THE PITHEOREM, SCALE MODELS, AND REPLICAS
Studies of the impacts between missiles and structures have been a long and continuing military activity. An early example is the evolution of squarecornered Norman castles into the rounded structures of Edward I so as to better resist the impact of large catapulted rocks. By the sixteenth century mathematics and chemical explosives had enabled the embryonic formation of modernstyle artillery units with development focussed on highvelocity kilogramsize ordnance for effective mobile deployment. During the English Civil War (16421651) success revealed itself in the form of 10 kg castiron cannon balls with sufficient kinetic energy to reduce stonebuilt castles to ruins. During operation “Desert Storm” starting January 16, 1991 US tanks fired projectiles of some 9 kg with supersonic muzzle velocities as great as 1900 m/s. Though the rotating machinery and pressurized components in nuclear power plants can produce potentially damaging missiles, their masses and velocities are radically different from the military. For instance a turbine failure at Calder Hall in 1958 created a number of subsonic missiles of order 1 tonne [278]. The probability of plant failures producing missiles with damage potential has been estimated as 10“4 to 10“5 per operating year [279]. Impacts on reactor structures from subsonic external sources such as crashing aircraft are also probable, and that for a heavy fighter (e. g.,Tornado) is judged to be about 10“6 per year. Though light aircraft pose virtually no hazard to reactor containments, they can potentially damage fuel stores or control rooms with the same probability of 10“6 per year. Large airliners are considered to have an impact probability of at least one order less than 10“6 per year. These power plant impacts produce far less local heating than do military projectiles, so that material properties like creep strength are far less adversely affected. Military data are therefore inappropriate for reactor safety assessments for which the relevant UK studies began in earnest [106] around 1977. In the context of a nuclear power plant, a missile is described as soft if a significant fraction of its deformation is orders of magnitude greater than that of the target. Missiles from disintegrating power plant items generally suffer a dissimilar deformation to their target, and are designated as hard. Table 6.1 summarizes the pertinent parameters of these radically different nonmilitary type impacts.
Table 6.1 Potential Missile Hazards to Reactor Plant [106,290]

Because experiments with fullsize missiles and nuclear plant structures are impractical, scale models are a necessity. Appropriate scaling rules can be developed either from the fundamental equations or by the presently more convenient route of dimensional analysis [280]. The essence of dimensional analysis is the Buckingham PiTheorem, which characterizes a physical process in terms of the minimum number of dimensionless combinations of its pertinent variables. If F denotes a finite polynomial in the variables {xi, x2,.. ..xng, then F is homogeneous of order integer m if and only if
where for all integer j
n
kjp = m and aj 2 R
p=1
Euler proved that the most general solution of
n @F
У>р@ = 0 is F(xi, X2. …Xn)= 0 (6.2)
p=i @xp
Table 6.2 Parameters of a LowVelocity Missile Impact

in which the constants and exponents of the Piterms are to be determined experimentally.
If the dynamics of a scale model are to replicate its prototype, then a constant scaling of the geometric lengths alone would be inappropriate in the present context. Specifically suppose the geometric lengths in equation (6.3) are scaled by l, and to assist visualization model strains are to match those of the actual structure. Arbitrary scaling of stresses and densities by say f and m then necessitates a functionally dependent scaling of the model velocity to achieve the same Piterm. Using primed variables for the model, it is therefore required that
= vVpJs
so the model velocity for dynamic similarity must be scaled according to
Vі = V/ f/m with f = pt/p’t and m — S/S’ (6.4)
By definition, a replica model[102] has scaled variables that reproduce the set of all dynamically characterizing Piterms of the prototype.
Early international experiments [68,106] to validate replica scaling techniques for the study of missileconcrete impacts involved microconcrete with an appropriately scaled aggregate mix and steel reinforcement mesh to represent a prototype. Dynamically similar tests[103] at AEEW employed the three masssized pairs in Table 6.3, and for each pair three different bonding reinforcement quantities of 1/8, 1/4 and 1/2% EWEF[104] were investigated for the concrete panels. Visually identical overall damage patterns were produced for each different reinforcement, and the excellent consistency of the measured target penetration velocities is shown in Figure 6.5. These tests adopted the replica scaling in Table 6.4 so that the reinforcement has identical strength, yield and elastic modulus as the prototype. Also the microconcrete target is manufactured to provide the same compressive and
Missile 
Target 

Diameter (mm) 
Mass (kg) 
Diameter (m) 
Thickness (mm) 
313 
490 
6.0 
640 
120 
27 
2.3 
246 
40 
1 
0.767 
82 
Table 6.3 Hard MissileTarget Combinations in Replica Scaling Studies 
Bending reinforcement quantity (% EWEF)
50 ——————— 1——————— 1——————— *——————— *
0 1/8 1/4 3/8 1/2
Figure 6.5 Experimental Validation of Critical Perforation Velocity with Bending Reinforcement Quantity for Three Sizes of Concrete Target
Table 6.4 A Consistent Set of Replica Scale Factors

tensile strength as a typical constructional concrete. Because crack widths and spacings in a concrete structure’s flexural response markedly depend [281] on the bonding strength between the concrete and its steel reinforcement, the production of such carefully scaled microconcrete required a dedicated laboratory facility. Actual impacts on a reactor containment induce strainrates in the range 0.01 to 1.0/s, so that the increased dynamic strength of a replica’s steel reinforcement becomes a major difficulty should it become much smaller than the prototype [283,284]. Figure 6.6 illustrates the variation in dynamic strengths of reinforced concrete materials at high straining rates.