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Defense and
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 DSS Vol. 1, No. 1, December 2020, pp.16-25 

17 

Data on the initial velocity of the fragments are needed to be able to estimate the elements of the trajectory of 
the fragments, and thus their kinetic energy at a given moment (during movement through the atmosphere). 
The measure of the explosive power is mainly expressed in the literature using the strength of the shock wave 
generated by that explosive, or on the total chemical energy that the explosive contains. In this way, the 
velocity of the shock wave, the detonation pressure, and the heat generated by the detonation of explosives 
can be expressed. Although this way of understanding and assuming the properties of explosives is accurate, it 
does not provide information on the initial velocity that an explosive can communicate during detonation of 
munition to fragments [1]. 
During World War II, physicist Ronald W. Gurney published several scientific papers explaining how the 
initial velocity of fragments could be calculated with relatively high accuracy. His scientific works thus 
created a method that is still used today to calculate the initial velocity of fragments. This method was 
developed to suit different systems and configurations of metal-explosive systems. Although the shock wave 
plays a very large role in the transfer of energy from explosives during detonation to metal, Gurney in his 
method does not take into account the properties of the shock wave itself. 
In his research, Gurney assumed [1] that during detonation, a final amount of energy is released by the 
explosives, which is converted into kinetic energy of fragments and kinetic energy of detonation gases. He 
also assumed that detonation gases have a uniform density and a linear one-dimensional velocity profile. 
The Gurney method can be used for all one-dimensional metal-explosive systems. 
The Gurney constant, which appears in his method, can be estimated experimentally (explosive cylinder 
expansion test), computer programs (in hydrocodes), and analytical models. Henry (1967), Jones (1980) and 
Kennedy (1970) reported Gurney’s constant values for certain explosives while Dobratz (1982) made the 
greatest contribution [2]. 
Some researchers (Kennedy, Randers-Pehrson, Lloyd, Odinstov) have proposed certain modifications of the 
Gurney model, and other authors (Hirsch, Chanteret, Chou-Flis, Kleinhanss, Hennequin) have applied the 
Gurney method to imploding configurations (configurations where the explosive is on the outside and the 
body on the inside; i.e. liner and explosive in HEAT warheads). 
Modifications of the Gurney base model mainly consisted of deriving formulas for geometric configurations 
of systems not covered by the original Gurney model, and for a larger range of M/C ratios. 
Henry (1967), Jones (1980), and Kennedy (1970) also used Gurney’s method for different metal-explosive 
configurations.  
Hirsch (1986) modified the basic Gurney formulas for exploding cylinders and spheres to extend their use to 
lower metal to explosive mass ratios M/C [2]. 
Another extension of the Gurney method was given by Chanteret (1983) who developed an analytical model 
for symmetric geometric configurations. 
Fucke et all. (1986) and Bol and Honcia (1977) measured fragment velocities for large M/C ratios [2]. 
Karpp and Predeborn (1974, 1975) showed that the assumptions about the initial velocity of the fragments 
obtained by the Gurney method are adequate for cases when the flow is one-dimensional and for practical 
C/M relations that can be encountered in reality (0,1 <C / M <2) [2]. 
Aziz et al. performed an analysis for an open sandwich configuration, where the metal was considered a solid, 
and where the gases were approximated by the ideal gas equation [1]. 
Butz et al. (1982) analyzed Gurney’s model for symmetric sandwich configuration. They found that the 
Gurney model fully follows the results obtained experimentally, and even for very small M/C ratios. The 
smallest ratio they considered was 0,05, and even for such a small ratio, the Gurney model gave a result that 
agrees with the experimental result [1]. 
Cooper states that the Gurney constant can be estimated using the detonation velocity of explosives [3]. 
Different authors [4-11] investigated the influence of rarefaction waves from two ends in cylindrical 
warheads, since the length of casing is not infinite, and consequently, fragment velocities are lower than the 
Gurney velocity prediction. Gao [4] concluded that the fragment velocities in the central part of the warhead 
are not influenced by rarefaction waves when the L/D ratio (slenderness) was more than 2. Huang et al. [11] 
proposed modified formula for the fragment velocity from the cylindrical casing, initiated at one end. Charron 
[10] also analysed rarefaction waves influence on fragment velocity. 
In recent times, numerical simulations in terminal ballistics are increasingly used. They are based on the 
formulas of continuum mechanics, using conservation equations, together with constitutive equations that 
define the behavior of the material, and specifying the initial and boundary conditions related to a given 
problem. 



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2. Gurney method 

The Gurney equation describes the initial velocity of the fragments as a function of the ratio of the mass of the 
explosive charge to the mass of the metal body and the empirically determined constant (Gurney constant). In 
deriving the Gurney formula, a hollow cylinder of mass M, of inner radius R, filled with an explosive of mass 
C and density  is considered. The initial velocity of the fragments for the cylindrical configuration of the 
metal body and explosive can be written in the form: 

 






 



2
1

2

C
M

E
vM   (1) 

where the parameter E2  represents the Gurney constant (the name characteristic Gurney velocity can also be 
found in the literature) and depends on the M/C ratio. The quantity E (so-called Gurney energy) has the 
dimension of energy per unit mass (J/kg), so the parameter E2  has the dimension of velocity (because the 
unit for energy is J = kgm2/s2). 
The generalized Gurney formula (for different body warhead shapes) can be written in the form: 

 
1

0 2
2

















g

g

n
n

C
M

Ev   (2) 

where the constant ng can have value of 1, 2 or 3, depending on whether the body is a flat, cylindrical or 
spherical configuration. A schematic representation of possible configurations is given in Fig. 2. A detailed 
derivation of these formulas can be found in references [2,14]. 

 
Fig. 2 Expressions for initial velocities of fragments in different geometric configurations [adopted from 2] 

The expression for the initial velocity of fragments in flat symmetrical sandwich configurations (expression 2 
at ng = 1) can be used to calculate the parameters of explosive-reactive armor on tanks. Expression 2 at ng = 3 
is applied to hand grenades and some types of cluster projectiles, while expression 2 at ng = 2 is reduced to 
expression (1) and is used to estimate the initial velocity of fragments in most HE warheads. Theoretically, the 
maximum initial velocity of the fragment is obtained when the mass of the body M approaches zero, and for 
cylindrical configurations it is Ev 22max0  . 
The Gurney constant can be estimated using tests, computer programs (using i.e. JWL equations of state for 
explosives), and analytical models (i.e. Kamlet and Finger use thermochemical parameters of explosive 
charge to estimate the Gurney constant; Kennedy gives the expression E  0,7ED, where ED is the detonation 
heat of the explosive). Cooper states that the Gurney constant can be estimated via the detonation velocity of 
explosives D (which, depending on the density and composition of the explosives, can be determined in 
computer programs, i.e. EXPLO5), by the expression: 

 97,2/2 DE    (3) 

The value of 2.97 from expression (3) was obtained by averaging the values of E2  for known explosives 
(Table 1). This relationship is capable of accurately predicting E2 of sensitive and moderately sensitive near 
ideal explosives; however, this equation is somewhat unsuitable for predicting E2 of insensitive and nonideal 
explosives (ie. NTO, PBX-9502, AFX-902), because it overestimates E2 of insensitive explosives [20]. 



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Table 1 shows the values of density, detonation velocity and Gurney constant for different types of explosive 
charges. Similar data for different types of explosives can be found in the NATO manual AASTP-1 [13]. 

Table 1. Experimental values of density, detonation velocity and characteristic Gurney velocities for different 
types of explosives [3] 

Explosive Density (kg/m3) 
Det. velocity 

(m/s) 
Gurney constant 

(m/s) ED 2/  
Composition A-3 1590 8140 2630 3,09 
Composition B 1710 7890 2700 2,92 
Composition C-3 1600 7630 2680 2,85 
Cyclotol (75/25) 1754 8250 2790 2,96 
H-6 1760 7900 2580 3,06 
Octogen 1835 8830 2800 3,15 
LX-14 1890 9110 2970 3,07 
Octol (75/25) 1810 8480 2800 3,03 
PBX 9494 1840 8800 2900 3,03 
PBX 9502 1885 7670 2377 3,23 
PETN 1760 8260 2930 2,82 
RDX 1770 8700 2830 2,97 
Tetril 1620 7570 2500 3,03 
TNT 1630 6860 2370 2,89 
Tritonal (80/20) 1720 6700 2320 2,89 

In real conditions, the initial velocity of the fragment represents the resultant of the initial velocity of the 
fragments (Gurney expression), the translational velocity of the projectile at the moment of impact on the 
target and the rotational velocity of the warhead (if the warhead rotates during the flight; gyroscopic 
stabilization). But since the translational and rotational velocity components are much smaller, only the initial 
velocity value, determined by the Gurney method, is usually used in the calculations. Using Gurney's 
methodology to estimate the initial velocity of the fragments ome does not make large errors in determining 
the total initial velocity of the fragments. By considering the values of the projectile's impact velocities (which 
are generally different for each angle of incidence), the calculation can be potentially closer to the real 
situation, with the spray of fragments moving in the direction of the projectile's movement. 
In an axisymmetric system with variable diameter, the C/M parameter is calculated depending on the axial 
position. This means that the initial velocity of the fragments will also vary, depending on the position on the 
projectile body. The initial velocities of the fragments, calculated by Gurney's formula, agree well with the 
test data for HE projectiles, as shown in Fig. 3 (left). Fig. 3 (right) shows a comparison of the initial velocities, 
calculated by the Gurney method, with the experimental data for a steel cylinder with ratio L/D = 2 (ratio of 
length and diameter; warhead slenderness), filled with composition B and initiated from the left side. Fig. 3 
illustrates how effects in the end part of the cylinder can affect the prediction of the initial velocities. 

 
Figure 3. Comparison of the Gurney method with experimental data [adopted from 2] 



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The standard cylinder test, developed at Lawrence Livermore National Laboratory (LLNL), is used in the 
research of the phenomenology of the expansion of metal cylinders due to the action of explosive charges. 
Several types of explosives can be used, as well as several types of cylinder materials (i.e. copper and two 
types of steel with different percentages of carbon and alloying elements). Karpp and Predebon observed that 
in most of these experiments the maximum initial velocity of the fragments was achieved at the expansion 
ratio of the body R/R0 = 2 (the diameter of the body in the process was expanded twice compared to the initial 
one). Even at the ratio R/R0 = 1,75 - about 92% of the maximum initial velocity of the fragments has already 
been achieved. Nevertheless, it seemed necessary to consider the effects of gas leakage through the end parts 
of the cylinders, and to include the obtained models in non-stationary numerical simulations, to obtain more 
accurate values of initial velocities. In this sense, Karp and Predebon used the HEMP program, based on the 
finite element method, to model the process of natural fragmentation. The standard JWL equation of state for 
explosives was used, the cylinder was modeled as an elasto-plastic material, and the effects of gas leakage 
were taken into account. A comparison was made with the experimental data, for cylinders with different 
types of steel and with geometric ratios L/D = 2. The explosive charges were TNT, composition B and Octol. 
The agreement between the experiments and the calculation in the program was great [15]. 
When estimating fragment initial velocity for HE projectiles, the Gurney model and CAD techniques for 
modeling can be used (Fig. 4). I.e. 3D models of projectiles can be divided into quasi-cylindrical segments, 
with characteristic front, center and rear part. For each segment of the projectile body, the initial velocity of 
the fragments is determined separately, using expression (1), and thus the profile of the initial velocities of the 
fragments along the projectile body can be obtained. It is usually assumed there is no expansion of the 
projectile body before fragmentation and that the initial velocity vector of the fragments is normal to the 
projectile body on each segment. Crull and Swisdak [17] follow a similar procedure in their research. 

 
Figure 4. A 3D model of a HE projectile, divided into segments [16] 

Numerical simulations are used often during the past 20 years, especially with the introduction of cheap and 
powerful processor clusters. These simulations are based on the conservation equations, with the use of 
equations of state, material strength, and fracture models. Figure 5 shows the results of a numerical simulation 
of the detonation of a 155mm HE M107 projectile, performed by Prytz et al. [18] in the AUTODYN, where 
the analysis of the initial velocities of the fragments was performed and compared with the values obtained in 
the Split-X program. The results show a relatively good agreement of the values of the initial velocities of the 
fragments, and potential for using the simulations for calculation of fragment initial velocities. 

 
Figure 5. Initial velocities fragments for 155mm M107 (Split-X and AUTODYN) [18]  



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3. Application of Gurney method to HE projectiles 

In our research (Zecevic et al. [19]), we analyzed the influence of explosive type and projectile design on the 
values of the initial velocities of fragments for different types of HE projectiles (artillery and mortar 
projectiles, as well as rocket projectiles warheads).  
Figures 6 and 7 show the influence of explosives type (TNT and composition B) on the initial velocity of 
fragments for several types of HE projectiles (152 mm M84 artillery projectile, 128 mm M87 missile warhead 
and 120 mm mortar projectiles, models W1 and W2). The abscissa on the diagrams represents the relative 
distance of individual projectile segments from the top of the projectile.  
The diagrams show that the use of explosives with better detonation characteristics (higher density, higher 
detonation velocity and higher detonation pressure of composition B compared to TNT charge) can 
significantly increase the initial velocity of the fragments. 

 
Figure 6. Variation of initial velocity of fragments, depending on the type of explosives (TNT and 

Composition B) for mortar HE projectiles 120mm (warheads W1 and W2) [19] 

 
Figure 7. Variation of initial velocity of fragments, depending on the type of explosives (TNT and 

Composition B charge) for rocket projectile warhead - model 128mm HE M87 and artillery projectile - model 
152mm HE M84 [19] 

Figure 8 shows the influence of projectile design on the values of the initial velocity of fragments for several 
types of HE projectiles (artillery projectiles 105mm, 122mm, 152mm and 155mm, rocket projectile warheads 
128mm and mortar projectiles 120mm).  
The diagrams in Fig. 8 show that by changing only the design of the projectile (i.e. increasing the ratio of the 
mass of the explosive to the mass of the projectile body C/M, reducing the thickness of a body) can also 
significantly increase the initial velocity of the fragments on individual segments of the projectile. The 
maximum initial velocities of the fragments are achieved in the central parts of the HE projectile, where the 
largest mass ratios C/M are present. 
Fig. 8 also shows the simultaneous influence of both the type of explosive and the design of the projectile on 
the initial velocity of the fragments for 122mm caliber artillery projectile. Thus, the 122mm HE M76 
projectile, which has a higher C/M ratio and explosive (composition B) with better detonation characteristics 
than in the case of 122mm OF-462 (TNT) projectile, has a higher initial velocity of fragments on all projectile 
segments.  
The higher initial velocity generally corresponds to the higher kinetic energies of the fragments leading to a 
larger lethal zone of the projectile.  



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If we look closely at initial velocities values of fragments for projectile 122mm HE OF-462 (maximum value 
of around 1200 m/s) in Fig. 8, the front part of the 122mm OF-462 projectile has somewhat higher initial 
velocities. On the other hand, the rear spray of fragments has a lower initial velocity, because of the smaller 
C/M ratio. 
 

 

 

 
Figure 8. Influence of projectile design on the values of the initial velocity of fragments for several types of 

HE projectiles [19] 

Catovic [20] provided a different way of displaying the initial velocities of fragments (comparing i.e. to Fig. 
8), presented via a polar diagram (Grapher software can be used) in which one axis represents the initial 
velocities of the fragment (units m/s) and the angular axis of the diagram represents the angles measured from 
the projectile axis to the centers of the individual projectile segments. The point of origin (lower part of Fig. 9) 
is represented by the projectile center of mass.  
The directions in the lower part of Fig. 9 (on the projectile) should not be confused with the initial velocity 
vectors; they serve here only to estimate the angles measured from the projectile axes to the centers of the 
individual projectile segments (in order to construct the diagram in Fig. 10). Such diagrams can be useful in 
visualization of the parameters for HE projectile terminal ballistics. We used polar diagrams also in our model 
for determination of HE projectile lethal zones [21]. 
In our research [20] we used the Gurney model also as an integral part of a new method for determination of 
lethal radius for high-explosive artillery projectiles (Fig. 10). In this model, the values of the initial velocity of 
fragments are required in order to accurately calculate fragment trajectory through the atmosphere on its way 
to the target. This method showed promising results confirming the applicability of the Gurney model. 
 



 DSS Vol. 1, No. 1, December 2020, pp.16-25 

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Figure 9. Variation of initial fragment velocity for 122mm HE OF-462 projectile as a function of segment 

position, presented using the polar diagram [adapted from 20] 

 
Figure 10. Body segments of artillery projectiles, for which Gurney method is applied, used in our model for 

determination of the lethal radius [20] 



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4. Limitations of Gurney model 

The scope of application of Gurney equations is limited due to certain assumptions taken when deriving them. 
Henry (1967), Jones (1980) et al. and Kennedy (1970) made a list of these limitations, and they are cited by 
Walters and Zukas [2]: 

- Range of M/C ratio. Henry (1967) claims that Gurney's equations give good results for the range 0,1 
< M/C < 5, while Kennedy (1970) claims that the range of application of Gurney's equations is 0,2 < 
M/C < 10. 

- Acceleration phase. The Gurney method in its basic form is not able to analyze motion during 
acceleration. Therefore, Henry (1967) and Jones (1980) et al. took advantage of the assumption of a 
linear velocity profile in uniformly dense gas and combined it with the gas state equation to obtain 
acceleration when moving a metal plate, and came to two conclusions. The first conclusion is that 
detonation gases must be fully allowed to expand to atmospheric pressure, and the second is that the 
fragment will reach the calculated velocity only if no external force acts on it in the acceleration 
phase. 

- Assumption of the gas velocity profile. Assumptions of the linear velocity profile and constant 
density of the expanding gas are essential assumptions. Gurney's method in its basic form neglects the 
effect of maximum pressure. On the other hand, although these assumptions give certain errors, they 
greatly facilitate and shorten the derivation of Gurney formulas. 

- Resistance to deformation. The forces in the metal that oppose the deformations are not taken into 
account, except for inertia. These forces reduce the assumed initial velocity of the fragment. 

- Metal fragmentation. Metal disintegration is possible when the M/C ratio is < 2 for high-density 
explosives. The explosion can in some cases be prevented if a cavity of a few millimeters is made 
between the explosive and the metal. 

- Early cracks on the projectile body. Leakage of detonation gases through cracks in the body can 
significantly reduce the assumed initial velocity of the fragment, depending on the geometry of the 
crack and the amount of gas released. 

5. Conclusions 

The literature related to the initial velocity of fragments for HE ammunition is presented. The basic formulas 
for fragment initial velocity, that can be used for different configurations of munition, are presented. In the 
last part, the research we performed using the Gurney method for different types of projectiles is given. 
The Gurney model for fragment initial velocity, which can be used for various ammunition setups, is 
described. The research we conducted using the Gurney method for various types of projectiles is given. 
The results  show that the use of explosives with better detonation characteristics and geometry with higher 
C/M ratio one can increase the initial velocity of the fragments. Limitations of Gurney model are also 
presented. 
 

References  

[1] J. A. Zukas, W. P. Walters, Explosive effects and applications, Springer, New York, 1993. 

[2] W. P. Walters, J. A. Zukas, Fundametals of shaped charges, Wiley-Interscience Publication, 1989. 

[3] P. P. W. Cooper, Explosives Engineering, Wiley, 1996. 

[4]  Y.G. Gao, S.S. Feng, B. Zhang, T. Zhou, "Effect of the length-diameter ratio on the initial fragment 
velocity of cylindrical casing", IOP Conf. Series: Materials Science and Engineering, doi:10.1088/1757-
899X/629/1/012020, 2019. 

[5]  T. Zulkoski, "Development of optimum theoretical warhead design criteria", Naval Weapons Center, 
China Lake CA, 1976. 

[6]  Z. Guo, G.-y. Huang, C. Liu, S. Feng, "Velocity axial distribution of fragments from non-cylindrical 
symmetry explosive-filled casing", International Journal of Impact Engineering, vol 118, pp. 1-10, 2018. 

[7]  E. Hennequin, "Influence of the edge effects on the initial velocity of fragments from a warhead", NASA 
STI/Recon Technical Report, 1986. 



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[8]  G. Randers-Pehrson, "An improved equation for calculating fragment projection angle", Proceedings of 
the 2nd International Symposium on Ballistics, pp. 223-226, 1976. 

[9]  D. Felix, I. Colwill, E. Stipidis, "Real-time calculation of fragment velocity for cylindrical warheads", 
Defence Technology, vol 15, pp. 264-271, 2019. 

[10]  Y. J. Charron, "Estimation of velocity distribution of fragmenting warheads using a modified Gurney 
method", Thesis, Faculty of the School of Engineering of the Air Force Institute of Technology Air 
Training Command, 1979. 

[11]  G. Huang, W. Li, S. Feng, "Axial distribution of fragment velocities from cylindrical casing under 
explosive loading". International Journal of Impact Engineering, vol 76, pp. 20-27, 2015. 

[12]  R. M. Lloyd, Conventional Warhead Systems Physics and Engineering Design, Progress in 
Astronautics and Aeronautics, Vol. 179, 1998. 

[13]  Manual of NATO Safety Principles for the Storage of Military Ammunition and Explosives, 
NATO/PFP UNCLASSIFIED, AASTP-1 (Edition 1), PART II, May 2006. 

[14]  J. Carleone, Tactical Missile Warheads, American Institute of Aeronautics and Astronautics, 
Washington, 1993. 

[15]  R. R. Karpp, W. W. Predebon, "Calculations of fragment velocities from naturaly fragmenting 
munitions", BRL, Aberdeen Proving Ground, Maryland, july 1975. 

[16]  A. Catovic, "Prediction of terminal-ballistic parameters for the natural fragmentating high-explosive 
warheads using experimental data and numerical methods", Thesis, Faculty of Mechanical Engineering 
Sarajevo, 2019. 

[17]  M. Crull, M. M. Swisdak, "Methodologies for calculating primary fragment characteristics", 
Department of Defense Explosives Safety Board, Alexandria, VA, October 2005. 

[18]  A. Prytz, G. Odegarstuen, E. Sogstad, B. Marstein, E. Smedtad, "Fragmentation of 155mm artillery 
gerande simulations and experiment", 26th International Symposium on Ballistics, Miami, Florida, 
September, 2011. 

[19]  B. Zecevic, J. Terzic, A. Catovic, S. Serdarevic-Kadic, "Influencing Parameters on HE Projectiles with 
Natural Fragmentation", 9th Seminar New Trends in Research of Energetic Materials, University of 
Pardubice, Pardubice, April 19–21, 2006. 

[20]  A. Catovic, E. Kljuno, "A novel method for determination of lethal radius for high-explosive artillery 
projectiles", Defence Technology, https://doi.org/10.1016/j.dt.2020.06.015, 2020. 

[21]  B. Zecevic, A. Catovic, J. Terzic, "Comparison of lethal zone characteristics of several natural 
fragmenting warheads", Central European Journal of Energetic Materials, ISSN 1733-7178, Vol 5(2), 
pp. 67-81, 2008.