Module 4: Combined Stresses in Timber & Connections - Theory & Concepts

Module 4: Combined Stresses in Timber & Connections

Combined Stresses

Many structural timber members are subjected to multiple load types simultaneously. The most common scenario is combined bending and axial compression (e.g., a truss chord resisting both its own weight and the compressive force of the truss, or a wall stud resisting wind loads laterally while supporting roof loads vertically).

Combined Axial and Bending Loads

To ensure the member does not fail under combined loading, designers use Interaction Equations. These equations calculate the ratio of the actual stress to the allowable stress for each load type individually. The sum of these ratios must be

\le 1.0

Checklist

Basic Interaction Equation (Tension + Bending): For members subjected to simultaneous tension and bending, the structural interaction is relatively straightforward:

$$
\\frac{f_t}{F_t'} + \\frac{f_b}{F_b'} \\le 1.0
$$

Checklist

Combined Bending and Compression ( $P-\Delta$ Effect): This scenario is far more complex. When a column is subjected to bending (or eccentric loading), the resulting lateral deflection ( $\Delta$ ) creates an additional bending moment ( $P \times \Delta$ ). This secondary moment must be amplified and included in the interaction equation. The NSCP dictates a highly specific interaction formula involving the Euler elastic buckling stress to account for this non-linear amplification.

$$
\left( \frac{f_c}{F_c'} \right)^2 + \frac{f_b}{F_b' \left( 1 - \frac{f_c}{F_{cE}} \right)} \le 1.0
$$

Key Takeaways

The interaction equation checks if a member can safely resist combined bending and axial loading simultaneously.
Combined compression and bending is governed by the $P-\Delta$ effect, which requires amplifying the bending stress due to lateral deflections.
The sum of stress ratios (Actual / Allowable) must never exceed 1.0.

Timber Fasteners and Connections

A timber structure is only as strong as its weakest link, which is almost invariably the connections. Unlike continuous steel or monolithic concrete, timber structures must be joined together.

The Weak Link: Connections

Connections introduce severe stress concentrations, require boring holes that reduce the net cross-sectional area, and rely on bearing stresses perpendicular to or parallel to the grain. Most catastrophic failures in timber structures initiate at a connection.

Nails and Spikes: Primarily resist lateral shear loads. They rely on the density of the wood (Specific Gravity) for holding power. Withdrawal resistance (pulling out) is low and should not be relied upon for primary structural loads.
Lag Screws: Provide substantially higher withdrawal and lateral resistance than nails. They require pre-drilled pilot holes to prevent splitting the wood, with different diameters for the unthreaded shank and the threaded portion.
Bolts: Essential for heavy timber framing. They transfer loads via bearing against the wood hole. Holes must be drilled slightly oversized (typically +1/16 inch) to allow for installation without forcibly driving the bolt, which can split the wood.
Split Rings and Shear Plates: Used in heavy roof trusses. These circular steel connectors are embedded in pre-cut grooves between two wood members. They provide an immense bearing area, transferring load far more effectively than the bolt shank alone, which only serves to hold the assembly together.

Nails, Screws, and Bolt Detailing

When detailing any timber connection, the design must carefully specify Edge Distance, End Distance, and Fastener Spacing (Gauge and Pitch). Insufficient edge/end distances will cause the wood to split or shear out long before the fastener itself yields or the wood fibers crush in bearing.

Design of Bolted Timber Joints

The design of a bolted connection relies on ensuring the bolt does not yield in bending and the wood does not crush in bearing.

Checklist

Yield Limit Model (European Yield Model): The modern NSCP approach evaluates several possible failure modes (e.g., wood crushing in the main member, wood crushing in the side member, bolt yielding in one hinge, bolt yielding in two hinges) and selects the lowest capacity as the nominal design value ( $Z$ ).
EYM Failure Modes: The six standard yield modes evaluated for dowel-type fasteners are:
- Mode $I_m$ : Wood crushing completely in the main member.
- Mode $I_s$ : Wood crushing completely in the side member.
- Mode $II$ : Wood crushing in both members with the fastener remaining straight (typical for identical wood pieces).
- Mode $III_m$ : Fastener yields in bending (forms one plastic hinge) inside the main member.
- Mode $III_s$ : Fastener yields in bending (forms one plastic hinge) inside the side member.
- Mode $IV$ : Fastener yields in bending in both members (forms two plastic hinges), often occurring in thick, dense wood with a slender bolt.
Hankinson's Formula: The bearing capacity of a bolt depends critically on the angle of the applied load relative to the wood grain. Wood is strong parallel to the grain ( $P$ ) and weak perpendicular ( $Q$ ). For loads applied at an intermediate angle ( $\theta$ ), Hankinson's empirical formula is used to calculate the allowable bearing stress ( $N$ ):

Checklist

Placement Geometry: Careful attention must be paid to minimum edge distances, end distances, and spacing between bolts to prevent block shear, splitting, or plug shear failures around the connection group.

Yield Limit Equations for Dowel Fasteners

Yield Limit Theory

The Yield Limit Theory (or European Yield Model, EYM) calculates the lateral load-carrying capacity of a dowel-type connection by evaluating all possible failure mechanisms mathematically. The theory assumes the connection will fail when either the fastener yields in bending (forming one or more plastic hinges) or the wood fibers crush (exceeding dowel bearing strength).

The fundamental design capacity is taken as the minimum value produced by all applicable yield limit equations. These equations depend on four key parameters:

Dowel Bearing Strength ( $F_e$ ): The compressive strength of the wood pushing against the side of the bolt. This is directly proportional to the wood's specific gravity ( $G$ ) and depends on whether the load is parallel or perpendicular to the grain.
Fastener Bending Yield Strength ( $F_{yb}$ ): The internal material strength of the steel bolt or nail before it permanently bends.
Fastener Diameter ( $D$ ): Larger diameters increase both bearing area and bending capacity.
Member Thicknesses ( $t_m, t_s$ ): The thickness of the main member and side members, dictating how deeply the fastener penetrates and where bending hinges might form.

Connection Adjustment Factors

Just like base stress values, connection design capacities must be adjusted for environmental and geometric conditions. Two critical factors are the Group Action Factor (

C_g

) and the Geometry Factor (

C_{\Delta}

Checklist

Group Action Factor ( $C_g$ ): When multiple fasteners (like bolts or split rings) are placed in a row parallel to the load, they do not share the load equally. The outer fasteners carry a disproportionately higher percentage of the total load due to the differential stretching of the wood and the steel side plates. As the number of fasteners in a row increases, $C_g$ decreases the average capacity per fastener.
Geometry Factor ( $C_{\Delta}$ ): The tabulated capacity of a fastener requires standard minimum edge distances, end distances, and spacing to prevent wood splitting. If these minimums cannot be met, the allowable capacity must be proportionally reduced by the Geometry Factor ( $C_{\Delta} \lt 1.0$ ). If absolute minimums (often half the standard requirements) are not met, the connection is invalid.

European Yield Model (EYM) Failure Modes

The EYM calculates the lowest load capacity among several possible failure scenarios for dowel-type fasteners.

Primary Failure Modes (EYM)

Mode I_m: The bearing capacity of the main member is exceeded, causing the wood to crush locally without bending the fastener.
Mode I_s: The bearing capacity of the side member is exceeded, leading to crushing of the side member wood.
Mode II: The fastener pivots slightly, causing crushing in both the main and side members, but the fastener does not permanently deform.
Mode III_m and III_s: The fastener develops a single plastic hinge (yields) inside either the main or side member, alongside localized wood crushing.
Mode IV: The fastener develops two plastic hinges (yields entirely) within the connection. This ductile failure is often considered the ideal capacity limit because it absorbs the most energy.

Key Takeaways

Members under combined loading (like beam-columns) must satisfy interaction equations ensuring the sum of the utilization ratios (actual/allowable) is $\le 1.0$ .
Combined bending and compression requires considering the non-linear $P-\Delta$ effect, where the axial load amplifies the bending moment.
Connections are the most critical points in a timber structure.
Bearing strength is highly dependent on grain orientation. The allowable capacity for loads applied at an angle $\theta$ to the grain is determined using Hankinson's formula.
Detailed fastener spacing (edge distance, end distance) is vital for nails, screws, and bolts to prevent premature wood splitting failures.

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