Bending and shear stresses across a beam section: identify the correct statements Consider the stress distributions in a prismatic beam under pure bending and shear. Which combination of statements is correct?

Introduction / Context:
Understanding how bending and shear stresses vary across a beam section is foundational for safe design. Bending stresses vary linearly with distance from the neutral axis, while shear stresses follow shape-dependent distributions (e.g., parabolic in rectangles), peaking at the neutral axis and vanishing at free surfaces.

Given Data / Assumptions:

• (a) The bending stress in a section is zero at its neutral axis and maximum at the outer fibres.
• (b) The shear stress is zero at the outer fibres and maximum at the neutral axis.
• (c) The bending stress at the outer fibres is a principal normal stress (in pure bending, shear is zero on the cross-section).
• (d) The planes of principal stresses are inclined at 45° to the neutral plane.

Concept / Approach:
In pure bending of a prismatic beam, normal stress varies linearly and no in-plane shear acts on the cross-section, making the normal stress itself principal at that section. Shear stress distribution is zero at the free surfaces and maximum at the neutral axis for common shapes. The statement about principal planes at 45° pertains to a pure shear state, not pure bending.

Step-by-Step Solution:
Evaluate (a): True—linear distribution from zero at neutral axis to maximum at extreme fibres.Evaluate (b): True—shear stress is highest at the neutral axis and zero at outer surfaces (for rectangular, I-section webs dominate).Evaluate (c): True—on the cross-section in pure bending, shear equals zero, so the normal stress equals a principal normal stress.Evaluate (d): False—45° principal planes occur under pure shear; not applicable to pure bending planes.

Verification / Alternative check:
Mohr’s circle for pure bending has a point on the normal-stress axis with zero shear, confirming that the normal stress at the section is principal. For a rectangular section in shear, the parabolic distribution confirms zero at the free boundaries and maximum at the neutral axis.

Why Other Options Are Wrong:
Only (a) and (b): Ignores that (c) is also true in pure bending.Only (a) or All (a)–(d): Either omits valid statements or includes the incorrect (d).None of these: Incorrect since multiple statements are true.

Common Pitfalls:

• Confusing pure shear behavior with bending behavior.
• Assuming the presence of shear on the cross-section under pure bending.

Final Answer:
Only (a), (b), and (c) are correct