Answer :
The deflection at point "a = 3 ft" is 0.639 inches.
To find the deflection at point "a," we use the Moment-Area Method. Firstly, we calculate the bending moment "M" due to the point load and the distributed load. Then, we construct the moment diagram and find the area "A" enclosed by the moment diagram, the tangent at point "a," and the vertical line from the origin to point "a." Finally, we compute the deflection "δ" using the formula: delta = (M * A) / (E * I).
1. Bending Moment Calculation:
The bending moment "M" at point "a" is the sum of moments due to the point load and the distributed load.
M = P * a + (w * a^2) / 2
Substituting the given values,
M = 5 k * 3 ft + (1.5 k * 3^2 ft^2) / 2 = 22.5 kft
2. Construction of Moment Diagram:
Using the calculated bending moment "M = 22.5 kft," we sketch the moment diagram. The diagram consists of a linear portion due to the point load and a parabolic portion due to the distributed load.
3. Deflection Calculation:
Next, we determine the area "A" enclosed by the moment diagram, the tangent at point "a," and the vertical line from the origin to point "a."
delta = (M * A) / (E * I)
Plugging in the values,
delta = (22.5 kft * 12 in/ft * 3 ft) / (29000 ksi * 76.8 in^4) = 0.639 in
Therefore, the deflection at point "a" is 0.639 inches.
