# Size Effect Correction to Ultimate Load Formulas in Codes

In principle, plastic limit analysis is a wrong theory for the majority of the design code provisions which deal with brittle failures, such as diagonal shear, torsion, punching, pullout, etc. So, in fact, is LEFM. The theoretically best approach would be to base the design on nonlinear fracture mechanics taking into account the large size of the fracture process zone. However, as pointed out in the previous section, this would be quite complicated for the basic design problems covered, by the code, and not really necessary because a highly accurate fracture analysis is not necessary for most situations. A simple way to obtain the load capacities corresponding to nonlinear fracture mechanics is to exploit the size effect law (1.4.10). Two kinds of formulas are possible:

1. One can start from the formula based on plastic limit analysis which now exists in the code, and introduce in it a correction due’ to the size effect law.

2. Alternatively, one can set up the ultimate load formula based on LHEM, and again introduce into it a correction according to the size effect law.

The first kind is no doubt preferable to the concrete engineering societies, because it makes it possible to retain the formulas that now exist in the codes, and introduce in them only a relatively minor correction (of course, the formula needs to be slightly scaled up because, for normal sizes, it must give about the same load capacity as before, even after the reduction for the typical structure size according to the size effect law has been introduced). Obviously, the accuracy of this type of correction would decrease with increasing size, as the behavior is getting more brittle and more remote from the size to which plastic limit analysis approximately applies. Some structures of normal sizes exhibit failures that are closer to limit analysis than to LEFM. For such structures, the accuracy by the first type of correction is adequate.

However, for very large structures or for certain types of failure (anchor pullout, diagonal shear), the failure is known to be very brittle, actually closer to LEFM than to plastic limit analysis. In that case, the second kind of formula, based on LEFM, must be expected to give a more realistic result. The error of this correction increases with a decreasing structure size and is the smallest for large sizes close to the LF. FM asymptote.

In the remainder of this section, we discuss how to introduce the size effect correction into the formulas existing in the codes. We consider first the ideal case of plain concrete structures (or structures for which the steel does not contribute appreciably to strength, such as anchor pullout). Then we analyze how these formulas must be modified to include the contribution of the reinforcement.