Project ForgeA cantilever beam of length L is loaded in the x-y plane by a uniformly distributed load q (fixed end at x=0), and by a concentrated tip load P parallel to the z-axis (applied at x=L). The resulting bending moment distributions about the y and the z axes are denoted My and Mz respectively. Which one of the options given depicts qualitatively CORRECT variations of My and Mz along the length of the beam?
- AMy: curved (parabolic) decrease from max at x=0 to 0 at x=L; Mz: straight-line decrease from max at x=0 to 0 at x=L
- My: straight-line decrease from max at x=0 to 0 at x=L; Mz: curved (parabolic) decrease from max at x=0 to 0 at x=L
- CMy: straight-line decrease from max at x=0 to 0 at x=L; Mz: straight-line decrease from max at x=0 to 0 at x=L, less steep than My
- DMy: curved increase from x=0 to x=L; Mz: curved increase from x=0 to x=L
Explanation
The tip point load P (parallel to z) produces bending about the y-axis: My(x)=P(L−x), a straight line from a maximum at the fixed end (x=0) to zero at the tip (x=L). The uniformly distributed load q (in the x-y plane) produces bending about the z-axis: Mz(x)=q(L−x)²/2, a convex (parabolic) curve also decreasing from a maximum at x=0 to zero at x=L, but dropping off increasingly gently near the tip rather than linearly.
Source: GATE ME 2023 Official Paper (2023)
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