In the present paper, the postcracking strength of an RC shear wall element which follows the construction practices applied in Greece during the 70s is examined by taking into account the complex geometry of the crack of the wall and the mixed friction-plastification mechanisms that develop in the vicinity of the crack. Due to the significance of the crack geometry, a multiresolution analysis based on fractal geometry is performed, taking into account the size of the aggregates of concrete. The materials (steel and concrete) are assumed to have elastic-plastic behaviour. For concrete, both cracking and crushing are taken into account in an accurate manner. On the interfaces of the crack, unilateral contact and friction conditions are assumed to hold. For every structure corresponding to each resolution of the interface, a classical Euclidean problem is solved. The obtained results lead to interesting conclusions concerning the influence of the simulation of the geometry of the fractal crack on the mechanical interlock between the two faces of the crack, a factor which seems to be very important to the postcracking strength of the lightly reinforced shear wall studied here.

Many RC structures are facing a number of challenges, that is, earthquakes, hurricanes, and so forth, which may threaten their safety and serviceability. Therefore, modern structures built in seismic prone areas are designed to have significant bending and shear strength and ductility. However, existing structures designed according to earlier versions of the seismic codes and constructed using low strength materials usually have inadequate shear strength. For that, shear cracks appear in the shear wall elements of these structures reducing their overall capacity.

Generally, cracks are of large interest in RC structures since their properties reflect not only the condition of concrete as material but also the condition of the entire system at structural level. Crack width is commonly used as a convenient indicator of damage to RC elements, but it should be noted that the distribution and the geometry of the cracks are also important in measuring the extent of damage presented in the structure [

It is well known that the geometry of the interfaces of a crack is of fundamental importance to the study of friction, wear, and also strength evaluation. The recent research of fractured surfaces of various materials provides a deeper insight into the geometry of cracks. The corresponding research on metals [

The multiscale nature of the surface roughness suggests the use of fractal geometry. The fractal approach adopted here for the simulation of the geometry of the cracks formed in a shear wall uses computer generated self-affine curves for the modelling of the interface roughness, which is strongly dependent on the values of the structural parameters of these curves. The computer generated fractal interfaces, which are based on a given discrete set of interface data, are characterized by a precise value of the resolution

The aim of this paper is to study how the resolution of a fractal crack

The fractal nature of material damage has been a matter of a very intense research during the last three decades. The fractal nature of fractured surfaces in metals was shown more than 30 years ago by Mandelbrot et al. [

Fractal sets are characterized by noninteger dimensions [

Many methods which are based on experimental or numerical calculations, such as the Richardson method [

In relation (

The idea of self-affinity is very popular in studying surface roughness because experimental studies have shown that, under repeated magnifications, the profiles of real surfaces are usually statistically self-affine to themselves [

Fractal interpolation functions give profiles which look quite attractive from the viewpoint of a graphic roughness simulation. In higher approximations, these profiles appear rougher as it is shown in the next section where the first to fifth approximations of a fractal interpolation function are presented. Moreover, the roughness of the profile is strongly affected by the free parameters

Another model of self-affine profiles, which can be used for roughness description, is the multilevel hierarchical profile. This profile has a hierarchical structure and is constructed by using a certain iterative scheme presented in [

It must be mentioned here that an important advantage of the fractal interpolation functions presented in [

In Figure

The considered shear wall.

The considered shear wall is divided into two parts by a crack which has been formed due to shear failure of concrete. It is important to be mentioned here that, in low strength concretes, as in the case examined here, the fractured surfaces are rougher compared to the fractured surfaces developed in high strength concretes [

The computer generated interfaces

Characteristics of the considered structures.

Iteration ( |
Resolution |
Interface length |
---|---|---|

1st |
1.404 |
4.888 |

2nd |
0.468 |
4.946 |

3rd |
0.156 |
5.080 |

4th |
0.052 |
5.373 |

5th | 0.017 |
5.935 |

The first five resolutions of the fractal crack.

The objective here is to estimate the capacity of the shear wall under an action similar to the one that has created the crack. For this reason, a horizontal displacement of 20 mm is applied on the upper side of the wall (see Figure

For the modelling of the above problem it is assumed that the opposite sides of the fracture are perfectly matching surfaces in a distance of 0.1 mm and the finite element method is used. In order to avoid a much more complicated three-dimensional analysis, two-dimensional finite elements were employed; however, special consideration was given to the incorporation of the nonlinearities that govern the response of the wall. More specifically, the mass of concrete was modelled through quadrilateral and triangular plain stress elements. The finite element discretization density is similar for all the considered problems [

The adopted materials laws: (a) C16 concrete, (b) S220 steel, and (c) S400 steel.

The steel rebars were modelled through two-dimensional beam elements, which were connected to the same grid of nodes as the plain stress elements simulating the concrete. At each position, the properties that were given to the steel rebars take into account the reinforcement that exists in the whole depth of the wall. For example, the horizontal and vertical elements that simulate the steel mesh are assigned an area of 100.48 mm^{2} that corresponds to the cross-sectional area of two 8 mm steel rebars. For simplicity, the edge reinforcements (^{2} (i.e., 4 ^{2}). For the steel rebars, a modulus of elasticity

Figure

FE discretizations for third, fourt, and fifth resolutions of the fractal interface.

In this paper, only the finite element models corresponding to the 3rd, 4th, and 5th resolutions of the fractal crack were considered, because 1st and 2nd resolutions do not have a meaning from the engineering point of view. On the other hand, the 5th resolution gives a good lower bound of

At the interfaces, unilateral contact and friction conditions were assumed to hold. The Coulomb’s friction model was followed with a coefficient equal to 0.6. At each scale, a classical Euclidean problem is solved by using a variational formulation [

For every value of the vertical loading

Two cases are considered:

in the first case, the wall is uncracked;

in the second case, where a fractal crack

The solution of the above problems is obtained through the application of the Newton-Raphson iterative method. Due to the highly nonlinear nature of the problem, a very fine load incrementation was used. The maximum value of the horizontal displacement (20 mm) was applied in 2000 loading steps, while the total vertical loading was applied in the 1st load step and was assumed as being constant in the subsequent steps.

Figure

Load-displacement (

In cases of the cracked walls, the beneficial effect of the normal compressive loading is once more verified. This result holds for the 3rd, the 4th, and the 5th resolutions of the fractal crack but for small displacement values only. For larger displacement values, the three variants of the cracked wall behave differently. The 4th and the 5th resolutions appear to have a stable behaviour without strength degradations. However, it is noticed that, in the case of the 3rd resolution and for heavy axial loading, significant strength degradation takes place.

The above results can be more easily understood if we compare in the same diagram the curves obtained for the four different structures studied here (uncracked, 3rd resolution, 4th resolution, and 5th resolution) for specific load levels. In Figure

Comparison of the behaviour of the four variants of the examined wall for specific values of the compressive axial loading.

However, the most interesting case is the case where heavy axial loading (

Significantly different is the behaviour of the 3rd resolution of the fractal crack. Although in the first loading steps the results are close to those of the 4th and 5th resolutions, after a displacement value of 2.5 mm, significant strength degradation appears, having the form of successive vertical branches. Moreover, the ultimate strength of this wall is significantly lower than the other variants.

It is interesting to try to explain this significantly different behaviour that appears between the walls corresponding to the 3rd and the higher resolutions of the fractal crack. For this reason, all the parameters affecting the behaviour of the wall will be comparatively studied in the sequel.

Figure

Cracking strains for various values of the vertical loading, for the cases of the uncracked wall (left column) and the cracked walls (3rd resolution: middle column and 4th resolution: right column).

The above results alone cannot explain the significantly different responses that the two cracked variants of the wall exhibit. For this reason, the plastic strains of concrete obtained for the same applied horizontal displacement are examined. Figure

Plastic strains for various values of the vertical loading, for the cases of the uncracked wall (left column) and the cracked walls (3rd resolution: middle column and 4th resolution: right column).

On the other hand, the cracked walls seem to deform significantly in the vicinity of the crack. This phenomenon is more pronounced in the case of the 3rd resolution of the fractal wall and especially in the case of heavy axial loading, where it can be noticed that the vicinity of the crack is in the crushed state; that is, the forces are transmitted solely by the steel mesh in this region. For the case of the 4th resolution, this phenomenon is rather limited; that is, it can be concluded that in this case the crack retains partially its ability to transfer shear and compressive forces through the contact and friction phenomena developed in the interface and through the mechanical interlocking that occurs between the two interface parts.

In the sequel, it is interesting to examine the deformations that have occurred at the steel mesh. Figure

Deformation of the steel mesh for various values of the vertical loading, for the cases of the uncracked wall (left column) and the cracked walls (3rd resolution: middle column and 4th resolution: right column).

Now, the difference in the response between the 3rd and the 4th resolutions of the fractal crack for the case of heavy vertical loading will be explained. As it has already been noticed, higher vertical loading leads to higher values of the horizontal loading. The increased horizontal forces have to be transferred from the upper part of the cracked wall to its lower part. In this respect, three mechanisms are developed in order to facilitate the horizontal load transfer:

exploitation of the tensile strength of the horizontal rebars;

development of friction on the part of the crack where contact forces occur;

mechanical interlock between the two interfaces of the crack.

The first two mechanisms are almost similar in both cracked walls. However, it is obvious from Figure

Figures

Forces developed in the steel mesh for various values of the vertical loading for the uncracked wall (left column:

Forces developed in the steel mesh for various values of the vertical loading for the 3rd resolution of the crack (left column:

Forces developed in the steel mesh for various values of the vertical loading for the 4th resolution of the crack (left column:

For the case of the uncracked wall (Figure

For the case of the 3rd resolution of the crack (Figure

The situation is rather different in the case of the 4th resolution of the fractal crack (Figure

In the paper, the finite element analysis of a typical shear wall element which follows the construction practices applied in Greece during the 70s was presented assuming that a certain crack has been developed as a result of an earthquake action. The crack was modelled following tools from the theory of fractals. Three different resolutions of the fractal crack were considered by taking into account the aggregate sizes of the concrete, and their results were compared to those of the initially uncracked wall. The main finding of the paper is that the cracked wall still has the capacity to sustain monotonic horizontal loading. For small axial loading values, this capacity is similar to that of the initially uncracked wall. However, for larger axial loading values where the demands increase, it seems that a more accurate simulation of the geometry of the fractal crack (i.e., considering higher values of the resolution of the interfaces) leads to better results. Using lower resolution values, the roughness of the interfaces is not taken into account, and therefore the mechanical interlock between the two faces of the crack is rather limited, leading the concrete in the vicinity of the crack to overstressing and gradually to a complete loss of its capacity to sustain any kind of forces. In this case the bending capacity of the wall is significantly limited.