How Deep Does a Projectile Go into Concrete?

1. Why penetration first?

A projectile striking a buried protective structure first penetrates the soil and the protective layers, then detonates. The shock it delivers to equipment inside depends very strongly on the distance between the burst point and the equipment: in ground-shock relations, acceleration falls roughly with the third to fourth power of distance. A metre of uncertainty in penetration depth therefore becomes a multiple of uncertainty in the shock the equipment sees. It is the first, and most sensitive, question of the design.

2. The Young equations

The equations compiled by C. W. Young for Sandia National Laboratories are an empirical summary of thousands of full-scale tests into soil, rock and concrete since the 1960s. The report (SAND97-2426, 1997) gives them in US units, with depth D (ft), projectile weight W (lb), cross-sectional area A (in²) and striking velocity V (ft/s):

Three factors carry the physics. S is the target's penetrability (large S, easily penetrated). N is the nose performance: for a tangent ogive N = 0.18 · L_n/d + 0.56, and the nose length follows from geometry, L_n = d · √(CRH − 0.25). W/A gives the projectile its "concentrated weight": the same mass in a slimmer body goes deeper. A further factor corrects light projectiles (in rock and concrete, K = 0.4 · W^0.15 for W < 400 lb).

Note the structure of the high-velocity equation: depth grows linearly with velocity. This is the origin of the systematic trend we will meet shortly.

3. The target's language: the S-number

For concrete, S is computed from compressive strength f_c, reinforcement ratio, cure time, target thickness and target width (edge effect): S = 0.085 · K_e · (11 − P) · (t_c · T_c)^−0.06 · (5000/f_c)^0.3, with f_c in psi. For rock, S = 12 · (f_c · Q)^−0.3, where Q reflects joints and fractures. For soils the report gives a table: dense cemented sand 2–4, fill 8–10, soft saturated clay 20–30. The spread is large: the same projectile goes about ten times deeper into fill than into concrete.

4. Limits of the equations

5. An inconsistency in the report itself

The report's appendix also gives the equations in SI units. Converting the main text ourselves, the high-velocity coefficient matches the appendix. The low-velocity coefficient does not: the conversion gives —, while the appendix states 0.0008 (ratio —). In US units the two equations join continuously at 200 ft/s; with the appendix coefficient there is a jump of — at 61 m/s, which disappears with the converted coefficient. Because the appendix value is widely quoted we do not declare it a definite error, but our engine uses the main text. Velocities above 61 m/s are unaffected.

6. Blind validation 1: inside the range

Frew, Forrestal and Cargile (2006) fired 13 kg, 76.2 mm, 3 CRH ogive-nose steel projectiles into 23 MPa plain concrete at 164–337 m/s, varying the target diameter between 0.91 m and 1.83 m. These projectiles are within Young's mass range. In the table below, each Young prediction is built only from values reported in the paper (cure time taken at the middle of the test dates, ~300 days); no parameter is tuned to the measurements.

7. Blind validation 2: outside the range

Forrestal and co-workers (1996) fired lighter rods (0.48 and 1.6 kg; 20.3 and 30.5 mm) into 51 and 63 MPa concrete at 405–1201 m/s. These projectiles are below Young's lower mass limit, so this is not a validation but an answer to the question "how do the equations behave outside their range?"

8. What the two data sets agree on

Two independent data sets point the same way: Young over-predicts at low velocity and under-predicts at high velocity. The cause is in the equation: in the high-velocity range depth grows linearly with velocity, whereas in the tests it grows faster. The equation is therefore most accurate in a "middle" band; for the 13 kg data this band is 280–337 m/s.

The Frew data test one more thing: the effect of target width. When the target diameter falls from 24 to 12 projectile diameters, penetration in the tests increased by —, while Young's edge-effect factor predicts —. The factor points the right way but exaggerates the effect. For wide targets (real structures) the factor is 1 and does not enter.

8a. Two more models: Forrestal, NDRC and the scale effect

The model of Forrestal and co-workers (1994) comes by another route: it takes the force on the nose from cavity-expansion theory, F = πa²(S·f_c + N·ρ·V²), N = (8ψ − 1)/(24ψ²). The target is described by one dimensionless constant, S; the papers fit S to each data set's own data and give the form S ∝ 1/√f_c. We fitted the constant to the published S values for unreinforced targets: S = 71.5/√f_c (f_c in MPa; projectile diameters 12.9–30.5 mm). The third model is the modified NDRC formula (Kennedy 1966, 1976), widely used in nuclear-plant engineering for hard-missile impact: G = (180/√f_c)·N·(W/d)·(V/1000d)^1.8; x/d = 2√G (G ≤ 1), x/d = G + 1 (G > 1). We took it from two independent reviews (Kennedy's paper was not opened); in one review the branches are printed the wrong way round in one equation — we derived the correct assignment from that review's own definition of G; with the swapped branches the prediction drops by a further factor. For the ogive nose the "very sharp" factor (N = 1.14) is used. Below, four predictions side by side; the last column is fitted, not blind, using the target resistance reported for these tests (R ≈ 162.5 MPa).

Two lessons. At this scale only Young does not under-predict blind; Forrestal (with the laboratory S) and NDRC give about half — both on the unsafe side. Scale: the S fitted from laboratory projectiles overstates the target resistance by about a factor of two for 76.2 mm projectiles; a larger projectile sees a lower resistance. Young, built on full-scale tests, transfers more reliably to large scale. Form: once the resistance is taken at the right scale, Forrestal's physically based velocity dependence does not carry Young's 30% error at low velocity. A very heavy threat class (0.8 m diameter) lies far outside the laboratory scale; there, Forrestal with the laboratory S badly under-predicts penetration — the unsafe side.

9. Figures for a heavy threat class

Protective design works with threat classes, not named weapons. Below is the penetration of a very heavy penetrator class — ~14 t, 0.8 m diameter, normal impact at 295 m/s — into different targets. The nose shape is unknown, so 3, 6 and 9 CRH are given separately.

The contrast is striking: the same projectile stops at 3–5 m in concrete and at 40–50 m in fill. Figures of "~60 m" often repeated in the press for this class approach that order only in weak soil, not in concrete.

10. In practice

• Use the Young equations only within their range; ceramic, metal, liquid and composite layers need another method or tests.
• If your design velocity lies outside the band where the equation is most accurate, widen your uncertainty: at low velocity Young is on the safe side, at high velocity on the unsafe side.
• Establish the soil class on site: a factor of two in S is a factor of two in depth.
• For layered targets a single-medium equation is not enough; see PRT-02.

Try your own scenario in the Penetration Explorer; its "compare with experiment" button plots the Frew data from this article.

11. Check yourself

1. Two projectiles have the same mass; one has half the diameter of the other. How do their depths compare according to Young?
2. If a target's S-number doubles, what happens to depth? In which kinds of target can S double?
3. Why does Young under-predict at high velocity? Which feature of the equation explains it?
4. A protective envelope contains a B₄C ceramic layer. Can you apply the Young equations to it? If not, what would you do?
5. Why is the edge-effect factor usually 1 in a real structure?