22-Mec-B1 Advanced Machine Design · December 2014
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Reference texts: Budynas & Nisbett, Shigley’s Mechanical Engineering Design, 10th ed. (shafts §7, bolted joints §8, journal bearings §12, clutches/brakes §16); Norton, Machine Design; Juvinall & Marshek, Fundamentals of Machine Component Design; Hibbeler, Mechanics of Materials.
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.
The material will not yield. Both accepted static-failure theories for ductile metals respond only to the differences among the principal stresses, i.e. to the deviatoric (distortional) part of the stress state, not to the hydrostatic mean. With $\sigma_1=\sigma_2=\sigma_3=S_y$ the stress state is purely hydrostatic, so there is no shear anywhere and no distortion energy is stored.
By the maximum-shear-stress (Tresca) theory the governing shear is $\tau_{\max}=\tfrac{1}{2}(\sigma_1-\sigma_3)=0$, which is far below the yield shear $S_y/2$. By the distortion-energy (von Mises) theory the equivalent stress is
$$\sigma' = \sqrt{\tfrac{1}{2}\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right]} = 0 \;\lt\; S_y.$$
Both criteria give a factor of safety that is effectively unbounded. Physically, a material can sustain hydrostatic pressures many times its yield strength without permanent deformation — deep-ocean and high-pressure-vessel experience confirms this. Yielding requires shape change, and equal triaxial stress produces only volume change.
A hollow shaft is preferred because it removes material from near the neutral axis, where that material carries very little bending or torsional stress but still adds weight. Because the polar (and area) second moments grow with the fourth power of radius while mass grows only with the square, moving the same material to a larger mean radius gives a markedly higher strength-to-weight and stiffness-to-weight ratio. For a given torque and allowable stress a hollow shaft is lighter, and for a given weight it is stiffer and stronger.
The disadvantages are practical: a hollow shaft costs more to manufacture (boring, or forming and welding a tube), its thinner wall is more vulnerable to local buckling/crippling and to denting, keyways and splines are harder to cut and weaken the thin wall more severely, and connecting to it (press fits, couplings) is more difficult than with a solid section.
The square section needs less material. For pure bending the strength of a section is set by its elastic section modulus $Z=I/c$; the efficiency of a shape is measured by the dimensionless ratio $Z/A^{3/2}$ (section modulus per unit of cross-sectional area, area being the direct measure of material used). For the two shapes:
$$\left(\frac{Z}{A^{3/2}}\right)_{\text{square}} = \frac{s^3/6}{(s^2)^{3/2}} = \frac{1}{6} = 0.1667,\qquad \left(\frac{Z}{A^{3/2}}\right)_{\text{circle}} = \frac{\pi d^3/32}{(\pi d^2/4)^{3/2}} = \frac{8}{32\sqrt{\pi}} = 0.1411.$$
The square is the more efficient shape. Equating the section moduli of the two ($s^3/6=\pi d^3/32$) and comparing areas gives $A_{\text{square}}/A_{\text{circle}}=0.894$: the square carries the same bending strength with about 11 % less cross-sectional area, hence less material and less weight. (An I-section would be better still, but among these two the square wins.)
Mean stress shifts the alternating stress a part can survive. A tensile mean stress is detrimental: it reduces the permissible alternating amplitude and shortens fatigue life, because it holds micro-cracks open and promotes their growth. The effect is captured by the mean-stress failure lines — Goodman, Gerber, or Soderberg — on which the safe alternating stress $\sigma_a$ falls steadily as the mean stress $\sigma_m$ rises toward the ultimate (or yield) strength. A compressive mean stress is beneficial, increasing fatigue strength, which is exactly why shot-peening, cold-rolling and case-hardening (all of which induce compressive residual surface stress) are used to improve fatigue performance.
