Wooden Beam Calculator

About Wooden Beam Deflection and Strength Calculation

This calculator provides an approximate check of a rectangular timber beam for deflection, bending strength, and shear strength. It is intended for quick assessment of floor beams, decking beams, and cantilever members under uniformly distributed or point loads.

The calculation is based on classical strength of materials formulas and the general Eurocode approach. The coefficients and formulas used are shown explicitly below so the user can see the assumptions and how the results are obtained.

Guidelines and recommendations

European standards references. The verification logic follows the common limit state design framework in EN 1990. The way loads are handled is consistent with EN 1991. For timber member design, the main reference is EN 1995-1-1 (Eurocode 5). In the current version, the calculator uses engineering simplifications and does not build design load combinations according to EN 1990.

Beam scheme and calculation factors. The selected support scheme defines factors used to compute deflection and the maximum bending moment for a uniformly distributed load:

• Pinned-pinned: deflection factor m = 0.0130208333, moment factor m1 = 0.125001 ≈ 1/8.
• Fixed-pinned: m = 0.0054054054, m1 = 0.125 = 1/8.
• Fixed-fixed: m = 0.0026041667 ≈ 1/384, m1 = 0.08333333 ≈ 1/12.
• Cantilever: m = 0.125 = 1/8, m1 = 0.5 = 1/2.

Dimensions and units. Section geometry is entered in millimetres: thickness t (mm) and depth h (mm). Span is entered in millimetres L (mm). Loads can be entered in kg/m or kN/m (for distributed load) and in kg or kN (for point load).

Section properties. From the entered t and h, the calculator computes:

• Area: A = t·h (mm²).
• Second moment of area: I = t·h3/12 (mm⁴).
• Section modulus: W = t·h2/6 (mm³).
• First moment for shear at the neutral axis (rectangle): Q = t·h2/8 (mm³).

Elastic modulus for deflection. Deflection is computed using a constant value E = 10000 MPa. This is a typical order of magnitude for structural softwood. In design practice under EN 1995-1-1, E depends on the strength class and service conditions, so deflection results here should be treated as approximate.

Timber strength class and design bending strength. For the normal-stress check, the design bending strength Ryd (MPa) is taken as:

• C16: Ryd = 8.62 MPa
• C24: Ryd = 12.92 MPa
• C30: Ryd = 16.15 MPa

These values are specified at the design level of strength (including typical duration and material safety effects). This simplification makes it possible to compare computed stresses with an allowable design level without additional inputs.

Unit conversion (kg ↔ kN). Conversion uses gravitational acceleration g = 9.81 m/s². In practical terms:

1 kN = 1000 N ≈ 1000/9.81 ≈ 101.97 kgf

When switching units, the numeric load value is recalculated so the physical load remains the same.

Self-weight of the beam. The calculator adds self-weight as an additional uniformly distributed load. The assumed timber density is ρ = 550 kg/m³. Self-weight is most noticeable for longer spans and comparatively light imposed loads.

Uniformly distributed load: deflection. Deflection uses the scheme factor m:

f = m·q·L4 / (E·I·100) + m·qsw·L4 / (E·I·100)

Here q is the applied distributed load, qsw is the distributed load from self-weight, L is the span, E is the elastic modulus, and I is the second moment of area. The scaling factors reflect internal unit conversion because the geometry is entered in mm.

Uniformly distributed load: bending and stresses. The maximum bending moment is computed using the scheme factor m1 and includes self-weight:

M = (q/100)·L2·m1 + (qsw)·L2·m1

Normal bending stress (MPa) is then:

σ = M / W

Bending strength condition:

σ ≤ Ryd

Shear force and shear. For the shear-stress check, the maximum shear force is used. For a uniformly distributed load, the design maximum (with internal unit conversion) is:

V = (q/100)·L/2 for most schemes and V = (q/100)·L for a cantilever

Shear stress is computed using the rectangular-section formula:

τ = V·Q / (I·t)

The shear strength limit is set in a simplified way as a fraction of design bending strength: τ ≤ 0.1·Ryd. This is a conservative guideline for a quick shear check without detailed inputs for grade, moisture, and service conditions.

Point load: deflection. For a point load P, a scheme-dependent factor is used. The adopted values are:

• Pinned-pinned: k = 0.020833
• Fixed-pinned: k = 0.00912
• Fixed-fixed: k = 0.0052
• Cantilever: k = 0.3333333

Deflection under a point load is computed as (as implemented, with internal scaling for mm):

f = (k·P·L3)/(E·I)·10 + m·qsw·L4/(E·I·100)

This includes deflection from the point load P and deflection from self-weight.

Point load: bending and stresses. The maximum bending moment from the point load P depends on the selected scheme. In general, the calculator applies the characteristic formula for that scheme, then computes σ = M/W and compares it to Ryd. This selection is necessary because moment distributions are fundamentally different for a cantilever versus a simply supported beam.

Deflection limit. The allowable deflection is defined as a span-to-factor ratio:

flim = Lx / k

For a cantilever, the effective length is taken as Lx = 2·L, for other schemes Lx = L. The factor k is selected from the span range Lx (mm) with smooth transitions:

• if Lx ≤ 1000 mm, then k = 120
• if 1000 < Lx ≤ 3000 mm, then k changes linearly from 120 to 150
• if 3000 < Lx ≤ 6000 mm, then k changes linearly from 150 to 200
• if 6000 < Lx ≤ 24000 mm, then k changes linearly from 200 to 250
• if 24000 < Lx ≤ 36000 mm, then k changes linearly from 250 to 300
• if Lx > 36000 mm, then k = 300

Serviceability condition: f ≤ flim. This approach applies stricter limits for short spans and more relaxed limits for long spans.

How to read the results. Deflection answers the question “how much the beam will sag”. The σ check shows the bending strength margin, and the τ check shows the shear margin. If any condition is not satisfied, common measures are increasing h, reducing the span, reducing the load, or changing the support scheme.

FAQs

Why is deflection calculated with a fixed elastic modulus of 10000 MPa?

This provides a simple, repeatable deflection estimate without requiring extra inputs. Under EN 1995-1-1, the elastic modulus depends on strength class and service conditions, so for design you should use an appropriate E. For preliminary sizing, a fixed value usually gives the correct order of magnitude.

Is the self-weight of the timber beam included?

Yes, self-weight is added as a uniformly distributed load along the beam. The assumed density is ρ = 550 kg/m³. For long spans, self-weight can noticeably affect both deflection and stresses.

What does the shear check mean and why is the limit set to 0.1·Ryd?

The shear check evaluates the shear stress τ caused by the shear force V. In this calculator, the limit is simplified as τ ≤ 0.1·Ryd to give a conservative quick guideline without additional inputs. EN 1995-1-1 treats shear in more detail, including timber parameters and service conditions.

Why is a doubled effective length used for cantilevers in the deflection limit?

Cantilever members are usually more sensitive to deflection from a usability and perception standpoint. Using Lx = 2·L makes the deflection criterion stricter for cantilevers, reducing the risk of visible or uncomfortable deformations.

Which units are better: kg/m or kN/m?

Both are equivalent and are converted using g = 9.81. In European engineering practice, kN and kN/m are more common because they align with Eurocodes (EN 1991). If your source data is in “kilograms”, kg and kg/m are often more convenient.