What is the maximum deflection (ymax) for a cantilever beam of span l subjected to a point load P acting at a distance a from the fixed end as shown in figure? (Take Young's modulus as E and moment of inertia of beam section as I)
- Aymax = [Pa² / (6EI)](3l - a)Correct
- Bymax = [Pa² / (3EI)](3l - a)
- Cymax = [3Pa² / (2EI)](3l - a)
- Dymax = [Pa² / (2EI)](3l - a)
Solution
Concept
Beyond the load point the cantilever has no moment, so it stays straight. The extra deflection is slope at load × remaining length.
Formula
ymax = Pa3/(3EI) + [Pa2/(2EI)](l - a) = [Pa2/(6EI)](3l - a)
Step-by-step solution
1. Treat the part from the fixed end to the load (length a) as a cantilever with end load P. 2. Deflection at load point: yC = Pa3/(3EI). 3. Slope at load point: θC = Pa2/(2EI). 4. From the load to the free end there is no bending moment, so this part stays straight. 5. Extra deflection at free end = θC × (l - a) = Pa2(l - a)/(2EI). 6. ymax = Pa3/(3EI) + Pa2(l - a)/(2EI) = [Pa2/(6EI)](2a + 3l - 3a) = [Pa2/(6EI)](3l - a).
Explanation
Maximum deflection of a cantilever occurs at the free end. The beam bends only up to the load. After the load it stays straight and tilts at the slope found at the load point. Adding the deflection at the load and slope × (l - a) gives Pa²(3l - a)/(6EI).
Why the other options are wrong
B: the denominator should be 6EI, not 3EI, so B is two times too large. C: the factor 3/2 is wrong and gives a value far too large. D: the denominator 2EI is wrong and gives three times the correct value.
Correct answer: A) y_max = [Pa² / (6EI)](3l - a)
