Module 3: Timber Beams

Module 3: Timber Beams

Timber Beams

Beams are structural members designed primarily to resist transverse (perpendicular) loads, which induce internal bending moments and shear forces. The design of a timber beam is typically an iterative process ensuring three primary criteria are met: Flexural Strength (moment capacity), Horizontal Shear (shear capacity), and Serviceability (deflection limits).

Bending Stress (Flexure)

The most fundamental check for a beam is whether it can safely resist the maximum bending moment (

M

) induced by the applied loads without failing in tension or compression at its extreme fibers.

Checklist

The actual bending stress ( $f_b$ ) must not exceed the adjusted allowable bending stress ( $F_b'$ ).
For a standard rectangular cross-section, the maximum bending stress is calculated using the flexure formula:

$$
f_b = \\frac{M}{S} = \\frac{6M}{b d^2}
$$

Key Takeaways

Flexure strength dictates that the actual bending stress ( $f_b$ ) must be less than the adjusted allowable bending stress ( $F_b'$ ).
The section modulus ( $S$ ) geometrically defines the bending capacity of the cross-section.

C_F

As timber members increase in size, the statistical probability of encountering a critical strength-reducing defect (like a large knot) increases. The NSCP accounts for this by reducing the allowable bending stress based on member dimensions.

Size Factor ( $C_F$ ) for Sawn Lumber

The Size Factor (

C_F

) applies primarily to visually graded dimension lumber and timbers. It is determined using empirical formulas that compare the depth of the actual member (

d

) to a standard baseline depth (often 300 mm or 12 inches).

C_F = \left( \frac{300}{d} \right)^{1/9}

Note: Exponent values can vary based on exact lumber grade and species grouping as per NSCP tables. Depth $d$ is in mm.

Volume Factor ( $C_V$ ) for Glulam

For Glued Laminated Timber (Glulam), the Volume Factor (

C_V

) replaces the Size Factor. It accounts for the member's width (

b

), depth (

d

), and length (

L

). If both

C_V

and

C_L

(Beam Stability Factor) apply to a Glulam beam, the designer must use the lesser of the two, not both simultaneously.

C_V = \left( \frac{130}{b} \right)^{1/x} \left( \frac{300}{d} \right)^{1/x} \left( \frac{6100}{L} \right)^{1/x} \le 1.0

Note: The exponent 'x' depends on the specific Glulam manufacturing standard (e.g., $x=20$ or $x=10$ ). Dimensions are in mm.

C_L

Just like a long column can buckle laterally under compression, the top edge of a simply supported beam is subjected to compressive bending stresses. If the beam is deep and narrow, and the compression edge is not adequately braced against lateral movement, the beam will buckle sideways before reaching its full bending capacity. This is known as Lateral-Torsional Buckling (LTB).

Calculating CL and Empirical Rules

The Beam Stability Factor (

C_L

) reduces the allowable bending stress (

F_b

) to account for this instability. It depends on the unbraced length (

L_u

), the beam dimensions (

b, d

), and the effective length (

l_e

). If a beam is fully laterally supported along its entire compression edge (e.g., continuous floor joists securely fastened to rigid subflooring), lateral-torsional buckling cannot occur, and

C_L = 1.0

Empirical Lateral Support Rules (NSCP/NDS): To avoid complex calculations, the code offers approximate lateral bracing rules based on the depth-to-width ratio (

d/b

) of a rectangular beam:

$d/b le 2$ : No lateral support required.
$3 le d/b le 4$ : Ends must be held in position to prevent rotation.
$d/b = 5$ : One edge must be continuously supported along its entire length.
$d/b = 6$ : Bridging, blocking, or cross-bracing required at intervals $\le 8$ feet (2.4m).
$d/b = 7$ : Both edges must be continuously supported.

$$
C_L = \\frac{1 + (F_{bE} / F_b^*)}{1.9} - \\sqrt{ \\left[ \\frac{1 + (F_{bE} / F_b^*)}{1.9} \\right]^2 - \\frac{F_{bE} / F_b^*}{0.95} }
$$

Key Takeaways

Deep, narrow beams are susceptible to lateral-torsional buckling.
The Beam Stability Factor ( $C_L$ ) accounts for the lateral instability of unbraced beams.
Full lateral bracing, such as a continuous floor deck, eliminates LTB and sets $C_L = 1.0$ .

Biaxial Bending

When a beam is loaded such that bending occurs simultaneously about both its strong axis (x-x) and weak axis (y-y), it undergoes biaxial bending. A classic example is a purlin resting on a sloped roof truss subjected to vertical gravity loads.

Biaxial Interaction Equation

To prevent flexural failure at the extreme corners of the cross-section where stresses from both axes add together, the combined stress ratio must not exceed 1.0.

$$
\frac{f_{bx}}{F_{bx}'} + \frac{f_{by}}{F_{by}'} \le 1.0
$$

C_L in Biaxial Bending

When calculating the adjusted allowable stress for the strong axis (

F_{bx}'

), the Beam Stability Factor (

C_L

) must be applied if the compression edge is unbraced. However,

C_L

is always 1.0 for weak axis bending (

F_{by}'

), as lateral-torsional buckling does not occur when bending about the weak axis.

Notched Beams

Timber beams are often notched at their ends to fit over supports, or along their span to accommodate piping and wiring. Notching a beam causes a severe reduction in its strength. The sudden change in cross-section creates immense Stress Concentrations at the re-entrant corner of the notch.

When a notch is located on the tension face of a beam, the stress concentration acts as a crack initiator. As the beam bends, the tensile forces will cause the wood to split horizontally starting from the notch corner, propagating along the grain and leading to catastrophic, brittle failure. The NSCP strongly restricts the depth and location of notches, particularly in the middle third of the span and on the tension face.

Horizontal Shear

While steel beams typically fail in vertical shear, timber beams almost exclusively fail in horizontal (longitudinal) shear. This is a direct consequence of wood being an orthotropic material.

Criticality of Horizontal Shear in Wood

When a beam bends, the layers of wood fibers tend to slide past one another longitudinally. Because the bond between parallel wood fibers (the lignin) is relatively weak (low shear strength parallel to the grain), the beam will split horizontally along the neutral axis near the supports where shear forces are highest.

Checklist

For a rectangular beam, the maximum horizontal shear stress ( $f_v$ ) occurs at the neutral axis (the center depth) and is $1.5$ times the average shear stress. This parabolic shear stress distribution is expressed by the fundamental shear equation for rectangular sections:

$$
f_v = \\frac{3V}{2A} = \\frac{3V}{2 b d}
$$

Checklist

The actual calculated shear stress must be less than or equal to the adjusted allowable shear stress ( $F_v'$ ).
When calculating the design shear force ( $V$ ), loads located within a distance ' $d$ ' (the depth of the beam) from the face of the support can generally be neglected, as they transfer directly into the support via compression, rather than inducing shear across the span.

Key Takeaways

Timber beams fail in horizontal shear along the neutral axis because shear strength parallel to the grain is exceptionally weak.
Maximum shear stress ( $f_v$ ) is 1.5 times the average shear stress ( $V/A$ ) for rectangular beams.
Loads within a distance 'd' from supports are ignored when determining maximum shear $V$ .

Bearing at Supports (Compression Perpendicular to Grain)

Bearing Failure

A beam must transfer its vertical loads safely into its supports (walls, columns, or girders) through its bearing area. This induces a compressive stress perpendicular to the grain (

f_{c\perp}

), which crushes the hollow wood fibers transversally. The required bearing area is dependent on the maximum reaction force (

R

) and the allowable bearing stress of the wood species (

F_{c\perp}'

If the bearing length is short (less than 150mm), a Bearing Area Factor ( $C_b$ ) is applied to increase the allowable stress, reflecting the fact that wood fibers extending beyond the bearing area act to distribute the load and resist crushing.

$$
f_{c\perp} = \frac{R}{A_{bearing}} \le F_{c\perp}'
$$

Deflection Criteria

Even if a beam is strong enough to resist bending and shear, it must also be stiff enough. Excessive deflection (sagging) can crack plaster ceilings below, cause doors to jam, or create an uncomfortable "bouncy" floor.

Checklist

Calculated actual deflections ( $\Delta_{actual}$ ) under service loads (unfactored loads) must not exceed code-specified allowable limits ( $\Delta_{allowable}$ ). For a simply supported beam under a uniform distributed load ( $w$ ), the maximum deflection is commonly calculated as:

Checklist

Common limits include $L/360$ for live loads only (to prevent cracking of finishes) and $L/240$ for total loads (dead + live).
Creep: Wood permanently deforms (sags) over time when subjected to sustained, long-term loads (like heavy dead loads). The total long-term deflection often includes an amplification factor (e.g., $1.5$ to $2.0$ times the immediate dead load deflection) depending on moisture conditions.

Key Takeaways

Serviceability (deflection limits) is critical for preventing cosmetic damage and ensuring comfort.
Standard deflection limits are $L/360$ for live load and $L/240$ for total load.
Long-term sustained loads cause creep, increasing the total deflection over time.

Special Design Considerations

Specific detailing and support conditions require special attention in timber design.

Checklist

Bearing (Compression Perpendicular to Grain, $F_{c\perp}$ ): Where a beam rests on a support (like a wall plate or column bracket), the heavy reaction force must be distributed over a sufficient bearing area. The allowable bearing stress ( $F_{c\perp}'$ ) can be increased using the Bearing Area Factor ( $C_b$ ) if the bearing length ( $l_b$ ) is less than 150 mm and not closer than 75 mm to the end of the member.

If the beam bears on a support at an angle (e.g., a rafter birdsmouth cut resting on a top plate), the allowable bearing stress must be calculated using Hankinson's Formula, interpolating between the strong parallel-to-grain capacity ( $F_c$ ) and the weaker perpendicular-to-grain capacity ( $F_{c\perp}$ ).
Notches: Notching a timber beam, especially on the tension face near the middle of the span or at the ends, is highly discouraged. Notches drastically reduce the effective depth ( $d_e$ ) and create severe stress concentrations that rapidly propagate horizontal shear cracks, often causing abrupt failure.

Creep and Deflection Limit Exceptions

Wood, like many natural materials, undergoes creep over long periods of sustained loading.

Time-Dependent Deflection (Creep)

When a timber beam is subjected to a permanent load (e.g., dead load from flooring or roofing materials), it initially deflects elastically. Over months or years, the wood fibers continue to stretch, causing the deflection to increase, sometimes doubling the initial elastic deflection. The NSCP accounts for this by requiring designers to calculate the total long-term deflection, typically by multiplying the immediate dead load deflection by a creep factor (e.g.,

1.5

for dry wood,

2.0

for wet wood) and adding it to the immediate live load deflection.

Key Takeaways

Timber beam design ensures adequate Bending ( $F_b$ ), Horizontal Shear ( $F_v$ ), and stiffness (Deflection limits).
The Beam Stability Factor ( $C_L$ ) reduces bending capacity if the compression edge is laterally unsupported.
Horizontal shear is a critical failure mode due to wood's inherent weakness parallel to the grain. It dictates that $f_v$ at the neutral axis must not exceed $F_v'$ .
Avoid notching beams, especially on the tension side, as it creates massive stress concentrations that initiate premature horizontal splitting.

PreviousModule 2: Timber Tension and Compression Members - Examples & Applications

NextModule 3: Timber Beams - Examples & Applications

Timber Beams

Bending Stress (Flexure)

Checklist

Size (CFC_FCF​) and Volume (CVC_VCV​) Factors

Size Factor (CFC_FCF​) for Sawn Lumber

Volume Factor (CVC_VCV​) for Glulam

Beam Stability Factor (CLC_LCL​)

Calculating CL and Empirical Rules

Biaxial Bending

Biaxial Interaction Equation

C_L in Biaxial Bending

Notched Beams

Horizontal Shear

Criticality of Horizontal Shear in Wood

Checklist

Checklist

Bearing at Supports (Compression Perpendicular to Grain)

Bearing Failure

Deflection Criteria

Checklist

Checklist

Special Design Considerations

Checklist

Creep and Deflection Limit Exceptions

Time-Dependent Deflection (Creep)

Size ( $C_F$ ) and Volume ( $C_V$ ) Factors

Size Factor ( $C_F$ ) for Sawn Lumber

Volume Factor ( $C_V$ ) for Glulam

Beam Stability Factor ( $C_L$ )