25-Nav-A4 Ship Structure and Strength of Ships · May 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Exams — May 2016 — 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: six compulsory questions, marks indicated per sub-part, totalling 100; some formulae (beam bending relations, section modulus, shear flow, deflection/slope tables) and a Normal (cumulative) distribution table are supplied at the end of the exam and are used directly below. All six 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–strain behaviour and fatigue basics; Muckle, W., Muckle’s Naval Architecture (2nd ed., Butterworths) — hydrostatics, Bonjean curves and structural terminology; Ang, A.H-S. & Tang, W.H., Probability Concepts in Engineering (2nd ed., Wiley) — structural reliability, load/resistance margin; IACS Common Structural Rules — steel grades, fatigue design (S–N curves, Paris Law).
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. Young's Modulus and Post Yield Modulus. Young's modulus $E$ is the slope of the initial linear-elastic portion of the stress–strain curve, $\sigma=E\varepsilon$, and is a true material constant (independent of yield strength). The post-yield modulus is the (much smaller, sometimes near-zero) slope of the stress–strain curve after yielding, in the strain-hardening region; for mild steel it is typically only a few percent of $E$, which is why the elastic–perfectly plastic idealization (post-yield modulus $=0$, Q1a.v) is such a good approximation for structural design.
ii. 3D state of stress. At any point in a loaded body, the general stress state is described by a symmetric $3\times3$ stress tensor with six independent components (three normal stresses $\sigma_x,\sigma_y,\sigma_z$ and three shear stresses $\tau_{xy},\tau_{yz},\tau_{zx}$) rather than the single normal stress of simple uniaxial tension; a 3D state can always be resolved into three mutually perpendicular principal stresses $\sigma_1\ge\sigma_2\ge\sigma_3$ with zero shear on the principal planes, and yield criteria such as von Mises (given on the formula sheet) are stated in terms of all three principal stresses.
iii. Engineering stress vs. true stress. Engineering stress divides the applied load by the original (undeformed) cross-sectional area, $\sigma_{eng}=P/A_0$; true stress divides by the actual, current (necked-down) area, $\sigma_{true}=P/A$. Because the area shrinks as the specimen elongates in tension, $\sigma_{true}>\sigma_{eng}$ once necking begins, and the engineering stress–strain curve shows an apparent post-ultimate softening that is purely a bookkeeping artifact of using the original area — the true stress in fact continues to rise (the material is still strain-hardening) right up to fracture.
iv. Single fillet weld. A weld deposited along the corner formed by two plates meeting at an angle (typically 90°, e.g. a stiffener web to a plate) on one side only, producing an approximately triangular weld cross-section; its throat thickness (the shortest distance from the weld root to the weld face) governs its shear capacity. Because it is welded on only one side, a single fillet weld is inherently eccentric to the joined plates' mid-planes, inducing a secondary bending stress in the joint that a double (both-sided) fillet weld avoids — a key reason single fillet welds are used only for lightly loaded or non-fatigue-critical connections.
i. Stress intensity. The stress intensity factor $K$ quantifies the magnitude of the stress field local to a crack tip, $K=Y\sigma\sqrt{\pi a}$, where $\sigma$ is the remote applied stress, $a$ the crack length, and $Y$ a geometry correction factor; it is the single parameter (from linear elastic fracture mechanics) that governs both the onset of unstable brittle fracture ($K$ reaching the material's fracture toughness $K_{IC}$) and, under cyclic loading, the rate of fatigue crack growth.
ii. Paris Law. The empirical fatigue-crack-growth relation $$\frac{da}{dN}=C(\Delta K)^m$$ giving the crack-growth increment per load cycle as a power law in the cyclic stress intensity range $\Delta K=K_{max}-K_{min}$, with material constants $C$ and $m$ (typically $m\approx3$ for structural steel); it describes the middle, roughly log-linear region of the crack-growth-rate curve and is integrated to predict fatigue life from an initial (e.g. inspection-detectable) crack size to a critical (fracture) size.
iii. Stress range. The stress range $\Delta\sigma=\sigma_{max}-\sigma_{min}$ is the peak-to-trough swing of a cyclic stress, and it is the controlling parameter (not the mean or the maximum stress alone) for fatigue crack initiation and growth in welded steel structures — the basis of the S–N (stress-range vs. cycles-to-failure) curves used in classification-society fatigue design.
