Beam Deflection Calculator

About Beam Deflection Calculation

The calculator computes beam deflection and checks bending strength for a uniformly distributed load (kg/m or kN/m) or a concentrated force (kg or kN). The calculation uses standard strength of materials formulas for the selected support scheme and cross section geometry.

The self weight of the beam is also included. The output contains the calculated deflection d (mm), the allowable deflection d_lim (mm), and stress checks (normal and shear). For some thin walled sections, simplified checks for the web and flange are also provided.

Guidelines and recommendations

Standards reference follows the engineering logic used in Eurocodes. Deflection and internal actions are obtained from linear elastic analysis. Material guidance and checks follow the approach of EN 1990 (basis), EN 1991 (actions), EN 1993-1-1 (steel), EN 1995-1-1 (timber).

Units and load conversion use the standard gravitational acceleration g = 9.80665. The following conversions are applied:

1 kN = 1000 N

1 kg ≈ 9.80665 N

Therefore, to convert: kg/m → kN/m the factor 9.80665 / 1000 is used. For the reverse conversion the factor is 1000 / 9.80665.

Material defines the modulus of elasticity E, the density for self weight, and the strength values used in checks.

• Steel (EN 1993-1-1). Modulus of elasticity: E = 200000 MPa. Density: 7850 kg/m³. Design strength used as a limit in this calculator: S235 → 197 MPa, S275 → 231 MPa, S355 → 298 MPa, S420 → 353 MPa. Shear coefficient: k_v = 0.58.
• Timber (EN 1995-1-1). Modulus of elasticity: E = 10000 MPa. Density for self weight: 700 kg/m³. Design strength used as a limit: C16 → 8.62 MPa, C24 → 12.92 MPa, C30 → 16.15 MPa. Shear coefficient: k_v = 0.10.

Beam self weight is added to the external load. The mass per meter is obtained from the cross sectional area A and density ρ:

G = ρ · A · g

where G is the distributed load from self weight (N/m), ρ is density (kg/m³), A is the area (mm², converted to m²), g = 9.80665. Then G is converted to kN/m or kg/m depending on the selected output units.

Section properties are computed from the input dimensions. The calculation uses:

• A area (mm²).
• I second moment of area about the bending axis (mm⁴).
• W section modulus (mm³), typically W = I / y, where y is the distance from the neutral axis to the extreme fiber (mm).

Support scheme affects the maximum bending moment and deflection through coefficients. For a uniformly distributed load q, the calculator uses the following numeric coefficients:

• Pinned-pinned: deflection coefficient k_f = 0.0130208333 (this equals 5/384). Moment coefficient k_M = 0.125001 (≈ 1/8).
• Fixed-pinned: k_f = 0.0054054054 (this equals 1/185). k_M = 0.125 (this equals 1/8).
• Fixed-fixed: k_f = 0.0026041667 (this equals 1/384). k_M = 0.08333333 (this equals 1/12).
• Cantilever: k_f = 0.125 (this equals 1/8). k_M = 0.5 (this equals 1/2).

Actions for a uniformly distributed load are computed as:

Mmax = kM · q · L²

where q is the total distributed load (kN/m or N/m) and L is the span (m or mm, converted to consistent units).

Deflection for a uniformly distributed load is computed as:

d = kf · q · L⁴ / (E · I)

where E is the modulus of elasticity (MPa), I is the second moment of area (mm⁴), and d is obtained in mm after unit conversion.

Concentrated force is calculated using standard formulas for a centrally applied force. For deflection, the coefficient k_p (instead of k_f) is used and depends on the support scheme:

• Pinned-pinned: k_p = 0.020833 (this equals 1/48).
• Fixed-pinned: k_p = 0.00912.
• Fixed-fixed: k_p = 0.0052.
• Cantilever: k_p = 0.3333333 (this equals 1/3).

Then the deflection from a force P (N or kN) is computed as:

d = kp · P · L³ / (E · I)

where L is the effective length for the selected scheme. For a cantilever, an increased length is used in the deflection limit check: L_eff = 2 · L.

Allowable deflection is defined using the divisor n in the rule d_lim = L_eff / n. The divisor n is chosen automatically based on length (mm):

• L_eff ≤ 1000: n = 120.
• 1000 < L_eff ≤ 3000: n linearly from 120 to 150.
• 3000 < L_eff ≤ 6000: n linearly from 150 to 200.
• 6000 < L_eff ≤ 24000: n linearly from 200 to 250.
• 24000 < L_eff ≤ 36000: n linearly from 250 to 300.
• L_eff > 36000: n = 300.

This selection matches common serviceability practice. For residential and public floors, the range L/200…L/300 is often used. For cantilevers, limits are usually stricter, therefore L_eff = 2·L is applied.

Normal stress check compares the calculated stress with the allowable value for the selected material and grade:

σ = Mmax / W

where σ is the normal stress (MPa). The criterion is: σ ≤ v, where v is the selected strength value (MPa). The reserve is shown as a percentage using v/σ − 1.

Shear stress check compares the calculated shear stress τ with the limit v · k_v:

τ ≤ v · kv

where for steel k_v = 0.58 and for timber k_v = 0.10. This provides a clear numeric shear limit without adding unnecessary complexity.

Combined stress effect for some sections is estimated using an equivalent stress and compared with the threshold 0.87 · v:

σeq ≤ 0.87 · v

This criterion is used as an engineering check when normal and shear stresses act together.

Simplified web and flange checks for thin walled elements use dimensionless criteria. The web check uses the limit: λ ≤ 2.5. For the flange, the actual ratio is compared with the limit:

w = 0.5 · √(206000 / v)

If the conditions are not met, a practical recommendation is to increase thickness or provide stiffeners.

FAQs

Why are deflection and allowable deflection based on different lengths for a cantilever?

For cantilever beams, deflections are more noticeable and serviceability criteria are often stricter. Therefore the allowable deflection check uses L_eff = 2·L instead of only the geometric cantilever length. This makes the criterion more conservative for the same span.

What is included in the total distributed load?

The total distributed load q is the sum of the external load and the beam self weight. Self weight is computed from material density and cross section area using g = 9.80665. As a result, deflection and stresses change even for the same external load when the material or geometry changes.

Which strength values are used for steel and timber?

For steel, the calculator uses fixed levels (MPa): S235 → 197, S275 → 231, S355 → 298, S420 → 353. For timber: C16 → 8.62, C24 → 12.92, C30 → 16.15. These values are used as limits in the normal stress check.

Why is a shear check needed if a bending check is already performed?

Bending governs stresses in extreme fibers, but shear can be critical near supports and in thin webs. The check τ ≤ v·k_v adds control for web behavior and regions with high shear. For steel k_v = 0.58 is used. For timber k_v = 0.10 is used.

What deflection limits are commonly used in practice?

The range L/200…L/300 is often used depending on the structure and finish sensitivity. In this calculator, the divisor n varies from 120 to 300 with beam length, covering common engineering targets. If stricter control is required, use the upper part of the range and apply cantilever limits.