Analysis of Shrinkage Cracking of a Slab on the Ground Using a Probabilistic and Deterministic Approach ^†

Abstract

This paper gives insight into the influences of spatially varying material properties on the non-linear finite element modeling of the rectangular and square-shaped floor slabs. Modeling the properties of the concrete using spatially distributed properties instead of using constant properties is more realistic. We analyzed the influence of probabilistic concrete property modeling on the development of cracks in floor slabs. A concrete floor subjected to a shrinkage load was analyzed using the material model with randomly distributed compressive strength. Consequently, the tensile strength and elastic modulus were also randomly distributed. The concrete properties were defined by the random field generator that uses the Fast Fourier Transformation method and the guidelines given by the Joint Committee on Structural Safety Probabilistic Model Code. The obtained results are compared with the values from modeling with constant concrete properties. The comparison shows that the crack development in the slabs with and without varying material properties is statistically significantly different.

1. Introduction

Floor slabs are concrete elements built as continuously supported thin plates on the ground. The issues often occurring in industrial floor slabs are related to cracks, which can develop even before applying any service loads on the slab. Those types of cracks generally arise from the shrinkage of concrete that largely progresses in the first weeks after concrete pouring. Friction to the ground, reinforcement, columns, foundations, perimeter beams, and changes in the slab thickness prevent free movements of the slab, which can induce the development of tensile stresses. If the stress level reaches the tensile strength of the concrete, then cracking will occur.

Concrete is a composite material built out of rock aggregates connected with hardened cement. The aggregates vary in shape and size and are not uniformly distributed during concrete production, which results in spatial variability of material properties such as strength, stiffness, and permeability. Therefore, it is of interest to include the spatially varied properties in the modeling of structural elements. Modeling structural elements using spatially varied properties is often referred to as the probabilistic approach modeling, while in typical everyday calculations, mechanical properties are set at a constant value, which presents the deterministic approach.

Random field (RF) modeling is a method used in a probabilistic approach since it includes the influence of the spatial variability of properties that vary in space (or time). RF is a collection of random variables at points in the considered space. The influence of RF in modeling and its impact on the structural behavior of concrete structures is presented in [1]. The RF model is defined by a correlation function that describes the correlation between two distinct spatial points. The correlation between two close points is larger than the correlation between two points that are far apart. To apply RF in a structural analysis, a correlation function and the corresponding correlation length need to be defined. Models with several correlation functions with different correlation lengths can be found in [2]. The definition of the parameters for a correlation function using in situ measurements of spatially distributed concrete strength is presented in [3,4]. The RF analysis has been employed on different types of concrete structures. An investigation on the impact of concrete’s spatial variability on structural response and failure analysis of a reinforced concrete chimney is presented in [5]. In [6], the structural reliability analysis considering the effect of RF is proposed and verified by numerical case studies on a prestressed concrete bridge. A study of surface deterioration analysis based on RF for marine reinforced concrete structures is presented in [7].

In this research, we employ the non-linear finite element method (FEM) to ascertain the shrinkage of the concrete slab on the ground. The analysis is conducted with and without spatially varied material properties. For the probabilistic approach, we analyzed parameters for the non-uniform mechanical properties in the concrete through the literature and decided to use the recommendations from the Joint Committee on Structural Safety Probabilistic Model Code (JCSS-Code) [8]. The analysis includes results of principal stresses, tensile strength of concrete, and crack development in the concrete slab on the ground.

The main goal of this research is a better understanding of how a chosen slab-on-ground design is affected in terms of shrinkage cracking with a probabilistic approach in the design of the slab-on-ground regarding the shrinkage of the concrete slab. Therefore, the thickness and reinforcement ratio could be calculated more economically. We simulate the development of cracking with and without varying material properties. Obtained results of the probabilistic approach are statistically compared with the values from the deterministic modeling. The statistical analysis of the differences between the results of probabilistic and deterministic FEM simulations is the novelty of the presented research in comparison to the existing literature. The analysis of those results showed that modeling structural elements using spatially varied properties statistically significantly differs from the deterministic modeling.

2. Slabs Geometry and Parameters in Probabilistic Modeling

The numerical study of the floor slabs is performed with and without spatially varying material properties. The main factors investigated are the development of stresses and strains induced by the shrinkage of the concrete considered on two slab dimensions—one rectangular and one square-shaped slab.

2.1. Properties of Slabs

A rectangular and square-shaped slab is investigated using non-linear FEM analysis. The slab is modeled as a flat concrete plate with steel mesh reinforcement. The square-shaped slab has dimensions of 30 m by 30 m, while the rectangular slab is 14 m by 30 m. A constant thickness of 20 cm and the same material properties were adopted for both slabs.

The behavior of a concrete slab on the ground under shrinkage is predominantly related to the friction between the slab and the supporting base [9]. The frictional behavior has to be defined with proper resistance criteria through the stress–slippage friction relationship. The interaction between the slab and the soil is simulated through an interface, i.e., contact plane. In this research, a frictional material of the interface was used with a normal stiffness modulus $k_z=30 \mathrm{N}/{\mathrm{m}\mathrm{m}}^3$, shear stiffness modulus $k_x=k_y=3 \mathrm{N}/{\mathrm{m}\mathrm{m}}^3$, cohesion $C=0.03 \mathrm{N}/{\mathrm{m}\mathrm{m}}^2$, and friction angle $\varnothing =35°$.

2.2. Material Properties

The probabilistic approach of determining the compressive strength of concrete means the generation of the RF, by an appropriate method, based on the chosen distribution that defines the compressive strength. Therefore, the basic compressive strength $f_{co}$ is a random variable, usually with normal or log-normal distribution [10,11]. In our models, we chose the log-normal (LN) distribution because it can better model (compared to other distributions) the concrete strength, which can vary significantly due to factors like mix proportions, curing conditions, and material properties. The precise choice of the LN distribution (defined by its mean value $\mu$ and standard deviation $\sigma$) significantly affects the realistic modeling of the properties of the concrete, as is shown in [11]. Hence, we followed the JCSS-Code to define the material properties of concrete with spatially varying properties. We used the concrete grade class C15 and “Ready mixed” as the concrete type. For this choice, the predefined values for the mean value and standard deviation of basic compressive strength are $\mu =30.52 \mathrm{M}\mathrm{P}\mathrm{a}$ and $\sigma =5.9 \mathrm{M}\mathrm{P}\mathrm{a}$, respectively.

The relations between the Young’s modulus $E_c$, tensile strength $f_{ct}$, and the compressive strength $f_c$ with the basic compressive strength $f_{co}$ are also defined in [8], as

$$\begin{gathered} f_c=\alpha f_{co}^{0.96}, f_{ct}=0.3f_c^{2/3}, \mathrm{and} E_c=10500f_c^{1/3}\left({\displaystyle \frac{1}{1+\beta }}\right), \\ \mathrm{where} \alpha =\beta =0.7. \end{gathered}$$

(1)

The Fast Fourier transformation (FFT) method was chosen as the RF generator since the method performs very well with respect to efficiency and accuracy when compared to other methods, such as the Covariance Matrix Decomposition method or Local Average Subdivision method, and its computation is the fastest [12,13]. With the FFT method, the RF can be defined in the global $xy$ plane, which is well-suited for the analysis of the slabs. The number of grid lines in the $x$ and $y$ directions has to be equal to a power of 2, i.e., $2^m, m\in \left\{\mathrm{0,1},2,\dots \right\}$ [14].

It is important to properly define an RF mesh, which is independent of the finite element mesh (FE-mesh). The grid lines of the RF mesh are defined in the global $x$ and $y$ directions with the distances $L_{RFX}$ and $L_{RFY}$, respectively. The values of $L_{RFX}$ and $L_{RFY}$ depend on the type of correlation function and the correlation length ($L_c$). For modeling spatial variability of concrete properties, the squared exponential correlation function is predominantly used because it provides a smooth transition between correlation points in the structure, allowing a more natural estimation of strength or other properties at different locations [15]. The correlation length quantifies how far apart two points can be before their values can be considered statistically independent. This parameter has a large influence on the RF because the fluctuations in the RF increase as the correlation length decreases. In practice, it is fairly hard to determine the correlation length because the scatter of the experimental data can be high, and the correlation length depends on many factors, such as the aggregate size or shape of the concrete element, etc. [4,15]. The visualization and impact of the correlation length in the RF are discussed in [16]. The proper choice of the value of the correlation length ensures a balanced change in the values of the considered property at two points. Therefore, in our analyses, we used $L_c=5 \mathrm{m}$, as it is suggested by the JCSS-Code for describing properties of concrete. According to [12], for the squared exponential correlation function, $L_{RFX}$ and $L_{RFY}$ should be taken between $L_c/4$ and $L_c/2$. For both analyzed slabs, we chose the same grid line distances $L_{RFX}=L_{RFY}=2 \mathrm{m}$. The chosen correlation length defines the exponent in the number of grid lines as 3 and 4 for the 14 m and 30 m edge lengths of the slab, respectively.

The correlation function will reach the threshold value ($c_1$) as the lag distance (the distance between two points) increases. Therefore, the value of the correlation function belongs to the interval $(c_1,1]$. That means, $c_1$ defines the smallest measure of correlation between two points in the RF. Since the property of concrete from a single batch defines its threshold value, the threshold value varies between batches. Hence, in our research, we use a threshold value $c_1=0.5$ suggested by the JCSS-Code.

The Poisson’s ratio ${\mu }_c=0.2$ and fracture energy in tension $G_F^I=0.143 \mathrm{N}/\mathrm{m}\mathrm{m}$ are independent of spatially varying properties, so they are the same in both models.

In the deterministic approach, the behavior of the concrete was described using a total strain crack-based model with rotating crack orientation. The linear elastic properties of the concrete included Young’s modulus $E_c=17.120 \mathrm{G}\mathrm{P}\mathrm{a}$ and mass density of 2350 kg/m³. The compressive part of the stress–strain curve is described using a parabolic curve with compressive cylinder strength $f_{ck}=21.3 \mathrm{M}\mathrm{P}\mathrm{a}$ and compressive fracture energy $G_c=33.52 \mathrm{N}/\mathrm{m}\mathrm{m}$. The exponential softening curve is used to describe the behavior in tension with the concrete tensile strength $f_{ct}=2.3 \mathrm{M}\mathrm{P}\mathrm{a}$ and fracture energy $G_F^I$. The tensile and compressive fracture energies are determined using the fib Model Code [17].

The above given values of compressive and tensile strength and Young’s modulus are determined using $f_{co}=\mu$ in expressions (1). In this way, the results of the probabilistic approach modeling and the results of the deterministic approach are comparable.

Shrinkage of the concrete is applied to the calculation model as a strain load of ${\epsilon }_x={\epsilon }_y={\epsilon }_z=-5\times {10}^{-4}$ which corresponds to the long-term shrinkage strain determined using the expressions from Eurocode EN1992-1-1 [18]. The shrinkage strains in the transversal directions are ${\epsilon }_{xy}={\epsilon }_{yz}={\epsilon }_{xz}=-1\times {10}^{-8}$.

The top and bottom zones of the slab are reinforced with steel mesh with bars with a 6 mm diameter and 150 mm spacing in two orthogonal directions. The position of the reinforcement mesh is 30 mm from the top and 25 mm from the bottom of the slab. The steel reinforcement is modeled using a non-hardening Von Mises plasticity model with an elasticity modulus of $E_s=200 \mathrm{G}\mathrm{P}\mathrm{a}$ and a yield strength of $f_s=500 \mathrm{M}\mathrm{P}\mathrm{a}$. The spatial variation in the steel’s material properties is very low, and therefore the steel is modeled as a material with constant properties.

3. FEM Modeling of the Slabs

This research is focused on the analysis of cracks that develop during the shrinkage of concrete in slab-on-ground. The calculation of stresses and strains in concrete, taking shrinkage into consideration, was performed using a non-linear FEM based analysis. The developed FEM model included the behavior of the concrete in the elastic and post-cracking stages, the yielding of reinforcement, and the frictional effect between the slab and the ground. A similar FEM model was verified by the optimization study presented in [19].

For the probabilistic approach, in the FEM based software, the material data needs to be specified in the preprocessing stage, where the random fields for the different material parameters must be generated before the analysis stage. The random field’s values are assigned to the model at the element level, and throughout the analysis, values are saved and retrieved from a database as needed.

3.1. FEM Model Description

The commercial finite element program DIANA FEA (version 10.7) was used to model a full-size slab continually supported on the ground. The slab was modeled using a twenty-node isoparametric solid brick element, which can account for both the linear and nonlinear post-cracking behavior of concrete. Based on numerical computations in [19], a mesh with an element size of 500 mm along the width and length of the slab was chosen to achieve a balance between the reliability of results and computational time. The thickness of the slab was modeled by three finite elements. The interface elements between the slab and the ground consist of two parallel planes that can slip or move relative to each other. The lower interface plane’s nodal displacements are fixed to the ground, prohibiting movement in all three directions, whereas the other interface plane is attached to the slab’s bottom side. The frictional behavior between the slab and ground is modeled using the Coulomb friction failure criterion.

The slab’s self-weight was applied during the first load phase, and shrinkage deformations were imposed progressively using a time-step approach. The shrinkage strain load was applied to the model in twenty equal steps with a factor of 0.05. A load factor of 0.8 was used in the analysis of the results. That load factor was chosen because it corresponds to a time of four years, which was the time when 80% of the shrinkage was achieved. The BFGS (secant quasi-Newton) iteration scheme was used in the nonlinear FEA. The maximal number of iterations was set to 150. The energy convergence norm with the value of the norm of 0.001 was used. More details of the used procedure can be found in [19].

3.2. FEM Results

For each slab dimension, the FEM analysis included three calculations with a probabilistic approach (denoted by ${RF}_1^{14}, {RF}_2^{14}, {RF}_3^{14}$ for a 14 × 30 slab and ${RF}_1^{30}, {RF}_2^{30}, {RF}_3^{30}$ for 30 × 30 slab) and one with deterministic (denoted by $D^{14}$ for a 14 × 30 slab and $D^{30}$ for a 30 × 30 slab). Results of probabilistic calculations are independent since they are generated by different random samples. Due to the similar conclusions that are obtained for both slab dimensions, we only present the results of the slab with dimensions of 30 m by 30 m.

The tensile strength of the concrete generated with RF distribution is presented in Figure 1a–c, while Figure 1d shows a constant value of tensile strength used in the deterministic modeling. The same color scale was used to indicate the tensile strength in all graphs in Figure 1. The maximum value (red) is the highest concrete tensile strength from the random analysis, while the minimum value (blue) is the lowest.

Since the basic compressive strength is defined as a random variable and it is related to compressive and tensile strength, and Young’s modulus by (1), the spatial distribution for the compressive and tensile strength, and Young’s modulus, follows a similar pattern as in Figure 1a–c.

When the concrete slab shrinks, every node is displaced to the center of the slab. Tensile strains are developed in the slab as a result of the bedding’s restriction of this displacement. The tensile strains are the highest at the center of the slab. Cracks develop in places where the value of the principal tensile stress is higher than the tensile strength. When an RF is applied, the initial cracks occur at a variable load factor. The principal tensile stress (${\sigma }_c$) in the considered slab, for the load factor 0.8, is presented in Figure 2, to show where the cracks will develop during the shrinkage of the concrete.

Based on the presented results, the RF implies that cracking starts near the middle of the slab, while, in the deterministic approach, the cracking starts in the center of the slab. Comparing the tensile strength of the slab to the stress plot, it can be observed that cracking starts at a weaker point of the slab. The stress patterns in deterministic analysis are symmetric, but in probabilistic analysis are not. Also, in the analysis with the spatial variables, secondary cracks develop in both radial and tangential directions. Figure 5.25 from [20] shows that the distribution of cracks obtained from probabilistic calculations is closer to the actual distribution of cracks that occur during the shrinkage of the concrete slab.

4. Comparison of the Probabilistic and Deterministic Results

The aim of this analysis is to compare the probabilistic and deterministic approaches in the prediction of cracks on slab-on-ground. Therefore, we considered the measure of safety against cracking determined by $SC=f_{ct}- {\sigma }_c$ and calculated it in every node and element from the FEM results. Since the first crack redistributes the stresses, we used results from the load factor 0.2, as it is the last load factor for which slabs remain without cracks for all considered models. Therefore, $SC>0$ for ${RF}_1^{14}$,${RF}_2^{14}$, ${RF}_3^{14}$, $D^{14}$, ${RF}_1^{30}$, ${RF}_2^{30}$, ${RF}_3^{30}$, and $D^{30}$ for load factor 0.2. The graph of $SC$ values for the slab with dimensions of 30 m by 30 m, in the nodes from FEM results, is presented in Figure 3. For easier visualization, the values for $SC$ are grouped in intervals that present the level of the measure of safety against cracking. Values in the orange interval are critical since those values indicate that principal stresses are close to the tensile strength of concrete. The measure of safety against cracking is the lowest at the center of the slab, which is consistent with the tensile stress graphs in Figure 2. Comparing the graphs of Figure 3, we can observe that they are different. It is a question of whether those differences are significantly different. Therefore, we statistically compared the probabilistic results with the deterministic ones, analyzing the mean value of their differences. The set of differences between $SC$ of ${RF}_i^a$ and $SC$ of $D^a$, in every FEM element, is taken as a sample of statistical data and denoted by $S_i^a$, for every $i\in \left\{1,2,3\right\}$ and $a\in \left\{30,14\right\}$. The size of the sample $S_i^{30}$ is $n=\mathrm{14,400}$, while the size of $S_i^{14}$ is $n=6720$. It is reasonable to assume the values in one sample are independent of each other. Also, the considered samples are mutually independent. The histograms of the considered data, presented in Figure 4, show that the samples have approximately normal distributions and do not have outliers. Hence, we can use a one-sample t-test for the mean to test the hypotheses:

H₀: 

A mean is equal to zero. H_A: A mean is not equal to zero.

Applying the one-sample t-test for the mean of $S_1^{30}$ yielded the p-value of 0. The same result is obtained for $S_1^{14}$. Those results imply the rejection of the null hypothesis as results for both slabs (samples $S_1^{30}$ and $S_1^{14}$) are statistically significant. Hence, we can say that the results of probabilistic and deterministic analysis statistically differ. Comparing the distribution of $SC$ intervals from probabilistic results—Figure 3a—with the $SC$ intervals from deterministic results—Figure 3d—the differences are clearly visible. Similar conclusions can be drawn for the other two presented probabilistic results.

In order to avoid mistakenly rejecting a true null hypothesis, we also checked the results for $S_2^{30}$, $S_3^{30}$, $S_2^{14}$, and $S_3^{14}$. All tests yielded a p-value of 0. The details of values obtained in testing and $99\%$ confidence intervals for means of the considered samples can be seen in

Table 1. The sample statistic.

Sample	n	Mean	Standard Deviation	${\mathit{t}}_{\mathit{n}-1}$	p-Value	99% Confidence Interval
$S_1^{30}$	14,400	−0.2553	0.1417	−216.18	0.0000	(−0.2583, −0.2522)
$S_2^{30}$	14,400	−0.1380	0.1589	−104.29	0.0000	(−0.1415, −0.1346)
$S_3^{30}$	14,400	0.2697	0.1768	183.05	0.0000	(0.2659, 0.2735)
$S_1^{14}$	6720	−0.1737	0.0887	−160.54	0.0000	(−0.1765, −0.1709)
$S_2^{14}$	6720	−0.0280	0.1695	−13.52	0.0000	(−0.033, −0.0226)
$S_3^{14}$	6720	0.1956	0.1530	104.79	0.0000	(0.1908, 0.2004)

, and details of the calculations in [21]. Notice that none of the $99\%$ confidence intervals contain 0.

5. Conclusions

We presented the results of the calculation of stresses and strains in concrete in slab-on-ground (square- and rectangular-shaped slab), considering shrinkage of concrete using non-linear FEM based analysis, with and without the spatially varying material properties of concrete. The analysis of the results of three FEM calculations with a probabilistic approach and one with a deterministic approach has given insight into the statistically significant influence of spatially varying material properties compared to the deterministic approach. The illustrations of the principal tensile stresses for the shrinkage concrete show that the probabilistic modeling gives a more realistic representation of the cracking process since the crack initiation does not occur symmetrically at the structural center of the slab, as given by the deterministic modeling, but rather at locations characterized by reduced tensile strength. The statistical comparison of the results of probabilistic and deterministic approaches, through the measure of safety against cracking before the first cracks in the slab appear, indicates that the results are statistically significantly different. The choice of parameters for the probabilistic approach is complex due to the sensitivity of the properties of the concrete to the variation in parameters. The presented results are limited to a specified choice of parameters. These pivot results suggest further considerations and analysis of the application of probabilistic modeling of concrete properties.