Draws the influence lines of a straight beam in any material, over any number of spans and cantilevers, for the bending moment, shear force, reactions, deflection and rotation at the places you choose, then puts a train of axles and a lane load where they act worst. Every step shows its formula.

What it checks

Indeterminacy: r − (3 + h), with r the reactions, springs included, and h the hinges. Above zero, the moments depend on the stiffness and the influence lines are curves: the deflected shapes of the Müller-Breslau principle.
Influence lines, up to six effects: the bending moment (sagging positive), the shear force just left or just right of a section, the vertical reaction (upward positive) or the reaction moment (anticlockwise positive) of a support, the deflection (downward positive) or the rotation (anticlockwise positive) of a section.
For each line: the largest and smallest ordinates y_max and y_min with their positions, and the areas A_+ and A_−, ∫ y dx over the positive and over the negative parts.
Axles: Σ Q_k·y(x_k) at the worst position, largest and smallest, with the position of the train.
Lane load: q_L·A_+ for the largest value and q_L·A_− for the smallest, or over the whole beam if you choose.
Permanent load: g·(A_+ + A_−), added to both the largest and the smallest value.
Largest and smallest value: the sums of the largest and of the smallest contributions.
Envelope along the beam, optional: the moving load at its worst for every section of a regular grid, both sides of every interior point for the shear force, with the absolute extremes M_max, M_min, V_max and V_min, refined between the sections.

What you enter and what you get

You enter up to twelve segments and the modulus E. A span between two supports is one segment, a cantilever one, a span with a hinge in it two. For each segment: its length, its second moment of area, and a constant section or straight or parabolic haunches. For each point: the support (pinned, roller, fixed, free, guided up and down, guided left and right, or springs) and a continuous or hinged joint. Then the load path and up to six effects, each at a section or a support.

For the moving load: up to ten axles, each with its load and its distance from axle 1, and whether the train also stands in the reverse order; the lane load q_L; the uniform permanent load g; and the envelope with 4 to 20 sections per segment.

You get each influence line drawn, the extreme values of every effect with the share of each load, and the envelope.

Assumptions and limits

A straight beam, with one modulus E along its whole length. Only E·I enters the forces on rigid supports.
In a haunch the width is constant, so I grows with the cube of the depth.
No factor is applied: enter characteristic or design values as you need them. Loads downward are positive.
The axles stand at fixed distances from one another: a vehicle, a tandem or a crane.
Through cross-beams, the loads stand on stringers spanning simply between them, and every influence line becomes straight between the cross-beams.
The permanent load here is uniform over the whole beam. Other permanent loads go to the continuous-beam module.
An analysis only: no section is verified.

Worked example

A beam 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

What does an influence line give me?

The value of one effect at one place when a load of 1 kN stands at position x. The effect of any set of loads is then Σ P·y(x) for forces and ∫ q·y(x) dx for distributed loads.

How do I enter a crane on a runway girder?

As a train of axles at their fixed distances, from left to right as it stands on the beam. If it can also stand in the reverse order, choose that, and the module tries both.

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

Influence lines and moving loads on a beam

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