Metallurgy, ASTM E10 Hardness Testing & Structural Steel Inspection
Brinell Hardness (HBW) Spherical-Cap & Steel UTS Formula
Calculate ASTM E10 Tungsten-Carbide Brinell Hardness HBW, Naive Projected-Circle Error HB, Estimated Carbon/Alloy Steel Ultimate Tensile Strength UTS (MPa), and d/D Validity Ratio.
Divide by curved spherical-cap area PI()*D*(D - SQRT(D^2 - d^2))/2—never projected circle PI()*d^2/4!
In ASTM E10 / ISO 6506 Brinell hardness testing on castings and forgings, a tungsten-carbide ball of diameter D (typically 10.0 mm) is pressed into metal under heavy force F_kgf (typically 3,000 kgf for steel), leaving a circular rim of diameter d (mm). Unlike Meyer hardness, true Brinell Hardness HBW divides force by the CURVED SPHERICAL-CAP surface area A_cap = 0.5 * PI() * D * (D - SQRT(D^2 - d^2))—giving HBW = (2 * F_kgf) / (PI() * D * (D - SQRT(D^2 - d^2)))! Dividing by the flat projected circle PI() * d^2 / 4 overstates hardness by ~5%–10%, while ASTM E10 restricts valid indents to 0.24 <= d / D <= 0.60 and relates wrought steel Ultimate Tensile Strength via UTS_MPa ≈ 3.45 * HBW (~500 * HBW in psi)!
Verify on a Forged 4140 Steel Crankshaft—Test_Load = 3,000 kgf (B2), Carbide_Ball_Diameter D = 10.0 mm (B3), and Mean_Impression_Diameter d = 4.20 mm (B4 -> d/D = 0.42 in E5): Exact Brinell Hardness equals (2 * 3,000) / (PI() * 10 * (10 - SQRT(100 - 17.64))) = 206.53 HBW (E2, vs 216.54 HB naive flat-circle error in E3), yielding an Estimated Steel UTS of 3.45 * 206.525 = 712.51 MPa (E4)!
| ASTM E10 Brinell Inspection KPI (3,000 kgf, D = 10.0 mm Ball, d = 4.20 mm) | Formula | Verified Output |
|---|---|---|
| Exact Brinell Hardness (HBW) | 2 * F_kgf / (PI() * D * (D - SQRT(D^2 - d^2))) | 206.53 HBW (142.55 @ d = 5.0 mm) |
| Naive Flat-Circle Error (kgf/mm²) | F_kgf / (PI() * d^2 / 4) | 216.54 (+10.01 Overstatement!) |
| Estimated Steel UTS (MPa) | 3.45 * Exact_Brinell_HBW | 712.51 MPa (491.81 @ d = 5.0 mm) |
| ASTM E10 Indent Ratio (d / D) | Impression_d_mm / Ball_D_mm (0.24..0.60) | 0.42 (0.50 @ d = 5.0 mm) |
ASTM E10 Exact Spherical-Cap HBW & Validity Formula
Validate Force_kgf > 0, Ball_D_mm > 0, and enforce the ASTM E10 valid geometric window 0.24 * D <= d <= 0.60 * D!
=LET(f_kg, B2, dm_b, B3, dm_i, B4, IF(OR(f_kg <= 0, dm_b <= 0, dm_i < 0.24 * dm_b, dm_i > 0.6 * dm_b), "Invalid ASTM E10 Indent", ROUND((2 * f_kg) / (PI() * dm_b * (dm_b - SQRT((dm_b ^ 2) - (dm_i ^ 2)))), 2)))Normalized Mild-Steel Wider Impression Check (Set Impression d = 5.00 mm)
Widen Impression_Diameter B4 from 4.20 mm to 5.00 mm (d/D E5 = 0.50): Exact Brinell Hardness E2 drops to 142.55 HBW and Estimated Steel UTS E4 falls to 491.81 MPa! Also test B4 = 7.00 mm (d/D = 0.70 > 0.60 ASTM limit) to verify the guard!
=IF(ISTEXT(E2), E2, ROUND(3.45 * ((2 * B2) / (PI() * B3 * (B3 - SQRT((B3 ^ 2) - (B4 ^ 2))))), 2))Try the example
Use the original values or change them to test your understanding.