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4.1 Internal Pressure Design & Cylindrical Shell Calculations

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API 510 Pressure Vessel Inspector Exam

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1 Chapter 1: Introduction, Scope & Certification Requirements

1.1 Exam Structure, Administration & Study Strategy 1.2 API 510 Code Scope, Exclusions & Jurisdictional Boundaries 1.3 Roles, Responsibilities & Inspector Qualification

2 Chapter 2: Inspection Planning, Data Evaluation & Remaining Life

2.1 Corrosion Rates, Remaining Life & Corrosion Allowance Calculations 2.2 Inspection Intervals, On-Stream Inspection & Interval Deferrals 2.3 Condition Monitoring Locations (CMLs) & Data Analysis

3 Chapter 3: Vessel Evaluation, Corrosion Averaging & Fitness-for-Service

3.1 Corrosion Averaging & Minimum Thickness Evaluation 3.2 Pitting Evaluation & Corroded Area Limits 3.3 Fitness-for-Service (API 579-1) & Risk-Based Inspection (API 580)

4 Chapter 4: ASME Section VIII Div 1 Design & Thickness Calculations

4.1 Internal Pressure Design & Cylindrical Shell Calculations 4.2 Ellipsoidal & Hemispherical Head Calculations 4.3 Spherical Shells, Static Head & Vessel MAWP 4.4 Openings, Nozzle Reinforcement & External Pressure Basics

5 Chapter 5: ASME Section VIII Fabrication, Welds & Joint Efficiency

5.1 Weld Joint Categories (UW-3) & Joint Efficiencies (UW-12) 5.2 Radiographic Examination (UW-11, UW-51, UW-52) 5.3 Post-Weld Heat Treatment (PWHT) & Impact Testing (UCS-56, UCS-66)

6 Chapter 6: Welding Inspection & Metallurgy (API 577 & ASME Sec IX)

6.1 Welding Procedures (WPS/PQR) & Essential Variables 6.2 Welder Performance Qualification (WPQ) & Testing 6.3 Welding Processes, Metallurgy & Weld Discontinuities (API 577)

7 Chapter 7: Nondestructive Examination (ASME Section V)

7.1 Radiographic Examination Techniques & Image Quality Indicators (Article 2) 7.2 Ultrasonic Thickness Measurement (Article 23, SE-797) 7.3 ASME Section V Articles 1, 6 & 7: General Requirements, PT & MT

8 Chapter 8: Damage Mechanisms (API RP 571)

8.1 Atmospheric Corrosion & Corrosion Under Insulation 8.2 HTHA, High-Temperature H2/H2S Corrosion & Sulfidation 8.3 Amine, Caustic & Chloride Stress Corrosion Cracking 8.4 Wet H2S Damage, Brittle Fracture & Erosion-Corrosion 8.5 Boiler-Water, Steam-Condensate & Hydrochloric Acid Corrosion 8.6 Caustic Corrosion & Naphthenic Acid Corrosion

9 Chapter 9: Pressure Vessel Inspection Practices (API RP 572 & API RP 578)

9.1 Internal & External Vessel Inspection Practices (API 572) 9.2 Structural Components, Internal Trays, Linings & Skirts 9.3 Material Verification Programs & Positive Material Identification (API 578)

10 Chapter 10: Pressure-Relieving Devices (API RP 576)

10.1 PRD Types, Construction & Operating Principles 10.2 PRD Causes of Failure, Shop Testing & Overhaul 10.3 PRD Inspection Intervals, Records & Rupture Disks

11 Chapter 11: Repairs, Alterations, Rerating & Pressure Testing (API 510 & PCC-2)

11.1 Repair vs. Alteration Authorization, Execution & Methods 11.2 PCC-2 Core Welded Repairs: Articles 101, 201, 202, 211 & 212 11.3 PCC-2 Special Welding: Articles 209, 210, 215 & 216 11.4 PCC-2 Excavation, Flange, Exchanger & Test Articles 304, 305, 312, 501 & 502 11.5 Vessel Rerating Requirements & Engineering Calculations 11.6 Pressure Testing: Hydrostatic, Pneumatic & NDE in Lieu of Testing

4.1 Internal Pressure Design & Cylindrical Shell Calculations

Key Takeaways Explain This Topic AI

Circumferential (hoop) stress governs longitudinal weld seams and requires twice the thickness of longitudinal stress for a given pressure.

The ASME UG-27(c)(1) circumferential stress formula is t = (P * R) / (S * E - 0.6 * P), valid for P ≤ 0.385 * S * E and t ≤ 0.5 * R.

The longitudinal stress formula is t = (P * R) / (2 * S * E + 0.4 * P), valid for P ≤ 1.25 * S * E.

Corrosion allowance treatment follows the problem statement: use the corroded dimension for pressure calculations, add CA when converting pressure thickness to nominal required thickness, and do not deduct or add it twice.

Allowable stress S is obtained from ASME Section II Part D Table 1A based on material specification, design temperature, and applicable design margin.

Last updated: August 2026

Internal Pressure Design & Cylindrical Shell Calculations

Pressure vessel design under ASME Section VIII, Division 1 relies on the membrane theory of thin-walled shells. In a thin-walled cylindrical vessel subjected to internal pressure, the vessel wall is in a state of biaxial tension. Understanding the mathematical derivations, code limitations, and parameter interactions defined in paragraph UG-27 is paramount for any API 510 Authorized Pressure Vessel Inspector.

1. Membrane Stress Theory in Cylindrical Shells

When a closed cylindrical vessel experiences internal pressure ( P P P ), the internal fluid exerts uniform outward normal pressure on the cylindrical walls and axial thrust on the end closures (heads). +-----------------------------------------------------------------------------------------+

  • | MEMBRANE STRESSES IN A CYLINDRICAL SHELL |
  • | |
  • | ^ Circumferential (Hoop) Stress (σ_h) |
  • | | Acts tangentially around the circumference |
  • | +-----+---------------------------------------------------+-----+ |
  • | | | | | |
  • | <-----+=====[=========== LONGITUDINAL WELD SEAM ===============]=====+-----> |
  • | | | (Subjected to Hoop Stress σ_h) | | Longitudinal |
  • | +-----+---------------------------------------------------+-----+ Stress (σ_L) |
  • | | |
  • | v Circumferential (Hoop) Stress (σ_h = P*R / t) |
  • | |
  • | GOVERNING PRINCIPLE: |
  • | - Hoop stress σ_h is TWICE longitudinal stress σ_L (σ_h = 2 * σ_L). |
  • | - σ_h acts across the LONGITUDINAL weld seam. |
  • | - σ_L acts across the CIRCUMFERENTIAL (girth) weld seam. |
  • | - Therefore, the longitudinal seam governs the minimum required shell thickness. |
  • +-----------------------------------------------------------------------------------------+
  • Principal Stress Equations (Thin-Wall Idealization)

Circumferential (Hoop) Stress ( σ h \sigma_h σ h ​ ): σ h = P ⋅ R t \sigma_h = \frac{P \cdot R}{t} σ h ​ = t P ⋅ R ​

Acts along the circumferential plane.

Tends to split the cylinder longitudinally from end to end.

Directly stresses the longitudinal weld seams (Category A).

Longitudinal (Axial) Stress ( σ L \sigma_L σ L ​ ): σ L = P ⋅ R 2 t = 1 2 σ h \sigma_L = \frac{P \cdot R}{2t} = \frac{1}{2} \sigma_h σ L ​ = 2 t P ⋅ R ​ = 2 1 ​ σ h ​

Acts along the longitudinal axis of the cylinder.

Tends to pull the vessel apart like a telescoping tube.

Directly stresses the circumferential girth seams (Category B).

Because hoop stress is twice as large as longitudinal stress ( σ h = 2 σ L \sigma_h = 2 \sigma_L σ h ​ = 2 σ L ​ ), the longitudinal weld seam is subjected to double the tensile stress of the circumferential seam. Consequently, circumferential stress governs the required wall thickness of cylindrical shells under internal pressure.

2. ASME Section VIII Div 1 Shell Formulas (UG-27)

ASME Section VIII, Division 1 modifies the classical thin-wall equations by adding empirical correction terms ( − 0.6 P -0.6P − 0.6 P and + 0.4 P +0.4P + 0.4 P ) to account for non-linear through-thickness stress distribution in moderately thick shells.

Circumferential Stress (Longitudinal Joints) — UG-27(c)(1)

When the thickness does not exceed one-half of the inside radius ( t ≤ 0.5 R t \le 0.5R t ≤ 0.5 R ) or the design pressure does not exceed 0.385 S E 0.385 S E 0.385 SE ( P ≤ 0.385 S E P \le 0.385 S E P ≤ 0.385 SE ): t = P ⋅ R S ⋅ E − 0.6 ⋅ P \mathbf{t = \frac{P \cdot R}{S \cdot E - 0.6 \cdot P}} t = S ⋅ E − 0.6 ⋅ P P ⋅ R ​ P = S ⋅ E ⋅ t R + 0.6 ⋅ t \mathbf{P = \frac{S \cdot E \cdot t}{R + 0.6 \cdot t}} P = R + 0.6 ⋅ t S ⋅ E ⋅ t ​

Where:

  • t t t = Minimum required thickness of shell (inches or mm)
  • P P P = Internal design pressure or MAWP (psi or kPa)

R R R = Inside radius of shell course under evaluation in corroded condition: R = R original + C A R = R_{\text{original}} + CA R = R original ​ + C A (inches or mm)

S S S = Maximum allowable stress value from ASME Section II, Part D, Table 1A (psi or kPa)

E E E = Joint efficiency of the longitudinal weld seam per UW-12 (dimensionless, 0.45 ≤ E ≤ 1.0 0.45 \le E \le 1.0 0.45 ≤ E ≤ 1.0 )

Longitudinal Stress (Circumferential Joints) — UG-27(c)(2)

When the thickness does not exceed one-half of the inside radius ( t ≤ 0.5 R t \le 0.5R t ≤ 0.5 R ) or P ≤ 1.25 S E P \le 1.25 S E P ≤ 1.25 SE : t = P ⋅ R 2 ⋅ S ⋅ E + 0.4 ⋅ P \mathbf{t = \frac{P \cdot R}{2 \cdot S \cdot E + 0.4 \cdot P}} t = 2 ⋅ S ⋅ E + 0.4 ⋅ P P ⋅ R ​ P = 2 ⋅ S ⋅ E ⋅ t R − 0.4 ⋅ t \mathbf{P = \frac{2 \cdot S \cdot E \cdot t}{R - 0.4 \cdot t}} P = R − 0.4 ⋅ t 2 ⋅ S ⋅ E ⋅ t ​

Where:

E E E = Joint efficiency of the circumferential girth seam per UW-12.

Outer Radius ( R o R_o R o ​ ) Formulas (ASME Appendix 1-1)

If the outside radius ( R o R_o R o ​ ) or outside diameter ( D o D_o D o ​ ) is known rather than the inside radius:

Stress Orientation Thickness Formula ( t t t ) Pressure Formula ( P P P )

Circumferential Stress (Hoop) t = P ⋅ R o S ⋅ E + 0.4 ⋅ P t = \frac{P \cdot R_o}{S \cdot E + 0.4 \cdot P} t = S ⋅ E + 0.4 ⋅ P P ⋅ R o ​ ​ P = S ⋅ E ⋅ t R o − 0.4 ⋅ t P = \frac{S \cdot E \cdot t}{R_o - 0.4 \cdot t} P = R o ​ − 0.4 ⋅ t S ⋅ E ⋅ t ​

Longitudinal Stress (Axial) t = P ⋅ R o 2 ⋅ S ⋅ E − 0.6 ⋅ P t = \frac{P \cdot R_o}{2 \cdot S \cdot E - 0.6 \cdot P} t = 2 ⋅ S ⋅ E − 0.6 ⋅ P P ⋅ R o ​ ​ P = 2 ⋅ S ⋅ E ⋅ t R o + 0.6 ⋅ t P = \frac{2 \cdot S \cdot E \cdot t}{R_o + 0.6 \cdot t} P = R o ​ + 0.6 ⋅ t 2 ⋅ S ⋅ E ⋅ t ​

Important

Outer Radius Rule with Corrosion Allowance: When calculating required thickness using the outer radius formula, the outside radius R o R_o R o ​ of a vessel subject to internal corrosion does not change because metal loss occurs on the inside surface. Therefore, do not add corrosion allowance to R o R_o R o ​ ; instead, calculate the pressure-required thickness t t t and then add the corrosion allowance ( t nominal = t + C A t_{\text{nominal}} = t + CA t nominal ​ = t + C A ).

3. ASME Section II Part D Allowable Stress ( S S S )

The allowable stress value ( S S S ) represents the maximum permissible tensile stress in the material at the design metal temperature.

Determination of Allowable Stress ( S S S )

Listed in ASME Section II, Part D, Subpart 1, Table 1A (ferrous materials) and Table 1B (non-ferrous materials).

Design Margin: Since the 1999 Addenda of ASME Section VIII Div 1, the design margin on tensile strength is 3.5 (i.e., S = min ⁡ ( S u 3.5 , 2 3 S y ) S = \min\left(\frac{S_u}{3.5},\; \frac{2}{3} S_y\right) S = min ( 3.5 S u ​ ​ , 3 2 ​ S y ​ ) at room temperature).

Prior to 1999, the design margin was 4.0 (i.e., S = min ⁡ ( S u 4.0 , 2 3 S y ) S = \min\left(\frac{S_u}{4.0},\; \frac{2}{3} S_y\right) S = min ( 4.0 S u ​ ​ , 3 2 ​ S y ​ ) ).

Typical Allowable Stress Values at Temperature (SA-516 Gr. 70 Carbon Steel)

  • Temperature (°F) Allowable Stress S S S (Current 3.5 Margin) Allowable Stress S S S (Pre-1999 4.0 Margin)
  • 20 to 500°F 20,000 psi (138 MPa) 17,500 psi (121 MPa)
  • 600°F 19,400 psi (134 MPa) 17,500 psi (121 MPa)
  • 650°F 18,800 psi (130 MPa) 17,500 psi (121 MPa)
  • 700°F 18,100 psi (125 MPa) 16,600 psi (114 MPa)
  • 750°F 14,800 psi (102 MPa) 14,800 psi (102 MPa)
  • 800°F 12,000 psi (82.7 MPa) 12,000 psi (82.7 MPa)

Note

Notice that allowable stress decreases significantly at elevated temperatures due to thermal softening and the onset of creep mechanisms. In API 510 evaluations, always verify the design metal temperature from the vessel nameplate or U-1A Form before selecting S S S .

4. Weld Joint Efficiency ( E E E ) Summary (UW-12)

Joint efficiency ( E E E ) is a numerical factor that reduces the allowable stress to compensate for potential flaws in welded joints.

Joint Type Weld Description Full Radiography (UW-11(a)) Spot Radiography (UW-11(b) / UW-52) Visual Inspection Only (UW-12(c))

Type 1 Butt weld with complete penetration & fusion (double welded or equivalent) E = 1.00 E = 1.00 E = 1.00 E = 0.85 E = 0.85 E = 0.85 E = 0.70 E = 0.70 E = 0.70

Type 2 Single-welded butt joint with backing strip remaining in place E = 0.90 E = 0.90 E = 0.90 E = 0.80 E = 0.80 E = 0.80 E = 0.65 E = 0.65 E = 0.65

Type 3 Single-welded butt joint without backing strip N/A N/A E = 0.60 E = 0.60 E = 0.60

Seamless Seamless pipe or formed head (in base metal) E = 1.00 E = 1.00 E = 1.00 E = 1.00 E = 1.00 E = 1.00 E = 0.85 E = 0.85 E = 0.85 (per UW-12(d))

5. Comprehensive Step-by-Step Shell Thickness Calculation

Let us work through an exam-level calculation covering all aspects of shell sizing, corrosion allowance, and joint efficiency.

Problem Statement

A horizontal flash drum is to be fabricated from SA-516 Grade 70 carbon steel plate. The vessel data is as follows:

  • Design Internal Pressure ( P P P ): 350 psig 350\text{ psig} 350 psig
  • Design Temperature: 500 ∘ F 500^\circ\text{F} 50 0 ∘ F

Inside Diameter ( D i D_i D i ​ ): 60.0 in. 60.0\text{ in.} 60.0 in. ( R i = 30.0 in. R_i = 30.0\text{ in.} R i ​ = 30.0 in. )

  • Corrosion Allowance ( C A CA C A ): 0.125 in. 0.125\text{ in.} 0.125 in. (1/8 in.)
  • Longitudinal Joint: Type 1 butt weld, Spot Radiographed per UW-52
  • Circumferential Joint: Type 1 butt weld, No Radiography (visual examination only)
  • Allowable Stress ( S S S ): 20 , 000 psi 20,000\text{ psi} 20 , 000 psi at 500 ∘ F 500^\circ\text{F} 50 0 ∘ F

Required:

Determine the minimum required thickness ( t circ t_{\text{circ}} t circ ​ ) based on circumferential stress.

Determine the minimum required thickness ( t long t_{\text{long}} t long ​ ) based on longitudinal stress.

Determine the governing required thickness in new condition ( t design t_{\text{design}} t design ​ ).

Calculate the MAWP of the new shell if nominal plate thickness selected is t nom = 0.750 in. t_{\text{nom}} = 0.750\text{ in.} t nom ​ = 0.750 in.

Step 1: Establish Corroded Inside Radius ( R R R ) R = R i + C A = 30.0 in. + 0.125 in. = 30.125 in. R = R_i + CA = 30.0\text{ in.} + 0.125\text{ in.} = \mathbf{30.125\text{ in.}} R = R i ​ + C A = 30.0 in. + 0.125 in. = 30.125 in.

Step 2: Determine Joint Efficiencies ( E E E )

For Circumferential Stress (acting across longitudinal joint): Type 1 Butt + Spot RT → E = 0.85 \rightarrow \mathbf{E = 0.85} → E = 0.85

For Longitudinal Stress (acting across circumferential joint): Type 1 Butt + Visual Only → E = 0.70 \rightarrow \mathbf{E = 0.70} → E = 0.70

Step 3: Calculate Required Thickness for Circumferential Stress (Hoop Stress) t circ = P ⋅ R S ⋅ E − 0.6 ⋅ P t_{\text{circ}} = \frac{P \cdot R}{S \cdot E - 0.6 \cdot P} t circ ​ = S ⋅ E − 0.6 ⋅ P P ⋅ R ​ t circ = 350 ⋅ 30.125 ( 20 , 000 ⋅ 0.85 ) − ( 0.6 ⋅ 350 ) = 10 , 543.75 17 , 000 − 210 = 10 , 543.75 16 , 790 = 0.6280 in. t_{\text{circ}} = \frac{350 \cdot 30.125}{(20,000 \cdot 0.85) - (0.6 \cdot 350)} = \frac{10,543.75}{17,000 - 210} = \frac{10,543.75}{16,790} = \mathbf{0.6280\text{ in.}} t circ ​ = ( 20 , 000 ⋅ 0.85 ) − ( 0.6 ⋅ 350 ) 350 ⋅ 30.125 ​ = 17 , 000 − 210 10 , 543.75 ​ = 16 , 790 10 , 543.75 ​ = 0.6280 in.

Step 4: Calculate Required Thickness for Longitudinal Stress (Axial Stress) t long = P ⋅ R 2 ⋅ S ⋅ E + 0.4 ⋅ P t_{\text{long}} = \frac{P \cdot R}{2 \cdot S \cdot E + 0.4 \cdot P} t long ​ = 2 ⋅ S ⋅ E + 0.4 ⋅ P P ⋅ R ​ t long = 350 ⋅ 30.125 ( 2 ⋅ 20 , 000 ⋅ 0.70 ) + ( 0.4 ⋅ 350 ) = 10 , 543.75 28 , 000 + 140 = 10 , 543.75 28 , 140 = 0.3747 in. t_{\text{long}} = \frac{350 \cdot 30.125}{(2 \cdot 20,000 \cdot 0.70) + (0.4 \cdot 350)} = \frac{10,543.75}{28,000 + 140} = \frac{10,543.75}{28,140} = \mathbf{0.3747\text{ in.}} t long ​ = ( 2 ⋅ 20 , 000 ⋅ 0.70 ) + ( 0.4 ⋅ 350 ) 350 ⋅ 30.125 ​ = 28 , 000 + 140 10 , 543.75 ​ = 28 , 140 10 , 543.75 ​ = 0.3747 in.

Step 5: Determine Governing Required Thickness

Comparing t circ = 0.6280 in. t_{\text{circ}} = 0.6280\text{ in.} t circ ​ = 0.6280 in. vs t long = 0.3747 in. t_{\text{long}} = 0.3747\text{ in.} t long ​ = 0.3747 in. , circumferential stress governs: t required, corroded = 0.6280 in. t_{\text{required, corroded}} = \mathbf{0.6280\text{ in.}} t required, corroded ​ = 0.6280 in.

Total minimum required thickness in new uncorroded condition: t design = t required, corroded + C A = 0.6280 in. + 0.125 in. = 0.7530 in. t_{\text{design}} = t_{\text{required, corroded}} + CA = 0.6280\text{ in.} + 0.125\text{ in.} = \mathbf{0.7530\text{ in.}} t design ​ = t required, corroded ​ + C A = 0.6280 in. + 0.125 in. = 0.7530 in.

Step 6: Calculate Shell MAWP in Corroded Condition for t nom = 0.750 in. t_{\text{nom}} = 0.750\text{ in.} t nom ​ = 0.750 in.

If nominal thickness is 0.750 in. 0.750\text{ in.} 0.750 in. , the corroded thickness available is: t = t nom − C A = 0.750 in. − 0.125 in. = 0.625 in. t = t_{\text{nom}} - CA = 0.750\text{ in.} - 0.125\text{ in.} = 0.625\text{ in.} t = t nom ​ − C A = 0.750 in. − 0.125 in. = 0.625 in. M A W P = S ⋅ E ⋅ t R + 0.6 ⋅ t = 20 , 000 ⋅ 0.85 ⋅ 0.625 30.125 + ( 0.6 ⋅ 0.625 ) = 10 , 625 30.125 + 0.375 = 10 , 625 30.50 = 348.36 psig MAWP = \frac{S \cdot E \cdot t}{R + 0.6 \cdot t} = \frac{20,000 \cdot 0.85 \cdot 0.625}{30.125 + (0.6 \cdot 0.625)} = \frac{10,625}{30.125 + 0.375} = \frac{10,625}{30.50} = \mathbf{348.36\text{ psig}} M A W P = R + 0.6 ⋅ t S ⋅ E ⋅ t ​ = 30.125 + ( 0.6 ⋅ 0.625 ) 20 , 000 ⋅ 0.85 ⋅ 0.625 ​ = 30.125 + 0.375 10 , 625 ​ = 30.50 10 , 625 ​ = 348.36 psig

(Since 348.36 psig < 350 psig, a standard 0.750 in. plate would require slight derating or selecting the next plate size of 0.8125 in. / 13/16 in.).

6. Common Exam Pitfalls & Calculation Traps

Trap Incorrect Method Correct Code Method

Diameter vs Radius Using D = 60 in. D = 60\text{ in.} D = 60 in. directly in UG-27 formula Formula requires Inside Radius R = D / 2 = 30 in. R = D/2 = 30\text{ in.} R = D /2 = 30 in. , then add C A CA C A .

Forgetting Corrosion Allowance in R R R Using R = 30.0 in. R = 30.0\text{ in.} R = 30.0 in. instead of R = 30.125 in. R = 30.125\text{ in.} R = 30.125 in. In corroded condition, the inside radius expands by C A CA C A : R = R i + C A R = R_i + CA R = R i ​ + C A .

Swapping Joint Efficiencies Using circumferential seam E E E for hoop stress calculation Longitudinal seam E E E resists hoop stress ( t circ t_{\text{circ}} t circ ​ ); circumferential seam E E E resists longitudinal stress ( t long t_{\text{long}} t long ​ ).

Outer Radius Confusion Adding C A CA C A to R o R_o R o ​ when using Appendix 1-1 Internal corrosion increases R i R_i R i ​ , but R o R_o R o ​ remains constant. Do not add C A CA C A to R o R_o R o ​ .

0.6 P 0.6P 0.6 P Sign Flip Using S E + 0.6 P S E + 0.6 P SE + 0.6 P in denominator for t t t Denominator is S E − 0.6 P S E - 0.6 P SE − 0.6 P for inside radius; S E + 0.4 P S E + 0.4 P SE + 0.4 P for outside radius.

Loading diagram... ASME Section VIII Div 1 Cylindrical Shell Thickness Workflow