Abstract:
A light weight explosive mass for a spherical charge of high explosives used to simulate nuclear bursts at ground or above ground level or underwater wherein the sphericity of the loaded explosive mass must be maintained within tight tolerance, exemplary of which are particulate ammonium nitrate/fuel oil (ANFO) explosive and liquid nitromethane explosive, contained in a flexible fabric shell that becomes spheroidal by reason of being filled with the explosive, is disclosed.

Description:
FIELD OF THE INVENTION 
     This invention relates to weapons and, more specifically, to weapons testing, namely the simulation of very high energy, e.g. nuclear, explosions using conventional explosives. More specifically, the present invention is embodied in a light weight container for a spherical charge of high explosives used to simulate nuclear bursts at ground or above ground level or underwater. Sphericity of the loaded container is maintained within tight tolerance. 
     BACKGROUND OF THE INVENTION 
     Ground level and above ground testing makes use of simulated nuclear bursts. These simulations detonate hundreds of tons of inexpensive non-nuclear explosives, typically ANFO, a mixture of ammonium nitrate and fuel oil, in a spherical geometry. 
     The recently used container for the granular ANFO produced undesirable effects when the explosive was detonated. The container was basically a spherical fiberglass shell, which caused reflected shocks that interfered with the desired spherical pressure wave propagation and, in addition, broke into fragments which damaged experimental setups. 
     Previous configurations have also included stacked bags of ANFO which do not permit above ground configurations, have non-uniform explosive densities, and present difficulties in generating a spherical detonation wave front. 
     An alternative system consists of a cast explosive sphere which then is suspended in a &#34;cargo net&#34; arrangement. This system is limited in size because of transportation problems and manufacturing problems of molds, etc. 
     It is an object of the present invention to provide an improved container and simulator that will simulate nuclear explosions more accurately and eliminate the hazards and inconvenience of the current practices and the prior art. 
     SUMMARY OF THE INVENTION 
     A light weight container for a spherical charge of high explosives used to simulate nuclear bursts at ground or above ground level or underwater wherein the sphericity of the loaded container must be maintained within tight tolerance is disclosed and claimed. 
     According to the present invention, the inert fiberglass structural shell of the prior art is replaced by a bag-like container consisting of fibrous, cloth-like material. Ideally, this material is very thin, strong, it is flexible and can be folded like cloth but does not stretch (i.e. is not rubbery), and it can be made to contain liquids without leaking (possibly by a rubberized coating process). 
     A design of a spherically shaped container system can be achieved such that the entire surface of the container, such as a coated woven KEVLAR® bag, remains in biaxial tension when tile container is filled with an emulsion, gelled, liquid or granular high explosive. (KEVLAR® is a registered trademark of E. I. duPont for aromatic polyamide (aramid) fibers of great strength and products manufactured from such fibers.) Analysis shows that the key to keeping all of the container&#39;s surface in biaxial tension is to support it via an attached girdle, and that tile line of attachment of the girdle must be a line of latitude on the spherical surface lying between +9° and +30° latitude, measured relative to the horizontal equator. The use of a high-strength fabric made of a fiber such as KEVLAR permits fabrication of a container having a very low inert mass fraction of the spherical charge (approx. 0.2%). The initial geometry of such a container can be adjusted so as to deform to the desired sphere upon loading. The container can be filled through a top entry port. The low mass of this container minimizes interference with the desired spherical shock wave propagation. 
     The flexible explosive charge container of this invention can be fabricated in a commercial shop and shipped to any desired test site. The container can be of any size desired for the test since the principles are scaleable. Explosives for the test can arrive at the test site in conventional transportation packaging and poured or pumped into the container for the test. This approach reduces overall high explosive charge costs for such tests, improves safety and handling procedures, and accomplishes the desired concept of standard, highly uniform, accurately spherical HE simulation techniques. Further, turnaround time between subsequent shots can be much shorter than with other methods. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 depicts the container and weapon simulation device of this invention. 
     FIG. 2 is a geometric depiction of the device of this invention for reference in explaining the design parameters and experimental data contained in the specification. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     It will be recognized that the present invention can easily be manufactured using ordinary methods and known materials once the concept and description of the invention are available. Accordingly, much of the specification and the drawings are devoted to providing a clear understanding of the invention. Accordingly, it is emphasized that the drawings and descriptions are exemplary and explanatory in nature and are not intended to provide specifics of the many well-known methods of manufacture that are available to those working in this art. 
     The overall concept of the invention can be best understood by reference to FIG. 1 which depicts, in somewhat simplified form, a weapons simulator constructed in accordance with this invention. The weapons simulator system 10 comprises a weapons simulator container 20 in the configuration approximating a sphere constructed of a fabric or film, the preferred material comprising coated KEVLAR® fabric sewn, adhesively bonded or otherwise formed in accordance with the criteria set forth hereinafter. The container 20 comprises defines a vertical axis 21 and has an opening 22 approximately in the top center to permit the container to be filled with explosive, typically a liquid such as nitromethane or an emulsion such as QM100, an emulsion consisting of ammonium nitrate, fuel oil, and water, although any explosive that can be made to flow may be used. The explosive may be in the form of a liquid, in which case the container comprises liquid proofed fabric or liquid proof film, or it may be in the form of a gel, an emulsion or particles, all of which assume a configuration that is a function of the shape of the container, the elasticity of the container and the effect of gravity, i.e. they act approximately as a liquid. 
     The container, when filled approximates the configuration of a sphere having a equatorial circumference 24. Additional latitudinal circumferential lines 26 and 28 are also shown in FIG. 1 to suggest the latitudinal limits at which the securement ring 30 must be defined. In use, the latitudinal circumferences, if marked on the sphere 20, would be circles defining horizontal latitudinal planes parallel to the ground and perpendicular to the vertical axis 21 of the spherical container 20. In the example depicted in FIG. 1, the securement ring 30 coincides with the latitudinal circumference 26 but, as discussed in detail below, such coincidence is not necessary. It is also pointed out here that the securement ring could lie below the plane defined by the diametrical circumference 24, e.g. at or above the latitudinal circumference 28. The securement ring may be a rigid ring secured to the fabric container or simply a latitudinal area, which may or may not be reinforced, to which support ropes or wires 40, 42, 44, 46, and 46, and additional supporting lines not shown may be secured. The supporting lines, e.g. 40, 42, 44, 46, and 48, are connected to the sphere 20 around the securement ring at spaced apart locations and extend tangentially from the sphere at the connection points, the tangential relationship being defined by any suitable means. In the example depicted in FIG. 1, a rigid support ring 50 supported by any desired number of lines, e.g. cables, wires or ropes, one of which is indicated at 52 connected to a hook or ring 60 suspended by a cable, for example, by a boom (not shown) or any other desired structure, e.g. a tower, lighter-than-air craft, a cable strung between tall trees, etc. to provide means for supporting the sphere a desired distance above the ground surface. The diameter of the hanger ring 50 is so related to the diameter of the sphere as to align the supporting lines 40-48 along tangents to the lateral circumference of the sphere to which the lines are attached. 
     The securement ring on the container may be positioned from approximately 30 degrees below the diameter to approximately 9 degrees above the diameter, as described more fully below. The lateral circumference 26 as depicted in FIG. 1 is not to scale and is spaced a greater distance from the diametric circumference 24 for clarity of illustration. 
     The collapsed and folded container is secured by ropes, wires, chains or cables to the support. Thereafter, the explosive is poured into the top of the container to fill up the container. During filling the container gradually assumes a substantially spherical configuration as described and depicted in FIG. 1. 
     The invention, in its embodiment as a weapon simulator, comprises a fabric (woven or non-woven), film or other flexible material in the general configuration of a sphere supported in the air, or other fluid (e.g. under water), substantially filled with an explosive each particle or molecule of which is acted separately upon by gravity, i.e., behaves as a liquid or approximately as a liquid, to expand the container to form an approximately spheroidal explosive mass supported at a multiplicity of points on the surface of the container, said support points defining generally a circle on said surface not more than about 30 degrees below nor more than about nine degrees above the diameter of the sphere. 
     The container comprises walls of flexible material and means for attaching the container to means for supporting the container in a fluid. The container is supported above the ground or floor in air to simulate an air burst and in water to simulate a water burst. The container walls are so constructed and configured as to define a spheroidal body when the container is substantially filled with a material that behaves approximately as a liquid. The attaching means comprises means for attaching the container at a multiplicity of points along a latitudinal line generally parallel to a diametrical circumference that is horizontal when the container is supported in use and not more than about 30 degrees below nor about 9 degrees above said diametrical circumference. The flexible material preferably comprises polyamide fiber fabric, woven or non-woven, such as is made from KEVLAR® fibers produced by E. I. dupont de Nemours, Inc. If desired, the flexible material may further comprise an organic polymeric film associated with the fabric, i.e. impregnated into, coated onto or bonded to the fabric. Rubber, natural or synthetic, is the preferred material because of its low cost, ready availability, ease of use and because it seals the fabric against leakage of typical liquid explosives such as nitromethane. The attaching means preferably comprises fabric forming a reinforcing ring around the circumference of the container as shown in FIG. 1. 
     In another embodiment, ready for use, the invention is an explosive mass comprising a normally non-spheroidal container, means supporting the container in fluid, the container comprising flexible walls and explosive substantially filling the container. The weight of the explosive is acted upon by gravity forcing the container into a substantially spherical configuration having a horizontal diametric circumference. The means supporting the container may comprise a multiplicity of supports secured to the container walls along latitudinal plane substantially parallel to the horizontal diametric circumference and not more than about 30 degrees below nor 9 degrees above said diametric circumference. 
     As described before, means supporting the container may comprise a multiplicity of elongate tensioned flexible strands, ropes, cables, wires, etc., secured to the container at their respective proximal ends and extending upwardly from the container. The distal ends of the strands may be secured to support structure of any kind. The preferred wall materials are also as described above. 
     FIG. 1 represents, of course, the result of a series of analyses, designs and experiments. A preliminary analysis of the feasibility of fabricating and utilizing a fabric or film structural shell was undertaken. While doubts remained even after the analysis, a tentative conclusion was reached that it would probably be technically possible to make such a structure and to use it for its intended function. 
     The design of the container draws on the technology of structural design and construction of large, low-pressure tires. The maximum stress in the wall would be comparable to that at its bottom, where it is just that required to contain a pressure equal to the pressure head generated by the weight of a column of explosive equal to the diameter of the spherical container. Since the density of the explosive is comparable to water and the diameter of a typical sphere is in the range 20 to 35 ft, the equivalent gas pressure to be contained by the container wall is between 10 and 16 psi. For a 20 ft.-diameter sphere, this results in a tensile stress in the wall of about 500 lb/in, specifying the required strength of the piles of fabric. 
     Contrary to a gas-filled tire, the stress in the wall in this case is a function of height, since the pressure is generated by the action of gravity on a dense medium. This will also tend to distort the shape of the shell from its initial shape; if the initial shape is spherical, elastic distortion of the fabric will result in a non-spherical shape. If a final spherical shape is desired, the initial fabric shape must be selected to be the one which will elastically distort into a sphere under the anticipated load. 
     The analytical investigation proceeded along the following lines, reference being made at a number of points to FIG. 2, wherein lines 41 and 43 are depicted to indicate multiple tangential support lines extending upwardly and, in this case, divergingly from a lateral circumference 27. To see what fabric stretch does to the shape of the sphere, assume that (1) the main explosive inside the shell behaves as a liquid, and (2) stretching of the fabric causes the sphere to assume an oblate spheroidal shape. In reality, neither assumption is quite correct. If the main explosive is granular and likely to behave more like sand, the distorted shape of the sphere will not be a perfect oblate spheroid. However, in order to gain some insight into the magnitudes involved, these assumptions will be used. 
     For a spherical shell, the volume, V s  and surface area, A s  are given in terms of the radius, r, as follows 
     
         V.sub.s =(4/3)πr.sup.3 ; A.sub.s =4πr.sup.2          (1) 
    
     For an oblate spheroid with major and minor semi-axes of a and b respectively and an eccentricity ε, the volume V and surface area A are given by 
     
         V=(4/3)πd.sup.2 b; A=2πa.sup.2 +(πb.sup.2 /ε)1n[(1+ε)/(1-ε)]                (2) 
    
     Assuming the volume remains the same as the sphere distends into a spheroid then 
     
         V=V.sub.s or a.sup.2 b=r.sup.3.                            (3) 
    
     Also by definition of eccentricity, 
     
         ε=(1/a)√(2.sup.2 -b.sup.2),                 (4) 
    
     so that 
     
         b=a √(1-ε.sup.2).                           (5) 
    
     From (3) and (5) 
     
         a/r=1/(1-ε.sup.2).sup.1/6 ; b/r=(1-ε.sup.2).sup.1/3,(6) 
    
     and from (1) and (2) 
     
         A/A.sub.s =(1/2)(a/r).sup.2 +(1/4)(b/r).sup.2 (1/ε)1n[(1+ε)/(1-ε)],             (7) 
    
     or substituting (6) in (7) 
     
         A/A.sub.s =(1/2)(1-ε.sup.2).sup.-1/3 +(1-ε.sup.2).sup.2/3 /4ε]1n[(1+ε)/(1-ε)].              (8) 
    
     Table 1 shows how the sphere is distorted with different area changes. 
     
                       TABLE 1______________________________________Area Change of Oblate Spheroid for Constant Volumeε  b/a(A/As).sup.-1______________________________________0.1        0.9950     4.493 × 10.sup.-60.2        0.9798     7.430 × 10.sup.-50.3        0.9539     3.983 × 10.sup.-40.4        0.9165     0.0013700.5        0.8660     0.0037630.6        0.8000     0.0091720.7        0.7141     0.021260.8        0.6000     0.050340.9        0.4359     0.14005______________________________________ 
    
     Thus, if an eccentricity of 0.3 is allowable, such that b/a=0.9539, then the allowed surface stretch is 0.04%. 
     For a sphere suspended at its equator the shape assumed is approximately a prolate spheroid (cigar shaped). In that case Eq (2) becomes 
     
         V=(4/3)πab.sup.2 ; A=2πb.sup.2 +(2πab/ε) sin.sup.- ε                                                 (2&#39;) 
    
     Eq (3) then becomes 
     
         ab.sup.2 =r.sup.3,                                         (3&#39;) 
    
     and Eq (8) becomes 
     
         A/a.sub.s =(1/2)(1-ε.sup.2).sup.1/3 +(1-ε.sup.2).sup.-1/6 (sin.sup.-1 ε)/(2ε).                      (8&#39;) 
    
     The corresponding table is Table 2 
     
                       TABLE 2______________________________________Area Change of Prolate Spheroid for Constant Volumeε  b/a(A/As).sup.-1______________________________________0.1        0.9950     4.486 × 10.sup.-60.2        0.9798     7.382 × 10.sup.-50.3        0.9539     3.923 × 10.sup.-40.4        0.9165     0.0013320.5        0.8660     0.0035690.6        0.8000     0.0085460.7        0.7141     0.019110.8        0.6000     0.042830.9        0.4359     0.10792______________________________________ 
    
     Comparing oblate and prolate spheroids, the area increases are quite similar for eccentricities E&lt;0.5. 
     Next consider a sphere suspended as shown in FIG. 2. Let the stresses along the longitudes and latitudes by N.sub.φ and N.sub.θ respectively, per unit length. These stresses are those required to generate the forces on the fluid of density, ρ, to support it in the earth&#39;s gravitational field. Let the shell portion above the line of suspension, (AB), be the Upper Shell and that below it, the Lower Shell. Then the forces are given as: 
     Upper Shell (o&lt;φ&lt;φ.sub.°): 
     
         N.sub.φ =(ρr.sup.2 /6)[(1-2 cos.sup.2 φ/(1+cosφ)](9) 
    
     
         N.sub.θ =(ρr.sup.2 /6)[5-6 cos φ+2 cos.sup.2 φ/(1+cosφ)].                                      (10) 
    
     Lower Shell (φ.sub.° &lt;φ&lt;180°): 
     
         N.sub.φ =(ρr.sup.2 /6)[5+2 cos.sup.2 θ/(1-cos φ)](11) 
    
     
         N.sub.74 =(ρr.sup.2 /6)[1-6 cos φ-2 cos.sup.2 φ/(1-cos φ)].(12) 
    
     Assume the weight of the explosive V=240,000 lb and its density is ρ=62.4 lb/ft 3  (same as water, then r≈116.6 in. and ρr 2  /6=81.9 lb/in. Using these numbers, Table 3 shows the N.sub.φ, N.sub.θ values for the Upper and Lower Shells for different values of φ. 
     
                       TABLE 3______________________________________Stress Distribution in Upper and Lower ShellsUPPER SHELL               LOWER SHELLφN.sub.φ           N.sub.θ   N.sub.φ                                 N.sub.θdeg  lb/in      lb/in           lb/in lb/in______________________________________ 0   0          0                     negative 10  1.86       5.60                  negative 20  7.33       22.30                 negative 30  16.06      49.76                 negative 40  27.47      87.48                 negative 50  40.70      134.81                negative 60  54.59      191.07                negative 70  67.61      255.66                negative 80  77.68      328.32  (≈81°)                           414.21                                 0 90  81.89      409.43          409.43                                 81.89100  75.91      500.72          413.64                                 162.99110  52.77      606.58          423.70                                 235.65120  0          736.97          436.72                                 300.25130  negative                   450.60                                 356.51140  negative                   463.85                                 403.84150  negative                   475.25                                 441.55160  negative                   483.98                                 469.02170  negative                   489.45                                 485.71180  negative                   491.32                                 491.32______________________________________ 
    
     Since the fabric cannot sustain compression without buckling, the value of φ.sub.°, i.e. the suspension latitude, must lie between φ.sub.° ≈81° and φ.sub.° ≈120°. Outside these limits either N.sub.φ or N.sub.θ become negative, signifying that the force is compressive in one dimension. 
     We now estimate the amount of fabric required, assuming it to be made of Kevlar with the following properties: 
     
                       TABLE 4______________________________________Specific tensile strength            =             9.5 × 10.sup.6 in.Density          ρk =      0.053 lb/in.sup.3Elastic Modulus  E =           27 × 10.sup.6 lb/in.sup.2Strain to failure            e.sub.f =     1.3%______________________________________ 
    
     Assuming an initial geometry with an eccentricity ε=0.5, designed to deform into a sphere, then the allowed area change is ˜0.4%, representing a unidirectional strain of e=0.2%. 
     This means that the working stress must be, assuming that Poisson&#39;s ratio is 0.5 
     
         σ.sub.w =2Ee=2×27×10.sup.6 ×0.002=108,000 psi. 
    
     Assuming maximum values for N.sub.φ and N.sub.θ of 500 lb/in, then the required thickness of a unidirectional Kevlar film is t 1  =(500)/(108,000) in.=0.00463 in. Applying a small safety factor and doubling the thickness to allow for two directional strength, then t≈0.01 in. 
     Assuming that this thickness is uniformly applied along the sphere&#39;s surface, then the weight of the Kevlar is 
     
         W.sub.k =4πr.sup.2 tρ.sub.k =90 lb. 
    
     Application of an additional safety factor and use of a woven fabric geometry rather than film will probably result in a fabric thickness of the order of 0.1 in. and a fabric weight of 400-500 lb. We envision the joining of pieces of fabric cut to appropriate patterns to give the required initial shape and joined together by sewing in the manner of fabricating a parachute or the skin fabric of a blimp or by adhesive bonding as is common in joining the fabric pieces of flexible liquid containers. The seams will, of course, have to be strong enough to sustain the 500 lb/in maximum stress value, but this is feasible. The total equivalent mass thickness of 0.020 to 0.040 n. of Kevlar film is considered to be a small enough quantity that it is likely to be vaporized or consumed and that it also will have a negligible effect on shock wave reflection. 
     In an initial test of the design concept, the container intended to be filled with nitromethane was tested for its sphericity. The container design specified that sphericity should be maintained to within plus or minus 2% of nominal. The test was designed to ascertain the radii of 25-35 points on the surface of the container to a &#34;best fit&#34; container center. 
     In a preliminary evaluation, a container as described was filled with water and elevated to waist height. A transit was set up about 50 feet from the container and a `witness board` was erected about 3 feet behind the container (from the transit) and surveyed to be normal to the transit. Readings were then taken by the transit to points on the circumference and the witness board was marked accordingly. After all readings were made, the container was rotated through 90 degrees and an additional series of readings was made and marked on a new witness board. It should be noted that although parallax was present and is somewhat significant at the transit standoff distance, it is irrelevant since it is relative differences in radii that are to be measured; also, there was some tendency for the container to swing in the breeze, accounting for some error. Readings were impossible at locations at which the suspension system interfered with the view and were meaningless at the filler port. A mark was placed on the witness board marking the vertical and top; this reference was surveyed. Additionally, in one viewing direction, the level of the filler port was marked. 
     The witness boards were recovered and each marked point was numbered. The conformance to the 2% requirement was made in the following manner: A circle was drawn such that the circle would be as close to as many points as possible. The distance of each point from the center of the circle was then calculated by measuring the x and y coordinates of each point relative to the circle center. X and y coordinates were measured rather than just the radius (z) so as to provide information on the location of each point. X and y coordinates were measured to the nearest 1/32 of an inch. The fractional part of each dimension was converted to decimal and is provided in the following table, along with the calculated radius, z. 
     
                       TABLE 5______________________________________ORIENTATION #1POINT                              RADIUS,#      X-COORDINATE  Y-COORDINATE  IN______________________________________1      5.938         17.875        18.842      7.000         17.188        18.563      7.875         16.688        18.454      8.844         16.125        18.395      11.469        14.375        18.396      14.281        11.906        18.597      15.313        10.281        18.448      18.000        3.438         18.339      18.250        -1.313        18.3010     17.500        -4.563        18.0911     16.313        -8.250        18.2812     14.063        -11.500       18.1713     11.188        -14.438       18.2714     6.750         -17.063       18.3515     2.969         -18.063       18.3116     -0.219        -18.250       18.2517     -4.188        -17.938       18.4218     -7.938        -16.438       18.2519     -11.094       -14.313       18.1120     -14.313       -11.000       18.1121     -16.625       -7.531        18.0522     -17.750       -3.813        18.1523     -18.219       0.594         18.2324     -17.813       4.500         18.3725     -15.250       9.750         18.1026     -12.500       13.375        18.3127     -9.188        15.875        18.3428     -6.875        16.750        18.1129     -6.250        17.000        18.1130     -5.563        17.250        18.1231     -5.250        17.219        18.00______________________________________ Mean radius = 18.26 in. Max allowable (+2%) = 18.63 in. Min allowable (-2%)  17.89 in. 
    
     
                       TABLE 6______________________________________ORIENTATION #2POINT                              RADIUS,#      X-COORDINATE  Y-COORDINATE  IN______________________________________1      5.813         17.625        18.562      10.031        15.063        18.103      13.188        12.563        18.214      15.813        9.125         18.265      16.188        8.250         18.176      18.000        3.000         18.257      18.000        -0.750        18.028      17.438        -4.750        18.079      15.938        -8.500        18.0610     13.563        -11.938       18.0711     10.625        -14.594       18.0512     5.813         -17.500       18.4413     0.813         -18.500       18.5214     3.438         -18.250       18.5715     7.875         -16.563       18.3416     -11.938       -13.625       18.1217     -15.313       -9.625        18.0918     -17.219       -5.438        18.0619     -18.125       -1.500        18.1920     -18.000       3.375         18.3121     -15.750       9.063         18.1722     -13.500       12.250        18.2323     -10.563       14.875        18.2424     -8.375        16.250        18.2825     -7.125        16.750        18.2026     -6.563        16.938        18.1727     -6.000        17.125        18.15______________________________________ Mean radius = 18.22 in. Max allowable (+2%) = 18.58 in. Min allowable (-2%)  17.86 in. 
    
     Sphericity Analysis 
     Given points x i , y i , i=1,2, . . . ,N that lie approximately on a circle. Determine by means of least squares the center of the circle, (a,b), and its radius, r. 
     Solution: The equation of a circle is 
     
         (x-a).sup.2 +(y-b).sup.2 =r.sup.2                          (1) 
    
     Let the deviation be defined as 
     
         d.sub.i =r.sup.2 -(x.sub.i -a).sup.2 -(y.sub.i -b).sup.2   (2) 
    
     We now set ##EQU1## We now expand Eqs. (4), (5) and (6) ##EQU2## Eliminating the r 2  -a 2  -b 2  factor between Eqs (7) and (8) and between Eqs (7) and (9) results in two equations with a and b as unknowns. ##EQU3## Note that B 1  =A 2  then Equations (10) and (11) become 
     
         aA.sub.1 +bB.sub.1 =C.sub.2                                (18) 
    
     
         aB.sub.1 +bB.sub.2 =C.sub.2                                (19) 
    
     Whose solution is ##EQU4## r is then obtained from Eq (7), namely ##EQU5## We used the deviation of the square of the radius, i.e. d i  in Eq (2), to solve for a, b and r. To obtain the deviations of the radius, we define ##EQU6## and use the following equation to obtain the standard deviation of the radius ##EQU7## 
     
                       TABLE 6______________________________________Results:    Orientation #1                   Orientation #2______________________________________a =         0.099       -0.026b =         0.057       -0.018r =         18.282      18.2182%xr =      0.366       0.364.sup.D std =       0.157       0.147______________________________________ The above results were computed on a spreadsheet as shown in Table 7 and Table 8. 
    
     
         __________________________________________________________________________SPHERICITY ANALYSIS__________________________________________________________________________POINTX    Y    X 2  X 3  Y 2  Y 3  X*Y   X 2*Y                                         X*Y 2 D 2  D__________________________________________________________________________ORIENTATION #1 1   5.938     17.875          35.260               209.37                    319.516                         5711.34                              106.142                                    630.27                                         1897.28                                               0.2188                                                    -0.4677 2   7.000     17.188          49.000               343.00                    295.427                         5077.81                              120.316                                    842.21                                         2067.99                                               0.0346                                                    -0.1861 3   7.875     16.688          62.016               488.37                    278.489                         4647.43                              131.418                                    1034.92                                         2193.10                                               0.0058                                                    -0.0765 4   8.844     16.125          78.216               691.75                    260.016                         4192.75                              142.610                                    1261.24                                         2299.58                                               0.0001                                                    -0.0110 5   11.469     14.375          131.538               1508.61                    206.641                         2970.46                              164.867                                    1890.86                                         2369.96                                               0.0000                                                    -0.0008 6   14.281     11.906          203.947               2912.57                    141.753                         1687.71                              170.030                                    2428.19                                         2024.37                                               0.0391                                                    -0.1979 7   15.313     10.281          234.488               3590.71                    105.699                         1086.69                              157.433                                    2410.77                                         1618.57                                               0.0023                                                    -0.0476 8   18.000     3.438          324.000               5832.00                    11.820                         40.64                              61.884                                    1113.91                                         212.76                                               0.0042                                                    0.0651 9   18.250     -1.313          333.063               6078.39                    1.724                         -2.26                              -23.962                                    -437.31                                         31.46 0.0064                                                    0.079910   17.500     -4.563          306.250               5359.38                    20.821                         -95.01                              -79.853                                    -1397.42                                         364.37                                               0.0776                                                    0.278611   16.313     -8.250          266.114               4341.12                    68.063                         -561.52                              134.582                                    -2195.44                                         1110.30                                               0.0041                                                    0.064312   14.063     -11.500          197.768               2781.21                    132.250                         -1520.88                              -161.725                                    -2274.33                                         1859.83                                               0.0244                                                    0.156213   11.188     -14.438          125.171               1400.42                    208.456                         -3009.69                              -161.532                                    -1807.22                                         2332.20                                               0.0010                                                    0.032114   6.750     -17.063          45.563               307.55                    291.146                         -4967.82                              -115.175                                    -777.43                                         1965.24                                               0.0071                                                    -0.084215   2.969     -18.063          8.815               26.17                    326.272                         -5893.45                              -53.629                                    -159.22                                         968.70                                               0.0040                                                    -0.063616   -0.219     -18.250          0.048               -0.01                    333.063                         -6078.39                              3.997 -0.88                                         -72.94                                               0.0008                                                    -0.027517   -4.188     -17.938          17.539               -73.45                    321.772                         -5771.94                              75.124                                    -314.62                                         -1347.58                                               0.0468                                                    -0.216418   -7.938     -16.4387          63.012               -500.19                    270.208                         -4441.68                              130.485                                    -1035.79                                         -2144.91                                               0.0044                                                    -0.066619   -11.094     -14.313          123.077               -1365.41                    204.862                         -2932.19                              158.788                                    -1761.60                                         -2272.74                                               0.0045                                                    0.067420   -14.313     -11.000          204.862               -2932.19                    121.000                         -1331.00                              157.443                                    -2253.48                                         -1731.87                                               0.0138                                                    0.117321   -16.625     -7.531          276.391               -4594.99                    56.716                         -427.13                              125.203                                    -2081.50                                         -942.90                                               0.0068                                                    -0.082722   -17.750     -3.813          315.063               - 5592.36                    14.539                         -55.44                              67.681                                    -1201.33                                         -258.07                                               0.0003                                                    0.018523   -18.219     0.594          331.932               -6047.47                    0.353                         0.21 -10.822                                    197.17                                         -6.43 0.0019                                                    -0.043624   -17.813     4.500          317.303               -5652.12                    20.250                         91.13                              -80.159                                    1427.86                                         -360.71                                               0.0297                                                    -0.172525   -15.250     9.750          232.563               -3546.58                    95.063                         926.86                              -148.688                                    2267.48                                         -1449.70                                               0.0166                                                    0.129026   -12.50     13.375          156.250               -1953.13                    178.891                         2392.66                              -167.188                                    2089.84                                         -2236.13                                               0.0026                                                    -0.050727   -9.188     15.875          84.419               -775.64                    252.016                         4000.75                              -145.860                                    1340.16                                         -2315.52                                               0.0036                                                    -0.060328   -6.875     16.750          47.266               -324.95                    280.563                         4699.42                              -115.156                                    791.70                                         -1928.87                                               0.0366                                                    0.191329   -6.250     17.000          39.063               -244.14                    289.000                         4913.00                              -106.250                                    664.06                                         -1806.25                                               0.0357                                                    0.189030   -5.563     17.250          30.947               -172.16                    297.563                         5132.95                              -95.962                                    533.84                                         -1655.34                                               0.0328                                                    0.181231   -5.250     17.219          27.563               -144.70                    296.494                         5105.33                              -90.400                                    474.60                                         -1556.59                                               0.0938                                                    0.3063SUM =6.718     55.716          4668.503               1957.11                    5700.441                         15588.75                              82.479                                    3701.50                                         1229.16                                               0.7606                                                    0.0208ORIENTATION #2 1   5.813     17.265          33.791               196.43                    310.641                         5475.04                              102.454                                    595.57                                         1805.75                                               0.1339                                                    -0.3659 2   10.031     15.063          100.621               1009.33                    226.894                         3417.70                              151.097                                    1515.65                                         2275.97                                               0.0083                                                    0.0913 3   13.188     12.563          173.923               2293.70                    157.829                         1982.81                              165.681                                    2185.00                                         2081.45                                               0.0007                                                    -0.0273 4   15.813     3.125          250.051               3954.06                    83.266                         759.80                              144.294                                    2281.72                                         1316.68                                               0.0050                                                    -0.0706 5   16.188     8.250          262.051               4242.09                    68.063                         561.52                              133.551                                    2161.92                                         1101.80                                               0.0003                                                    0.0174 6   18.000     3.000          324.000               5832.00                    9.000                         27.000                              54.000                                    972.00                                         162.00                                               0.0035                                                    -0.0593 7   18.000     -0.750          324.000               5832.00                    0.563                         -0.42                              -13.500                                    -243.00                                         10.13 0.0312                                                    0.1767 8   17.438     -4.750          304.084               5302.61                    22.563                         -107.17                              -82.831                                    -1444.40                                         393.44                                               0.0153                                                    0.1237 9   15.938     -8.500          254.020               4048.57                    72.250                         -614.13                              -135.473                                    -2159.17                                         1151.52                                               0.0196                                                    0.139910   13.563     -11.938          183.955               2494.98                    142.516                         -1701.35                              -161.915                                    -2196.05                                         1932.94                                               0.0199                                                    0.141211   10.265     -14.594          112.891               1199.46                    212.985                         -3108.30                              -155.061                                    -1647.53                                         2262.96                                               0.0271                                                    0.164512   5.813     -17.500          33.791               196.43                    306.250                         -5359.38                              -101.728                                    -591.34                                         1780.23                                               0.0458                                                    -0.214013   0.813     -18.500          0.661               0.54 342.250                         -6331.63                              -15.041                                    -12.23                                         278.25                                               0.0804                                                    -0.283614   -3.438     -18.200          11.820               -40.64                    331.240                         - 6028.57                              62.572                                    -215.12                                         -1138.80                                               0.0794                                                    -0.281815   -7.875     -16.563          62.016               -488.37                    274.333                         -4543.78                              130.434                                    -1027.16                                         -2160.37                                               0.0090                                                    0.094716   -11.938     -13.625          142.516               -1701.35                    185.641                         -2529.35                              162.655                                    -1941.78                                         -2216.18                                               0.0178                                                    0.133417   -15.313     -9.625          234.488               -3590.71                    92.641                         -891.67                              147.388                                    -2256.95                                         -1418.61                                               0.0265                                                    0.162918   -17.219     -5.438          296.494               -5105.33                    29.572                         -160.81                              93.637                                    -1612.33                                         -509.20                                               0.0365                                                    0.191119   -18.125     -1.500          328.516               -5954.35                    2.250                         -3.38                              27.188                                    -492.77                                         -40.78                                               0.0034                                                    0.058720   -18.000     3.375          324.000               -5832.00                    11.391                         38.44                              -60.750                                    1093.50                                         -205.03                                               0.0053                                                    -0.073021   -13.500     12.250          182.250               -2460.38                    150.063                         1838.27                              -165.375                                    2232.56                                         -2025.84                                               0.0000                                                    -0.003823   -10.563     14.875          111.577               -1178.59                    221.0266                         3291.33                              -157.125                                    1659.71                                         -2337.23                                               0.0006                                                    -0.025124   -8.375     16.250          70.141               -587.43                    264.063                         4291.02                              -136.094                                    1139.79                                         -3211.52                                               0.0045                                                    0.066825   -7.125     16.750          50.766               -361.71                    280.563                         4699.42                              -119.344                                    850.32                                         -1999.01                                               0.0001                                                    0.009726   -6.563     46.938          43.073               -282.69                    286.896                         4859.44                              -111.164                                    729.57                                         -1882.90                                               0.0021                                                    0.046127   - 6.000     17.125          36.000               -216.00                    293.266                         5022.17                              -105.450                                    616.50                                         -1759.59                                               0.0041                                                    0.0644SUM =1.439     30.769          4499.556               4895.67                    4460.387                         5628.44                              -285.942                                    442.16                                         -465.61                                               0.5842                                                    0.0160__________________________________________________________________________ A1 = -289357 a = 0.099 B1 = -4365.08 b = 0.057 C1 = -28929.9 r = 18.282 B2 = -347219 2%*r = 0.366 C2 = -20231.7 D std = 0.157 A1 = -242972 a = -0.026 B1 = 15529.43 b = -0.018 C1 = 6141.715 r = 18.218 B2 = -238967 2%*r = 0.364 C2 = 3782.134 D std = 0.147 
    
     The invention is embodied in a container that, when empty, is not in a spheroidal configuration, i.e. in a non-spheroidal configuration, and means for securing support structure to the container generally circumferentially around the container, the container being so constructed and configured as to define a spheroidal body when filled with material that behaves generally as a liquid, the means for securing support structure defining a ring not more than approximately 30 degrees below nor more than about 9 degrees above the diameter of said spheroidal body. Geometric terms are used to describe and define the configuration of the container when full with full recognition that while the terms are geometrically descriptive as applied such terms are not rigorous geometric definitions in the pure mathematical sense. Thus, the terms are used in a qualified manner. &#34;Spheroidal&#34; is used in the normal sense to mean shaped approximately as a sphere but not necessarily forming a perfect sphere. The empty container is described as being non-spheroidal meaning that without being filled as described the container would not be sufficiently spheroidal to function effectively and efficiently in a weapon simulator. Obviously, the container would if inflated with air, for example, have some resemblance to a sphere but would not, in that configuration, define an efficient weapon simulator explosive mass. &#34;Circumference&#34; and such derivatives of that term as &#34;latitudinal plane&#34; are used to describe the circle defined by slicing a spheroidal body at any plane, including but not limited to the diametrical plane. Materials are described as behaving generally like a liquid when they conform to the shape of the container and are constrained in the bottom of the container and exert different forces upon different portions of the container at different levels of the material as a result of gravity. Each particle, physical molecule in the case of true liquids, globules or micella in the case of gels and the like, grains in the case of sand-like materials, is said to be acted upon separately by gravity such as to seek the lowest level available to the particle. The density of water, 62.4 lb/ft 3 , is used as a general reference in defining the materials that, when filling the container, cause the container to become spheroidal. Other densities may be accommodated with little or no redesign and only minor design changes, in accordance with the analysis and design criteria described, are required to accommodate lighter or heavier materials. 
     The invention is also embodied in a weapon simulator that comprises the container as defined above filed with an explosive as defined and supported in the manner defined. 
     INDUSTRIAL APPLICATION 
     This invention is useful in evaluating the effects of nuclear and large, high energy explosions in fluid, air or water, using low-cost readily available explosive materials.