Calculating Collapse Pressure in Unwelded Cylindrical Tubing
Predicting Rupture Pressure in Seamless Cylindrical Pipes for Gas Storage Cylinders Using Limit State Design Approach
Seamless steel pipes, necessary to prime-force fuel cylinders (e.g., for CNG, hydrogen, or business gases), have to stand up to internal pressures exceeding 20 MPa (as much as 70 MPa in hydrogen garage) although guaranteeing protection margins against catastrophic burst failure. These cylinders, almost always conforming to ISO 9809 or DOT 3AA standards and constructed from high-energy steels like 34CrMo4 or AISI 4130 (σ_y ~seven hundred-one thousand MPa), face stringent calls for: burst pressures (P_b) needs to exceed 2.25x provider power (e.g., >forty five MPa for 20 MPa running power), without leakage or fracture under cyclic or overpressure circumstances. Burst failure, driven through plastic instability in the hoop direction, is stimulated by means of wall thickness (t), greatest tensile strength (σ_uts), and residual ovality (φ, deviation from circularity), alongside residual stresses from production (e.g., chilly drawing, quenching). Plastic restrict load theory, rooted in continuum mechanics, affords a powerful framework to mannequin the connection among P_b and these parameters, allowing desirable protection margin handle for the period of construction. By integrating analytical units with finite issue research (FEA) and empirical validation, Pipeun ensures cylinders meet safe practices reasons (SF >2.25) when optimizing textile use. Below, we aspect the modeling frame of mind, parameter impacts, and manufacturing controls, ensuring compliance with necessities like ASME B31.3 and ISO 9809.
Plastic Limit Load Theory for Burst Pressure Prediction
Plastic minimize load concept assumes that burst occurs whilst the pipe reaches a kingdom of plastic instability, wherein hoop pressure (σ_h) exceeds the textile’s drift ability, ultimate to uncontrolled thinning and rupture. For a thin-walled cylindrical stress vessel (D/t > 10, D=outer diameter), the hoop rigidity beneath inner rigidity P is approximated by the Barlow equation: σ_h = P D / (2t). Burst drive P_b corresponds to the aspect wherein σ_h reaches or exceeds σ_uts, adjusted for plastic flow and geometric imperfections like ovality. The classical restriction load answer, depending on von Mises yield criterion, predicts P_b for an excellent cylinder as:
\[ P_b = \frac2 t \sigma_uts\sqrt3 D \]
This assumes isotropic, solely plastic waft at σ_uts (more commonly 900-1100 MPa for 34CrMo4) and no geometric defects. However, residual ovality and stress hardening introduce deviations, necessitating refined fashions.
For thick-walled cylinders (D/t < 10, not unusual in top-rigidity cylinders, e.g., D=2 hundred mm, t=five-10 mm), the Lamé equations account for radial stress (σ_r) and hoop stress gradients throughout the wall:
\[ \sigma_h = P \left( \fracr_o^2 + r_i^2r_o^2 - r_i^2 \precise) \]
in which r_o and r_i are outer and interior radii. At burst, the equal strain σ_e = √[(σ_h - σ_r)^2 + (σ_r - σ_a)^2 + (σ_a - σ_h)^2]/√2 (σ_a=axial rigidity, ~P/2 for closed ends) reaches σ_uts at the inner floor, yielding:
\[ P_b = \frac2 t \sigma_utsD_o \cdot \frac1\sqrt3 \cdot \left( 1 - \fractD_o \precise) \]
For a two hundred mm OD, 6 mm wall cylinder (t/D_o=zero.03), this predicts P_b~47 MPa for σ_uts=one thousand MPa, conservative as a consequence of neglecting stress hardening.
Ovality, defined as φ = (D_max - D_min) / D_nom (most often zero.5-2% post-manufacture), amplifies neighborhood stresses because of stress focus aspects (SCF, K_t~1 + 2φ), reducing P_b by 5-15%. The modified burst rigidity, per Faupel’s empirical correction for ovality, is:
\[ P_b = \frac2 t \sigma_uts\sqrt3 D_o \cdot \frac11 + ok \phi \]
wherein k~2-three relies upon on φ and pipe geometry. For φ=1%, P_b drops ~5%, e.g., from forty seven MPa to 44.5 MPa. Strain hardening (n~zero.1-0.15 for 34CrMo4, in line with Ramberg-Osgood σ = K ε^n) elevates P_b with the aid of 10-20%, as plastic glide redistributes stresses, modeled simply by Hollomon’s regulation: σ_flow = K (ε_p)^n, with K~1200 MPa.
Influence of Key Parameters
1. **Wall Thickness (t)**:
- P_b scales linearly with t in step with the prohibit load equation, doubling t (e.g., 6 mm to twelve mm) doubles P_b (~47 MPa to 94 MPa for D=2 hundred mm, σ_uts=a thousand MPa). Minimum t is set by means of ISO 9809: t_min = P_d D_o / (2 S + P_d), the place P_d=design force, S=2/3 σ_y (~six hundred MPa). For P_d=20 MPa, t_min~4.eight mm, yet t=6-eight mm guarantees SF>2.25.
- Manufacturing tolerances (API 5L, ±12.5%) necessitate t_n>t_min+Δt, with Δt~zero.five-1 mm for seamless pipes, confirmed simply by ultrasonic gauging (ASTM E797, ±0.1 mm).
2. **Ultimate Tensile Strength (σ_uts)**:
- Higher σ_uts (e.g., 1100 MPa for T95 vs. 900 MPa for C90) proportionally boosts P_b, essential for light-weight designs. Quenching and tempering (Q&T, 900°C quench, 550-600°C mood) optimize σ_uts at the same time as asserting ductility (elongation >15%), ensuring plastic crumple precedes brittle fracture (K_IC>100 MPa√m).
- Low carbon equal (CE<0.forty three) prevents martensite, protecting longevity in welds (Charpy >27 J at -20°C).
three. **Residual Ovality (φ)**:
- Ovality from bloodless drawing or spinning (φ~0.five-2%) introduces SCFs, decreasing P_b and accelerating fatigue. FEA fashions (ANSYS, shell features S4R) teach φ=2% raises σ_h by 10% at oval poles, losing P_b from 47 MPa to 42 MPa.
- Hydrostatic sizing put up-manufacture (1.1x P_d) reduces φ to <0.5%, restoring P_b within 2% of just right.
Modeling with FEA for Enhanced Accuracy
FEA refines analytical predictions by way of shooting nonlinear plasticity, ovality results, and residual stresses (σ_res~50-150 MPa from Q&T). Pipeun’s workflow uses ABAQUS:
- **Geometry**: A 200 mm OD, 6 mm t cylinder, meshed with 10^five C3D8R materials, with φ=0.5-2% mapped from laser profilometry (ISO 11496).
- **Material**: Elasto-plastic type with von Mises yield, σ_uts=a thousand MPa, n=zero.12, calibrated by the use of ASTM E8 tensile checks. Residual stresses from Q&T are enter as preliminary situations (σ_res~a hundred MPa, consistent with hollow-drilling, ASTM E837).
- **Loading**: Incremental P from 0 to failure, with burst outlined at plastic instability (dε/dP→∞). Boundary situations simulate closed ends (σ_a=P/2).
- **Output**: FEA predicts P_b=48.five MPa for φ=0.five%, t=6 mm, σ_uts=a thousand MPa, with σ_e peaking at 1050 MPa on the interior floor. Ovality of two% reduces P_b to forty five MPa, aligning with Faupel’s correction.
Sensitivity analyses fluctuate t (±10%), σ_uts (±5%), and φ (±50%), producing P_b envelopes (43-50 MPa), with Monte Carlo simulations (10^4 runs) yielding ninety five% trust SF>2.3 for P_d=20 MPa.
Safety Margin Control in Production
Pipeun’s manufacturing integrates minimize load predictions to make sure SF=P_b/P_d>2.25:
- **Wall Thickness Control**: Seamless pipes are chilly-drawn with t_n=t_min+1 mm (e.g., 7 mm for t_min=6 mm), proven by UT (ASTM E213). Hot rolling ensures uniformity (±0.2 mm), with rejection for t
- **Testing**: Burst exams (ISO 9809, 1.5x P_d minimum) validate P_b, with 2025 trials on two hundred mm OD cylinders attaining P_b=49 MPa (t=6.2 mm, φ=0.4%), 10% above FEA. Hydrostatic exams (1.5x P_d, no leak) and fatigue biking (10^four cycles at P_d) be certain SF.
- **NDT**: Ultrasonic (UT, ASTM E213) and magnetic particle inspection (MPI, ASTM E709) locate flaws (a
Challenges comprise residual stress variability (σ_res±20%) from Q&T, addressed by way of inline tempering (six hundred°C, 2 h), and ovality creep in skinny partitions, mitigated with the aid of multi-level sizing. Emerging AI-driven FEA optimizes t and φ in true-time, cutting back security margins to two.3 whereas reducing cloth by way of five%.
In sum, plastic restrict load conception, augmented by means of FEA, maps the interaction of t, σ_uts, and φ to predict P_b with
