🎓 Lesson 11
D4
Shear Capacity Components: Vc, Vs, and φVn
Vc is the shear strength provided by the concrete itself, Vs is the extra shear strength added by steel stirrups, and φVn is the total safe shear capacity of the beam after applying a safety factor.
🎯 Learning Objectives
- ✓ Calculate Vc using ACI 318-19 equations for different cross-section types
- ✓ Design minimum and maximum spacing of stirrups to satisfy Vs requirements
- ✓ Analyze whether a given beam section satisfies φVn ≥ Vu (factored shear demand)
- ✓ Explain the physical mechanisms behind diagonal tension cracking and how Vc and Vs resist it
- ✓ Apply ACI 318 limits on maximum shear capacity and reinforcement ratios
📖 Why This Matters
Shear failure is sudden, brittle, and catastrophic—unlike flexural failure, it gives no warning. In mining infrastructure (e.g., headframe foundations, crusher pedestals, haul road retaining walls), underestimating Vc, Vs, or φVn can lead to collapse during dynamic loading from blasting vibrations or heavy equipment impact. Mastering these components ensures life-safety and serviceability in high-consequence RC structures.
📘 Core Principles
Shear resistance in reinforced concrete arises from three primary mechanisms: (1) aggregate interlock across cracks, (2) dowel action of longitudinal bars, and (3) vertical component of cracked concrete struts — collectively modeled as Vc. Transverse steel (Vs) resists shear by acting like tension ties in a truss analogy, bridging diagonal cracks. As shear demand increases, Vc alone becomes insufficient; Vs must be added to prevent brittle diagonal cracking. The strength reduction factor φ accounts for variability in material properties and modeling uncertainty — especially critical in seismic or blast-loaded environments where dynamic effects reduce ductility.
📐 Key Calculation
ACI 318-19 provides multiple expressions for Vc depending on member type and loading conditions. For nonprestressed, normal-weight concrete beams with shear reinforcement, the simplified equation is most commonly used. Vs is calculated directly from stirrup area, yield strength, spacing, and effective depth.
💡 Worked Example
Problem: A rectangular RC beam has f'c = 35 MPa, fy = 420 MPa, b = 300 mm, d = 550 mm, and uses double-leg #10 stirrups (Asv = 2 × 100 mm² = 200 mm²) spaced at 150 mm. Calculate φVn.
1.
Step 1: Compute Vc = 0.17√f'c × b × d = 0.17 × √35 × 300 × 550 = 0.17 × 5.916 × 300 × 550 ≈ 167.3 kN
2.
Step 2: Compute Vs = (Asv × fy × d) / s = (200 × 420 × 550) / 150 = 308,000 N = 308.0 kN
3.
Step 3: Compute Vn = Vc + Vs = 167.3 + 308.0 = 475.3 kN; then φVn = 0.75 × 475.3 = 356.5 kN
Answer:
The design shear capacity φVn is 356.5 kN, which exceeds typical service-level shear demands (Vu ≤ 300 kN) for this beam size and grade.
🏗️ Real-World Application
At the Bingham Canyon Mine (Utah), RC foundation beams supporting gyratory crushers were retrofitted after vibration-induced diagonal cracking was observed post-blast. Original design assumed Vc-only resistance but neglected dynamic amplification of Vu. Engineers recalculated φVn using updated Vu (1.4× static + 0.5× blast-induced dynamic component), increased stirrup density (s reduced from 200 mm to 100 mm), and verified Vs contribution satisfied ACI 318-19 §22.5.6.2 — preventing recurrence over 12+ years of operation.