effect and g = 180 kN/m, with the envelope at 10 sections per segment. These are the module's own results for its worked example, computed when this page was built.

Inputs
Input	Value	Ref.
Number of segments n	3
Modulus of elasticity E	35000 N/mm²
Length l₁	30 m
Second moment of area I₁	4500000 10⁶ mm⁴
Section along the segment	Constant
Length l₂	40 m
Second moment of area I₂	4500000 10⁶ mm⁴
Section along the segment	Constant
Length l₃	30 m
Second moment of area I₃	4500000 10⁶ mm⁴
Section along the segment	Constant
Support	Pinned
Support	Roller
Joint	Continuous
Support	Roller
Joint	Continuous
Support	Roller
Load path	Directly on the beam
Number of effects	5
Effect	Bending moment
Position of the section x₁	12 m
Effect	Bending moment
Position of the section x₂	30 m
Effect	Shear force, right of the section
Position of the section x₃	30 m
Effect	Support reaction
At point	1
Effect	Deflection
Position of the section x₅	50 m
Number of axles	2
Axle load Q₁	300 kN
Distance from axle 1 a₂	1.2 m
Axle load Q₂	300 kN
Order of the axles	Also in the reverse order
Distributed moving load q_L	27 kN/m
Where it stands	Wherever it increases the effect
Uniform permanent load on the whole beam g	180 kN/m
Largest and smallest M and V along the whole beam	Yes
Sections per segment	10

Calculation: The beam
Step	Value	Ref.
Length of the beamΣ l_i	l_tot = 100.00 m
Reactions against equations of equilibriumr = 5 reactions (4 vertical and rotational, springs included; 1 horizontal); 3 equations of equilibrium + 1 for each of the h = 0 hingesstatically indeterminate, degree 2; for bending, degree 2: the moments depend on the stiffness, and settlements produce forces	r − (3 + h) = 2

Step	1Segment 10.00 → 30.00 m	2Segment 230.00 → 70.00 m	3Segment 370.00 → 100.00 m	Ref.
Length l m	30.00	40.00	30.00
Bending stiffness E·I MN·m²E·I between the haunches	157,500.0	157,500.0	157,500.0

Calculation: The influence lines
Step	Value	Ref.
Load path	directly on the beam
Shape of the linesstatically indeterminate beam: the influence lines are curves (the deflected shapes of the Müller-Breslau principle)	curves

Step	1Effect 1M at 12.00 m	2Effect 2M at 30.00 m	3Effect 3V_r at 30.00 m	4Effect 4R at point 1
Effect	bending moment	bending moment	shear force right of the section	support reaction
Unit of the ordinates	kN·m per kN	kN·m per kN	kN per kN	kN per kN
Largest ordinate y_max	6.259	0.770	1.000	1.001
its position x m	12.00	82.60	30.00	30.80
Smallest ordinate y_min	-1.437	-3.594	-0.087	-0.112
its position x m	45.20	45.20	82.60	82.60
Area of the positive parts A_+∫ y dx where y > 0 (ordinate units · m)	93.001	14.999	21.687	41.400
Area of the negative parts A_−∫ y dx where y < 0 (ordinate units · m)	-35.552	-141.375	-1.687	-2.187

Step	5Effect 5w at 50.00 m	Ref.
Effect	deflection
Unit of the ordinates	mm per kN
Largest ordinate y_max	0.0042
its position x m	50.00
Smallest ordinate y_min	-0.0012
its position x m	82.60
Area of the positive parts A_+∫ y dx where y > 0 (ordinate units · m)	0.0988
Area of the negative parts A_−∫ y dx where y < 0 (ordinate units · m)	-0.0476

Calculation: The moving load
Step	Value	Ref.
AxlesQ₁ = 300.0 kN, Q₂ = 300.0 kN at 1.20 m	n = 2
Total of the axles and its lengthfrom the first to the last axle: 1.20 mtried in both orders	ΣQ = 600.0 kN
Lane loadwherever it increases the effect	q_L = 27.00 kN/m
Permanent load on the whole beam	g = 180.00 kN/m

Calculation: Largest and smallest values of the effects
Step	1Effect 1M at 12.00 m	2Effect 2M at 30.00 m	3Effect 3V_r at 30.00 m
Axles, largestΣ Q_k·y(x_k) at the worst position	3,595.7	461.0	594.3
position	axle 1 at 12.00 m	axle 1 at 82.00 m	axle 1 at 30.00 m
Axles, smallest	-861.5	-2,153.6	-51.9
position	axle 1 at 44.40 m	axle 1 at 44.40 m	axle 1 at 82.00 m
Lane load, largestq_L·A_+ (or q_L·A_− for a negative q_L)	2,511.0	405.0	585.6
Lane load, smallestq_L·A_−	-959.9	-3,817.1	-45.6
Permanent loadg·(A_+ + A_−)	10,340.9	-22,747.7	3,600.0
Largest value maxsum of the largest contributions	16,447.7	-21,881.7	4,779.8
Smallest value minsum of the smallest contributions	8,519.5	-28,718.5	3,502.6
Unit	kN·m	kN·m	kN

Step	4Effect 4R at point 1	5Effect 5w at 50.00 m
Axles, largestΣ Q_k·y(x_k) at the worst position	600.4	2.53
position	axle 1 at 30.40 m	axle 1 at 49.20 m
Axles, smallest	-67.2	-0.73
position	axle 1 at 82.00 m	axle 1 at 82.00 m
Lane load, largestq_L·A_+ (or q_L·A_− for a negative q_L)	1,117.8	2.67
Lane load, smallestq_L·A_−	-59.1	-1.29
Permanent loadg·(A_+ + A_−)	7,058.3	9.21
Largest value maxsum of the largest contributions	8,776.5	14.41
Smallest value minsum of the smallest contributions	6,932.0	7.19
Unit	kN	mm

Calculation: Envelope along the beam
Step	Value	Ref.
Sections analysed10 per segment, both sides of every interior point for the shear force; the absolute extremes of M are refined between them	31
Absolute largest bending momentat x = 50.018 m	M_max = 20,077.7 kN·m
Absolute smallest bending momentat x = 30.000 m	M_min = -28,716.9 kN·m
Largest shear forceat x = 30.00 m	V_max = 4,779.8 kN
Smallest shear forceat x = 70.00 m	V_min = -4,779.8 kN

Step	0Point 0x = 0.00 m	1Point 1x = 30.00 m	2Point 2x = 70.00 m	3Point 3x = 100.00 m
Largest bending moment M_max kN·msagging positive	0.0	-21,880.5	-21,880.5	0.0
Smallest bending moment M_min kN·m	0.0	-28,716.9	-28,716.9	0.0
Largest shear force, left of the point V_max,l kN	·	-3,429.4	-3,502.6	-1,790.0
Smallest shear force, left of the point V_min,l kN	·	-4,583.7	-4,779.8	-2,898.2
Largest shear force, right of the point V_max,r kN	2,898.2	4,779.8	4,583.7	·
Smallest shear force, right of the point V_min,r kN	1,790.0	3,502.6	3,429.4	·

How this module is verified

Verified against independent calculations used on real projects, before release.

Questions

Where does the lane load stand?

With "wherever it increases the effect", on every part where the influence line has the sign sought: the positive parts for the largest value, the negative parts for the smallest.

What changes when the deck rests on cross-beams?

The loads stand on stringers that span simply between cross-beams, so the girder receives them only at the cross-beams. Every influence line then becomes straight between them.

Can I print the calculation?

The full calculation is shown on screen, free and without an account. A print-ready A4 calculation note is part of the Basic plan.

Page updated: 14 November 2026

Bridge deck influence lines and moving loads

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