Subtopic: Shear Flow in Thin-Walled Closed Sections
Question Statement:
The idealized cross section of a thin-walled wing box structure has two stringers carrying boom areas $B = 100$ mm$^2$ separated by height $h = 200$ mm and width $w = 400$ mm. If a vertical shear load $V_y = 10$ kN acts at the shear center, the shear flow (in N/mm) in the vertical webs is _________ (round off to the nearest integer).
Official Answer Key:
25.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.54)**
**1. Intuitive Physical Setup & Given Parameters**:
- Vertical shear force: $V_y = 10\text{ kN} = 10000\text{ N}$.
- Web height: $h = 200\text{ mm}$.
- Box width: $w = 400\text{ mm}$.
**2. Step-by-Step Structural Derivation**:
1. **Vertical Equilibrium of Shear Flows**:
In a symmetric two-web closed section subjected to vertical shear force $V_y$ through its shear center:
The total vertical resultant of shear flows across both webs must equilibrate the applied vertical shear force $V_y$:
$\int_{web 1} q\,dy + \int_{web 2} q\,dy = V_y$
2. By symmetry, both vertical webs carry equal shear flow:
$2 q h = V_y \implies q = \frac{V_y}{2h}$
3. **Evaluating Numerical Shear Flow**:
$q = \frac{10000\text{ N}}{2 \times 200\text{ mm}} = \frac{10000}{400} = 25\text{ N/mm}$
The shear flow in each vertical web is exactly **25 N/mm**.
**3. Comprehensive Options & Distractor Analysis**:
- **Question Type**: Numerical Answer Type (NAT).
- **Exact Analytical Evaluation**: **25.0 N/mm** (Acceptable Range: [24, 26]).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **25.0**
- **Key Formula**: $q = \frac{V_y}{2h}$ for symmetric two-web box.
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