🔬 Stress Linearization in Pressure Vessel FEA
When you use FEA (like with Calculix or OpenRadioss) on a pressure vessel, you get a full, complex distribution of stress through the wall thickness. However, global design codes like ASME BPVC Section VIII, Division 2 (Design-by-Analysis) and EN 13445 classify stresses into categories based on how they contribute to potential failure modes like plastic collapse or fatigue.
Stress linearization is the mathematical post-processing technique used to decompose the total elastic stress tensor calculated by FEA along a specific line (the Stress Classification Line) into a simplified, through-thickness distribution of membrane and bending stresses.
1. The Stress Classification Line (SCL)
- Definition: An SCL is a straight line segment drawn through the thickness of the vessel wall, perpendicular to the mid-surface, usually in areas of high-stress concentration (e.g., nozzle junctions, support skirts, head-to-shell transitions).
- Purpose: The FEA results (the total stress) are sampled at points along this line.
2. Decomposing the Total Stress
Along the SCL, the complex stress profile $\sigma_{total}(t)$ is mathematically broken down into three fundamental components:
| Stress Component | Through-Thickness Profile | Failure Mechanism | Design Limit Governed By |
| Membrane ($\sigma_m$ or $P_m$) | Constant (average stress) | Gross plastic deformation (rupture) | Average force acting on the cross-section. |
| Bending ($\sigma_b$ or $P_b$) | Linear (varies from face to face) | Plastic collapse (yielding across the thickness) | Bending moment acting on the cross-section. |
| Peak ($\sigma_p$ or $F$) | Non-linear (the difference) | Low-cycle fatigue, brittle fracture | Stress concentration at the surface. |
The Process:
- Integration: The raw stress component values ($\sigma_{xx}, \sigma_{yy}, \tau_{xy}$, etc.) along the SCL are mathematically integrated to find the equivalent Membrane (average) stress and Bending stress components.
- Classification: The resulting linearized stresses are then classified by the engineer as Primary ($P_m$, $P_L$, $P_b$), Secondary ($Q$), or Peak ($F$).
- Primary Stresses (16$P$): Required to satisfy equilibrium with external forces and moments. They are non-self-limiting and can lead to immediate collapse.
- Secondary Stresses ($Q$): Self-limiting stresses (e.g., from thermal expansion). They cause distortion but cannot lead to immediate collapse in a ductile material.
- Peak Stresses ($F$): Highly localized, non-linear stresses that only cause local yielding or are related to stress concentration effects (like fatigue).
- Comparison: The classified stresses (e.g., $P_m$, $P_L + P_b$, $P_L + P_b + Q$, etc.) are finally compared against the specific allowable stress limits ($S_m$, $1.5 S_m$, etc.) defined in the relevant design code (e.g., ASME Part 5, Figure 5.1).
3. Connection to ISO 16528
ISO 16528 is a performance standard. It requires that the vessel design prevents critical failure modes (plastic collapse, fatigue, etc.).
By using FEA, performing the Stress Linearization, and verifying the classified stresses against the stringent limits of codes like ASME BPVC Div. 2, the engineer is providing the required rigorous proof that the vessel meets the essential safety and performance requirements mandated by the international ISO standard.
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