Degrees of freedom at a member end in a 2D (plane) frame — bending moment components In a two-dimensional plane frame (bending occurs about the out-of-plane axis), how many independent bending moment components can act at a single member end (i.e., how many possible end bending moments about the relevant axis)?

Introduction / Context:
Plane (2D) frame analysis considers in-plane forces and a single bending moment component about the out-of-plane axis. Understanding end actions and degrees of freedom clarifies what internal forces/moments may be transferred at joints and which releases change the behavior.

Given Data / Assumptions:

• Structure is a 2D frame (no torsion about the member longitudinal axis).
• Forces considered: axial force N, shear force V, and bending moment M_z (about z, perpendicular to the plane).

Concept / Approach:
At any member end in a 2D frame, the internal resultant consists of N, V, M_z. Only one independent bending moment component exists (about the out-of-plane axis). Therefore, the maximum number of possible bending moment components at a single end is one.

Step-by-Step Solution:

Verification / Alternative check:
Finite element beam elements for 2D frames have three DOFs per node (u, v, theta), where theta corresponds to the single bending rotation, confirming one bending moment component conjugate to theta.

Why Other Options Are Wrong:

• Two/three/four: would imply additional bending axes or torsion considered; that is 3D frame behavior.
• Zero: only true if the end is fully released (pin), but the question asks maximum possible.

Common Pitfalls:
Confusing 2D frame action with 3D beam behavior; counting axial or torsional effects as separate bending components.

Final Answer:
One