A solid beam with a rectangular cross-section of breadth 0.5 m and depth 0.12 m, experiences a vertical shear force of 50 kN at a section.
The maximum shear stress in that section is ______ MPa (answer in two decimal places).
The maximum shear stress in that section is ______ MPa (answer in two decimal places).
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Correct Answer: 1.25
Solution and Explanation
Shear stress across a solid rectangular beam is not the same everywhere on the section. It is zero at the top and bottom edges and rises to a peak at the neutral axis, following a parabolic curve, so simply dividing the shear force by the area only gives an average value, not the true peak.
The cross-sectional area here is $A = b \times d = 0.5 \times 0.12 = 0.06$ square metres. With a shear force of $V = 50$ kN, the average shear stress works out to $\tau_{avg} = V/A = 50000/0.06 = 833333.33$ pascals.
For a rectangle, the peak shear stress at the neutral axis is exactly one and a half times this average value, a standard result that comes from the parabolic shear stress formula for a rectangular section. So $\tau_{max} = 1.5 \times 833333.33 = 1250000$ pascals.
Converting to megapascals by dividing by one million gives $\tau_{max} = 1.25$ MPa, which is the maximum shear stress carried at that section.
