Sources of Size Effect on Structural Strength
There are six different size effects that may cause the nominal strength to depend on structure size:
I. Boundary layer effect, also known as the wall effect. This effect is due to the fact that the concrete layer adjacent to the walls of the formwork has inevitably a smaller relative content of large aggregate pieces and a larger relative content of cement and mortar than the interior of the member. Therefore, the surface layer, whose thickness is independent of the structure size and is of the same order of magnitude as the maximum aggregate size, has different properties. The size effect is due to the fact that in a smaller member, the surface layer occupies a large portion of
Figure 1.3.1 The essence of ihe difference between statistical and fracture size effect (adapted from AC1 Committee 446 1992).
the cross-section, while in a large member, it occupies a small part of the cross-section. In most situations, this type of size effect does not seem to be very strong. A second type of boundary layer effect arises because, under normal stress parallel to the surface, the mismatch between the clastic properties of aggregate and mortar matrix causes transverse stresses in the interior, while at the surface these stresses are zero. A third type of boundary layer size effect arises from the Poisson effect (lateral expansion) causing the surface layer to nearly be in plane stress, while the interior is nearly in plane strain. This causes the singular stress field at the termination of the crack front edge at the surface to be different from that at the interior points of the crack front edge (Bazant and Estenssoro 1979). A direct consequence of this, easily observable in fatigue crack growth in metals, is that the termination of the front edge of a propagating crack cannot be orthogonal to the surface. The second and third types exist even if the composition of the boundary layer and the interior is the same.
2. Diffusion phenomena, such as heat conduction or pore water transfer. Their size effect is due to the fact that the diffusion half-times (i. e., half-times of cooling, heating, drying, etc.) are proportional to the square of the size of the structure. At the same time, the diffusion process changes the material properties and produces residual stresses which in turn produce inelastic strains and cracking. For example, drying may produce tensile cracking in the surface layer of the concrete member. Due to different drying times and different stored energies, the extent and density of cracking may be rather different in small and large members, thus engendering a different response. For long-time failures, it is important that drying causes a change in concrete creep properties, that creep relaxes these stresses, and that in thick members the drying happens much slower than in thin members.
3. Hydration heat or other phenomena associated with chemical reactions. This effect is related to the previous one in that the half-time of dissipation of the hydration heat produced in a concrete member is proportional to the square or the thickness (size) of the member. Therefore, thicker members heat to higher temperatures, a well-known problem in concrete construction. Again, the nonuniform temperature rise may cause cracking, induce drying, and significantly alter the material properties.
4. Statistical size effect, which is caused by the randomness of material strength and has traditionally been believed to explain most size effects in concrete structures. The theory of this size effect, originated by Weibull (1939), is based on the model of a chain. The failure load of a chain is determined by the minimum value of the strength of the links in the chain, and the statistical size effect is due to the fact that the longer the chain, the smaller is the strength value that is likely to be encountered in the chain. This explanation, which certainly applies to the size effect observed in the failure of a long concrete bar under tension (Fig. 1.3.1), is described by Weibull’s weakest-link statistics. However, as we will see in Chapter 12, on closer scrutiny, this explanation is found to be inapplicable to most types of failures of reinforced concrete structures. In contrast to metallic and other structures, which fail at the initiation of a macroscopic crack (i. e., as soon as a microscopic flaw or crack reaches macroscopic dimensions), concrete structures fail only after a large stable growth of cracking zones or fractures. The stable crack growth causes large stress redistributions and a release of stored energy, which, in turn, causes a much stronger size effect, dominating over any possible statistical size effect. At the same time, the mechanics of failure restricts the possible locations of the decisive crack growth at the moment of failure to a very small zone. This causes the random strength values outside this zone to become irrelevant, thus suppressing the statistical size effect. We will also see that some recent experiments on diagonal shear failure of reinforced concrete beams contradict the prediction of the statistical theory.
5. Fracture mechanics size effect, due to the release of stored energy of the structure into the fracture front. This is the most important source of size-effect, and will be examined in more detail in the next section and thoroughly in the remainder of the book.
6. Fractal nature of crack suifaces. If fractality played a significant role in the process of formation of new crack surface, it would modify the fracture mechanics size effect. However, such a role is not indicated by recent studies (Chapter 12 and Bazant 1997d). Probably this size effect is only a hypothetical conjecture.
In practical testing, the first 3 sources of size effect can be, for the most part, eliminated if the structures of different sizes are geometrically similar in two rather than three dimensions, with the same thickness for all the sizes. Source 1 becomes negligible for sufficiently thick structures. Source 2 is negligible if the specimen is scaled and is at constant tenrperature. Source 3 is significant only for very massive structures. The statistical siz. c effect is always present, but its effect is relatively unimportant when the fracture siz. e effect is important. Let us now give a simple explanation of this last and dominant size effect.