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September 8th, 2015

The size effect is, for design engineers, the most compelling reason for adopting fracture mechanics. Therefore, we discuss it more thoroughly now, and we will return to it in considerable detail in future

Figure 1.2.4 Influence of the structure size on the length of the yielding plateau in a punched slab (from ACI Committee 446 1992). |

chapters. By general convention, the load capacity predicted by plastic limit analysis or any (deterministic) theory in which the material failure criterion is expressed in terms of stress or strain (or both) are said to exhibit no size effect. The size effect represents the deviation from such a prediction, i. e., the size effect on the structural strength is the deviation, engendered by a change of structure size, of the actual load capacity of a structure from the load capacity predicted by plastic limit analysis (or any theory based on critical stresses or strains).

The size effect is rigorously defined through a comparison of geometrically similar structures of different sizes. It is conveniently characterized in terms of the nominal strength, <7yv„, representing the value of the nominal stress, сгдг, at maximum (ultimate) load, Pu. The nominal stress, which serves as a load parameter, may, but need not, represent any actual stress in the structure and may be defined simply as сгдг = P/bD when the similarity is two-dimensional or as Р/D2 when the similarity is three-dimensional; b = thickness of a two-dimensional structure, and D = characteristic dimension of the structure, which may be chosen as any dimension, e. g., the depth of the beam, or the span, or half of the span, since only the relative values of cryv matter. The nominal strength is then a^u ~ Pu/bD or Pu/D2 (see Section

1.4.1 for more details).

According to the classical failure theories, such as the elastic analysis with allowable stress, plastic limit analysis, or any other theory that uses some type of a strength limit or failure surface in terms of stress or strain (e. g., viscoelasticity, viscoplasticity), одru is constant, i. e., independent of the structure size, for any given geometry, notched or not. We can, for elhmple, illustrate it by considering the elastic and plastic formulas for the strength of beams in bending shear and torsion. These formulas are found to be of the same form except for a multiplicative factor. Thus, if we plot log o^u vs. log D, we find the failure states, according to strength or yield criteria, to be always given by a horizontal line (dashed line in Fig. 1.2.5). So, the failures according to the strength or yield criteria exhibit no size effect.

By contrast, failures governed by linear elastic fracture mechanics exhibit a rather strong size effect, which in Fig. 1.2.5 is described by the inclined dashed line of slope — 1 /2, as we shall justify in Chapter 2. The reality for concrete structures is a transitional behavior illustrated by the solid curve in Fig. 1.2.5. This curve approaches a horizontal line for the strength criterion if the structure is very small, and an inclined straight line of slope —1/2 if the structure is very large.

There is another size effect that calls for the use of fracture mechanics. It is the size effect on ductility

Figure 1.2.5 Size effects: (a) on the curves of nominal stress vs. relative deflection, and (b) on the strength in a bilogarithmic plot (adapted from ЛС1 Committee 446 1992). |

Figure 1.2.6 Size effect on the structural ductility (adapted from ACI Committee 446 1992). |

of the structure, which is the opposite of brittleness, and may be characterized by the deformation at which the structure fails under a given type of loading. For loading in which the load is controlled, structures fail (i. e., become unstable) at their maximum load, while for loading in which the displacement is controlled, structures fail in their post-peak, strain-softening range. In a plot of одг vs. the deflection, the failure point is characterized by a tangent (dashed line in Fig. 1.2.6) of a certain constant inverse slope – Cs where Cs is the compliance of the loading device (see e. g., Bazant and Cedolin 1991, Sec. 13.2). Geometrically similar structures of different sizes typically yield load-deflection curves of the type shown in Fig. 1.2.6. As illustrated, failure occurs closer to the peak as the size increases. This effect is again generally predicted by fracture mechanics, and is due to the fact that in a larger structure more strain energy is available to drive the propagation of the failure zone.

The well-known effect of structure size or member size on crack spacing and crack width is, to a large extent, also explicable by fracture mechanics. It may also be noted that the spurious effect of mesh size (Reason 2, Section 1.2.2) can be regarded as a consequence of the structural size effect.