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Mechanical Engineering · 400 Level
This is the complete lecture note for MEE406 Advanced mechanics of materials, prepared for Mechanical Engineering students at 400 Level. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
Course code: MEE 406 Course title: Advanced Mechanics of Materials Course credit load: 2 Units Series 1: Thick Cylinders and Compound Cylinders • Part 1.1: Stress Analysis in Thick Cylinders o Sub Part 1.1.1: Lamé's Equations for Thick Cylinders o Sub Part 1.1.2: Radial and Hoop Stress Distribution o Sub Part 1.1.3: Thick Cylinders Under Internal and External Pressure • Part 1.2: Design Considerations for Thick Cylinders o Sub Part 1.2.1: Maximum Shear Stress in Thick Cylinders o Sub Part 1.2.2: Wall Thickness Selection o Sub Part 1.2.3: Yielding and Factor of Safety in Thick Cylinder Design • Part 1.3: Compound Cylinders o Sub Part 1.3.1: Principle of Compounding Cylinders o Sub Part 1.3.2: Shrink-Fit Interference and Initial Stresses o Sub Part 1.3.3: Stress Distribution in a Compound Cylinder Under Pressure • Part 1.4: Design and Application of Compound Cylinders o Sub Part 1.4.1: Determining Optimum Shrink-Fit Allowance o Sub Part 1.4.2: Autofrettage as an Alternative to Compounding o Sub Part 1.4.3: Practical Applications of Compound Cylinders Introduction A thin-walled pressure vessel, treated in earlier coursework using a simplified uniform stress assumption across its wall thickness, no longer gives an accurate picture once that wall becomes genuinely thick relative to the cylinder's radius, and this opening Series examines the more
rigorous thick cylinder theory this situation demands, before extending that theory to compound cylinders, an assembly of two cylinders deliberately fitted together to withstand considerably higher pressure than either cylinder alone could safely contain. The aim of this Series is to equip you with the ability to derive and apply Lamé's equations for stress distribution in a thick cylinder, to select an appropriate wall thickness for a given design pressure, and to analyse and design compound cylinders using shrink-fit interference or autofrettage. We shall commence this Series with stress analysis in thick cylinders, before turning to design considerations for thick cylinders. We will then examine compound cylinders, and conclude this Series by considering the design and application of compound cylinders. Series Learning Outcomes Upon completing this Series, you should be able to: • SLO 1.1: derive Lamé's equations for a thick cylinder, • SLO 1.2: determine radial and hoop stress distribution in a thick cylinder, • SLO 1.3: apply the maximum shear stress criterion to thick cylinder design, • SLO 1.4: select wall thickness for a thick cylinder under given pressure and safety requirements, • SLO 1.5: explain the principle of compounding cylinders, • SLO 1.6: determine stresses in a compound cylinder arising from shrink-fit interference, • SLO 1.7: analyse the combined stress distribution in a compound cylinder under working pressure, • SLO 1.8: compare compounding and autofrettage as methods of improving cylinder strength, 1.1 Stress Analysis in Thick Cylinders 1.1.1 Lamé's Equations for Thick Cylinders Lamé's equations express the radial and hoop, or circumferential, stress at any radius within a thick-walled cylindrical vessel subjected to internal or external pressure, derived by considering the equilibrium of a thin cylindrical element at that radius together with the compatibility
requirement that the material's radial and circumferential strains remain consistent with a single, continuous radial displacement field throughout the cylinder's wall thickness. These equations assume the cylinder is sufficiently long that end effects can be neglected, that the material behaves elastically, and that plane sections remain plane, assumptions that hold well for a genuinely thick cylinder but that a thin-walled cylinder's simplified uniform stress treatment does not require, since a thin wall's stress variation across its thickness is negligible enough to ignore entirely. Worked example: Deriving Lamé's equations for a thick cylinder begins by considering the radial equilibrium of a thin cylindrical shell element at an arbitrary radius, then introducing the compatibility condition linking radial and hoop strain through a single displacement field. Application: Recognising when a cylinder is genuinely thick enough to require Lamé's equations, rather than the simpler thin-walled treatment, is the essential first judgement before analysing any pressurised cylindrical vessel. Fill in the blank: Lamé's equations are derived from equilibrium together with a compatibility condition linking radial and hoop ________. 1.1.2 Radial and Hoop Stress Distribution The radial stress within a thick cylinder under internal pressure varies from a maximum compressive value at the bore, equal in magnitude to the internal pressure itself, to zero at the outer surface if that surface is unpressurised, while the hoop stress varies in the opposite sense, reaching its maximum tensile value at the bore and decreasing, though remaining tensile, toward the outer surface, this hoop stress at the bore being the value that typically governs a thick cylinder's design against yielding. This non-uniform stress distribution across the wall thickness, considerably more pronounced the thicker the cylinder relative to its bore radius, is precisely why a thin-walled treatment, which assumes uniform hoop stress throughout the wall, becomes progressively less accurate, and eventually genuinely unsafe if relied upon, as a cylinder's wall thickness grows relative to its radius. Worked example: Plotting the radial and hoop stress distributions across a thick cylinder's wall thickness under internal pressure confirms that hoop stress is highest at the bore, precisely where a thin-walled analysis would instead assume a uniform, and therefore understated, stress value. Application: Recognising that hoop stress peaks at the bore, and that this peak value typically governs design, focuses design attention correctly on the cylinder's inner surface rather than an average value across the wall. Fill in the blank: Hoop stress in a thick cylinder under internal pressure reaches its maximum value at the ________.
1.1.3 Thick Cylinders Under Internal and External Pressure A thick cylinder subjected to internal pressure alone develops tensile hoop stress throughout its wall, decreasing from the bore outward, whereas a cylinder subjected to external pressure alone instead develops compressive hoop stress throughout its wall, a fundamentally different stress state that must be checked separately against buckling as well as yielding, since a sufficiently thin cylinder under external pressure can fail by elastic instability well before the material itself yields. Lamé's equations readily extend to the more general case of a cylinder subjected to both internal and external pressure simultaneously, a situation frequently arising in compound cylinder analysis, examined later in this Series, where the inner cylinder experiences both the working internal pressure and the external pressure imposed on it by the surrounding outer cylinder. Worked example: Applying Lamé's equations to a cylinder subjected to internal pressure on its bore and a separate external pressure on its outer surface calculates the resulting combined radial and hoop stress distribution directly, superimposing the two pressure effects correctly within the same equations. Application: Recognising that Lamé's equations extend directly to combined internal and external pressure prepares the ground for the compound cylinder analysis examined later in this Series, where exactly this combined loading arises. Fill in the blank: A thick cylinder under external pressure alone must be checked against buckling as well as against ________. Figure 1.1: Radial and hoop stress distribution across a thick cylinder wall under internal pressure.
Pilot questions 1.1 1. Describe how Lamé's equations are derived. (Linked to SLO 1.1) 2. Explain the assumptions underlying Lamé's equations. (Linked to SLO 1.1) 3. Describe how radial and hoop stress vary across a thick cylinder's wall under internal pressure. (Linked to SLO 1.2) 4. Explain why the thin-walled assumption becomes inaccurate for a genuinely thick cylinder. (Linked to SLO 1.2) 5. Explain why a cylinder under external pressure must be checked against buckling. (Linked to SLO 1.2) 1.2 Design Considerations for Thick Cylinders 1.2.1 Maximum Shear Stress in Thick Cylinders The maximum shear stress at any point within a thick cylinder is half the difference between the two principal stresses, radial and hoop, acting at that point, and since both stresses reach their most extreme values at the bore under internal pressure, the maximum shear stress within the entire cylinder likewise occurs at the bore, making this location the critical section governing yielding according to the maximum shear stress, or Tresca, failure criterion widely used in thick cylinder design. Applying the maximum shear stress criterion to thick cylinder design sets this bore maximum shear stress equal to half the material's yield stress, divided by an appropriate factor of safety, providing the governing design equation from which either an acceptable working pressure for a given cylinder geometry, or a required wall thickness for a given design pressure, can be calculated directly. Worked example: Calculating the maximum shear stress at a thick cylinder's bore from its radial and hoop stresses there, then comparing this value against half the material's allowable stress, applies the maximum shear stress criterion directly to check that cylinder's safety under a specified working pressure. Application: Focusing the maximum shear stress check specifically at the bore, where both principal stresses reach their extreme values, correctly identifies the critical section governing a thick cylinder's design against yielding. Fill in the blank: The maximum shear stress at any point in a thick cylinder is half the difference between the radial and ________ stresses. 1.2.2 Wall Thickness Selection
Selecting a thick cylinder's wall thickness for
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