Question 1 of 6: Short-Answer Set — Proof Stress, Hollow Shafts, Mean Stress & Biaxial Yield

Question 1: Short-Answer Set — Proof Stress, Hollow Shafts, Mean Stress & Biaxial Yield (10 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.

(a) The 0.2 % offset proof (yield) stress. For ductile metals such as aluminium alloys and many stainless steels the stress–strain curve has no sharp yield point, so a yield strength must be defined rather than read off directly. $\sigma_{0.2}$ is the stress at which a line drawn parallel to the initial elastic slope, but offset by a plastic strain of $0.2\% = 0.002$, intersects the curve. It is the conventional yield strength $S_y$ used in design for these materials, marking the onset of appreciable permanent deformation.

(b) Hollow versus solid shaft. In torsion and bending, material near the neutral axis carries very little stress because stress varies linearly with radius. Removing that lightly-stressed core gives a hollow shaft a far higher strength- and stiffness-to-weight ratio: for the same outer diameter the polar and second moments of area fall only slightly, while the mass falls in proportion to the removed area. A hollow shaft therefore transmits nearly the same torque and resists nearly the same bending as a solid one of equal outer size but weighs much less — important for rotating machinery, drivelines and marine propulsion shafting, where inertia and weight matter. The disadvantages: it is more expensive to manufacture (boring, or seamless-tube processes), it has a thin wall that is prone to local buckling and denting, it is harder to key, press-fit or machine features onto, and stress concentrations at the bore or at cross-holes are less forgiving.

(c) Effect of mean stress on fatigue. Fatigue strength is governed primarily by the alternating stress $\sigma_a$, but a superimposed tensile mean stress $\sigma_m$ is damaging: it holds fatigue cracks open, accelerates their growth and lowers the alternating stress the part can survive for a given life. This trade-off is captured by the mean-stress lines — Goodman, Gerber and Soderberg — on a $\sigma_a$–$\sigma_m$ diagram. The modified-Goodman relation is

so as $\sigma_m$ rises toward $S_{ut}$ the permissible $\sigma_a$ falls to zero. A compressive mean stress is by contrast beneficial — the reason shot-peening and surface rolling, which induce compressive residual stresses, improve fatigue life.

(d) Higher yield capacity in plane-strain equi-biaxial tension. Yielding depends on the difference between principal stresses (shear), not their absolute size. In plane-strain equi-biaxial tension the in-plane stresses are equal, $\sigma_1=\sigma_2=\sigma$, and the constraint against through-thickness contraction develops a third stress $\sigma_3=\nu(\sigma_1+\sigma_2)$. Because the principal stresses are close together the maximum shear stress is small for a given $\sigma$, so a larger applied stress is needed to reach the yield shear. By the Tresca criterion the axial stress at yield rises to

i.e. about 15 % more load-carrying capacity than the uniaxial case, where the full applied stress appears directly as the yielding shear.