In strength of materials: The shape of the bending-moment diagram along a beam that carries a uniformly increasing (linearly varying) load over its entire span is always:

Introduction / Context:
Bending-moment (BM) and shear-force (SF) diagrams reveal how internal actions vary along a beam. When the external load varies linearly (uniformly increasing or triangular loading), it is important to recall the integration relationships between load, shear, and moment.

Given Data / Assumptions:

• Prismatic beam with constant flexural rigidity assumed for action–effect reasoning.
• Load w(x) varies linearly along the span.
• Static equilibrium; small deflection theory.

Concept / Approach:
The fundamental relationships are: dV/dx = w(x) and dM/dx = V(x). If w(x) is linear in x, integrating once gives V(x) as a quadratic function (parabola). Integrating the quadratic shear once more gives M(x) as a cubic function.

Step-by-Step Solution:
Assume w(x) = a + b x (linear).Integrate: V(x) = ∫ w(x) dx = a x + (b/2) x^2 + C1 → quadratic (parabolic).Integrate again: M(x) = ∫ V(x) dx = (a/2) x^2 + (b/6) x^3 + C1 x + C2 → cubic.Therefore, the bending-moment diagram is a cubic (often called “cubical” in exam options).

Verification / Alternative check:
Special cases confirm the rule: for uniform load (constant w), V is linear and M is parabolic; increasing the load order by one increases the order of V and M by one in turn.

Why Other Options Are Wrong:
Linear/Parabolic: correspond to constant or uniform loads, not linearly varying loads.Circular/Exponential: not produced by integrating a linear function twice.

Common Pitfalls:
Confusing the shape of shear with moment; under a linearly varying load, shear is parabolic but moment is cubic.

Final Answer:
Cubical.