Structural Engineering, ACI 318 Reinforced Concrete & Flexural Design
ACI 318 Whitney Stress Block Concrete Beam Moment Formula
Calculate Whitney Compression Block Depth a (mm), Net Tensile Rebar Strain epsilon_t, Nominal Flexural Capacity M_n (kN·m), and Tension-Controlled Design Moment phi*M_n (kN·m).
Why ACI 318 checks net tensile strain epsilon_t >= 0.005 BEFORE granting phi = 0.90 flexural capacity!
At ultimate crushing strain epsilon_cu = 0.003 in a singly reinforced concrete beam of width b (mm), effective rebar depth d (mm), cylinder strength f_c_prime (MPa), and tension steel area A_s (mm²) at yield f_y (MPa), Whitney’s Equivalent Rectangular Stress Block replaces nonlinear parabolic concrete stress with uniform 0.85 * f_c_prime over depth a = (A_s * f_y) / (0.85 * f_c_prime * b)! Neutral axis depth c = a / beta_1 (where beta_1 = 0.85 for f_c_prime <= 28 MPa, stepping down by 0.05 per 7 MPa to a 0.65 floor) governs Net Tensile Steel Strain epsilon_t = 0.003 * ((d - c) / c): ONLY when epsilon_t >= 0.005 (ductile tension-controlled failure with visible warning cracks before concrete crushes!) does ACI 318 grant full strength reduction phi = 0.90 on M_n = A_s * f_y * (d - a / 2)!
Try this 300 x 600 mm Floor Girder first—Beam_Width b = 300.0 mm (B2), Effective_Depth d = 520.0 mm (B3), Tension_Rebar_Area A_s = 1,800.0 mm² (B4), Concrete_Strength f_c_prime = 28.0 MPa (B5), and Steel_Yield f_y = 420.0 MPa (B6): Whitney Stress Block Depth a hits (1,800 * 420) / (0.85 * 28 * 300) = 105.88 mm (E2), Neutral Axis c = 105.88 / 0.85 = 124.57 mm gives ductile Tensile Strain epsilon_t = 0.0095 (>= 0.005!, E3), Nominal Moment M_n equals 353.10 kN·m (E4), and Design Capacity phi * M_n reaches 317.79 kN·m (E5)! In a shared Arte Sheets structural calculation workbook, a `suggester` can test higher-strength concrete in Suggest Mode while asking a hint-first AI prompt ("Why does raising f_c_prime from 28 MPa to 42 MPa barely increase phi*M_n even though concrete strength jumped 50%?") before the `owner` signs off.
| ACI 318 Flexural KPI (b=300 mm, d=520 mm, As=1,800 mm², fc'=28, fy=420) | Formula | Verified Output |
|---|---|---|
| Whitney Block Depth a (mm) | (As * fy) / (0.85 * fc_prime * b) | 105.88 mm (70.59 @ fc' = 42 MPa) |
| Net Tensile Rebar Strain eps_t | 0.003 * ((d - (a / beta_1)) / (a / beta_1)) | 0.0095 (0.0136 @ fc' = 42 MPa) |
| Nominal Moment Capacity Mn (kN·m) | (As * fy * (d - a / 2)) / 1e6 | 353.10 kN·m (366.44 @ fc' = 42 MPa) |
| Design Moment phi*Mn (kN·m) | 0.90 * Mn (when eps_t >= 0.005) | 317.79 kN·m (329.79 @ fc' = 42 MPa) |
ACI 318 Tension-Controlled Whitney Design Moment Formula
Validate b > 0, d > 50, A_s > 0, f_c_prime in [17, 70] MPa, f_y in [280, 550] MPa, and enforce ductile tension-controlled strain epsilon_t >= 0.005!
=LET(bw_m, B2, dp_m, B3, as_m, B4, fc_m, B5, fy_m, B6, IF(OR(bw_m <= 0, dp_m <= 50, as_m <= 0, fc_m < 17, fc_m > 70, fy_m < 280, fy_m > 550), "Invalid Beam Input", LET(ad_p, (as_m * fy_m) / (0.85 * fc_m * bw_m), b1_f, IF(fc_m <= 28, 0.85, MAX(0.65, 0.85 - 0.05 * ((fc_m - 28) / 7))), cd_p, ad_p / b1_f, ep_t, 0.003 * ((dp_m - cd_p) / cd_p), IF(ep_t < 0.005, "OVER-REINFORCED BRITTLE", ROUND((0.9 * as_m * fy_m * (dp_m - ad_p / 2)) / 1000000, 2)))))50% Stronger Concrete Lever-Arm Insensitivity Check (Set f_c_prime = 42.0 MPa)
Boost Concrete_Strength B5 by +50% from 28.0 MPa to 42.0 MPa (beta_1 = 0.75): Whitney Block Depth E2 shrinks from 105.88 mm to 70.59 mm, nudging internal lever arm (d - a/2) from 467.06 mm to 484.71 mm—so Design Capacity E5 rises only +3.78% to 329.79 kN·m!
=IF(ISTEXT(E5), E5, LET(ad_p, (B4 * B6) / (0.85 * B5 * B2), b1_f, IF(B5 <= 28, 0.85, MAX(0.65, 0.85 - 0.05 * ((B5 - 28) / 7))), cd_p, ad_p / b1_f, ROUND(0.003 * ((B3 - cd_p) / cd_p), 4)))Try the example
Use the original values or change them to test your understanding.