51
International RILEM Conference on Materials, Systems and Structures in Civil Engineering
Conference segment on Service Life of Cement-Based Materials and Structures
22-24 August 2016, Technical University of Denmark, Lyngby, Denmark
parts the stiffness of the foundation will reduce the crack width considerably. Only if the
macrocrack proceeds over the height of the wall, they become visible. For an efficient design
it is very desirable to know whether these macrocracks will stop at a certain height or proceed
over the whole wall as well as what is the distance to the next macrocrack. Thus, significant
factors influencing these aspects are discussed in the following section.
1.3 Relevant influences on the final macrocrack pattern
The material, technological and environmental conditions determine mostly the magnitude of
strains and strain rate, and as such define whether the cracks form or not. The final pattern of
cracks depends mostly on geometry, dimensions and restraint conditions.
In general, the maximum tensile stresses in a base-restrained element occur in the plane of
symmetry in length direction. This is also where first cracks are formed and where they reach
the greatest heights. For the same material, technological and environmental conditions the
height of this crack would depend solely on the restraint situation dependent on the EA and EI
as well as L/H ratios.
Depending on the cracking potential of hardening concrete and geometrical characteristics of
the wall, further primary cracks can successively develop in the wall. In shorter walls, these
cracks reach lower heights due to a smaller effective L/H ratio. The cracks are usually vertical
in the central part of the wall and slanted near the edges where the rotational restraint
becomes more significant. Horizontal cracks can be formed at the joint if shear stresses at the
joint exceed the bond strength. A comprehensive description of cracking pattern in walls on
foundations is presented in chapter 2 of [4].
1.4 Modelling
The modelling of hardening-induced macrocrack patterns of walls on foundations is a
complex matter. The major challenge is to combine complex time- and stress-dependent
material behaviour with crack formation on structural level. Only a modest number of
contributions exist, whereby the fundamental work by Rostasy and Henning [5] is certainly to
be seen as one of the most important ones. Next to this, the authors of this paper proposed two
approaches independently of each other. Other pertinent proposals are not known.
2. Numerical prediction of hardening-induced macrocrack formation in walls
2.1 Model used
The model used was based on the proposal of Knoppik-Wróbel and Klemczak [1].
Calculations were performed with a computer implementation of this phenomenological
model that allows for thermo–mechanical analysis of walls on foundation taking into account
the effect of hydration heat, temperature development, ageing, creep, soil–structure
interaction and behaviour of concrete after damage.
The analysis was performed in two steps. In the first step non-linear and non-stationary
thermal fields were determined in concrete elements and subsoil, respectively:
(1)
(2)
52
International RILEM Conference on Materials, Systems and Structures in Civil Engineering
Conference segment on Service Life of Cement-Based Materials and Structures
22-24 August 2016, Technical University of Denmark, Lyngby, Denmark
where is temperature, K;
is specific heat, kJ/(kg·K); is density, kg/m
3
; is thermal
conductivity, W/(m·K) and
is the rate of hydration heat generated per unit volume of
concrete, W/m
3
. The function of hydration heat time-development was described with the
approximation function of equivalent age, :
(3)
where
is the
total amount of hydration heat, J/g, and
are calibration coefficients
dependent on the type of cement. 3
rd
type boundary conditions were used. The aim of this
study was to investigate mechanical behaviour of the wall, thus physical analysis was limited
to thermal analysis. The authors are, however, aware that other influences such as autogenous
and drying shrinkage as well as coupling of these phenomena are not less important.
The imposed thermal strains were treated as volumetric strains and they were calculated based
on the changes of temperature:
(4)
(5)
where
is the coefficient
of thermal expansion, 1/K.
Viscoelasto–viscoplastic material model with the modified 3-parameter Willam–Warnke
failure criterion (MWW3) was used for hardening concrete following Klemczak [7] and
elasto–plastic material model with the modified Drucker–Prager failure criterion was used for
soil (see [4]). Detailed formulations of these models are given in [1] and [4]. The possibility
of crack occurrence was defined with the damage intensity factor (
):
(6)
Graphical interpretation of
is shown in Fig. 3. When
DIF = 1, it is equivalent to
formation of a crack in the direction perpendicular to the direction of the principal tensile
stress. Smeared cracking pattern was used. When failure is reached, material exhibits
softening behaviour. In the model, deviatoric and volumetric softening was applied with
hardening and softening laws adopted following Majewski [8].
Figure 3. Graphical interpretation of damage intensity factor (
DIF)