Beam Deflection Calculator

Beam deflection formula and units

For a simply supported beam with a single point load at the center, the maximum deflection occurs at midspan and is given by the standard elastic beam formula:

where:

• Δ is the maximum deflection at midspan (meters).
• P is the point load at the center of the span (newtons).
• L is the clear span between supports (meters).
• E is Young’s modulus or modulus of elasticity (pascals).
• I is the second moment of area of the beam cross-section (m³·m = m⁴).

In the calculator, you enter:

• Span length in meters (m).
• Point load in kilonewtons (kN). The calculator converts kN to newtons internally (1 kN = 1,000 N).
• Young’s modulus in gigapascals (GPa). The calculator converts GPa to pascals (1 GPa = 10⁹ Pa).
• Moment of inertia in cm³·cm = cm⁴. The calculator converts cm⁴ to m⁴ using 1 cm⁴ = 10⁻⁸ m⁴.
• Allowable deflection ratio in the form L/x (for example, enter 360 for L/360).

This unit handling lets you work with the values commonly used in structural tables and design guides while keeping the underlying calculation consistent in SI units.

Understanding Young’s modulus (E)

Young’s modulus is a measure of material stiffness. A higher E means a material deforms less under the same stress. In the deflection formula, E appears in the denominator, so increasing E reduces the predicted deflection.

Typical ranges for E in gigapascals (GPa) for common structural materials are:

• Structural steel: about 200–210 GPa.
• Reinforced concrete (uncracked, short-term): about 25–35 GPa.
• Engineered wood (glulam, LVL): about 10–16 GPa depending on grade.
• Softwood framing lumber: about 8–12 GPa.

Because deflection is inversely proportional to E, switching from a timber member with E ≈ 10 GPa to a steel member with E ≈ 200 GPa (a 20× increase) dramatically reduces deflection if all other parameters stay the same.

Moment of inertia (I) and section stiffness

The second moment of area, I, reflects how the cross-sectional area of the beam is distributed relative to the neutral axis. Deep sections with more material far from the neutral axis have higher I and are therefore stiffer in bending.

For simple shapes, I can be computed analytically. For example, for a rectangular cross-section with width b and height h (in consistent units):

In the calculator interface, you enter I in cm⁴, which is how many structural shape tables report section properties. Internally, the value is converted to m⁴ using:

Im⁴ = Icm⁴ × 10−8

This conversion is important: if you forget it and try to enter I directly in m⁴, the result will be off by a factor of 10⁸. As long as you input I in cm⁴ as requested by the field label, the calculator takes care of the unit change.

Deflection limits and L/x ratios

In many building codes and design guides, serviceability criteria are expressed as a limit on deflection, typically written as L/x. The idea is that longer spans are allowed more absolute deflection, but only in proportion to their length.

For a span length L, an L/x limit corresponds to an allowable deflection:

When you enter a value like 240, 360, or 480 into the calculator’s “Allowable Deflection Ratio (L/x)” field, it computes this allowable deflection and compares it to the predicted deflection Δ.

Common examples include:

• L/240: often used for roofs without brittle finishes.
• L/360: common limit for floors supporting plaster or drywall ceilings, aiming for acceptable vibration and crack control.
• L/480: stricter limit sometimes used where higher stiffness or minimal visible sag is important.

Interpreting the calculator is straightforward:

• If Δ ≤ Δ_allow, the beam meets the selected deflection criterion.
• If Δ > Δ_allow, the beam exceeds the limit and may feel too flexible or cause finish damage, even if strength is adequate.

Worked example using the default values

This section walks through the calculation using typical default inputs similar to those shown in the form. These numbers are illustrative only, but they demonstrate how the formula and units come together.

Step 1 – Inputs

• Span length, L = 2.0 m
• Point load at center, P = 10 kN
• Young’s modulus, E = 200 GPa
• Moment of inertia, I = 800 cm⁴
• Allowable ratio, L/x = L/360

Step 2 – Unit conversions

Convert each value to the base units used in the formula:

• P = 10 kN = 10 × 1,000 N = 10,000 N
• E = 200 GPa = 200 × 10⁹ Pa = 2.0 × 10¹¹ Pa
• I = 800 cm⁴ = 800 × 10⁻⁸ m⁴ = 8.0 × 10⁻⁶ m⁴
• L is already in meters: 2.0 m

Step 3 – Apply the deflection formula

Using Δ = (P L³) / (48 E I):

1. Compute L³: 2.0³ = 8.0 m³.
2. Compute the numerator: P L³ = 10,000 N × 8.0 m³ = 80,000 N·m³.
3. Compute the denominator: 48 × 2.0 × 10¹¹ Pa × 8.0 × 10⁻⁶ m⁴.

First combine the scalar factors: 48 × 2.0 × 8.0 = 768.

Combine the powers of ten: 10¹¹ × 10⁻⁶ = 10⁵.

So the denominator is 768 × 10⁵ N/m² · m⁴ = 7.68 × 10⁷ N·m².

The deflection is then:

Δ = 80,000 / (7.68 × 10⁷) ≈ 1.04 × 10⁻³ m

This is approximately 0.00104 m, or about 1.04 mm of deflection at midspan.

Step 4 – Compare to allowable L/x

For L = 2.0 m and an L/360 limit:

Δ_allow = L / 360 = 2.0 / 360 ≈ 0.00556 m

Converting to millimeters: 0.00556 m ≈ 5.56 mm.

Comparing:

• Actual Δ ≈ 1.04 mm
• Allowable Δ_allow ≈ 5.56 mm

Since 1.04 mm < 5.56 mm, the beam satisfies the L/360 serviceability criterion for this load case. In the live calculator, the results panel will make this comparison for you automatically once you enter or adjust the inputs.

Comparison: materials and deflection for the same span

The formula shows that deflection is inversely proportional to both E and I. To illustrate how material stiffness and section properties affect performance, the table below compares example midspan deflections for the same span and load but with different representative values of E and I.

These values are approximate, but they highlight the strong influence of both material choice and section geometry on midspan deflection. For any configuration you are considering, plug in your actual E and I values to obtain more precise predictions.

Assumptions and limitations

This calculator uses a simplified analytical model. It is helpful for understanding trends and making quick checks, but it does not replace full structural analysis or code-compliant design. The key assumptions and limitations are:

• Linear elastic behavior: The material is assumed to follow Hooke’s law with a constant Young’s modulus E. Effects such as cracking in concrete, yielding in steel, or non-linear timber behavior are not included.
• Small deflections: The derivation assumes deflections are small relative to the span, so geometric nonlinearity (change in geometry under load) is neglected.
• Prismatic member: The beam is assumed to have constant cross-section and constant E and I along its length, with no tapers, splices, or stiffness changes.
• Simply supported ends: Both supports are modeled as simple (pinned) supports with no fixity and no rotational restraint. Fixed or continuous supports produce different deflection patterns and formulas.
• Single central point load only: The formula is for one concentrated load at midspan. Uniformly distributed loads, multiple point loads, or eccentric loads require other expressions.
• No time-dependent effects: Creep, shrinkage, temperature effects, and long-term stiffness reduction are not considered. For concrete or timber, long-term deflections can be substantially larger than short-term predictions.
• No dynamic or vibration analysis: The tool treats loads as static. Vibration serviceability (for floors, footbridges, etc.) needs separate checks.
• Not a full code check: Strength (bending, shear, bearing) and detailed code-specific serviceability criteria are outside the scope. Use this calculator as a supplement to, not a substitute for, proper design to the relevant standard.

For more complex loadings or boundary conditions, or for final design, consult a qualified structural engineer and use appropriate structural analysis tools and code provisions.