A numerical approach for modeling the fire induced restraint effects in reinforced concrete beams

ARTICLE IN PRESS

Fire Safety Journal 43 (2008) 291–307 www.elsevier.com/locate/firesaf

A numerical approach for modeling the fire induced restraint effects in reinforced concrete beams M.B. Dwaikat, V.K.R. Kodur Civil and Environmental Engineering, Michigan State University, East Lansing, MI 48824-1126, USA Received 19 March 2007; received in revised form 23 June 2007; accepted 21 August 2007 Available online 29 September 2007

Abstract In this paper, a model to predict the influence of fire induced restraints on the fire resistance of reinforced concrete (RC) beams is presented. The three stages, associated with the fire growth, thermal and structural analysis, for the calculation of fire resistance of the RC beams are explained. A simplified approach to account for spalling under fire conditions is incorporated into the model. The validity of the numerical model is established by comparing the predictions from the computer program with results from full-scale fire resistance tests. The program is used to conduct two case studies to investigate the influence of both the rotational and the axial restraint on the fire response of the RC beams. Through these case studies, it is shown that the restraint, both rotational and axial, has significant influence on the fire resistance of the RC beams. r 2007 Elsevier Ltd. All rights reserved. Keywords: Fire resistance; Computer program; Fire induced restraint; High temperature; High-strength concrete; Reinforced concrete beams; Spalling; Numerical model

1. Introduction Reinforced concrete (RC) structural systems are quite frequently used in high-rise buildings and other built infrastructure due to a number of advantages they provide over other materials. When used in buildings, the provision of appropriate fire safety measures for structural members is an important aspect of design since fire represents one of the most severe environmental conditions to which structures may be subjected in their life time. The fire resistance of the RC members is generally established using prescriptive approaches which are based on either the standard fire resistance tests or the empirical calculation methods. The RC beams can develop significant restraint under fire exposure. The degree of restraint is mainly dependent on the support conditions and will determine the fire behavior and resistance of the RC beams. The end restraints can be rotational, axial or both. Rotational restraints, under fire conditions, can improve the fire Corresponding author. Tel.: +1 517 353 5107; fax: +1 517 432 1827.

E-mail address: [email protected] (V.K.R. Kodur). 0379-7112/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2007.08.003

response of an RC beam through moment redistribution between span (positive moment) and support (negative moment) sections within the length of the beam. Therefore, rotational restraints increase the fire resistance of the RC beams. Even if the moment redistribution has been considered in the design at room temperature, rotational restraints are expected to have positive effect on the fire resistance of the RC beams. This is because the critical span section will experience higher strength loss (due to tension steel being closer to the fire exposure surface) than the critical support section (due to low thermal conductivity of the concrete which keeps the temperatures of the tensile reinforcement at the support section lower). Furthermore, the curvature ductility of an RC beam under fire conditions is higher than that at room temperature, which allows for higher moment redistribution under fire exposure. The effect of the axial restraints on the fire resistance of the RC beams depends on the vertical location of the restraint force. Generally, the axial restraint force in an RC beam is expected to be below the neutral axis of the RC beam section as a result of the higher temperature rise in the bottom fibers of the beam as shown in Fig. 1. This can

ARTICLE IN PRESS M.B. Dwaikat, V.K.R. Kodur / Fire Safety Journal 43 (2008) 291–307

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Nomenclature b c Ct d f c;20 f c;T fy k2 Kg L Ls Pf Ps t T T0 th

beam width clear concrete cover total compressive force effective depth of the beam concrete strength at room temperature concrete strength at temperature T yield strength of steel a constant for the transient strain calculations and it ranges between 1.8 and 2.35 global stiffness matrix for strength analysis length of the beam length of the beam segment equivalent nodal load vector due to applied loading equivalent nodal load vector due to P–d effect time temperature initial temperature time, h

improve the fire resistance of the RC beam through the arch action associated with the axial restraints, which increases the strength and the stiffness of the beam under fire exposure. However, axial restraints may lead to spalling of the concrete which in turn might reduce the fire resistance of the RC beams. Thus, it is essential to investigate and quantify the influence of the fire induced restraint forces and moments on the fire response of the RC beams. At present, there are limited approaches or methodologies to properly model the fire induced restraint effects in the RC beams. This paper presents the development of a computer model for tracing the fire response of the restrained RC beams under the realistic fire, loading and failure scenarios. The model is capable of predicating the fire induced restraint effects in the RC beams for any material properties, fire scenario, loading and support conditions. The model is based on a macroscopic finite element approach and uses a series of moment–curvature relationships for tracing the fire response of the RC beams. The model is verified against experimental

Tf Tt TN

fire temperature total tensile force fire or ambient temperature depending on the boundary, G w applied distributed load y the distance from the geometrical centroid of the beam Z Zener–Hollomon parameter for the creep strain d nodal displacements Deth change in the thermal strain Detr change in the transient strain ec strain at the topmost fibers of concrete ecr, eme, et, eth, etr creep strain, mechanical strain, total strain, thermal strain and transient strain ecrs, emes, eths, ets creep strain, mechanical strain, thermal strain and total strain in steel et0 creep strain parameter k curvature y temperature-compensated time s current stress in concrete or steel

data by comparing the predicted temperatures, deflections and fire resistance times with the measured ones from fire tests. Results from case studies are presented to illustrate the influence of the fire induced restraint forces and moments on the fire resistance of the RC beams. 2. Fire response of RC beams A review of literature indicates that limited fire resistance tests have been conducted on the RC beams. Most of these tests were conducted on unrestrained beams and thus there is limited information on the fire response of the restrained RC beams in the literature. The most notable fire tests are those carried out by Lin et al. [1] and by Dotreppe and Franssen [2]. Both the tests were conducted on the axially unrestrained beams. Lin et al. [1] tested 11 305 mm  355 mm RC beams under ASTM E119 [3] standard fire exposure. Their study investigated the influence of a number of factors including beam continuity, moment redistribution and aggregate type on the behavior

Neutral axis after 15 minutes 60 minutes

h/2

Geometrical centroidal axis

120 minutes P h/2

Cross section

Temperature distribution after 15 minutes

P

Temperature distribution after 60 minutes

P

Temperature distribution after 120 minutes

Fig. 1. Schematic diagram showing the thermal gradients and the position of axial force as a function of time of fire exposure.

ARTICLE IN PRESS M.B. Dwaikat, V.K.R. Kodur / Fire Safety Journal 43 (2008) 291–307

of the RC beams under the standard fire conditions. One of the tested beams had simple support conditions at both the ends, while the remaining beams had cantilever spans on one or both the ends. The applied loading on the cantilever was chosen to reflect the continuity effect in the RC beams. This study concluded that the fire resistance of the continuous beams is much higher than that of the simply supported (SS) beams due to the occurrence of redistribution of bending moments and shear forces in fire conditions. The second experimental study, reported by Dotreppe and Franssen [2], involved testing an SS RC beam of a rectangular cross-section under ISO 834 [4] fire exposure. The main objective of this fire resistance test was to assess the fire resistance rating of an RC beam. The above-reported fire tests did not consider important factors such as axial restraint, fire exposure, specimen size, loading and failure conditions. A review of the literature also shows that there have been limited numerical studies to account for the fire induced restraint effect on the behavior of the RC beams. The analytical studies reported by Dotreppe and Franssen [2], and Ellingwood and Lin [5], have a number of limitations. Specifically, they were not validated in the whole range of behavior and do not account for important factors such as the fire induced restraint effects, the fire exposure scenario, the failure criterion, the concrete strength and the load level. Also, the above analytical studies focused only on the behavior of the RC beams fabricated with NSC and cannot be used for the HSC beams. This is because, in the case of the HSC, spalling under fire situations is to be accounted for. Some researchers theorized spalling to be caused by the build-up of pore pressure during heating [6,7]. The HSC is believed to be more susceptible to this pressure build-up because of its low permeability compared to the NSC. The extremely high water vapor pressure, generated during exposure to fire, cannot escape due to the high density of the HSC and this pressure often reaches the saturation vapor pressure. At 300 1C, the pressure reaches about 8 MPa. Such internal pressures are often too high to be resisted by the HSC mix having a tensile strength of about 5 MPa [6]. Data from various studies show that predicting fire performance of HSC, in general, and spalling, in particular, is very complex since it is affected by a number of factors [6–8]. A recent study has developed an approach for modeling the response of the RC beams under the realistic fire and loading scenarios [9]. However, the model does not account for fire induced restraint effects. This paper presents an investigation into the influence of the fire induced restraint forces and moments on the behavior of the RC beams under fire and loading conditions.

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beam in the entire range of loading up to collapse under fire. The RC beam is divided into a number of segments along its length (as shown in Fig. 2) and the mid-section of the segment is assumed to represent the behavior of the whole segment. The fire resistance analysis is carried out by incrementing time steps. At each time interval, the numerical model performs the analysis through three main steps:

  

establishing temperatures due to fire exposure, carrying out heat transfer analysis to determine temperature distribution across each segment and performing the strength analysis, which is carried out through three sub-steps: J calculating the axial restraint force in the RC beam, J generation of the M–k relationships (utilizing the axial restraint force computed above) for each beam segment, J performing structural analysis of the beam to compute the deflections and the internal forces.

The fire induced restraint is accounted for in the analysis through the axial restraint force, which involves significant computational effort. The axial restraint force, at each time interval, is computed based on an iterative approach by satisfying the equilibrium and the compatibility criteria along the span of the beam. The deflections and the curvatures, which are required to calculate the axial force, are assumed to be equal to those calculated in the preceding time step. At room temperature, the analysis starts by assuming the axial force, due to fire induced restraint, to be zero. Once the axial restraint force in the beam is computed, the next step is the generation of the M–k relationships for each segment of the beam. It has been well established that the M–k relationships appropriately represent the behavior of an RC beam at ambient conditions. In the current model, such M–k relationships are established as a function of time for various segments in the beam and they are in turn used to trace the response of the beam under fire conditions. The effect of the restraint is included in the generated M–k relationships through the axial force computed above. The M–k relationships, at various time steps, are generated using the changing properties of the constituent materials, namely concrete and reinforcement. In this way, the material nonlinearity and the fire induced restraint effect will be implicitly accounted for in the analysis. Li

∗Segment

w

3. Numerical model 1∗

2

3

4

5

6

7

8

9

10

11

12

3.1. General approach The numerical model, proposed here, uses moment– curvature relationships to trace the response of an RC

L Fig. 2. Layout of a typical RC beam and its idealization for analysis.

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Using the M–k relationships, the load (moment) the beam can carry at a particular time step is evaluated. Also, the deflection of the beam at that time step can be calculated through a stiffness approach by evaluating the average stiffness of the beam. The average stiffness of the beam is computed using segmental stiffness, which is estimated by means of the known M–k relationships. A flowchart showing the numerical procedure for fire resistance calculations is given in Fig. 3. The temperatures and the strength capacities for each segment, and the computed deflections in the beam, are used to evaluate failure of the beam at each time step. At every time step, each beam segment is checked against four pre-determined sets of failure criteria, which include prescriptive thermal criterion, and performance-based strength and deflection considerations. The analysis is continued until the strength failure of the beam is reached. Based on the four sets of failure criterion outlined by Kodur and Dwaikat [9], the failure of an RC beam is said to occur when: (1) The temperature in steel rebars (tension reinforcement) exceeds the critical temperature which is 593 1C for reinforcing steel. (2) The beam is unable to resist the specified applied service load. (3) The maximum deflection of the beam exceeds L/20 at any fire exposure time, where L is the span length. (4) The rate of deflection exceeds the limit given by the following expression: L2 ðmm=minÞ, (1) 9000d where L is the span length of the beam (mm) and d the effective depth of the beam (mm).

ture relationship specified in ASTM E1529 [11] is incorporated into the model. 3.3. Heat transfer analysis The fire temperatures computed above are used to calculate the temperatures within the beam cross-section of each segment using a finite element method. The crosssectional area of each segment is subdivided into a number of elements and temperature rise in a beam segment is derived by establishing a heat balance for each element. Detailed equations for the calculation of segment temperatures are derived. The temperature is assumed to be uniform along the length of the segment and thus the calculations are performed for a unit length of each segment. Steel reinforcement is not specifically considered in the thermal analysis because it does not significantly influence the temperature distribution in the beam cross-section [12]. Generally, beams are exposed to fire from three sides as shown in Fig. 4. However, the heat transfer model is capable of predicting the cross-sectional temperature distribution for any type of boundary conditions. More details about the heat transfer analysis are provided in Ref. [13]. 3.4. Strength analysis 3.4.1. General analysis procedure The third step is the strength analysis at the mid-section of each segment. The cross-sectional temperature distribution generated from thermal analysis is used as an input to the strength analysis. For the strength analysis, the following assumptions are made:



Plane sections before bending remain plane after bending. Tensile strength of the concrete, at elevated temperatures, is accounted for based on the reduction factors proposed in Eurocode 2 [14]. There is no bond-slip between the steel reinforcement and the concrete. Concrete of strength higher than 70 MPa is considered to be HSC. For the HSC, the concrete strength in each element is computed based on the extent of spalling in that element.

The model generates various critical output parameters, such as temperatures, stresses, strains, axial restraint force, deflections and moment capacities, at various fire exposure times.



3.2. Fire temperatures



The fire temperatures are calculated by assuming that the three sides of the beam are exposed to the heat of a fire, whose temperature follows that of the standard fire exposure such as ASTM E119 [3] or any other design fire scenario [10]. The time–temperature relationship for the ASTM E119 standard fire can be approximated by the following equation:   pffiffiffiffi pffiffiffiffi T f ¼ T 0 þ 750 1  exp 3:79553 th þ 170:41 th , (2)

Additionally, for the axial force calculations, the following assumptions are made:

where th is the time (hours), T0 the initial temperature (1C) and Tf the fire temperature (1C). For design fires, the time–temperature relationship specified in the SFPE [10] is built into the model. Also, to simulate hydrocarbon fire scenarios, the time–tempera-



 

At any time step, the fire induced axial force is constant along the span of the beam. At any time step, the curvature and the displacements for each segment from the preceding time step are used to compute the fire induced axial restraint force.

Following the thermal analysis, and at every time step, the strength analysis is performed in three sub-steps, namely

ARTICLE IN PRESS M.B. Dwaikat, V.K.R. Kodur / Fire Safety Journal 43 (2008) 291–307

estimating the axial force in each segment of the restrained beam, generating the moment curvature (M–k) relationship for each segment and carrying out nonlinear stiffness

295

analysis to trace the response of the beam under fire conditions. The axial force, due to the fire induced restraint, is computed based on the curvature distribution

Start Input data, including spalling parameters Calculation of fire temperature

Thermal properties

Calculation of crosssectional temperature

Calculate the extent of spalling

Subroutine to calculate fire induced axial restraint force

Initial central total of concrete

Initial curvature

Stress-strain relations and deformation properties

Calculate strains and stresses

No

Internal axial force = applied force (0.0)

Increment curvature

Yes

Write curvature and internal moment (point on the momentcurvature curve)

∗ either crushing of concrete orrupture of steel ∗∗ n = the total number of segments in the beam

Ultimate curvature∗

No

Increment strain

No

Segment = i+1

Yes Segment = n∗∗ Yes Nonlinear stiffness analysis of beam

Calculate deflection and rate of deflections

Failure limit states

No

Increment time step

Yes End

Fig. 3. (a) Flowchart showing the steps associated with the analysis of an RC beam exposed to fire (main program), (b) flowchart showing the steps associated with the axial restraint force calculations (subroutine).

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Start subroutine Curvatures and deflections from preceding time

Generate axial-force-central-strain relationship for each segment

Initial axial force

Calculate central strain in each element

Check Compatibility and equilibrium at node B

No

Yes

End subroutine

Fig. 3. (Continued)

along the span of the beam. This axial force is assumed to be zero at the first time step (room temperature). For the subsequent time steps, the curvature distribution from the preceding time step is utilized to compute the axial force. The use of the curvatures from the preceding time step does not have a significant influence on the restraint force provided small time steps are used. The M–k relationships, at various time steps, are generated using the calculated axial restraint force and the constitutive relationships for the concrete and the reinforcement. In the model, uniaxial bending is assumed and the bending moment is assumed to be about the horizontal axis. However, the model can be extended to account for biaxial bending cases. The interaction between the axial force and the bending moment capacity (and stiffness of the beam) is automatically accounted for in the generated M–k relationships. In the nonlinear stiffness analysis, the deflection of the beam at each time step can be calculated through a stiffness approach which utilizes the generated M–k relationships to compute the stiffness for each beam segment. Various strain components including the mechanical strain, the thermal strain and the creep strain for both the concrete and the reinforcing steel and the transient strain for concrete are accounted for in the model. The creep and the transient strains, which are often not accounted for, might play a major role in predicting the fire behavior of the RC beams, particularly the deflection and the rate of deflection. Spalling of concrete

is incorporated into the model through a simplified approach proposed by Kodur et al. [15]. This approach is based on detailed experimental studies and considers various material and structural parameters that influence spalling. The strength calculations, at elevated temperatures, are carried out using the same rectangular network described above and shown in Fig. 4(b). The temperatures, the deformations and the stresses in each element are represented by those at the center of the element. The temperature at the center of the concrete element is obtained by averaging the temperatures of the nodes of that element in the network. The same procedure is used for the steel reinforcement, with the values of the stress, strain components and the temperature of each bar represented by those at the center of the bar. The temperature at the center of a steel bar is approximated by the temperature at the location of the center of the bar cross-section (in concrete). The total strain in a concrete element, at any fire exposure time, is given as the sum of the thermal expansion, the mechanical strain, the creep strain and the transient strain: t ¼ th þ me þ cr þ tr ,

(3)

where et is the total strain, eth the thermal strain, eme the mechanical strain, ecr the creep strain and etr the transient strain. The thermal strain is directly dependent on the temperature in the element and can be obtained by knowing the temperature and the thermal expansion of the concrete. The creep strain is assumed to be a function of time, temperature and stress level, and is computed based on Harmathy’s [16] approach using the following expression: cr ¼ b1

s pffiffi dðT293Þ te ,

f c;T

(4)

where b1 ¼ 6.28  106 s0.5, d ¼ 2.658  103 K1, T is the concrete temperature (K) at time t (s), fc,T the concrete strength at temperature T and s the stress in the concrete at the current temperature. The transient strain, which is specific for the concrete under fire conditions, is computed based on the relationship proposed by Anderberg and Thelandersson [17]. The transient strain is related to thermal strain as follows: Dtr ¼ k2

s Dth , f c;20

(5)

where k2 is a constant which ranges between 1.8 and 2.35 (a value of 2 will be used in the analysis), Deth the change in the thermal strain, Detr the change in the transient strain and fc,20 the concrete strength at room temperature. For the steel reinforcement, the total strain, at any fire exposure time, is calculated as the sum of three

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components, as given by the following equation: ts ¼ ths þ mes þ crs ,

(6)

where ets, eths, emes and ecrs are the total strain, the thermal strain, the mechanical strain and the creep strain in the steel reinforcement, respectively. Similar to the concrete, the thermal strain in steel can be directly calculated from the knowledge of the rebar temperature and the thermal expansion coefficient of the reinforcing steel. The creep strain is computed based on Dorn’s theory and the model proposed by Harmathy [18] with some modifications made to account for different values of yield strength of steel. According to Harmathy’s model, the creep strain in steel is given by the following expression: crs ¼ ð3Z2t0 Þ1=3 y1=3 þ Zy,

(7)

where 8 !4:7 > > s > 19 > ; > < 6:755  10 fy Z¼ > > > > 1:23  1016 ðe10:8ðs=f y Þ Þ; > :

297

9 > s 5 > > p > > f 12 = y

s 5 > > > 4 > ; f y 12 >

,

R y ¼ eDH=RT dt, DH=R ¼ 38; 900 K, t is the time (h), t0 ¼ 0:016ðs=f y Þ1:75 , s the stress in steel and fy the yield strength of steel. Fig. 5 shows the distributions of the total strain, the stress and the internal forces for the beam cross-section at any fire exposure time. The total strain in any element (concrete or rebar) can be related to the curvature of the beam by the following expression: t ¼ 0 þ ky,

(8) Ambient conditions

c b

Fire

Fire

D

c Fire Fig. 4. Cross-section of an RC beam and its discretization for analysis. (a) Cross-sectional details, (b) discretization, (c) boundary conditions for heat transfer analysis.

Element area Am

h/2

Geometrical centroidal axis σs,C

εs,C Neutral axis y

Element stress σm

ε0

Axis of zero mechanical strain

1

h/2

Cs Cc

κ

εs,T

Ct = C s + C c

P Tc Tt = Ts + Tc

σs,T Ts

Cross section

Total strain diagram

Stress diagram

Internal forces

Force Equilibrium P = Tt + Ct

Fig. 5. Variation of the strain, the stress and the internal forces in a typical beam cross-section exposed to fire.

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where e0 is the total strain at the geometrical centroid of the beam cross-section, k the curvature and y the distance from the geometrical centroid of the beam cross-section. In the model, Eqs. (3)–(8) can be used to carry out the strain computation of a segment at any given fire exposure time. At any time step, and for an assumed value of e0 and k, the total strain in each element (concrete or rebar) can be determined using Eq. (8). Then the thermal, the transient (for concrete only), and the creep strains in the concrete and the rebars are evaluated using known temperatures and the corresponding equations derived above. Using the knowledge of the total strain, and the thermal, the transient and the creep strains, the mechanical strain in the element can be expressed by rearranging Eqs. (3) and (6): me ¼ t  th  cr  tr mes ¼ ts  ths  crs

for concrete;

for steel:

(9) (10)

Then for the estimated mechanical strain, the stress in the element can be established using the stress strain relationships for steel and concrete. 3.4.2. Calculation of fire induced axial force The calculation of the fire induced axial force (P) in a beam segment is analogous to the analysis of the prestressed concrete beams, and is the summation of the compressive and the tensile forces. Thus, the axial force can be written as Z P ¼ T þ C ¼ s dA, (11) where T is the resultant tensile force in the concrete and the reinforcement, C the resultant compressive force in the concrete and the reinforcement and s the stress in the infinitesimal area dA within the cross-section of each segment. The integration in Eq. (11) is difficult to be computed analytically. Therefore, for an idealized cross-section, where the cross-sectional area is divided into a number of elements, Eq. (11) can be approximated by X P¼T þC ¼ sm A m , (12) where sm is the stress at the center of each element and Am the area of each element. However, the stress distribution in the cross-sectional area of the segment can be computed for a given central total strain (e0) and the curvature (k) using Eqs. (3)–(10) and the constitutive relationship of the constituent materials. Thus, the axial force can be related to the central total strain (e0) and the curvature (k) in mathematical formulation as P ¼ fð0 ; kÞ.

(13)

Eq. (13) shows that the axial force in a beam segment can be calculated if k and e0 are known for that segment. For the beam segment i, the axial force can be written as Pi ¼ fð0i ; ki Þ,

(14)

where Pi, e0i, ki are the axial force, the central total strain and the curvature in segment i. As stated in the assumptions above, the axial restraint forces for all segments are equal. Hence, the axial force in segment i (Pi) equals constant value (P). Based on the assumptions above, the curvature in segment i, for the nth time step (kin), is equal to the curvature in the same segment computed in the (n1)th time step (kin1). Thus, for an assumed value of P, the central total strain in each segment is computed by solving Eq. (14), which can be written as P ¼ fð0i ; kni Þ  fð0i ; kn1 Þ. i

(15)

The total central strain in each segment is used to check the compatibility conditions. The value of the axial force, P, is modified until the compatibility and the equilibrium conditions are satisfied. Fig. 6 shows an RC beam exposed to fire and its deformed shape at the current and the preceding time steps of fire exposure. The columns restraining the beam are modeled as a spring of stiffness (k). The compatibility conditions along the span of the beam require that the following equation is satisfied: X l i  L  D ¼ 0, (16) where li is the projected length of the deformed segment i, L the beam length and D the total expansion in the beam length. Fig. 6(b) shows that the projected length of a beam segment is given by the following formula: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2 l i ¼ ðsi Þ2  ðwni2  wni1 Þ2  ðsi Þ2  ðwn1 (17) i2  wi1 Þ , and where si is the length of the deformed segment i, wn1 i1 n1 wi2 the deflections at the beginning and the end of the beam segment which were computed in the (n1)th time step and wni1 and wni2 the deflections at the beginning and the end of the beam segment in the nth time step. Also, the length of the deformed segment i (si) can be computed from the definition of strain as si ¼ ð1 þ 0i ÞLi ,

(18)

where Li is the length of segment i in the undeformed beam. Combining Eqs. (16)–(18), the compatibility conditions require that X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2 ð1 þ 0i Þ2 L2i  ðwn1 (19) i2  wi1 Þ  L  D ¼ 0. Further, the equilibrium of node B in Fig. 6 requires that P ¼ kD.

(20)

Thus, for an assumed value of axial force (P), an iterative procedure is used to solve Eq. (15) in order to compute the axial strain in each segment. Then, Eq. (19) is checked to ensure the compatibility requirements with D being calculated using Eq. (20). The value of P is modified until Eq. (19) is satisfied within a pre-determined tolerance. The error involved in the estimation of Eqs. (17) and (15) becomes smaller if shorter time steps are used.

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Segment i k A

B

Li L

n−1

k

li

wi1n−1

wi2n−1

 n−1

sin−1 n

k

li

wi1n wi1n

(n-1)th time step

nth time step

sin n lin

lin−1 wi1n−1

wi2n−1 sin−1 (n-1)th time step

wi1 wi2n sin nth time step

Fig. 6. Illustration of the axial restraint force calculations. (a) Deflected shape at the (n1)th and the nth time step, (b) typical beam segment.

3.4.3. Generation of M–k relationships The moment–curvature generation, at elevated temperatures, is carried out using the same rectangular network described above and shown in Fig. 4. Once the axial restraint force in the beam is computed, the M–k relationships are generated through an approach analogous to the method used for the prestressed concrete beam. In this approach the M–k relationships are established by iterating the central total strain (e0) and the curvature (k). At the beginning of the analysis, values for the curvature and the central total strain (in concrete) are assumed. Then, the total strain in each of the rebars and the concrete elements is computed from the assumed strain and curvature. The stresses in the rebars and the concrete elements are determined using the constitutive laws of the materials. The temperature of a rebar is assumed to be equal to the temperature at the location of the center of the rebar. Once the stresses are known, the forces are computed in the concrete and the rebars. The curvature is iterated until equilibrium of forces is satisfied (internal force equal to the fire induced axial restraint force). Once the equilibrium is satisfied, the moment and the corresponding curvature are calculated. Thus, the values of the moment and the curvature are stored to represent a point on the moment–curvature curve. The value of the central total strain is incremented to generate subsequent points on the moment–curvature curve. This procedure is repeated for each time step of fire exposure. The generated moment–curvature curves are used for tracing the behavior of the beam through nonlinear structural analysis. The

generation of M–k relationships is an important part of the proposed numerical model since these relationships form the basis for the analysis of an RC beam exposed to fire. 3.4.4. Beam analysis The moment–curvature relationships and the axial restraint force generated for various segments are used to trace the response of the whole beam exposed to fire. At each time step the deflection of the beam is evaluated through stiffness approach. The secant stiffness for each segment is determined from the M–k relationships, based on the moment level reached in that particular segment. The second order effect that results from the axial restraint force is accounted for in the model. The effect of the second order moments, developed due to the axial restraint force, is calculated and added to the external loads in the stiffness analysis. To compute the deflection of the beam at any time step and for a given loading condition, the stiffness matrix and the loading vector are computed for each longitudinal segment. Then they are assembled in the form of nonlinear global stiffness equation, which can be written as K g d ¼ Pf þ Ps ,

(21)

where Kg is the global stiffness matrix, d the nodal displacements, Pf the equivalent nodal load vector due to applied loading and Ps the equivalent nodal load vector due to the P–d effect. Thus, for any given time step, the temperatures (in concrete and steel), the moment capacity and the curvatures

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as well as the deflection and the rate of deflection of the beam are known for a given fire exposure. These output parameters are used to evaluate the failure of the beam at local (in each segment) and global (whole beam) levels. 3.5. Spalling Spalling of concrete is incorporated into the model through a simplified approach proposed by Kodur et al. [15]. This approach is based on some experimental studies and considers various material and structural parameters that influence spalling. According to their model, the extent of spalling in the HSC can be computed based on the following guidelines:

 

  

Spalling occurs when the temperatures in an element exceed 350 1C. Spalling is influenced by the stirrup configuration adopted for the beam as follows: J Spalling occurs throughout the cross-section when the stirrups are bent in a conventional pattern (no hooks). J Spalling occurs only outside the reinforcement cage when the stirrups are bent at 1351 into the concrete core. No spalling occurs when polypropylene fibers are present in the concrete mix. No spalling occurs inside the reinforcement core when the stirrup spacing is 0.7 times the standard spacing. The extent of spalling depends on the following factors: J Type of aggregate: the extent of spalling is higher (100%) in the siliceous aggregate HSC than that in the carbonate aggregate HSC (40%). J Presence of steel fibers in the concrete mix: the extent of spalling in the HSC beams with steel fiber is about 50%.

In addition, it is assumed that spalling does not occur in the RC beams fabricated with NSC. Thus, the proposed numerical model is capable of accounting for the nonlinear high-temperature material characteristics, the complete structural (beam) behavior, various fire scenarios, the fire induced restraint effects, the concrete types (such as different aggregate types, fiber mixes and concrete strengths), the spalling mechanisms and the failure criteria. 4. Validation of the model As indicated in the literature review, there have been no well-reported fire tests that have been carried out on RC beams under restraint conditions. Consequently, there is a lack of test data for comparing the predictions from the proposed model. Therefore, the validity of the model was established by comparing predicted results from the model with the measured values from fire tests for SS beams tested by Lin et al. [1] and Dotreppe and Franssen [2]. In addition

the model was validated against the results of a finite element program, SAFIR, for a pin–pin beam. More discussion about the validation of the model is presented in the following sections. 4.1. Comparison with the beam tested by Lin et al. [1] The model was validated against the fire test reported by Lin et al. [1]. The geometric and the material properties of the tested beam are taken from the literature and are given in Table 1 [1]. The axial restraint stiffness (k), for the beam, is assumed to be zero. The beam was tested under ASTM E119 standard fire exposure and hence the beam was also analyzed by exposing its three sides to the standard time–temperature curve specified in ASTM E119. The fire resistance of the beam is calculated based on the four sets of failure criteria previously discussed and the results are summarized in Table 1. Predicted results from the analysis are compared to measured values from the fire tests in Figs. 7 and 8. In Fig. 7, the calculated average temperatures in the rebars are compared with the measured values for Beam I, reported by Lin et al. [1]. It can be noted that there is good agreement between the predicted and the measured values in the entire range of fire exposure. The steep increase in the rebar temperature in the early stages of fire exposure is due to the occurrence of high thermal gradient at the beginning of the fire exposure time as a result of faster increase in fire temperature (see ASTM E119 fire curve in Fig. 7). A review of the predicted temperatures in concrete at various depths indicated that the model predictions follow the expected trend with lower temperatures at larger depths from the fire exposed surface. However, the predicted concrete temperatures could not be compared with test data since the measured temperatures were not Table 1 Properties and analysis results for the RC beam tested by Lin et al. [1] Property Description Cross-section (mm) Length (m) Reinforcement

Beam I

Tested by Lin et al. [1] 305  355 6.1 2 +19 mm top bars 4 +19 mm bottom bars 30 f0 c (MPa) fy (MPa) 435.8 Loading ratio 0.42 Applied total load (kN) 80 Concrete cover thickness (mm) 25 (bottom) 38 (side) Aggregate type Carbonate Fire resistance based on the failure criterion (min) Rebar temperature 110 Strength 140 Deflection (BS 476) 102 Rate of deflection (BS 476) 105 Fire resistance based on ACI 216.1 (min) 180 Fire resistance based on test (min) 80

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Temperature (°C)

1000 800

Critical temperature limit for rebars = 593 °C

600 400 ASTME119 fire

200

Test data Model

0 0

30

60

90

120

150

Time (min)

Fig. 7. Predicted and measured rebar temperatures for test beam, Beam I.

Time (min) 0

30

60

90

120

150

301

the strength failure criterion. Overall, the predicted fire resistance from the deflection failure criteria (102 min) is a reasonable estimate to the measured value in the fire test (when the test was terminated). The fire resistance of the beam was also estimated using ACI 216.1 prescriptive criterion and accordingly it was 191 min. This fire resistance value is higher than that predicted by the computer model for the four failure criteria. This is mainly because the fire resistance predicted by ACI 216.1 is based on the concrete cover thickness to the rebar (and minimum cross-sectional dimensions) of the beam and does not account for important factors such as the load level, the strength and the deflection criteria [19]. 4.2. Comparison with the beam reported by Dotreppe and Franssen [2]

0.0

-500.0 Deflection limit = 305 mm

-750.0

Test data Model

-1000.0 -1250.0 -1500.0

Fig. 8. Predicted and measured deflections for test beam, Beam I.

reported by Lin et al. [1]. Fig. 8 shows the predicted and the measured midspan deflections as a function of the fire exposure time for Beam I tested by Lin et al. [1]. It can be seen that the model predictions are in close agreement with the measured deflections, throughout the fire exposure time. The fire resistance of this beam was evaluated based on four sets of failure criteria and its values are given in Table 1. In addition, the fire resistance of the beam was also evaluated based on ACI 216.1 specifications [19]. The measured fire resistance, for Beam I, is lower than that predicted by the computer model for all failure criteria. This is mainly because the test was terminated after 80 min of fire exposure, and before the beam attained complete failure (common practice in fire resistance tests to limit the possible damage to equipment from sudden literature), probably due to the severe conditions experienced towards the final stages in the fire tests [1]. The fire resistance of this beam would have been slightly higher if the test had been continued till the complete failure. The fire resistance predicted based on the rebar temperature (110 min) is much lower than that for the strength failure criterion (140 min). This is because the rebar temperature failure criteria are based on load level of 50% of the room temperature capacity of the beam; however, the load level on this beam is lower than 50% (about 42%) and this results in higher fire resistance from

The model was also validated against the fire test results reported by Dotreppe and Franssen [2]. The predicted rebar temperatures and the deflections from the analysis show good agreement with the measured values throughout the test. Fig. 9 shows the variation of deflection as a function of the fire exposure time for the beam reported by Dotreppe and Franssen [2]. It can be seen that the deflections predicted by the model are quite close to the measured deflection throughout the fire exposure time. The fire resistance values of the beam tested by Dotreppe and Franssen were evaluated based on the thermal, the strength, the deflection and the rate of deflection failure criteria and were found to be 120, 145, 123 and 115 min, respectively. The measured fire resistance for this beam was 120 min. For this beam, the thermal failure criterion predicts lower fire resistance than the strength failure criterion because the load ratio in the test was lower than 50%. 4.3. Comparison with SAFIR for a pin–pin beam Results from the computer model were compared with the results from the finite element program, SAFIR, for a pin–pin RC beam. The beam has the same material and Time (min) 0

30

60

90

120

150

0 -50 Deflection (mm)

Deflection (mm)

-250.0

-100 -150 -200 -250

Test Data Model

-300 -350

Fig. 9. Predicted and measured deflections for test beam reported by Dotreppe and Franssen [2].

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beams with sufficient degree of accuracy for practical applications. It should be noted that the model developed in this paper accounts for the creep and the transient strain components as well as the fire induced spalling of the HSC which are not accounted for in SAFIR.

cross-sectional properties as those shown in Fig. 10(a). The applied loading on the beam is assumed to be uniformly distributed with a value of 22.5 kN/m. The beam is assumed to be exposed to ASTM E119 standard fire from three sides. The obtained results are shown in Figs. 11 and 12. The variation of the axial restraint force as a function of the fire exposure time for the two models is shown in Fig. 11. It can be seen that both the curves are in good agreement. The minor discrepancies between the two curves can be attributed to the differences in the constitutive relationships and the numerical procedures used in the two models. Fig. 12 shows the variation of the deflection of the beam as a function of fire exposure time based on the results of the two models. It can be seen that the predicted deflections (using the model developed in this paper) are pretty close to SAFIR predictions. The results also show that the model, developed in this paper, predicted close fire resistance to that computed by SAFIR. SAFIR predicted fire resistance of 73 min for this beam, while the model, developed here, predicted fire resistance of 70 min. This points out that the model (developed here) is capable of predicting the fire response of the restrained RC

40 mm

2500

Axial force (kN)

2000

1500 SAFIR Model

1000

500

0 0

15

30

45

60

75

90

Time (min)

Fig. 11. Predicted axial restraint force from SAFIR and the developed model.

150 mm

Point A

f'c = 30 MPa f y = 400 MPa

6 φ 20 mm

500 mm

φ 10 mm stirrups 40 mm 300 mm 22.5 kN/m k

6m 38 kN/m k

6m Fig. 10. Elevation and cross-section of the RC beam used in case studies. (a) Cross-section, (b) elevation for a simply supported beam, (c) elevation for a fixed ended beam.

ARTICLE IN PRESS M.B. Dwaikat, V.K.R. Kodur / Fire Safety Journal 43 (2008) 291–307

5. Case studies To investigate the influence of the fire induced restraint forces on the flexural response of the RC beams exposed to fire, two sets of RC beams were analyzed. The first set of beams was analyzed to study the effect of axial restraint only, while the second set was analyzed to study both the rotational restraint and the axial restraint. The beams in the first set (BSS1, BSS2, BSS3, BSS4 and BSS5) are all SS with different values of axial restraint stiffness (k). The values of the axial restraint stiffness are varied in such a way that they cover various types of boundary conditions commonly encountered in practice. As an illustration, the axial restraint stiffness of 200 kN/mm for beam BSS5 represents very stiff boundary conditions (shear wall), while a value of 5 kN/mm for beam BSS2 represents very soft columns on the beam boundaries. The beams in the second set (BFE1, BFE2, BFE3, BFE4 and BFE5) are rotationally fixed from both the ends (FE), but allowed to expand axially with different values of axial restraint stiffness. All the analyzed beams have the same length and the cross-sectional dimensions as shown in Fig. 10, and they were analyzed under the exposure of ASTM E119 standard fire. The beams are assumed to be made of Time (min) 0

15

30

45

60

75

90

0

Deflection (mm)

-20 -40 SAFIR

-60

Model

303

concrete with compressive strength of 30 MPa and reinforced with steel rebars having yield strength of 400 MPa. The room temperature capacity of the analyzed beams was calculated based on ACI 318 [20]. The applied loading, shown in Fig. 10, is calculated for a dead-loadto-live-load ratio of 2, based on ASCE 07 [21] (1.2 dead load+1.6 live load for room temperature calculation and 1.2 dead load+0.5 live load under fire conditions). For the load calculations, the ultimate load at room temperature is equated to the room temperature capacity of each beam. Further, the allowable redistribution of moment of 20% at room temperature (resulting from the support conditions), as per ACI 318 provisions, is accounted for in the load calculations. The fire resistance is evaluated based on the four sets of failure criteria (one thermal, one strength and two deflection limit states). All the beams are analyzed using 5-min time steps. A summary of the results from the analysis is presented in Table 2. 5.1. SS beams Results from the analysis of the fire induced axial restraint force on the flexural response of five SS RC beams, namely BSS1, BSS2, BSS3, BSS4 and BSS5, are presented in Figs. 13–15 and in Table 2. The assumed axial restraint stiffness for these beams are 0.0, 5.0, 20.0, 50.0 and 200.0 kN/mm, respectively. Fig. 13 shows that except between 5 and 15 min of the fire exposure time, the axial restraint force for the four analyzed restrained beams (beam BSS1 has zero axial restraint stiffness) increases with time till failure. This is because the development of the fire induced axial restraint force in the RC beams is influenced by two main factors:

-80

 -100 -120

Fig. 12. Predicted deflections from SAFIR and the developed model.

Thermal expansion of the beam which is restricted by the axial restraint boundaries. Thermal expansion of the beam increases with the fire exposure time due to the increase in fire (ASTM E119 standard fire) and beam temperatures throughout the fire exposure time.

Table 2 Properties and results for the RC beams used in the analysis Beam designation

Boundary conditions

BSS1 BSS2 BSS3 BSS4 BSS5

Simply supported (SS)

BFE1 BFE2 BFE3 BFE4 BFE5

Fixed end (FE)

a

No failure.

Axial restraint stiffness (kN/mm)

Fire resistance based on the failure criterion (in min) Rebar temperature

Strength

Deflection

Rate of deflection

0 5 20 50 200

180 180 180 180 180

145 130 155 175 235

127 127 154

129 125 150

a

a

a

a

0 5 20 50 200

180 180 180 180 180

250 340 415 435 420

a

a

a

a

a

a

a

a

a

a

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leads to higher axial expansion (D), for relatively small deflections. This increases the developed axial restraint force and thus makes the developed arch action more effective.

1400 1200 Axial force (kN)

4

1000 800

3

600 2 1-SS (k = 5 kN/mm)

400

2-SS (k = 20 kN/mm)

200

3-SS (k = 50 kN/mm)

1

4-SS (k = 200 kN/mm)

0 0

50

100

150

200

250

Time (min)

Fig. 13. Variation of the axial restraint force as a function of the fire exposure time for a simply supported RC beam with different values of k.

200 1

Moment (kN.m)

160

2

4 5

120 3

80

1-SS and SS + k = 5 kN/mm (Time = 0.0) 2-SS (Time = 60 min) 3- SS (Time = 120 min)

40

4 - SS + k = 5 kN/mm (Time = 60) 5- SS + S = 5 kN/mm (Time = 120 min)

0 0

50

100

150

200

520

300

350

400

Curvature (1/km)

Fig. 14. Moment–curvature curves at various times for beams BSS1 and BSS2 under fire exposure. Time (min) 0

50

100

150

200

250

0

Deflection (mm)

-100

5

-200

4 3

-300 -400

Deflection limit = 300 mm

1-SS (k = 0.0) 2-SS (k = 5 kN/mm) 3-SS (k = 20 kN/mm)

2

-500 1

4-SS (k = 50 kN/mm) 5-SS (k = 200 kN/mm)

-600

Fig. 15. Variation of deflection as a function of the fire exposure time for a simply supported RC beam with different values of k.



The vertical location of the restraint axial force in the RC beams is generally below the neutral axis of any cross-section along the span of the beam as a result of the thermal gradients. This will develop an arch action which counters the effect of the applied loading, as shown in Fig. 16, and reduces the downward deflection of the beam. Increased downward deflection of the beam

At early stages of fire exposure, the location of developed axial force is quite below the neutral axis at the critical span section. This creates countermoments that reduce the downward deflection of the beam. The reduction in the deflection of the beam reduces the developed axial restraint force in the beam (particularly for high axial restraint stiffness) as can be seen from Fig. 13. This explains the kink in the development of the axial restraint force shown in Fig. 13 between 5 and 15 min of fire exposure. However, with the fire exposure time, the beam experiences higher thermal expansion which will increase the developed axial restraint force. As expected, the axial restraint force, developed in the beam, increases by increasing the axial restraint stiffness as shown in Fig. 13. This is mainly because of the high restrictions on the beam expansion which are provided by the restraining boundaries. The moment–curvature curves at various times of fire exposure are shown in Fig. 14 for axially restrained SS beams. Fig. 14 shows that the moment–curvature curve at room temperature is the same for the restrained and the unrestrained beams. This is because the axial restraint force is assumed to be zero at room temperature. The figure also shows that, for both the restrained and the unrestrained beams, the moment capacity decreases, and the curvature increases with the fire exposure time. However, lower rate of reduction of moment capacity is observed for the restrained beams. This is mainly due to the effect of the developed arch action which opposes the applied loading and increases the capacity of the restrained beams. The variations of deflection as a function of the fire exposure time for the analyzed beams are illustrated in Fig. 15. The figure shows that, for the beam BSS1 (with zero axial restraint stiffness), the deflection of the beam increases throughout the fire exposure time. However, for the other four beams (BSS2, BSS3, BSS4 and BSS5), the same trend is observed except between 5 and 15 min of the fire exposure time. Within this time interval, the deflection of the beam decreases and then increases again. This is because of the development of arch action as discussed above in this section. At the early stages of fire exposure, the arch action is effective in reducing the deflection of the beam because the vertical distance between the axial restraint force and the neutral axis is relatively large due to high thermal gradients across the beam depth. However, the arch action becomes less effective in reducing the downward deflection of the beam at the later stages of fire exposure. The reasons for this are:



The neutral axis of any section along the span of the beam moves downward because of the increase in the axial restraint force which requires larger area of

ARTICLE IN PRESS M.B. Dwaikat, V.K.R. Kodur / Fire Safety Journal 43 (2008) 291–307

305

Line pass through the neutral axes of different sections along the span of the beam

Axial restraint stiffness k

Load

Axial restraint force

Reaction

P = kΔ

Reaction Δ Fig. 16. Illustration of the arch action mechanism in the RC beams.



compressive concrete zone. This will reduce the distance between the axial force and the neutral axis, and therefore reduce the effect of the arch action. The P–d effect which creates an additional moment that increases the deflection.

The fire resistance of the five beams is shown in Table 2. The table shows that there is a difference between the fire resistance values obtained based on the strength failure criterion and those obtained based on the thermal failure criterion. This is because the applied load ratio (the ratio of the applied load under fire conditions to the capacity of the section at room temperature), used in the analysis, is quite higher than 50%, and also because the thermal limit state is based on the critical temperatures which are computed for 50% load ratio. Further, the table shows that, for most of the beams, neither the deflection, nor the rate of deflection limit states, is exceeded (see Fig. 15). This is because of t he influence of the developed arch action which reduces the deflection and the rate of deflection of the beams. Furthermore, the table shows that, except for beam BSS2, the fire resistance of the RC beams increases by increasing the axial restraint stiffness. This is due to the fact that increasing the axial restraint stiffness will increase the axial restraint force which in turn will enhance the arch action effect in countering the applied loading (increase the moment capacity of the beam) and that will in turn increase the fire resistance. However, the lower fire resistance obtained for beam BSS2 (which has a very small axial restraint stiffness), than that for beam BSS1, is due to the P–d effect dominating the arch action effect at the later stages of fire exposure. In other words, certain level of the degree of restraint is required for the fire induced restraint effect to be fully beneficial. Based on ACI 216.1 tables, the fire resistance of beam BSS1 is about 3.5 h and it is more than 4 h for the other

beams. The large difference between the fire resistance values obtained from the analysis and those given in ACI 216.1 is mainly because the ACI 216.1 tables are computed based on the prescriptive conditions that do not account for the variations of important factors, such as the load level and the restraint conditions.

5.2. Rotationally restrained beams To investigate the effect of the fire induced axial restraint force on the flexural response of the rotationally restrained RC beams, five RC beams, namely BFE1, BFE2, BFE3, BFE4 and BFE5, were analyzed. The results of the analysis are shown in Figs. 17–19 and in Table 2. The variation of the axial restraint force as a function of time is shown in Fig. 17 for four beams (BFE1 has zero axial restraint stiffness). It can be seen from Fig. 17 that the axial restraint force increases throughout the fire exposure time for low axial restraint stiffness. However, for high axial restraint stiffness, different trend is observed. For the highest axial restraint stiffness used in the analysis (k ¼ 200 kN/mm), the axial force decreases within the time period between 180 and 390 min. This is because, during this period, most of the cross-sectional area of the beam gains high temperature which results in the softening of the constituent materials (concrete and steel) and thus reducing the axial stiffness and the axial force capacity of the beam. The sharp increase in the axial force just prior to failure, particularly for high axial restraint stiffness, is due to the high increase in deflection which increases the arch action effect (discussed above) prior to the failure of the beam. This axial force reduction is not observed for the SS beams because the failure occurs, in this case, before the constituent materials attain large loss of strength and stiffness (as a result of no redistribution of moments).

ARTICLE IN PRESS M.B. Dwaikat, V.K.R. Kodur / Fire Safety Journal 43 (2008) 291–307

306 1200

1-FE (k = 5 kN/mm) 2-FE (k= 20 kN/mm)

Axial force (kN)

1000

3-FE (k= 50 kN/mm) 4-FE (k= 200 kN/mm)

800

4 3

600 2

400 1

200 0 0

60

120

180

240

300

360

420

480

Time (min)

Fig. 17. Variation of the axial force as a function of the fire exposure time for a fixed ended RC beam with different values of k.

Time (min) 0

60

120

180

240

300

360

420

480

0

Deflection (mm)

-50 5

-100 1-FE (k = 0) 2-FE (k = 5 kN/mm)

-150

3-FE (k = 20 kN/mm)

3 4

4-FE (k = 50 kN/mm)

-200

2

5-FE (k = 200 kN/mm) 1

-250

Fig. 18. Variation of deflection as a function of the fire exposure time for a fixed ended RC beam with different values of k.

Axial force eccentricity (mm)

25

1-FE (k = 5 kN/mm) 2-FE (k = 20 kN/mm)

1

20

3-FE (k = 50 kN/mm) 4-FE (k = 200 kN/mm)

15 Geometrical centroidal axis 2

10

P

3

P

4

e

5

0 0

60

120

180

240

300

360

420

480

Time (min)

Fig. 19. Variation of the eccentricity, e, as a function of the fire exposure time for a fixed ended RC beam with different values of k.

Fig. 18 illustrates the variation of the midspan deflection of the analyzed beams as a function of the fire exposure time. The figure shows that the deflection increases throughout the fire exposure time with lower increase of deflection during the early stages of the fire exposure time.

This is mainly because the beam has not yet lost significant stiffness due to fire exposure and also because of the arch action effect which plays a major role in reducing the deflection in the early stages of the fire exposure time. However, the softening of the beam material and the P–d effect reduce the arch action effect at the later stages of fire exposure and thus higher values of deflection are observed. It can also be seen from Fig. 18 that, at the later stages of fire exposure and by increasing the axial restraint stiffness, the deflection of the beam decreases until a certain value of the axial restraint stiffness, then the deflection increases. This could be attributed to the fact that, for low axial restraint stiffness, the arch action effect increases with increasing the axial restraint stiffness which in turn reduces the deflection of the beam. However, for high axial restraint stiffness, high axial force is developed in the beam. This moves the neutral axis of the critical span section downward in order to increase the area of the concrete in the compressive zone. Thus, the arch action effect becomes less effective due to the reduction in the eccentricity between the axial restraint force and the neutral axis of the critical span section. Further, the higher developed axial restraint force will increase the second order moment resulting from the P–d effect, particularly at later stages of fire exposure. The P–d effect is influenced by the distance between the geometrical centroid of the beam cross-section and the axial restraint force. This eccentricity decreases with the fire exposure time as illustrated in Fig. 19. The eccentricity is calculated by dividing the support bending moment by the axial restraint force developed in the beam. Positive eccentricity is assumed when the axial force is below the geometrical centroid. It can be seen that the highest eccentricity is observed at the beginning of fire exposure, then the eccentricity decreases with the fire exposure time. This is due to the high thermal gradients developed in the beam cross-section at the early stages of the fire exposure time. However, the thermal gradients in the beam crosssection, decrease with time and thus the eccentricity decreases asymptotically to zero as can be seen from Fig. 19. The figure also shows that increasing the axial restraint stiffness will reduce the eccentricity at any fire exposure time. This can be attributed to the fact that increasing the axial restraint stiffness increases the axial restraint force developed in the beam and thus the eccentricity decreases to limit the bending moment at the end supports of the beam. The fire resistance of the five analyzed beams is shown in Table 2. The table shows that there is a large difference between the fire resistance values obtained based on the strength failure criterion and those obtained based on the thermal failure criterion. This could be attributed to the higher moment redistribution (under the fire exposure conditions) taking place between the support and the midspan moments, and this will increase the capacity of the beam. Further, under fire exposure, the negative moment capacity at the beam supports decreases with relatively

ARTICLE IN PRESS M.B. Dwaikat, V.K.R. Kodur / Fire Safety Journal 43 (2008) 291–307

lower rates when compared to the span moment capacity. This is because the tension rebars at support sections are relatively far from the fire exposed surface of the beam. The table also shows that for all the analyzed beams neither the deflection, nor the rate of deflection limit states, is exceeded. This is because of the influence of the developed arch action and support moments which reduce the deflection and the rate of deflection of the beams. Further, the table shows that the fire resistance of the RC beams increases by increasing the axial restraint stiffness till a certain value then it decreases. This is due to the fact that increasing the axial restraint stiffness will increase the axial restraint force which will increase both the arch action effect and the P–d effect. For low axial stiffness, the arch action effect will dominate the behavior and thus the fire resistance increases with increasing the axial restraint stiffness. However, for large axial restraint stiffness, the P–d effect dominates the behavior and thus the fire resistance decreases with increasing the axial restraint stiffness as shown in Table 2. Based on ACI 216.1 tables, the fire resistance is more than 4 h for the five analyzed beams. However, the predicted fire resistance for the five beams is higher than 4 h as can be seen from Table 2. It should be noted that the ACI tables do not account for important factors such as the load ratio and the axial restraint stiffness which dramatically influence the fire resistance of the RC beams as shown in Table 2. 6. Conclusions Based on the results of this study, the following conclusions can be drawn:









There is very limited information on the fire induced restraint effects in the RC beams and on the overall fire response of the RC beams especially under design fire and realistic loading scenarios. The moment–curvature-based numerical concept, presented in this study, is capable of predicting the fire behavior of restrained RC beams in the entire range: from pre-fire stage to collapse stage, with an accuracy that is adequate for practical purposes. Using the model the fire resistance of an RC beam can be evaluated for any value of significant parameters such as the axial restraint stiffness, the load level, the concrete strength, the beam length and the section dimensions, without the necessity of testing. Fire induced rotational and axial restraint effects have major influence on the fire resistance of the RC beams. Generally, the fire resistance of the RC beams increases when the beam is axially and/or rotationally restrained. The effect of fire induced restraint on the fire resistance of the RC beams is largely affected by the axial restraint stiffness. Therefore, the fire resistance of the RC beams should be evaluated based on the realistic axial restraint stiffness.

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