5.3 Geometry of pile caps: deep three-dimensional
5.3.5 Size effect in deep elements and in pile caps
“Size effect” is the common expression that refers to the decrease of the nominal shear strength of beams with increasing size.
Figure 5.22 Dimension of slender beams loaded in shear (Walraven 1978)
The beams shown in Figure 5.22 all have the same aspect ratio a/d=3. Figure 5.23 (a) shows the variation of the relative shear capacity divided by the concrete tensile strength (vu=Vu/bdfct) of the beams with increasing dimension. It can be seen that, the bigger the beams are the smaller the relative shear capacity is.
Amean
Farc
Figure 5.23 (a) Relative shear capacity of gravel and lightweight concrete beams with constant aspect ratio and increasing effective depth, (b) Crack patterns of the beams, reduced to equal proportions, at the same relative shear force (Walraven 1978)
Historically different hypotheses have been presented to justify this behaviour.
In 1939, Weibull (1939) argued that the size effect could be explained as the increased probability to find a weaker section with increasing size of the beams. Although interesting for members subjected to pure tension, this approach has been disregarded by researchers over time for shear.
In 1972, Taylor (1972) advanced that the reduction of shear strength could be explained by the increasing size of cracks in deep members. He considered that the interlock phenomenon contributed with about 35 up to 50 percent of the total shear capacity and argued that, if the aggregate size was kept constant, the interlock phenomenon would become less effective in deep members. However Taylor’s proposition was proven wrong by Walraven (1978). Indeed, in lightweight aggregate, the cracks intersect the aggregate particles and do not form mainly in the cement paste like in gravel concrete. Therefore very little aggregate interlocking occurs.
Nevertheless, as shown in Figure 5.23 (a), lightweight concrete is also prone to size effects.
The most accepted explanation up to now was proposed by Reinhardt (1981) and further developed by Bažant (1994) and is based on linear elastic fracture mechanics.
As can be seen in Figure 5.23 (b), for the same relative shear force, a beam shows more critical crack pattern if its dimensions are bigger. Indeed, linear elastic fracture mechanics theory pointed out that the crack propagation was more important in larger members because of the greater energy-release rate. The comparison between strength predictions of deep members according to both static and fracture mechanics is shown in Figure 5.24.
(a) (b)
Figure 5.24 Size effect influence on shear strength prediction by static and fracture mechanics (Bažant 1994)
It is now accepted by most of the researchers that the more critical cracking pattern in larger members results in concrete softening. This fact was acknowledged in the Modified Compression Field Theory developed by Vecchio and Collins (1986), were the strength of cracked concrete in compression is dependent on the principal tensile strains in uncracked concrete, on the cracks width at crack interface and on the average tensile strain in cracked concrete.
However, the case of pile caps is different and no relevant information was found in current literature about the actual importance of size effects in pile caps. Indeed, the conclusions drawn above cannot be easily extended to pile caps. Stocky pile caps do not develop the same cracking pattern as deep beams. For instance, Adebar, Kuchma and Collins (1990) showed that pile caps without stirrups had very few cracks before failure compared to deep beams, which means that the softening of the concrete struts is less developed. This is due to the large importance of confinement by inactive concrete which allow highly loaded bottle-shaped concrete struts to carry high stresses without severe cracking. If the dimensions of a specimen are increased, the confinement is increased way more in a pile cap than in a deep beam where the width is kept constant. For these reasons, it was assumed in this thesis work that no correction factor for the size effect should be considered in the model. However it would be very interesting to carry out some experimental study on the subject in order to get a better understanding of the influence of size in three-dimensional structures.
It should be pointed out that ignoring size effect, as it is proposed in the model developed in this thesis, is in contradiction with the sectional approach according to design codes. For instance in Eurocode, and respectively in BBK, the factors k and ξ are used to take into account the strength reduction due to size effect. These factors are implemented in both the formulas to evaluate beam shear capacity and punching shear capacity.