Consider the midship section of a vessel with the centerline (CL) and neutral axis (NA) as shown in the figure. Assume that the cross-section is symmetric about the centerline, the plate thickness is uniform throughout the section and \(h_1<h_2\).When the vessel is subjected to a vertical bending moment in its upright condition, which of the following statements are TRUE?
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Bending stress is maximum at farthest points from the neutral axis, while shear stress is maximum at or near the neutral axis.
Magnitude of shear stress is maximum at points P and S
Magnitude of shear stress is minimum at points Q and U
Magnitude of bending stress is maximum at points S and T
Magnitude of bending stress is minimum at points Q and U
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The Correct Option isC, D
Solution and Explanation
Step 1: Bending stress distribution.
Bending stress varies linearly with distance from the neutral axis according to
\[
\sigma = \frac{My}{I}
\]
Thus, fibers farthest from the neutral axis carry the maximum bending stress.
In the given section:
- Points \(S\) and \(T\) lie far above the NA → largest value of \(y\).
Therefore, bending stress is maximum at S and T. (Statement C is TRUE)
Points \(Q\) and \(U\) lie exactly on the NA → \(y = 0\).
Hence, bending stress is zero at Q and U. (Statement D is TRUE)
Step 2: Shear stress distribution.
Shear stress is given by
\[
\tau = \frac{VQ}{Ib}
\]
It is:
- Zero at the extreme top and bottom (points S, T, P).
- Maximum near the neutral axis.
Thus:
- Shear stress is NOT maximum at P or S → (A) is FALSE.
- Shear stress is NOT minimum at Q and U (it is actually maximum there) → (B) is FALSE. Conclusion:
Only statements C and D are TRUE.
