A modified layered-section method for responses of fire-damaged reinforced concrete beams under static and blast loads

Abstract

Reinforced concrete structures are currently under the threat of both fire and blast. The absence of theoretical methods demonstrates a drawback in the assessment of blast-resistant structures after exposure to fire. A modified layered-section method was developed in this article, which was not only able to determine the complete static resistance–deflection curves of fire-damaged reinforced concrete beams but also able to predict the responses of reinforced concrete beams subjected to blast after fire exposure. The high-temperature effects and the strain-rate effects were included in the concrete and steel material models in the proposed method. A corresponding calculation program FBBA was also compiled based on the explicit Newmark algorithm on the platform of Maple software. The developed method and program were validated by the existing test results. Analytical results showed that after fire exposure, the reinforced concrete beams show significant degradation in the residual bearing capacity, but increase in the ductility. The higher the steel reinforcement ratio, the more degradation the bearing capacity of reinforced concrete beams after fire exposure suffers. The blast resistance of the fired reinforced concrete beams was underestimated without considering the strain-rate effects or just considering the average strain-rate effects.

Keywords

Reinforced concrete fire explosion layered-section method single degree of freedom

Introduction

In addition to the normal design loads, accidental explosion and fire hazard may be prone to terrible threatens to the safety of reinforced concrete (RC) structures. Moreover, explosion and fire frequently ensue along with each other in the city area where inflammable and explosive materials are concentrated. Concern on combined effects of fire and blast, which significantly aggravate the damage of the structures, was greatly raised by scientists and engineers ever since the 9/11 terrorist attacks. Fire exposure usually induces severe micro-structural changes and impacts the degradation of mechanical properties of steel and concrete material, as well as the static and dynamic resistance of RC components. Blast-resistant performances of fire-damaged RC structures are far different from those under the ambient temperature. Thus, it is very important and even imperative to understand the failure mechanism of RC structures subjected to fire and sequent blast loads, in order to assess and predict the static and blast performances of fire-damaged key RC components.

It is well known that the fire exposure deteriorates the elastic modulus and the strength of concrete and steel material. Numerous models were developed during the last two decades to represent the concrete behavior under high temperatures. Lie and Lin (1985) proposed an instantaneous stress–strain relation for concrete with parabolic ascending and descending branches. The residual compressive strength and the peak strain were temperature dependent. The Eurocode 2 (2004) gave the strength and deformation properties at elevated temperatures. The proposed stress–strain relations were defined by two parameters: the compressive strength, $f_{c, θ}$ , and the strain, $ε_{c 1, θ}$ , corresponding to $f_{c, θ}$ . Values of these parameters were also determined. Chang et al. (2006) conducted an experimental research on the complete compressive stress–strain relationships for concrete after being heated to temperatures of 100°C–800°C. The equations for the complete stress–strain curves of fire exposed concrete were developed. The influences of specimen shape were also considered in these equations. The properties including residual compressive, peak strain, and elastic modulus in the proposed equation fit the test results well.

During the past several years, researchers have paid much attention to study the static residual bearing capacity of RC components under fire. Kodur and Dwaikat (2008) reported a validated macroscopic finite element (FE) model that was capable of simulating the effects of fire scenario, axial restraint, and failure criteria on the fire resistance of RC beams. It was revealed that the axial restraint benefited, reducing the deflection of the beam under ASTM E119 standard fire, and failure criteria of deformation rate should be considered in the assessment. El-Fitiany and Youssef (2009, 2012) introduced a sectional method to predict the axial bearing capacity, axial deformation, and sectional moment–curvature curve of RC columns after exposure to ASTM E119 standard fire. Both the strength reduction and the thermal strain of the concrete and steel bars resulting from fire exposure were taken into account. In this method, however, the section was divided into layers in which the mechanical properties were simplified as uniform. Further study was carried out to derive an explicit formula to predict the residual flexural bearing capacity based on numerical analyses. Han and Huo (2003) carried out tests on 12 concrete-filled hollow steel columns subjected to axial loads after ISO 834 standard fire exposure. It was found that the axial residual bearing capacity of the tested columns significantly decreased with the fire duration.

However, the responses of RC structures subjected to both blast and fire loads were much more complex when compared to single blast or fire. Heretofore, the available literatures focusing on the material properties of concrete at both high temperature and high strain rate were limited. Huo et al. (2013) experimentally investigated the dynamic behaviors of concrete after exposure to elevated temperatures up to 700°C. The results revealed that the strain-rate enhancement effects of concrete decreased with the elevated temperature. Chen et al. (2015) systematically studied the combined effects of high strain rate and high temperature using a specially manufactured microwave-heating automatic time-controlled Split Hopkinson Pressure Bar (MATSHPB) device in which the temperature can reach as high as 950°C. An explicit formula for the concrete strength dynamic increase factor (DIF) was proposed in this article.

The limited literatures on the performances of buildings under blast and fire mainly focused on the steel structures. An integrated nonlinear analysis method was reported by Song et al. (2000) and Izzuddin et al. (2000) to study the behaviors of steel members and frames subjected to scenarios of blast loads followed by fire. Liew and Chen (2004) and Chen and Liew (2005) proposed a mixed-element approach that was embedded into FE software ABAQUS. Local buckling and ultimate resistance of steel frames and columns under combined action of fire and explosion were studied by this approach. Fang et al. (2010, 2013) developed a three-dimensional numerical approach in ABAQUS to analyze the influences of blast loads on fire resistance of steel components, including the temperature-induced rate sensitivity and softening effects. Full-scale blast experiments on fire-damaged hollow steel slabs were conducted by Kakogiannis et al. (2013) to study the dynamic responses and crack patterns. Corresponding numerical analyses were also carried out in the commercial available software LS-DYNA. Only Ruan et al. (2015) proposed a numerical approach in ABAQUS to predict the dynamic behaviors and failure modes of RC columns subjected to gas explosion after fire exposure, although it still requires numerous experimental validation. However, these developments are not sufficient to understand the dynamic responses of RC structures under blast and fire. A simplified, rational, and practical theoretical method that is competent for quick assessment is very necessary.

In this study, the traditional layered-section method was modified to estimate the behaviors of fire-damaged RC beams under static and blast loads. A temperature distribution model across the section was introduced with a new section-dividing methodology. The equivalent single-degree-of-freedom (SDOF) model was applied to solve the dynamic response by using the explicit Newmark algorithm. The corresponding solving programs are compiled on the Maple platform. Moreover, the static residual capacity and structural ductility of fire-damaged RC beam, as well as the strain-rate effects, were discussed by using the proposed method.

Modified layered-section method for static loads

The traditional layered-section method is usually applied to calculate the moment–curvature curves of cross-section, which is important to predict the resistance–deflection curves of RC beams (Chen et al., 2011; Wu et al., 2009; Zhu and Dong, 1985). However, there were a number of challenges to introduce this method into analyses on fire-damaged RC beams. First, the existing method assumes dividing the section into layers with uniform properties. However, fire exposure leads to non-uniform temperature distribution not only along the depth but also along the width. Temperature distribution along two directions results in different material properties across the section. Second, the existing extrapolation method is used in the traditional layered-section method to find the appropriate strain at the middle axis of cross-section. However, this method usually leads to the endless loop in algorithm at large section curvature, which actually hampers the calculation of the complete moment–curvature curve that is important for the dynamic analyses.

Modified layer dividing

In the traditional layered-section method, the rectangle beam section is divided into n layers according to the section depth H. Nevertheless, the properties of concrete and steel in each layer were non-uniform after fire exposure. It accounted for the low thermal inertia of concrete during the heat transfer process. Thus, a new mesh dividing method was proposed, as shown in Figure 1. The temperature in each mesh grid was assumed to be uniform, which was represented by the temperature at the center of the grid. The accuracy of temperature distribution across the beam section can be adjusted by the grid density. Thus, the stress–strain curve of fire-damaged material can be determined by the temperature in each mesh grid.

Figure 1.

Section mesh dividing.

It is obvious that the grid size n depends on the section dimension, and a smaller grid size usually leads to more accurate results, but a longer computational time. The empirical grid size of 4 mm × 4 mm was used in the square cross-section of RC beams in this study.

Temperature distribution model

It was assumed that the RC beam was heated in a uniform way along the longitudinal direction; thus, the temperatures along the beam were consistent. Wu (2003) proposed a simplified model to predict the temperature at any point across the section of RC components under fire. It was demonstrated to be suitable for the rectangle concrete section under fire history recommended by ISO 834, and the suggested fire time ought to be longer than 30 min.

For a square section, it was expressed by equation (1).

T (x, y) = T_{f} - \frac{(T_{f} - T_{x}) (T_{f} - T_{y})}{T_{f} - T_{c}}

(1)

where $T_{x}$ is the temperature x distance away from the boundary along the X direction; $T_{y}$ is the temperature y distance away from the boundary along Y direction. They are determined by equations (2) and (3)

\frac{T_{x} - T_{c}}{T_{f} - T_{c}} = (1 - \frac{2 x}{B}) \exp (- 2.7 \frac{2 x}{B})

(2)

\frac{T_{y} - T_{c}}{T_{f} - T_{c}} = (1 - \frac{2 y}{B}) \exp (- 2.7 \frac{2 y}{B})

(3)

where $T_{f}$ denotes the temperature on the boundary; $T_{c}$ denotes the temperature at the center of the section. B is the size of the section; $T_{f}$ and $T_{c}$ are calculated by equations (4) and (5), respectively

T_{f} = T_{g 0} + 345 \lg [8 (0.91 t - 17) + 1]

(4)

T_{c} = {\begin{cases} 100 + β_{1} (t - t_{100}) t_{30} \leq t \leq t_{100} \\ 100 + 345 \lg [β_{2} (t - t_{100}) + 1] t > t_{100} \end{cases}}

(5)

where $T_{g 0}$ is the ambient temperature; $t_{30}$ and $t_{100}$ are the fire times when the temperature at the center of the section reaches 30°C and 100°C, respectively. $t_{30}$ and $t_{100}$ can be determined by equations (6) and (7), respectively

t_{30} = 702.2 B^{2}

(6)

t_{100} = 700 B^{2} + 326 B - 37

(7)

where $β_{1}$ and $β_{2}$ are parameters determined by equations (8) and (9), respectively

β_{1} = - 3.026 \times 10^{- 4} B^{- 3.5}

(8)

β_{2} = 4.72 B^{2} - 3.6 B + 0.713

(9)

For a rectangle section, 1 < H/B < 2, where B and H denote the dimension of rectangle section in the X and Y directions, respectively. As shown in Figure 2, along the Y direction, the whole section was divided into two parts. In part 1, the temperature distribution is determined by equation (10)

\frac{T_{y} - T_{c y}}{T_{f} - T_{c y}} = (1 - \frac{2 y}{B}) \exp (- 2.7 \frac{2 y}{B})

(10)

where $T_{c y}$ is the temperature B/2 distance away from the boundary along the Y direction and is determined by equation (11)

T_{c y} = {\begin{cases} 100 + (B / H)^{1.5} β_{1} (t - {(H / B)}^{0.43} t_{100}) 1.1 t_{30} \leq t \leq (H / B)^{0.43} t_{100} \\ 100 + 345 \lg [(B / H) β_{2} (t - {(H / B)}^{0.43} t_{100}) + 1] t > (H / B)^{0.43} t_{100} \end{cases}}

(11)

Figure 2.

The division of the rectangle section.

In part 2, the temperature distribution is determined by equation (12)

\frac{T_{y} - T_{c}}{T_{c y} - T_{c}} = (1 - \frac{2 (y - B / 2)}{H - B}) \exp (- 2.7 \frac{2 (y - B / 2)}{H - B})

(12)

Material model including temperature effects

Steel bars

The stress–strain relationship of steel bars at the ambient temperature was usually modeled as linear elastic and linear strain hardening curve. It was reported that the high temperature led to a significant degradation of the strength of steel bars, and the degree of degradation depended on the ever-experienced highest temperature (Yang et al., 2008). Thus, a two-linear curve model proposed by Han et al. (2002) was introduced to describe the stress–strain curve of steel bars after fire exposure. It is described as follows

σ_{s}^{T} = {\begin{cases} E_{s} ε_{s}^{T} ε_{s}^{T} \leq ε_{y}^{T} \\ f_{y}^{T} + 0.01 E_{s} (ε_{s}^{T} - ε_{y}^{T}) ε_{s}^{T} > ε_{y}^{T} \end{cases}}

(13)

where $σ_{s}^{T}$ and $ε_{s}^{T}$ denote the stress and strain, respectively; $f_{y}^{T}$ and $ε_{y}^{T}$ are the yielding stress and corresponding critical strain, respectively; $f_{y}^{T}$ depends on the ever-experienced highest temperature T, as described in equation (14); $E_{s}$ is the modulus of elasticity; $ε_{y}^{T}$ is defined as $ε_{y}^{T} = f_{y}^{T} / E_{s}$ The typical stress–strain curves of post-fire steel bars after different temperature exposure are shown in Figure 3

f_{y}^{T} = {\begin{cases} 1.0 f_{y} T \leq 400 ° C \\ [1.0 + 2.33 \times 10^{- 4} (T - 20) - 5.88 \times 10^{- 7} (T - 20)] f_{y} T > 400 ° C \end{cases}}

(14)

Figure 3.

Stress–strain curves of steel bars after exposure to different high temperatures.

Concrete

It was reported that the mechanical properties of normal strength concrete after fire exposure also mainly depended on the ever-experienced highest temperature, and hardly recovered after that (Naus, 2006). Thus, the stress–strain curve under fire can be used to determine the mechanical properties of post-fire concrete. A concrete model considering high-temperature effects $T$ T proposed by Lie and Lin (1985) was adopted in this study. It includes parabolic ascending and descending branches, which is expressed as follows

σ_{c}^{T} = {\begin{cases} f_{c}^{T} [1 - {(\frac{ε_{c}^{T} - ε_{p}^{T}}{ε_{p}^{T}})}^{2}] ε_{c}^{T} \leq ε_{p}^{T} \\ f_{c}^{T} [1 - {(\frac{ε_{c}^{T} - ε_{p}^{T}}{3 ε_{p}^{T}})}^{2}] ε_{c}^{T} > ε_{p}^{T} \end{cases}}

(15)

ε_{p}^{T} = 0.0025 + (6 T + 0.004 T^{2}) \times 10^{- 6}

(16)

f_{c}^{T} = {\begin{cases} f_{c} 0 ° C \leq T \leq 200 ° C \\ f_{c} [1.0 - 0.0015 (T - 200) 200 ° C \leq T \leq 500 ° C \\ f_{c} [0.25 - 0.003 (600 - T) 500 ° C \leq T \leq 600 ° C \\ f_{c} [0.25 - 7.5 \times 10^{- 4} (T - 600) 600 ° C \leq T \leq 800 ° C \end{cases}}

(17)

where $σ_{c}^{T}$ and $ε_{c}^{T}$ represent the stress and corresponding strain of concrete, respectively; $f_{c}^{T}$ is the compressive strength, and $ε_{p}^{T}$ is the corresponding critical strain. The typical stress–strain curves of concrete after exposure to different elevated temperatures are shown in Figure 4.

Figure 4.

Stress–strain curves of concrete after exposure to different high temperatures.

In addition to the high-temperature effects, the temperature gradient due to thermal conductivity is another effect which leads to concrete degradation. It should be considered by using the thermodynamic analyses of RC components under fire, which for now was not considered in the proposed method yet. However, the proposed concrete material properties were obtained by using the way of thermal conduction in which the temperature gradient effect was partially considered in the degradation of concrete.

Moment–curvature curve

The moment–curvature curve of the cross-section was determined by sectional force equilibrium along normal direction to the section. The section equilibrium analysis was conducted based on the following three assumptions:

Plane section assumption

Boundary slip between concrete and steel bars was not considered. The section remains plane after bending.

Flexure deformation assumption

Only flexural deformation occurred during the loading process. Shear deformation was not considered in this study.

Tensile stress assumption

The tensile stress of the concrete was neglected in the calculation.

The strain and force distribution across the rectangle section is shown in Figure 5, where $N_{s}$ represents the tensile force of steel bars; $N_{s^{'}}$ represents the compressive force of steel bars; $N_{c}$ is the compressive force of concrete; $N_{c i}$ represents the compressive force of concrete in each layer at the height of $H i / n + H / 2 n$ in the section depth; $H$ is divided into n layers; $M$ is the external moment.

Figure 5.

Strain distribution and forces across the section.

The calculation of moment–curvature curve is carried out through the following steps:

Input the dimensions of the section, the heating time, the initial curvature $φ$ , and the initial strain at the mid-depth of the section $ε_{0}$ .

Material strain $ε_{i j}$ in each grid is determined by $ε_{i j} = ε_{0} - (H i / n + H / 2 n) \cdot (φ / H)$ based on the plane section assumption. Material stress $σ_{i j}$ in each grid is determined according to the stress–strain curves of concrete or steel bars at different temperatures.

Calculate the total force across the section by summing up three parts, which are compressive force of concrete, tensile force, and compressive force of steel bars.

Adjust the strain at mid-depth of the cross-section until the sectional horizontal forces achieve equilibrium. It can be described as

0 = N_{c} + N_{s} = \sum_{i = 0}^{H / Δ H} \sum_{j = 0}^{B / Δ B} [σ_{c i j} \times ε_{i j} \times Δ B \times Δ H] + \sum_{i = 1}^{n} \sum_{j = 1}^{m} [σ_{s i j} \times ε_{i j} \times 0.25 π d^{2}]

(18)

The equilibrium of moments to the mid-depth of cross-section is described as

M = M_{c} + M_{s} = \sum_{i = 0}^{H / Δ H} \sum_{j = 0}^{B / Δ B} [σ_{c i j} \times ε_{i j} \times Δ B \times Δ H \times z_{i}] + \sum_{i = 1}^{n} \sum_{j = 1}^{m} [σ_{s i j} \times ε_{i j} \times 0.25 π d^{2} \times z_{i}]

(19)

Output and store the curvature and corresponding sectional moment in a matrix.

Increase the section curvature $φ$ by step $Δ φ$ , and repeat the steps from (2) to (4). It is assumed that the clockwise curvature is positive.

The extrapolation method was adopted in finding out the appropriate strain at mid-depth of cross-section. It benefited in avoiding numerical oscillation before the sectional moment reaches the yield moment of M_y; more details can be found in Zhu and Dong (1985). However, the extrapolation method might fall into “endless loop” after section yielding. Therefore, the extrapolation method was replaced by the interpolation method in the softening stage of section in this study. The ultimate capacity M_u and the complete moment–curvature curve can thus be obtained. Figure 6 gives the flowchart for the calculation of the complete section moment–curvature curve, and a corresponding program was compiled on the platform of Maple software.

Figure 6.

Flowchart of the calculation of the section moment–curvature curve.

Static resistance–deflection curve

The static resistance–deflection curve of RC beam thus can be calculated based on the sectional relationship between moment and curvature. It was generally accepted that there were two loading methods in the quasi-static experiments on RC beams. Those are force-controlled ramp and displacement-controlled ramp. The deflection increased significantly in the softening stage of RC beam, but with little increase in section moment. To avoid the iteration difficulty in force convergence, a calculation procedure based on the displacement-controlled ramp was proposed to calculate the quasi-static resistance–deflection curve of fire-damaged RC beam. The detailed computational steps are presented as follows:

Calculate the complete moment–curvature curve according to section “Moment–curvature curve.”

Increase the curvature $φ$ by $Δ φ$ to obtain the corresponding moment $M_{m}$ in each step based on the stored moment–curvature curve.

Obtain the distributed load $P_{m}$ or symmetrical concentrated load $Q_{m}$ for each section moment $M_{m}$ . $P_{m}$ is calculated by $(8 M_{m}) / L^{2}$ , where L represents the length of the beam. $Q_{m}$ is calculated by $M_{m} / L_{a}$ , where $L_{a}$ is the distance between the load point and support.

Divide the total length of the fire-damaged RC beam $L$ into $2 K$ parts. Thus, there are $2 K + 1$ sections along the beam.

Calculate the moment of each cross-section $M_{k}$ based on the predetermined load $P_{m}$ or $Q_{m}$ . For the distributed load $P_{m}$ , $Q_{m}$ is calculated by $[(P_{m} L) \cdot (k L / K)] / 2 + [P_{m} {(k L / K)}^{2}] / 2$ ; for the concentrated load $Q_{m}$ , $M_{k}$ equals to $M_{m}$ in the pure bending part; and $M_{k}$ is calculated by $Q_{m} k \cdot (L / K)$ between the load point and the support, where $k$ denotes the kth section to the end of the beam.

Find out the corresponding curvature $φ_{k}$ of each cross-section according to the stored moment–curvature curve.

Calculate the rotation angle $ω_{k}$ and the deflection $δ_{k}$ of each section along the whole beam by superposition. Thus, the mid-span displacement could be finally obtained.

Figure 7 shows the flowchart for the calculation of complete resistance–deflection curve of RC beams.

Figure 7.

Flowchart of the calculation of the complete resistance–deflection curve.

Verification of the modified method for static resistance

Zheng et al. (2008) conducted a series of tests to study the residual bearing capacity of simple-supported beams after fire exposure. The test RC beams were loaded after the fired beam had been cooled down to the ambient temperature. The tested beam B-1 was chosen to validate the proposed method.

The test RC beam B-1 was 190 mm in width, 300 mm in depth, with a total length of 4900 mm. It was reinforced with 2 ∅ 20 HRB 400 longitude steel bars at the tensile zone, and was reinforced with 2 ∅ 20 HRB 400 longitude steel bars at the compressive zone. The shear stirrups were ∅ 6.5 HRB235 bars arranged with an even space of 150 mm. The thickness of the concrete cover was 25 mm. The average compressive strength of concrete was 52.1 MPa, and the yield strength of longitude steel bars was 410 MPa. The modulus of elasticity of steel bars was 2 × 10⁵ MPa. All the parameters were obtained by the laboratory material tests.

The test beam B-1 was exposed to fire for 34 min on four faces without any preloading, according to the temperature time history suggested by ISO 834. After that, the fired beam was loaded with a pair of symmetrical concentrated quasi-static loads. The distribution of external loads between the two concentrated loads can be considered as the pure bending. Details of the test beam B-1 and experiments are shown in Figure 8.

Figure 8.

Configuration of the test beam B-1.

Figure 9(a) shows the calculated moment–curvature curve at the mid-span of beam B-1. It is obvious that the section moment increased with the curvature linearly at the elastic stage, and stabilized in the later softening stage while the moment reached to 58.67 kN m, which indicates that the bottom steel bars had been yielding. The ultimate mid-span moment of the fire-damaged beam B-1 was 61 kN m, which was 4.4% higher than the experimental data (Zheng et al., 2008). Figure 9(b) presents the comparison between the predicted resistance–deflection curve and the test data.

Figure 9.

Comparison between the predicted results and test data: (a) moment–curvature curve and (b) resistance–deflection curve.

It is found that the calculated ultimate deflection is a little smaller than the maximum deflection captured at the static loading test. The little difference is attributed to the symmetrical longitudinal reinforcement bars in the test RC beam. Actually, the calculation program was first set to automatically cease while there was no appropriate curvature for the sectional horizontal forces to achieve equilibrium in the iterative process. In that case, the flexure deformation of RC beam was too large, and the plane section assumption could not satisfy the section equilibrium. It means that the ultimate curvature was calculated in the program, but not predefined, and it depended on the individual specimen. However, the steel reinforcement bars in the test RC beam were arranged symmetrically. In the proposed method, the development of the cracks and the slip between bars and concrete were not considered. It means that the compressive reinforcement was always taken into account in the calculation. If the test beam was reinforced with symmetric longitudinal bars, the calculation program was unable to automatically cease and was unable to capture the maximum deflection exactly and automatically. As the symmetric longitudinal bars were helpful to achieve section force equilibrium even at very large deformation, an empirical limitation of compressive strain of 0.015 for the compressive bars was added into the program for the symmetrically reinforced RC beam.

The little discrepancy of the bearing capacity was attributed to the error of the temperature distribution in beam section predicted by Wu’s (2003) model. Because of the low thermal conductivity and high specific heat of concrete, the heat is not only transferred from the fired boundary to the inner part during heating but also continues after that, in the cooling phase. It leads to a little discrepancy in the ever-experienced highest temperature between the heating phase and the cooling phase. The temperature distribution model proposed by Wu was able to predict the ever-experienced highest temperature during the heating phase. However, the heating transfer during the cooling phase was not considered; therefore, it underestimated the ever-experienced highest temperature distributed in the RC beam. This error overestimated the residual load bearing capacity of RC beams after fire exposure.

Effects of heating duration and steel reinforcement ratio

A total of six RC beams with different fire duration and steel ratio were calculated using the compiled programs to discuss the influences of heating duration and steel reinforcement ratio. Table 1 lists details of the calculated beam specimens.

Table 1.

Parameters of calculated beam specimens.

Beam no.	Size (m)	Steel bars	Steel ratio	Fire duration (min)
b-1-1	0.2 × 0.2 × 2.5	2 ∅ 16	1.0%	20
b-1-2	0.2 × 0.2 × 2.5	2 ∅ 25	2.45%	20
b-2-1	0.2 × 0.2 × 2.5	2 ∅ 16	1.0%	40
b-2-2	0.2 × 0.2 × 2.5	2 ∅ 25	2.45%	40
b-3-1	0.2 × 0.2 × 2.5	2 ∅ 16	1.0%	60
b-3-2	0.2 × 0.2 × 2.5	2 ∅ 25	2.45%	60

The calculated non-dimensional moment–curvature curves and static resistance–deflection curves at mid-span of the RC beams after different fire durations are shown in Figure 10. The non-dimensional section moment was defined as the ratio of the section moment after fire $M^{T}$ to the ultimate moment at the ambient temperature $M_{u}$ . The non-dimensional static resistance was defined as the ratio of distributed resistance of RC beam after fire $P^{T}$ to the ultimate distributed resistance at the ambient temperature $P_{u}$ . $φ$ is the section curvature after fire. $H$ is the depth of the RC beam section. $D$ represents the deflection at mid-span of the beam. L represents the length of the RC beam.

Figure 10.

Influences of the fire duration and steel ratio: (a) moment–curvature curves and (b) resistance–deflection curves.

Figure 11 presents the relationships between the non-dimensional ultimate section moment and fire duration, where t represents the fire duration and the non-dimensional ultimate section moment was defined as the ratio of the ultimate moment after different fire duration $M_{u}^{T}$ to $M_{u}$ . Figure 12 presents the curve of non-dimensional ultimate section curvature versus fire duration, where the non-dimensional ultimate curvature was defined as the ratio of the ultimate curvature after fire duration $φ_{u}^{T}$ to that at the ambient temperature $φ_{u}$ .

Figure 11.

Influences of steel ratio on ultimate moment.

Figure 12.

Influences of steel ratio on ductility.

It is generally accepted that the RC beams with higher steel reinforcement ratio usually possess higher load bearing capacities but lower ductility at the ambient temperature. As shown in Figures 9(a) and 10, it is obvious that the section moment significantly decreases with the fire durations. The longer the fire duration, the more decrease the ultimate section moment suffers. The ultimate moment of b-3-1 after 60 min fire exposure is 80.1% of that of beam b-1-1 at the ambient temperature. The ultimate moment of b-3-2 after 60 min fire exposure is 76.2% of that of beam b-1-2 at the ambient temperature. As shown in Figure 11, the RC beams with high steel reinforcement ratio of 2.45% suffer much heavier decrease in the ultimate section moment than those with low reinforcement ratio of 1.00% after exposure to fire. Similar trend is also found on the corresponding non-dimensional static resistance–deflection curves as the fire duration increases, as shown in Figure 10(b).

The ductility of the RC beams was also affected by the fire duration, as observed in Figure 10(a) and (b). It is found that the RC beams suffering fire exposure basically endure larger deflection. As shown in Figure 12, it is found that the ultimate section curvature of the fire-damaged beam increased with the fire duration, even up to 300%. It indicates that better ductility occurs as the fire duration increases. It is mainly attributed to the effects of fire exposure on the stress–strain curve of materials. As shown in Figure 4, the ultimate compressive strength of concrete decreases with the exposure temperature; however, the corresponding critical strain increases with the temperature, which under 600°C is about twice of that at the ambient temperature.

However, the steel reinforcement ratio has few influences on the ductility of RC beams after fire exposure, since the two curves with different reinforcement ratios, as observed in Figure 12, are close to each other.

Dynamic resistance model under blast loads

Equivalent SDOF method

The aim of analyzing the dynamic response is to obtain the dynamic deflections and stresses of RC structures after fire exposure. The above proposed method for the static analysis was able to predict the static resistance–deflection curves of the fire-damaged RC beams, which was considered as a static resistance model. The shear failure of fire-damaged RC beam was not considered in this study; thus, the dynamic resistance model here was developed based on a combination of the static resistance model with the rate-sensitive material model under the framework of the SDOF system (Henrych, 1979; Hinton and Owen, 1980; Morison, 2005). The equation of motion for an equivalent SDOF system can be written as follows

M_{e} \overset{\cdot\cdot}{Y} + R_{e} (Y, \overset{\cdot}{ε}) = P_{e} (t)

(20)

where $M_{e}$ is the equivalent mass, $R_{e} (Y, \overset{\cdot}{ε})$ is the equivalent dynamic resistance determined by introducing material strain-rate effects into the static resistance model proposed in section “Modified layered-section method for static loads” and $P_{e} (t)$ is the equivalent external dynamic loads. Damping was not considered in this study. Equation (20) was solved using the explicit predictor-corrector Newmark algorithm (Hinton and Owen, 1980). This algorithm established predicted values of displacement, velocity, and acceleration in each step time, and then adjusted the predicted values by using the corresponding equivalent resistance–deflection curve. This method is very efficient in the blast analysis since the equilibrium iteration is avoided in each time step. The detailed steps for solving the algorithm are shown in Figure 13.

Figure 13.

Steps of the explicit predictor-corrector Newmark algorithm.

In Figure 13, β and δ are two free parameters that control the accuracy and stability of the method. In this calculation, β and δ were determined as 0.25 and 0.5, respectively. $\begin{array}{l} {\tilde{d}}_{n + 1} \end{array}$ , ${\tilde{v}}_{n + 1}$ , and ${\tilde{a}}_{n + 1}$ are the predictors of displacement, velocity, and acceleration, respectively. $d_{n + 1}$ , $v_{n + 1}$ , and $a_{n + 1}$ are the corresponding correctors of ${\tilde{d}}_{n + 1}$ , ${\tilde{v}}_{n + 1}$ , and ${\tilde{a}}_{n + 1}$ , respectively.

Strain-rate effects

It is demonstrated that the strain-rate effects significantly affect the responses of RC structures under blast loads (Headquarters, Department of the Army, 1986; Malvern, 1951). Both concrete and steel are strain-rate dependent after fire exposure, and high strain rates enhance the strengths of these two materials. Code for design of civil air defense basement (GB50038-2005, 2005) simply provides a set of constant DIFs to consider the strain-rate effects, where the DIF for concrete is 1.5 and the DIF for HRB 400 steel bars is 1.2. It is designated as the average strain-rate effects.

Actually, the local strain rate along the RC beam is different at each time step, which depends on the development of displacement. The constant DIF is obviously not accurate enough. A modified elasto-viscoplastic rate-sensitive model proposed by Fang and Izzuddin (1997a, 1997b) was introduced into the static resistance model to calculate the dynamic resistance. They denoted the parameter X as the overstress which represented the difference between the dynamic stress and the corresponding static stress. It is described as

X = σ - g (ε)

(21)

where $σ$ is the value of dynamic stress, and $g (ε)$ is the corresponding reference static stress. The function $f 〈 X 〉$ is used to describe the strain rate. It is proposed that $f 〈 X 〉$ can be expressed as

f 〈 X 〉 = E (1 - μ) {\overset{\cdot}{ε}}_{*} {[\exp (\frac{X}{S N}) - 1]}^{N}

(22)

where E is the modulus of elasticity; µ is the hardening parameter of the material; $S$ , $N$ , and ${\overset{\cdot}{ε}}_{*}$ are the model parameters, which can be determined by the tests of constant strain rates. Since the study of the strain-rate effects of concrete and steel bars after fire exposure are still not sufficient, the parameters here were determined based on the suggestion by the CEB Secretariat (1988), which is actually for the ambient temperature. The parameters were determined as

S = 0.0066 (1 - μ_{c}) f_{c}, N = 2, {\overset{\cdot}{ε}}_{*} = 6.66 \times 10^{- 4} s^{- 1} for concrete

(23)

S = 6 MPa, N = 1, {\overset{\cdot}{ε}}_{*} = 5 \times 10^{- 5} s^{- 1} for steel bar

(24)

This rate-sensitive model is able to update the material dynamic stress associated with the local strain-rate effects in each time step. The local material strain rate is determined by the development of displacement in time step, which is considered as constant in each time step Δt.

In order to meet the modified elasto-viscoplastic rate-sensitive model, the bilinear model for steel bars after fire described in section “Modified layered-section method for static loads” was adopted as the reference static stress–strain curve for steel. The concrete model considering high temperature effects (Lie and Lin, 1985) proposed in section “Modified layered-section method for static loads” is simplified to a trilinear reference model as the reference static stress–strain curve for concrete, which is expressed as follows

σ_{c}^{T} = {\begin{cases} \frac{f_{c}^{T}}{ε_{p}^{T}} ε_{c}^{T} 0 \leq ε_{c}^{T} \leq ε_{p}^{T} \\ f_{c}^{T} [1 - \frac{0.5}{(3 + 0.29 f_{c}^{T}) / (145 f_{c}^{T} - 1000)} (ε_{c}^{T} - ε_{p}^{T}) ε_{p}^{T} < ε_{c}^{T} \leq ε_{40 c}^{T} \\ 0.4 f_{c}^{T} ε_{c}^{T} > ε_{40 c}^{T} \end{cases}}

(25)

where $f_{c}^{T}$ is the compressive strength of concrete related to the fire duration; $ε_{p}^{T}$ is the corresponding critical strain; $σ_{c}^{T}$ and $ε_{c}^{T}$ are the stress and strain, respectively; $ε_{40 c}^{T}$ is the residual strain when the residual strength of concrete is $0.4 f_{c}^{T}$ . Details of the material model are shown in Figure 14.

Figure 14.

Reference static stress–strain curves: (a) concrete and (b) steel bars.

Based on the explicit predictor-corrector Newmark solving algorithm and the elasto-viscoplastic rate-sensitive model described above, a corresponding program designated as FBBA was compiled on the platform of Maple software. Detailed procedure could be found on literature (Chen et al., 2011). The time step $Δ t$ applied in this study is 0.1 ms, which is small enough to meet the convergence criterion.

Experimental validation and discussion

A series of in situ blast tests on fire-damaged RC beams were conducted to validate the compiled program FBBA (Zhai et al., in press). The RC beams were first heated in a furnace, and then blast loaded after cooling to the ambient temperature.

All the test RC beams were 2500 mm long with a square cross-section of 200 mm × 200 mm. They were reinforced with 2 ∅ 16 HRB 400 tensile reinforcement bars and 2 ∅ 10 HRB 400 compressive reinforcement bars. The shear stirrups were ∅6 HRB 235 bars with an even space of 150 mm. The thickness of the concrete cover was 20 mm. Figure 15 shows the details of the test beam. Laboratory tests were conducted to get basic material parameters. The average compressive strength of concrete was 32.40 MPa. The average yield strength of the steel bars was 540 MPa (∅10), 451.4 MPa (∅16), and 492.2 MPa (∅6).

Figure 15.

Details of test RC beam.

The RC beams were heated without any preloading in the testing furnace at Southeast University in Nanjing, China, as shown in Figure 16. The thermal energy was provided and adjusted by natural gas burners located in the furnace. During the test, the column was exposed to the average temperature followed as closely as the ISO 834 fire standard curve. The RC beam in the real fire is usually exposed to three-face fire. Thus, the upper surface and two ends of the tested beam were wrapped by the refractory cotton during the fire process, as shown in Figure 17.

Figure 16.

Gas fire furnace.

Figure 17.

Test beams in the furnace.

After fire duration, the test beams were cooled down naturally to the ambient temperature. Field blast tests on the RC beams after fire were conducted to validate the theoretical method. The blast tests were carried out by using a specially designed setup, as shown in Figure 18. The RC beam in the blast pit was supported by brackets at the two ends. The blast loads were generated by the rock emulsion explosive hung 1.5 m exactly above the mid-span of the beam. The overpressures on the test RC beam were recorded by the pressure transducers mounted at the mid-span of the RC beam. The displacement gauges were mounted on the frame under the test beam. Table 2 lists the details of blast test scenarios.

Figure 18.

Blast test device.

Table 2.

Blast test scenarios.

Beam no.	Charge weight (kg)	Fire duration (min)
B-2-1	1	0
B-2-2	1	90
B-2-3	7	0
B-2-4	7	90

Actually, in order to apply the recorded blast loads into the compiled program, the overpressure time histories were simplified, and the multiple peaks due to the reflection were neglected, as shown in Figure 19(b). The rising branch of blast loads was simplified as a straight line, and the declining branch was replaced by sectional conics, although the blast load was commonly simplified as a triangle load in the dynamic analysis (GB50038-2005, 2005).

Figure 19.

Blast overpressure time histories: (a) 1 kg and (b) 7 kg.

Figure 20 shows the calculated resistance–deflection curves of the test RC beams using different rate-sensitive models. It is obvious that the resistance-deflection relationships using different rate-sensitive models show similar curve shapes. However, the blast-resistant capacity of the RC beams was significantly underestimated without considering the strain-rate effects or just considering average strain-rate effects. The capacity of RC beam without considering strain-rate effects is 37.5% lower than the blast resistance with the elasto-viscoplastic rate-sensitive model at the ambient temperature. The capacity of RC beam without considering strain-rate effects is 40.1% lower than that of the RC beam with the elasto-viscoplastic rate-sensitive model after 90 min of fire exposure. The calculation results just considering average strain-rate effects are also added for comparison.

Figure 20.

Resistance–deflection curves of test RC beams: (a) ambient temperature and (b) 90-min fire duration.

The calculated displacement time histories at mid-span of the four beams are plotted in Figure 21. The dynamic mid-span displacement $D$ is normalized by division by overall depth of the beam $H$ . The results without strain-rate effects and just considering average strain-rate effects are also added into Figure 21 for comparison.

Figure 21.

Calculated displacement time histories of test RC beams: (a) B-2-1, (b) B-2-2, (c) B-2-3, and (d) B-2-4.

As shown in Figure 21, it is evident that the calculated results by incorporating the modified elasto-viscoplastic rate model basically agree well with the test data on both the peak displacement and the free oscillation phase. The accuracy of proposed method and corresponding program FBBA was validated by the test data. The errors between the test results and the calculated results might be attributed to several reasons. First reason is the simplification of blast loads. Actually, in order to apply the recorded blast loads into the compiled program, the multiple peaks due to the reflection in the overpressure time history were neglected, as shown in Figure 19(b). It inevitably over-predicted the rising time of blast overpressure on the beams, which actually underestimated the blast loads. Moreover, the errors in the measurement of the overpressure were inevitable in the explosion with such a small scaled distance. In addition, the measurement errors are inevitable in the blast tests. The pressure and the deformation at the mid-span may not be exactly recorded by the transducer, especially with a higher blast pressure. The calculated results also indicate that the anti-blast capacity of the fire-damaged RC beams would be significantly underestimated without considering the strain-rate effects or just considering average strain-rate effects.

Conclusion

A modified layered-section method was first proposed to predict the complete resistance–deflection curve of RC beams after fire exposure. The calculated results showed that the static load bearing capacity of fire-damaged RC beams decreased significantly with the fire duration. RC beams with high steel reinforcement ratio suffered much heavier decrease in ultimate resistance than those with low reinforcement ratio after exposure to fire. However, the ductility of RC beams was improved after being exposed to fire, while the steel ratio had few influences on the ductility of fire-damaged RC beams.

The proposed modified layered-section method was further developed to predict the dynamic responses of RC beams subjected to blast loads after fire. The elasto-viscoplastic rate-sensitive material model was incorporated with the calculated static resistance model based on the equivalent SDOF theory. In situ blast experiments were carried out on the fire-damaged beams to validate the proposed method and corresponding program FBBA. The calculated results demonstrated that the blast resistance of the fired RC beams would be significantly underestimated without considering the strain-rate effects or just considering average strain-rate effects.

The proposed calculation method and the corresponding program are helpful to assess and predict the static and blast performances of key RC components after fire exposure. However, the distribution model of ever-experienced highest temperature in the RC components and the strain-rate effects of concrete after fire exposure were needed to be further studied.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This work was supported by the National Basic Research Program of China (No. 2015CB058003) and National Natural Science Foundation of China (No. 51378016, No. 51210012, No. 51238007).

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