# Prediction of Thermal Cracking During Construction of Massive Monolithic Structures
## 2 Materials and Methods  
The object of the study in this article is a fragment of a dock wall with a length of 21,980 mm. The cross-section of the calculated structure is shown in Figure 1.  
Concreting of this structure was performed at average daily ambient temperature \(T_{\infty}=10\)\({}^{\circ}\)C. The initial temperature of the concrete mix was 15 \({}^{\circ}\)C. The concrete had a class B22.5 in compressive strength according to Russian standards. An inspection of the block upon completion of the concrete work revealed the presence of numerous cracks, the location of which is shown in Figure 2. Most cracks start at the bottom of the wall and extend to the top. The cracks are vertical, meaning they are caused by normal stresses acting along the longitudinal axis of the wall. To analyze the causes of early crack formation, temperature fields and stresses were calculated using a program developed by the authors in the MATLAB R2024b environment,  
Figure 1: Cross-section of the calculated structure (dimensions are given in mm). Figure 2: Location of cracks. based on the finite element method. The analysis was performed in a three-dimensional formulation. Due to symmetry, half of the structure was considered. The structure was divided into 6-node volumetric prismatic finite elements (Figure 3). At the first stage, a mesh of triangles in the cross-section was generated using the PDE Toolbox package, and then three-dimensional bodies were formed by extrusion (Figure 4). The reinforcement of the structure was not taken into account when analyzing temperature stresses, since in a massive structure the percentage of reinforcement is usually small,  
Figure 4: Finite element mesh. Figure 3: Finite elements used. and the reinforcement effect occurs mainly after cracking. Also, taking into account the reinforcement leads to a strong complication of the calculation model. The temperature field was determined by solving the differential equation of heat conduction, which for a three-dimensional problem has the following form [21]:  
\[\lambda\left(\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y ^{2}}+\frac{\partial^{2}T}{\partial z^{2}}\right)+W=\rho c\frac{\partial T}{ \partial t}, \tag{1}\]  
where \(\lambda\) is the thermal conductivity coefficient, \(T\) is the temperature, \(\rho\) is the density, \(c\) is the specific heat capacity, \(t\) is the time, \(W\) is the power of internal heat sources (\(W/\text{m}^{3}\)), \(W=\frac{\partial Q}{\partial t}\). In determining the temperature field, two types of boundary conditions were used. In the presence of convective heat exchange on any of the surfaces, the boundary conditions on it are written as:  
\[\lambda\frac{\partial T}{\partial n}+h(T-T_{\infty})=0, \tag{2}\]  
where \(\frac{\partial(...)}{\partial n}\) is the derivative along the normal to the surface, \(h\) is the heat transfer coefficient, \(T_{\infty}\) is the ambient temperature. At points of the foundation made of old concrete, very far from the block of fresh concrete, the temperature is assumed to be constant and specified as:  
\[T(t)=T_{0}. \tag{3}\]  
With finite element discretization, the differential Equation (1) is reduced to a system of linear differential equations of the following form [22]:  
\[[C]\frac{\partial T}{\partial t}+[K]\{T\}+\{F\}=0, \tag{4}\]  
where \([C]\) is the damping matrix, \([K]\) is the thermal conductivity matrix, \(\{F\}\) is the load vector. Matrices \([K]\), \([C]\) and the vector \(\{F\}\) for individual finite elements were determined by the following formulas:  
\[ \begin{gathered}\left[C^{(e)}\right]=\rho c\int\limits_{V}{[N]^{T }[N]dV};\\ \left[K^{(e)}\right]=\int\limits_{V}{[B]^{T}[D][B]dV}+\int\limits_{S}{h[N]^ {T}[N]dS};\\ \left\{F^{(e)}\right\}=-\int\limits_{V}{W[N]^{T}dV}-\int\limits_{V}{ ht_{\infty}[N]^{T}dS},\end{gathered}  \tag{5}\]  
where \([N]\) is the matrix of shape functions, \([B]\) is the matrix of gradients of shape functions, \([D]= \begin{bmatrix}\lambda&0&0\\ 0&\lambda&0\\ 0&0&\lambda\end{bmatrix}