To find the bending moment at the mid-point of the beam, we will analyze the equilibrium of the beam and then apply the method of sections.

1. Identify Parameters and Support Reactions

The beam is simply supported at its ends $A$ and $B$ over a total length $L = 1 \text{ m}$. It is subjected to a uniformly distributed bending moment $m = 1 \text{ N m/m}$ acting in the clockwise direction throughout its length.

The total external moment applied to the beam is: 
$M_{\text{ext}} = \int_{0}^{L} m \, dx = 1 \text{ N m/m} \times 1 \text{ m} = 1 \text{ N m}$ (clockwise)

Let $R_A$ and $R_B$ be the vertical reactions at the supports. For equilibrium, the reactions must provide a counter-clockwise couple to balance the applied moment. Taking moments about point $A$: 
$\sum M_A = 0 \implies (R_B \times 1 \text{ m}) - 1 \text{ N m} = 0 \implies R_B = 1 \text{ N}$ (acting upwards) 
From vertical equilibrium ($\sum F_y = 0$): 
$R_A + R_B = 0 \implies R_A = -1 \text{ N}$ (acting downwards)

2. Calculate the Bending Moment at the Mid-point

Now, we consider a section at the mid-point $x = 0.5 \text{ m}$ from the left support $A$. We analyze the left portion of the beam to find the internal bending moment $M$.

Using the standard sign convention (sagging is positive):

• A downward force $R_A = 1 \text{ N}$ at the left end creates a negative bending moment: $M_R = -R_A \times x = -1 \text{ N} \times 0.5 \text{ m} = -0.5 \text{ N m}$.
• A clockwise distributed moment $m = 1 \text{ N m/m}$ to the left of the section creates a positive bending moment: $M_m = \int_{0}^{0.5} m \, dx = 1 \text{ N m/m} \times 0.5 \text{ m} = +0.5 \text{ N m}$.

The total internal bending moment at the mid-point is the sum of these contributions: 
$M(0.5) = M_R + M_m = -0.5 \text{ N m} + 0.5 \text{ N m} = 0 \text{ N m}$

In fact, for this specific loading condition on a simply supported beam, the internal bending moment is zero at every point along the length of the beam because the reaction couple exactly balances the distributed moment at any given section.

Final Answer:

The bending moment at the mid-point of the beam is 0 N m.