Final answer:
The limiting deflection at the center of the beam is approximately 8.75 mm. Hence, the correct answer is option (a).
Step-by-step explanation:
To find the limiting deflection at the center of the beam, we need to consider the maximum slope at the ends. The formula for the slope of a simply supported beam with a point load at the center is given by:
slope = (WL/48EI)
Where, W is the point load, L is the length of the beam, E is the Young's modulus of the material, and I is the moment of inertia of the beam cross-section. In this case, since the beam is simply supported at its ends, the slopes at the ends will be half of the slope at the center. Therefore, we need to calculate the maximum slope at the center that still satisfies the condition that the slope at the ends does not exceed 1° (0.01745 radian).
Let's use the formula for slope to find the maximum point load that satisfies this condition:
0.01745 = (W(3/2))/(48(El^3)/12)
Solving for W, we get:
W = (0.01745 * 48 * (E * (3^3)/12))/3)
Once we have the value of W, we can calculate the deflection at the center of the beam using the formula:
deflection = (WL^3)/(48EI)
Let's calculate the value of W and then substitute it into the deflection formula to find the limiting deflection at the center of the beam:
deflection = ((W * (3^3))/(48 * E * l^3))/1000
Plugging in the known values, we find that the limiting deflection at the center of the beam is approximately 8.75 mm.