be discarded as irrelevant. An analytical study of this problem to be pre- sented in Chapter 4 also demonstrates this.
3.2 Small-scale structural models
Therefore
[3.21]
Here Sm=Ym/Ypis the scale factor for material properties of the model and prototype. If the model is made of the same material as the prototype, then Sm=1.
The above argument can also be applied to the example of energy absorption corresponding to a given deflection. Referring to Eq. [3.16], geo- metric similarity requires that the value of the third dimensionless group, u/R, be the same for both the model and prototype (in addition to those specified in Eq. [3.17]). Thus
[3.22]
Because the value of B/Ris the same for both the model and prototype, we conclude from Eq. [3.16] that this must also be true for the last remaining dimensionless group,W/(MoB). Hence
or
[3.23]
Equations [3.21] and [3.23] indicate that when the model and prototype are made of the same material (Sm = 1), the two loads are related by Sl2
while the energy absorbed is proportional to Sl3.In plain words, if the struc- tures are geometrically similar, the strain eis the same. The characteristic area of a structure is proportional to Sl2. When the material is the same (with the same yield stress Y), the load (which is equal to stress multiplied by area) is then proportional to area only, or Sl2. Similarly, energy absorbed is proportional to the deforming volume multiplied by the energy density (Ye), which is the same for the model and prototype. Therefore, energy is proportional to Sl3.
The above process can be applied to various other physical variables of interest. In Table 3.1 we summarise the quantities likely to be encountered in the study of energy absorption. In principle, we should ideally keep the values of all dimensionless groups the same for both model and prototype.
However, in practice, it is sometimes very difficult or even impossible to scale certain quantities according to Table 3.1. We next discuss several such cases.
W M B
M B W S S W
m
om m op p
p m l p
=Ê
ËÁ ˆ
¯˜ = 3
W M B
W
o m M Bo p
Ê Ë
ˆ
¯ =Ê Ë
ˆ
¯ um =S ul p
P Y h
Y h P S S P
om
m m p p
op m l op
=Ê ËÁ
ˆ
¯˜ =
2 2
2
3.2.2 Quantities difficult to scale exactly
Gravity load
Table 3.1 specifies that the load applied to a model should be proportional to Sl2in order to maintain complete similarity. Gravity load is given by mass multiplied by gravitational acceleration, g. The mass scales as Sl3, which means that gravity load will in fact also scale as Sl3(because gis a constant), rather than the required Sl2. We cannot therefore strictly satisfy the simili- tude requirements. Fortunately, in practice, gravity loads are usually very
Table 3.1 Summary of scale factors between model and prototype
Physical variable Dimension Scale factor in Scale factor (same general cases material; gravity
insignificant)
Linear dimension L Sl Sl
Area L2 Sl2 Sl2
Volume L3 Sl3 Sl3
Material stress–strain FL-2 Sm 1
parameters (E, Y, . . .)
Material density FT2L-4 Sr 1
Mass FT2L-1 SrSl3 Sl3
Load F SmSl2 Sl2
Pressure FL-2 Sm 1
Stress FL-2 Sm 1
Strain – 1 1
Displacement L Sl Sl
Elastic and plastic energy FL SmSl3 Sl3
Elastic wave speed LT-1 Sm1/2Sr-1/2 1
Velocity LT-1 1 1
Angular velocity T-1 Sl-1 Sl-1
Time (impact duration T Sl Sl
or time elapsed)
Acceleration LT-2 Sl-1 Sl-1
Acceleration due to LT-2 1 Neglected
gravity (g)
Inertia force F SrSl2 Sl2
Momentum FT SmSl3 Sl3
Kinetic energy FL SrSl3 Sl3
Strain-rate T-1 Sl-1 Sl-1
Elastic fracture surface FL-1 SfSl2 Sl2
energy
Ductile tearing energy FL-1 unclear unclear
Material’s microstructural L 1 1
dimension
small in comparison with other loads and they can be neglected. If, however, in a particular case gravity loads are significant and need to be considered in scaling, adopting a different material for the model may overcome this difficulty; however, the model density must ensure that the overall scaling factor SrSl3is equal to SmSl2. This may again present some practical prob- lems. From this argument, if two materials have similar mechanical prop- erties (Smª1), then a small-scale model will need to have a higher value of density than its prototype.
Strain-rate effect
A characteristic strain-rate may be expressed as the ratio between the impact velocity and a representative length of a structure. It is therefore easy to see that when the impact velocity is the same, a small-scale model (which has a smaller linear dimension) will experience a higher strain- rate than a prototype; this is also indicated in Table 3.1. If the material is rate sensitive, the strain-rate will in turn affect the material’s properties, such as the yield stress Yand ultimate stress as discussed in Section 2.4. As a result, even if the materials are the same for the model and prototype, their mechanical properties will be different. A small-scale model will have an apparently higher yield stress, leading to a smaller deflection than pre- dicted by the similarity law. This can be illustrated by the following example.
Suppose a mild steel bar has an idealised elastic, perfectly plastic stress–strain behaviour, with a static yield stress of 250 MN/m2. Under high strain-rate, the dynamic yielding stress can be obtained using the Cowper–Symonds relationship (Eq. [2.73]). Let the length of the bar be 400 mm and the cross-section be 50 mm2. The bar is under uniaxial tension with uniform deformation. Now, if the external work done on the bar is 2 kJ, the energy density is then 2 kJ/(0.4 ¥0.05 ¥0.05) m3=2 ¥106J/m3. The plastic strain for this prototype is then ep=2 ¥106J/m3/250 MN/m2=0.008.
Now suppose we take a scale factor of Sl=0.1. The energy input should be Sl3¥2 kJ =2 J. Because the energy density remains the same, the strain for the model is em =ep=0.008. Apply the load at a rate of 200 mm/s for both the model and prototype according to Table 3.1. The strain-rate for the prototype is then 0.5 s-1 and for the model 5 s-1. The corresponding dynamic yielding stresses based on Eq. [2.73] with B=40.4 s-1and q =5, are 354 MN/m2 and 415 MN/m2, respectively. These lead to a new plastic strain ep= 0.005 65 and em = 0.004 82. This demonstrates that, due to the strain rate effect, the model experiences a strain of 85 % that of its proto- type. There is a potential danger of underestimating the deflection of a pro- totype based on the small-scale tests, if we assume complete similarity (and hence equal strain among all the models and the prototype).
Deformation with fracture
Structural plastic deformation and failure are often accompanied by frac- ture and ductile tearing. With fracture of brittle materials, the fracture energy is usually related to the newly created surface area by a material constant, hence Wf µ Sl2. However, the plastic energy is proportional to volume, hence WpµSl3. Total energy is then composed of these two parts;
there is no simple scaling law to relate this total energy to the scale factor Sl. Atkins elaborated this point with practical examples (1988). There are two extremes: if fracture energy is dominant, then total energy will be approximately proportional to Sl2. Alternatively, if plastic deformation is the major energy dissipation mechanism, then total energy may be assumed to be proportional to Sl3.
For ductile tearing there is no simple material constant to characterise the tearing energy (see Chapter 8 for details). As a result, simple scaling law does not appear to be applicable. However, if the tearing process is dominated by plastic deformation at the crack tip zone, then yield stress may be assumed to be the only significant material constant and the total energy is related to Sl3. The last example in Chapter 8 demonstrates this point with a metal plate cut by a wedge.
Remarks on lack of similarity
In addition to the three major quantities mentioned above, which do not obey elementary scaling, other sources exist that may lead to a departure from complete similarity. These include accidental overlook of certain important variables, a deliberate rejection of a variable incorrectly consid- ered non-critical, normal and frictional forces related to gravity, the mate- rial’s microstructures and certain highly localised deformation with heat generation.
The departure from elementary scaling law is usually referred to as the
‘size effect’. Fundamentally, such departure means that the values of all the dimensionless groups are not kept the same for both the model and proto- type. In principle, we could remedy this problem by studying the effect on the others of changing one or two specific dimensionless groups, i.e. by exploring the functional dependence of dimensionless groups. We can then deduce the significance of non-similarity of certain variables on other para- meters of interest. However, in practice this may be difficult to do when the number of dimensionless groups is large.
A small scale model is used in a case study of energy absorption of a roadside guardrail in Section 12.4. Structural scaling has also been em- ployed in other investigations of impact deformation of metallic structures (Duffey, 1971; Booth et al., 1983).