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 is the one depicting a linear variation for _My (which is positive) and a parabolic (quadratic) variation for _Mz (which is negative) along the length of the beam.

Here is the detailed step-by-step logical and mathematical derivation:

1. Analysis of the Bending Moment _Mz (caused by load in the xy-plane):
The beam is loaded in the xy-plane by a uniformly distributed load of intensity q acting downwards (in the -y direction) as shown in the main diagram (Image 0).
Let x be the distance measured from the fixed end at the left support (x = 0) to the free end (x = L).
The relation between the distributed load w_y(x) and the bending moment M_z(x) is given by:
$$\frac{d^2{M}_z}{d{x}^2}=-q$$
Integrating once with respect to x gives the shear force V_y(x):
$$\frac{d{M}_z}{dx}=V_y\left(x\right)=-qx+C_1$$
At the free end (x = L), the shear force is zero (V_y(L) = 0), which yields:
$$C_1=qL\Rightarrow V_y\left(x\right)=q(L-x)$$
Integrating a second time to find the bending moment M_z(x):
$$M_z\left(x\right)=-\frac{1}{2}q{(L-x)}^2$$
This shows that:
• The bending moment M_z(x) varies quadratically (parabolically) along the length of the beam.
• At the free end (x = L), M_z(L) = 0.
• At the fixed support (x = 0), M_z(0) = -qL²/2, which is negative due to hogging.
Therefore, the plot of M_z is a parabolic curve starting at a negative value at the fixed end and ending at zero at the free end.

2. Analysis of the Bending Moment _My (caused by concentrated tip load P parallel to the z-axis):
The concentrated load P acts at the free tip of the beam (x = L) parallel to the z-axis (pointing in the -z direction).
At any cross-section at a distance x from the support, the bending moment about the y-axis is due to the force P acting with a lever arm of (L - x):
$$M_y\left(x\right)=P(L-x)$$
This shows that:
• The bending moment M_y(x) varies linearly along the length of the beam.
• At the free end (x = L), M_y(L) = 0.
• At the fixed support (x = 0), M_y(0) = PL, which is positive.
Therefore, the plot of M_y is a straight line starting at a positive value at the fixed end and decreasing to zero at the free end.

Conclusion:
Comparing these derivations with the options, the graph showing a positive linear variation for M_y and a negative parabolic variation for M_z represents the qualitatively correct bending moment diagrams.