Calculation
wF = F·a²·(L−a)²/(3·E·I·L)
For one load between simple supports, the result gives beam deflection exactly where the load acts.
Elastische Biegelinie des einfach gelagerten Balkens · wF = F·a²·(L−a)²/(3·E·I·L)
Calculate Beam Deflection Under an Off-Centre Point Load
For one load between simple supports, the result gives beam deflection exactly where the load acts.
MINTSI
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Result
Select a target and calculate.
Understand the inputs
- Deflection at load point wF — Elastic displacement exactly under the downward point load, positive downward; compare it with serviceability limits. This is not necessarily the maximum deflection along the beam.
- Point load F — Single downward transverse force from load specification or measurement; self-weight and other loads are excluded.
- Support span L — Distance between the two ideal pin/roller supports along the straight beam axis, from drawing or measurement.
- Load distance a from left support — Distance from support A to the force along the beam axis; 0 ≤ a ≤ L. Distance to the right support is L−a.
- Young modulus E — Bending elastic modulus from material data, for example about 210 GPa for steel.
- Second moment of area I — Second moment of area of the constant section about the bending axis, from a section table or section calculation.
Example
F = 1,000 N, L = 3 m, a = 1 m, E = 210 GPa and I = 100 cm⁴ give wF = 2.116 mm. At a = 1.5 m, deflection is 2.679 mm.
Assumptions and limits
Straight prismatic Euler–Bernoulli beam on two simple supports, small elastic deflection, one transverse point load and constant EI. Shear deformation, self-weight and other loads are excluded. Deflection at the load point is not generally the maximum beam deflection for an off-centre load.
Understand Deflection at an eccentric point load
A single force loads a beam between two supports. This calculator estimates elastic deflection directly beneath the force, even when the force is off centre.
What does this quantity describe?
A simply supported beam can rotate at both ends; one support also fixes axial movement while the other permits it. The deflection curve is the deformed beam axis. For a straight beam of constant flexural rigidity E·I and one transverse force F at distance a from support A, deflection at the load is wF = F·a²·b²/(3·E·I·L), where b = L−a is distance to the right support and L is the span.
Formula and variables
wF = F·a²·(L−a)²/(3·E·I·L)
Distance to second support: b = L−aDeflection at load: wF = F·a²·b²/(3EIL)Midspan a = L/2: wF = F·L³/(48EI)
| Symbol / input | Meaning |
|---|---|
| Deflection at load point wF | Elastic displacement exactly under the downward point load, positive downward; compare it with serviceability limits. This is not necessarily the maximum deflection along the beam. |
| Point load F | Single downward transverse force from load specification or measurement; self-weight and other loads are excluded. |
| Support span L | Distance between the two ideal pin/roller supports along the straight beam axis, from drawing or measurement. |
| Load distance a from left support | Distance from support A to the force along the beam axis; 0 ≤ a ≤ L. Distance to the right support is L−a. |
| Young modulus E | Bending elastic modulus from material data, for example about 210 GPa for steel. |
| Second moment of area I | Second moment of area of the constant section about the bending axis, from a section table or section calculation. |
Choose the inputs correctly
F is the downward point load in newtons from loading specifications or measurement. L is support span in metres and a is distance from support A to the force; 0 ≤ a ≤ L. E is material Young modulus from a data sheet, about 210 GPa for steel. I is the second moment of area about the bending axis, from a section table or calculator. wF is displacement at the load, reported in millimetres and positive downward; compare it with the permitted serviceability deflection.
How to use the calculator
Measure span and load position along the same beam axis. Find I for the actual bending direction, take E from material data and enter only the transverse point load F. Read the result as deflection at the load point, not a general maximum.
Worked example
A steel beam with L = 3 m, E = 210 GPa and I = 100 cm⁴ carries F = 1,000 N at a = 1 m. Thus b = 2 m and wF = 1000·1²·2²/(3·210·10⁹·10⁻⁶·3) m = 0.002116 m = 2.116 mm. At midspan, a = b = 1.5 m, it is 2.679 mm.
Understand the result and units
At fixed load position, deflection scales with F and inversely with E and I. Exchanging a and b leaves the result unchanged. At an ideal support (a = 0 or a = L), displacement is zero.
Internal units are N for F, m for L and a, Pa for E and m⁴ for I; wF results in m. Available input units convert to SI first. 100 cm⁴ equal 10⁻⁶ m⁴.
Typical applications
Early estimates for beams and machine frames with a local load before serviceability and strength checks.
Assumptions, limits and common mistakes
Only a straight prismatic Euler–Bernoulli beam with two simple supports, small elastic deflection and one transverse point load. Shear deformation, self-weight, variable rigidity and other loads are omitted. With an off-centre force the greatest beam deflection may occur elsewhere.
Common mistake: Do not measure load distance from the wrong support or enter a larger than L. The result is displacement at the load, not automatically the largest displacement. Using I about the wrong bending axis understates deflection.
Frequently asked questions
What is “Deflection at an eccentric point load” used for?
Early estimates for beams and machine frames with a local load before serviceability and strength checks.
Where do the input values come from?
F is the downward point load in newtons from loading specifications or measurement. L is support span in metres and a is distance from support A to the force; 0 ≤ a ≤ L. E is material Young modulus from a data sheet, about 210 GPa for steel. I is the second moment of area about the bending axis, from a section table or calculator. wF is displacement at the load, reported in millimetres and positive downward; compare it with the permitted serviceability deflection.
What does the result not cover?
Only a straight prismatic Euler–Bernoulli beam with two simple supports, small elastic deflection and one transverse point load. Shear deformation, self-weight, variable rigidity and other loads are omitted. With an off-centre force the greatest beam deflection may occur elsewhere.