A thin-walled open channel section of uniform wall thickness $t$ has a web of height $h$ and two identical horizontal flanges of width $b$. If $h = 2b$, the distance $e$ of the shear center from the vertical web (measured outwards away from the flanges) is:

(A) $e = \frac{3 b^2}{6b + h} = \frac{3}{8} b$

(B) $e = \frac{b}{2}$

(C) $e = \frac{3}{4} b$

(D) $e = \frac{b}{3}$

Answer Key:

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### 1. Strategy & Theoretical Principles For an open thin-walled channel section subject to a vertical shear force $V_y$: 1. The shear flow in the flanges varies linearly from zero at the free edge to maximum at the web junction: $$q_f(s) = \frac{V_y}{I_{xx}} \int_0^s y t \, ds = \frac{V_y}{I_{xx}} \left(\frac{h}{2}\right) t s$$ 2. The total horizontal flange force in each flange is: $$F_f = \int_0^b q_f(s) \, ds = \frac{V_y t h}{2 I_{xx}} \left[\frac{s^2}{2}\right]_0^b = \frac{V_y t h b^2}{4 I_{xx}}$$ 3. Taking moments about the center of the web: $$V_y e = F_f h \implies e = \frac{F_f h}{V_y} = \frac{t h^2 b^2}{4 I_{xx}}$$ ### 2. Analytical Formulation for Moment of Inertia The second moment of area about the horizontal centroidal axis $I_{xx}$ comprises the web and both flanges: $$I_{xx} = \frac{t h^3}{12} + 2 \left[ b t \left(\frac{h}{2}\right)^2 \right] = \frac{t h^3}{12} + \frac{t b h^2}{2} = \frac{t h^2}{12} (h + 6b)$$ ### 3. Step-by-Step Calculation Substitute $I_{xx}$ into the shear center formula: $$e = \frac{t h^2 b^2}{4 \left[ \frac{t h^2}{12} (h + 6b) \right]} = \frac{3 b^2}{6b + h}$$ Given the geometric condition $h = 2b$: $$e = \frac{3 b^2}{6b + 2b} = \frac{3 b^2}{8b} = \frac{3}{8} b = 0.375 b$$ ### 4. Distractor Forensics - **Option A:** Correct standard closed-form derivation. - **Option B ($b/2$):** Assumes shear center coincides with flange half-width. - **Option C ($3b/4$):** Omitted factor of 2 in moment arm $h$. - **Option D ($b/3$):** Neglected flange moment of inertia contribution to $I_{xx}$. ### 5. Exam Trap & Standard Reference - **Trap Alert:** Always measure $e$ outward from the web. The line of action of the resultant flange forces creates a counter-clockwise couple that must be balanced by $V_y e$. - **Standard Reference:** Megson, T.H.G., *Aircraft Structures for Engineering Students*, 6th Edition, Butterworth-Heinemann, Chapter 17 (Bending and Shear of Thin-Walled Open Beams).

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