Deflection of Simply Supported Beams: Point Load vs. UDL
(A) 58
(B) 85
(C) 54
(D) 45
Explanation
The maximum deflection of a simply supported beam depends on the type and distribution of the load. Below, we calculate the deflections for both beams and find their ratio.
Beam A: Central Point Load
For a simply supported beam with a central point load W, the maximum deflection (δ_A) occurs at the center and is given by:
δ_A = W l³48 EI
Where:
- W = Point load
- l = Length of the beam
- E = Modulus of elasticity
- I = Moment of inertia
Beam B: Uniformly Distributed Load (UDL)
For a simply supported beam with a uniformly distributed load (UDL) where the total load is W, the load intensity is w = Wl. The maximum deflection (δ_B) occurs at the center and is given by:
δ_B = 5 w l⁴384 EI = 5 (W/l) l⁴384 EI = 5 W l³384 EI
Ratio of Deflections
The ratio of maximum deflections is:
δ_A / δ_B = W l³ / 48 EI5 W l³ / 384 EI = 3845 × 48 = 384240 = 85
Thus, the ratio of maximum deflection of Beam A to Beam B is 85.
Summary Table
| Beam | Loading Condition | Maximum Deflection |
|---|---|---|
| Beam A | Central Point Load (W) | W l³48 EI |
| Beam B | UDL (Total Load W) | 5 W l³384 EI |
Key Notes
- Maximum deflection for a central point load is higher than for a uniformly distributed load of the same total load.
- The comparison assumes both beams have the same span (l) and material properties (E and I).
- Deflection formulas are derived from standard beam theory for simply supported beams.
Note: Understanding beam deflection is crucial for designing structures that can withstand applied loads without excessive deformation.
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