🎓 Lesson 6
D2
Design Example: Singly Reinforced Rectangular Beam
A singly reinforced rectangular beam is a concrete beam with steel bars only in its bottom part to resist bending when it’s loaded from above.
🎯 Learning Objectives
- ✓ Calculate the required area of tension steel (As) for a given factored moment and beam dimensions
- ✓ Design a singly reinforced rectangular beam by selecting appropriate dimensions, steel grade, and concrete strength per ACI 318 strength reduction factor requirements
- ✓ Analyze whether a given beam section is ductile (tension-controlled) using strain compatibility and εt verification
- ✓ Explain the role of the strength reduction factor (ϕ) and justify its value based on the strain in the extreme tension steel
- ✓ Apply ACI 318 limits on reinforcement ratio (ρmin and ρmax) to ensure safety and serviceability
📖 Why This Matters
Singly reinforced beams form the backbone of nearly all low- to mid-rise building frames, mine ventilation structures, portal supports, and conveyor foundations in mining infrastructure. Understanding how to design them correctly ensures structural integrity under dynamic loads (e.g., equipment vibration, blast-induced ground motion) while optimizing material use—critical for cost-sensitive, remote mining projects where logistics constrain concrete and rebar delivery.
📘 Core Principles
Flexural design hinges on equilibrium and strain compatibility. Under bending, concrete resists compression in the top region (assumed to follow a rectangular stress block per ACI 318 §22.2.2.4), while steel in the bottom yields to resist tension. The neutral axis depth (c) separates these zones. Ductility is ensured when the tension steel strain (εt) exceeds 0.005 at nominal moment capacity—making the section 'tension-controlled' and eligible for ϕ = 0.90. Reinforcement ratios must stay between ρmin (to prevent brittle failure) and ρmax (to avoid sudden compression failure). ACI 318 also mandates minimum beam dimensions, cover, and bar spacing—especially important in aggressive mining environments with sulfate exposure or abrasive dust.
📐 Nominal Moment Capacity (Mn)
The nominal moment capacity of a singly reinforced rectangular beam is derived from force equilibrium and moment arm about the centroid of the compression block. It assumes a simplified Whitney rectangular stress block with coefficient β1 dependent on concrete strength.
💡 Worked Example
Problem: Design the tension steel for a simply supported beam with b = 300 mm, d = 550 mm, f'c = 28 MPa, fy = 420 MPa, and Mu = 225 kN·m. Use ϕ = 0.90.
1.
Step 1: Compute β1 = 0.85 (since f'c = 28 MPa ≤ 28 MPa, per ACI 318 Table 22.2.2.4.3)
2.
Step 2: Solve for required 'a' using ϕMn ≥ Mu → 0.90·[As·420·(0.55 − a/2)] = 225×10⁶ N·mm; but As = (0.85·f'c·b·a)/fy → substitute to get quadratic in 'a': a² − 1294.1a + 223,600 = 0
3.
Step 3: Solve → a ≈ 192 mm → then As = (0.85·28·300·192)/420 = 3,299 mm² → select 4–DB29 bars (As = 3,322 mm²)
4.
Step 4: Verify ductility: c = a/β1 = 192/0.85 = 226 mm; εt = 0.003·(550 − 226)/226 = 0.0043 < 0.005 → borderline; increase d to 570 mm → recalculate: a = 187 mm, c = 220 mm, εt = 0.0047 → still < 0.005; try 5–DB25 (As = 2,454 mm²) → a = 115 mm → εt = 0.0061 > 0.005 → OK. Final As = 2,454 mm².
Answer:
The final design uses 5–DB25 bars (As = 2,454 mm²), yielding εt = 0.0061 (> 0.005), confirming tension-controlled behavior and ϕ = 0.90 is valid. Mn = 251 kN·m → ϕMn = 226 kN·m ≥ Mu = 225 kN·m — acceptable.
🏗️ Real-World Application
At the Red Dog Mine (Alaska), a 6-m span conveyor gallery beam was designed as a singly reinforced rectangular section (300 × 600 mm, f'c = 35 MPa, fy = 420 MPa) to support dynamic belt loads and occasional blast-induced vibrations. Due to chloride-laden air and freeze-thaw cycles, 45 mm concrete cover (ACI 318 Table 20.6.1.3.1 for severe exposure) and epoxy-coated DB25 bars were specified. The design followed ρ = 0.012 (within ρmin = 0.0033 and ρmax = 0.0181), ensuring both strength and crack control—verified via field strain gauges during load testing showing εt = 0.0058 at ultimate load.