CELE Strength of Materials — Thin-Walled Pressure VesselsConcept Map

Concept

Thin-Wall Criterion

Sub Concepts

• Wall thickness (t)
• Inner radius (r)
• Ratio t/r ≤ 1/10
• Uniform stress assumption
• Applicability limits

Relationship To Central

Foundational validity condition; all formulas apply only when this criterion is satisfied

Concept

Cylindrical Pressure Vessels

Sub Concepts

• Hoop (circumferential) stress σh = pD/2t = pr/t
• Longitudinal (axial) stress σl = pD/4t = pr/2t
• Stress ratio: σh = 2σl
• Governing stress (hoop)
• Failure mode (longitudinal seam splitting)
• In-plane shear stress
• Absolute maximum shear stress

Relationship To Central

Primary vessel type; exhibits two distinct principal membrane stresses

Concept

Spherical Pressure Vessels

Sub Concepts

• Radial symmetry
• Single principal stress σ = pD/4t = pr/2t
• Equivalence to cylinder longitudinal stress
• Stress efficiency (50% of cylinder hoop)
• High-pressure applications (LPG, gas storage)
• Comparison to cylindrical design

Relationship To Central

Alternative vessel geometry; exhibits single membrane stress in all directions

Concept

Membrane Stresses

Sub Concepts

• Definition: average stress through wall thickness
• Principal stresses (two in cylinder, one in sphere)
• Radial stress (negligible, ≈ 0 in thin-wall)
• Mohr's circle analysis
• Maximum shear stress determination

Relationship To Central

Core stress components governing vessel design and failure

Concept

Design for Allowable Stress

Sub Concepts

• Allowable stress σallow (material strength ÷ safety factor)
• Minimum wall thickness formula
• Gauge pressure vs. absolute pressure
• Inner diameter vs. outer diameter distinction
• Verification of thin-wall assumption post-design
• Safety factor selection

Relationship To Central

Practical engineering methodology for sizing walls and selecting materials

Concept

Joint Efficiency (η)

Sub Concepts

• Definition: 0 < η ≤ 1
• Reduces effective allowable stress at seams
• Longitudinal seam (hoop stress governs)
• Circumferential seam (longitudinal stress governs)
• Typical values (0.85–1.0 for welded/riveted vessels)
• Integration into wall thickness formula

Relationship To Central

Empirical reduction factor accounting for weld/rivet seam weakness

Concept

Equilibrium Analysis

Sub Concepts

• Free-body diagram methodology
• Longitudinal half-cut equilibrium (hoop derivation)
• Transverse cut equilibrium (longitudinal derivation)
• Hemispherical cap equilibrium (sphere derivation)
• No thick-wall (Lamé) theory required

Relationship To Central

Mathematical foundation; derives stress formulas from force and moment balance

Concept

Practical Applications in Civil Engineering

Sub Concepts

• Water penstocks (hydroelectric schemes)
• Elevated water storage tanks
• Boilers and steam systems
• Compressed air reservoirs
• Underground/above-ground pipelines
• Liquefied gas (LPG) storage (spherical)
• Pressure distribution from hydraulics (Bernoulli connection)

Relationship To Central

Real-world relevance and context for Philippine civil engineering practice

Concept

Material Selection and Properties

Sub Concepts

• Tensile yield strength (σy)
• Ultimate tensile strength (σult)
• Ductility (for safety)
• Corrosion resistance (internal + external environment)
• Fatigue considerations (if cyclic pressure)
• Impact resistance (low temperature)

Relationship To Central

Links vessel design to material behavior and strength characteristics

Concept

Critical Design Pitfalls

Sub Concepts

• Confusing hoop and longitudinal stresses
• Using outer diameter instead of inner
• Neglecting joint efficiency
• Skipping thin-wall validity check
• Applying cylinder formulas to a sphere
• Confusing gauge vs. absolute pressure
• Oversizing due to unjustified safety factor

Relationship To Central

Common errors that lead to undersized/incorrect vessels or analysis mistakes

To

Cylindrical Pressure Vessels

From

Thin-Wall Criterion

Strength

strong

Relationship

Validity prerequisite; all cylindrical stress formulas assume t/r ≤ 1/10

To

Spherical Pressure Vessels

From

Thin-Wall Criterion

Strength

strong

Relationship

Validity prerequisite; all spherical stress formulas assume t/r ≤ 1/10

To

Membrane Stresses

From

Cylindrical Pressure Vessels

Strength

strong

Relationship

Cylinders exhibit two principal membrane stresses (hoop and longitudinal)

To

Membrane Stresses

From

Spherical Pressure Vessels

Strength

strong

Relationship

Spheres exhibit a single isotropic membrane stress in all directions

To

Cylindrical Pressure Vessels

From

Equilibrium Analysis

Strength

strong

Relationship

Mathematical foundation; longitudinal and transverse cuts derive hoop and longitudinal formulas

To

Spherical Pressure Vessels

From

Equilibrium Analysis

Strength

strong

Relationship

Mathematical foundation; hemispherical cap equilibrium derives the single stress formula

To

Cylindrical Pressure Vessels

From

Design for Allowable Stress

Strength

strong

Relationship

Practical application; hoop stress governs cylinder wall thickness sizing

To

Spherical Pressure Vessels

From

Design for Allowable Stress

Strength

strong

Relationship

Practical application; single stress governs sphere wall thickness sizing

To

Design for Allowable Stress

From

Joint Efficiency

Strength

strong

Relationship

Reduction factor η reduces allowable stress at welded or riveted seams during design

To

Cylindrical Pressure Vessels

From

Joint Efficiency

Strength

strong

Relationship

Hoop stress governs longitudinal seams; longitudinal stress governs circumferential seams

To

Design for Allowable Stress

From

Material Selection and Properties

Strength

strong

Relationship

Material strength and safety factor determine allowable stress σallow used in design formulas

To

Cylindrical Pressure Vessels

From

Practical Applications in Civil Engineering

Strength

moderate

Relationship

Water penstocks, pipelines, boiler tubes, and elevated tanks commonly use cylindrical geometry

To

Spherical Pressure Vessels

From

Practical Applications in Civil Engineering

Strength

moderate

Relationship

LPG storage, high-pressure gas vessels, and nuclear containment favor spherical shapes

To

Cylindrical Pressure Vessels

From

Critical Design Pitfalls

Strength

moderate

Relationship

Common errors include confusing hoop/longitudinal stress ratio and missing thin-wall check

To

Design for Allowable Stress

From

Critical Design Pitfalls

Strength

moderate

Relationship

Errors include forgetting joint efficiency η, using wrong diameter, and unjustified safety margins

To

Mohr's Circle Analysis

From

Membrane Stresses

Strength

moderate

Relationship

Mohr circle construction determines in-plane and absolute maximum shear stresses from principal stresses

To

Spherical Pressure Vessels

From

Cylindrical Pressure Vessels

Strength

strong

Relationship

Comparative analysis: sphere requires half the wall thickness of cylinder for same conditions (efficiency gain)

To

Material Selection and Properties

From

Design for Allowable Stress

Strength

moderate

Relationship

Material choice determines σallow; must balance strength, corrosion resistance, and cost

To

Equilibrium Analysis

From

Thin-Wall Criterion

Strength

moderate

Relationship

Thin-wall assumption simplifies equilibrium (no Lamé thick-wall theory needed)

To

Practical Applications in Civil Engineering

From

Joint Efficiency

Strength

moderate

Relationship

Real vessels have welds/rivets; η values depend on inspection and fabrication methods