5.5 Pressure Vessels, Welds, Fatigue, GD&T, and Design Safety

Pressure vessels: identify the wall model

A pressure-vessel question begins by deciding whether the thin-wall approximation applies. The common FE cue is that wall thickness is small relative to radius — typically t < r/10. For a thin-walled cylinder of internal pressure p, mean radius r, and thickness t:

• Hoop (circumferential) stress: σh = pr/t
• Longitudinal (axial) stress: σl = pr/(2t)

Hoop stress is twice the longitudinal stress, which is why pressurized pipes split along their length, not around their circumference. For a thin sphere, both directions equal pr/(2t), making spheres the most material-efficient pressure container. When t is not small relative to r, the thin-wall formulas break down and thick-wall (Lamé) equations apply, giving a radial stress gradient through the wall.

Worked example. A thin cylinder has p = 2 MPa, inner radius r = 0.5 m, wall t = 10 mm. Hoop stress σh = pr/t = (2×10⁶)(0.5)/(0.010) = 100 MPa; longitudinal σl = 50 MPa. Compare σh to the allowable (Sy/n) to size the wall.

Fatigue: alternating and mean stress

Parts under cyclic loading can fail far below the static yield strength. Fatigue questions resolve the cyclic load into a mean stress σm = (σmax + σmin)/2 and an alternating stress σa = (σmax − σmin)/2. For steels, the endurance limit Se is the stress below which infinite life is expected; a rough starting estimate is Se ≈ 0.5·Su, then reduced by Marin factors for surface finish, size, loading type, temperature, and reliability, plus the fatigue stress-concentration factor Kf.

Mean stress shifts the safe alternating stress. The three FE mean-stress lines connect σa and σm:

Worked example. With Se = 200 MPa, Su = 500 MPa, σa = 100 MPa, σm = 150 MPa, the Goodman left side = 100/200 + 150/500 = 0.5 + 0.3 = 0.8 < 1, so the part survives, and the factor of safety is 1/0.8 = 1.25. Soderberg, using Sy < Su, would give a larger number and predict a smaller margin — that is why it is the conservative choice.

GD&T and the factor of safety

Geometric dimensioning and tolerancing (GD&T) is standardized by ASME Y14.5. It controls form, orientation, location, profile, and runout. Form controls such as flatness and circularity do not require a datum; orientation, location, and runout controls use the applicable datum references. A feature control frame gives the geometric symbol, tolerance and applicable modifiers, followed by datum references where required. Maximum Material Condition (MMC) allows a tolerance to grow (a bonus tolerance) as the feature departs from its maximum-material size — useful for clearance fits. FE questions usually test what a symbol controls and what a datum is, not a full tolerance stack-up.

The factor of safety is the unifying design-safety idea:

n = (material strength) / (applied stress)

Use Sy for a ductile-yielding check. Fatigue requires the applicable endurance or finite-life strength and any mean-stress correction; Su alone is not a fatigue limit. The number chosen reflects load uncertainty, material variability, consequence of failure (life-safety vs. nuisance), inspection access, and applicable codes/standards (e.g., ASME BPVC for pressure vessels, AWS for welds). A higher factor is justified when loads are poorly known or failure is catastrophic. Many FE problems stop at asking you to identify the governing failure mode (yield, fatigue, buckling, fracture) rather than complete a long calculation — recognize the mode, then apply the right strength and factor.

Welds, buckling, and combined-load checks

Welded joints deserve a closer look because the FE tests them often. A fillet weld carries load across its throat, the narrowest section of the triangular bead. For an equal-leg 45° fillet with leg size h, the throat is h/√2 ≈ 0.707h. For effective length L, A = 0.707hL, and direct shear is τ = F/A. When a load is applied eccentrically to a weld group, the weld sees a primary (direct) shear F/A plus a secondary shear from the moment, τ″ = Mc/J with J based on the effective throat area; the two are combined as vectors and the critical location depends on the vector sum. Complete-joint-penetration groove welds, by contrast, can develop load through the full effective joint thickness and are checked in tension or compression like the base metal.

Two more design-safety checks round out the topic. For an ideal slender elastic column, Euler buckling gives Pcr = π²EI/(KL)², using the least relevant flexural stiffness EI. Ideal effective-length factors are K = 1 for pinned-pinned, 0.5 for fixed-fixed, approximately 0.7 for fixed-pinned, and 2.0 for fixed-free. The end restraints and Euler assumptions must match the problem. A column can buckle far below its yield stress, so slenderness — not material strength — governs. Finally, combined loading (bending plus axial, or bending plus torsion on a shaft) is resolved with principal stresses and a yield theory; the distortion-energy (von Mises) stress σ′ = √(σ² + 3τ²) for the common bending-plus-torsion case is a frequent FE result.

The capacity used in a safety-factor check depends on the failure mode: yield strength for ductile yielding, an appropriate fracture criterion for brittle failure, an endurance/fatigue criterion for cyclic loading, and critical load for buckling. Select the mode before forming the capacity-to-demand ratio.

Thin-wall pressure-vessel quick recap

• Hoop (circumferential) stress σ_h = pr/t is twice the longitudinal stress σ_l = pr/2t, so a cylinder fails along a longitudinal seam first.
• A spherical vessel carries σ = pr/2t in every direction, half the cylinder hoop stress for the same p, r, t.
• Thin-wall assumptions hold when r/t ≥ 10; below that, use thick-wall (Lamé) equations.