Tapered Steel Beam Calculator (AISC DG 25) – Accurate LTB, Shear & Deflection Analysis

The taper ratio measures how much the web depth varies relative to the shallow end:

\[\gamma = \frac{d_2 - d_1}{d_1}\]

DG 25 requires γ ≤ 0.268 for direct application of the design guide provisions without additional rigorous analysis. The web depth at any station along the span is:

\[d(x) = d_1 + (d_2 - d_1)\cdot\frac{x}{L}\]

For a doubly-symmetric welded I-section with web depth d(x), flange width b_f, flange thickness t_f, and web thickness t_w:

\[A(x) = 2\,b_f\,t_f + t_w\,d(x)\] \[I_x(x) = 2\left[b_f\,t_f\left(\frac{d(x)}{2} + \frac{t_f}{2}\right)^2 + \frac{b_f\,t_f^3}{12}\right] + \frac{t_w\,d(x)^3}{12}\] \[S_x(x) = \frac{I_x(x)}{d(x)/2 + t_f}\] \[Z_x(x) = b_f\,t_f\left(\frac{d(x)}{2} + \frac{t_f}{2}\right)\times 2 + \frac{t_w\,d(x)^2}{4}\]

Warping constant (doubly symmetric):

\[C_w(x) = \frac{I_y\,h_o^2}{4} \approx \frac{t_f\,b_f^3}{24}\cdot h_o^2\]

St. Venant torsion constant:

\[J(x) = \frac{1}{3}\left(2\,b_f\,t_f^3 + d(x)\,t_w^3\right)\]

LRFD governing combinations (ASCE 7-22 Section 2.3):

\[\begin{aligned} U_1 &= 1.4D \\ U_2 &= 1.2D + 1.6L + 0.5(L_r \text{ or } S) \\ U_3 &= 1.2D + 1.6(L_r \text{ or } S) + L \\ U_4 &= 1.2D + 1.0W + L + 0.5S \\ U_5 &= 0.9D + 1.0W \end{aligned}\]

ASD governing combinations (ASCE 7-22 Section 2.4):

\[\begin{aligned} D + L,\quad D + S,\quad D + L_r,\quad D + 0.6W + L + 0.5S \end{aligned}\]

For a simply supported beam of span L with uniform factored load w_u:

\[R_A = R_B = \frac{w_u\,L}{2}\] \[V(x) = R_A - w_u\,x\] \[M(x) = R_A\,x - \frac{w_u\,x^2}{2}\] \[M_{\max} = \frac{w_u\,L^2}{8} \quad \text{at }x = L/2\]

C_b accounts for the non-uniform moment within the unbraced length:

\[C_b = \frac{12.5\,M_{\max}}{2.5\,M_{\max} + 3\,M_A + 4\,M_B + 3\,M_C}\]

Where M_A, M_B, M_C are absolute moments at the quarter, midpoint, and three-quarter points of the unbraced length, and M_max is the maximum moment within the segment.

C_b = 1.0 for uniform moment (most conservative). For simply supported beam with UDL, C_b = 1.14.

Properties evaluated at the critical section within the unbraced length per DG 25 (2nd edition). Limiting unbraced lengths:

\[L_p = 1.76\,r_y\,\sqrt{\frac{E}{F_y}}\] \[L_r = 1.95\,r_{ts}\,\frac{E}{0.7F_y}\sqrt{\frac{J\,c}{S_x\,h_o} + \sqrt{\left(\frac{J\,c}{S_x\,h_o}\right)^2 + 6.76\left(\frac{0.7F_y}{E}\right)^2}}\]

Plastic range (L_b ≤ L_p): M_n = M_p = F_y · Z_x

Inelastic LTB (L_p < L_b ≤ L_r):

\[M_n = C_b\left[M_p - (M_p - 0.7F_y S_x)\frac{L_b - L_p}{L_r - L_p}\right] \le M_p\]

Elastic LTB (L_b > L_r):

\[F_{cr} = \frac{C_b\,\pi^2 E}{\left(L_b/r_{ts}\right)^2}\sqrt{1 + 0.078\,\frac{J\,c}{S_x\,h_o}\left(\frac{L_b}{r_{ts}}\right)^2}\] \[M_n = F_{cr}\,S_x \le M_p\]

LRFD design moment: φ_bM_n, where φ_b = 0.90. ASD: M_n/Ω_b, Ω_b = 1.67.

Flange compactness classification:

\[\lambda_f = \frac{b_f}{2\,t_f} \le \lambda_{pf} = 0.38\sqrt{\frac{E}{F_y}} \quad \text{(compact)}\] \[\lambda_{rf} = 1.0\sqrt{\frac{E}{F_y}} \quad \text{(noncompact limit)}\]

Web compactness limit:

\[\lambda_{pw} = 3.76\sqrt{\frac{E}{F_y}}, \qquad \lambda_{rw} = 5.70\sqrt{\frac{E}{F_y}}\]

\[A_w(x) = d(x)\,t_w\] \[\phi_v V_n = \phi_v \cdot 0.6 F_y A_w C_{v1}\]

Where φ_v = 1.0 for h/t_w ≤ 2.24√(E/F_y), else φ_v = 0.90. C_v1 = 1.0 for compact webs (h/t_w ≤ 2.24√(E/F_y)).

Since I_x varies along the span, deflection cannot be computed with standard closed-form formulas. This calculator uses numerical double integration (Simpson's rule at n stations):

\[\delta(x) = \iint \frac{M(x)}{E\,I(x)}\,dx^2\]

Equivalently, the curvature κ(x) = M(x) / [E I(x)] is integrated twice using the moment-area / conjugate beam method via numerical quadrature.

Maximum deflection limit check:

\[\delta_{\max} \le \frac{L}{N} \quad (N = 360 \text{ for floors, } 240 \text{ for roofs})\]

Total weight by trapezoidal integration over varying cross-sectional area:

\[W = \rho \cdot L \cdot \frac{A_1 + A_2}{2}\]

Where ρ = 490 lb/ft³ = 0.2836 lb/in³ and A₁, A₂ are areas at shallow and deep ends. For more accuracy, Simpson's rule is used over n stations.

Equivalent prismatic W-section weight uses the average section:

\[W_{\text{prismatic}} = \rho \cdot L \cdot A_{\text{avg}}\]

Savings: ≈ 25–40% for optimally tapered members (AISC DG 25 commentary).

When axial load is present (P_u > 0):

\[\text{If } \frac{P_r}{P_c} \ge 0.2:\quad \frac{P_r}{P_c} + \frac{8}{9}\left(\frac{M_{rx}}{M_{cx}}\right) \le 1.0 \quad \text{(H1-1a)}\] \[\text{If } \frac{P_r}{P_c} < 0.2:\quad \frac{P_r}{2P_c} + \left(\frac{M_{rx}}{M_{cx}}\right) \le 1.0 \quad \text{(H1-1b)}\]

Where P_c = φ_cP_n (LRFD compression capacity), M_cx = φ_bM_n.