Timber Beam and Column: Method

Timber Beam and Column

What the tool does

The tool checks a rectangular sawn-lumber or glulam beam and a rectangular column the way an allowable-stress design (ASD) calculation does: the reference design values of the species and grade are adjusted by the factors of the National Design Specification (NDS), the member forces come from closed-form statics, and each stress is compared with its adjusted allowable as a plain ratio f/F′ ≤ 1. The mechanics are those of the USDA Forest Products Laboratory's Wood Handbook (FPL-GTR-190, chapter 9): the bending and shear stress equations 9-13 and 9-14, Euler's column formula 9-25, the FPL fourth-power parabola 9-26 of Newlin and Gahagan, Ylinen's column equation 9-28, the lateral-torsional buckling equations 9-32 and 9-33 with Table 9-2, and the interaction of buckling modes, equations 9-35 to 9-40. The NDS is the design standard that writes those mechanics as adjustment factors; the tool uses its factor names and the few numerical facts the standard states (the load-duration values, the 1.20 and 0.822 coefficients, the constant c, the effective-length rules, the size and bearing-area formulas, the slenderness limits of 50), and cites it for them. The NDS Supplement's tables of reference design values are copyrighted and are not shipped: the analyst enters the values for the species, grade and size class; the form's defaults are illustrative.

Conventions and units

b is the thickness (width) of the member and d its depth; the beam bends about its strong axis (loads act in the plane of d) and the column's eccentricity e₁ and applied moment M₁ act in the plane of d. Nominal sizes are dressed to the actual sizes of PS 20: nominal 2 to 4 in → ½ in less (1.5, 2.5, 3.5 in), dimension-lumber widths of 8 in and more → ¾ in less (7.25, 9.25, 11.25, 13.25 in), timbers 5 in and thicker → ½ in less in both dimensions (5.5, 7.5, 9.5, 11.5 in); glulam, rough or green sizes are entered as actual dimensions. The beam is a simple span; the uniform load and the point loads act downward at the service level (allowable-stress design, no load factors); the column's axial load is compression. Lengths, loads, stresses and moduli pass through the unit menus; the nominal sizes, the adjustment factors and the eccentricity term are plain numbers. The whole analysis is in SI base units inside and the page prints the display units.

Reference design values and where they come from

The Wood Handbook chapter 7 explains how a visually graded design value is derived: the clear-wood strength of the species (chapter 5, Table 5-3; ASTM D 2555) at its 5 % exclusion limit is multiplied by the strength ratio of the grade (the ratio of the strength of a piece with the largest permitted knot or slope of grain to the strength without it, ASTM D 245) and divided by a factor for ten-year load duration and safety, 2.1 for bending (1.9 for compression parallel to grain). The chapter's own example: a 5 % exclusion value of 7,000 lb/in² in bending with a strength ratio of 40 % gives 2,800 lb/in² for the grade, and 2,800/2.1 = 1,333 lb/in² of the kind a design table lists. The modulus of elasticity is the clear-wood average times a quality factor (1.8 × 10⁶ × 0.8 = 1.44 × 10⁶ lb/in² in the chapter's example). The handbook's Table 5-3b lists clear Coast Douglas-fir at 12 % moisture: modulus of rupture 12,400, modulus of elasticity 1.95 × 10⁶, compression parallel 7,230, compression perpendicular 800 and shear 1,130 lb/in². The form's defaults (F_b 1,000, F_v 180, F_c 1,500, F_c⊥ 625, E 1.7 × 10⁶, E_min 0.62 × 10⁶ lb/in²) are illustrative values of that order for a Douglas-fir grade; the design values of record are in the NDS Supplement (Tables 4A to 5D) and the analyst enters them.

E_min is the modulus used for stability: a 5 % exclusion value of E with a 1.66 safety factor, E_min = 1.03 E (1 − 1.645 COV_E)/1.66 (NDS Appendix D), which is 0.365 E for the 0.25 coefficient of variation of visually graded sawn lumber and 0.528 E for the 0.10 of glulam (the Timber Bridges manual quotes the same coefficients of variation). A blank E_min is computed from E and the material's COV_E.

Adjustment factors

Every strength is adjusted as F′ = F × (product of the applicable factors) and the page prints each product with its numbers. C_D is the load-duration factor of the shortest-duration load in the combination (0.9 permanent, 1.0 ten years, 1.15 two months, 1.25 seven days, 1.6 ten minutes, 2.0 impact; the Wood Handbook chapter 7 and its Figure 7-9 describe the same ten-year basis, and the 1990 Timber Bridges manual prints the pre-1991 set with 1.33 for wind); it applies to F_b, F_v and F_c, not to F_c⊥, E or E_min. C_M (wet service) and C_t (temperature; the Wood Handbook Table 7-9 values fill the fields from the temperature select) are entered separately for the strengths and for the moduli; C_i (incising) likewise. C_F is the size factor: entered for dimension lumber (NDS Supplement Table 4A), or computed as (12/d)^1/9 for sawn timbers deeper than 12 in when the box is checked (NDS 4.3.6.2; the Wood Handbook chapter 7 gives the same ninth-power depth rule of ASTM D 245). C_fu is the flat-use factor on F_b, C_r = 1.15 the repetitive-member factor (checkbox). C_b = (l_b + 0.375)/l_b (l_b in inches) is the bearing-area factor for a bearing shorter than 6 in and at least 3 in from the member end (NDS 3.10.4). C_L and C_P are the stability factors computed below.

Beam: forces and deflection

The simple span under a uniform load w and point loads P_k at a_k is solved by superposing the textbook closed forms: M(x) = w x (L − x)/2 + Σ P b x/L (left of a) or P a (L − x)/L (right of a); V(x) = w (L/2 − x) + Σ P b/L or − P a/L; δ(x) = w x (L³ − 2 L x² + x³)/(24 EI) + Σ P b x (L² − b² − x²)/(6 L EI) (left of a) or P a (L − x)(2 L x − x² − a²)/(6 L EI) (right of a), with b = L − a. The maximum moment and deflection are located by a scan refined by golden-section search; the Wood Handbook Table 9-1 constants (5/384, 1/48, 11/768, 1/96) are the special cases and the test script checks them, together with the Beam Deflection tool's singularity-function engine and a numpy double integration. The deflection uses E′ = E C_M C_t C_i and omits the shear term of eq. 9-2 (k_s W L/(G A′), a few percent for a span-to-depth ratio of 15). The deflection limit is a span ratio chosen on the form (L/180, L/240, L/360, L/480); the Wood Handbook chapter 9 recommends designing dry timber under long-time load to about two-thirds of the long-time limit because of creep.

Beam: bending, shear and bearing

f_b = M/S with S = b d²/6 (eq. 9-13); f_v = 3V/(2 b d) (eq. 9-14, k = 3/2 for a rectangle), with V at the support or, when the box is checked, at a distance d from the support with the uniform load within d neglected (NDS 3.4.3.1); f_c⊥ = R/(b l_b) at the bearing. The allowables are F_b′ = F_b C_D C_M C_t C_L C_F C_fu C_i C_r, F_v′ = F_v C_D C_M C_t C_i and F_c⊥′ = F_c⊥ C_M C_t C_b.

Beam stability factor C_L

A beam whose compression edge is not continuously supported can buckle sideways and twist. The Wood Handbook (eqs 9-32, 9-33) gives the elastic critical stress of a rectangular beam, f_bcr = π² E/α² with the slenderness factor α = √(2π) (EI_y/GK)^1/4 √(L_e h)/b, L_e from its Table 9-2 (0.887 L/(1 − 2h/L) for a uniform load, 0.742 L/(1 − 2h/L) for a centre load, with the load at the top edge); the page prints it for comparison with G = E′/16 and the thin-strip torsion constant. The NDS writes the same physics as a stability factor on the ratio F_bE/F_b*: the beam slenderness R_B = √(l_e d/b²) (at most 50), the critical stress F_bE = 1.20 E′_min/R_B², F_b* = F_b with every factor except C_fu and C_L, and

C_L = [1 + F_bE/F_b*]/1.9 − √{[(1 + F_bE/F_b*)/1.9]² − (F_bE/F_b*)/0.95},

which is Ylinen's column equation (below) with c = 0.95: C_L → 1 as R_B → 0 and C_L → F_bE/F_b* (the Euler branch) for slender beams. The effective length l_e follows the NDS Table 3.3.3 single-span rules chosen on the form (uniform load: 2.06 l_u for l_u/d < 7, else 1.63 l_u + 3d; centre load 1.80 l_u / 1.37 l_u + 3d; any load 2.06 l_u / 1.63 l_u + 3d; equal end moments 1.84 l_u; the Timber Bridges manual prints the same forms as its eqs 5-4 to 5-7), the Wood Handbook Table 9-2 lengths, or an entered value. A beam with d ≤ b or with its compression edge continuously supported takes C_L = 1 (NDS 3.3.3.1). For comparison the page also prints the pre-1991 NDS form of the Timber Bridges manual: C_s = √(l_e d/b²), C_k = 0.811 √(E′/F_b*) (0.956 for glulam), C_L = 1 − (1/3)(C_s/C_k)⁴ for 10 < C_s ≤ C_k and F_b′ = 0.438 E′/C_s² beyond (eqs 5-3, 5-9 to 5-14).

Column stability factor C_P

A long column buckles at Euler's stress f_cr = π² E/(L/r)² (eq. 9-25, r = b/√12 for a rectangle); a short one crushes at F_c. Newlin and Gahagan (USDA Technical Bulletin 167, 1930) joined the two with the FPL fourth-power parabola, eq. 9-26, tangent to Euler's curve at two-thirds of the crushing strength (the mark 0.667 on the handbook's Figure 9-7). Ylinen (1956) instead fitted a nonlinear stress-strain law, ε = (F_c/E)[c f/F_c − (1 − c) ln(1 − f/F_c)] (eq. 9-27), and used its tangent modulus in Euler's formula, which gives the closed form

f_cr = (F_c + f_e)/(2c) − √{[(F_c + f_e)/(2c)]² − F_c f_e/c} (eq. 9-28),

where f_e is Euler's stress; c = 0.8 fits the solid-sawn and glulam test data of the handbook (0.9 for strand lumber), c = 0.97 reproduces the fourth-power curve. The test script solves the tangent-modulus equation from eq. 9-27 directly and recovers eq. 9-28 to 10⁻⁷. The NDS (3.7.1, since 1991; Zahn 1992) writes the same equation as C_P = f_cr/F_c* with F_c* = F_c with every factor except C_P, f_e replaced by F_cE = 0.822 E′_min/(l_e/d)² (the handbook's eq. 9-38; 0.822 = π²/12 for a rectangle), and c = 0.8 for sawn lumber, 0.85 for round timber poles, 0.9 for glulam and structural composite lumber. The effective length is l_e = K_e l with the buckling-length factor of the six idealised end conditions (NDS Appendix G, printed in the Timber Bridges manual Table 5-13: recommended 0.65, 0.80, 1.2, 1.0, 2.10, 2.4 against the theoretical 0.5, 0.7, 1.0, 1.0, 2.0, 2.0); the larger of K_e l₁/d and K_e l₂/b governs and may not exceed 50. The page prints Euler's and the fourth-power stresses on the mean E′ and the pre-1991 NDS column (K = 0.671 √(E′/F_c*), C_P = 1 − (1/3)((l_e/d)/K)⁴, F_c′ = 0.30 E′/(l_e/d)² beyond K; Timber Bridges eqs 5-28 to 5-31) for comparison, and draws the column curve with the design point.

Combined bending and axial load

For a column with an eccentric load or an applied moment the Wood Handbook's interaction of buckling modes (eq. 9-35, without the ponding term) is used:

(f_c/F_c′)² + [f_b1 + 6 (e₁/d) f_c (1.234 − 0.234 θ_c1)]/(θ_c1 F_b1′) + [f_b2 + 6 (e₂/b) f_c (1.234 − 0.234 θ_c2)]/(θ_c2 F_b2′) ≤ 1.0,

with the moment-magnification factors θ_c1 = 1 − f_c/F_cE1 (eq. 9-36) and θ_c2 = 1 − f_c/F_cE2 − [f_b1 + 6 (e₁/d) f_c]/F_bE (eq. 9-37), F_cE1 and F_cE2 the Euler stresses of the two axes (eqs 9-38, 9-39) and F_bE the beam-buckling stress of the column bent about its strong axis (the handbook's F″_b1, eq. 9-40, which as printed lacks a power of d₂; the tool uses the NDS form 1.20 E′_min/R_B² with l_e = 1.84 l₂, the equal-end-moment rule). The factor (1.234 − 0.234 θ) equals the NDS (1 + 0.234 f_c/F_cE) of its eccentric-column equation 15.4-1, and the whole expression is the NDS 3.9.2 interaction when e = 0. f_b1 = M₁/S is the stress of the applied strong-axis moment; F_b1′ = F_b* C_L and F_b2′ = F_b* C_fu. When f_c reaches F_cE of either axis the magnification is unbounded and the column fails by buckling, which the page says.

Reading the results

Each check is a line "f ≤ F′, ratio, satisfied / NOT satisfied"; the ratio is the fraction of the allowable used, so 1/ratio is the reserve on the service load (the NDS factors of safety are inside the reference values and the stability coefficients). The charts show the moment, shear and deflected shape of the beam, C_L against R_B and the column curve against l_e/d with the design points. Not covered: notched and tapered beams (Wood Handbook eqs 9-16 to 9-21), shear deflection, built-up and spaced columns (eq. 9-29), round columns (enter the equivalent square), glulam's volume factor C_V (enter it as C_F), biaxial applied moments, connections and masonry; a masonry tool is planned for this section.