🎓 Lesson 17
D5
Generating Interaction Curves Using Strain Compatibility
An interaction curve shows how much axial load and bending moment a reinforced concrete column can safely carry together, like a strength map for combined forces.
🎯 Learning Objectives
- ✓ Calculate axial load and moment capacity for at least three distinct neutral axis positions using strain compatibility
- ✓ Construct a complete P–M interaction curve for a rectangular tied column with symmetric reinforcement
- ✓ Analyze whether a given factored load combination (Pu, Mu) falls inside, on, or outside the interaction curve
- ✓ Explain the physical significance of key points on the curve: pure compression, balanced failure, pure bending, and tension-controlled limit
- ✓ Apply phi-factors (ϕ) from ACI 318 to adjust nominal capacities for design strength
📖 Why This Matters
In mining infrastructure—like headframes, crusher foundations, and ventilation shaft linings—columns often resist heavy vertical loads *plus* significant lateral forces from equipment vibration, blast-induced ground motion, or wind. Assuming columns handle only axial load or only bending leads to dangerous underdesign. Interaction curves are the only reliable way to verify safety under realistic combined loading—and they’re required by all major design codes. Skipping them risks catastrophic brittle failure.
📘 Core Principles
Strain compatibility assumes plane sections remain plane after bending—a foundational assumption for RC analysis. At ultimate strength, concrete strains are capped at ε_c = 0.003 (per ACI 318), while steel strains vary linearly with distance from the neutral axis. By systematically varying the neutral axis depth (c), we compute corresponding concrete compression resultant (C_c), steel forces (T_s or C_s), axial force (P_n), and moment (M_n). Each (P_n, M_n) pair plots one point on the interaction curve. The curve transitions through four critical regions: pure compression (c → ∞), compression-controlled (c > c_bal), balanced (c = c_bal, ε_t = 0.002), tension-controlled (c < c_bal), and pure bending (P_n = 0). Understanding this progression reveals how ductility, strength, and failure mode shift with loading path.
📐 Nominal Axial Load and Moment Capacity
The core calculation solves equilibrium (ΣF_x = 0, ΣM = 0) and strain compatibility simultaneously. For a rectangular section with symmetric reinforcement, the nominal axial load P_n and moment M_n are computed for a given neutral axis depth c using Whitney’s equivalent rectangular stress block.
💡 Worked Example
Problem: Given: 400 mm × 600 mm rectangular column; f'_c = 28 MPa; f_y = 420 MPa; 4–25M bars (A_s = A'_s = 1963 mm² each side); d' = 60 mm; d = 540 mm; assume c = 220 mm.
1.
Step 1: Compute β₁ = 0.85 − 0.05(f'_c − 28)/7 = 0.85 (since f'_c ≤ 28 MPa); a = β₁c = 0.85×220 = 187 mm.
2.
Step 2: Concrete compression resultant: C_c = 0.85f'_c·a·b = 0.85×28×187×400 = 1,777,840 N ≈ 1778 kN.
3.
Step 3: Steel strains: ε_s' = 0.003(c−d')/c = 0.003(220−60)/220 = 0.00218 > ε_y = f_y/E_s = 420/200,000 = 0.0021 → compression steel yields: C_s' = A_s'·f_y = 1963×420 = 824,460 N.
4.
Step 4: ε_s = 0.003(d−c)/c = 0.003(540−220)/220 = 0.00436 > ε_y → tension steel yields: T_s = A_s·f_y = 1963×420 = 824,460 N.
5.
Step 5: P_n = C_c + C_s' − T_s = 1777840 + 824460 − 824460 = 1777840 N ≈ 1778 kN.
6.
Step 6: M_n = C_c·(d − a/2) + C_s'·(d − d') = 1777840×(540 − 93.5) + 824460×(540 − 60) = 1777840×446.5 + 824460×480 = 793,800,000 + 395,740,000 = 1,189,540,000 N·mm ≈ 1190 kN·m.
Answer:
For c = 220 mm, P_n = 1778 kN and M_n = 1190 kN·m — a compression-controlled point near balanced condition. This falls within typical P_n range (1200–2200 kN) and M_n range (800–1400 kN·m) for this section.
🏗️ Real-World Application
At the Red Lake Mine (Ontario, Canada), a 12-m-tall headframe column supporting hoist machinery was subjected to combined axial load (Pu = 1450 kN) and seismic moment (Mu = 980 kN·m) per ASCE 7-22 load combinations. Designers generated a full ϕP_n–ϕM_n interaction curve per ACI 318-19 using strain compatibility. The (ϕPu, ϕMu) point plotted 12% inside the curve boundary—confirming adequacy. Field instrumentation later verified peak strains matched predicted tension-controlled behavior during a 4.1-magnitude local blast event, validating the curve’s predictive accuracy.