Beam Displacement Calculator: Deflection and Formula

A beam displacement calculator estimates how far a beam bends under a specified load, using its span, support conditions, material stiffness, cross-section, and loading pattern. Enter those values to calculate the maximum vertical deflection; the result is an idealized elastic estimate, not a design approval.

Formula

This calculator uses the standard Euler–Bernoulli beam formula for a simply supported beam carrying a uniformly distributed load across its full span:

δ_max = 5wL⁴ / (384EI)

• δ_max is the maximum vertical displacement at midspan, in metres before conversion to millimetres.
• w is the uniform load per unit length, in newtons per metre (N/m).
• L is the distance between supports, in metres (m).
• E is Young’s modulus, in pascals (Pa).
• I is the cross-section’s second moment of area about the bending axis, in metres to the fourth power (m⁴).

The calculator accepts E in gigapascals (GPa), I in centimetres to the fourth power (cm⁴), and w in kilonewtons per metre (kN/m), then converts them to SI units within the expression. Its first result is millimetres; the second converts that displacement to inches. The model assumes a straight, prismatic beam, small elastic deformation, and ideal simple supports.

Worked Example

Using the default values—span 3 m, Young’s modulus 200 GPa, second moment of area 1,000 cm⁴, and uniform load 5 kN/m—convert the inputs first:

• w = 5 kN/m = 5,000 N/m.
• E = 200 GPa = 200,000,000,000 Pa.
• I = 1,000 cm⁴ = 0.00001 m⁴.

Substitute into the formula: δ_max = 5 × 5,000 × 3⁴ / (384 × 200,000,000,000 × 0.00001) = 0.00439453125 m. That is 4.39453125 mm, which rounds to 4.39 mm. In inches, 4.39453125 ÷ 25.4 = 0.173013…, or 0.173 in to three decimal places.

What Beam Displacement Means

Displacement, often called deflection in beam engineering, describes the movement of a point on a beam from its unloaded position. It is not the same as stress, strain, or the load-bearing capacity of the beam. A beam can have a modest calculated displacement yet still need checks for strength, stability, vibration, connections, or other conditions.

The example’s maximum occurs at midspan because the load and supports are symmetric. Other load patterns can shift the point of maximum displacement. For instance, a single point load at the centre of a simply supported beam uses δ_max = PL³/(48EI), where P is the point load in newtons. A cantilever with an end point load uses δ_max = PL³/(3EI). These are different boundary and loading cases, so do not use the calculator’s uniform-load formula for them.

When to Use This Estimate

This calculation is useful for understanding how span, stiffness, and load affect bending in a basic beam model. It can support early comparisons: for otherwise identical conditions, a longer span produces much more deflection because L is raised to the fourth power in this formula. A larger E or I reduces deflection in direct proportion.

Use consistent definitions for all inputs. Span is the support-to-support distance, not necessarily the beam’s overall cut length. The load must be expressed per unit length and distributed uniformly over the whole span. If you have a total load rather than a line load, it must be converted appropriately before using this formula; a concentrated load is not interchangeable with a uniform load.

Choosing the Material and Section Values

Young’s modulus measures elastic stiffness: it describes the relationship between stress and elastic strain, not the beam’s strength. A material value should match the material and relevant condition. For example, a nominal steel value may be around 200 GPa, but material specifications and design standards should govern actual engineering work. Do not treat a generic value as a verified property for an unknown member.

The second moment of area I is a geometric property of the cross-section, not the mass moment of inertia. It depends on the axis about which the beam bends. Rotating an asymmetric section can substantially change I and therefore the predicted displacement. For a rectangle of width b and vertical depth h bending about its centroidal horizontal axis, I = bh³/12. Use the actual dimensions and correct orientation; the depth is cubed.

Assumptions, Accuracy, and Common Mistakes

• Wrong support model: A simply supported beam can rotate at its supports. A fixed end, cantilever, continuous beam, or flexible support has different behavior and usually a different formula.
• Unit mismatch: Mixing millimetres with metres or using GPa as though it were Pa can make the result wrong by large factors. Convert the whole equation consistently.
• Wrong section axis: Enter I for the actual bending axis, rather than a value calculated for another orientation.
• Relying on rounded inputs: Since span is raised to the fourth power, small span errors can matter. Keep adequate precision until the final display.
• Assuming the model covers every effect: The basic formula omits effects such as shear deformation, material nonlinearity, large deflection, dynamic loading, and local or connection behavior.

Real structures also require load combinations and checks beyond displacement. Applicable allowable deflection criteria depend on the structure, use, materials, and governing design requirements; there is no single limit that applies to every beam. For construction or safety-critical decisions, have the assumptions, loads, section properties, and results reviewed by a qualified structural professional.

Frequently Asked Questions

What does a beam displacement calculator calculate?
It estimates how far a beam moves under a specified load. This calculator reports maximum elastic displacement for a simply supported beam under a uniform load across its full span.

Where is maximum deflection for a uniformly loaded simply supported beam?
For a symmetric, full-span uniform load, the maximum vertical deflection is at midspan. Different load arrangements or supports can produce a different location and value.

How do I calculate beam deflection by hand?
For the case covered here, use δ_max = 5wL⁴/(384EI), with all values converted to consistent SI units. Convert the resulting metres to millimetres by multiplying by 1,000, or to inches by dividing by 0.0254.

Is second moment of area the same as moment of inertia?
In beam calculations, I commonly means the area’s second moment of area, with units of length to the fourth power. It is different from mass moment of inertia, which describes resistance to angular acceleration.

Does a low deflection prove that a beam is safe?
No. Deflection is only one behavior to assess. A complete evaluation may also need strength, stability, vibration, connections, support conditions, and applicable design requirements checked.

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in L | Beam span | m | 3 | 0
in E | Young's modulus | GPa | 200 | 0
in I | Second moment of area | cm^4 | 1000 | 0
in w | Uniform load | kN/m | 5 | 0
out delta | Maximum displacement | mm | 5 * (w * 1000) * (L ^ 4) / (384 * (E * 1000000000) * (I * 0.00000001)) | 2
out delta_in | Maximum displacement | in | delta / 25.4 | 3