11.1 Pressure Vessels & Piping: Thin and Thick Walls, ASME VIII and B31

11.1 Thin-Walled & Thick-Walled Pressure Vessels & Piping

Pressure containment vessels and high-energy piping systems represent mission-critical components across power generation, chemical processing, aerospace propulsion, and fluid power hydraulics. Ensuring structural integrity requires a rigorous understanding of multiaxial stress states, the geometric boundary separating thin-shell membrane behavior from thick-walled continuum behavior, and compliance with governing consensus safety codes such as the ASME Boiler and Pressure Vessel Code (BPVC).

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Geometric Classification: Thin-Walled vs. Thick-Walled Threshold

The fundamental distinction between thin-walled and thick-walled vessels hinges on whether the radial stress gradient across the vessel wall can be safely neglected.

                  Thin-Walled Shell (r/t > 10)                 Thick-Walled Cylinder (r/t <= 10)
            +---------------------------------------+    +---------------------------------------+
            | - Wall thickness t < 0.1 * r_i        |    | - Wall thickness t >= 0.1 * r_i       |
            | - Uniform membrane stress assumption  |    | - Steep non-linear stress gradient    |
            | - Radial normal stress sigma_r ~ 0    |    | - Radial stress sigma_r != 0 (varies) |
            | - Negligible bending across thickness |    | - Peak stress at inner bore (r = r_i) |
            +---------------------------------------+    +---------------------------------------+

The $10\%$ Thickness Threshold

A vessel is classified as thin-walled when the ratio of its internal radius to wall thickness exceeds $10$:

$$\frac{r_i}{t} > 10 \quad \Longleftrightarrow \quad t < 0.10 r_i$$

In terms of inside diameter $D_i$, this corresponds to $D_i / t > 20$. In thin-walled shells:

• The circumferential hoop stress and longitudinal stress are assumed uniform across the wall thickness.
• Radial stress $\sigma_r$ ranges from $-p$ on the inner wetted surface to $0$ on the unpressurized exterior surface; because $\sigma_r \ll \sigma_h$, it is commonly neglected in elementary structural checks.

When $r_i / t \le 10$ ($t \ge 0.10 r_i$), the vessel must be analyzed as a thick-walled cylinder. Membrane theory becomes dangerously non-conservative, as peak tangential stresses at the inner bore significantly exceed the average membrane stress.

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Thin-Walled Cylindrical & Spherical Shells

Consider a thin-walled cylindrical vessel with inside radius $r$, wall thickness $t$, and uniform internal gage pressure $p$.

                                  Thin-Walled Cylindrical Vessel Stresses
                                  
                                 ^ Hoop Stress sigma_h (Circumferential)
                                 |
                        +--------|--------+
                       /         |         \
                      /    +-----v-----+    \
                     |     |  p (int)  |     | ----> Longitudinal Stress sigma_l (Axial)
                      \    +-----------+    /
                       \                   /
                        +-----------------+

1. Circumferential (Hoop) Stress ($\sigma_1 = \sigma_h$)

Cutting the cylinder along a diametral longitudinal plane and enforcing static equilibrium between fluid pressure and wall tensile forces:

$\sum F_y = 0 \implies 2 \sigma_h (t \cdot L) - p (2 r \cdot L) = 0$ $\sigma_h = \frac{p r}{t} = \frac{p D}{2t}$

2. Longitudinal (Axial) Stress ($\sigma_2 = \sigma_l$)

For a closed-end cylinder, cutting a transverse cross section perpendicular to the vessel axis:

$\sum F_x = 0 \implies \sigma_l (2 \pi r t) - p (\pi r^2) = 0$ $\sigma_l = \frac{p r}{2t} = \frac{p D}{4t}$

Key relationship: in a thin-walled cylinder, hoop stress is exactly twice the longitudinal stress:

$\sigma_h = 2 \sigma_l$

This explains why over-pressurized thin-walled tubes or sausages burst along a longitudinal split rather than snapping transversely.

3. Radial Stress ($\sigma_3 = \sigma_r$)

Radial stress acts perpendicular to the shell surface:

• At the inner bore: $\sigma_r(r_i) = -p$ (compressive contact pressure).
• At the outer surface: $\sigma_r(r_o) = 0$.

In thin-walled theory, since $\sigma_h / p = r / t > 10$, $\sigma_r$ is small compared to $\sigma_h$, but it governs the critical maximum shear stress at the inner bore.

4. Thin-Walled Spherical Shells

By spherical symmetry, all planes passing through the center are identical:

$\sum F = 0 \implies \sigma (2 \pi r t) - p (\pi r^2) = 0$ $\sigma_{\text{sphere}} = \frac{p r}{2t} = \frac{p D}{4t}$

A spherical vessel requires only half the wall thickness of a cylindrical shell of the same radius and design pressure to sustain the same membrane stress.

5. Maximum Shear Stress & Failure Criteria

The principal stresses on the inner surface of a thin-walled cylinder are:

$\sigma_1 = \sigma_h = \frac{p r}{t}, \quad \sigma_2 = \sigma_l = \frac{p r}{2t}, \quad \sigma_3 = -p$

According to the Tresca Maximum Shear Stress Theory (MSST):

$\tau_{\max} = \frac{\sigma_1 - \sigma_3}{2} = \frac{\frac{pr}{t} - (-p)}{2} = \frac{p}{2}\left(\frac{r}{t} + 1\right) \approx \frac{pr}{2t}$

Under von Mises Distortion Energy Theory (DET), the equivalent stress $\sigma'$ is:

$\sigma' = \sqrt{\sigma_1^2 - \sigma_1 \sigma_2 + \sigma_2^2} = \sqrt{\left(\frac{pr}{t}\right)^2 - \left(\frac{pr}{t}\right)\left(\frac{pr}{2t}\right) + \left(\frac{pr}{2t}\right)^2} = \frac{\sqrt{3}}{2} \frac{pr}{t} \approx 0.866 \sigma_h$

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Thick-Walled Cylinders: The Lamé Equations

When wall thickness $t > 0.1 r_i$, stresses vary significantly with radius $r$. Gabriel Lamé derived the exact elastic continuum solution in 1852 by considering force equilibrium on an infinitesimal cylindrical element:

$\frac{d\sigma_r}{dr} + \frac{\sigma_r - \sigma_t}{r} = 0$

                       Thick-Walled Cylinder Stress Distributions (Internal Pressure Only)
     Stress
       ^
       |   +-----------------------+ <-- Tangential Hoop Stress sigma_t (Peak at r = r_i)
       |   | \                     |
       |   |  \                    |
       |   |   \-------------------+ <-- Tangential Hoop Stress at r = r_o
       |   |                       |
    0 -+---+-----------------------+-------------------> Radius r
       |   |                       |
       |   |   /-------------------+ <-- Radial Stress sigma_r = 0 at r = r_o
       |   |  /                    |
       |   +-v---------------------+ <-- Radial Stress sigma_r = -p_i at r = r_i
           r_i                     r_o

General Lamé Equations

For a thick-walled cylinder subjected to internal pressure $p_i$ and external pressure $p_o$:

$\sigma_r(r) = \frac{p_i r_i^2 - p_o r_o^2}{r_o^2 - r_i^2} - \frac{(p_i - p_o) r_i^2 r_o^2}{r^2 (r_o^2 - r_i^2)}$ $\sigma_t(r) = \frac{p_i r_i^2 - p_o r_o^2}{r_o^2 - r_i^2} + \frac{(p_i - p_o) r_i^2 r_o^2}{r^2 (r_o^2 - r_i^2)}$

For closed-end cylinders, the longitudinal stress $\sigma_l$ is uniform across the wall:

$\sigma_l = \frac{p_i r_i^2 - p_o r_o^2}{r_o^2 - r_i^2}$

Special Case: Internal Pressure Only ($p_o = 0$)

Setting external pressure $p_o = 0$ simplifies the expressions to:

$\sigma_r(r) = \frac{p_i r_i^2}{r_o^2 - r_i^2} \left( 1 - \frac{r_o^2}{r^2} \right)$ $\sigma_t(r) = \frac{p_i r_i^2}{r_o^2 - r_i^2} \left( 1 + \frac{r_o^2}{r^2} \right)$ $\sigma_l = \frac{p_i r_i^2}{r_o^2 - r_i^2}$

Peak Bore Stresses & Inefficiency of Wall Thickening

The critical peak stress state occurs at the innermost bore ($r = r_i$):

$\sigma_r(r_i) = -p_i$ $\sigma_t(r_i) = p_i \left( \frac{r_o^2 + r_i^2}{r_o^2 - r_i^2} \right)$ $\tau_{\max}(r_i) = \frac{\sigma_t(r_i) - \sigma_r(r_i)}{2} = p_i \left( \frac{r_o^2}{r_o^2 - r_i^2} \right)$

At the outer surface ($r = r_o$):

$\sigma_r(r_o) = 0$ $\sigma_t(r_o) = p_i \left( \frac{2 r_i^2}{r_o^2 - r_i^2} \right)$

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The Autofrettage Principle & Residual Stress Conditioning

To overcome the elastic limitation of monobloc thick-walled cylinders, engineers use autofrettage (French for "self-hooping"). Autofrettage is widely employed in artillery gun barrels, ultra-high-pressure waterjet intensifiers ($60,000\text{ psi}$), and diesel common-rail fuel injection manifolds.

                                  The Autofrettage Process Cycle
                                  
    Phase 1: Overpressurization             Phase 2: Elastic Unloading             Phase 3: Residual State
    +---------------------------+          +---------------------------+          +---------------------------+
    | Plastic zone at bore      |          | Outer elastic jacket      |          | High compressive residual |
    | Elastic outer mantle      |  ----->  | contracts and springs     |  ----->  | hoop stress at inner bore |
    | Fully yields inner fibers |          | back elastically          |          | prevents fatigue crack tip|
    +---------------------------+          +---------------------------+          +---------------------------+

Mechanics of Autofrettage

1. Overpressurization: The vessel is subjected to an internal autofrettage pressure $p_{\text{auto}}$ that drives the inner bore past its yield strength, creating an internal plastic deformation zone up to an intermediate radius $r_p$ ($r_i < r_p < r_o$), while the outer mantle remains elastic.
2. Elastic Recovery: When $p_{\text{auto}}$ is released, the outer elastic layer attempts to contract back to its original dimensions. The permanently stretched inner plastic layer resists this contraction.
3. Resulting Residual Stress Field: The outer layer clamps down on the inner core, locking in a large compressive residual hoop stress at the inner bore ($\sigma_{t,\text{res}} < 0$).
4. Service Advantage: Under subsequent operating service pressure $p_{\text{op}}$, the tensile hoop stress induced at the bore must first overcome this pre-existing compressive residual stress. Consequently, the net mean stress is shifted downward, significantly extending fatigue life and preventing bore crack propagation.

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ASME BPVC Section VIII Division 1 Code Rules

Industrial pressure vessels operating above $15\text{ psig}$ are governed by the ASME Boiler and Pressure Vessel Code (BPVC) Section VIII Division 1. ASME BPVC adopts a "design-by-rule" methodology that modifies thin-shell equations to account for thick-shell curvature, weld joint quality, and material allowable stresses.

Code Sizing Variables

• $P$: Maximum Allowable Working Pressure (MAWP) in $\text{psi}$ (or $\text{MPa}$).
• $R$: Inside radius in $\text{in}$ (corroded condition: $R = R_{\text{initial}} + CA$).
• $S$: Maximum allowable stress of the material in $\text{psi}$ (from ASME Section II Part D, typically $S_{ut} / 3.5$ or $2/3 S_y$).
• $E$: Weld joint efficiency ($0.70 \le E \le 1.00$):
  • $E = 1.00$: Full radiography (Type 1 butt welds with 100% volumetric NDE per UW-11(a)).
  • $E = 0.85$: Spot radiography (per UW-11(b) and UW-52).
  • $E = 0.70$: Visual inspection only (no radiography per UW-11(c)).
• $CA$: Corrosion allowance specified by the process engineer (typically $1/16\text{ in}$ to $1/4\text{ in}$, or $1.5\text{ mm}$ to $6.0\text{ mm}$).

Cylindrical Shell Thickness Formulas (ASME UG-27)

Circumferential stress (longitudinal joint) governs shell thickness:

$$t = \frac{P R}{S E - 0.6 P}$$

Longitudinal stress (circumferential joint) governs axial capacity:

$$t = \frac{P R}{2 S E + 0.4 P}$$

(The empirical $0.6 P$ and $0.4 P$ factors correct the thin-shell membrane equations for non-linear stress distribution through the thickness up to $t \le 0.5 R$ or $P \le 0.385 S E$.)

Formed Head Geometries & Sizing Rules (ASME UG-32)

Vessel heads enclose the ends of cylindrical shells. The three standard ASME formed head geometries are:

            Hemispherical Head                     2:1 Semi-Ellipsoidal Head              ASME Torispherical Head
                 (UG-32f)                                   (UG-32d)                              (UG-32e)
                 
                   ___                                        ____                                  _____
                .-'   '-.                                  .-'    '-.                            .-'     '-.
              .'         '.                              .'          '.                        .'           '.
             /             \                            /              \                      /               \
            |               |                          |                |                    |                 |
            |               |                          |                |                    |                 |
          Radius R = D/2                             Depth h = D/4                        Crown L = D, Knuckle r_k = 0.06D
          Lowest thickness                           Matches shell thickness              Prone to knuckle stress peak

1. Hemispherical Head (UG-32f):

$$t = \frac{P R}{2 S E - 0.2 P}$$

A hemispherical head requires approximately half the thickness of the adjacent cylindrical shell, making it the most material-efficient head geometry, though more costly to fabricate.

1. 2:1 Semi-Ellipsoidal Head (UG-32d): Major-to-minor axis ratio is $2:1$ (inside depth $h = D/4$):

$$t = \frac{P D}{2 S E - 0.2 P} = \frac{P R}{S E - 0.1 P}$$

Because the thickness formula closely matches the cylindrical shell formula, the head thickness is identical to the shell thickness, simplifying fabrication and welding transition.

1. ASME Flanged and Dished (Torispherical) Head (UG-32e): Consists of a spherical crown of radius $L$ (typically $L = D$) and a toroidal knuckle of radius $r_k$ ($r_k \ge 0.06 D$ and $r_k \ge 3 t$):

$$t = \frac{0.885 P L}{S E - 0.1 P}$$

Torispherical heads are easier to press than ellipsoidal heads, but high localized bending stresses at the knuckle mandate roughly $77\%$ greater thickness than 2:1 ellipsoidal heads.

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Formed Head Geometry Comparison Table

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Process & Power Piping (ASME B31)

Pressure piping is designed to the ASME B31 code sections, chiefly B31.1 Power Piping (boiler external piping, steam and power plant systems) and B31.3 Process Piping (chemical, petroleum and similar plants). For straight pipe under internal pressure with $t < D/6$, ASME B31.3 gives the pressure design thickness

$$t = \frac{P D}{2\,(S E W + P Y)}$$

where:

• $D$ is the pipe outside diameter and $P$ the internal design gage pressure.
• $S$ is the code allowable stress at the design temperature, and $E$ is the longitudinal-joint quality factor ($1.0$ for seamless pipe).
• $W$ is the weld joint strength reduction factor ($1.0$ below the creep range).
• $Y$ is a coefficient equal to $0.4$ for ferritic steels below $900^\circ\text{F}$ ($482^\circ\text{C}$).

The minimum required thickness adds mechanical, corrosion and erosion allowances: $t_m = t + c$. Seamless pipe may be supplied up to $12.5\%$ under its nominal wall, so the selected nominal wall must satisfy $t_{\text{nom}} \ge t_m / 0.875$. A traditional estimate of the needed schedule number is $\text{Sch} \approx 1{,}000 P/S$.

Piping design also checks thermal expansion and flexibility (anchors, guides, loops and expansion joints keep the displacement stress range within code limits), support and hanger spacing, and water hammer: a sudden valve closure raises pressure by $\Delta p = \rho\, a\, \Delta v$, where $a$ is the pressure-wave speed (about $1{,}400\,\text{m/s}$ in water in a stiff pipe), so a $2\,\text{m/s}$ velocity change can add roughly $2.8\,\text{MPa}$ ($400\,\text{psi}$).

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Realistic Worked Engineering Scenario: Chemical Reactor Vessel Sizing

                                Chemical Reactor Pressure Vessel
                     
                     +----------------- 2:1 Ellipsoidal Head -----------------+
                     |                                                         |
                     +=========================================================+
                     |  Internal Design Pressure P = 450 psig                  |
                     |  Design Temperature = 300 deg F                         |
                     |  Inside Diameter D_i = 48.0 in (R = 24.0 in)            |
                     |  Material: SA-516 Grade 70 Steel                        |
                     |  Allowable Stress S = 20,000 psi                        |
                     |  Weld Joint Efficiency E = 0.85 (Spot RT)               |
                     |  Corrosion Allowance CA = 0.125 in (1/8 in)             |
                     +=========================================================+
                     |                                                         |
                     +------------------- Hemispherical Head ------------------+

Problem Statement

A chemical process engineering facility requires a new carbon steel autoclave reactor vessel. The design specifications are:

• Design internal pressure: $P = 450\text{ psig}$
• Inside diameter: $D_i = 48.0\text{ in}$ (initial uncorroded inside radius $R_0 = 24.0\text{ in}$)
• Design temperature: $300^\circ\text{F}$
• Vessel material: ASME SA-516 Grade 70 plate ($S = 20,000\text{ psi}$, $S_y = 38,000\text{ psi}$, $S_{ut} = 70,000\text{ psi}$)
• Joint efficiency: $E = 0.85$ (spot radiographed Category A and B butt welds)
• Corrosion allowance: $CA = 0.125\text{ in}$ ($1/8\text{ in}$)

Calculate:

1. The minimum required cylindrical shell thickness $t_{\text{shell}}$ and nominal plate specification.
2. The minimum required thickness for the top 2:1 semi-ellipsoidal head and bottom hemispherical head.
3. Verify whether the vessel satisfies the thin-walled threshold ($r/t > 10$).
4. The peak bore stresses and maximum shear stress at the inner radius using thick-walled Lamé theory, and compute the static factor of safety against yielding.

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Step-by-Step Mathematical Solution

Step 1: Cylindrical Shell Thickness Sizing (UG-27)

Under corroded conditions, the active inside radius is:

$$R = R_0 + CA = 24.0 + 0.125 = 24.125\text{ in}$$

Applying the ASME UG-27 circumferential hoop equation:

$$t_{\text{calc}} = \frac{P R}{S E - 0.6 P} = \frac{450\text{ psi} \times 24.125\text{ in}}{(20,000\text{ psi} \times 0.85) - (0.6 \times 450\text{ psi})}$$ $$t_{\text{calc}} = \frac{10,856.25\text{ lbf/in}}{17,000 - 270\text{ psi}} = \frac{10,856.25}{16,730} = 0.6489\text{ in}$$

Adding the corrosion allowance to obtain total nominal thickness:

$$t_{\text{nominal}} = t_{\text{calc}} + CA = 0.6489 + 0.125 = 0.7739\text{ in}$$

Selecting the next standard commercial plate size: $13/16\text{ in}$ plate ($0.8125\text{ in}$), or commonly $7/8\text{ in}$ plate ($0.875\text{ in}$).

Step 2: Formed Heads Thickness Sizing (UG-32)

1. Top 2:1 Semi-Ellipsoidal Head: Corroded inside diameter: $D = 2 R = 2 \times 24.125 = 48.25\text{ in}$.

$$t_{\text{ellip}} = \frac{P D}{2 S E - 0.2 P} = \frac{450 \times 48.25}{2(20,000)(0.85) - 0.2(450)} = \frac{21,712.5}{34,000 - 90} = \frac{21,712.5}{33,910} = 0.6403\text{ in}$$

Total nominal thickness:

$$t_{\text{nom, ellip}} = 0.6403 + 0.125 = 0.7653\text{ in} \implies \text{specify } 13/16\text{ in } (0.8125\text{ in}) \text{ plate}$$

1. Bottom Hemispherical Head:

$$t_{\text{hemi}} = \frac{P R}{2 S E - 0.2 P} = \frac{450 \times 24.125}{34,000 - 90} = \frac{10,856.25}{33,910} = 0.3201\text{ in}$$

Total nominal thickness:

$$t_{\text{nom, hemi}} = 0.3201 + 0.125 = 0.4451\text{ in} \implies \text{specify } 1/2\text{ in } (0.500\text{ in}) \text{ plate}$$

(Notice that the hemispherical head requires only $58\%$ of the nominal thickness of the 2:1 ellipsoidal head).

Step 3: Thin-Wall Ratio Verification

Evaluating the thin-walled criterion on the corroded shell ($t = 0.6489\text{ in}$, $r = 24.125\text{ in}$):

$$\frac{r}{t} = \frac{24.125\text{ in}}{0.6489\text{ in}} = 37.18 \gg 10$$

Because $r/t = 37.2$ is well above $10$, the vessel operates in the thin-walled shell regime.

Step 4: Lamé Thick-Walled Stress Analysis & Factor of Safety

To verify peak bore stress, apply Lamé's formulation using corroded dimensions: $r_i = 24.125\text{ in}$, $t = 0.6489\text{ in} \implies r_o = r_i + t = 24.7739\text{ in}$.

1. Tangential hoop stress at inner bore ($r = r_i$):

$$\sigma_t(r_i) = P \left( \frac{r_o^2 + r_i^2}{r_o^2 - r_i^2} \right) = 450 \left( \frac{24.7739^2 + 24.125^2}{24.7739^2 - 24.125^2} \right)$$ $$r_o^2 = 613.746, \quad r_i^2 = 582.016 \implies r_o^2 + r_i^2 = 1,195.762, \quad r_o^2 - r_i^2 = 31.730$$ $$\sigma_t(r_i) = 450 \times \left( \frac{1,195.762}{31.730} \right) = 450 \times 37.685 = 16,958\text{ psi}$$

(Compare this to the thin-wall formula: $\sigma_h = P R / t = 450 \times 24.125 / 0.6489 = 16,730\text{ psi}$. The Lamé bore stress is $1.4\%$ higher due to curvature).

1. Radial stress at inner bore: $\sigma_r(r_i) = -P = -450\text{ psi}$.
2. Longitudinal stress: $\sigma_l = \frac{P r_i^2}{r_o^2 - r_i^2} = \frac{450 \times 582.016}{31.730} = 8,254\text{ psi}$.
3. Maximum shear stress at inner bore:

$$\tau_{\max} = \frac{\sigma_t(r_i) - \sigma_r(r_i)}{2} = \frac{16,958 - (-450)}{2} = \frac{17,408}{2} = 8,704\text{ psi}$$

1. Static Factor of Safety against yield ($S_y = 38,000\text{ psi}$): Under Tresca MSST ($S_{sy} = 0.5 S_y = 19,000\text{ psi}$):

$$FS_{\text{yield}} = \frac{S_{sy}}{\tau_{\max}} = \frac{19,000\text{ psi}}{8,704\text{ psi}} = 2.18$$

Under von Mises DET:

$$\sigma' = \sqrt{\frac{1}{2}\left[ (16,958 - 8,254)^2 + (8,254 - (-450))^2 + (-450 - 16,958)^2 \right]} = 15,076\text{ psi}$$ $$FS_{\text{DET}} = \frac{S_y}{\sigma'} = \frac{38,000}{15,076} = 2.52$$

The vessel has substantial reserve against yielding, consistent with the margins built into the design-by-rule formulas.