An I-section connecting rod of a tractor diesel engine is designed to have equal buckling resistance as a strut in both the plane of oscillation ($x-x$, pinned ends) and the perpendicular plane ($y-y$, fixed ends). According to Rankine-Gordon column theory, the theoretical ratio of area moment of inertia about the major axis to that about the minor axis ($I_{xx} / I_{yy}$) is ________ (answer in integer).
Answer Key:
4
Step-by-Step Solution & Explanation
```
Connecting Rod Buckling Equilibrium:
x-x plane (plane of motion): Pin-ended at both wrist pin and crank pin
Effective length L_ex = L
y-y plane (perpendicular): Fixed-ended due to broad bearing bushes
Effective length L_ey = L / 2
```
**Step-by-step Calculation:**
1. Buckling load in $x-x$ plane ($P_{cr, x}$):
$P_{cr, x} = \frac{\pi^2 E I_{xx}}{L_{ex}^2} = \frac{\pi^2 E I_{xx}}{L^2}$
2. Buckling load in $y-y$ plane ($P_{cr, y}$):
$P_{cr, y} = \frac{\pi^2 E I_{yy}}{L_{ey}^2} = \frac{\pi^2 E I_{yy}}{(L/2)^2} = \frac{4 \pi^2 E I_{yy}}{L^2}$
3. Equating buckling strengths for equal resistance ($P_{cr, x} = P_{cr, y}$):
$\frac{\pi^2 E I_{xx}}{L^2} = \frac{4 \pi^2 E I_{yy}}{L^2} \implies I_{xx} = 4 I_{yy}$
$\frac{I_{xx}}{I_{yy}} = 4.0$