A beam of length 𝐿 is loaded in the 𝑥𝑦 −plane by a uniformly distributed load, and by a concentrated tip load parallel to the 𝑧 −axis, as shown in the figure. The resulting bending moment distributions about the 𝑦 and the 𝑧 axes are denoted by 𝑀_𝑦 and 𝑀_𝑧 , respectively.

Which one of the options given depicts qualitatively CORRECT variations of 𝑀𝑦 and 𝑀𝑧 along the length of the beam?

The correct option depicting the qualitative variations of the bending moments along the length of the beam is the second option:
where $M_y$ is positive and varies linearly from a maximum value at $x=0$ to zero at $x=L$, and $M_z$ is negative and varies quadratically (curved) from a maximum magnitude at $x=0$ to zero at $x=L$.

1. Analysis of Bending Moment about the z-axis ($M_z$):
The beam is subjected to a uniformly distributed load $q$ acting in the vertical $xy$-plane downwards (negative $y$-direction).
Consider a section at a distance $x$ from the fixed support. The length of the segment from this section to the free end is $(L-x)$.
The bending moment about the $z$-axis at this section is due to the distributed load acting on the segment $(L-x)$:

$$M_z\left(x\right)=-\frac{q{(L-x)}^2}{2}$$

This equation demonstrates that:
• $M_z\left(x\right)$ is negative (causing hogging, which is standard for downward loading on a cantilever beam).
• The variation is quadratic (parabolic) because of the ${(L-x)}^2$ term.
• At the support $(x=0)$, the magnitude is maximum:
$M_z\left(0\right)=-\frac{q{L}^2}{2}$.
• At the free end $(x=L)$, the moment is zero:
$M_z\left(L\right)=0$.

2. Analysis of Bending Moment about the y-axis ($M_y$):
The beam is subjected to a concentrated tip load $P$ acting at the free end $(x=L)$ parallel to the $z$-axis in the negative $z$-direction (pointing into the plane of the page).
The bending moment about the vertical $y$-axis at a distance $x$ from the fixed support is:

$$M_y\left(x\right)=P(L-x)$$

This equation demonstrates that:
• $M_y\left(x\right)$ is positive.
• The variation is linear with respect to $x$.
• At the support $(x=0)$, the magnitude is maximum:
$M_y\left(0\right)=PL$.
• At the free end $(x=L)$, the moment is zero:
$M_y\left(L\right)=0$.

Comparing these derivations with the options shown in the figures:
• $M_y$ must be represented by a straight line in the positive region.
• $M_z$ must be represented by a parabolic curve in the negative region.
This matches the second option.