Thick-Wall Cylinder & Pressure Vessel Lamé Stress Calculator
Calculate non-linear hoop, radial, and longitudinal stress distributions, Von Mises equivalent bore stress, autofrettage yield margins, and Faupel burst pressure for high-pressure cylinders and hydraulic barrels.
Lamé Stress Tensor & Yield Audit
Live Lamé Stress Distribution & Cylinder Cross-Section
First-Principles Lamé Elasticity & Failure Derivations
For a thick cylinder of inner radius $r_i = 3.00$ in, outer radius $r_o = 5.00$ in, wall ratio $k = r_o / r_i = 1.667$, subject to internal pressure $P_i = 7,500$ psi and external pressure $P_o = 0$:
Evaluating at the inner bore ($r = r_i$):
Equilibrium of axial fluid thrust across the cross-sectional steel wall area:
Combining the 3 principal stresses ($sigma_1 = sigma_t$, $sigma_2 = sigma_z$, $sigma_3 = sigma_r$):
Faupel's semi-empirical model incorporates both material yield ($S_y = 95.0$ ksi) and ultimate tensile strength ($S_u = 125.0$ ksi):
With an operating pressure of 7,500 psi, the design offers a burst safety factor of 6.38x and a yield safety factor of 4.67x against material yield strength.
ASME Section VIII Thick Cylinder Compliance Report
Generating ASME thick cylinder stress audit...
5 Fatal Thick-Wall Cylinder Engineering Traps
1. Using Thin-Wall Formula (Pr/t) on High-Pressure Cylinders (k > 1.10)
The thin-wall hoop formula ($sigma = P r / t$) assumes stress is uniformly distributed across wall thickness. For cylinders with wall ratio $k = r_o / r_i > 1.10$, stress is highly non-linear. The inner bore experiences 30% to 150% higher stress than average thin-wall equations predict. Designing high-pressure hydraulic barrels (3,000+ psi) with thin-wall math results in premature bore yield, permanent bulging, and piston seal blow-by.
2. The Law of Diminishing Returns: Over-Thickening Wall Beyond k = 2.0
In thick-walled cylinders, the maximum hoop stress asymptotes to $P_i$ as $k o infty$ ($sigma_{t,min} ge P_i$). Increasing wall thickness beyond $k = 2.0$ adds massive steel weight and cost but yields almost zero reduction in inner bore peak stress. If working pressure exceeds $0.5 imes S_y$, single-wall thickening cannot prevent bore yield; autofrettage, wire-winding, or multi-layer shrink-fit sleeves must be employed.
3. Neglecting Radial Compressive Stress in Triaxial Yield (σr = -Pi)
Engineers frequently compare only hoop stress $sigma_t$ directly against yield strength $S_y$. However, the fluid exerts direct radial compressive stress $sigma_r = -P_i$ on the bore surface. Under Tresca or Von Mises criteria, the principal stress difference is $sigma_1 - sigma_3 = sigma_t - (-P_i) = sigma_t + P_i$. Neglecting radial compression underestimates equivalent bore shear stress by thousands of psi.
4. Fatigue Bore Micro-Cracking under Cyclic Pressure Pulsations
Even if static peak stress satisfies ASME safety factors, high-pressure cyclically pressurized vessels (e.g. waterjet intensifiers, gas booster accumulators) fail by low-cycle fatigue. Fluid penetration into microscopic bore surface inclusions creates local stress concentration ($K_t > 3.0$), propagating fast brittle fatigue cracks that fracture the cylinder well below its theoretical burst pressure.
5. External Pressure Collapse (Buckling / Ovalization Instability)
When thick cylinders operate in subsea deepwater environments or inside shrink-fit cooling jackets with high external pressure ($P_o > P_i$), failure shifts from tensile yield to compressive shell buckling. Initial out-of-roundness (ovalization $ge 0.5%$) exponentially amplifies bending moments across the wall, causing sudden elastic snap-through collapse.