Thick-Walled Cylinder — von Mises and Tresca Pressure Limits: 22-Mec-A7 Advanced Strength of Materials, December 2013, Q5

Question 5: Thick-Walled Cylinder — von Mises and Tresca Pressure Limits (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A closed thick cylinder with a fixed pressure ratio p_i = 4.5 p_e, sized by yield at the bore.

Given data
Radii	a = 60 mm (bore), b = 85 mm (outer)
Material	Elastic limit σ_Y = 300 MPa, ν = 0.3
Loading	p_i = 4.5 p_e (internal / external pressure)
Assumption	Closed ends ⇒ σ_z = (σ_r+σ_θ)/2 (intermediate principal stress)

Find. The allowable internal pressure p_i from (a) von Mises and (b) maximum-shear (Tresca).

Bore radius a, outer radius b; internal pressure p_i, external p_e = p_i/4.5. Yield governs at r = a.

Approach. Lamé’s solution gives σ_r, σ_θ, and σ_z = (σ_r+σ_θ)/2 as multiples of p_i at the most-stressed point (the bore). Setting each yield criterion equal to σ_Y gives the allowable p_i.

Lamé constants. With p_e = p_i/4.5, a² = 3600, b² = 7225, $$A=\frac{p_ia^{2}-p_eb^{2}}{b^{2}-a^{2}}=0.5502\,p_i,\qquad B=\frac{(p_i-p_e)a^{2}b^{2}}{b^{2}-a^{2}}=5582\,p_i.$$
Principal stresses at the bore (r = a). $$\sigma_\theta=A+\tfrac{B}{a^{2}}=2.100\,p_i,\qquad \sigma_r=A-\tfrac{B}{a^{2}}=-p_i,\qquad \sigma_z=A=0.550\,p_i.$$
Maximum-shear (Tresca) criterion (part b). The extreme principal stresses are σ_θ and σ_r: $$\sigma_\theta-\sigma_r=3.100\,p_i=\sigma_Y\ \Longrightarrow\ \boxed{p_i=\frac{300}{3.100}=96.8\ \text{MPa}.}$$
von Mises criterion (part a). With all three principal stresses, $$\sigma_{vM}=\sqrt{\tfrac12\big[(\sigma_\theta-\sigma_r)^2+(\sigma_r-\sigma_z)^2+(\sigma_z-\sigma_\theta)^2\big]}=2.685\,p_i=\sigma_Y,$$ $$\boxed{p_i=\frac{300}{2.685}=111.7\ \text{MPa}.}$$
Compare. von Mises permits about 15 % more pressure than Tresca (111.7 vs 96.8 MPa); the maximum-shear criterion is the more conservative (safe) design value. The corresponding external pressures are p_e = 24.8 MPa and 21.5 MPa.

Results — Question 5
Criterion	Allowable p_i	Corresponding p_e
von Mises	111.7 MPa	24.8 MPa
Maximum shear (Tresca)	96.8 MPa	21.5 MPa

Topic: Elastic thick-walled pressure vessel (Lamé theory) combined with the two classical yield criteria.

Key relations: σ_r,θ = A ∓ B/r²; for closed ends σ_z = A = (σ_r+σ_θ)/2. Tresca: σ_max − σ_min = σ_Y. von Mises: the root-mean-square of the principal-stress differences equals σ_Y.

Why this works: the hoop stress is largest at the bore, so yield initiates there. Tresca ignores the intermediate stress σ_z and therefore predicts yielding at a lower pressure; von Mises accounts for all three and gives a higher (less conservative) limit.

Common pitfall: evaluating the stresses at the outer wall (they are smaller there), omitting the axial stress for a closed vessel, or dropping the external pressure — here p_e is a fixed fraction of p_i and must be carried in both Lamé constants.

Source: Boresi & Schmidt, Advanced Mechanics of Materials 6th ed., Ch. 11 (thick-wall cylinders) with the von Mises / Tresca criteria of Ch. 4.

Question 5 of 7: Thick-Walled Cylinder — von Mises and Tresca Pressure Limits

Question 5: Thick-Walled Cylinder — von Mises and Tresca Pressure Limits (20 marks)