A thin-walled open I-beam has two horizontal flanges of width $b$ and thickness $t_f$, and a vertical web of height $h$ and thickness $t_w$. According to St. Venant's torsion theory for thin-walled open sections ($b \gg t_f, h \gg t_w$), the torsional constant $J$ of the cross-section is given by:

(A) $J = \frac{1}{3} \left( 2 b t_f^3 + h t_w^3 \right)$

(B) $J = \frac{1}{3} (2b + h) (t_f + t_w)^3$

(C) $J = \frac{2 b h t_f t_w}{t_f + t_w}$

(D) $J = \frac{b t_f^3 + h t_w^3}{12}$

Answer Key:

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### 1. Strategy & Theoretical Principles For any thin-walled open cross-section composed of $N$ narrow rectangular segments of length $s_i$ and thickness $t_i$ ($s_i \gg t_i$): By St. Venant's torsion theory, the total torsional stiffness constant $J$ is the direct sum of the individual segments: $$J = \sum_{i=1}^N \frac{1}{3} s_i t_i^3$$ ### 2. Analytical Application to I-Beam The I-beam comprises 3 thin rectangular segments: 1. Top flange: length $b$, thickness $t_f \implies J_1 = \frac{1}{3} b t_f^3$ 2. Bottom flange: length $b$, thickness $t_f \implies J_2 = \frac{1}{3} b t_f^3$ 3. Central web: length $h$, thickness $t_w \implies J_3 = \frac{1}{3} h t_w^3$ Summing all segments: $$J = J_1 + J_2 + J_3 = \frac{1}{3} \left( 2 b t_f^3 + h t_w^3 \right)$$ ### 3. Distractor Forensics - **Option A:** Correct summation. - **Option D:** Divided by 12 (confusing torsion constant with second moment of area). ### 4. Exam Trap & Standard Reference - **Trap Alert:** Thin open section torsion constants scale with the *cube of the thickness* ($t^3$), summed linearly across all segments. - **Standard Reference:** Megson, T.H.G., *Aircraft Structures for Engineering Students*, Chapter 18.

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