Bending Stress MCQ with answer and solution

Concept

First moment of area depends on the position of the axis and can be +ve, -ve or zero. Second moment of area is always +ve.

Formula

Q = ∫y dA = A×ȳ ; I = ∫y² dA ; I for circle about centroidal axis = πR⁴/4

Step-by-step solution

1. First moment Q = ∫y dA = A×ȳ, where ȳ is the distance of the centroid from the axis. 2. ȳ can be +ve or -ve (centroid on either side of the axis), or zero (axis through the centroid). So Q can be +ve, -ve or zero. 3. Second moment I = ∫y² dA. y² is never negative and dA is positive, so I is always +ve. 4. For a circle about a centroidal axis, I = πR⁴/4 > 0. 5. So option (A) is correct.

Explanation

The first moment of area uses distance y, which can be positive or negative depending on the reference axis. The second moment uses y², which is always positive. So the second moment of area is always positive.

Why the other options are wrong

B: The second moment can never be negative because y² ≥ 0. C: The first moment is not always negative. It changes sign with the axis. D: The first moment can be negative or zero, for example zero about a centroidal axis.

Shortcut

y can change sign, so Q can be + or -. y² cannot change sign, so I is always +.

Common mistake

Thinking the first moment is always positive. About a centroidal axis it is zero.