Subtopic: Kutta-Joukowski Theorem & Thin Airfoil Theory
Question Statement:
A cantilever beam with an unsymmetric cross-section is subjected to a transverse
shear force (P) at its free end. P acts at the shear center of the beam cross-section.
Which one of the following statements is TRUE about the deformation of this
beam?
Options:
(A) The beam undergoes only torsion
(B) The beam undergoes only bending
(C) The beam undergoes both torsion and bending
(D) The beam undergoes neither torsion nor bending
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.17)**
**1. Intuitive Physical Setup & Given Parameters**:
A cantilever beam with an unsymmetric cross-section is subjected to a transverse shear force $P$ acting precisely through its **shear center**.
**2. Step-by-Step Mathematical Derivation**:
1. **Definition of Shear Center**:
The shear center (also known as the center of flexure) is that unique point in the cross-section plane through which a transverse shear load must be applied so that the resultant internal shear flow produces **zero net twisting moment** about the longitudinal axis.
2. **Load Passing Through Shear Center**:
- When load line passes through the shear center: $\sum M_{\text{torsion}} = 0$. The beam undergoes flexural deformation (**bending alone**) without any twist/torsion.
- When load line does not pass through the shear center: A twisting couple $T = P \cdot e$ is generated, causing coupled bending and torsion.
3. **Conclusion**:
Because $P$ passes directly through the shear center, no torsional moment is induced; the beam experiences only bending.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) The beam undergoes only torsion**: ❌ **INCORRECT**. Transverse force produces bending moments along the span.
- **(B) The beam undergoes only bending**: ✅ **CORRECT**. By definition of the shear center, any transverse load acting through it induces pure flexure with zero twisting.
- **(C) The beam undergoes both torsion and bending**: ❌ **INCORRECT**. This occurs only when the load line is offset from the shear center.
- **(D) The beam undergoes neither torsion nor bending**: ❌ **INCORRECT**. A transverse load definitely produces flexural bending.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: Load through centroid $\implies$ pure bending only for doubly symmetric sections. Load through shear center $\implies$ pure bending for ANY arbitrary unsymmetric section.
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