Subtopic: Shear Flow in Open and Closed Thin-Walled Beams
Question Statement:
A thin-walled tube of circular cross-section with mean radius $r$ and wall thickness $t$ has a central vertical web of thickness $t$ which divides it into two equal semicircular cells. Under a twisting moment $T$, the shear flow in the outer walls and central web are related by:
Options:
(A) equal shear flow in both cells, zero shear flow in the central web
(B) shear flow in the central web is double the outer wall
(C) shear flow in one cell is zero
(D) shear flow in the central web equals outer wall
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2008 Q.54)**
**1. Multi-Cell Thin-Walled Torsion Theory**:
By symmetry of the two identical semicircular cells:
- Rate of twist $\frac{d\theta}{dz}$ must be equal in both cells: $\theta_1' = \theta_2'$.
- By geometric and load symmetry under pure torsion: $q_1 = q_2 = q$.
- The net shear flow in the common interior web between cell 1 and cell 2 is the difference:
$$q_{\text{web}} = q_1 - q_2 = q - q = 0$$
Thus, the central web carries **zero shear flow**, and shear flow is identical in both outer semicircular walls.
**2. Official Master Key**:
- **Official Answer Key**: **A**