25-Nav-A4 Ship Structure and Strength of Ships · May 2013
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Exams — May 2013 — 98-Nav-A4 Ship Structure and Strength of Ships. Three-hour, closed-book exam (no notes permitted); Casio/Sharp non-programmable calculator and simple drawing equipment allowed. Format: five compulsory questions, marks indicated per sub-part, totalling 100; some formulae (fixed-end loads, beam deflection/slope tables, 2D beam-element stiffness) are supplied at the end of the exam and are used directly below. All five are solved in full.
Reference texts: Hughes, O.F. & Paik, J.K., Ship Structural Analysis and Design (2nd ed., SNAME, 2010) — hull-girder strength, panel/plate structure, section properties and shear flow in thin-walled hull sections; Hibbeler, R.C., Mechanics of Materials (10th ed., Pearson) — beam bending/deflection, stress transformation and Mohr's circle; Muckle, W., Muckle's Naval Architecture (2nd ed., Butterworths) — structural terminology and conventions; IACS Common Structural Rules / classification-society rules — steel grades, structural detail classification and fatigue design (S–N curves).
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.
i. Von Mises failure criterion. The von Mises (distortion-energy) criterion predicts that a ductile material begins to yield when the distortion (shape-change) component of strain energy at a point reaches the distortion energy stored at yield in a simple uniaxial tension test. Equivalently, yielding is predicted once a single scalar — the von Mises equivalent stress — reaches the material's uniaxial yield strength $\sigma_y$. It is the most widely used criterion for ductile metals (including shipbuilding steels) because it correctly predicts that yielding is insensitive to hydrostatic (all-round) stress and depends only on the deviatoric (shape-distorting) part of the stress state.
ii. Von Mises equivalent stress. For general principal stresses $\sigma_1,\sigma_2,\sigma_3$, the equivalent (von Mises) stress is $$\sigma_{eq}=\sqrt{\tfrac12\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right]},$$ or, for a general 2D (plane) stress state with in-plane shear $\tau_{xy}$, $\sigma_{eq}=\sqrt{\sigma_x^2-\sigma_x\sigma_y+\sigma_y^2+3\tau_{xy}^2}$. It collapses any multiaxial stress state into one number directly comparable with $\sigma_y$ from a simple tension test, which is exactly why it is the standard yield check for a multiaxially-loaded ship structural detail (e.g. plating under combined bending and shear).
iii. Mohr's circle, uniaxial stress. With only $\sigma_x=\sigma$ applied (all other stress components zero), the circle is centred at $(\sigma/2,0)$ with radius $\sigma/2$, so it passes through the origin (representing the stress-free plane perpendicular to loading) and through $(\sigma,0)$ (the loaded plane). The maximum shear stress, at the top of the circle, is $\tau_{max}=\sigma/2$ — the classical result that a uniaxial tension test also produces a 45° shear plane at half the applied normal stress.
iv. Mohr's circle, pure shear. With only $\tau_{xy}=\tau$ applied ($\sigma_x=\sigma_y=0$), the circle is centred at the origin with radius $\tau$. The principal stresses, read at the circle's intersections with the $\sigma$-axis, are $\sigma_1=+\tau$ and $\sigma_2=-\tau$, occurring on planes rotated 45° from the shear planes — the classic result that pure shear is equivalent to equal-and-opposite biaxial tension/compression at 45°.
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.
i. The S–N curve. A plot (typically log–log) of applied cyclic stress range or amplitude $S$ against the number of cycles to failure $N$ for a given material or, more usefully in ship structures, for a given welded structural detail. Welded-steel S–N curves are usually a straight line on log–log axes of slope $m\approx3$ (per IIW/DNV/classification-society classification), sometimes with a fatigue (endurance) limit or a change of slope at very high cycle counts; the curve is the basic input to any fatigue-life calculation.
ii. Miner's Rule. The linear cumulative-damage rule: for a load history broken into blocks of $n_i$ applied cycles at stress range $S_i$, each block causes fatigue damage $n_i/N_i$ (where $N_i$ is the S–N-curve life at $S_i$), and total damage accumulates linearly, $D=\sum n_i/N_i$; fatigue failure is predicted once $D=1$. It is the standard (if approximate — it ignores load-sequence effects) way to combine a realistic, variable-amplitude stress spectrum into a single life prediction.
iii. Connection details. In fatigue design, ship structural connections (weld toes, bracket toes, cut-outs, stiffener terminations) are not all treated the same: each geometry is assigned to a standard "detail class" (e.g. Class B through W in the DNV/IIW/class-society system) based on the local stress concentration and weld quality it produces, and each class has its own S–N curve. This matters because fatigue life is governed by the local (hot-spot) stress at the detail, not by the nominal gross-section stress, so two members carrying identical nominal stress can have very different fatigue lives if their connection details differ.
