Consider a cantilever beam AB of span L fixed at A and free at B and subjected to a uniformly w throughout the span. According to the moment-area method, the deflection at free end B of the beam will be
- A(1/EI) x area under the bending moment diagram of the beam (BMD)
- B(1/EI) x moment of the area under the BMD about point A
- C(1/EI) x moment of the area under the BMD about point BCorrect
- D(1/EI) x moment of the area under the BMD about mid-span of beam
Solution
Concept
Mohr's Second Moment-Area Theorem states that the tangential deviation of a point on the elastic curve from the tangent at another point is equal to the first moment of the M/EI diagram between the two points about the point where the deflection is desired.
Formula
δB = (1/EI) x (Area of BMD) x (xbar)
Step-by-step solution
1. Identify the beam type: Cantilever fixed at A and free at B. 2. Recall Mohr's second moment-area theorem for deflection: The deflection at the free end B relative to the fixed end A is equal to the moment of the M/EI diagram between A and B taken about B. 3. Therefore, the deflection equals (1/EI) times the moment of the area under the bending moment diagram about point B.
Explanation
According to the second Mohr's moment-area theorem, the vertical deviation of any point B relative to the tangent at another point A (which is the fixed support where slope is zero) is equal to the moment of the area of the B/EI diagram between A and B taken about point B. For a cantilever fixed at A and free at B, the deflection at B is directly given by this moment of area about point B.
Why the other options are wrong
A: This gives the slope, not the deflection. B: Taking the moment about A gives the deviation of the tangent at B relative to A, which is the slope at B. D: Taking the moment about the mid-span is incorrect for finding the free end deflection.
Shortcut
For a cantilever, deflection at the free end is always calculated as the moment of the area of the BMD about the free end (point B).
Common mistake
Taking the moment about the fixed end (point A) instead of the free end (point B) where the deflection is actually being measured.
Correct answer: C) (1/EI) x moment of the area under the BMD about point B