Wooden Beam Calculator

About Wooden Beam Deflection and Strength Calculation

The Wooden Beam Deflection and Strength Calculator performs calculations for a rectangular wooden beam for deflection, bending strength, and shear strength. The online wooden beam calculator takes into account the cross-section dimensions, span length, timber strength class, support condition, distributed or concentrated load, and the beam's self-weight.

The results include beam deflection, allowable deflection, normal and shear stresses, as well as cross-sectional area, second moment of area, section modulus, beam mass, self-weight load, maximum bending moment, and maximum shear force.

Guidelines and recommendations

Standards and timber properties

European standards. The properties of structural timber are based on the strength classes defined in EN 338 “Structural timber — Strength classes”. The general approach to checking timber members follows the principles of EN 1995-1-1 “Eurocode 5: Design of timber structures — Part 1-1: General — Common rules and rules for buildings”. Related standards for loads and limit-state design are EN 1991 “Eurocode 1: Actions on structures” and EN 1990 “Eurocode: Basis of structural design”.

Timber strength class properties. Each strength class uses its own mean stiffness, mean density, characteristic bending strength, and characteristic shear strength:

• C16: E0,mean = 8000 MPa, ρ = 370 kg/m3, fm,k = 16 MPa, fv,k = 3.2 MPa.
• C24: E0,mean = 11000 MPa, ρ = 420 kg/m3, fm,k = 24 MPa, fv,k = 4.0 MPa.
• C30: E0,mean = 12000 MPa, ρ = 460 kg/m3, fm,k = 30 MPa, fv,k = 4.0 MPa.

Design strength. Characteristic strength is converted to design strength using the general Eurocode 5 relationship:

fd = kmod·fk/γM

The calculator uses fixed values of kmod = 0.7 and γM = 1.3. This gives design bending strengths of 8.62 MPa for C16, 12.92 MPa for C24, and 16.15 MPa for C30. The corresponding design shear strengths are 1.72 MPa, 2.15 MPa, and 2.15 MPa.

Modification factor kmod. In EN 1995-1-1, this factor depends on the load-duration class and timber service conditions. The calculator does not vary this factor automatically and uses the fixed value 0.7, which is one of the main assumptions of the calculation.

Applied load. The calculator treats the load entered by the user as an already defined design action. Load combination factors from EN 1990 and EN 1991 are not applied automatically to the entered value.

Beam section properties

Cross-sectional area. For a rectangular beam with width t and height h, the area is calculated as:

A = t·h

When the dimensions are entered in millimetres, the area is obtained in mm². It is used to determine the beam mass, self-weight, and shear stress.

Second moment of area. The bending stiffness of the rectangular section is characterised by:

I = t·h3/12

The value of I is expressed in mm⁴ and directly affects beam deflection. Therefore, increasing the beam height has a much greater effect on stiffness than the same relative increase in width.

Section modulus. The section modulus used for the bending-stress check is:

W = t·h2/6

The section modulus is expressed in mm³ and relates the maximum bending moment to the stress in the outermost timber fibres.

First moment of area. For the maximum shear stress in a rectangular section, the calculator uses:

Q = t·h2/8

Load units and self-weight

Load conversion. The acceleration due to gravity g = 9.80665 m/s2 is used to convert values given in kilograms into force. A distributed load in kg/m is converted into force per unit length, while a concentrated load in kg is converted into newtons. Values entered directly in kN/m or kN are treated as force quantities.

Self-weight. The mass per metre is determined from the cross-sectional area and the mean density of the selected timber strength class. The distributed self-weight load is calculated as:

qsw = ρ·A·g·10-9

With A in mm², qsw is obtained in kN/m. Self-weight is always included as a uniformly distributed load, including when the main external load is concentrated.

Beam mass. For a beam length L in millimetres, the mass is calculated as:

m = ρ·A·L·10-9

The result is expressed in kilograms.

Bending moment and shear force

Total distributed load. For an external distributed load, the force effects are calculated using q = qext + qsw. For an external concentrated force P, the distributed component q consists of the beam self-weight.

Design force effects. In the formulas below, q is in kN/m, P is in kN, and L is in metres. The maximum bending moment is obtained in kN·m and the maximum shear force in kN:

• Simply supported: Mmax = q·L2/8 + P·L/4, Vmax = q·L/2 + P/2.
• Fixed-pinned: Mmax = q·L2/8 + 3·P·L/16, Vmax = 5·q·L/8 + 11·P/16.
• Fixed-fixed: Mmax = q·L2/12 + P·L/8, Vmax = q·L/2 + P/2.
• Cantilever: Mmax = q·L2/2 + P·L, Vmax = q·L + P.

For beams supported at both ends, the concentrated force is positioned in the central part of the span according to the selected calculation scheme. For a cantilever, the force is applied at the free end. The beam self-weight acts along the entire beam length.

Beam deflection calculation

Bending stiffness. Deflection is governed by the product E·I: the greater the timber modulus of elasticity or the second moment of area, the smaller the deformation. In the deflection formulas, the calculator uses q in N/mm, P in N, L in mm, E in MPa, and I in mm⁴, so the result is obtained directly in millimetres.

Simply supported beam. Deflection from the distributed and concentrated loads is calculated using superposition:

f = 5·q·L4/(384·E·I) + P·L3/(48·E·I)

Fixed-fixed beam. For a beam fixed at both ends:

f = q·L4/(384·E·I) + P·L3/(192·E·I)

Cantilever beam. For a distributed load acting along the full length and a force applied at the free end:

f = q·L4/(8·E·I) + P·L3/(3·E·I)

Fixed-pinned beam. For a distributed load acting alone, the maximum-deflection coefficient is 0.00541612:

fq = 0.00541612·q·L4/(E·I)

For a concentrated force acting alone, the coefficient is √5/240 ≈ 0.00931695:

fP = 0.00931695·P·L3/(E·I)

When self-weight and a concentrated force act simultaneously, the location of maximum deflection does not necessarily coincide with midspan. Therefore, for the fixed-pinned scheme, the calculator combines the elastic curves from both loads and determines the maximum absolute deflection along the beam length.

Allowable deflection

Deflection criterion. The allowable value is defined as the span length divided by a coefficient n:

flim = L/n

When the coefficient is selected automatically, it varies smoothly with span length:

• up to 1000 mm — n = 120;
• from 1000 to 3000 mm — linearly from 120 to 150;
• from 3000 to 6000 mm — linearly from 150 to 200;
• from 6000 to 24000 mm — linearly from 200 to 250;
• from 24000 to 36000 mm — linearly from 250 to 300;
• above 36000 mm — n = 300.

The automatically selected value is used as a practical serviceability guideline rather than a universal code limit for every structure. If a specific project requirement is known, a custom L/n ratio can be entered; increasing n reduces the allowable deflection and makes the check more stringent.

Strength checks and result evaluation

Normal stress. The maximum bending stress is calculated from the section modulus:

σm = Mmax/W

The bending strength requirement is satisfied when:

σm ≤ fm,d

Shear stress. The maximum shear stress is determined from the maximum shear force:

τ = Vmax·Q/(I·t)

For a rectangular section, this relationship is equivalent to τ = 1.5·Vmax/A. The shear strength check is satisfied when:

τ ≤ fv,d

Strength reserve. When a strength check is satisfied, the displayed reserve shows the relative ratio between the design strength and the calculated stress:

Reserve = (fd/σ - 1)·100%

For the shear check, τ is used instead of σ, and fv,d is used instead of fd. Deflection, bending, and shear are checked independently: passing one condition does not compensate for exceeding another.

FAQs

Why does the same C24 wooden beam deflect less than a C16 beam?

The timber strength class affects not only the allowable stress but also the stiffness of the wood. The calculator uses E = 8000 MPa for C16 and 11000 MPa for C24, so with the same cross-section, span, and load, the calculated deflection of a C24 wooden beam is lower.

Why can a wooden beam pass the strength check but fail the deflection check?

Strength and stiffness describe different aspects of beam behaviour. The bending stress may remain well below the design strength of the timber, while low E·I stiffness over a long span can still produce excessive deflection. For wooden beams, deflection can therefore become the governing condition even when the load-bearing capacity is sufficient.

Why is self-weight included when the main load is concentrated?

A concentrated force describes only the external action at one point. The wooden beam itself has mass along its full length, so its self-weight simultaneously acts as a uniformly distributed load qsw and contributes to deflection, bending moment, and shear force.

What does the strength reserve in the result mean?

The reserve shows how much higher the design strength of the selected timber class is than the calculated stress in relative terms. For example, a reserve of 20% represents the ratio between the design limit and the current stress; it does not automatically mean that the external load can be increased by exactly 20%, because the beam self-weight is also included in the calculation.

How should I choose the allowable deflection ratio L/n?

If no specific requirement is known, the Wooden Beam Deflection and Strength Calculator uses an automatic guideline based on the span length. If a project specification, design requirement, or applicable rule gives a specific limit, the custom L/n value should be used. The larger the denominator n, the stricter the deflection requirement.