A 10 kN axial load is applied eccentrically on a rod of square cross-section (1 cm $\times$ 1 cm) as shown in the figure.
The strains measured by the two strain gages attached to the top and bottom surfaces at a distance of 0.5 m from the tip are \( \varepsilon_1 = 0.0016 \) and \( \varepsilon_2 = 0.0004 \), respectively.
The eccentricity in loading, \( e \), is _________ mm.
Show Hint
The eccentricity in loading can be calculated using strain measurements and the bending stress distribution. Ensure to account for the geometry of the section when calculating bending moments.
The strain at a distance from the neutral axis in bending is given by the formula:
\[
\varepsilon = \frac{M y}{E I},
\]
where \( M \) is the bending moment, \( y \) is the distance from the neutral axis, \( E \) is the modulus of elasticity, and \( I \) is the second moment of area.
For the square cross-section, the moment of inertia \( I \) is:
\[
I = \frac{b^4}{12} = \frac{(1\, \text{cm})^4}{12} = 0.0000833 \, \text{cm}^4.
\]
The bending moment at the point of interest is given by:
\[
M = F e = 10000 \, \text{N} \times e \, \text{m}.
\]
From the strain difference equation:
\[
\varepsilon_1 - \varepsilon_2 = \frac{M (y_1 - y_2)}{E I}.
\]
Solving for \( e \), we get:
\[
e \approx 0.95 \, \text{mm}.
\]
Thus, the eccentricity in loading is:
\[
\boxed{0.95 \, \text{mm}}.
\]
