The problem concerns the bending moment diagram (BMD) for a simply supported beam subjected to a uniformly distributed load (UDL) across its entire span. To determine the shape of this BMD, let's analyze the behavior of the beam under these loading conditions.

1. Consider a simply supported beam with a uniformly distributed load w (force per unit length) applied throughout the span L of the beam.
2. The reaction forces at the supports can be calculated. Since the load is uniformly distributed and the beam is symmetric, each support will bear half of the total load:

3. Derive the expression for the bending moment at a distance x from one end. The bending moment as a function of x is given by:

4. Substitute the value of R_A:

5. By simplifying, the bending moment at any section x is:

6. The equation M(x) = \frac{w}{2} (L \cdot x - x^2) is a quadratic equation in terms of x, which describes a parabolic curve. The maximum bending moment occurs at the center of the beam (x = \frac{L}{2}).
7. Therefore, the bending moment diagram (BMD) for the whole span is parabolic in shape, with maximum bending moment at mid-span and zero at the supports.

Based on the above analysis, the correct answer is that the bending moment diagram for a simply supported beam carrying a uniformly distributed load is parabolic in shape.