Thermal Expansion Behavior of Prefabricated Box Culverts and Its Impact on Structural Stability

With the rapid advancement of infrastructure development, prefabricated box culverts, as a novel type of precast concrete structure, have been widely applied in urban drainage systems, underground passages, and other engineering projects, owing to their high construction efficiency and easily controllable quality [1-5]. However, in practical engineering applications, these culvert structures often face complex thermal environments. The thermal expansion behavior, in particular, plays a decisive role in the integrity and stability of the structures [6-8]. Therefore, an in-depth study of the thermal expansion behavior of prefabricated box culverts is of significant importance to ensure their reliability and safety under various environmental conditions.

Although existing research has explored the thermal expansion properties of precast concrete components, there is a scarcity of literature focused specifically on prefabricated box culverts [9-11]. The interaction between the temperature and stress fields in these culverts is complex, with general temperature changes potentially causing uneven thermal expansion, thereby affecting structural stability [12-15]. Consequently, research on the thermal expansion behavior of prefabricated box culverts and its impact on structural stability is both urgent and crucial.

Existing studies have primarily focused on the thermal expansion effects under single conditions or have used simplified models for analysis [16-18], largely overlooking the complex three-dimensional stress states that prefabricated box culverts experience in actual applications and the influence of such states on thermal expansion behavior [19-22]. Moreover, current research methods often lack precision in predicting and validating thermal expansion coefficients, which, to a certain extent, limits the application of theoretical research findings in engineering practice.

This article systematically tests and validates the thermal expansion behavior of prefabricated box culverts. Initially, the triaxial linear thermal expansion strain, average linear thermal expansion coefficient, and thermal expansion stress of the culverts under three-dimensional stress states are determined experimentally. The experimental results are then compared and analyzed against theoretical predictions to validate the accuracy of the thermal expansion coefficients under three-dimensional stress conditions. Subsequently, a temperature-stress coupling model is constructed to analyze in-depth the variation of thermal stress in prefabricated box culverts under different temperature conditions and its impact on structural stability. This research not only extends the design theory of prefabricated box culverts but also provides a reference for the safety analysis of other precast concrete structures in complex environments, possessing significant academic value and practical significance.

To accurately describe and calculate the thermal stress distribution of prefabricated box culvert specimens under varying temperature conditions and effectively assess the structure's performance under thermal loads, this paper has developed a temperature-stress coupling model based on thermoelasticity theory. By analyzing the patterns of thermal stress changes under different fire scenarios, weak points in the structure can be identified, enabling targeted reinforcement measures or design improvements to enhance the structure's fire resistance and overall stability. Figure 4 presents the schematic diagram of the temperature-stress testing system for prefabricated box culvert specimens.

4.png

Figure 4. Temperature-stress testing system for prefabricated box culvert specimens

This paper employs the principle of superposition to decompose the temperature-stress coupling analysis problem of prefabricated box culverts into two parts: external loads and temperature loads. External loads typically include gravity, traffic load, earth pressure, etc., while temperature loads involve thermal expansion or contraction due to temperature changes. This decomposition method simplifies the solution process of complex problems and helps to clearly understand the individual impacts of different types of loads on the stress response of prefabricated box culvert structures.

The paper first establishes the basic physical equations of thermoelasticity for temperature-stress coupling analysis of prefabricated box culverts. It begins by defining the displacement changes in each direction of a microelement under a three-dimensional stress state. These displacement changes, compared to the size of the microelement, yield strain components in three orthogonal directions. Based on the geometrical relationships of the microelement, linear and shear strain components in each direction can be further calculated. If the temperature changes from s₁ to s₂, with the temperature change of the microelement being s=s₁-s₂, and the thermal expansion coefficient represented by β, then the length changes of da, db, dc are (1+βs)da, (1+βs)db, (1+βs)dc, respectively. The expressions for strain components are:

$\left\{ \begin{align}  & {{\gamma }_{a0}}={{\gamma }_{b0}}={{\gamma }_{c0}}=\beta s \\ & {{\varepsilon }_{ab0}}={{\varepsilon }_{bc0}}={{\varepsilon }_{ca0}}=0 \\\end{align} \right.$                   (7)

Utilizing Hooke's Law, the linear and shear strains obtained in the previous step are related to the corresponding stress components. Hooke's Law provides a linear relationship between stress and strain in elastic materials, used to calculate the stress components generated inside a microelement under a given external load. Assuming the shear modulus is represented by H, the elastic modulus by R, Poisson's ratio by ω, and the volumetric stress by Φ, the calculation formulas are:

$\left\{ \begin{align}  & {{\gamma }_{a}}=\frac{\partial i}{\partial a}=\frac{1}{R}\left[ {{\delta }_{a}}-\omega \left( {{\delta }_{b}}+{{\delta }_{c}} \right) \right]+\beta s \\ & {{\gamma }_{b}}=\frac{\partial i}{\partial b}=\frac{1}{R}\left[ {{\delta }_{b}}-\omega \left( {{\delta }_{a}}+{{\delta }_{c}} \right) \right]+\beta s \\ & {{\gamma }_{c}}=\frac{\partial i}{\partial c}=\frac{1}{R}\left[ {{\delta }_{c}}-\omega \left( {{\delta }_{b}}+{{\delta }_{a}} \right) \right]+\beta s \\\end{align} \right.$               (8)

${{\varepsilon }_{ab}}=\frac{{{\pi }_{ab}}}{H},{{\varepsilon }_{bc}}=\frac{{{\pi }_{bc}}}{H},{{\varepsilon }_{ca}}=\frac{{{\pi }_{ca}}}{H}$                      (9)

$H={R}/{2\left( 1+\omega  \right)}\;$                        (10)

$\Phi ={{\delta }_{a}}+{{\delta }_{b}}+{{\delta }_{c}}$                        (11)

Under the influence of temperature, the generalized Hooke's Law, which incorporates temperature effects, was applied. This law not only relates stress to mechanical strain but also includes thermal strain caused by temperature changes. Thermal strain is typically represented by the product of temperature change and the material's linear thermal expansion coefficient. Through this step, an expression of strain that includes temperature-stress coupling effects is obtained, namely:

$\left\{ \begin{align}  & {{\delta }_{a}}=2H{{\gamma }_{a}}+\frac{\omega }{1+\omega }\Phi -2H\beta ds \\ & {{\delta }_{b}}=2H{{\gamma }_{b}}+\frac{\omega }{1+\omega }\Phi -2H\beta s \\ & {{\delta }_{c}}=2H{{\gamma }_{c}}+\frac{\omega }{1+\omega }\Phi -2H\beta s \\\end{align} \right.$                           (12)

${{\pi }_{ab}}=H{{\varepsilon }_{ab}},{{\pi }_{bc}}=H{{\varepsilon }_{bc}},{{\pi }_{ca}}=H{{\varepsilon }_{ca}}$                         (13)

Finally, the volumetric strain caused by temperature changes needs to be considered. Volumetric strain results from the combined effect of linear thermal expansion in three orthogonal directions. By adding up the linear thermal expansion strains in the three directions, the volumetric strain can be determined. In the temperature-stress coupling analysis of prefabricated box culverts, volumetric strain is an important parameter for assessing the overall stability of the structure under thermal loads, namely:

$\begin{align}  & r={{\gamma }_{a}}+{{\gamma }_{b}}+{{\gamma }_{c}}=\frac{1-2\omega }{1+\omega }\frac{\Phi }{2H} \\ & +3\beta s=\frac{1-2\omega }{R}\Phi +3\beta s \\\end{align}$                  (14)

Consequently, there is:

$\begin{align}  & r={{\gamma }_{a}}+{{\gamma }_{b}}+{{\gamma }_{c}}=\frac{1-2\omega }{1+\omega }\frac{\Phi }{2H} \\ & +3\beta s=\frac{1-2\omega }{R}\Phi +3\beta s \\\end{align}$                   (15)

Assuming the thermal stress parameter is represented by α and the Lamé parameter by η, the relationship using Lamé's constants in elasticity theory is given as:

$\begin{aligned} & \left\{\begin{array}{l}\delta_a=2 H \gamma_a+\eta r-\alpha_s \\ \delta_b=2 H \gamma_b+\eta r-\alpha_s \\ \delta_c=2 H \gamma_c+\eta r-\alpha_s\end{array}\right. \\ & \alpha=\frac{\beta R}{1-2 \omega}=\beta(3 \eta+2 H) \\ & \eta=\frac{R \omega}{(1+\omega)(1-2 \omega)} \\ & \end{aligned}$                    (16)

Further, based on continuum mechanics and thermoelastic theory, the equilibrium differential equations are derived. These equations describe how mechanical balance and energy balance within materials are expressed at a microscopic level through the interrelationships between stress, strain, and temperature. These equations combine Newton's second law and Hooke's law to describe the stress distribution inside a microelement. Assuming the components of body force on each axis are represented by A, B, and C, the equations are:

$\left\{ \begin{align}  & \frac{\partial {{\delta }_{a}}}{\partial a}+\frac{\partial {{\pi }_{ba}}}{\partial b}+\frac{\partial {{\pi }_{ca}}}{\partial c}+A=0 \\ & \frac{\partial {{\delta }_{b}}}{\partial b}+\frac{\partial {{\pi }_{cb}}}{\partial c}+\frac{\partial {{\pi }_{ab}}}{\partial a}+B=0 \\ & \frac{\partial {{\delta }_{c}}}{\partial c}+\frac{\partial {{\pi }_{ac}}}{\partial a}+\frac{\partial {{\pi }_{bc}}}{\partial a}+C=0 \\\end{align} \right.$                      (17)

By substituting the above constitutive relations into the equilibrium differential equations, the displacement components of the prefabricated box culvert structure under specific temperature and stress conditions can be solved. These displacement components typically appear in the form of partial differential equations and can be solved using numerical methods such as finite element analysis. The expressions for displacement components directly link the structure's deformation with internal stress, external load, and temperature change, providing an important tool for assessing the behavior of structures under actual operating conditions. The expressions are as follows:

$\left\{ \begin{align}  & \left( \eta +H \right)\frac{\partial \gamma }{\partial a}+H{{\nabla }^{2}}i-\alpha \frac{\partial s}{\partial a}+A=0 \\ & \left( \eta +H \right)\frac{\partial \gamma }{\partial b}+H{{\nabla }^{2}}i-\alpha \frac{\partial s}{\partial b}+B=0 \\ & \left( \eta +H \right)\frac{\partial \gamma }{\partial c}+H{{\nabla }^{2}}i-\alpha \frac{\partial s}{\partial c}+C=0 \\\end{align} \right.$                          (18)

${{\nabla }^{2}}=\frac{{{\partial }^{2}}}{\partial {{a}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{b}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{c}^{2}}}$                       (19)

Thermoelastic geometric equations can clearly describe the deformation of box culvert structures under the influence of temperature changes and stress. To construct these geometric equations, the basic relationship between strain and displacement within the framework of thermoelasticity must first be established. This usually begins with the definition of strain components, i.e., strain is a function of the gradient of the displacement field. For prefabricated box culverts, this means considering the displacement of each point of the culvert under the influence of temperature and external loads. Subsequently, to account for temperature effects, the thermal expansion coefficient and how the temperature field affects the displacement of material points are introduced. The product of the thermal expansion coefficient and temperature change represents the thermal strain component caused by temperature change, which must be combined with the strain component caused by mechanical forces to obtain the total strain component. Finally, the strain-displacement relationship and compatibility conditions are combined to form a complete geometric equation, expressed as follows:

$\begin{align}  & \left\{ \gamma  \right\}={{\left\{ {{\gamma }_{a}},{{\gamma }_{b}},{{\gamma }_{c}},{{\varepsilon }_{ab}},{{\varepsilon }_{bc}},{{\varepsilon }_{ca}} \right\}}^{S}} \\ & ={{\left\{ \frac{\partial i}{\partial a},\frac{\partial i}{\partial b},\frac{\partial i}{\partial b},\frac{\partial n}{\partial a}+\frac{\partial i}{\partial b},\frac{\partial q}{\partial b}+\frac{\partial n}{\partial c},\frac{\partial i}{\partial c}+\frac{\partial q}{\partial a} \right\}}^{S}} \\\end{align}$                (20)

To ensure the continuity and consistency of internal displacement distribution in box culvert structures under the combined action of temperature changes and external loads, this study constructs the displacement coordination equations of thermoelasticity. Initially, considering thermal strains caused by temperature, displacement vectors are taken into account, typically achieved by combining temperature gradients with the material's thermal expansion coefficient. Thermal strain directly contributes to displacement components and thus must be reflected in the displacement coordination equations. This combines thermal strain with elastic strain caused by external loads on the prefabricated box culverts. To ensure that the material does not undergo physically impossible deformations, compatibility conditions are used to establish coordinated relationships between various displacement components. These conditions ensure the continuity of the displacement field, allowing the displacement vector to naturally transition along different directions. Integrating all the above considerations forms the displacement coordination equation, a set of partial differential equations that describe the interrelationships between displacement components. These equations often involve second-order derivatives of displacement to ensure the continuity of strain, expressed as:

$\left\{ \begin{align}  & \frac{{{\partial }^{2}}{{\gamma }_{a}}}{\partial {{b}^{2}}}+\frac{{{\partial }^{2}}{{\gamma }_{b}}}{\partial {{a}^{2}}}=\frac{{{\partial }^{2}}{{\varepsilon }_{ab}}}{\partial a\partial b} \\ & \frac{{{\partial }^{2}}{{\gamma }_{b}}}{\partial {{c}^{2}}}+\frac{{{\partial }^{2}}{{\gamma }_{z}}}{\partial {{b}^{2}}}=\frac{{{\partial }^{2}}{{\varepsilon }_{bc}}}{\partial b\partial c} \\ & \frac{{{\partial }^{2}}{{\gamma }_{z}}}{\partial {{c}^{2}}}+\frac{{{\partial }^{2}}{{\gamma }_{x}}}{\partial {{b}^{2}}}=\frac{{{\partial }^{2}}{{\varepsilon }_{ca}}}{\partial c\partial a} \\ & \frac{\partial }{\partial z}\left( \frac{\partial {{\varepsilon }_{ca}}}{\partial b}+\frac{\partial {{\varepsilon }_{ab}}}{\partial c}+\frac{\partial {{\varepsilon }_{bc}}}{\partial a} \right)=2\frac{{{\partial }^{2}}{{\gamma }_{a}}}{\partial b\partial c} \\ & \frac{\partial }{\partial b}\left( \frac{\partial {{\varepsilon }_{ab}}}{\partial c}+\frac{\partial {{\varepsilon }_{bc}}}{\partial a}+\frac{\partial {{\varepsilon }_{ca}}}{\partial b} \right)=2\frac{{{\partial }^{2}}{{\gamma }_{b}}}{\partial c\partial a} \\ & \frac{\partial }{\partial c}\left( \frac{\partial {{\varepsilon }_{bc}}}{\partial a}+\frac{\partial {{\varepsilon }_{ca}}}{\partial b}+\frac{\partial {{\varepsilon }_{ab}}}{\partial c} \right)=2\frac{{{\partial }^{2}}{{\gamma }_{c}}}{\partial a\partial b} \\\end{align} \right.$                  (21)

The solution of the temperature-stress coupling model for prefabricated box culverts involves setting a series of physically reasonable boundary conditions to simulate the behavior of culverts in actual engineering. These boundary conditions typically include: 1) Mechanical boundary conditions: These define the stress or displacement distribution of the culverts under the action of external loads. 2) Thermal boundary conditions: These describe the behavior of culverts in thermal environments, such as exposure to different temperatures or fire impact. 3) Coupled boundary conditions: Considering temperature-stress coupling effects, boundary conditions that involve both temperature and stress impacts are needed. 4) Initial conditions: These significantly impact the solution process, especially in dynamic analysis or simulating the time response of culverts. Assuming the surface stress components of the boundary microelement are represented by $\bar{A}, \bar{B}, \bar{C}$ and the normal direction cosines are 1, l, v, the expressions for stress boundary conditions are:

$\left\{ \begin{align}  & \bar{A}={{\delta }_{a}}m+{{\pi }_{ba}}l+{{\pi }_{ca}}v \\ & \bar{B}={{\delta }_{b}}l+{{\pi }_{cb}}v+{{\pi }_{ab}}m \\ & \bar{C}={{\delta }_{z}}v+{{\pi }_{ac}}m+{{\pi }_{bc}}l \\\end{align} \right.$                      (22)

Finite element analysis is used to solve the coupled model, obtaining the culvert structure response under various temperature and stress conditions, including temperature distribution, displacement field, and stress field. Particularly, attention is paid to the locations of maximum stress and strain in the structure and the trends of these values with temperature changes. Next, it's analyzed whether these stresses and strains exceed the permissible limits of the material, especially considering material performance degradation at high temperatures. Additionally, the expansion or contraction caused by temperature leading to functional defects, such as crack opening or closure or failure of connectors, is evaluated.

Combining the actual usage conditions of prefabricated box culverts, such as load history, construction quality, and material aging, the potential performance of the structure in real working environments is assessed. For extreme events like dynamic loading or fire, the structure's stability under rapid temperature changes in a short period is specifically considered. Further, through sensitivity analysis and parameter studies of the solution results, key factors affecting the stability of box culvert structures, such as support conditions, material thermal expansion coefficients, and thermal conductivity, need to be identified. With this information, potential issues under different working conditions can be predicted, and mitigation measures can be formulated.

Finally, based on the solution results, the safety margin of the structure can be assessed, design optimization can be carried out, or recommendations for reinforcement or repair can be made. If the model predicts stability risks under certain conditions, the structural safety performance can be improved by adjusting design parameters or changing construction processes.