The midship section of a barge of breadth \( W \) and depth \( H \) is shown in the figure. All plate thicknesses are equal. The barge is subjected to a longitudinal bending moment in the upright condition. Which one of the following statements is correct?
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Remember that bending stress is maximum at the extreme fibers farthest from the neutral axis, while shear stress is maximum at the neutral axis for a rectangular section. The location of the neutral axis is crucial in determining these maximum stresses.
Magnitude of longitudinal bending stress is maximum at point P and magnitude of vertical shear stress is maximum at point Q.
Magnitude of longitudinal bending stress is maximum at point S and magnitude of vertical shear stress is maximum at point R.
Magnitude of longitudinal bending stress is maximum at point Q and magnitude of vertical shear stress is maximum at point S.
Magnitude of longitudinal bending stress is maximum at point R and magnitude of vertical shear stress is maximum at point P.
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The Correct Option isA
Solution and Explanation
Step 1: Understanding Longitudinal Bending Stress.
The longitudinal bending stress (\(\sigma_x\)) in a beam is given by the flexure formula:
\[
\sigma_x = -\frac{My}{I_z}
\]
where \(M\) is the bending moment, \(y\) is the distance from the neutral axis, and \(I_z\) is the second moment of area about the neutral axis. The magnitude of the bending stress is directly proportional to the distance from the neutral axis.
From the figure, the neutral axis is located at a distance of \(H/3\) from the bottom. The distances of the points P, Q, R, and S from the neutral axis are:
Point P (top): \(H - \frac{H}{3} = \frac{2H}{3}\)
Point Q (at the neutral axis level on the side): \(\frac{H}{3}\)
Point R (at the neutral axis level on the side): \(\frac{H}{3}\)
Point S (bottom): \(\frac{H}{3}\)
The maximum distance from the neutral axis is at Point P. Therefore, the magnitude of longitudinal bending stress is maximum at Point P.
Step 2: Understanding Vertical Shear Stress.
The vertical shear stress (\(\tau_{xy}\)) in a beam with a rectangular cross-section is given by:
\[
\tau_{xy} = \frac{VQ}{I_z b}
\]
where \(V\) is the shear force, \(Q\) is the first moment of area of the section above or below the point where stress is calculated, \(I_z\) is the second moment of area, and \(b\) is the width of the section at that point. For a rectangular section, the shear stress distribution is parabolic with the maximum value at the neutral axis and zero at the top and bottom surfaces.
In this case, the neutral axis is at a level corresponding to points Q and R. Therefore, the magnitude of vertical shear stress is maximum at points Q and R.
Step 3: Combining the results.
From Step 1, the magnitude of longitudinal bending stress is maximum at point P.
From Step 2, the magnitude of vertical shear stress is maximum at points Q and R.
Looking at the options, option (A) states: "Magnitude of longitudinal bending stress is maximum at point P and magnitude of vertical shear stress is maximum at point Q." This matches our findings.
