Reinforced Concrete Structures – Full Detail Notes
Design of Beams: Singly and Doubly Reinforced Beams, with Complete Design Steps and Checks
PSC Subjective Note · Nepal Engineering Service (Civil), Second Paper · Lok Sewa Aayog
Definition of Beam
Beam: A beam is a horizontal structural member that carries loads perpendicular to its longitudinal axis. It resists loads applied laterally to the beam's axis primarily through bending and shear.
Beams are designed for two main forces:
- Bending Moment (M): Maximum at mid-span for simply supported beams
- Shear Force (V): Maximum at supports
Key Point: Beams are also known as flexural members since they are designed for both shear force and bending moment.
Classification Based on Reinforcement
Based on the location of longitudinal reinforcement, beams are classified into two types:
| Basis | Singly Reinforced Beam (SRB) | Doubly Reinforced Beam (DRB) |
|---|---|---|
| Definition | Longitudinal reinforcement provided only in tension zone | Longitudinal reinforcement provided in both tension and compression zones |
| Reinforcement Location | Tension zone only | Both tension and compression zones |
| When Used | When Mu ≤ Mlim | When Mu > Mlim |
| Economy | More economical | Less economical but necessary for high moments |
Singly Reinforced Beam (SRB)
Singly Reinforced Beam: If longitudinal reinforcement is provided only in the tension zone, then it is called a Singly Reinforced Beam.
Examples
Simply Supported Beam
When a simply supported beam carries load from top, it deflects downward creating a sagging moment (positive moment). The bottom fibers are in tension and top fibers are in compression.
Cantilever Beam
In a cantilever beam, negative hogging moment occurs. The top fibers are in tension and bottom fibers are in compression. Hence, main reinforcement is provided at the top.
Doubly Reinforced Beam (DRB)
Doubly Reinforced Beam: If longitudinal reinforcement is provided in both tension and compression zones, then it is called a Doubly Reinforced Beam.
When is DRB Required?
A doubly reinforced beam is required when:
- The factored bending moment (Mu) exceeds the limiting moment of resistance (Mlim)
- Depth of beam is restricted due to architectural or other constraints
- Both sagging and hogging moments occur (e.g., fixed beams)
- To reduce long-term deflection
Condition for DRB:
If Mu > Mlim → Provide Doubly Reinforced Beam
Example: Fixed Beam
In fixed beams, both sagging (positive) and hogging (negative) moments occur. At mid-span, positive moment causes tension at bottom, while at supports, negative moment causes tension at top.
Design Steps for Singly Reinforced Beam
Exam Tip: For 10 marks questions, write all 8 steps. For 5 marks questions, write up to Step 6 and draw the figure. Checking steps may be omitted for 5-mark questions.
Step 1: Calculate Maximum Bending Moment (Mu)
For simply supported beam with UDL:
Mu = (wu · L2) / 8
Where: wu = Factored UDL, L = Effective span
Step 2: Calculate Limiting Moment of Resistance (Mlim)
Mlim = k · fck · b · d2
Where:
- k = 0.148 for Fe 250 steel
- k = 0.138 for Fe 415 steel
- k = 0.133 for Fe 500 steel
- fck = Characteristic strength of concrete
- b = Width of beam
- d = Effective depth of beam
Step 3: Compare Mu with Mlim
- If Mu ≤ Mlim → Design as Singly Reinforced Beam
- If Mu > Mlim → Design as Doubly Reinforced Beam
Step 4: Calculate Area of Steel Required (Ast,req)
For singly reinforced beam:
Mu = 0.87 fy Ast d (1 − (Ast fy) / (b · d · fck))
Or use approximate formula:
Ast,req = Mu / (0.87 · fy · j · d)
Where j ≈ 0.9 (lever arm factor)
Step 5: Calculate Number of Bars
Number of bars = Ast,req / (Area of one bar) = Ast,req / ((π/4) · φ2)
Where φ = diameter of bar
Note: Always round up to next whole number. Minimum 2 bars required.
Step 6: Calculate Provided Area of Steel (Ast,prov)
Ast,prov = Number of bars × (Area of one bar)
Step 7: Check the Beam
The designed beam must be checked for:
- Percentage of steel
- Deflection
- Shear reinforcement (stirrups/rings)
Step 8: Draw Detailing Figure
Draw the longitudinal section and cross-section showing reinforcement details.
Important Checks
Check Percentage of Steel
Minimum Steel:
Ast,min = (0.85 · b · d) / fy
Maximum Steel:
Ast,max = 0.04 · b · D = 4% of gross cross-sectional area
Condition: Ast,min ≤ Ast,prov ≤ Ast,max
Check Deflection
(L/d)provided ≤ (L/d)basic × Kt × Kc × Kf
Where:
- (L/d)basic = 20 for simply supported beam (span ≤ 10m)
- (L/d)basic = 26 for continuous beam
- (L/d)basic = 7 for cantilever beam
- Kt = Tension modification factor (depends on % of tension steel and stress in steel)
- Kc = Compression modification factor (depends on % of compression steel)
- Kf = Flange modification factor (for T-beam and L-beam)
Note: Kt varies from 0.5 to 2.0, Kc varies from 1.0 to 1.5. Values are obtained from IS 456:2000 curves.
Shear Reinforcement
Stirrups (rings) must be provided to resist shear force. The spacing and diameter of stirrups are calculated based on the shear force diagram.
Summary
Key Points to Remember
- Beam is a horizontal structural member carrying transverse loads
- Singly Reinforced Beam: Steel only in tension zone
- Doubly Reinforced Beam: Steel in both tension and compression zones
- Use DRB when Mu > Mlim or depth is restricted
- Simply supported beam → Sagging moment → Tension at bottom
- Cantilever beam → Hogging moment → Tension at top
- Always check: % of steel, deflection, and shear reinforcement
- Minimum 2 bars must be provided in beams
For Exams:
- Define beam first before defining SRB/DRB
- Draw clear diagrams with proper labeling
- Write all formulas even if numerical is not asked
- Mention IS code provisions where applicable
Long / Descriptive Questions and Answers
Difference between Singly and Doubly Reinforced Beam (5)
| Basis | SRB | DRB |
|---|---|---|
| Full Form | Singly Reinforced Beam | Doubly Reinforced Beam |
| Reinforcement Location | Tension zone only | Both tension and compression zones |
| When Used | When Mu ≤ Mlim | When Mu > Mlim, depth is restricted, or both sagging and hogging moments occur |
| Typical Example | Simply supported beam (sagging moment, steel at bottom) | Fixed beam (sagging and hogging moments, steel at top and bottom) |
| Economy | More economical | Less economical but necessary for high moments |
Define Beam and Explain Sagging and Hogging Moments (5)
Beam: A beam is a horizontal structural member that carries loads perpendicular to its longitudinal axis. It resists loads applied laterally to the beam's axis primarily through bending and shear. Beams are designed for two main forces: bending moment (M) — maximum at mid-span for simply supported beams — and shear force (V) — maximum at supports.
Types of bending moment:
- Sagging moment (positive): Occurs in simply supported beams; the beam deflects downward, bottom fibers are in tension and top fibers are in compression, so main reinforcement is provided at the bottom.
- Hogging moment (negative): Occurs in cantilever beams; the top fibers are in tension and bottom fibers are in compression, so main reinforcement is provided at the top.
Key Point: Beams are also known as flexural members since they are designed for both shear force and bending moment.
Explain the Design Steps for Singly Reinforced Beam (5) or (10)
The design steps for a singly reinforced beam are:
- Calculate maximum bending moment: Mu = (wu · L2) / 8 for simply supported beam with UDL.
- Calculate limiting moment of resistance: Mlim = k · fck · b · d2 (k = 0.148 for Fe 250, 0.138 for Fe 415, 0.133 for Fe 500).
- Compare Mu with Mlim: If Mu ≤ Mlim → design as singly reinforced beam; if Mu > Mlim → design as doubly reinforced beam.
- Calculate area of steel required: Ast,req = Mu / (0.87 · fy · j · d), where j ≈ 0.9.
- Calculate number of bars: Number of bars = Ast,req / ((π/4) · φ2); always round up, minimum 2 bars.
- Calculate provided area of steel: Ast,prov = Number of bars × (Area of one bar).
- Check the beam: Percentage of steel, deflection and shear reinforcement (stirrups/rings).
- Draw detailing figure: Longitudinal section and cross-section showing reinforcement details.
Exam Tip: For 10 marks questions, write all 8 steps. For 5 marks questions, write up to Step 6 and draw the figure. Checking steps may be omitted for 5-mark questions.
Write Down the Important Checks in Beam Design (5)
The important checks in beam design are given below:
Check Percentage of Steel
- Minimum steel: Ast,min = (0.85 · b · d) / fy
- Maximum steel: Ast,max = 0.04 · b · D = 4% of gross cross-sectional area
- Condition: Ast,min ≤ Ast,prov ≤ Ast,max
Check Deflection
- (L/d)provided ≤ (L/d)basic × Kt × Kc × Kf
- (L/d)basic = 20 for simply supported beam (span ≤ 10m), 26 for continuous beam, 7 for cantilever beam
- Kt = Tension modification factor (varies from 0.5 to 2.0)
- Kc = Compression modification factor (varies from 1.0 to 1.5)
- Kf = Flange modification factor (for T-beam and L-beam)
- Values are obtained from IS 456:2000 curves.
Shear Reinforcement
- Stirrups (rings) must be provided to resist shear force.
- The spacing and diameter of stirrups are calculated based on the shear force diagram.