Concrete Penetration Depth — Analytical Pipeline (Li & Chen 2003)

International Journal of Impact Engineering - Dimensionless formulae for penetration depth of concrete target impacted by a non-deformable projectile

Over six decades of ballistic testing have produced dozens of empirical formulae for concrete penetration — NDRC, Barr (UKAEA), ACE. All three share the same structural defects: unit-dependent coefficients that break when switching from SI to imperial, nose shape parameters assigned as discrete lookup values rather than computed from geometry, and calibration ranges limited to shallow impacts (\(0.6 < X/d < 2.0\) ) that collapse at deep penetration (\(X/d > 5\) ) with errors exceeding 20%.

Li & Chen (2003) resolved these defects by building on two independent pillars. The Buckingham Pi theorem reduces the 10 physical variables of the problem to three dimensionless groups: the impact factor \(I^*\) , the mass ratio \(\lambda\) , and the nose factor \(N^*\) . Dynamic cavity expansion theory (Forrestal & Luk 1988) then provides the axial force law — static confinement resistance plus inertial resistance — and introduces the empirical parameter \(S\) that bridges uniaxial compressive strength to effective confined resistance. Combining the two theories recombines the three Pi groups into two operational numbers, \(I\) and \(N\) , which govern the result entirely.

The outcome is two closed-form formulae — Eq. (15a) for shallow penetration, Eq. (15b) for deep — validated against approximately 130 data points covering \(X/d\) from 0.07 to 92.8.

Quick Example

4340-steel ogive projectile, CRH = 2, impacting 35 MPa concrete at 277 m/s. Data: Forrestal et al. (1994), Table 3, shot 14.

Penetration depth: \(X = 167\,\text{mm}\) (\(X/d = 6.21\) , deep regime) Test measurement: 173 mm. Model error: 3.4%. NDRC prediction: 137 mm. NDRC error: 20.5%.

The 26.9 mm ogive projectile at 277 m/s penetrates 167 mm into 35 MPa concrete — about 6.2 calibers. At this velocity the static resistance term accounts for 94% of the retarding force; the dynamic inertia term contributes only 6%. NDRC, calibrated on shallow impacts, underestimates by a factor that grows with penetration depth.

Pipeline summary:

Pipeline Overview

The pipeline is sequential with one bifurcation. Nodes 0–3 are always executed. Node 4 selects the depth formula based on whether \(I\) exceeds the crater threshold \(\pi k/4\) . Node 5a (shallow) or 5b (deep) computes the dimensionless depth; Node 6 converts to metres or millimetres.

📄 Download: The A4 Ballistic Pipeline — slide deck (PDF) - A visual representation of the calculation

📄Download The Theory Behind the Calculation (PDF) — A visual walkthrough of the physics: from Buckingham Pi to cavity expansion to the final dimensionless equations.

Node 0 — Validity Pre-Check

Purpose. Verify that the model assumptions hold before computing. If a condition is violated, the pipeline emits a warning but does not block execution — the engineering judgement remains with the user.

Node 1 — Nose Geometry

Purpose. Compute the nose shape factor \(N^*\) and the dimensionless crater depth \(k\) .

Nose factor \(N^*\) — Eqs. (2), (3a)–(3c)

\(N^*\) is defined by a surface integral over the nose profile. For standard geometries the integral has closed form:

\(N^*\) is a continuous function of geometry, not a discrete lookup. A lower value means a sharper, more penetration-efficient nose: flat nose \(N^* = 1.0\) , hemispherical \(N^* = 0.5\) , ogive CRH=2 \(N^* = 0.156\) , ogive CRH=4.5 \(N^* = 0.076\) .

Nose height \(H/d\)

Crater depth parameter \(k\) — Eq. (25)

The crater–tunnel transition depth is the sum of the Prandtl plastic slip depth for a flat punch (\(0.707d\) ) and the nose height \(H\) :

Node 2 — Target Resistance

Purpose. Compute the empirical constant \(S\) that scales the uniaxial compressive strength \(f_c\) to the effective resistance under dynamic triaxial confinement. \(S\) cannot be derived analytically; it is back-calculated from penetration tests and then fitted as a function of \(f_c\) .

Two correlations are provided (\(f_c\) in MPa for both):

Original — Forrestal et al. (1994), Eq. (12):

Simplified — Li & Chen (2003), Eq. (21):

The simplified form makes \(I\) proportional to \(f_c^{-1/2}\) , aligning with the \(f_c\) dependence in NDRC, Hughes and Chang and enabling direct comparison. For \(f_c > 30\,\text{MPa}\) the two correlations are practically equivalent.

For shot 14 (\(f_c = 35.2\,\text{MPa}\) , \(S = 12\) from Table 2), both correlations agree to within 0.7 units.

Node 3 — Dimensionless Numbers

Purpose. Compute the two numbers that govern the penetration depth.

Intermediate quantities

Impact factor (Eq. 5):

Mass ratio (Eq. 6):

Johnson damage number (Eq. 7, informational):

\(\Phi_J\) depends only on velocity and target — not on the projectile. It classifies impact severity independently of the launcher.

Operational numbers

\(I\) is the ratio of projectile kinetic energy to the concrete’s effective absorption capacity (resistance under confinement, scaled by \(S\) , over volume \(d^3\) ).

\(N\) combines relative projectile mass and nose sharpness. A heavy, sharp projectile has large \(N\) and penetrates more for equal \(I\) .

Node 4 — Regime Selection

Purpose. Determine whether the projectile stops inside the crater or advances into the tunnel.

The projectile exits the crater zone with residual velocity \(V_1 > 0\) if and only if \(I > \pi k / 4\) :

Node 5a — Shallow Penetration — Eq. (15a)

Condition: \(X/d \leq k\) (projectile stops within the crater zone)

Derived from the energy balance \(X^2 = MV_0^2/c\) with the crater force constant \(c\) from the continuity condition at \(x = kd\) :

Consistency checks. At the regime boundary \(I = \pi k/4\) : the formula gives \(X/d = k\) , matching the deep formula below. For \(N \gg 1\) : reduces to Eq. (16a) of the paper. For \(N \gg 1\) and \(I/N \ll 1\) : reduces to \(X/d = \sqrt{4kI/\pi}\) , Eq. (17a).

Note on shallow correction. For \(X/d < 0.5\) the formula systematically overestimates (paper Fig. 11). The empirical correction Eq. (27) — \(\left(X/d\right)_{\text{corr}} = 1.628\,(X/d)^{2.789}\) — is available in the notebook as an opt-in flag. It is off by default: the correction was fitted on flat-nosed, low-energy data where scatter is large, and its applicability to other nose types is uncertain.

Node 5b — Deep Penetration — Eq. (15b)

Condition: \(X/d > k\) (projectile traverses the crater and bores a tunnel)

Derived by integrating the equation of motion \(MV\,dV/dx = -(\pi d^2/4)(Sf_c + N^*\rho_c V^2)\) from \(x = kd\) to \(x = X\) , then non-dimensionalising through \(I\) and \(N\) :

Penetration depth grows logarithmically with \(I\) at fixed \(N\) — not linearly. Each additional unit of velocity buys progressively less depth.

Consistency checks. At the regime boundary \(I = \pi k/4\) : \(\ln(1) = 0\) , giving \(X/d = k\) — exact continuity with Eq. (15a). For \(N \gg 1\) : reduces to Eq. (16b). For \(N \gg 1\) and \(I/N \ll 1\) : reduces to \(X/d = k/2 + 2I/\pi\) , Eq. (17b).

Node 6 — Dimensional Output

Purpose. Convert \(X/d\) to physical depth and summarise all intermediate values.

Report also: regime, \(I\) , \(N\) , \(k\) , \(S\) , \(I^*\) , \(\lambda\) , \(\Phi_J\) , and the semi-infinite check (\(3X\) minimum target thickness).

Numerical Verification — Shot 14

Forrestal et al. (1994), Table 3. Ogive CRH=2, \(f_c = 35.2\,\text{MPa}\) .

Input

Node 1

Note: paper Tables 2–4 use \(k = 2\) (empirical value from Forrestal 1994). The tool uses \(k = 2.030\) from Eq. (25). For this verification, \(k = 2\) is used to reproduce the paper tables.

Node 2

Node 3

Node 4

Node 5b

Paper Table 2, shot 14: \(X/d_{\text{anal}} = 6.21\) ✓ — \(X/d_{\text{test}} = 6.43\) — \(X/d_{\text{NDRC}} = 5.11\)

Node 6

Test: 173 mm. Error: 3.4%. Semi-infinite check: \(3X = 502\,\text{mm}\) minimum target thickness.

Figures

For fixed \(N\) , \(X/d\) grows logarithmically with \(I\) in the deep regime. The shallow-to-deep transition (knee of each curve) shifts with \(k\) and \(N\) . The current case is marked in red.

For fixed \(I\) , deeper penetration results from a heavier, sharper projectile (large \(N\) ). The sensitivity to \(N\) decreases as \(I/N \to 0\) — for shot 14 (\(I/N = 0.067\) ) geometry contributes marginally and \(X/d\) is driven almost entirely by \(I\) .

How to Use the Notebook

The Python notebook implements the full pipeline (Nodes 0–6) and runs directly on Google Colab. No installation required: numpy and matplotlib are preinstalled.

Open in Colab

Modify only the INPUT PARAMETERS cell. Parameters:

Each node prints its full output — all intermediate values are visible.

Known limitations. The model requires: rigid (non-deformable) projectile — erosion becomes significant above ~800 m/s for hardened steel; semi-infinite target — rear boundary effects (scabbing, perforation) not modelled, verify thickness \(\geq 3X\) ; normal incidence — oblique impact not covered; unreinforced or lightly reinforced concrete (\(< 1.5\%\) per direction); aggregate size small relative to diameter (\(d/a > 5\) recommended, continuum assumption breaks down below \(d/a \approx 2\) ).

References

[1] Li QM, Chen XW (2003). Dimensionless formulae for penetration depth of concrete target impacted by a non-deformable projectile. Int. J. Impact Eng. 28, 93–116.

[2] Forrestal MJ, Altman BS, Cargile JD, Hanchak SJ (1994). An empirical equation for penetration depth of ogive-nose projectiles into concrete targets. Int. J. Impact Eng. 15(4), 395–405.

[3] Forrestal MJ, Luk VK (1988). Dynamic spherical cavity-expansion in a compressible elastic-plastic solid. ASME J. Appl. Mech. 55, 275–279.

[4] Sliter GE (1980). Assessment of empirical concrete impact formulas. ASCE J. Struct. Div. 106(ST5), 1023–1045.