25-Nav-A5 Ship Design · May 2016

Question 7 of 8: Cantilevered Square Bar — Combined Axial Compression and Torsion

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams, May 2016 — 98-Mar-A5 Advanced Strength of Materials, 3 hours, open book, non-communicating calculator permitted (any five of the eight problems constitute a complete paper, all equal value; all eight answered below for full study coverage).

Reference texts: Boresi & Schmidt, Advanced Mechanics of Materials, 6th ed.; Hibbeler, Mechanics of Materials, 10th ed.; Timoshenko & Goodier, Theory of Elasticity, 3rd ed.

It is solved as the strength-of-materials exam it actually is.

Question 7: Cantilevered Square Bar — Combined Axial Compression and Torsion (equal value)

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. Solid square bar, side $b$ (unknown), combined compression $P$ and torque $T$.

Given data
Quantity	Value
Axial force $P$	75 kN (compressive)
Torque $T$	11 kN·m
Yield stress $\sigma_y$	330 MPa
Safety factor	2

Find. Minimum $b$ by (a) Tresca and (b) von Mises.

Cantilevered square bar under compressive axial load P and torque T at the free end.

Approach. On the outer surface of the square bar the governing point sees a uniform compressive normal stress from $P$ and the maximum torsional shear stress from $T$ (solid-square Saint-Venant torsion coefficient $\alpha=0.208$); combine them via the standard axial-plus-torsion equivalent-stress formulas for Tresca and von Mises and solve each for $b$.

Stress state as a function of $b$. $$\sigma=\frac{P}{b^2}\qquad \tau=\frac{T}{\alpha b^3},\ \ \alpha=0.208\ \text{(solid square, Saint-Venant torsion table)}$$
Allowable stress. $\sigma_{\text{allow}}=\sigma_y/\text{SF}=330/2=165\text{ MPa}$.
Part (a) — Tresca. For a uniaxial normal stress combined with shear, $\sigma_{\max}-\sigma_{\min}=\sqrt{\sigma^2+4\tau^2}$: $$\sqrt{\left(\frac{P}{b^2}\right)^2+4\left(\frac{T}{\alpha b^3}\right)^2}=\sigma_{\text{allow}}$$ Solving numerically for $b$: $$\boxed{b_{\text{Tresca}}=86.28\text{ mm}}\qquad(\sigma=10.08\text{ MPa},\ \tau=82.35\text{ MPa}\ \text{at this }b)$$
Part (b) — Von Mises. $\sigma_{VM}=\sqrt{\sigma^2+3\tau^2}=\sigma_{\text{allow}}$: $$\boxed{b_{\text{von Mises}}=82.25\text{ mm}}\qquad(\sigma=11.09\text{ MPa},\ \tau=95.05\text{ MPa}\ \text{at this }b)$$ Von Mises permits a smaller section because it allows a higher effective shear ($\sigma_{\text{allow}}/\sqrt3$) than Tresca ($\sigma_{\text{allow}}/2$) for a shear-dominated stress state.

Question 7 — final results
Criterion	Minimum $b$
Maximum shear stress (Tresca)	86.28 mm
Von Mises	82.25 mm

Topic: Combined axial and torsional loading of a non-circular (solid square) bar, sized by failure theory.

Key relations: solid-square torsion $\tau_{\max}=T/(\alpha b^3)$, $\alpha\approx0.208$; combined-loading Tresca $\sigma_{\max}-\sigma_{\min}=\sqrt{\sigma^2+4\tau^2}$; von Mises $\sqrt{\sigma^2+3\tau^2}$.

Why this works: A point on the surface of the bar under uniform axial stress $\sigma$ and surface shear $\tau$ from torsion is in a 2-D stress state whose principal stresses are $\sigma/2\pm\sqrt{(\sigma/2)^2+\tau^2}$; substituting these into each failure criterion's general form collapses it to the compact $\sqrt{\sigma^2+4\tau^2}$ (Tresca) / $\sqrt{\sigma^2+3\tau^2}$ (von Mises) combined-stress formulas used directly here, both nonlinear in $b$ because $\sigma\propto b^{-2}$ and $\tau\propto b^{-3}$.

Common pitfall: using the circular-shaft torsion formula $\tau=Tc/J$ for a square section (there is no exact closed-form $J$ for a non-circular bar); the correct approach uses the tabulated Saint-Venant coefficient $\alpha$ (and a companion $\beta$ for twist, not needed here) for a rectangular/square cross-section.

Source: Boresi & Schmidt, Advanced Mechanics of Materials, Ch.6 (torsion of noncircular members, rectangular-section table); Hibbeler, Mechanics of Materials, Ch.9 (combined loadings, failure theories).

← Question 6 Next: Question 8 →