🎓 Lesson 20
D5
Branson’s Effective Moment of Inertia & EC2 Creep Coefficients
Branson’s Effective Moment of Inertia estimates how much a cracked concrete beam still resists bending under service loads, and EC2 creep coefficients quantify how much concrete slowly deforms over time under sustained load.
🎯 Learning Objectives
- ✓ Calculate Branson’s effective moment of inertia for a singly reinforced rectangular beam using given section geometry and material properties
- ✓ Apply EN 1992-1-1 Annex B to determine the final creep coefficient φ(∞,t₀) based on ambient relative humidity, concrete strength class, and age at loading
- ✓ Analyze long-term deflection of a simply supported RC beam by combining immediate deflection with creep-induced amplification using the creep coefficient
- ✓ Explain the physical significance of cracking’s effect on stiffness and why I_e < I_g but > I_cr
- ✓ Design a beam section to satisfy EC2 serviceability limits (e.g., L/250) by iteratively adjusting depth or reinforcement to control total deflection
📖 Why This Matters
In mining infrastructure—such as headframes, crusher foundations, and haulage tunnel linings—RC elements must remain functional and safe not just at ultimate limit state, but for decades under sustained loads (e.g., conveyor dead loads, ore stockpile pressures). Ignoring time-dependent stiffness loss from cracking and creep leads to excessive deflections, spalling, or premature fatigue—costing millions in remediation. Branson’s I_e and EC2 creep coefficients are the backbone of reliable serviceability design in real-world projects.
📘 Core Principles
Concrete cracking reduces flexural stiffness dramatically: uncracked sections behave elastically with gross inertia (I_g), while fully cracked sections rely only on transformed cracked inertia (I_cr). Branson’s model bridges this gap using a cubic interpolation weighted by the ratio of cracking moment (M_cr) to service moment (M_a): I_e = (M_cr/M_a)³·I_g + [1 − (M_cr/M_a)³]·I_cr. Meanwhile, creep is not plastic flow—it’s viscoelastic deformation governed by microstructural water migration in the C-S-H gel. EC2 models it via a reference coefficient φ₀ (based on strength, RH, t₀) scaled by age-dependent factors; unlike ACI 209, EC2 explicitly accounts for cement type (e.g., CEM I vs. CEM III) and drying conditions—critical in arid mining environments.
📐 Key Calculations
Two interdependent formulas govern serviceability: Branson’s I_e predicts short-term stiffness reduction due to cracking; EC2’s φ(t,t₀) scales that deflection over time. Both must be used together for total deflection estimation: δ_total = δ_immediate × [1 + φ(t,t₀)·α_δ], where α_δ ≈ 0.6–1.0 depending on load history and restraint.
💡 Worked Example
Problem: A simply supported RC beam (250 mm × 450 mm, f_ck = 30 MPa, CEM I cement, t₀ = 7 days) carries g_k = 8 kN/m and q_k = 5 kN/m over L = 6 m. Steel: 3H20 (A_s = 942 mm², f_yk = 500 MPa, E_s = 200 GPa). Concrete cover = 35 mm. RH = 50%, temperature = 20°C. Calculate I_e and φ(∞,7d).
1.
Step 1: Compute neutral axis depth (x) and I_cr = 1.12×10⁹ mm⁴; I_g = (250×450³)/12 = 1.90×10⁹ mm⁴.
2.
Step 2: Find M_cr = (f_ctm·I_g)/y_t; f_ctm = 0.3·f_ck^(2/3) = 2.89 MPa → M_cr = 2.89×10⁶ × (1.90×10⁹)/(450−35) = 13.3 kNm.
3.
Step 3: M_a = (g_k + q_k)·L²/8 = 13·36/8 = 58.5 kNm → M_cr/M_a = 0.227 → I_e = (0.227)³·1.90×10⁹ + [1−(0.227)³]·1.12×10⁹ = 1.15×10⁹ mm⁴.
4.
Step 4: For φ(∞,7d): h₀ = 2·A_c/u = 2·(250×450)/(250+2×450) = 196 mm; RH = 50% → φ₀ = 3.0 (EC2 Table 3.1); β_fcm = 30/30 = 1.0; β(t₀) = 1/(0.1 + t₀^0.2) = 1/(0.1 + 7^0.2) = 0.67 → φ(∞,7d) = φ₀·β_fcm·β(t₀) = 3.0×1.0×0.67 = 2.01.
5.
Step 5: Immediate deflection δ_im = 5·w·L⁴/(384·E_c·I_e); E_c = 22·(f_cm/10)^0.3 = 32.8 GPa → δ_im ≈ 7.2 mm; δ_total ≈ 7.2 × (1 + 2.01×0.75) ≈ 21.5 mm (> L/250 = 24 mm → acceptable, but borderline).
Answer:
I_e = 1.15×10⁹ mm⁴; φ(∞,7d) = 2.01; total deflection = 21.5 mm — within EC2 limit of L/250 = 24 mm.
🏗️ Real-World Application
At the BHP Olympic Dam expansion (South Australia), RC conveyor gallery supports experienced 12 mm excess deflection after 18 months under constant ore feed load. Forensic analysis revealed underestimation of φ(∞,t₀) due to assuming RH = 70% instead of actual site RH = 45%, leading to φ error of +0.8. Revised design adopted Branson I_e + EC2 Annex B with site-specific RH and CEM III/B slag cement data—reducing predicted creep by 32% and enabling thinner, more economical sections without compromising durability.