CELE Steel & Timber Design — Timber DesignCheat Sheet

Formula

F′ = F × C_D × C_M × C_t × C_F × C_L × C_P × C_r (and others as applicable)

Meaning

F = reference design value (species/grade); C_* = adjustment factors; F′ = adjusted allowable stress

Watch Out

Do NOT apply all factors blindly. Some factors interact (e.g., C_F and C_L for bending interact; use the controlling one, not the product). Read NSCP carefully for which factors apply to each stress type.

When To Use

Always — this is the backbone of every timber check; multiply reference value by the relevant adjustment factors for the loading and member type.

Formula

f ≤ F′

Meaning

f = actual stress (from loads); F′ = adjusted allowable; design is OK if actual ≤ allowable

Watch Out

Forgetting to compute f correctly (e.g., using gross area instead of net area for tension; using the wrong section modulus formula for bending).

When To Use

Final check for every bending, shear, and compression member

Value

0.9

Symbol

C_D,perm

Quantity

Load duration factor C_D (permanent load)

Value

1.0

Symbol

C_D,standard

Quantity

Load duration factor C_D (standard 10-year load)

Value

1.15

Symbol

C_D,const

Quantity

Load duration factor C_D (2-month construction load)

Value

1.25

Symbol

C_D,7day

Quantity

Load duration factor C_D (7-day load)

Value

1.6

Symbol

C_D,wind

Quantity

Load duration factor C_D (wind or seismic load)

Value

2.0

Symbol

C_D,impact

Quantity

Load duration factor C_D (impact load)

Term

Reference Design Value (F)

Example

F_b = 16.5 MPa for a specific grade of Mahogany or Pine — before any adjustments.

Definition

Tabulated stress for a timber species and grade under standard conditions; reduced by adjustment factors for real design.

Term

Adjustment Factor (C)

Example

C_D = 1.6 for wind/seismic (short duration); C_D = 0.9 for permanent loads — same timber, different load.

Definition

Multiplier reflecting load duration, moisture, temperature, member size, stability, and other conditions.

Term

Allowable Stress Design (ASD)

Example

Unlike LRFD steel (load factor on loads), timber ASD divides the safe load by a safety factor into the allowable.

Definition

Working-stress method: design stress (actual) must not exceed allowable (reference × factors); safety is in the allowable, not in load factors.

Formula

f_b = M / S ≤ F′_b

Meaning

f_b = actual bending stress (MPa); M = bending moment (N⋅mm); S = section modulus (mm³); F′_b = adjusted allowable bending stress

Watch Out

Using gross section modulus when the section is notched or perforated; use net S. Also, many students forget to apply C_L (beam stability factor) for lateral-torsional buckling of slender beams.

When To Use

Every timber beam under bending load; M is max moment and S is for the governing orientation.

Formula

S = b × h² / 6 (rectangular cross-section)

Meaning

b = width (smaller dimension); h = depth (larger dimension); S in units of width × depth².

Watch Out

Some codes define b and h differently; always use the formula S = I/c where I = moment of inertia and c = distance to extreme fiber. For rectangles, S = bh²/6 is correct.

When To Use

Standard rectangular timber sections (sawn lumber, rough lumber).

Formula

F′_b = F_b × C_D × C_M × C_t × C_F × C_L × C_r (and others as applicable)

Meaning

F_b = reference bending stress; each C is an adjustment; for bending, C_r (repetitive member) may apply if three or more members spaced ≤600 mm.

Watch Out

C_F (size factor) and C_L (beam stability) often compete for the same failure mode in sawn lumber. NSCP may require using the more conservative (lower) value or a specific interaction formula, not the product.

When To Use

Compute allowable bending stress before comparing to f_b.

Formula

C_L = (1 + λ_b) / 1.9 − √[(1 + λ_b)² / 3.61 − λ_b / 0.95] (for sawn lumber, slenderness formula)

Meaning

λ_b = slenderness ratio for lateral-torsional buckling; reduces C_L for deep, unsupported beams.

Watch Out

This formula is approximate and varies by code edition; NSCP may use a simplified lookup or a different formula. Verify in the applicable code edition. If the beam is well-braced (e.g., joists on a deck), C_L ≈ 1.0.

When To Use

When a beam is slender in the lateral direction and may buckle sideways.

Formula

I = b × h³ / 12 (moment of inertia for rectangular section)

Meaning

b = width; h = depth; I governs deflection and lateral buckling.

Watch Out

Ensure you use the correct h (the dimension perpendicular to the bending axis).

When To Use

Computing deflection or designing for lateral stability.

Value

1.15 (for three or more spaced ≤600 mm); 1.0 otherwise

Symbol

C_r

Quantity

Repetitive member factor C_r

Value

1.0 (no lateral-torsional buckling concern)

Symbol

C_L,braced

Quantity

Fully braced beam stability factor C_L

Term

Section Modulus (S)

Example

A 150 × 300 mm sawn beam has S = (150 × 300²) / 6 = 2.25 × 10⁶ mm³.

Definition

Geometric property S = I / c (moment of inertia divided by distance to extreme fiber); larger S = stronger beam in bending.

Term

Lateral-Torsional Buckling

Example

A long, deep timber beam with no lateral bracing along its span may buckle to the side under a bending moment.

Definition

Sideway buckling of a beam's compression flange when not braced perpendicular to bending; C_L accounts for this.

Term

Repetitive Member Factor (C_r)

Example

A deck with joists on 400 mm centers gets C_r = 1.15, increasing allowable bending stress.

Definition

Upward adjustment (often 1.15) when three or more identical members are spaced ≤600 mm and act together (e.g., floor joists).

Note

Do not confuse this with shear stress or compression; f_b is purely bending-induced normal stress on fibers.

Equation

f_b = M / S

Conditions

M in N⋅mm, S in mm³; both computed at the section of maximum moment.

Formula

f_v = (3 V) / (2 A) ≤ F′_v (rectangular cross-section)

Meaning

f_v = actual shear stress (MPa); V = shear force (N); A = gross cross-sectional area (mm²); F′_v = adjusted allowable shear stress; factor 1.5 accounts for parabolic shear distribution.

Watch Out

Forgetting the 1.5 factor. Also, at notched supports, shear stress is amplified significantly — use the reduced net depth and an amplification formula from NSCP (often the d / d_n ratio is large, making shear critical).

When To Use

Short, deep beams where shear governs (high V, low M). Always check; wood is weak in parallel shear.

Formula

F′_v = F_v × C_D × C_M × C_t × C_H (and others per NSCP)

Meaning

F_v = reference shear stress; C_D, C_M, C_t are standard adjustments; C_H may account for notches or reduced section.

Watch Out

Shear adjustment factors differ slightly from bending factors; C_L does not apply to shear (shear is a different failure mode). Some factors like C_H (notch factor) are specific to shear and compression.

When To Use

Compute allowable shear before comparing to f_v.

Formula

f_v,notch = f_v × (d / d_n) (amplified shear at notched supports)

Meaning

d = full beam depth; d_n = net depth at the notch; amplification can be 1.5 to 3× for deep notches.

Watch Out

Notched beams are a classic CELE trap. Do NOT forget this check. Many students ignore the notch entirely and fail shear. Always examine supports for notches or other reductions.

When To Use

When a beam is notched at a support to rest on a ledger or pocket; shear capacity drops sharply.

Value

0.8–1.5 MPa (species and grade dependent)

Symbol

F_v

Quantity

Typical reference shear stress F_v (timber)

Term

Horizontal Shear (Parallel to Grain)

Example

In a timber beam spanning 2 m under 10 kN point load, shear is high near the supports and may govern design.

Definition

Shear stress acting along grain direction within a beam; timber is weak here (low F_v).

Term

Notched Beam

Example

A beam notched 50 mm deep on a 200 mm beam (d_n = 150 mm) at a support sees 1.33× shear amplification.

Definition

Beam with a rectangular cut (notch) at a support to reduce bearing area or fit on a ledger; drastically reduces shear and bearing capacity.

Note

Shear is zero at mid-span for symmetric loading, maximum at reactions.

Equation

V = w L / 2 (for uniform load on simply supported beam)

Conditions

w = load per unit length; L = span; max shear occurs at supports.

Formula

f_c = P / A ≤ F′_c

Meaning

f_c = actual axial compression stress (MPa); P = axial load (N); A = cross-sectional area (mm²); F′_c = adjusted allowable compression stress.

Watch Out

Forgetting to account for slenderness (C_P factor). A column is not just A × F_c; buckling reduction via C_P is essential for slender members.

When To Use

For timber columns or axially loaded posts; also used for bearing on supports (perpendicular to grain uses different formula).

Formula

F′_c = F_c* × C_P, where F_c* = F_c × C_D × C_M × C_t × C_F

Meaning

F_c* = adjusted reference compression stress (all factors except C_P); C_P = column stability factor (wood buckling reduction); product gives allowable.

Watch Out

Do NOT apply C_P directly to the reference F_c; it applies to F_c* (which includes D, M, t, F factors). Mistaking the order is a common error.

When To Use

Always compute F_c* first, then apply C_P based on slenderness.

Formula

F_{cE} = (0.822 × E′_{min}) / (ℓ_e / d)²

Meaning

F_{cE} = critical Euler buckling stress for the column; E′_{min} = adjusted minimum modulus (adjusted for wet, temp, etc.); ℓ_e = effective length; d = least dimension.

Watch Out

Use the MINIMUM modulus, not the average; E′_{min} is typically E × 0.66 or given in tables. Also, ℓ_e (effective length) is K × L, where K depends on end conditions (fixed, pinned, etc.).

When To Use

Used within the C_P formula to compute the buckling reduction factor.

Formula

C_P = [(1 + β) / (2c)] − √{[(1 + β) / (2c)]² − (β / c)}, where β = F_{cE} / F_c*, c = 0.8 (sawn), 0.90 (glulam)

Meaning

C_P computes the wood analog of the buckling reduction (ϕ_b for steel or χ in ACI); β is the Euler-to-yield ratio; c is a material constant.

Watch Out

This formula is algebraically messy; a calculator error is easy. Also, c = 0.8 for sawn lumber (typical), 0.90 for glulam — using the wrong c gives wrong C_P. If β > ~1, C_P ≈ 1 (short column); if β << 1, C_P drops (slender column).

When To Use

Once F_{cE} and F_c* are found, use this to get C_P.

Formula

ℓ_e / d = K L / d (slenderness ratio for wood columns)

Meaning

K = effective length factor (0.5 fixed, 1.0 pinned, 2.0 cantilever); L = actual unsupported length; d = least width/depth of section.

Watch Out

For timber, end conditions matter: pinned = K = 1.0 is the baseline. Fixed ends reduce K. Do not use the same K table as steel without checking timber code.

When To Use

Computing the slenderness ratio to find F_{cE}.

Value

0.8

Symbol

c

Quantity

Material constant c (sawn lumber)

Value

0.90

Symbol

c

Quantity

Material constant c (glulam)

Value

1.0

Symbol

K_pinned

Quantity

Effective length factor K (both ends pinned)

Value

0.5

Symbol

K_fixed

Quantity

Effective length factor K (both ends fixed)

Value

2.0

Symbol

K_cantilever

Quantity

Effective length factor K (one end fixed, one free)

Term

Column Stability Factor (C_P)

Example

A 150 × 150 mm sawn timber column with ℓ_e / d = 20 and F_c* = 10 MPa has C_P ≈ 0.78, reducing allowable to 7.8 MPa.

Definition

Reduction factor for buckling; similar to ϕ_b in steel or χ in concrete; accounts for slenderness (ℓ_e / d) and material behavior.

Term

Effective Length (ℓ_e)

Example

A 3 m column with pinned ends: ℓ_e = 1.0 × 3000 = 3000 mm. If fixed at both ends: ℓ_e = 0.5 × 3000 = 1500 mm.

Definition

ℓ_e = K × L; accounts for end conditions (fixed, pinned, free) and determines buckling behavior.

Term

Least Dimension (d)

Example

A 100 × 300 mm beam used as a column has d = 100 mm (least), so ℓ_e / d is based on 100, not 300.

Definition

Smaller of width or depth of a rectangular cross-section; used in slenderness ratio because buckling occurs about the weaker axis.

Note

This is the fundamental timber column design equation; if P > P_{allow}, the column fails in buckling.

Equation

P_{allow} = F′_c × A = (F_c* × C_P) × A

Conditions

P_{allow} is the maximum axial load the column can support; must not exceed actual load P.

Formula

f_c⊥ = P / (A bearing) ≤ F′_c⊥

Meaning

f_c⊥ = actual bearing stress perpendicular to grain (MPa); P = concentrated load (N); A_bearing = actual contact area perpendicular to grain (mm²); F′_c⊥ = adjusted allowable.

Watch Out

Ensure you use the actual bearing area (not the full cross-section). If a beam rests on a 100 mm wide ledge, use only that 100 mm length in calculating A_bearing. Also, F_c⊥ is much weaker than F_c (parallel), typically 1/3 to 1/2 the value.

When To Use

Checking timber members resting on ledges, bearing plates, or other supports where load is perpendicular to grain.

Formula

F′_c⊥ = F_c⊥ × C_D × C_M × C_t (typically; no C_F, C_P, or C_L for perpendicular compression)

Meaning

F_c⊥ = reference value perpendicular to grain; adjusted by D, M, t only (no size or stability factors).

Watch Out

Some codes include a bearing area factor C_b (length of bearing divided by depth of member); NSCP may or may not include this — check the edition.

When To Use

Compute the allowable perpendicular compression stress; usually fewer adjustments than parallel.

Value

2–4 MPa (varies by species)

Symbol

F_c⊥

Quantity

Typical reference bearing stress F_c⊥ (perpendicular to grain)

Term

Bearing Perpendicular to Grain

Example

A 150 × 300 mm beam (300 mm depth) resting on a 100 mm ledge has bearing area = 150 × 100 = 15,000 mm².

Definition

Load applied normal to the wood grain direction (e.g., a beam pressing down on a ledger); wood is much weaker here than parallel.

Formula

f_t = P / A_net ≤ F′_t

Meaning

f_t = actual tensile stress (MPa); P = axial tensile load (N); A_net = net cross-sectional area accounting for holes/knots (mm²); F′_t = adjusted allowable tension stress.

Watch Out

USE THE NET AREA, not gross. Any bolt holes, knots, or defects reduce A. Also, ensure the net section is at the critical plane (where a hole or defect is largest or where the grain runs out).

When To Use

For tied members, tension chords of trusses, or strut members in tension.

Formula

F′_t = F_t × C_D × C_M × C_t × C_g

Meaning

F_t = reference tension stress; C_g = grain angle factor (reduces strength if grain is not parallel to load direction).

Watch Out

Timber members in tension must have a relatively straight grain (C_g = 1 for 0° angle). If grain is at an angle, C_g < 1 and tension capacity drops. This is less commonly tested than bending and compression but can appear on exams.

When To Use

Compute the adjusted tension allowable; C_g is unique to tension.

Value

5–10 MPa (varies by species and grade)

Symbol

F_t

Quantity

Typical reference tension stress F_t (parallel to grain)

Term

Net Section (Tension)

Example

A 100 × 300 mm beam with a 25 mm diameter bolt hole has A_net = 100 × 300 − 100 × 25 = 27,500 mm².

Definition

Cross-sectional area minus the area of holes, knots, or defects in the critical failure plane.

Value

1.0

Symbol

C_M,dry

Quantity

C_M (dry service < 12% moisture)

Value

0.80–0.97 (varies by stress type)

Symbol

C_M,wet

Quantity

C_M (wet service > 20% moisture)

Value

1.0

Symbol

C_t,normal

Quantity

C_t (normal temperature, < 100°C)

Value

1.0

Symbol

C_r,single

Quantity

C_r (single member or wide spacing)

Value

1.15

Symbol

C_r,repetitive

Quantity

C_r (three+ members, ≤600 mm spacing)

Term

C_D — Load Duration Factor

Example

Permanent load: C_D = 0.9 (10% reduction). Wind/seismic: C_D = 1.6 (60% increase). Impact: C_D = 2.0.

Definition

Multiplier reflecting how long the load acts; wood is stronger for brief loads, weaker for permanent loads.

Term

C_M — Wet Service Factor

Example

C_M = 1.0 for dry service (< 12% MC). C_M = 0.85–0.97 for wet service (e.g., outdoor, always damp).

Definition

Reduction for moisture content > 20% (equilibrium moisture during service).

Term

C_t — Temperature Factor

Example

C_t = 1.0 for normal temperature. C_t = 0.8 at 150°C (not common in structural design).

Definition

Reduction for service temperatures > 100°C.

Term

C_F — Size Factor (Sawn Lumber)

Example

C_F = 1.5 for shallow beams. C_F = 1.0 for large glulam beams. Decreases as member size increases.

Definition

Upward adjustment for small cross-sections (smaller members often stronger per unit area).

Term

C_L — Beam Lateral Stability Factor

Example

C_L = 1.0 fully braced. C_L = 0.6–0.9 for longer unsupported spans (varies with slenderness).

Definition

Reduction for lateral-torsional buckling of unsupported compression flange.

Term

C_P — Column Stability Factor

Example

C_P = 1.0 for short, stocky columns. C_P = 0.3–0.7 for slender columns with high ℓ_e/d.

Definition

Reduction for axial column buckling (Euler-type).

Term

C_r — Repetitive Member Factor

Example

Floor joists 400 mm on center: C_r = 1.15. Single beam or joists > 600 mm apart: C_r = 1.0.

Definition

15% upward adjustment when three or more members are spaced ≤ 600 mm and act together (load sharing).

Term

C_b — Bearing Area Factor

Example

C_b = 1.25 + (0.375 × L_b / d) where L_b = bearing length, d = member depth (if NSCP includes it).

Definition

May apply to compression perpendicular to grain based on bearing length vs. member depth (NSCP variation).

Formula

E′ = E × C_M × C_t (adjusted modulus for deflection and buckling)

Meaning

E = reference modulus (from tables); E′ = adjusted for wet service and temperature; no other modulus factors.

Watch Out

For buckling (C_P formula), use E′_{min} (typically E × 0.66), not the average modulus. For deflection, use the standard adjusted E′. Do not apply C_D to modulus — only for stress.

When To Use

Computing deflection (Δ = 5wL⁴ / 384 E′I) or Euler buckling stress (F_cE = 0.822 E′_min / (ℓ_e/d)²).

Value

9650 MPa

Symbol

E_SouthernPine

Quantity

Reference modulus E (Southern Pine, typical)

Value

7550 MPa

Symbol

E_SPF

Quantity

Reference modulus E (Spruce-Pine-Fir, typical)

Value

0.66 × mean E

Symbol

E_min

Quantity

Minimum modulus (rule of thumb)

Term

Modulus of Elasticity (E)

Example

Southern Pine: E ≈ 9650 MPa. Spruce-Pine-Fir: E ≈ 7550 MPa. Varies by species.

Definition

Stiffness measure; higher E means less deflection and less buckling for a given slenderness.

Term

Minimum Modulus (E_min)

Example

For a species with mean E = 10,000 MPa, E_min ≈ 6,600 MPa is used in the F_cE formula.

Definition

Conservative estimate (often 0.66 × mean E) used in Euler buckling formula to be safe; accounts for material variability.

Note

This is the critical stress above which the column buckles elastically (wood analog of Euler buckling in steel).

Equation

F_cE = (0.822 × E′_min) / (ℓ_e / d)²

Conditions

Euler buckling stress for timber columns; E′_min is adjusted minimum modulus; ℓ_e/d is slenderness ratio.

Term

Design vs. Check

Example

Design: M = 20 kN·m, F′_b = 12 MPa → find S_req = M / F′_b = 1.67 × 10⁶ mm³ → choose beam. Check: given 150×300 beam, verify f_b ≤ F′_b.

Definition

Design: given load and limits, find required size/section. Check: given section and load, verify it is safe.

Values

• f_b = M / S
• F_b
• C_D, C_M, C_t, C_F, C_L, C_r (interaction per NSCP)
• 8–25 (varies by grade, species, adjustments)

Property

Bending (Flexure)

Values

• f_v = 3V / 2A
• F_v
• C_D, C_M, C_t, C_H (notch reduction if applicable)
• 0.5–2.0

Property

Shear (Parallel to Grain)

Values

• f_c = P / A
• F_c
• C_D, C_M, C_t, C_F (then × C_P for buckling)
• 5–12 (before C_P); ↓ by C_P for slender columns

Property

Compression Parallel

Values

• f_c⊥ = P / A_bearing
• F_c⊥
• C_D, C_M, C_t (rarely C_F; may include C_b bearing length factor)
• 1–4

Property

Compression Perpendicular

Values

• f_t = P / A_net
• F_t
• C_D, C_M, C_t, C_g (grain angle factor)
• 5–10

Property

Tension Parallel

Values

• 1.0
• Free to rotate at both ends; classic column case.
• Two columns sitting on bases with pin connections.

Property

Both ends pinned (standard)

Values

• 0.5
• Moments resist rotation; much stiffer, reduces effective length.
• Columns rigidly embedded in concrete or welded to rigid frames.

Property

Both ends fixed

Values

• 0.7
• Intermediate case; one end is rigid, one free to rotate.
• Column welded at base but pin-jointed at top.

Property

One end fixed, one pinned

Values

• 2.0
• One end is rigid, the other is unsupported; worst case, large effective length.
• Cantilever posts or flagpoles.

Property

One end fixed, one free (cantilever)

Values

• 0.9
• > 10 years continuously
• Dead load, long-term sustained load; reduces allowable because wood degrades under prolonged stress.

Property

Permanent Load

Values

• 1.0
• Standard design load (baseline)
• Normal floor live load, typical design case (baseline for reference values).

Property

Standard (10-year)

Values

• 1.15
• ~ 2 months (construction phase, temporary loads)
• Formwork load during construction; higher allowable due to short duration.

Property

2-month Load

Values

• 1.25
• ~ 1 week (short-term)
• Emergency repairs, temporary barriers; higher allowable.

Property

7-day Load

Values

• 1.6
• Brief (seconds to minutes)
• Wind gusts, earthquake; wood is temporarily stronger in short bursts.

Property

Wind or Seismic Load

Values

• 2.0
• Instantaneous (fraction of second)
• Dropped object, sudden impact; maximum allowable boost; rarely used in structural design.

Property

Impact Load

Values

• 1.3–1.5
• 1.1–1.2
• Smaller sections get size bonus; higher strength per unit area.

Property

2×4, 2×6 (small sawn)

Values

• 1.1–1.2
• 1.05–1.1
• Medium sections; intermediate C_F.

Property

2×8 to 2×10

Values

• 1.0
• 1.0
• Large sawn lumber; no size bonus (or very small).

Property

2×12 and larger sawn

Values

• 1.0 (approx.)
• 1.0 (approx.)
• Engineered product; size factor typically 1.0 for most glulam beams.

Property

Glued-laminated (Glulam)

Values

• < 12%
• 1.0
• 1.0
• 1.0

Property

Dry Service (typical indoor)

Values

• > 20% (equilibrium during service)
• 0.8–0.85
• 0.67–0.8
• 0.80–0.97 (varies)

Property

Wet Service (outdoor, damp)

Values

• —
• 200
• 200
• 1.0
• Standard shear; no amplification.

Property

0 (no notch)

Values

• 25
• 200
• 175
• 1.14
• Minor notch; 14% increase in shear stress.

Property

25

Values

• 50
• 200
• 150
• 1.33
• Moderate notch; 33% increase — significant.

Property

50

Values

• 75
• 200
• 125
• 1.6
• Deep notch; 60% increase — critical risk of shear failure.

Property

75