If a rectangular section simply supported beam is designed as a singly reinforced section, it needs an overall depth ‘Ds ’, however, keeping the other parameters the same, if it is designed as doubly reinforced, then the revised overall depth of beam becomes ‘Dd ’. If the mid-span defection for singly reinforced beam and doubly reinforced beam are ∆s and ∆d respectively, then, in general
- ADs > Dd and ∆s > ∆dCorrect
- BDs > Dd and ∆s < ∆d
- CDs < Dd and ∆s > ∆d
- DDs < Dd and ∆s < ∆d
Solution
Concept
Comparison of depth and deflection between singly and doubly reinforced rectangular beams under the same loading and parameters.
Formula
Mu = 0.36 fck b xu (d - 0.42 xu); Δ ∝ (1 / I)
Step-by-step solution
1. For a given bending moment, a singly reinforced beam requires a larger effective depth if the steel percentage is restricted. 2. By introducing compression steel (doubly reinforced section), the moment of resistance increases, allowing a smaller overall depth (Dd < Ds). 3. Deflection in beams decreases with an increase in effective depth and effective moment of inertia, combined with the presence of compression reinforcement which reduces long-term creep and shrinkage deflections. 4. Thus, Ds > Dd and Δs > Δd.
Explanation
A doubly reinforced beam provides compression steel, which increases the moment of resistance of the section for the same dimensions or allows a smaller overall depth compared to a singly reinforced beam for the same loading. Therefore, the required overall depth of a doubly reinforced beam (Dd) is less than that of a singly reinforced beam (Ds). Since deflection is inversely proportional to the moment of inertia and depth, the smaller depth and higher steel reinforcement in doubly reinforced beams control the deflection, making the deflection of the singly reinforced beam greater than that of the doubly reinforced beam (Δs > Δd).
Why the other options are wrong
B: Incorrect deflection relation since singly reinforced beam has larger depth but higher flexibility or lesser effective stiffness contribution without compression steel, or incorrect depth relation. C: Incorrect depth relation (Ds cannot be less than Dd for the same moment capacity without increasing width or steel immensely). D: Incorrect depth relation.
Shortcut
Doubly reinforced beams are used when depth is restricted; hence Dd < Ds. Compression reinforcement also helps reduce long-term deflections.
Correct answer: A) Ds > Dd and ∆s > ∆d