Steel Beam and Column Design: Method

Steel Beam and Column Design: Method, Conventions and the Specification Equations Used

What the tool does

The tool checks one doubly symmetric steel I-shape (a rolled W-shape or three welded plates) as a beam, a column, a tension member or a beam-column: an axial force P, a strong-axis moment M_x, a weak-axis moment M_y and a shear V are compared with the strengths of ANSI/AISC 360-16, each provision written out as mathematics with its section number, and the result is reported as the design strength φR_n (LRFD) or the allowable strength R_n/Ω (ASD) beside the required strength, with the demand-to-capacity ratio and a margin of safety for every limit state and for the interaction of axial force and bending. The Specification and the Steel Construction Manual are cited; none of their tables is reproduced. Every strength on the page is computed from the section dimensions, the steel properties and the lengths entered.

Conventions

The strong axis is x (the axis of the flanges, bending in the plane of the web) and the weak axis is y. Required strengths are magnitudes: P is compression or tension by the radio, moments and shear are entered positive. Under LRFD the required strengths come from the factored load combinations (ASCE 7 Section 2.3, UFC 3-301-01); under ASD from the service-load combinations (ASCE 7 Section 2.4). The tool does not apply load factors: enter P_u, M_u, V_u for LRFD and P_a, M_a, V_a for ASD. Second-order effects (the B₁ and B₂ amplifiers of Appendix 8 or a second-order analysis) must already be in the entered moments. Lengths: L_x and L_y are the unbraced lengths for buckling about each axis, L_z the length between points held against twist, L_b the length between points that brace the compression flange against lateral movement or the section against twist.

Section properties

A W-shape chosen from the list fills five dimensions of fact (d, b_f, t_f, t_w and k, the distance from the outer face of the flange to the web toe of the fillet, as the Manual's k_des); the designations are those of the AISC Manual Table 1-1. The properties are then the closed forms of three rectangles with the fillets ignored:

A = 2 b_f t_f + (d − 2t_f) t_w,   I_x = [b_f d³ − (b_f − t_w)(d − 2t_f)³]/12,   I_y = [2 t_f b_f³ + (d − 2t_f) t_w³]/12,

S_x = 2I_x/d,   S_y = 2I_y/b_f,   Z_x = b_f t_f (d − t_f) + t_w (d − 2t_f)²/4,   Z_y = t_f b_f²/2 + (d − 2t_f) t_w²/4,

J = [2 b_f t_f³ + (d − 2t_f) t_w³]/3,   h_o = d − t_f,   C_w = t_f b_f³ h_o²/24,   r_ts² = √(I_y C_w)/S_x.

For the 40 rolled shapes in the list these forms sit within about 1–3 % of the Manual's A, I, S and Z (below them, because the fillets add material at the web-flange junctions) and 5–20 % below the Manual's J, which the fillets stiffen most. For the W12x26 of the default case the plate forms give A 7.56 in² (Manual 7.65), I_x 201 in⁴ (204), Z_x 36.7 in³ (37.2), J 0.284 in⁴ (0.30) and C_w 605 in⁶ (607); the strong-axis strength at L_b = 10 ft comes out 1.3 % below the value from the Manual's properties. When the difference matters, enter the Manual's values through the typed-properties option, or load a saved section drawn in the Section Properties tool (which computes J and C_w exactly for the drawn outline). The slenderness ratios are b_f/2t_f for the flange and h/t_w for the web with h = d − 2k (the clear distance between the fillets) when k is given, else d − 2t_f.

Classification (B4)

Each element is compared with the limits of Table B4.1a (compression: nonslender when λ ≤ λ_r) and Table B4.1b (flexure: compact when λ ≤ λ_p, noncompact up to λ_r, slender beyond). For the flanges of a rolled I-shape (unstiffened, case 1 and case 10) λ_r = 0.56 √(E/F_y) in compression, λ_p = 0.38 √(E/F_y) and λ_r = 1.0 √(E/F_y) in flexure; for the web (stiffened, case 5 and case 15) λ_r = 1.49 √(E/F_y) in compression, λ_p = 3.76 √(E/F_y) and λ_r = 5.70 √(E/F_y) in flexure. At F_y = 50 ksi these are 13.5, 9.15 and 24.1 for the flange and 35.9, 90.6 and 137 for the web. The built-up flange limit of case 11 (0.95 √(k_c E/F_L)) is not used: a welded section with a flange between 1.0 √(E/F_y) and that limit is treated as slender, on the safe side.

Compression (E3, E4, E7)

For each axis the effective length is L_c = K L and the elastic buckling stress F_e = π² E/(L_c/r)² (E3-4). The one column curve gives F_cr = 0.658^F_y/F_e F_y when L_c/r ≤ 4.71 √(E/F_y), which is the same as F_y/F_e ≤ 2.25 (E3-2, inelastic buckling with residual stresses and initial crookedness), and F_cr = 0.877 F_e beyond it (E3-3, elastic buckling reduced for crookedness). Torsional buckling of the doubly symmetric member uses F_e = [π² E C_w/L_cz² + G J]/(I_x + I_y) (E4-2) on the same curve; it rarely governs a W-shape but can for a short, wide-flange built-up section. The lowest F_cr of the three modes gives P_n = F_cr A_g (E3-1) with φ_c = 0.90 and Ω_c = 1.67.

When an element is slender in compression, the 2016 effective-width method of E7 replaces the Q factors of earlier editions: for every element with λ > λ_r √(F_y/F_cr), b_e = b [1 − c₁ √(F_el/F_cr)] √(F_el/F_cr) (E7-3) with F_el = (c₂ λ_r/λ)² F_y (E7-5), c₁ = 0.22 and c₂ = 1.49 for the flange outstands and 0.18 and 1.31 for the web (Table E7.1); A_e is the gross area less (b − b_e) t of each reduced element and P_n = F_cr A_e (E7-1). The F_cr in these equations is the gross-section value of E3 or E4. The effective length factors of the six idealised cases of Commentary Table C-A-7.1 are offered as a select that fills K with the recommended design value (0.65, 0.80, 1.2, 1.0, 2.1, 2.0); K stays editable for the alignment-chart or analysis value. The Specification's recommended slenderness limit of 200 (E2) is reported, not enforced.

Tension (D2)

Tensile yielding on the gross section P_n = F_y A_g (D2-1, φ = 0.90, Ω = 1.67) and tensile rupture on the effective net area P_n = F_u A_e (D2-2, φ = 0.75, Ω = 2.00) with A_e = U A_n (D3-1); the net area at the connection and the shear-lag factor of Table D3.1 are entered (the AISC Connection Checks tool computes them for a bolted end). The lesser available strength is the P_c of the interaction.

Strong-axis flexure (F2, F3)

For a compact section the nominal strength is the lesser of the plastic moment M_p = F_y Z_x (F2-1) and the lateral-torsional buckling strength of the unbraced length: M_n = M_p for L_b ≤ L_p = 1.76 r_y √(E/F_y) (F2-5); M_n = C_b [M_p − (M_p − 0.7 F_y S_x)(L_b − L_p)/(L_r − L_p)] ≤ M_p for L_p < L_b ≤ L_r (F2-2); and M_n = F_cr S_x ≤ M_p beyond L_r (F2-3), where F_cr = C_b π² E/(L_b/r_ts)² √[1 + 0.078 (J c/(S_x h_o)) (L_b/r_ts)²] (F2-4), L_r = 1.95 r_ts (E/(0.7F_y)) √[J c/(S_x h_o) + √((J c/(S_x h_o))² + 6.76 (0.7F_y/E)²)] (F2-6) and c = 1 for the doubly symmetric I. The moment-gradient factor is entered or computed from the quarter-point moments, C_b = 12.5 M_max/(2.5 M_max + 3 M_A + 4 M_B + 3 M_C) (F1-1). The elastic part of this curve is Timoshenko's critical moment for a simply supported beam under uniform moment, M_cr = (π/L_b) √(E I_y G J + π² E² I_y C_w/L_b²), rewritten in r_ts with G = E/2.6 (the Lateral-Torsional Buckling tool in the Structures area draws both).

A noncompact flange adds the flange local buckling strength M_n = M_p − (M_p − 0.7 F_y S_x)(λ − λ_pf)/(λ_rf − λ_pf) (F3-1); a slender flange M_n = 0.9 E k_c S_x/λ² with k_c = 4/√(h/t_w) between 0.35 and 0.76 (F3-2). The lesser of lateral-torsional and flange local buckling is M_n, with φ_b = 0.90 and Ω_b = 1.67. F2 and F3 require a compact web; a noncompact or slender web (F4, F5, with the web plastification and bend-buckling factors) is flagged and not built, so the strength shown for such a section is an upper bound.

Weak-axis flexure (F6)

M_n = M_p = F_y Z_y ≤ 1.6 F_y S_y (F6-1; the cap limits the shape factor that may be used), reduced for a noncompact flange by F6-2 (the same linear interpolation as F3-1 with S_y) or a slender flange by F6-3 with F_cr = 0.69 E/(b_f/2t_f)² (F6-4). There is no lateral-torsional buckling about the weak axis.

Shear (G2.1)

V_n = 0.6 F_y A_w C_v1 on A_w = d t_w (G2-1). For the web of a rolled I-shape with h/t_w ≤ 2.24 √(E/F_y) the web yields in shear before buckling, C_v1 = 1.0 and the Specification allows φ_v = 1.00 and Ω_v = 1.50 (G2.1(a)). For other webs, C_v1 = 1.0 up to h/t_w = 1.10 √(k_v E/F_y) and 1.10 √(k_v E/F_y)/(h/t_w) beyond it (G2-3, G2-4) with k_v = 5.34 for an unstiffened web, φ_v = 0.90 and Ω_v = 1.67 (G2.1(b)). Tension-field action with transverse stiffeners (G2.2) is not offered.

Interaction (H1)

With P_c the available axial strength (compression or tension) and M_cx, M_cy the available flexural strengths: P_r/P_c + (8/9)(M_rx/M_cx + M_ry/M_cy) ≤ 1.0 when P_r/P_c ≥ 0.2 (H1-1a) and P_r/(2P_c) + (M_rx/M_cx + M_ry/M_cy) ≤ 1.0 otherwise (H1-1b). The ratios are printed, the interaction sum is treated as a demand-to-capacity ratio of 1.0 and its margin is 1/R − 1. The option of H1.2 to raise the lateral-torsional buckling strength by the tension is not used (conservative).

Deflection

A serviceability comparison at service load: Δ = 5 w L⁴/(384 E I_x) for a simple span under a uniform service load against L/n with n entered (360 for plastered floors, 240 unplastered, 180 roofs are the usual values; the governing building code sets them). No resistance factor applies.

Charts and figures

Figure 1 is the section with its plate dimensions. Figure 2 draws the nominal strong-axis strength against the unbraced length for the section, steel and C_b entered (the plateau at M_p, the straight inelastic line to L_r, the elastic curve beyond), the design or allowable strength dashed and the required moment as a point: a point above the dashed curve fails. Figure 3 is the column curve F_cr/F_y against KL/r for the entered steel with the Euler curve, the 4.71 √(E/F_y) split and the member's slenderness about each axis.

Checks

The test script reproduces the printed values of the Purdue CE 470 lecture notes (Example 4.1 Euler loads of a W12x50; Example 4.2 W14x74 A36 at 20 ft: KL/r 96.77, F_e 30.56 ksi, F_cr 21.99 ksi, φP_n 431.4 kips; Examples 4.3 and 4.4 slenderness limits and the W14x132 column; Example 5.1 plastic girder Z_x 189.26 in³; Examples 5.4 and 5.6 beam strengths read from the Manual charts (328, 312, 335, 376 kip-ft) recovered from the F2 and F3 arithmetic within 0.5 %; Example 5.2 deflections; Examples 6.1 and 6.2 beam-columns: φ_cP_n 685 and 663 kips, φ_bM_n 345 and 340 kip-ft, φ_bM_p 356 kip-ft of the noncompact-flange W12x65, interaction 0.819, 0.95 and 0.966), the FHWA Steel Bridge Design Handbook plate-girder section (Table 3: A 75.25 in², I 62,658 in⁴, S 1,581 and 1,973 in³, r_t 3.90 in, L_p 7.83 ft and L_r 29.39 ft by the AASHTO forms), the Lateral-Torsional Buckling tool's W12x26 values, and every limit state of twenty seeded sections against an independent numpy implementation.

Not covered

Singly symmetric sections, channels, tees and angles; noncompact or slender webs in flexure (F4, F5); built-up flange limits with k_c in B4; stiffened webs and tension-field action; web local yielding and crippling under concentrated loads (J10); torsion and combined torsion (H3); the B₁ / B₂ amplification and frame stability (Chapter C, Appendix 7 and 8); flange and web holes in flexure (F13); composite members; connections (the AISC Connection Checks tool).