Calculation
Mmax = F · a · (L−a) / L
For one perpendicular point load, the bending-moment diagram reaches its maximum directly beneath the load. The supports share the load according to its position.
Einfach gelagerter Balken mit Einzellast · Mmax = F a(L−a)/L
Calculate Maximum Bending Moment for an Off-Centre Point Load
For one perpendicular point load, the bending-moment diagram reaches its maximum directly beneath the load. The supports share the load according to its position.
MINTSI
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Result
Select a target and calculate.
Understand the inputs
- Maximum bending moment Mmax — Largest magnitude of internal bending moment at the load. A stress check also needs section modulus and material properties.
- Perpendicular point force F — One downward concentrated load, for example a wheel load; convert mass to weight using g if necessary.
- Support span L — Distance between the two ideal supports along the beam, from structural drawing or measurement.
- Load distance a from left support — Distance of the force from the left support along the beam. It must lie between zero and L; b = L−a is the distance to the right support.
Example
For F = 1 kN on L = 3 m at a = 1 m, b = 2 m and Mmax = 1·1·2/3 = 0.667 kN·m directly under the load.
Assumptions and limits
Straight statically determinate beam with two ideal supports and exactly one static perpendicular point load within the span. Self-weight, other loads, applied couples, lateral instability and deformation effects are excluded.
Understand Maximum beam moment under an off-centre point load
A wheel load on a beam is often away from midspan. This calculator finds the internal bending moment directly below the load.
What does this quantity describe?
A simply supported beam rests on two ideal supports and carries one perpendicular force F. L is the span, a its distance from the left support and b = L−a the distance to the right. Moment equilibrium gives left reaction FA = Fb/L. Moment rises up to the load and falls afterward; its maximum is Mmax = FA·a = Fab/L.
Formula and variables
Mmax = F · a · (L−a) / L
Right distance: b = L−aLeft reaction: FA = Fb/LMaximum bending moment: Mmax = Fab/L
| Symbol / input | Meaning |
|---|---|
| Maximum bending moment Mmax | Largest magnitude of internal bending moment at the load. A stress check also needs section modulus and material properties. |
| Perpendicular point force F | One downward concentrated load, for example a wheel load; convert mass to weight using g if necessary. |
| Support span L | Distance between the two ideal supports along the beam, from structural drawing or measurement. |
| Load distance a from left support | Distance of the force from the left support along the beam. It must lie between zero and L; b = L−a is the distance to the right support. |
Choose the inputs correctly
F is one downward point force in newtons, such as a wheel or machine load; convert mass to weight first if necessary. L is support spacing and a is measured from the left support to the force line. This model requires 0 ≤ a ≤ L. Obtain all distances from the same beam drawing.
How to use the calculator
Enter force, span and load location. The result is the largest internal moment magnitude. Divide by actual section modulus for bending stress; add self-weight and other loads to the moment diagram separately.
Worked example
A 1000 N force acts on a 3 m span 1 m from the left support. The right distance is b = 2 m. Mmax = 1000·1·2/3 = 666.7 N·m or 0.667 kN·m at the force.
Understand the result and units
At midspan, moment is largest for fixed force and span: F·L/4. Directly at an ideal support it is zero. For this single force the maximum occurs exactly where the force acts.
The calculation uses newtons and metres internally, yielding N·m. Force and length inputs are converted to SI before evaluation.
Typical applications
Early checks of beams under a single wheel, machine or suspended load; selecting a section before detailed stress and deflection analysis.
Assumptions, limits and common mistakes
Straight simply supported beam with one static perpendicular point force inside the span only. Self-weight, other forces, distributed loads, applied moments, support flexibility and instability are excluded. This is a moment, not a stress or load rating.
Common mistake: Do not measure a from the opposite support while using b = L−a. Distinguish force in N from mass in kg. F·L/4 applies only at midspan.
Frequently asked questions
What is “Maximum beam moment under an off-centre point load” used for?
Early checks of beams under a single wheel, machine or suspended load; selecting a section before detailed stress and deflection analysis.
Where do the input values come from?
F is one downward point force in newtons, such as a wheel or machine load; convert mass to weight first if necessary. L is support spacing and a is measured from the left support to the force line. This model requires 0 ≤ a ≤ L. Obtain all distances from the same beam drawing.
What does the result not cover?
Straight simply supported beam with one static perpendicular point force inside the span only. Self-weight, other forces, distributed loads, applied moments, support flexibility and instability are excluded. This is a moment, not a stress or load rating.