ASME VIII Div. 3 High-Pressure Vessels & Diesel Rails, Thick-Cylinder Autofrettage
Thick-Cylinder Autofrettage & Residual Bore Stress Formula
Calculate Wall Radius Ratio K (b/a), Tresca Bore Yield-Onset Pressure Py (MPa), Autofrettage Overpressure Paf (MPa), and Elastic-Unloading Residual Bore Hoop Stress (MPa).
Pre-yielding a high-pressure diesel common-rail bore to c = 31 mm locks in -693 MPa of crack-closing compressive hoop stress—unless K > 2.22 triggers Bauschinger reverse yielding!
Without pre-stressing, a thick-walled forged cylinder (`Inner Bore Radius a` mm, `Outer Radius b` mm -> **Wall Ratio `K = b / a`**, `Alloy Yield Strength sigma_y` MPa) begins yielding at its inner bore as soon as internal pressure hits **Tresca Elastic Limit `P_y = (sigma_y / 2) * (1 - (1 / K^2))`** (`MPa`, never exceeding `0.5 * sigma_y` no matter how thick you forge the wall!)! **ASME Section VIII Division 3 Autofrettage** deliberately pumps hydraulic oil or pulls an oversize carbide swage to **Autofrettage Pressure `P_{af} = (sigma_y / 2) * (1 - (c / b)^2 + 2 * ln(c / a))`** (`MPa`), expanding a plastic core out to **Elastic-Plastic Junction Radius `c`** (`a <= c <= b`) while the outer elastic shell (`c -> b`) squeezes back upon depressurization! Subtracting full Lamé unloading from loaded bore stress (`sigma_y - P_{af}`) leaves **Compressive Residual Bore Hoop Stress `sigma_{theta,res} = sigma_y - P_{af} * ((2 * K^2) / (K^2 - 1))`** (`MPa`)—clamp-shutting cyclic fatigue micro-cracks on `250` MPa diesel fuel rails, `300` MPa LDPE autoclave tubes, and `400` MPa waterjet intensifiers!
Verify on a `4340` Martensitic Steel High-Pressure Intensifier Barrel—Inner_Bore `a` = 20.0 mm (B2), Outer_Radius `b` = 42.0 mm (B3), Target_Plastic_Front `c` = 31.0 mm (B4), and Yield_Strength `sigma_y` = 960 MPa (B5): Wall Ratio `K` equals `2.10` (`<= 2.22` elastic-unloading limit! E2), Initial Bore Yield Pressure `P_y` equals `371.2` MPa (E3), Required Autofrettage Pressure `P_{af}` equals **`639.2` MPa** (E4), locking in **`-693.4` MPa** of protective compressive bore hoop stress upon unloading (E5)!
| ASME VIII Div. 3 Tresca Autofrettage KPI (a=20 mm, Sy=960 MPa) | Formula | Verified Output |
|---|---|---|
| Cylinder Wall Ratio K (b / a) | Outer_Radius_b_mm / Inner_Bore_a_mm | 2.10 (2.60 > 2.22 thick-wall regime) |
| Initial Bore Yield Py (MPa) | (Yield_Sy / 2) * (1 - (1 / (K ^ 2))) | 371.2 MPa (409.0 MPa) |
| Autofrettage Pressure Paf (MPa) | (Sy / 2) * (1 - (c/b)^2 + 2 * LN(c/a)) | 639.2 MPa @ c=31 mm (904.0 MPa @ c=46 mm) |
| Elastic Residual Bore Hoop (MPa) | Sy - Paf * ((2 * K^2) / (K^2 - 1)) | -693.4 MPa SAFE (-1,161.8 < -960 REVERSE YIELD!) |
Hill / Tresca Elastic-Plastic Autofrettage Pressure & Residual Hoop Formula
Validate `Inner_Radius a` in `[2, 500]` mm, `Outer_Radius b` in `(1.2 * a, 5.0 * a]`, `Plastic_Radius c` in `[a, b]`, and `Yield_Strength sigma_y` in `[150, 2,200]` MPa!
=LET(ra_m, B2, rb_m, B3, rc_m, B4, sy_m, B5, kw_r, rb_m / ra_m, pf_m, (sy_m / 2) * (1 - ((rc_m / rb_m) ^ 2) + 2 * LN(rc_m / ra_m)), IF(OR(ra_m < 2, ra_m > 500, rb_m <= 1.2 * ra_m, rb_m > 5 * ra_m, rc_m < ra_m, rc_m > rb_m, sy_m < 150, sy_m > 2200), "Invalid Autofrettage Input", ROUND(sy_m - pf_m * ((2 * (kw_r ^ 2)) / ((kw_r ^ 2) - 1)), 1)))Over-Thick Wall (`b = 52` mm -> `K = 2.60 > 2.22`, `c = 46` mm) Bauschinger Reverse-Yield Check
Thicken the forging to `B3 = 52.0` mm (`K = 2.60` in E2) and over-autofrettage out to `B4 = 46.0` mm (`P_{af}` E4 = `904.0` MPa): Purely elastic unloading predicts **`-1,161.8` MPa** in E5—exceeding `-sigma_y = -960` MPa compressive yield (`and firing even earlier around -650 MPa due to the Bauschinger effect`), proving why ASME `KD-5` caps useful residual compression above `K = 2.22`!
=IF(ISTEXT(E5), E5, ROUND((B5 / 2) * (1 - ((B4 / B3) ^ 2) + 2 * LN(B4 / B2)), 1))Try the example
Use the original values or change them to test your understanding.