CELE Steel & Timber Design — Timber DesignSummary

The cornerstone of timber design: the allowable stress is the reference design value ($F$) multiplied by a product of adjustment factors ($C$). The actual stress ($f$) must not exceed this allowable: $F' = F \times \prod C$, then verify $f \le F'$. This differs fundamentally from LRFD (used in steel and concrete) because it does not apply load and resistance factors separately; instead, it recognizes that wood material properties vary with duration, moisture, and geometry, and the factors reflect these variations directly.

Unique to timber: reflects wood's ability to carry higher stresses for brief loads. Values range from 0.9 (permanent/dead load) to 2.0 (impact). Standard cases: 1.0 for 10-year design life, 1.15 for 2-month loads (construction), 1.25 for 7-day loads, 1.6 for wind/seismic. Wood gains strength when load is brief because the matrix creeps less; applied to all stress types (bending, shear, compression). NSCP 2015 Table 6.2.1 specifies values.

Timber absorbs water and loses strength when wet or when moisture content (MC) exceeds 12–15%, depending on species. $C_M$ is typically 1.0 for dry service (MC $\le 12\%$, common in covered buildings) and 0.8–0.9 for wet service (outdoor, exposed, MC $>20\%$). Glulam and engineered products may use different thresholds. Applied to bending, compression, and shear.

$C_F$ (size factor) accounts for smaller sections being relatively stronger than larger ones — a 50×100 beam is proportionally stiffer than a 300×450 because the failure process is more brittle in large members. $C_L$ (beam lateral-torsional stability factor) reduces bending allowable for long, slender sections prone to buckling sideways. $C_L$ depends on unbraced length and depth; fully braced beams have $C_L \approx 1.0$; long, unsupported beams have $C_L < 1.0$. In practice, the governing (smaller) effect is often used rather than multiplying both.

For a section under bending moment $M$, the bending stress is $f_b = M/S$, where $S$ is the section modulus. For a rectangular section, $S = bh^2/6$. The actual stress must not exceed the allowable: $f_b \le F'_b = F_b \cdot C_D \cdot C_M \cdot C_t \cdot C_F \cdot C_L \cdot C_r$. All adjustment factors are applied; $C_r$ (repetitive member, e.g., floor joists spaced 400 mm) is unique to bending and can increase the allowable by up to 15% if three or more members share the load.

Horizontal shear perpendicular to the grain is the weakness in wood beams. For a rectangular section, $f_v = \frac{3V}{2A}$, where $V$ is shear force and $A$ is the cross-sectional area. The allowable is $F'_v = F_v \cdot C_D \cdot C_M \cdot C_t$ (fewer factors than bending; no $C_F$ or $C_L$ for shear). Short, deep beams (high $V/M$ ratio) and notched supports are critical. A **notched beam** (e.g., a timber column resting in a notch cut into a beam) undergoes stress concentration and shear amplification: effective shear stress is $f_v = \frac{3V}{2A} \cdot \frac{d}{d_n}$, where $d_n$ is the reduced depth at the notch. This is a classic CELE trap.

For a timber column or post under axial load $P$, the compression stress is $f_c = P/A$. The allowable is $F'_c = F_c^{*} \cdot C_P$, where $F_c^{*} = F_c \cdot C_D \cdot C_M \cdot C_t \cdot C_F$ includes all factors except $C_P$. The **column stability factor** $C_P$ (analog to the Euler buckling factor in steel) accounts for slenderness and is computed iteratively or from tables. It depends on the slenderness ratio $\ell_e/d$ (effective length to dimension perpendicular to buckling axis), the reference compression stress $F_c^{*}$, and the elastic buckling stress $F_{cE} = \frac{0.822 E'_{\min}}{(\ell_e/d)^2}$.

Given $F_c^{*}$ (reference compression stress with all factors except $C_P$), compute $\beta = F_{cE}/F_c^{*}$ where $F_{cE} = \frac{0.822 E'_{\min}}{(\ell_e/d)^2}$. Then: $$C_P = \frac{1 + \beta}{2c} - \sqrt{\left(\frac{1 + \beta}{2c}\right)^2 - \frac{\beta}{c}}$$ with $c = 0.8$ for sawn lumber and $c = 0.90$ for glulam. As $\ell_e/d$ increases, $\beta$ increases and $C_P$ decreases toward zero. For very short, stocky columns, $C_P \approx 1.0$. This is the wood equivalent of AISC Eq. (E3-3) for steel, but with timber-specific coefficients.

Each wood species (Douglas fir, pine, hardwoods, etc.) and grade (e.g., No. 1, No. 2, Select Structural) has published reference values from the National Design Specification (NDS) or Philippine timber standards. These include $F_b$ (bending), $F_v$ (shear parallel to grain), $F_c$ (compression parallel), $F_{c\perp}$ (compression perpendicular to grain), $F_t$ (tension), and $E$ (modulus). NSCP 2015 Table 6.2.2 or local timber grading agencies provide these. The reference values are the **baseline** before any adjustment factors are applied.

When three or more identically loaded, parallel members (e.g., floor joists, roof rafters) are spaced no more than 600 mm and share a common load (via a shared deck or rafter tie), the allowable bending stress increases by up to 15% (typical $C_r = 1.15$). This recognizes that a localized failure in one member is less critical when load can redistribute to neighbors. **Critical rule:** $C_r$ applies **only to bending**, not to shear or compression.

When a timber member is subjected to combined bending and compression (e.g., a column with eccentric load), or bending and shear, the actual stresses must satisfy individual checks **and** an interaction check to account for coupling effects. NSCP 2015 Section 6.10 provides equations like: $$\frac{f_b}{F'_b} + \frac{f_c}{F'_c} \le 1.0$$ (bending + compression). More complex forms include moment amplification for buckling. These are less common on CELE but appear in advanced problems.

All adjustment factors ($C_D$, $C_M$, $C_t$, $C_F$, $C_L$, $C_P$, $C_r$) are **multiplicative** and reduce (or occasionally increase) the reference design value $F$ to yield the allowable $F'$. A single missing or misapplied factor breaks the check. The product $F \times \prod C$ is the **foundation** of timber ASD.

$C_D$ directly controls how generous the allowable is: a permanent load (0.9) tightens the allowable; a 7-day load (1.25) or wind/seismic (1.6) relaxes it. Exam problems often vary the load duration to test understanding of this inverse relationship.

For design (given $M$, find section), rearrange $f_b = M/S \le F'_b$ to $S \ge M/F'_b$. This is the **required section modulus**. Choose a standard timber size with $S \ge S_{\text{req}}$. This is the typical CELE design problem flow.

Shear stress $f_v = 3V/(2A) = 3V/(2bh)$ is **inversely proportional to depth $h$**. Increasing depth (area) decreases shear stress strongly. A deep beam governed by shear can be fixed by increasing depth much more than widening. Notched beams break this by reducing the effective depth $d_n$, causing sharp shear rise.

Higher slenderness ratio $\ell_e/d$ reduces the elastic buckling stress $F_{cE}$, which increases $\beta = F_{cE}/F_c^{*}$, causing $C_P$ to fall. Lower $C_P$ directly reduces allowable compression stress $F'_c = F_c^{*} C_P$. In the extreme (very slender columns), $C_P \to 0$ and the member is ineffective, mirroring Euler buckling in steel.

Both $C_M$ (moisture) and $C_D$ (duration) reduce allowable stresses, but independently. A wet, short-duration member uses $C_M < 1.0$ (wet service) **and** $C_D > 1.0$ (brief load). The **product** determines the net allowable. Exam problems may combine these to test understanding.

The species and grade determine $F_b$, $F_v$, $F_c$, $E_\min$, etc. Different grades (e.g., No. 2 vs. Select Structural) have vastly different reference values. All downstream checks depend on this input. Exams always specify or provide a reference table.