Shear in Ships: 25-Nav-A4 Ship Structure and Strength of Ships, May 2016, Q6

Question 6: Shear in Ships (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.

Given. An open channel section (centreline dimensions): top flange 200 mm long × 7 mm thick, web 150 mm tall × 5 mm thick, bottom flange 200 mm long × 5 mm thick; vertical shear $V=100\text{ kN}$. Point B is the top of the web, at the flange-web junction (marked on the source figure).

Check

Dimensions are taken as the printed (centreline) lengths directly, per standard thin-wall shear-flow practice; the shear stress at B uses the web thickness (5 mm), since the marked point sits on the web immediately below the flange-web corner.

Find. Shear flow $q(s)$ around the open section and the shear stress at B.

Approach. Locate the section's neutral axis and $I_{NA}$ (composite-section method, as in Q3a), then apply the open-section shear-flow formula $q=VQ/I$ starting from each free edge (where $q=0$) and building up toward the web.

(a) Section properties. $A_{top}=200(7)=1400$, $A_{bot}=200(5)=1000$, $A_{web}=150(5)=750\ \text{mm}^2$; $A_{tot}=3150\ \text{mm}^2$. Neutral axis (from the bottom flange centreline): $$\bar y=\frac{1400(150)+1000(0)+750(75)}{3150}=\boxed{84.5\text{ mm}}$$ Moment of inertia (parallel-axis theorem on all three pieces): $$\boxed{I_{NA}=14.63\times10^6\ \text{mm}^4}$$
Shear flow entering the web from the top flange (at B). Sweeping the full top flange from its free tip to the web junction, $Q_B=A_{top}\,d_{top}=1400(65.5)=91{,}667\ \text{mm}^3$ (with $d_{top}=150-84.5=65.5\text{ mm}$): $$q_B=\frac{VQ_B}{I_{NA}}=\frac{100{,}000(91{,}667)}{14.63\times10^6}=\boxed{626.6\text{ N/mm}}$$
Shear flow entering the web from the bottom flange. $Q_C=A_{bot}\,d_{bot}=1000(84.5)=84{,}524\ \text{mm}^3$: $$q_C=\frac{VQ_C}{I_{NA}}=\frac{100{,}000(84{,}524)}{14.63\times10^6}=\boxed{577.8\text{ N/mm}}$$
Maximum shear flow (at the neutral axis, within the web). Adding the web area below the N.A. to $Q_C$: $Q_{max}=102{,}385\ \text{mm}^3$, giving $$q_{max}=\frac{V\,Q_{max}}{I_{NA}}=\boxed{699.9\text{ N/mm}}$$ — consistent with the general result that shear flow peaks at the neutral axis. Shear flow is zero at both free flange tips and rises monotonically toward the web in each flange, then continues rising through the web to its maximum at the N.A.
(b) Shear stress at B. Dividing the shear flow entering the web at B by the web thickness: $$\tau_B=\frac{q_B}{t_{web}}=\frac{626.6}{5}=\boxed{125.3\text{ MPa}}$$

Figure 6.1 — channel-section shear flow diagram: zero at both free flange tips, rising through each flange to the web, peaking at the neutral axis.

Quantity	Value
Neutral axis (from bottom flange)	84.5 mm
Moment of inertia, I_NA	14.63 × 10&sup6; mm⁴
Shear flow at B (top flange/web junction)	626.6 N/mm
Shear flow at bottom flange/web junction	577.8 N/mm
Max shear flow (at N.A.)	699.9 N/mm
Shear stress at B	125.3 MPa

Concept & Theory

Topic. Shear flow and shear stress in open thin-walled sections under transverse shear.

Key relations. $q(s)=VQ(s)/I_{NA}$, where $Q(s)$ is the first moment (about the N.A.) of the area between a free edge and the point $s$; $\tau=q/t$.

Why this works. In an open section, shear flow must start at zero at every free edge (there is nothing beyond the edge to transmit force to), which is exactly why the sweep in each flange begins at its tip; the flow accumulates additively as more area is swept, so it necessarily peaks where the first moment of the swept area is largest — which for a single-N.A. section is always at the N.A. itself.

Common pitfall. Using the flange thickness instead of the web thickness (or vice versa) when converting the shear flow at a flange-web junction into a shear stress — the flow $q$ is continuous across the junction, but the associated stress $\tau=q/t$ is not, because the two members meeting there generally have different thicknesses.

Source. Hibbeler, Mechanics of Materials, Ch. 7 (transverse shear, thin-walled open sections); Hughes & Paik, Ch. 8 (shear flow in ship hull cross-sections).

Question 6 of 6: Shear in Ships

Question 6: Shear in Ships (10 marks)