THE PI-THEOREM, 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 square-cornered 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 modern-style artillery units with develop­ment focussed on high-velocity kilogram-size ordnance for effective mobile deployment. During the English Civil War (1642-1651) success revealed itself in the form of 10 kg cast-iron cannon balls with sufficient kinetic energy to reduce stone-built 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 non-military type impacts.

Table 6.1

Potential Missile Hazards to Reactor Plant [106,290]

Missile Category

Example

Mass (tonne)

Velocity (m/s)

Soft

Military aircraft

20-50

150-300

Civil light aircraft

1-25

60-90

Boeing airliner 707

100-320

100

Steam-drum end

25

80

Semi-hard

Pipe-line end cap

0.03

170

Hard

Valve stem or bonnet

0.15

150

Turbine disc fragment

91.5

150

Because experiments with full-size 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 Pi-Theorem, 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

image213

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 Low-Velocity Missile Impact

Variable

Symbol

Dimensions

Missile diameter

d

L

Missile length

h

L

Nose radius of missile

r

L

Angle of nose

a

Missile density

pm

ML-[101]

Yield stress of missile

s

ML-1T-2

Missile velocity

V

LT-3

Angle of impact

b

Target thickness

L

L

Target width

w

L

Target density

Pt

ML-3

Yield stress of target

S

ML-1T-2

Strain

e

in which the constants and exponents of the Pi-terms 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 Pi-term. 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 Pi-terms of the prototype.

Early international experiments [68,106] to validate replica scaling techniques for the study of missile-concrete 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 mass-sized 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 Missile-Target Combinations in Replica Scaling Studies

Подпись: 120

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

Variable

Length

Velocity

Density

Stress

Strain

Scale Factor

1

1

1

1

1

Variable

Mass

Time

Force

Frequency

Strain-Rate

Scale Factor

13

1

12

1/1

1/1

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 strain-rates 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.

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