Nominal Stress and Nominal Strength
The size effect is understood as the dependence of the structure strength on the structure size. The strength is conventionally defined as the value of the so-called nominal stress at the peak load. The nominal stress is a load parameter defined as proportional to the load divided by a typical cross-sectional area:
(7N=c. fj;— for 2D similarity, ап—сптё> for 3D similarity (1.4.1)
in which P = applied load, b = thickness of a two-dimensional structure (which, for certain reasons, in experiments should be preferably chosen the same for all structure sizes); D — characteristic dimension of the structure or specimen; and ед = coefficient introduced for convenience, which can be chosen as c, v “ 1, if desired. For P = Pu = maximum load, Eq. (1.4.1) gives the nominal strength, oyu.
Coefficient Сдг can be chosen to make Eq. (1.4.1) coincide with the formula for the stress in a certain particular point of a structure, calculated according to a certain particular theory. For example, consider the simply supported beam of span S and depth h, loaded at mid-span by load P, as shown in Fig. 1.4.1 a. Now wc may choose, for example, an to coincide with the elastic bending formula for the maximum normal stress in the beam (Fig. 1.4. lb), and the beam depth as the characteristic dimension (£) = h), in which case we have
3 PS P S
<JN^2bh^=CNbD’W’ihCN = L5h
It appears that Cn depends on the span-to-depth ratio which can vary for various beams. It is thus important to note that the size effect may be consistently defined only by considering geometrically similar specimens or structures of different sizes, with geometrically similar notches or initial cracks. Without geometric similarity, the siz. e effect would be contaminated by the effects of varying structure shape. With this restriction (most often implicitly assumed), coefficient сц is constant because, for geometrically similar structures, S/h is constant by definition.
The foregoing definition of аn is not the only one possible. Alternatively, we may choose an to
Figure 1.4.1 (a) Three-point bent beam, (b) Elastic stress distribution, (c) Plastic stress distribution, (d) Elastic shear stress distribution, (e) Plastic shear stress distribution. (0 Shaft subjected to torsion, (g) Elastic shear stress distribution, (h) Plastic shear stress distribution, (i) Cantilever beam with linearly distributed load.
coincide with the plastic bending formula for the maximum stress (Fig. 1.4.1c), in which case we have
PS P S
on ~ ~b! i? = CNbD ’ withCiv = (= constam)
Alternatively, we may choose as the characteristic dimension the beam span instead of the beam depth (D = S), in which case we have
We may also choose on to coincide with the formula for the maximum shear stress near the support according to the elastic bending theory (Fig. 1.4.1 d), in which case we have, with D = ft,
On = i witheyy =0.75 (= constant) (1.4.5)
Alternatively, using the span as the characteristic dimension (D — S), we may write
3 p p 35
aN = 46ft = °NbD ’ wilhCiv = 4^ (= constant) (1.4.6)
All the above formulae are valid definitions of the nominal strength for three-point bent beams, although the first one (1.4.2) is the most generally used (and that used throughout this book). Other examples are given next.
Example 1.4.1 Consider torsion of a circular shaft of radius r, loaded by torque T = 2Pr where P is the force couple shown in Fig. 1.4. If. Using Сдг to coincide with the clastic formula for the maximum shear stress, we may write, taking D — 2r = diameter,
4P 16P P 16 ,
On = —5 = —рї = Слгку, with Cn = — (= constant) (1.4.7)
7ГГ1 7tD1 Vі ф 7Г
If, instead, we chose the radius as the characteristic dimension (D = r), we may write 4 P P 4
cn = —x = CN~rr, r, withejv = — (= constant) (1.4.8)
■КГ1 Dl 7Г
Note that in this case we have a three-dimensional similarity. 0
Example 1.4.2 Consider the cantilever of span і and cross-section depth ft shown in Fig. 1.4. li, which is loaded by distributed load p(x) increasing linearly from the cantilever end. We chose the value of the distributed load at the fixed end to be denoted as P/l (.£ is used to achieve the correct dimension). Now,
choosing on to coincide with the elastic bending formula for the maximum stress, and the characteristic dimension to coincide with the beam depth (D = h), we may write
P£ P l
°N ~ b№~ °NVd ‘ Wlthc’v = Ц (constant) (1-4.9)
which is again of the same form. D
To sum up, the nominal stress can be defined by the simple equation (1.4.1) regardless of the complexity of structure shape and material behavior, and can be used as a load parameter having the dimension of stress.