Effect of Transient Creep on the Structural Performance of Reinforced Concrete Walls under Fire

Abstract

This paper investigates and reveals the effect of the high-temperature transient creep on the structural performance of RC walls under fire. A theoretical model is established, which explicitly includes the transient creep and accounts for the explosive spalling, the material, and geometric nonlinearities under fire. The effects of the transient creep on the structural response and fire resistance of RC walls with little spalling and with explosive spalling are investigated, respectively, with elucidation of the mechanisms. The influences of wall geometries, concrete properties, and the eccentricity of load on the effect of the transient creep are quantitatively studied. Finally, the results are validated through comparison with tests in the literature. It is revealed that the transient creep significantly reduces the fire resistance of RC walls with little spalling by up to and greater than 60%, by decreasing the deflection toward the heated side. However, it increases the fire resistance of RC walls with explosive spalling by up to about 40% by reducing the spalling extent due to the stress relaxation effect. The stress relaxation effect of the transient creep has a crucial role in determining the spalling manner. The load level, the eccentricity of load, and the wall geometries are key influencing factors which have contrary influences on the effect of the transient creep on the fire resistance of RC walls with little spalling and with explosive spalling.

1. Introduction

Reinforced concrete (RC) walls are widely used as load-bearing structural members in medium-rise and high-rise buildings. The performance of RC walls under fire is crucial for structural stability. Under fire conditions, RC walls are usually exposed to fire on one side. The mechanical properties of concrete and steel bars are degraded by high temperature [1,2], which can remarkably reduce the load-bearing capacity of RC walls. Due to the high thermal mass of concrete, the temperature-dependent mechanical properties of steel and concrete are degraded unevenly along the thickness of the RC walls, which changes the eccentricity of the load and increases the probability of out-of-plane instability. Furthermore, the explosive spalling [3] of RC walls made with high-strength concrete can damage and lead to failure of the walls under fire.

Under high-temperature exposure, concrete undergoes large strains including thermal and stress-related strains. Specifically, different from other structural materials, concrete has a unique ‘transient creep’ which generates under first-time heating [4]. It mainly results from the rapid collapse of the concretes microstructure due to loss and the movement of water within concrete [5]. The dependence of the transient creep on the temperature and stress histories can have a profound influence on the fire-resistant properties of RC walls. In addition, understanding the effect of transient creep on the structural performance of RC walls is also important for situations where concrete has experienced an elevated temperature cycle before fire.

Several studies have reported on the effects of transient creep on RC members under fire. Sadaoui and Khennane [6] studied the influence of transient creep on the behavior of RC columns in fire by finite element modeling. Transient creep was modeled using two different approaches by considering it explicitly as an additional strain component or taking it implicitly in the constitutive relations of concrete. The results revealed that accounting transient creep explicitly can provide an insight into the structural behavior of the RC columns. The transient creep causes additional compressive stresses in the RC columns, which magnifies the moments and results of an abrupt decrease in the stiffness that is important for the rapid failure of RC columns in fire. Kodur and Alogla [7] studied the impact of transient creep on the performance of fire-exposed RC columns by three-dimensional finite-element modeling using ABAQUS. The transient creep was explicitly considered. The model also accounted for the mechanical degradations of concrete and reinforcement, and geometrical and material nonlinearities. The results indicated that transient creep significantly influenced the structural deformations when the temperatures in concrete were above 500 °C. Neglecting the transient creep can result in an underestimation of the structural deformations and a non-conservative fire resistance estimation of RC columns. However, these previous studies are limited to the RC columns exposed to four-sided fire and the explosive spalling of high-strength concrete under fire are not accounted for in the models. Furthermore, currently there are few studies on the effects of transient creep on RC walls under fire.

In this study, the influences of transient creep on the performance of RC walls under fire are investigated and revealed by a theoretical model which explicitly considers the transient creep according to the stress and temperature histories. The model takes into account the cracking, strain softening and tension stiffening of concrete, the yielding of steel reinforcement at elevated temperatures and geometric nonlinearity and is capable of predicting the explosive spalling, the displacement, and failure of RC walls in fire. The prediction of explosive spalling and the solving of structural models are alternately processed, and the structural model is solved by means of a nonlinear shooting method. The numerical examples are presented first, followed by model validation and parametric studies.

2. Modeling

The effect of transient creep on the structural performance of RC walls under fire is studied based on a theoretical model established by Chen et al. [8], but with further consideration of explosive spalling of high-strength concrete. The model includes a heat transfer analysis through finite element approximation, and a structural model which considers the cracking, strain softening and tension stiffening, the yielding of reinforcement at elevated temperatures, and geometric nonlinearity. The transient creep is explicitly considered and obtained by the classical model of Anderberg and Thelandersson [9]:

$$\left\{\begin{array}{l}{\epsilon }_{tr,c}=k_{tr}\left(\frac{{\sigma }_c}{f_c}\right){\epsilon }_{th,c} T\le 550 °\mathrm{C}\\ \frac{\partial {\epsilon }_{tr,c}}{\partial T}=0.0001\frac{{\sigma }_c}{f_c} T>550 °\mathrm{C}\end{array}\right.$$

(1)

$${\epsilon }_{cr,c}=0.53\times {10}^{-3}\frac{{\sigma }_c}{f_{c,T}}{\left(\frac{t}{t_r}\right)}^{0.5}\times e^{0.00304(T-20)}$$

(2)

where ${\epsilon }_{tr,c}$ is the transient strain, ${\epsilon }_{cr,c}$ is the creep strain, ${\epsilon }_{th,c}$ is the thermal strain, ${\sigma }_c$ is the thermal-mechanical stress, f_c and f_c,T are the room-temperature and temperature-dependent compressive strengths of concrete, respectively, k_tr is a dimensionless constant depending on aggregate type, t is the fire exposure time, and t_r = 10,080 s. The transient creep of each element is incremented at each time step. The increment of the thermal strain is calculated using the increment of temperature of each element within the current time step and the thermal–mechanical stress of each element at the end of the previous time step is saved and used.

Due to the possible influence of explosive spalling, the heat flux through the length and width of the wall also needs to be considered. Thus, the governing equation for the time-dependent temperature distribution inside the wall is revised as:

$${\lambda }_c{\nabla }^{2}T+Q=\rho C\frac{\partial T}{\partial t}$$

(3)

where ${\lambda }_c$, $\rho$, $C$ are the thermal conductivity, the density and the specific heat of concrete, respectively, $Q$ is the heat source due to phase changes in water, $T$ is temperature and $t$ is time. The determination of these above parameters at high temperatures and the coefficients of the boundary conditions can refer to the study by Chen et al. [8,10].

The explosive spalling is assumed to result from the thermal–mechanical stress in addition to stress due to vapor pressure. Thus, it occurs if:

$$F(f_{c,T},f_{t,T},{\sigma }_c,f_q)>1$$

(4)

where F is the failure function determined by the Menetrey–Willam failure criterion [11], $f_{c,T}$ and $f_{t,T}$ are the compressive and tensile strengths of concrete at elevated temperatures, and $f_q$ is the hygro-thermal stress resulting from vapor pressure and computed by:

$$f_q=\varphi (T)\cdot P_V$$

(5)

where $\varphi (T)$ is the porosity of concrete at elevated temperatures [12] and $P_V$ is the vapor pressure that can be predicted based on the principles of thermodynamics. In this study, the governing equations derived by Dwaikat and Kodur [13] are adopted for the prediction of the vapor pressure with a further consideration of the variation in the permeability of concrete due to the varying temperature, pressure, and the development of cracking during fire exposure following the approach proposed by Kodur and Banerji [14].

The solution procedure of the model is by discretizing the wall into two-dimensional finite elements, followed by a time-incrementing analysis including the heat transfer analysis by Equation (3), the pore vapor pressure and spalling analysis by Equations (4) and (5), and solving the structural model with iterations at each time step. A nonlinear shooting method [15] is used in solving the governing equations of the structural model through in-house coding by Matlab 7.0 software. If explosive spalling occurs at any time step, the spalled elements are removed, and the locations of the heated boundaries are renewed before the next time step. Failure of the wall is identified once the axial shortening and the deflection increase in the wall become acute and infinite, which indicates a loss in the load-bearing capacity.

3. Numerical Example

The RC wall depicted in Figure 1 is studied. The wall is under an eccentrically axial load P (with an eccentricity e) and is simply supported at the ends, as can be seen in Figure 1. The wall is subjected to a one-sided ISO-834 standard fire [16] at the inner face. The cross section of the wall is demonstrated in Figure 1b. A single layer of steel reinforcement is located at the middle depth of the wall. The concrete is made with calcareous aggregates. The room-temperature Young’s modulus of the steel is 200 GPa. The variations in the specific heat, thermal conductivity, and density at elevated temperatures are calculated according to Eurocode 2 [2]. The Young’s moduli and strengths of steel and concrete at elevated temperatures are determined by the tabulated data in Eurocode 2 [2]. The wall shown in Figure 1 is investigated for the cases with little spalling and with explosive spalling, respectively, where the room-temperature compressive strengths (tensile strengths) of concrete are 30 MPa (2.4 MPa) and 80 MPa (6.4 MPa), respectively. The ultimate axial loads of the wall are P_n = 1691.9 kN and 4511.7 kN for the investigated two cases, respectively, according to ACI 318-11 [17]. The porosity and the intrinsic permeability of the concrete with compressive strength of 80 MPa at room temperature are 7.5% and $1.0\times {10}^{-19}$ m², respectively. The initial relative humidity is assumed to be 80%.

3.1. Effect on a RC Wall with Little Spalling

The effect of the transient creep on the structural performance of the RC wall shown in Figure 1 with the concrete compressive strength of 30 MPa is investigated. The wall is assumed to have little spalling under fire. Figure 2 shows the influence of transient creep on the structural deformations of the investigated wall without a load under fire. As shown in Figure 2, the transient creep has a notable effect on the structural deformations even without a load. The transient creep would first decrease and then increase the deflection of the wall, and it decreases the axial elongation of the wall with the increasing fire exposure time. The mechanism can be understood by looking at the influence of transient creep on the strain and stress distributions through the depth of the wall (Figure 3). As shown in Figure 3, resulting from the sharply nonlinear thermal strain distribution (Figure 3b), remarkable self-equilibrating stresses occur within the section, with compressive stresses generated on the heated and unheated sides, and tensile stresses formed in the middle portion of the section. Thus, at the beginning of fire exposure, the transient creep strain would decrease the total strain on the heated side (see the comparison at t = 60 min in Figure 3a) and result in a decrease in the deflection of the wall. As fire exposure time increases, the temperature increases notably on the unheated side. The thermal strain and the transient creep strain increase in the unheated side (see the strain distributions at t = 120 min in Figure 3b,d), while the stress becomes tension, and the transient creep strain decreases in the heated side (see Figure 3c,d). Thus, the deflection of the wall would increase by the development of the transient creep strain with the increasing fire exposure time. Nevertheless, both the transient creep strains developed on the heated and unheated sides would counteract the thermal strain and decrease the axial elongation of the wall. It is noted that the transient creep can reach up to several thousand micro strains even without load (Figure 3d), which can result in a significant stress relaxation (Figure 3e).

Figure 4 shows the influence of transient creep on the structural deformations of the studied RC wall under eccentric loads of various load levels (P = 10–50% P_n, e = h/6) under fire. It can be seen from Figure 4 that the transient creep significantly shortens the structural failure time of the wall under fire by 15.8%, 57.1%, and 67.4%, under P = 10%, 30%, and 50% P_n, respectively. The structural failure time is determined as the critical time when the axial shortening and the deflection of the wall become acute and infinite (Figure 4), which indicates a buckling response of the wall. Under higher load levels (P = 30–50% P_n), the wall first develops a deflection towards the heated face due to thermal bowing but shifts to bending and buckling towards the unheated face mainly due to the moving of the neutral axis towards the unheated face. Thus, the significant reduction in the deflection of the wall towards the heated face resulting from the transient creep developed on the heated side would lead to much earlier buckling of the wall under these load levels (Figure 4a). However, under a lower load level (P = 10% P_n), the bending towards the unheated face is remarkably reduced and the wall fails much later by buckling towards the heated face (Figure 4a). Thus, the transient creep develops on the unheated side, increases the deflection towards the heated face, and thus, also shortens the failure time of the wall under fire.

3.2. Effect on a RC Wall with Explosive Spalling

The influence of transient creep on the structural performance of the RC wall shown in Figure 1 with the concrete compressive strength of 80 MPa is also investigated. Figure 5 demonstrates the predicted temperature gradients and pore pressure distributions through the thickness of the investigated wall without load at 10 min and 30 min fire times. It can be seen from Figure 5b that notable pore pressure formed on the heated side, resulting from rapid evaporation and dehydration of water due to the significant increase in temperature. The pore pressure increases rapidly in depth until it reaches the maximum at a certain distance inward from the heated face, and then it decreases. This is because the migration of water vapor inward to the lower-temperature region is impeded by a complete saturated layer/front formed by condensation, which is referred to as the ‘moisture clog’ phenomenon [18]. As can be seen in Figure 5b, due to the low permeability of the high-strength concrete, the pore pressure formed can be greater than 1 MPa, and it increases with the increasing fire exposure time. Meanwhile, the strength of concrete degrades significantly, which leads to explosive spalling at the heated side. It can be learned from Figure 5 that the transient creep can have influences on the extent of spalling and subsequently on the temperature gradient and pore pressure, which would in turn influence the spalling and structural behavior of RC walls under fire.

The influence of transient creep on the spalling behavior of the studied RC wall under different load levels (P = 0–50% P_n, e = h/6) is shown in Figure 6. The evolution of the wall thickness with the increasing fire exposure time is shown for the cases without load, and the evolutions of the wall thicknesses at the edge and at mid-height are shown for the cases under loading. As shown in Figure 6, explosive spalling occurs after several minutes of fire exposure, and it proceeds with the increasing fire exposure time. The transient creep has postponed the occurrence of explosive spalling, and the postponing effect enhances with the increasing load level. This should be mainly attributed to the stress relaxation caused by the transient creep, which significantly decreases the compressive stress on the heated side (Figure 3e). The increase in the load increases the compressive stress and the transient creep, which enhances the postponing of explosive spalling. Furthermore, it can be seen from Figure 6 that the transient creep reduces the spalling rate, and the reduction increases with the increasing load level. This is also attributed to the stress relaxation effect. It can also be learned from Figure 6 that due to the stress relaxation effect of the transient creep, the spalling process becomes much smoother and the variation in spalling along the height caused by geometrical nonlinearity is notably mitigated.

Figure 7 shows the influence of transient creep on the structural deformations of the studied RC wall under eccentric loads of various load levels (P = 10–50% P_n, e = h/6) under fire. It can be seen that, different from its effect on RC walls with little spalling (Figure 4), the transient creep prolongs the structural failure time of RC walls with explosive spalling. As shown in Figure 7, the deflection of the wall shifts towards the unheated side once the explosive spalling occurs and the explosive spalling leads to an early failure of the wall. This is mainly attributed to the loss of the cross-section of the wall and a further shifting of the centroid of the wall towards the unheated side. Nevertheless, as can be seen from Figure 7, the stress relaxation effect of the transient creep delays the occurrence and reduces the rate of explosive spalling and can to some extent prolong the failure time of the wall under fire.

4. Model Validation

Comparisons with test results reported in the literature are conducted to validate the proposed model. Figure 8 shows the comparison between the measured and predicted evolutions of the deflection at mid-height of NSC3 wall specimen in the study by Ngo et al. [19]. The height, width, and thickness of the wall are 2400 mm, 1000 mm, and 150 mm, respectively, and it is exposed to a one-sided ISO 834 standard fire [16] for two hours. The wall is under an axial load of 485 kN with an eccentricity of 10 mm towards the heated face and it experienced little spalling in the test. The compressive strength of concrete is 35.6 MPa at room temperature on the test day. Thermocouples were located at different depths to record the temperature evolutions. The evolutions of the deflections were measured by linear variable displacement transducers, which have an accuracy of 0.002 mm. It can be seen from Figure 8 that the transient creep significantly reduces the deflection toward the heated side with the increasing fire exposure time. It plays the role in enabling the magnitude of deflection to agree with the test results and correctly predicting the transformation of the deflection from toward the heated side to toward the unheated side.

Figure 9 shows the comparison between the measured and predicted results of wall specimen 7 in the study by Mueller et al. [20]. The height, width, and thickness of the wall are 3050 mm, 1020 mm and 203 mm, respectively, and the base of the wall is monolithically supported by a foundation that is tied to the floor. The wall is subjected to a concentric axial load of 2400 kN and is exposed to an ASTM-E119 standard fire [21] on one side. A horizontal out-of-plane restraint hydraulic actuator is set at 230 mm below the top at the mid-length of the wall. The compressive strength of concrete is 123 MPa under room temperature on the test day. The temperature evolutions at various depths are recorded by thermocouples embedded inside the wall and the average values at the mid-depth and the depths of 20 mm and 140 mm are taken for the comparison. The first 6 mm depth of the wall was measured to reach 100% relative humidity on the test day. The porosity and the intrinsic permeability of the concrete at room temperature were not provided and are assumed to be 4.5% and $5.0\times {10}^{-21}$ m², respectively. It can be seen from Figure 9 that the model reasonably predicts the temperature evolution and the structural deformation of the wall. Ignoring the transient creep could lead to a significant overestimation of the explosive spalling and the structural deformation of the wall.

5. Parametric Studies

Fire resistance of RC walls is defined as the time from the beginning of fire to failure of the wall. The effects of the main parameters on the influence of transient creep on the fire resistance of RC walls with little spalling and RC walls with explosive spalling are investigated, respectively.

5.1. RC Walls with Little Spalling

The wall demonstrated in Figure 1 with e = h/6, with the concrete strength of 30 MPa is used as the reference wall. The studied parameters include wall thickness, wall height, concrete strength, and the eccentricity of load. When one parameter is investigated, the other parameters are kept constant.

5.1.1. Effect of Wall Thickness

As shown in Figure 10a, the reducing effect of the transient creep on the fire resistance of the wall increases with the decreasing wall thickness. This should be mainly attributed to the increase in thermal strain within the wall with the decreasing wall thickness, which increases the transient creep of the wall. It can also be seen from Figure 10a that this effect enhances with the increasing load level.

5.1.2. Effect of Wall Height

As shown in Figure 10b, the reducing effect of the transient creep on the fire resistance of the wall increases with the increasing wall height. This should be mainly attributed to the increase in geometrically nonlinear effect with the increasing wall height.

5.1.3. Effect of Concrete Strength

As can be seen from Figure 10c, for all load ratios (the ultimate axial load changes with concrete strength), the influence of transient creep on the fire resistance of the wall demonstrates negligible change with varying concrete strength.

5.1.4. Effect of Eccentricity of Load

As can be seen from Figure 10d, the reducing effect of the transient creep on the fire resistance of the wall increases notably with the increasing eccentricity of load, which is mainly attributed to the increase in transient creep due to a greater compressive stress on the heated side with the increasing eccentricity of load.

5.2. RC Walls with Explosive Spalling

The wall demonstrated in Figure 1 with e = h/6, with the concrete strength of 80 MPa is used as the reference wall to investigate the effects of the main parameters on the influence of transient creep on the fire resistance of RC walls with explosive spalling. The studied parameters include wall thickness, wall height, the intrinsic permeability of concrete, and the eccentricity of load. When one of the parameters is investigated, the other parameters are kept constant.

5.2.1. Effect of Wall Thickness

As shown in Figure 11a, different from the wall with little spalling (Figure 10a), the effect of the transient creep on the fire resistance of the wall decreases with the decreasing wall thickness. This should be mainly attributed to the decreasing wall thickness and the thermal strain, and the associated transient creep near the heated face decreasing. Thus, the effect of the stress relaxation caused by the transient creep on the explosive spalling and subsequently the fire resistance of the wall decreases with the decreasing wall thickness.

5.2.2. Effect of Wall Height

As shown in Figure 11b, different from the wall with little spalling (Figure 10b), the effect of the transient creep on the fire resistance of the wall decreases with the increasing wall height. This should be mainly attributed to the increasing wall height, and the compressive stress near the heated face caused by the load is reduced due to an increased thermal bowing towards the heated side. The different trend under P = 10% P_n is due to the fact that the wall fails towards the heated side under this lower-load level.

5.2.3. Effect of Intrinsic Permeability of Concrete

The intrinsic permeability of concrete can vary due to the varying pore size distribution and initial damage state. Figure 11c shows the effect of the variation in the intrinsic permeability of concrete on the influence of the transient creep on the fire resistance of the wall. The intrinsic permeability of the concrete varies within the range of 10⁻²¹ m² to 10⁻¹⁷ m². It can be seen from Figure 11c that the influence of the transient creep on the fire resistance of the wall decreases with the decreasing intrinsic permeability of concrete. This should be mainly attributed to the earlier occurrence of explosive spalling with the decreasing intrinsic permeability of concrete. The transient creep would be smaller at the location of spalling, and thus, the stress relaxation effect and subsequently the fire resistance of the wall would be smaller with the decreasing intrinsic permeability of concrete.

5.2.4. Effect of Eccentricity of Load

As shown in Figure 11d, different from the wall with little spalling (Figure 10d), the effect of the transient creep on the fire resistance of the wall decreases with the increasing eccentricity of load, which is also resulting from the earlier occurrence of explosive spalling with the increasing eccentricity of load. The different trend under P = 10% P_n is due to the wall failing towards the unheated side when the eccentricity of load increases to h/3 under this lower load level.

6. Conclusions

This paper investigates the effect of transient creep on the structural performance of RC walls under fire through a theoretical model that accounts for the explosive spalling of high-strength concrete along with geometric and material nonlinearities at high temperatures and can reasonably predict the deformation behavior and fire resistance of RC walls under fire. The influences of the transient creep on the temperature and pore pressure gradients, structural response, and fire resistance of RC walls with little spalling and with explosive spalling are investigated and revealed, respectively. The conclusions on the results can be drawn as follows:

• The transient creep has a notable effect on the structural deformations of RC walls under fire even without load. The transient creep significantly reduces the fire resistance of RC walls with little spalling, and the effect increases with the increasing load level and/or the increasing eccentricity of load.
• The transient creep delays the occurrence, reduces the rate of explosive spalling, and increases the fire resistance of RC walls with explosive spalling. The stress relaxation effect of transient creep has a crucial role in determining the spalling manner, and it mitigates the variation in spalling along the wall height.
• The influence of the transient creep on the fire resistance of RC walls with little spalling increases with the decreasing wall thickness and/or the increasing wall height, while the effect decreases with the similar changes for RC walls with explosive spalling. The variation in the intrinsic permeability of concrete can have an effect on the influence of transient creep on the fire resistance of RC walls.
• The high thermal mass of concrete and the low permeability of high-strength concrete are the key factors leading to the transient creep’s significant influence on the fire resistance of RC walls.