1. Cohesive Soil – Alm & Hamre (2001)

Unit Shaft Friction:

• q_f,i = f_s   (initial, from CPT sleeve friction)
• q_f,res = 0.004 · q_c · (1 − 0.0025 · q_c / σ'_v0)
• k = (q_c / σ'_v0)^0.5 / 80
• q_f = q_f,res + (q_f,i − q_f,res) · exp(k · (d − p))

Unit End Bearing: q_b = 0.6 · q_c

2. Non-Cohesive Soil – Alm & Hamre (2001)

Unit Shaft Friction:

• K = 0.0132 · (q_c/σ'_v0) · (σ'_v0/100)^0.13
• q_f,i = K · σ'_v0 · tan(δ)
• q_f,res = 0.2 · q_f,i
• q_f = q_f,res + (q_f,i − q_f,res) · exp(k · (d − p))

Unit End Bearing: q_b = 0.15 · q_c · (q_c/σ'_v0)^0.2

3. Glauconite – NGI Modified (RAM)

Unit Shaft Friction: q_f = a · f_s · max(1, z/D_ext)^c · (1 − (D_int/D_ext)²)^b

Unit End Bearing: q_b = 0.25 · q_c

4. Rock

End Bearing (Stevens 1982): q_b = UCS · N_c,r

Shaft Friction – COWI Clay-like: Same as cohesive (f_si = f_s,CPT) with friction fatigue, applied to both internal and external surfaces.

Shaft Friction – COWI Sand-like: Same as non-cohesive (K-based) with friction fatigue, applied to external surface only.

Shaft Friction – Kadivar UCS-based:

• Slightly weathered: f_s = 2.62·UCS / (0.467+2.088·UCS)^0.945
• Moderately weathered: f_s = 1.196·UCS / (0.5+2.088·UCS)^0.83
• Highly to Moderately: f_s = 0.696·UCS / (0.75+2.088·UCS)^0.75
• Highly weathered: f_s = 0.127·UCS^0.54

5. Friction Fatigue

As the pile penetrates deeper, soil above the tip experiences friction fatigue. The exponential decay factor exp(k · (d − p)) reduces the shaft friction from its initial value towards the residual value. At the pile tip (d=p), there is no reduction. Further above the tip, the reduction increases exponentially.

6. Total SRD

SRD = Q_f,external + Q_f,internal + Q_p,coring

• Q_f,ext = Σ(π · D_ext · q_f · Δz) for all soil types
• Q_f,int = Σ(π · D_int · q_f · Δz) for cohesive + glauconite (+ rock if clay-like)
• Q_p,coring = π/4 · (D_ext² − D_int²) · q_p

7. Blow Count Analysis – Smith (1960) 1D Wave Equation

This tool implements the Smith (1960) one-dimensional wave equation — the same fundamental method used by GRLWEAP. The pile is discretized into lumped-mass segments connected by springs, and the stress wave propagation from each hammer blow is simulated in the time domain using explicit finite-difference integration.

Unlike simplified energy-balance formulas (Hiley, Gates, ENR), the Smith model correctly captures wave propagation effects: the stress wave travels at ~5100 m/s through the steel pile, reflects off soil resistance and the pile toe, and the pile segments move independently rather than as a rigid body. This is critical for long offshore piles where the pile length greatly exceeds the stress wave pulse length.

7.1 Input Parameters — What Each Input Does

Understanding each input parameter is essential for producing realistic blow count predictions. Below is a detailed explanation of every input, what it physically represents, and how it feeds into the calculation.

7.2 The Smith Model

The hammer-pile-soil system is idealized as a chain of lumped masses connected by springs:

The simulation runs in time steps of ~0.1 ms. At each step: spring forces are computed between adjacent masses, soil resistance is evaluated, and velocities/displacements are updated. The permanent set emerges naturally from the simulation — no energy balance formula is needed.

7.3 Equation of Motion (Smith, 1960)

At each time step Δt, the equation of motion is solved for every segment:

Where:

• m_i = mass of segment i = ρ · A · ΔL
• k = spring stiffness = E · A / ΔL (connects adjacent segments)
• u_i = displacement of segment i (m)
• R_soil,i = total soil resistance at segment i (static + dynamic)
• g = gravitational acceleration

Time integration (explicit Euler):

• v_i^t+1 = v_i^t + a_i^t · Δt
• u_i^t+1 = u_i^t + v_i^t+1 · Δt

CFL stability condition: Δt < ΔL / c_wave. We use Δt = ΔL / (c · 1.6) per GRLWEAP recommendation (safety factor 1.6).

Ram impact: At t=0, the ram has velocity v₀ = √(2 · E_delivered / M_ram). All other elements are at rest. The ram is connected to the helmet via a cushion spring (compression only — no tension, allowing ram separation after rebound).

7.4 Soil Resistance Model (at each pile segment):

At each embedded pile segment, the soil resistance has two components (GRLWEAP Eq. 3.21-3.22):

Where Q = quake (max elastic deformation), R_u = ultimate static resistance at that segment, J = Smith damping factor, and v_i = velocity of that pile segment (computed from the wave equation, NOT the ram velocity).

Because the Smith model computes the actual pile velocity at each segment and each time step, the damping is automatically correct — no need for the simplified Newtonian impact velocity approximation. This is the fundamental advantage over the Hiley formula.

Recommended Smith damping values (GRLWEAP 2010):

Sensitivity: decreasing damping by one-third changes blow count by 20–30% (Rausche et al., 2004).

7.5 Blow Count Determination (GRLWEAP Eq. 3.39–3.41):

After the simulation completes for one blow, the permanent set is computed:

The maximum toe displacement is the largest downward displacement reached by the pile toe during the simulation. The weighted average quake represents the elastic rebound. The permanent set is the difference — the net advance of the pile. If set ≤ 0, the pile rebounds fully: refusal.

7.6 Self-Weight Penetration

Before hammering begins, the pile penetrates under its own weight (plus any hammer weight sitting on the pile and crane vertical load). The tool compares the total driving weight against the SRD at each depth. Where SRD < driving weight, the pile sinks freely with zero blow count. Hammering only starts at the depth where SRD first exceeds the driving weight. The driving weight increases with depth as more pile steel enters the ground.

7.7 Driving Stress

When the ram strikes the pile, a compressive stress wave propagates down the pile at the speed of sound in steel (~5120 m/s). The peak stress from this impact wave is:

• σ_impact = E_steel × v_impact / c_wave
• c_wave = √(E_steel / ρ_steel) ≈ 5120 m/s for steel

The pile impedance Z = E×A/c relates force and velocity in the pile: F = Z×v. The driving stress must remain below the pile steel yield stress (typically 0.9 × f_y for S355 steel = ~320 MPa) to avoid pile damage.

7.8 How the Smith Simulation Works (Step by Step)

7.9 What Drives the Blow Count Up?

Understanding which factors increase blow count helps with hammer selection and risk assessment:

7.10 Quake Values (GRLWEAP 2010)

7.11 Refusal Criteria (GRLWEAP Practice)

• Practical refusal: >800 blows/m (240 bl/ft, 20 bl/in) — driving becomes inefficient and risks pile/hammer damage
• Hard refusal / driving stop: >1200 blows/m (360 bl/ft, 30 bl/in) — driving must stop to prevent equipment damage
• Fatigue limit: Total cumulative blow count limited (typically 5000–15000 blows) to control pile fatigue damage from repeated stress cycles

7.12 Hammer Efficiency (GRLWEAP 2010)

8. Limitations of Simplified Approach

• No stress wave propagation – uses single-degree-of-freedom energy balance vs. full Smith discretised model
• No cushion/helmet dynamic interaction modelled explicitly
• No residual stress analysis (RSA) between consecutive blows
• No soil setup or relaxation effects during driving interruptions
• Blow count sensitivity: ±30% variation is typical for simplified methods (Rausche et al., 2004)
• Always validate with full GRLWEAP analysis for final design

9. References

• Alm, T. and Hamre, L. (2001) – “Soil model for pile driveability predictions based on CPT interpretations”
• Stevens, R.S. (1982) – Bearing capacity of rock
• Kadivar et al. – UCS-based rock shaft friction
• COWI DR_SU-GNR-GN-CRI-1010 – Rock shaft friction treatment
• Smith, E.A.L. (1960) – “Pile-driving analysis by the wave equation”, ASCE J. Soil Mechanics, 86(4), 35–61
• Hiley, A. (1925) – “A rational pile driving formula and its application in practice explained”, Engineering, 119, 657–658
• GRLWEAP 2010 Background Report – Pile Dynamics Inc.
• Rausche, F., Likins, G., et al. (2004) – “Correlations and applications of GRLWEAP”, Proc. 7th Intl. Conf. on Application of Stress Wave Theory to Piles