Neutral axis and centroid in symmetric beams:\nState whether the following is true or false: “For a symmetrical, homogeneous beam section under pure bending, the neutral axis does not pass through the centroid.”

Introduction / Context:
The neutral axis (NA) is the locus of zero normal stress in bending. For many common engineering sections, quickly locating the NA is essential for stress calculation using the flexure formula. This item checks conceptual understanding of where the NA lies relative to the centroid for symmetric, homogeneous sections.

Given Data / Assumptions:

• Homogeneous, isotropic material.
• Symmetric prismatic cross-section.
• Pure bending without shear.

Concept / Approach:
Under Euler–Bernoulli assumptions, plane sections remain plane and strain varies linearly with distance y from the NA: epsilon = y / rho. Enforcing zero resultant normal force over the cross-section in pure bending makes the NA pass through the centroid for homogeneous sections. Hence, for symmetric homogeneous beams, the NA coincides with the centroidal axis.

Step-by-Step Solution:

Verification / Alternative check:
Standard shapes (rectangular, circular, I-section with symmetric flanges/web) have NA through centroid in homogeneous materials. Composite or transformed sections are the cases where NA shifts away from geometric centroid.

Why Other Options Are Wrong:

• Agree: contradicts beam theory fundamentals for homogeneous, symmetric sections.
• Conditional options apply to unsymmetrical or composite sections, not to the stated case.

Common Pitfalls:
Confusing centroidal axis with NA in cracked or composite sections, where transformed area methods are needed.

Final Answer:
Disagree