Fixed Beams MCQ with answer and solution

Concept

Bending Moment in Fixed Beam Under UDL

Formula

M(x) = -ql²/12 + (ql/2)x - (qx²)/2

Step-by-step solution

1. Use the bending moment formula: M(x) = -ql²/12 + (ql/2)x - (qx²)/2 2. Substitute x = l/4 for the quarter span location 3. Calculate: M(l/4) = -ql²/12 + ql²/8 - ql²/32 4. Find a common denominator (96): M(l/4) = (-8ql² + 12ql² - 3ql²) / 96 = ql²/96

Explanation

For a beam of span l fixed at both ends and subjected to a uniformly distributed load q:1. Fixed End Moments: The hogging bending moment at both fixed supports is -ql²/12. 2. Mid-span Moment: The sagging bending moment at the center (mid-span) is +ql²/24. 3. Quarter-Span Moment: The moment at any point x from a support is given by M(x) = -ql²/12 + (q·l/2)x - (q·x²)/2.Substituting x = l/4 into the formula: M(l/4) = -ql²/12 + (q·l/2)(l/4) - (q)(l/4)²/2 M(l/4) = -ql²/12 + ql²/8 - ql²/32 M(l/4) = (-8ql² + 12ql² - 3ql²) / 96 = ql²/96. Thus, the magnitude of the bending moment at the quarter span is ql²/96.

Shortcut

Remember key locations using 96: standard BMD denominator for quarter-span of fixed beam with UDL is 96 (ql²/96). Quick Key Points: - Fixed Ends: -ql²/12 - Mid-Span: +ql²/24 - Quarter-Span: ql²/96

Common mistake

Many students mistakenly treat the fixed beam as a simply supported beam and take the quarter-span bending moment from a simply supported UDL diagram (which is 3ql²/32), or they forget to include the negative fixed-end moment (-ql²/12) when calculating the net moment, leading them to select incorrect options like ql²/32 or ql²/16.