Question

Consider the following statement(s) for its correctness.
A. At neutral axis there is no stress of any kind.
B. Neutral axis of a section always passes through its centroid.
C. Moment of any area about an axis passing through its centroid is equal to zero.
D. According to theory of simple bending, all layers above the neutral axis are subjected to compression.
E. In case of symmetrical section like square, rectangle and circle the neutral axis passes through geometric centre.
Choose the correct answer from the options given below:

• A. A and C only
• B. B and D only
• C. B, C and E only
• D. A, B, C, D and E

Hint:

Remember that in real beams, bending is often accompanied by shear.
While bending stress is zero at the neutral axis, shear stress reaches its maximum value at this axis.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Bending theory (or beam theory) describes the stresses and strains developed in a structural element subjected to transverse loads.
The neutral axis is the longitudinal plane in a beam where the fibers undergo no changes in length, resulting in zero longitudinal bending stress.

Step 2: Detailed Explanation:
Let us analyze each statement:
- A. At neutral axis there is no stress of any kind: This is incorrect.
While the normal bending stress (\( \sigma \)) is zero at the neutral axis, the shear stress (\( \tau \)) in a beam subjected to transverse shear loads is typically at its maximum value at the neutral axis.
- B. Neutral axis of a section always passes through its centroid: This is correct.
For a beam subjected to pure bending (with no axial force), the net axial force must be zero.
This condition requires the first moment of area about the neutral axis to be zero:
\[ \int y \, dA = 0 \]
This equation is only satisfied if the neutral axis passes through the centroid of the cross-section.
- C. Moment of any area about an axis passing through its centroid is equal to zero: This is correct.
By mathematical definition, the first moment of an area about its centroidal axis is:
\[ Q = \int y \, dA = \bar{y} A \]
Since the distance from the centroid to the centroidal axis (\( \bar{y} \)) is zero, the first moment of area is zero.
- D. According to theory of simple bending, all layers above the neutral axis are subjected to compression: This is incorrect.
This statement is only true for a positive (sagging) bending moment.
Under a negative (hogging) bending moment, the layers above the neutral axis are in tension while those below are in compression.
- E. In case of symmetrical section like square, rectangle and circle the neutral axis passes through geometric centre: This is correct.
For symmetric sections, the centroid is located at the geometric center of the section.
Since the neutral axis passes through the centroid, it must also pass through the geometric center.
Therefore, statements B, C, and E are correct.

Step 3: Final Answer:
The correct statements are B, C, and E only.