Definition check — bending moment at a section of a beam Which statement correctly defines the bending moment at a given cross-section of a beam in terms of the forces acting on the beam?

Introduction / Context:
Bending moment (BM) is a fundamental concept in structural analysis. It represents the tendency of external forces to cause rotation about a point on the beam's axis, producing curvature. Correctly defining BM is essential for drawing shear force and bending moment diagrams, sizing members, and checking stresses using flexure formulas.

Given Data / Assumptions:

• Beam loaded by general external loads (point loads, UDLs, etc.).
• Sign convention is consistent (e.g., sagging positive).
• Section of interest cuts the beam at some location x.

Concept / Approach:
At a given section, the internal bending moment equals the algebraic sum of the moments of all external forces to one side of the section about that section. The word algebraic emphasizes that sign conventions must be used (e.g., counterclockwise positive), unlike a mere arithmetic (unsigned) summation. Shear force is the algebraic sum of lateral forces, whereas bending moment is the algebraic sum of their moments.

Step-by-Step Solution:

Verification / Alternative check:
In differential form, dM/dx = V, where V is shear force. This reinforces that moments must be treated algebraically, consistent with sign.

Why Other Options Are Wrong:

• Arithmetic sum (without signs) ignores direction, yielding incorrect results.
• Sum of forces (not moments) defines shear, not bending moment.
• “None of these” is incorrect because option (c) is the standard definition.
• Resultant shear force is a different internal action and not a definition of bending moment.

Common Pitfalls:
Mixing sign conventions between left and right free bodies; forgetting to include reactions; using absolute values instead of algebraic sums.

Final Answer:
Algebraic sum of the moments of all the forces on either side of the section