In steel design, consider a built-up beam. If y1 is the distance from the neutral axis to the extreme compression fiber (edge), what is the expression for the actual bending compressive stress fbc at that edge?

Introduction / Context:
Bending stress distribution in beams is a fundamental topic in strength of materials and steel design. Whether the beam is a rolled section or a built-up section, the flexure formula connects internal bending moment, section properties, and fiber distance from the neutral axis to compute compressive or tensile stresses.

Given Data / Assumptions:

• M is the bending moment at the section under consideration.
• I is the second moment of area (about the neutral axis for the bending plane).
• y1 is the distance from the neutral axis to the extreme compression fiber (edge).
• Plane sections remain plane and the material is linearly elastic within the working range.

Concept / Approach:
The elementary flexure (bending) formula states: sigma = M * y / I. Stress varies linearly with y; it is compressive above the neutral axis (in sagging) and tensile below. For the extreme compression fiber, y = y1, so the actual bending compressive stress is computed directly by substituting y1 into the formula.

Step-by-Step Solution:

Verification / Alternative check:
Cross-check by computing Z about the compression edge (Z = I / y1). Then fbc = M / Z. Both forms are algebraically identical and yield the same stress.

Why Other Options Are Wrong:
Forms such as I / (M * y1), M / (I * y1), or (I * y1) / M invert or scramble the flexure relation. They do not match the linear-elastic bending theory. M / Z is correct only if Z corresponds to the same edge, which is implicit but often not stated in quick formulas.

Common Pitfalls:
Using the wrong axis for I; mixing up y for tension vs. compression side; applying plastic section modulus instead of elastic section modulus when the question clearly assumes elastic behavior.

Final Answer:
fbc = (M * y1) / I