2.2.2 Two-ways shear forces transfer in reinforced concrete
2.2.2.5 Survey and classification of the different punching shear models
Different models, based on different assumptions and mechanical theories, have been developed during the last fifty years in order to describe punching failure in slabs. A survey and classification of these different approaches is proposed here.
Classifying different mechanical model approaches to punching shear is controversial.
Indeed, some models combine different theories. The reference in this part is mainly made to Reineck (fib 2001) whose classification has been followed.
The model by Kinnunen and Nylander was early developed, in 1960. This model proposed a comprehensible force equilibrium of the punching cone and gave the basics to most research works carried out afterwards. Therefore, the Kinnunen and Nylander approach is extensively described in this part.
a) The model proposed by Kinnunen and Nylander
The Kinnunen and Nylander (1960) approach is often considered as the first sound model proposed to represent punching shear.
The model developed by the Swedish researchers is based on a series of tests on slabs they carried out. During those tests, Kinnunen and Nylander noticed that the crack pattern of slabs over interior columns was always quite similar and that punching failure always happened after that cracking pattern had developed. The vertical tangential crack that propagates into an inclined shear crack at the column face combined to the vertical radial cracks at the side cut sectors from the slab, as shown in Figure 2.27 b.
Figure 2.27 a) and b) Geometry and equilibrium of the model (Kinnunen et al. 1960).
In addition, they noticed that the curvature of the slab in the radial direction at some distance from the load was almost constant. Therefore, the sectors defined before could be considered as rigid. The displacement of the rigid sectors is then simplified to a rotation of an angle ψ around a centre of rotation located at the root of the shear crack. The centre of rotation is indicated as C.R. in Figure 2.27 d).
Figure 2.28 c) and d) Geometry and equilibrium of the model (Kinnunen et al. 1960).
Kinnunen and Nylander also considered the concrete expansion under loading in their analysis.
Based on these hypotheses and under some simplifying assumption based on their experiments, Kinnunen and Nylander established the equilibrium of the conical shape shown in Figure 2.27 c).
The solving process is a trial-and-error method: a compressive web height has to be found, that fulfils both the moment and force equilibrium equations formulated.
The failure criterion chosen is the ultimate compressive strain at the bottom of the slab, in the conical shape submitted to a triaxial state of stresses (εc = -1.96‰).
b) Flexural capacity approach
Practice showed that, for many common slabs, the punching failure load was quite close to the flexural failure load. Based on that statement, different models have been developed. Although it is not the first one, the model developed by Moe (1961) is significant as it laid the basics for ACI design code in 1963.
Moe’s empirical approach is based on the assumption that the ultimate capacity of a slab is linked to its flexural and one way shear capacities:
1
= +
flex u shear
u
V A V V
V (2.30)
Where A is derived and calibrated by tests results.
These kinds of empirical flexural capacity approaches have disappeared from most codes although it can still be found in section 6.9 of BBK04.
c) Plasticity approach
The perfect plasticity theory, associated with limit theorems, is rather new and promising. Unlike elastic and elasto-plastic stress and strain equilibrium methods that require trial-and-error procedure, a direct evaluation of the bearing capacity of a structure is possible with the perfect plasticity theory. The theorems of limit analyses
(i.e. lower bound and upper bound) proposed by Drucker and Prager (1952) gave a simple and intuitive way to assess the ultimate strength of slabs.
It is important to note that, when using an upper bond method relying on the yield line theory, it is very convenient to know from the beginning either the yielding surface or the plastic flow displacement direction. In the case of punching, due to the symmetry of the problem, the displacement is orthogonal to the slab. The rate of internal work at failure is then dependant on the angle of the punching cone. In the case of direct strut action from the load to the support, the yield surface is known and the energy to develop a mechanism is dependent on the direction of the displacement at failure.
Upper bound solutions were proposed by Braestrup (1976) and Marti and Thürlimann (1977). Lower bound solutions were suggested by Braestrup in 1985 (CEB 1985) and Pralong (1982). The latest involves the tension strength of concrete as well.
d) Failure mechanism approaches with concrete tensile stresses in failure surface
Failure mechanism approaches, for example the Kinunnen and Nylander model, assume the location of the critical cracks that will lead to the failure mechanism. The accuracy of a failure mechanism model relies on a good choice of critical cracks locations. For example, Kinnunen and Nylander defined the shape of the critical cracks by a comparison with experimental tests. Since, other methods have been developed, like non-linear finite element analyses.
The recent development of fracture mechanism also encouraged researchers to include additional shear transfer forces mechanism in their models (refer to Chapter Mechanisms of shear transfer). Accounting for the tensile strength of concrete was proved very promising and studied, among others, by Menétrey (1994).
e) Truss models or strut and tie model
Strut-and tie models used in design practice and in codes are smeared models. They do not define the exact location of cracks. These smeared strut-and-tie models account for cracking and the associated reduction of the different possible shear transfer mechanisms, by limiting some parameters:
Muttoni et al. (2008), among other authors, considered more local strut-and-ties models to explain the transfer of shear forces at a macro level. Truss models were developed to explain shear transfer mechanisms in cracked flexural elements. Local truss models shows how shear force are transferred above the cracks, in elements without shear reinforcement (direct arch action, shear transfer in the compression membrane), between cracks (cantilever effect, dowel effect) and at the crack interface (residual tensile stresses, interface shear transfer). These local truss models are described extensively in the chapter Mechanisms of shear transfer. These local models are relevant for understanding but still have not been rationalized and simplified enough to provide the basics of a design method.
Some authors proposed interesting smeared strut-and-tie model with concrete ties.
The one way shear model by Reineck (2010) and the punching model by Pralong (1982) belong to the few propositions that were made on smeared truss models with concrete ties.
In this study, the issue of concrete tensile strength and its integration into a design procedure based on a strut-and-tie model has to be dealt with. Indeed, as explained in details previously, neglecting the tensile strength of concrete in pile caps is too
conservative when assessing the shear and local punching shear capacity to provide economically satisfying steel reinforcement quantities.
However, it was chosen not to specify concrete ties in the strut-and-tie model.
Concrete tensile positive contribution is considered through the effect of confinement on the capacity of the strut to carry load by arch action. Details of the procedure are found in chapter 5: Description of aspects specific to pile caps and implementation in the strut-and-tie model developed.
f) Fracture mechanics
Some recent models, like the one by Hallgren (2002), rely only on the fracture mechanics. They are usually coupled with heavy numerical analyses with a failure criterion derived from fracture mechanics.