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Size effect on fire resistance of structural concrete
Engineering Structures 99 (2015) 468–478
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Size effect on fire resistance of structural concrete Dronnadula V. Reddy a, Khaled Sobhan a,⇑, Lixian Liu b, Jody D. Young Jr. c a
Center for Marine Structures and Geotechnique, Civil, Environmental and Geomatics Engineering, Florida Atlantic University, Boca Raton, FL 33431, United States Kunming University of Science and Technology, Yunan Province, China c McCall Engineering, 6389 Tower Lane, Sarasota, FL 34240, United States b
a r t i c l e
i n f o
Article history: Received 5 March 2014 Revised 13 March 2015 Accepted 11 May 2015
Keywords: Size effect Fire resistance Reinforced concrete Column Beam
a b s t r a c t Research on the size effect of reinforced concrete members at ambient temperatures includes only limited studies on the size effect of their fire resistance, integrating thermal and structural similitudes. Preliminary testing on concrete cylinders and unreinforced beams showed a 1=4 ASTM E-119 exposure time for the fire rating of small scale specimens compared to the full scale ones. Based on numerical and experimental analyses at Florida Atlantic University, the following quantitative relationships were developed for columns and beams: (i) equivalent fire duration and specimen size; (ii) fire endurance and cross-section size, and (iii) cross-section size, concrete cover thickness, and load capacity after fire exposure. The fire endurance at similar load ratios increases with an increase of member cross-section size and concrete cover thickness. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Size has an effect on nominal strength of specimens made with quasibrittle materials such as concrete, rock, ice, ceramic, and composite materials [3,4,12]. In compressive and flexural failures of quasibrittle materials, the size effect is quite apparent. Investigations of the size effect on nominal strength have become a focus of interest to many researchers [4,12]. Gonnerman [10] experimentally showed that the compressive strength decreases as the specimen size increases. This phenomenon of reduction in strength, dependent on specimen size, is called the ‘‘reduction phenomenon,’’ due to the statistical effect of an inherent larger number of ‘‘flaws’’ in the larger sample. However, rare attention has been paid to the size effect on mechanical properties of RC members in the fire condition. Some recent relevant papers have addressed the constitutive relationships for structural concrete members at elevated temperatures [24,7,6]. Fire testing, especially full scale, is expensive and time-consuming for large complicated structures, and the values from small scale laboratory tests differ from those of real large structures due to the scale effect. The primary objective of this paper was to study the size effect on fire performance of axially loaded square RC columns and simply supported reinforced beams using the numerical modeling developed by Liu [19]. From a preliminary investigation comprising
⇑ Corresponding author. E-mail addresses: [email protected] (D.V. Reddy), [email protected] (K. Sobhan). http://dx.doi.org/10.1016/j.engstruct.2015.05.015 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.
concrete cylinders and unreinforced beams under direct fire exposure, a reduction in fire resistance rating in the reduced cross-section was observed (based on Heisler/Gröber charts for the heat transfer analysis for thermal similitude), i.e., 1=4 exposure time of that for the full-scale specimens, based on small scale specimen fire exposure testing in a furnace [21]. The current study comprises: (i) Fire endurance of a reference column as a function of the load ratio, based on fire test data or validated numerical modeling; (ii) Relationship between the load capacity of RC columns/beams in the fire condition and the cross-section size; (iii) Relationship between the fire endurance of RC columns/beams and the cross-sectional size; and (iv) Relationship between the fire endurance of RC columns/beams and the concrete cover thickness.
2. Research significance In view of the concerns related to increasing frequencies, and catastrophic effects of fire on the infrastructure, as evidenced by recent events, man-made and natural, considerable attention is warranted to fire-resistant structural design. This study, incorporating the integrated effect of thermal and structural similitudes, addresses the numerical modeling and validation by laboratory testing of certain critical structural and material parameters, i.e. cross-section sizes, cover thickness, and concrete properties. The outcomes will significantly impact the design criteria, and reduce the need for expensive and time-consuming large-scale fire testing.
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on the nominal strength of large structures failing in compression was derived by Bazant and Xiang [4] and has the form:
3. Methodology 3.1. Preliminary fire tests
rn ¼ C 1 d2=5 þ C 0
A series of fire tests was performed to develop empirical relationships for the direct fire exposure time to small-scale specimens, in accordance with American Society for Testing and Materials [2] for a 2-h fire rating. The ASTM time-temperature curve was followed during all pilot tests, which included the testing of seven 6 in. 6 in. 21 in. (152.4 mm 152.4 mm 533.4 mm) concrete beams and mix sets of three 6 in. 12 in. (152.4 mm 304.8 mm) concrete cylinders in direct fire exposure for durations of 120, 60, and 30 min. It was observed that when the small-scale specimens were exposed to the direct fire for the duration of 2 h, they suffered extreme deterioration during testing and after cooling. The prolonged exposure time initiated cracking, spalling, and internal stresses with the integrity of the member declining drastically throughout testing. In cooling, even more deterioration occurred to the point where the small-scale specimens were incapable of resisting any load. In various instances, some specimens completely disintegrated after cooling. Construction materials exposed to fire will experience changes in material properties and, in some cases, may undergo creep, decomposition, dehydration, and loss of material [20]. When the duration of the fire exposure was reduced to 60 min (1700°F or 927 °C) and 30 min (1550°F or 843 °C), the effects on the concrete specimens were not so detrimental. The specimens exposed to direct fire for 30 min retained integrity better than those exposed for 60 min (based on visual inspection); the 30 min exposure results are shown in Table 1. 3.2. Size effect on fire resistance of axially loaded RC columns
ð2Þ
where C1, C0 are constants. 3.2.2. Numerical modeling 3.2.2.1. Reference and studied columns. The following terminology is used: (i) Nominal Load Capacity: Calculated load capacity which decreases with fire exposure (Fig. 1); (ii) Design Load Capacity: Nominal Load Capacity at room temperature; (iii) Reference Column: 305 mm 305 mm column tested under fire by Lie [15], and Lie and Woollerton [17]; (iv) Studied Columns: The sizes of the studied columns tested by Liu [19] range from 300 mm 300 mm to 600 mm 600 mm (Table 2); (v) Liu [19] Values: These are the calculated values for the reference column 300 mm 300 mm. Consider a column 3600 mm (12 ft.) long with square cross section of 300 mm (12 in.), as the reference column. Four longitudinal reinforcing bars with a diameter 19 mm (0.75 in.) are used (q = 1.13%). The bars are tied with 8 mm (0.3 in.) ties. The main reinforcing bars have a specified yield strength of 420 MPa (61 ksi). The compressive strength of the concrete is 35 MPa (5000 psi), and the concrete cover thickness is 30 mm (1.18 in.). The studied columns with different cross-section sizes, shown in Table 2, have the identical load ratio, slenderness ratio, and approximately the same reinforcement ratio (1.13–1.27%). The fire duration (tr) and load capacity (Pr) of the reference column in fire condition are taken as the reference values. The parametric studies conducted by Liu [19] indicate that con0 crete strength (f c ¼ 30—50 MPa or 4000–7000 psi) steel reinforcement ratio (q = 1–4.5%), and moisture content (0–3% by weight) do
3.2.1. Theoretical investigation of the size effect In Solid Mechanics, the size effect is understood as the effect of the characteristic structure size on the nominal strength of structure, with geometrical similarity. Bazant [3] derived the size effect law from dimensional analysis and similitude arguments for geometrically similar structures of different sizes (with initial cracking), considering the energy balance at crack propagation as follows: 0
rN ¼
Pu Bf t ¼ bd ½1 þ ðd=k0 da Þ0:5
ð1Þ
where rN is nominal strength, Pu is maximum load, b is thickness of 0 specimen, d is characteristic dimension, f t is the direct tensile strength of concrete cylinder, da is maximum aggregate size, and B and k0 are empirical constants. rN is not a real stress but a load parameter having the dimension of stress. According to Bazant [3], the definition of ‘‘d’’ can be arbitrary (e.g. the column depth or half-depth, the column effective length, etc.), because it does not affect geometrically similar structures. The resulting size effect
Fig. 1. Nominal load capacity of the reference column during fire.
Table 1 Material and structural properties before and after fire exposure. Mix proportions – 1(cement Type I):1.68(silica sand):2.58 (3/8-in. FL pearock):0.48 (water). Material properties
Not exposed to fire
30-min. fire exposure
Compressive strength, MPa Splitting tensile strength, MPa Flexural strength, MPa
28.5 4.1 3.1
8.5 1.5 0.65
Not exposed to fire 105 8 230
30-min. fire exposure 58 4.5 86
Structural properties RC beams RC columns
Maximum load capacity (kN) Ultimate moment capacity (kN m) Maximum load capacity (kN)
1 MPa = 145 psi; 1 kN = 225 lb; 1 kN m = 738 lb ft.
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Table 2 Summary of the details of studied columns. Square cross section Size, d (mm)
Concrete strength (MPa)
300 350 400 450 500 550 600
35
Longitudinal reinforcement
Diameter (mm)
No. Steel ratio (%)
Strength (MPa)
18 22 25 20 22 20 22
4 4 6 8 8 12 12
420
1.13 1.24 1.23 1.24 1.22 1.25 1.27
Load capacity of column at room temperature Pa (kN)
3577 4927 6428 8150 10,039 12,189 14,542
Column length = 3600 mm. 1 MPa = 145 psi; 1 mm = 0.039 in.; 1 kN = 225 lb. a P = the total strength of the column at room temperature.
Society for Testing and Materials [2] fire on all sides. The column cross section is divided into 10 or more layers with equivalent depth, shown in Fig. 2(a). The criteria of similar temperature distributions can be defined as that of the temperatures at every geometrically equivalent layer, Fig. 2(a), being the same or as close as possible, and the average temperatures in the cross-sections with different dimensions being the same. The temperatures of each layer at given fire exposure time are shown in Table 3, and the equivalent fire durations are plotted in Fig. 2(b). Based on the calculated data in Table 3, the relationship between relative equivalent fire duration (t/tr) and the relative size (d/dr) is shown in Fig. 2(b) (dr = 300 mm or 11.8 in.; tr = 120 min). This indicates that the equivalent fire duration of any size column, which is geometrically similar to the reference one, can be determined in terms of the fire duration of reference column. Fire exposure can be regarded as one special type of load, with the nominal fire exposure density, tN, described as:
tN ¼ not have a significant effect on the fire resistance of axially loaded RC columns. These values represent the characteristic values in practical RC columns. So, any column, with similar details as mentioned above, can be taken as the reference column, since the slenderness ratio is the same. Bazant and Xiang [4] established that the size effect disappears asymptotically for small sizes of d. However, according to Bazant and Xiang [4], for larger sizes, the asymptotic effect can be approximated to d2/5. The nominal load capacity (normalized to design load capacity) of the reference column in the fire condition is shown in Fig. 1, together with the fire test results of Lie and Woollerton [17] and the computed values by Liu [19]. For the reference column, its fire endurance at a given load level or the load capacity at a given fire duration can be directly obtained from Fig. 1. 3.2.2.2. Equivalent fire duration. For geometrically similar concrete columns with similar details (Fig. 2), if the temperature distributions are similar, the nominal strengths of these columns should be similar, or a function of cross-section size. Taking one column as the reference column, the fire durations of other columns required to reach the same temperature distribution, as in the reference column, are called ‘equivalent fire durations.’ For convenience, in this study, the temperature distribution in the cross-section of the reference column exposed to standard fire for two hours is taken as the reference temperature distribution. It is assumed that all the columns are exposed to the American
t 2
d
ð3Þ
Model fire testing of half-scale specimens i.e. 6 6 in., compared to full-scale 12 12 in. indicated an exposure time of one-quarter of the ASTM 2-h exposure. This clearly indicates that the exposure time is inversely proportional to the cross sectional area, validating Eq. (3). The time exposure was regarded as one special type of load induced by fire. The plot of Fire Exposure Density vs. Relative Size is given in Fig. 3. Based on the predicted equivalent fire duration in Table 3 and the Bazant Law [4], the relationship between equivalent fire duration and cross-section size is a power function, Fig. 3, and the regression equation is:
y ¼ Cx0:54
ð4Þ
The power, 0.54, in Eq. (4), is a little less than that in Eq. (2), 0.4, i.e. obtained from the Bazant Law [4], by expressing the equivalence of fire exposure time per unit area to the stress. This accounts for the modification of size effect for temperature dependence. The regression equation based on Fig. 2(b) expressing relative fire duration – relative size relationship has the following form:
1:46 32 t d d ¼ tr dr dr
ð5Þ
where t is the equivalent fire endurance of a square column; tr is the fire endurance of reference column, d is the cross section size of a square column; and dr is the cross-section size of reference column.
Fig. 2. (a) Geometrically equivalent layers of the cross-section; (b) relative equivalent fire duration (t/tr) vs. the relative size (d/dr) (tr = the reference fire duration, dr = the reference size), Eq. (5).
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D.V. Reddy et al. / Engineering Structures 99 (2015) 468–478 Table 3 Temperatures of geometrically equivalent layers at given fire durations. Size (mm)
Equivalent fire duration (min)
Tf (°C)
T1 (°C)
T2 (°C)
T3 (°C)
T4 (°C)
T5 (°C)
T6 (°C)
Average (°C)
300 350 400 450 500 550 600
120 150 182 216 252 290 330
1007 1038 1066 1093 1119 1144 1170
967 1002 1035 1065 1094 1122 1149
665 675 684 691 698 704 709
447 442 442 433 428 423 418
318 306 295 285 275 267 259
247 233 220 209 199 190 181
226 211 198 187 176 168 160
478 478 479 478 478 479 479
T1, T2, T3, T4, T5, T6 are the temperatures of points 1, 2, 3, 4, 5, 6 (see Fig. 2), respectively; 1 mm = 0.039 in.
Fire Exposure Density (t / d2)
0.0014
Y = 0.0013X
0.0013
given in Table 2 for the equivalent fire duration. The finite difference model is similar to Lie and Irwin [18] and Kodur and Lie [13], but the moisture content effect on the model, and the material thermal and mechanical properties at elevated temperatures are different. The numerical procedure is performed in three steps: (i) fire scenario analysis: calculation of the fire temperature to which the column is exposed; (ii) heat transfer analysis: calculation of the transient temperature distribution in the column cross-section; and (iii) strength analysis: determination of the mechanical response due to thermal and mechanical loading.
-0.54
0.0012
0.0011
0.001
0.0009 1
1.2
1.4
1.6
1.8
2
2.2
Relative Size (d/d r) Fig. 3. Relationship between fire exposure density and relative size.
Using Eq. (5), when the fire endurance of the reference column is known, the equivalent fire endurance of any geometrically similar columns, can be determined, at the same load level. To illustrate Eq. (5), consider a column tested by Lie [15] as the reference column (d = 305 mm or 12 in.). The measured fire endurance was 187 min, for a load ratio of 0.34, Fig. 1 (square symbols). The fire endurance of a square column of 406 mm (15.8 in.) at the same load ratio is as follows:
32 3 t d 406 2 ¼ ¼ ¼ 1:535 ) t ¼ 187 1:535 ¼ 287 min : tr dr 305 From Fig. 1, when the load ratio = 0.34, the corresponding fire endurance of reference column (dr = 300 mm or 11.8 in.) is about 160 min (the solid curve), so the fire endurance of the square column of 406 mm (15.8 in.) is:
32 3 t d 406 2 ¼ ¼ ¼ 1:574 ) t ¼ 160 1:574 ¼ 252 min tr dr 300 It must be pointed out that the tested reference column size was 305 mm 305 mm (square symbols in Fig. 1), compared to the reference column used for the numerical analysis, 300 mm 300 mm (solid curve in Fig. 1). Furthermore, the values predicted for the 406 mm 406 mm column, based on the 300 mm 300 mm (numerical analysis) and 305 mm 305 mm (measured) are close to 266 min, actually measured for 406 mm 406 mm column by Lie [15]. The numerical modeling was based on the 300 mm 300 mm column, instead of the 305 mm 305 mm for the sake of computational convenience with geometric similarity. 3.2.2.3. Load capacity of RC columns at the equivalent fire duration. The load capacities are calculated for the RC columns
3.2.2.4. Fire temperature. The fire temperature is assumed to follow the American Society for Testing and Materials [2] standard fire, which can be approximated by the following equation:
pffiffi pffiffi T f ¼ T 0 þ 750 1 expð0:49 tÞ þ 22 t
ð6Þ
where t = time (min); T0 = initial temperature (°C); and Tf = fire temperature (°C). It is important to note that any time-temperature relationship can be used in the model. 3.2.2.5. Heat transfer. The following assumptions are made for the heat transfer analysis: The temperatures of the water and the concrete are the same at each location. Convection is ignored as a mechanism for heat flow from the fire surroundings into the column. Previous work in this area [18] has led to the conclusion that convection is responsible for less than 10% of the heat transfer at the surface of the column in standard fire endurance tests, and can thus be neglected for the purposes of numerical fire modeling. Free water in concrete evaporates in the range of 100–140 °C [5]. The moisture content function in concrete by volume is defined as follows:
8 T 6 100 C > < /0 /ðTÞ ¼ /0 ½1 ðT 100Þ=40 100 C < T 6 140 C > : 0 T 6 140 C
ð7Þ
where /0 is the initial moisture content. 3.2.2.6. Finite difference equations. The cross-sectional area of the original column is modeled as a rectangular mesh with nodal points spaced Dx and Dy apart in x- and y-directions, respectively, as shown in Fig. 4(a) and (b). Internal Nodes: Now consider a volume element of size, Dx Dy 1, identified as a general interior node (m, n) in a region in which no heat is generated, as shown in Fig. 5(a). Again assuming the direction of heat conduction to be toward the surface nodes, the energy balance on the volume element can be expressed as follows:
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Fig. 4. (a) Nodal network for heat transfer analysis; (b) element network for strength analysis.
(a) Internal node
(b) Corner node
(c) Surface node
Fig. 5. Schematic for energy balance on the volume element of typical nodes (m, n), (1, 1) and (m, 1).
X
Dt
Q_ ¼ DEelement
ð8Þ
all sides
where Q_ , is the rate of heat transfer, normally consisting of conduction terms for interior nodes, but may involve convection, heat flux, and radiation for boundary nodes, and DEelement , the rate change of the energy content in the element, can be expressed as:
DEelement ¼ mC DT ¼ ðqc C c þ qw C w /ÞV element DT
ð9Þ
Substituting Eq. (9) into Eq. (8) and dividing by Dt, gives
X
T Q_ ¼ qc C c V element
all sides
iþ1
Ti T iþ1 T i þ qw C w /ðiÞV element Dt Dt
ð10Þ
where qc = the concrete density, Cc and Cw = the specific heat capacities of concrete and water, respectively, Velement = the element volume, and /(i) = the moisture content at time t = i Dt. Assuming the temperatures between the adjacent nodes vary linearly and the thermal properties between the adjacent nodes as constants, the energy balance relation in Eq. (10) becomes:
i
kðm;nÞ Dy
T iðm1;nÞ T iðm;nÞ
Dx i
þ kðm;nÞ Dy ¼ ðqc C c Þiðm;nÞ
i
þ kðm;nÞ Dx
T iðmþ1;nÞ T iðm;nÞ
Dx iþ1 T ðm;nÞ T iðm;nÞ
þ ðqw C w Þiðm;nÞ
Dt
Dy i
þ kðm;nÞ Dx
T iðm;n1Þ T iðm;nÞ
Dy
Dx Dy
i T iþ1 ðm;nÞ T ðm;nÞ
Dt
T iðm;nþ1Þ T iðm;nÞ
Dx Dy/iðm;nÞ
ð11Þ
where T iðm;nÞ and T iþ1 ðm;nÞ are the temperature of nodal (m, n) at time t = i Dt and t = (i + 1)Dt, respectively. While the thermal properties between the adjacent nodes are considered constant, the dependence of the material properties on the temperature is accounted for by a step-wise function for the successive nodes. Taking a square i
mesh (Dx ¼ Dy ¼ l), and dividing each term by kðm;nÞ , the equation above can be solved explicitly for a new temperature, T iþ1 ðm;nÞ (thus the name explicit method), to give
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T iþ1 ðm;nÞ
¼ s T iðm1;nÞ þ T iðnþ1;mÞ þ T iðmþ1;nÞ þ T iðn1;mÞ þ ð1 4sÞT iðm;nÞ
e0;T ¼ ð50k2 15k þ 1Þe01 þ 20ðk 5k2 Þe02 þ 5ð10k2 kÞe03 ð19aÞ
ð12Þ where
where s is the dimensionless Fourier number as follows:
e01 ¼ 2:05 103 þ 3:08 106 T þ 6:17 109 T 2 þ 6:59 1012 T 3
i
s¼h
kðm;nÞ
i Dt 2 ðqc C c Þiðm;nÞ þ ðqw C w Þiðm;nÞ /iðm;nÞ l
ð13Þ
Boundary Nodes: For an element at the surface of the column, the change in energy content of the element, DEelement , must be equal to the sum of the heat inflow into the element due to radiation, Q rad , and the heat inflow due to conduction, Q cond , as follows:
DEelement ¼ Q
rad
þQ
cond
ð14Þ
Heat transfers into the surface element due to radiation can be approximated as [25]:
h i Q rad ¼ Aresurf ðT f þ 273Þ4 ðT surf þ 273Þ4
ð15Þ
where r is the Stephan–Boltzman constant (5.67 108 W/ (m2 K4)), esurf is the emissivity of the concrete surface, T f is the temperature of the surroundings, and T surf is the surface temperature of the object being heated. For the surface element, taking (m, 1) (1 < m < M) as an example, as shown in Fig. 5(c), equation Eq. (11) becomes:
i i i i T iþ1 ðm;1Þ ¼ s T ðm1;1Þ þ 2T ðm;2Þ þ T ðmþ1;1Þ þ ½1 4sT ðm;1Þ 4 4 2rec l i þ i s T f þ 273 T iðm;1Þ þ 273 kðm;1Þ
ð19bÞ
e02 ¼ 2:03 103 þ 1:27 106 T þ 2:17 109 T 2 þ 1:64 1012 T 3 ð19cÞ
e03 ¼ 0:002
eth , the free thermal expansion, is directly dependent on the temperature in the element. Traditionally, it is expressed by a linear function of element temperature by employing a thermal expansion coefficient, a:
eth ¼ aðT 20Þ
ð20Þ
For concrete with siliceous or carbonate aggregates, a, is given by Lie [16].
a ¼ ð0:008T þ 6Þ 106
ð21Þ
The transient creep strain (eth ), is a function of temperature (Lie et at., 1992):
etr ¼ ð16Þ
ð19dÞ
f cT h 3:3 1010 ðT 20Þ3 1:72 107 ðT 20Þ2 0 fc i þ0:0412 103 ðT 20Þ2
ð22Þ
0
Corner Nodes: Similarly, for elements at the corner, the temperature at time t = (i + 1)Dt, taking node (1, 1) as an example, is given by
i i i T iþ1 ð1;1Þ ¼ 2s T ð2;1Þ þ T ð1;2Þ þ ð1 4sÞT ð1;1Þ 4 4 4rec l þ i s T if þ 273 T ið1;1Þ þ 273 kð1;1Þ
ð17Þ
The European Code EVN 1992-1-2 [8] provides the thermal properties of concrete in fire conditions, given in Appendix A. In this paper, the conductivity of concrete with siliceous aggregate is the average of the upper and lower limits given in Eurocode 2 [8]. 3.2.2.7. Strength analysis. The cross sectional temperatures generated from thermal analysis are used as input to the strength analysis. For each element in the column cross-section, the temperature, stress, and strain are assumed to be represented by those at the centroid of the element. The temperature at the center of each element (Fig. 4b) is obtained by averaging the nodal temperatures of the elements in the heat transfer analysis network, shown in Fig. 4(a). 3.2.2.8. Concrete and reinforcement strains at elevated temperature. The total strain (et ) in concrete at elevated temperature consists of three terms: instantaneous stress–related strain (er ), unstrained thermal strain (eth ), and thermal transient creep strain (etr ). Thus, the strain that causes stress, at any fire exposure time, is calculated from:
er ¼ et eth etr
ð18Þ
The value of er at the peak stress (e0;T ) is the function of initial stress level, given by Terro [23]. The formula to account for the initial stress level, k, which is found by reducing the effect of fire temperature on e0;T at elevated temperature is as follows:
where f c and f cT are concrete strength at room and elevated temperature, respectively. Fig. 6(b) shows the distributions of total strain, stress, and internal forces for the column cross-section at any fire exposure time. The total strain in element (m, n) (concrete or rebar) can be related to the curvature of the column by the following expression:
et;ðm;nÞ ¼ u yðm;nÞ
ð23Þ
where u = curvature of the cross-section at the mid height of the column, yðm;nÞ = the distance of concrete element (m, n) to the neutral axis. The strain and temperature in any of the longitudinal reinforcing bars is assumed to be the same as that for the concrete element that the bar lies within. The stresses in concrete and reinforcement are then determined using the stress–strain relationship given in Appendixes A and B. Using the above equations, the curvature of the mid-height cross-section and the overall axial strain in the column are varied until the internal moment and force at mid-height, due to the contributions of each of the concrete elements and reinforcing bars, are equal to the external moment and applied load. The internal moment at mid-height is calculated using:
Miint ¼
M 1X N1 X m¼1 n¼1
where
riðm;nÞ Am;n dm;n þ
J X
rij Aj dj
ð24Þ
j¼1
riðm;nÞ = the stress in any concrete element of the
cross-section at time t = i Dt; Aðm;nÞ = the area of the element; dðm;nÞ = the distance from the center of the element to the centerline of the column; rij = the stress in any longitudinal rebar at t = i Dt; Aj = the cross-section area of rebar; and dj = the distance from the center of the rebar to the centerline of the column. The external moment is calculated as the product of the total vertical force at mid-height, Pi , times the horizontal deflection, plus the eccentricity of the column at that location, Fig. 6(a). The cross-sectional stress– strain distribution is shown in Fig. 6(b).
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P P (e 0+ea)
L/2
P
s
e 0+e a
As )
neutral axis
P (e 0+ea )
yn
e0 +ea+
n
m
t(m,n)
(m,n)
P
ly/2
L/2
central line
d (m,n)
L0
P (e 0+ea
s
As /
P
/
s
/
s
c
P
(a)
(b)
Fig. 6. (a) Moments in slender member with compression plus bending moment, for an equivalent pin-ended column; (b) stress–strain distribution in cross section due to axial load and bending.
M iext ¼ Pi ðD þ e0 þ ea Þ
where Pi is calculated as a summation of the force contribution from the individual elements. Thus,
Pi ¼
M1 N1 XX
J X
m¼1 n¼1
j¼1
riðm;nÞ Am;n þ
rij Aj
ð26Þ
Using Eqs. (6)–(26), and the temperature field, the strength of the RC column at each time step can be evaluated, until the strength reaches the applied moment value. The corresponding time is the fire endurance of the column. From Table 4 it can be seen that the relative load capacity contributed by concrete, PTc =PTcr , is almost the same as the relative cross-section area, 2
A=Ar ¼ ðd=dr Þ . There is a slight difference between the total relative load capacity, P T =PTr , and the relative cross-section area. This is because it is difficult to make all columns have the same reinforcement ratio, and the same arrangement of reinforcement. Furthermore, in the studied cases, the concrete cover to the longitudinal steel is the same, and does not change proportionally to the cross section size. The total relative strength of a square column under fire condition can be approximately given by:
PT PTr
¼
1:875 d dr
ð27Þ
where PT is the load capacity of RC column at equivalent fire duration given by Eq. (5), PTr is the load capacity of reference RC column at given fire duration. Table 4 Strength of concrete columns at the given durations, tr = 120 min. D (mm)
d/dr
(d/dr)2 te (min) t/tr
300 (ref) 1 1 350 1.167 1.361 400 1.333 1.778 450 1.500 2.250 500 1.667 2.778 550 1.833 3.361 600 2 4
P T and PTr = the total load capacity of the column and reference
ð25Þ
120 150 182 216 252 290 330
1 1.250 1.517 1.800 2.100 2.417 2.75
PT (kN) P Tc (kN) P T =P Tr
P Tc =P Tc;r
1165 1610 2046 2487 3053 3718 4324
1 1.362 1.753 2.210 2.729 3.301 3.930
1050 1430 1841 2321 2865 3466 4127
1 1.382 1.756 2.135 2.621 3.191 3.712
d = the depth of the square cross-section, te = the equivalent fire duration; 1 kN = 225 lb; 1 mm = 0.039 in.
column at elevated temperature, respectively. P Tc and PTc;r = the contribution of concrete to total load capacity of the column and reference column at elevated temperature, respectively. The load capacity contributed by the concrete of a square column under fire condition can be given by (regression analysis):
PTc PTc;r
1:9699 2 T d d PT Pc;r ¼ 0:993 ) 2c ¼ 2 dr dr d dr
ð28Þ
Eq. (28) indicates that the average stresses of plain concrete columns, with different cross-section sizes, are identical for equivalent fire duration. This result confirms that the criterion of ‘‘the equivalent fire duration’’ is reasonable. 3.3. Size effect on the fire endurance of simple flexural beams To enhance the fire resistance of RC members, the conventional design method in most publications, including building codes (Eurocode 2 [1], is to increase the concrete cover thickness of RC members. The temperature corresponding to the member failure must, among other factors, be dependent on the loading type and load level [22]. As specified in American Society for Testing and Materials [2], the fire endurance of a RC beam has been traditionally defined based on thermal and strength failure criteria as specified in American Society for Testing and Materials [2]. The following two sets of failure criteria are incorporated into this study to define the failure of the beam in the fire condition: (1) The applied service load exceeds the strength of the beam i.e. moment at first crack; and (2) the temperature in tension steel exceeds 593 °C. The computer model developed by Liu [19], based on the integrated thermal and strength criteria for failure, was used to evaluate the effects of the size of beam cross-section and the thickness of concrete cover on the fire endurance of RC beams. The results are described below. 3.3.1. Properties of studied beams The studied beams were simply-supported with a span of 6 m, shear span of 2 m, and a cross section of (200–300) mm (300– 0 600) mm; f c ¼ 30 MPa; tension rebar 2 £ 20 to 3 £ 20; f y ¼ 400 MPa.
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1200
3.3.2. Effects of beam depth and width The effects of the overall beam depth on the fire endurance of RC beams, based on tensile rebar temperature and structural strength (load level = 0.5), are given in Fig. 7. It is evident that the cross-section depth almost has no effect on the fire endurance, based on the thermal and strength failure criteria, and the fire endurance increases slightly with the increase in width. The effect on fire endurance based on the strength criterion is greater than that based on the temperature criterion.
Temperature, oC
1000
3.3.3. Effect of concrete cover thickness Figs. 8–10 show that the fire endurance increases with the concrete cover thickness for both criteria of strength and steel temperature. For example, for a simply supported beam cross section, shown in Fig. 9, the fire endurances based on rebar temperature are 104, 131, and 159 min, and based on strength criterion, the fire endurances are 99, 125, and 152 min, for concrete cover thickness, C = 30, 40, and 50 mm, respectively. The difference based on the two criteria is small. Fig. 10 shows that during the first 30 min fire exposure, the smaller concrete cover thickness results in the higher load capacity due to larger effective depth. However, when the rebar temperature is larger than 300 °C, as shown in Figs. 8 and 10, the reduction rate of the strength decreases with the increase in concrete cover thickness during fire exposure. After 120 min fire exposure, the strengths of the beam were 23%, 35%, 52%, and 69% of the strength at room temperature for C = 20, 30, 40, and 50 mm (0.78, 1.17, 1.56, and 1.95 in.), respectively. By properly increasing the concrete cover thickness, the fire resistance of RC flexural members can be improved significantly. This increase in the concrete cover thickness not only reduces the load capacity of a beam of fixed depth and reinforcement ratio at room temperature, but also possibly increases the width of surface cracks, and hence decreases the serviceability. Therefore, it is not practical to excessively increase the concrete cover thickness for improving the behavior of RC flexural members under fire conditions.
800 600 ASTM E119 fire
400
C=20mm C=30mm
200
C=40mm C=50mm
0
0
30
60
90
120
150
180
Time, min Fig. 8. Temperatures of corner rebar (1 mm = 0.039 in.).
3.4. Validation The validity of the computer model was established by comparing predicted results from the model with the measured values from fire tests for columns, Lie and Woollerton [17]. The geometric and material properties of the tested columns used in the analysis are given in Table 5. Taking Column I-2 as an example, the
Fig. 9. Details and properties of the studied beam with different concrete cover thickness (1 mm = 0.039 in.; 1 MPa = 145 psi; 1 kN m = 738 lb ft).
180
160 150 140
Fire endurance, min
Fire endurance, min
160 140 120 100 C=30mm
80
C=40mm
200
250
120 110 100 90
C=30mm
80
C=40mm
70
C=50mm
60
130
300
60
C=50mm
200
250
Cross-section width, mm
Cross-section width, mm
(a)
(b)
300
Fig. 7. Effects of beam cross-section width and concrete cover thickness (C) on fire endurance of RC beams (1 mm = 0.039 in.). (a) Based on the critical temperature of tensile steel. (b) Based on the strength of the beam.
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180
1
160 0.75
Relative strength
Strength, kN.m
140 120 100 80 C=20mm
60
C=30mm
40
C=40mm
20
C=50mm
0
0.5
0 0
30
60
90
120
150
C=20mm C=30mm C=40mm C=50mm
0.25
180
0
30
60
Time, min
90
120
150
180
Time, min
Fig. 10. Load capacity with different concrete cover thicknesses (1 kN m = 738 lb ft; 1 mm = 0.039 in.).
Table 5 Summary of relevant parameters and results. Column no.
Size (square) (mm)
Reinf. Qty-Size
Comp. strength MPa
Applied load (kN)
Design load (kN)
Load level*
I-2 I-3 I-4 I-7 I-8 I-9 II-1 II-2 II-3 II-4 II-5 II-6 II-9 III-1 III-2 III-14
305 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305
4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M
36.9 34.2 35.1 36.1 34.8 38.3 41.6 43.6 35.4 52.9 49.5 46.6 37.1 39.6 39.3 37.9
1333 800 711 1067 1778 1333 342 1044 916 1178 1067 1076 1333 800 1000 1178
3726 3517 3587 3665 3564 3835 4090 4245 3610 4965 4702 4477 4550 3935 3912 3804
0.358 0.227 0.198 0.291 0.499 0.348 0.084 0.246 0.254 0.237 0.227 0.240 0.293 0.203 0.256 0.310
Fire endurance (min) Measured
Predicted
170 218 220 208 146 187 340 201 210 227 234 188 225 242 220 183
167 221 234 193 117 171 304 213 209 217 221 215 192 233 208 185
1 kN = 225 lb; 1 mm = 0.039 in.; 1 MPa = 145 psi. Aggregate type: Siliceous; Strength of steel = 444 MPa; The clear cover of the main steel = 38 mm; relative humidity: 65–75%; M = mm. 0 a load level = applied load=ð0:85f c Ac þ f y As Þ, where fc = concrete strength at room temperature, Ac = area of concrete, fy = main steel strength at room temperature, As = area of steel.
Fig. 11. Temperature of concrete at various depths along centerline as function of exposure time (1 mm = 0.039 in.).
predicted values of the results (temperatures and deformations) from the analysis are compared with the measured values from fire tests in Figs. 11 and 12. The predicted load level effect on fire endurance is shown in Fig. 13, together with the measured values.
Fig. 12. Axial deformation of columns as a function of exposure time (1 mm = 0.039 in.).
It can be seen from Fig. 11 that, with the exception of the temperatures measured at 64 mm (2.5 in.) depth from 70 to 140 min, there is good agreement between the predicted and measured values. The temperatures measured at the center of the column show
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360 330
Fire Rsistance, min
300 270 240 210 180 150 120 90 60 30 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Load Level Fig. 13. Effect of load level on the fire endurance of columns.
initially a relatively slow rise before 100 °C. This temperature behavior may be the result of thermal-induced migration of the moisture toward the center of the column, where as shown in the figures, the influence of the migration is most pronounced Lie and Irwin [18]. Fig. 12 shows the calculated and measured axial deformations of the columns during exposure to fire. The numerical analysis predicts reasonably well, the trend in the progression of the deformation with time. The fire endurance decreased with load level, Fig. 13, until the column could no longer support the applied load, Table 5. It is easy to determine the fire resistance of column for given applied load level, and vice versa.
Acknowledgements The authors gratefully acknowledge the National Science Foundation (NSF), Grant No. 0425699 (Contract Monitor: Dr. O. Shinaishin), and the United States Department of Agriculture (USDA), Agreement No. 58-5148-7-175 for the interest, encouragement, and financial support.
Appendix A. Properties of concrete (siliceous aggregate) A.1. stress–strain relationships [14]
4. Conclusions Based on the current study, the following conclusions can be drawn: The numerical model used in this paper is capable of predicting the fire resistance of square or rectangular concrete columns, with an accuracy that is adequate for practical purposes. Using the curve developed in this study for the effect of load level on the fire endurance of columns, the fire resistance of square RC columns with the same similitude conditions can be evaluated easily for any given applied load, and vice versa. For axially loaded square columns, the fire endurance increases with the cross-section size at the same load level. Two relationships for axially load column have been developed: (i) fire endurance with cross-section size and, (ii) load capacity in the fire condition with cross-section size. The fire resistance of RC columns can be determined based on these two relationships, and the test data or numerical results of the reference column. Because the fire resistance of RC beams mainly depends on the load ratio and the temperature of the reinforced bars, the cross-section depth has almost no effect on the fire endurance of RC beams. The fire endurance increases only slightly with the increase of cross-section width. By properly increasing the concrete cover thickness, the fire resistance of RC beams can be improved significantly. However, it is not practical to excessively increase the concrete cover thickness for improving the fire resistance of RC beams. Although the cover thickness of 50 mm was observed to be the most suitable for the 2-h ASTM fire exposure time, further studies are needed to establish an optimal cover thickness related to effective depth.
y¼
8 > < 2x x2
06x61
x > 2ðx1Þ2 þx
x>1
:
0
0
where y ¼ r=f cT ; x ¼ e=e0T , and f cT , concrete strength at elevated temperature, is given by Hertz [11]:
" , 0
0
f cT ¼ f c 1
1þ
2 8 64 !# T T T T þ þ þ T1 T1 T1 T1
For siliceous aggregate, the proposed values for T1, T2, T8, T64 are: Siliceous aggregate: T1 = 15,000, T2 = 800, T8 = 570, and T64 = 100,000. A.2. Specific heat (J/kg °C) [23]
C c ðTÞ ¼ 900
for 20 C 6 T 6 100 C
C c ðTÞ ¼ 900 þ ðT 100Þ for 100 C < T 6 200 C C c ðTÞ ¼ 1000 þ ðT 200Þ=2 for 200 C < T 6 400 C C c ðTÞ ¼ 1100
for 400 C < T 6 1200 C
A.3. Thermal conductivity (W/m °C) [8] Lower limit:
T
T 2 kðTÞ ¼ 1:36 0:136 100 þ 0:0057 100
for 20 C 6 T 6 1200 C
Upper limit:
T
T 2 kðTÞ ¼ 2 0:2451 100 þ 0:0107 100
for 20 C 6 T 6 1200 C
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Appendix B. Properties of steel reinforcement B.1. Stress–strain relationship [23]
r¼
8 e EsT > > < f yT þ ðf uT > > :
0:62 e e eeyT 3 f yT Þ 1:5 euT yT 0:5 eyT euT eyT
jej 6 eyT
euT P jej > eyT
h i h i f yT ¼ f y = 1 þ 24ðT=1000Þ4:5 ; f uT ¼ f u = 1 þ 36ðT=1000Þ6:2 EsT ¼ f yT =eyT ; EsT ¼ f yT =eyT
euT ¼ 0:18 0:23ðT=1000Þ P 0:04; eyT ¼ 0:18 0:23ðT=1000Þ P 0:04 in which r, e = the stress and mechanical strain of steel reinforcement, respectively; fy, fu = the yield strength and ultimate strength of steel reinforcement at room temperature, respectively; f yT , f uT = the yield strength and ultimate strength of steel reinforcement at temperature T, respectively; eyT = the yield mechanical strain of steel reinforcement at temperature T, and taken as 0.001874 [9]; euT = the ultimate mechanical strain of steel reinforcement at temperature T. B.2. Thermal strain
es;th ¼ 16=ðT=1000Þ1:5 103 References [1] ACI/TMS 216.1-14. Code requirements for determining the fire resistance of concrete and masonry construction assemblies. ACI Committee 216; 2014. [2] American Society for Testing and Materials. Standard test methods for fire tests of building construction and materials, Designation E 119-95a. ASTM, Philadelphia; 1995. [3] Bazant ZP. Size effect in blunt fracture: concrete, rock, metal. J Eng Mech 1984;110(4):518–35.
[4] Bazant ZP, Xiang Y. Size effect in compression fracture: splitting crack band propagation. J Eng Mech, ASCE 1997;123(2):162–72. [5] Di Capua D, Mari AR. Nonlinear analysis of reinforced concrete cross sections exposed to fire. Fire Saf J 2007;42:139–49. [6] El-Fitiani SF, Youssef MA. Stress block parameters for reinforced concrete beams during fire events, ACI/TMS Committee 216 Report SP-279, Innovations in Fire Design of Concrete; 2011. [7] El-Fitiani SF, Youssef MA. Assessing the flexural and axial behavior of reinforced concrete members at elevated temperatures using sectional analysis. Fire Saf J 2009;44(5):691–703. [8] Eurocode 2. Design of Concrete Structures, Part 10: Structural fire design. Eurocode Committee for Standardization, CEN/TC 250/SC2/92/N77, Brussels; 1992. [9] Guo ZH, Shi XD. Behavior of reinforced concrete at elevated temperature and its calculation. Tsinghua University Press; 2003 [in Chinese]. [10] Gonnerman HF. Effect of size and shape of test specimen on compressive strength of concrete. In: Proc ASTM, vol. 25; 1925. p. 237–50. [11] Hertz KD. Concrete strength for fire safety design. Mag Concrete Res 2005;57:445–53. [12] Kim JK, Yi ST, Kim JHJ. Effect of specimen sizes on flexural compressive strength of concrete. ACI Struct J 2001;98:416–24. [13] Kodur VR, Lie TT. A computer program to calculate the fire resistance of rectangular reinforced concrete columns. In: Third Canadian conference on computing in civil and building engineering, Ottawa, Canada; 1996. p. 11–20. [14] Li L, Purkiss JA. Stress–strain constitutive equations of concrete material at elevated temperatures. Fire Saf J 2005;40:669–86. [15] Lie TT. Fire resistance of reinforced concrete columns: a parametric study. J Fire Prot Eng 1989;1(4):121–30. [16] Lie, TT. editor. Structural fire protection, Manuals and Report on Engineering Practice, No, 78, American Society of Civil Engineers (ASCE), New York, NY; 1992. [17] Lie TT, Woollerton JL. Fire resistance of reinforced concrete columns: Test Results. Internal Report IR, Institute of Research in Construction, No. 569, Ottawa; 1988 [302p]. [18] Lie TT, Irwin RJ. Method to calculate the fire resistance of reinforced concrete columns with rectangular cross section. ACI Struct J 1993;90(1):52–60. [19] Liu LX. Fire performance of high strength concrete materials and structural concrete. Ph.D Dissertation, Florida Atlantic University; 2009. [20] Milke JA. Analytical methods to evaluate fire resistance of structural members. J Struct Eng 1999;125(10):1179–87. [21] Reddy DV, Sobhan K, Young J. Effect of elevated temperature and fire on structural elements retrofitted by carbon fiber reinforced polymer composites. In: Proceeding of the 31st conference on ‘‘Our World in Concrete and Structures’’, Singapore; 2006. p. 325–36. [22] Shi XD, Tan TH, Tan KH, Guo ZH. Influence of concrete cover on fire resistance of reinforced concrete flexural members. J Struct Eng 2004;130(8):1225–32. [23] Terro MJ. Numerical modeling of the behavior of concrete structures in fire. ACI Struct J 1998;95(2):183–93. [24] Youssef MA, Moftah M. General stress–strain relationship for concrete at elevated temperatures. Eng Struct 2007;29(10):2618–34. [25] Yunus AC. Heat transfer – a practical approach. McGraw-Hill; 2003.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Size effect on fire resistance of structural concrete Dronnadula V. Reddy a, Khaled Sobhan a,⇑, Lixian Liu b, Jody D. Young Jr. c a
Center for Marine Structures and Geotechnique, Civil, Environmental and Geomatics Engineering, Florida Atlantic University, Boca Raton, FL 33431, United States Kunming University of Science and Technology, Yunan Province, China c McCall Engineering, 6389 Tower Lane, Sarasota, FL 34240, United States b
a r t i c l e
i n f o
Article history: Received 5 March 2014 Revised 13 March 2015 Accepted 11 May 2015
Keywords: Size effect Fire resistance Reinforced concrete Column Beam
a b s t r a c t Research on the size effect of reinforced concrete members at ambient temperatures includes only limited studies on the size effect of their fire resistance, integrating thermal and structural similitudes. Preliminary testing on concrete cylinders and unreinforced beams showed a 1=4 ASTM E-119 exposure time for the fire rating of small scale specimens compared to the full scale ones. Based on numerical and experimental analyses at Florida Atlantic University, the following quantitative relationships were developed for columns and beams: (i) equivalent fire duration and specimen size; (ii) fire endurance and cross-section size, and (iii) cross-section size, concrete cover thickness, and load capacity after fire exposure. The fire endurance at similar load ratios increases with an increase of member cross-section size and concrete cover thickness. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Size has an effect on nominal strength of specimens made with quasibrittle materials such as concrete, rock, ice, ceramic, and composite materials [3,4,12]. In compressive and flexural failures of quasibrittle materials, the size effect is quite apparent. Investigations of the size effect on nominal strength have become a focus of interest to many researchers [4,12]. Gonnerman [10] experimentally showed that the compressive strength decreases as the specimen size increases. This phenomenon of reduction in strength, dependent on specimen size, is called the ‘‘reduction phenomenon,’’ due to the statistical effect of an inherent larger number of ‘‘flaws’’ in the larger sample. However, rare attention has been paid to the size effect on mechanical properties of RC members in the fire condition. Some recent relevant papers have addressed the constitutive relationships for structural concrete members at elevated temperatures [24,7,6]. Fire testing, especially full scale, is expensive and time-consuming for large complicated structures, and the values from small scale laboratory tests differ from those of real large structures due to the scale effect. The primary objective of this paper was to study the size effect on fire performance of axially loaded square RC columns and simply supported reinforced beams using the numerical modeling developed by Liu [19]. From a preliminary investigation comprising
⇑ Corresponding author. E-mail addresses: [email protected] (D.V. Reddy), [email protected] (K. Sobhan). http://dx.doi.org/10.1016/j.engstruct.2015.05.015 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.
concrete cylinders and unreinforced beams under direct fire exposure, a reduction in fire resistance rating in the reduced cross-section was observed (based on Heisler/Gröber charts for the heat transfer analysis for thermal similitude), i.e., 1=4 exposure time of that for the full-scale specimens, based on small scale specimen fire exposure testing in a furnace [21]. The current study comprises: (i) Fire endurance of a reference column as a function of the load ratio, based on fire test data or validated numerical modeling; (ii) Relationship between the load capacity of RC columns/beams in the fire condition and the cross-section size; (iii) Relationship between the fire endurance of RC columns/beams and the cross-sectional size; and (iv) Relationship between the fire endurance of RC columns/beams and the concrete cover thickness.
2. Research significance In view of the concerns related to increasing frequencies, and catastrophic effects of fire on the infrastructure, as evidenced by recent events, man-made and natural, considerable attention is warranted to fire-resistant structural design. This study, incorporating the integrated effect of thermal and structural similitudes, addresses the numerical modeling and validation by laboratory testing of certain critical structural and material parameters, i.e. cross-section sizes, cover thickness, and concrete properties. The outcomes will significantly impact the design criteria, and reduce the need for expensive and time-consuming large-scale fire testing.
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on the nominal strength of large structures failing in compression was derived by Bazant and Xiang [4] and has the form:
3. Methodology 3.1. Preliminary fire tests
rn ¼ C 1 d2=5 þ C 0
A series of fire tests was performed to develop empirical relationships for the direct fire exposure time to small-scale specimens, in accordance with American Society for Testing and Materials [2] for a 2-h fire rating. The ASTM time-temperature curve was followed during all pilot tests, which included the testing of seven 6 in. 6 in. 21 in. (152.4 mm 152.4 mm 533.4 mm) concrete beams and mix sets of three 6 in. 12 in. (152.4 mm 304.8 mm) concrete cylinders in direct fire exposure for durations of 120, 60, and 30 min. It was observed that when the small-scale specimens were exposed to the direct fire for the duration of 2 h, they suffered extreme deterioration during testing and after cooling. The prolonged exposure time initiated cracking, spalling, and internal stresses with the integrity of the member declining drastically throughout testing. In cooling, even more deterioration occurred to the point where the small-scale specimens were incapable of resisting any load. In various instances, some specimens completely disintegrated after cooling. Construction materials exposed to fire will experience changes in material properties and, in some cases, may undergo creep, decomposition, dehydration, and loss of material [20]. When the duration of the fire exposure was reduced to 60 min (1700°F or 927 °C) and 30 min (1550°F or 843 °C), the effects on the concrete specimens were not so detrimental. The specimens exposed to direct fire for 30 min retained integrity better than those exposed for 60 min (based on visual inspection); the 30 min exposure results are shown in Table 1. 3.2. Size effect on fire resistance of axially loaded RC columns
ð2Þ
where C1, C0 are constants. 3.2.2. Numerical modeling 3.2.2.1. Reference and studied columns. The following terminology is used: (i) Nominal Load Capacity: Calculated load capacity which decreases with fire exposure (Fig. 1); (ii) Design Load Capacity: Nominal Load Capacity at room temperature; (iii) Reference Column: 305 mm 305 mm column tested under fire by Lie [15], and Lie and Woollerton [17]; (iv) Studied Columns: The sizes of the studied columns tested by Liu [19] range from 300 mm 300 mm to 600 mm 600 mm (Table 2); (v) Liu [19] Values: These are the calculated values for the reference column 300 mm 300 mm. Consider a column 3600 mm (12 ft.) long with square cross section of 300 mm (12 in.), as the reference column. Four longitudinal reinforcing bars with a diameter 19 mm (0.75 in.) are used (q = 1.13%). The bars are tied with 8 mm (0.3 in.) ties. The main reinforcing bars have a specified yield strength of 420 MPa (61 ksi). The compressive strength of the concrete is 35 MPa (5000 psi), and the concrete cover thickness is 30 mm (1.18 in.). The studied columns with different cross-section sizes, shown in Table 2, have the identical load ratio, slenderness ratio, and approximately the same reinforcement ratio (1.13–1.27%). The fire duration (tr) and load capacity (Pr) of the reference column in fire condition are taken as the reference values. The parametric studies conducted by Liu [19] indicate that con0 crete strength (f c ¼ 30—50 MPa or 4000–7000 psi) steel reinforcement ratio (q = 1–4.5%), and moisture content (0–3% by weight) do
3.2.1. Theoretical investigation of the size effect In Solid Mechanics, the size effect is understood as the effect of the characteristic structure size on the nominal strength of structure, with geometrical similarity. Bazant [3] derived the size effect law from dimensional analysis and similitude arguments for geometrically similar structures of different sizes (with initial cracking), considering the energy balance at crack propagation as follows: 0
rN ¼
Pu Bf t ¼ bd ½1 þ ðd=k0 da Þ0:5
ð1Þ
where rN is nominal strength, Pu is maximum load, b is thickness of 0 specimen, d is characteristic dimension, f t is the direct tensile strength of concrete cylinder, da is maximum aggregate size, and B and k0 are empirical constants. rN is not a real stress but a load parameter having the dimension of stress. According to Bazant [3], the definition of ‘‘d’’ can be arbitrary (e.g. the column depth or half-depth, the column effective length, etc.), because it does not affect geometrically similar structures. The resulting size effect
Fig. 1. Nominal load capacity of the reference column during fire.
Table 1 Material and structural properties before and after fire exposure. Mix proportions – 1(cement Type I):1.68(silica sand):2.58 (3/8-in. FL pearock):0.48 (water). Material properties
Not exposed to fire
30-min. fire exposure
Compressive strength, MPa Splitting tensile strength, MPa Flexural strength, MPa
28.5 4.1 3.1
8.5 1.5 0.65
Not exposed to fire 105 8 230
30-min. fire exposure 58 4.5 86
Structural properties RC beams RC columns
Maximum load capacity (kN) Ultimate moment capacity (kN m) Maximum load capacity (kN)
1 MPa = 145 psi; 1 kN = 225 lb; 1 kN m = 738 lb ft.
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Table 2 Summary of the details of studied columns. Square cross section Size, d (mm)
Concrete strength (MPa)
300 350 400 450 500 550 600
35
Longitudinal reinforcement
Diameter (mm)
No. Steel ratio (%)
Strength (MPa)
18 22 25 20 22 20 22
4 4 6 8 8 12 12
420
1.13 1.24 1.23 1.24 1.22 1.25 1.27
Load capacity of column at room temperature Pa (kN)
3577 4927 6428 8150 10,039 12,189 14,542
Column length = 3600 mm. 1 MPa = 145 psi; 1 mm = 0.039 in.; 1 kN = 225 lb. a P = the total strength of the column at room temperature.
Society for Testing and Materials [2] fire on all sides. The column cross section is divided into 10 or more layers with equivalent depth, shown in Fig. 2(a). The criteria of similar temperature distributions can be defined as that of the temperatures at every geometrically equivalent layer, Fig. 2(a), being the same or as close as possible, and the average temperatures in the cross-sections with different dimensions being the same. The temperatures of each layer at given fire exposure time are shown in Table 3, and the equivalent fire durations are plotted in Fig. 2(b). Based on the calculated data in Table 3, the relationship between relative equivalent fire duration (t/tr) and the relative size (d/dr) is shown in Fig. 2(b) (dr = 300 mm or 11.8 in.; tr = 120 min). This indicates that the equivalent fire duration of any size column, which is geometrically similar to the reference one, can be determined in terms of the fire duration of reference column. Fire exposure can be regarded as one special type of load, with the nominal fire exposure density, tN, described as:
tN ¼ not have a significant effect on the fire resistance of axially loaded RC columns. These values represent the characteristic values in practical RC columns. So, any column, with similar details as mentioned above, can be taken as the reference column, since the slenderness ratio is the same. Bazant and Xiang [4] established that the size effect disappears asymptotically for small sizes of d. However, according to Bazant and Xiang [4], for larger sizes, the asymptotic effect can be approximated to d2/5. The nominal load capacity (normalized to design load capacity) of the reference column in the fire condition is shown in Fig. 1, together with the fire test results of Lie and Woollerton [17] and the computed values by Liu [19]. For the reference column, its fire endurance at a given load level or the load capacity at a given fire duration can be directly obtained from Fig. 1. 3.2.2.2. Equivalent fire duration. For geometrically similar concrete columns with similar details (Fig. 2), if the temperature distributions are similar, the nominal strengths of these columns should be similar, or a function of cross-section size. Taking one column as the reference column, the fire durations of other columns required to reach the same temperature distribution, as in the reference column, are called ‘equivalent fire durations.’ For convenience, in this study, the temperature distribution in the cross-section of the reference column exposed to standard fire for two hours is taken as the reference temperature distribution. It is assumed that all the columns are exposed to the American
t 2
d
ð3Þ
Model fire testing of half-scale specimens i.e. 6 6 in., compared to full-scale 12 12 in. indicated an exposure time of one-quarter of the ASTM 2-h exposure. This clearly indicates that the exposure time is inversely proportional to the cross sectional area, validating Eq. (3). The time exposure was regarded as one special type of load induced by fire. The plot of Fire Exposure Density vs. Relative Size is given in Fig. 3. Based on the predicted equivalent fire duration in Table 3 and the Bazant Law [4], the relationship between equivalent fire duration and cross-section size is a power function, Fig. 3, and the regression equation is:
y ¼ Cx0:54
ð4Þ
The power, 0.54, in Eq. (4), is a little less than that in Eq. (2), 0.4, i.e. obtained from the Bazant Law [4], by expressing the equivalence of fire exposure time per unit area to the stress. This accounts for the modification of size effect for temperature dependence. The regression equation based on Fig. 2(b) expressing relative fire duration – relative size relationship has the following form:
1:46 32 t d d ¼ tr dr dr
ð5Þ
where t is the equivalent fire endurance of a square column; tr is the fire endurance of reference column, d is the cross section size of a square column; and dr is the cross-section size of reference column.
Fig. 2. (a) Geometrically equivalent layers of the cross-section; (b) relative equivalent fire duration (t/tr) vs. the relative size (d/dr) (tr = the reference fire duration, dr = the reference size), Eq. (5).
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D.V. Reddy et al. / Engineering Structures 99 (2015) 468–478 Table 3 Temperatures of geometrically equivalent layers at given fire durations. Size (mm)
Equivalent fire duration (min)
Tf (°C)
T1 (°C)
T2 (°C)
T3 (°C)
T4 (°C)
T5 (°C)
T6 (°C)
Average (°C)
300 350 400 450 500 550 600
120 150 182 216 252 290 330
1007 1038 1066 1093 1119 1144 1170
967 1002 1035 1065 1094 1122 1149
665 675 684 691 698 704 709
447 442 442 433 428 423 418
318 306 295 285 275 267 259
247 233 220 209 199 190 181
226 211 198 187 176 168 160
478 478 479 478 478 479 479
T1, T2, T3, T4, T5, T6 are the temperatures of points 1, 2, 3, 4, 5, 6 (see Fig. 2), respectively; 1 mm = 0.039 in.
Fire Exposure Density (t / d2)
0.0014
Y = 0.0013X
0.0013
given in Table 2 for the equivalent fire duration. The finite difference model is similar to Lie and Irwin [18] and Kodur and Lie [13], but the moisture content effect on the model, and the material thermal and mechanical properties at elevated temperatures are different. The numerical procedure is performed in three steps: (i) fire scenario analysis: calculation of the fire temperature to which the column is exposed; (ii) heat transfer analysis: calculation of the transient temperature distribution in the column cross-section; and (iii) strength analysis: determination of the mechanical response due to thermal and mechanical loading.
-0.54
0.0012
0.0011
0.001
0.0009 1
1.2
1.4
1.6
1.8
2
2.2
Relative Size (d/d r) Fig. 3. Relationship between fire exposure density and relative size.
Using Eq. (5), when the fire endurance of the reference column is known, the equivalent fire endurance of any geometrically similar columns, can be determined, at the same load level. To illustrate Eq. (5), consider a column tested by Lie [15] as the reference column (d = 305 mm or 12 in.). The measured fire endurance was 187 min, for a load ratio of 0.34, Fig. 1 (square symbols). The fire endurance of a square column of 406 mm (15.8 in.) at the same load ratio is as follows:
32 3 t d 406 2 ¼ ¼ ¼ 1:535 ) t ¼ 187 1:535 ¼ 287 min : tr dr 305 From Fig. 1, when the load ratio = 0.34, the corresponding fire endurance of reference column (dr = 300 mm or 11.8 in.) is about 160 min (the solid curve), so the fire endurance of the square column of 406 mm (15.8 in.) is:
32 3 t d 406 2 ¼ ¼ ¼ 1:574 ) t ¼ 160 1:574 ¼ 252 min tr dr 300 It must be pointed out that the tested reference column size was 305 mm 305 mm (square symbols in Fig. 1), compared to the reference column used for the numerical analysis, 300 mm 300 mm (solid curve in Fig. 1). Furthermore, the values predicted for the 406 mm 406 mm column, based on the 300 mm 300 mm (numerical analysis) and 305 mm 305 mm (measured) are close to 266 min, actually measured for 406 mm 406 mm column by Lie [15]. The numerical modeling was based on the 300 mm 300 mm column, instead of the 305 mm 305 mm for the sake of computational convenience with geometric similarity. 3.2.2.3. Load capacity of RC columns at the equivalent fire duration. The load capacities are calculated for the RC columns
3.2.2.4. Fire temperature. The fire temperature is assumed to follow the American Society for Testing and Materials [2] standard fire, which can be approximated by the following equation:
pffiffi pffiffi T f ¼ T 0 þ 750 1 expð0:49 tÞ þ 22 t
ð6Þ
where t = time (min); T0 = initial temperature (°C); and Tf = fire temperature (°C). It is important to note that any time-temperature relationship can be used in the model. 3.2.2.5. Heat transfer. The following assumptions are made for the heat transfer analysis: The temperatures of the water and the concrete are the same at each location. Convection is ignored as a mechanism for heat flow from the fire surroundings into the column. Previous work in this area [18] has led to the conclusion that convection is responsible for less than 10% of the heat transfer at the surface of the column in standard fire endurance tests, and can thus be neglected for the purposes of numerical fire modeling. Free water in concrete evaporates in the range of 100–140 °C [5]. The moisture content function in concrete by volume is defined as follows:
8 T 6 100 C > < /0 /ðTÞ ¼ /0 ½1 ðT 100Þ=40 100 C < T 6 140 C > : 0 T 6 140 C
ð7Þ
where /0 is the initial moisture content. 3.2.2.6. Finite difference equations. The cross-sectional area of the original column is modeled as a rectangular mesh with nodal points spaced Dx and Dy apart in x- and y-directions, respectively, as shown in Fig. 4(a) and (b). Internal Nodes: Now consider a volume element of size, Dx Dy 1, identified as a general interior node (m, n) in a region in which no heat is generated, as shown in Fig. 5(a). Again assuming the direction of heat conduction to be toward the surface nodes, the energy balance on the volume element can be expressed as follows:
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Fig. 4. (a) Nodal network for heat transfer analysis; (b) element network for strength analysis.
(a) Internal node
(b) Corner node
(c) Surface node
Fig. 5. Schematic for energy balance on the volume element of typical nodes (m, n), (1, 1) and (m, 1).
X
Dt
Q_ ¼ DEelement
ð8Þ
all sides
where Q_ , is the rate of heat transfer, normally consisting of conduction terms for interior nodes, but may involve convection, heat flux, and radiation for boundary nodes, and DEelement , the rate change of the energy content in the element, can be expressed as:
DEelement ¼ mC DT ¼ ðqc C c þ qw C w /ÞV element DT
ð9Þ
Substituting Eq. (9) into Eq. (8) and dividing by Dt, gives
X
T Q_ ¼ qc C c V element
all sides
iþ1
Ti T iþ1 T i þ qw C w /ðiÞV element Dt Dt
ð10Þ
where qc = the concrete density, Cc and Cw = the specific heat capacities of concrete and water, respectively, Velement = the element volume, and /(i) = the moisture content at time t = i Dt. Assuming the temperatures between the adjacent nodes vary linearly and the thermal properties between the adjacent nodes as constants, the energy balance relation in Eq. (10) becomes:
i
kðm;nÞ Dy
T iðm1;nÞ T iðm;nÞ
Dx i
þ kðm;nÞ Dy ¼ ðqc C c Þiðm;nÞ
i
þ kðm;nÞ Dx
T iðmþ1;nÞ T iðm;nÞ
Dx iþ1 T ðm;nÞ T iðm;nÞ
þ ðqw C w Þiðm;nÞ
Dt
Dy i
þ kðm;nÞ Dx
T iðm;n1Þ T iðm;nÞ
Dy
Dx Dy
i T iþ1 ðm;nÞ T ðm;nÞ
Dt
T iðm;nþ1Þ T iðm;nÞ
Dx Dy/iðm;nÞ
ð11Þ
where T iðm;nÞ and T iþ1 ðm;nÞ are the temperature of nodal (m, n) at time t = i Dt and t = (i + 1)Dt, respectively. While the thermal properties between the adjacent nodes are considered constant, the dependence of the material properties on the temperature is accounted for by a step-wise function for the successive nodes. Taking a square i
mesh (Dx ¼ Dy ¼ l), and dividing each term by kðm;nÞ , the equation above can be solved explicitly for a new temperature, T iþ1 ðm;nÞ (thus the name explicit method), to give
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T iþ1 ðm;nÞ
¼ s T iðm1;nÞ þ T iðnþ1;mÞ þ T iðmþ1;nÞ þ T iðn1;mÞ þ ð1 4sÞT iðm;nÞ
e0;T ¼ ð50k2 15k þ 1Þe01 þ 20ðk 5k2 Þe02 þ 5ð10k2 kÞe03 ð19aÞ
ð12Þ where
where s is the dimensionless Fourier number as follows:
e01 ¼ 2:05 103 þ 3:08 106 T þ 6:17 109 T 2 þ 6:59 1012 T 3
i
s¼h
kðm;nÞ
i Dt 2 ðqc C c Þiðm;nÞ þ ðqw C w Þiðm;nÞ /iðm;nÞ l
ð13Þ
Boundary Nodes: For an element at the surface of the column, the change in energy content of the element, DEelement , must be equal to the sum of the heat inflow into the element due to radiation, Q rad , and the heat inflow due to conduction, Q cond , as follows:
DEelement ¼ Q
rad
þQ
cond
ð14Þ
Heat transfers into the surface element due to radiation can be approximated as [25]:
h i Q rad ¼ Aresurf ðT f þ 273Þ4 ðT surf þ 273Þ4
ð15Þ
where r is the Stephan–Boltzman constant (5.67 108 W/ (m2 K4)), esurf is the emissivity of the concrete surface, T f is the temperature of the surroundings, and T surf is the surface temperature of the object being heated. For the surface element, taking (m, 1) (1 < m < M) as an example, as shown in Fig. 5(c), equation Eq. (11) becomes:
i i i i T iþ1 ðm;1Þ ¼ s T ðm1;1Þ þ 2T ðm;2Þ þ T ðmþ1;1Þ þ ½1 4sT ðm;1Þ 4 4 2rec l i þ i s T f þ 273 T iðm;1Þ þ 273 kðm;1Þ
ð19bÞ
e02 ¼ 2:03 103 þ 1:27 106 T þ 2:17 109 T 2 þ 1:64 1012 T 3 ð19cÞ
e03 ¼ 0:002
eth , the free thermal expansion, is directly dependent on the temperature in the element. Traditionally, it is expressed by a linear function of element temperature by employing a thermal expansion coefficient, a:
eth ¼ aðT 20Þ
ð20Þ
For concrete with siliceous or carbonate aggregates, a, is given by Lie [16].
a ¼ ð0:008T þ 6Þ 106
ð21Þ
The transient creep strain (eth ), is a function of temperature (Lie et at., 1992):
etr ¼ ð16Þ
ð19dÞ
f cT h 3:3 1010 ðT 20Þ3 1:72 107 ðT 20Þ2 0 fc i þ0:0412 103 ðT 20Þ2
ð22Þ
0
Corner Nodes: Similarly, for elements at the corner, the temperature at time t = (i + 1)Dt, taking node (1, 1) as an example, is given by
i i i T iþ1 ð1;1Þ ¼ 2s T ð2;1Þ þ T ð1;2Þ þ ð1 4sÞT ð1;1Þ 4 4 4rec l þ i s T if þ 273 T ið1;1Þ þ 273 kð1;1Þ
ð17Þ
The European Code EVN 1992-1-2 [8] provides the thermal properties of concrete in fire conditions, given in Appendix A. In this paper, the conductivity of concrete with siliceous aggregate is the average of the upper and lower limits given in Eurocode 2 [8]. 3.2.2.7. Strength analysis. The cross sectional temperatures generated from thermal analysis are used as input to the strength analysis. For each element in the column cross-section, the temperature, stress, and strain are assumed to be represented by those at the centroid of the element. The temperature at the center of each element (Fig. 4b) is obtained by averaging the nodal temperatures of the elements in the heat transfer analysis network, shown in Fig. 4(a). 3.2.2.8. Concrete and reinforcement strains at elevated temperature. The total strain (et ) in concrete at elevated temperature consists of three terms: instantaneous stress–related strain (er ), unstrained thermal strain (eth ), and thermal transient creep strain (etr ). Thus, the strain that causes stress, at any fire exposure time, is calculated from:
er ¼ et eth etr
ð18Þ
The value of er at the peak stress (e0;T ) is the function of initial stress level, given by Terro [23]. The formula to account for the initial stress level, k, which is found by reducing the effect of fire temperature on e0;T at elevated temperature is as follows:
where f c and f cT are concrete strength at room and elevated temperature, respectively. Fig. 6(b) shows the distributions of total strain, stress, and internal forces for the column cross-section at any fire exposure time. The total strain in element (m, n) (concrete or rebar) can be related to the curvature of the column by the following expression:
et;ðm;nÞ ¼ u yðm;nÞ
ð23Þ
where u = curvature of the cross-section at the mid height of the column, yðm;nÞ = the distance of concrete element (m, n) to the neutral axis. The strain and temperature in any of the longitudinal reinforcing bars is assumed to be the same as that for the concrete element that the bar lies within. The stresses in concrete and reinforcement are then determined using the stress–strain relationship given in Appendixes A and B. Using the above equations, the curvature of the mid-height cross-section and the overall axial strain in the column are varied until the internal moment and force at mid-height, due to the contributions of each of the concrete elements and reinforcing bars, are equal to the external moment and applied load. The internal moment at mid-height is calculated using:
Miint ¼
M 1X N1 X m¼1 n¼1
where
riðm;nÞ Am;n dm;n þ
J X
rij Aj dj
ð24Þ
j¼1
riðm;nÞ = the stress in any concrete element of the
cross-section at time t = i Dt; Aðm;nÞ = the area of the element; dðm;nÞ = the distance from the center of the element to the centerline of the column; rij = the stress in any longitudinal rebar at t = i Dt; Aj = the cross-section area of rebar; and dj = the distance from the center of the rebar to the centerline of the column. The external moment is calculated as the product of the total vertical force at mid-height, Pi , times the horizontal deflection, plus the eccentricity of the column at that location, Fig. 6(a). The cross-sectional stress– strain distribution is shown in Fig. 6(b).
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P P (e 0+ea)
L/2
P
s
e 0+e a
As )
neutral axis
P (e 0+ea )
yn
e0 +ea+
n
m
t(m,n)
(m,n)
P
ly/2
L/2
central line
d (m,n)
L0
P (e 0+ea
s
As /
P
/
s
/
s
c
P
(a)
(b)
Fig. 6. (a) Moments in slender member with compression plus bending moment, for an equivalent pin-ended column; (b) stress–strain distribution in cross section due to axial load and bending.
M iext ¼ Pi ðD þ e0 þ ea Þ
where Pi is calculated as a summation of the force contribution from the individual elements. Thus,
Pi ¼
M1 N1 XX
J X
m¼1 n¼1
j¼1
riðm;nÞ Am;n þ
rij Aj
ð26Þ
Using Eqs. (6)–(26), and the temperature field, the strength of the RC column at each time step can be evaluated, until the strength reaches the applied moment value. The corresponding time is the fire endurance of the column. From Table 4 it can be seen that the relative load capacity contributed by concrete, PTc =PTcr , is almost the same as the relative cross-section area, 2
A=Ar ¼ ðd=dr Þ . There is a slight difference between the total relative load capacity, P T =PTr , and the relative cross-section area. This is because it is difficult to make all columns have the same reinforcement ratio, and the same arrangement of reinforcement. Furthermore, in the studied cases, the concrete cover to the longitudinal steel is the same, and does not change proportionally to the cross section size. The total relative strength of a square column under fire condition can be approximately given by:
PT PTr
¼
1:875 d dr
ð27Þ
where PT is the load capacity of RC column at equivalent fire duration given by Eq. (5), PTr is the load capacity of reference RC column at given fire duration. Table 4 Strength of concrete columns at the given durations, tr = 120 min. D (mm)
d/dr
(d/dr)2 te (min) t/tr
300 (ref) 1 1 350 1.167 1.361 400 1.333 1.778 450 1.500 2.250 500 1.667 2.778 550 1.833 3.361 600 2 4
P T and PTr = the total load capacity of the column and reference
ð25Þ
120 150 182 216 252 290 330
1 1.250 1.517 1.800 2.100 2.417 2.75
PT (kN) P Tc (kN) P T =P Tr
P Tc =P Tc;r
1165 1610 2046 2487 3053 3718 4324
1 1.362 1.753 2.210 2.729 3.301 3.930
1050 1430 1841 2321 2865 3466 4127
1 1.382 1.756 2.135 2.621 3.191 3.712
d = the depth of the square cross-section, te = the equivalent fire duration; 1 kN = 225 lb; 1 mm = 0.039 in.
column at elevated temperature, respectively. P Tc and PTc;r = the contribution of concrete to total load capacity of the column and reference column at elevated temperature, respectively. The load capacity contributed by the concrete of a square column under fire condition can be given by (regression analysis):
PTc PTc;r
1:9699 2 T d d PT Pc;r ¼ 0:993 ) 2c ¼ 2 dr dr d dr
ð28Þ
Eq. (28) indicates that the average stresses of plain concrete columns, with different cross-section sizes, are identical for equivalent fire duration. This result confirms that the criterion of ‘‘the equivalent fire duration’’ is reasonable. 3.3. Size effect on the fire endurance of simple flexural beams To enhance the fire resistance of RC members, the conventional design method in most publications, including building codes (Eurocode 2 [1], is to increase the concrete cover thickness of RC members. The temperature corresponding to the member failure must, among other factors, be dependent on the loading type and load level [22]. As specified in American Society for Testing and Materials [2], the fire endurance of a RC beam has been traditionally defined based on thermal and strength failure criteria as specified in American Society for Testing and Materials [2]. The following two sets of failure criteria are incorporated into this study to define the failure of the beam in the fire condition: (1) The applied service load exceeds the strength of the beam i.e. moment at first crack; and (2) the temperature in tension steel exceeds 593 °C. The computer model developed by Liu [19], based on the integrated thermal and strength criteria for failure, was used to evaluate the effects of the size of beam cross-section and the thickness of concrete cover on the fire endurance of RC beams. The results are described below. 3.3.1. Properties of studied beams The studied beams were simply-supported with a span of 6 m, shear span of 2 m, and a cross section of (200–300) mm (300– 0 600) mm; f c ¼ 30 MPa; tension rebar 2 £ 20 to 3 £ 20; f y ¼ 400 MPa.
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1200
3.3.2. Effects of beam depth and width The effects of the overall beam depth on the fire endurance of RC beams, based on tensile rebar temperature and structural strength (load level = 0.5), are given in Fig. 7. It is evident that the cross-section depth almost has no effect on the fire endurance, based on the thermal and strength failure criteria, and the fire endurance increases slightly with the increase in width. The effect on fire endurance based on the strength criterion is greater than that based on the temperature criterion.
Temperature, oC
1000
3.3.3. Effect of concrete cover thickness Figs. 8–10 show that the fire endurance increases with the concrete cover thickness for both criteria of strength and steel temperature. For example, for a simply supported beam cross section, shown in Fig. 9, the fire endurances based on rebar temperature are 104, 131, and 159 min, and based on strength criterion, the fire endurances are 99, 125, and 152 min, for concrete cover thickness, C = 30, 40, and 50 mm, respectively. The difference based on the two criteria is small. Fig. 10 shows that during the first 30 min fire exposure, the smaller concrete cover thickness results in the higher load capacity due to larger effective depth. However, when the rebar temperature is larger than 300 °C, as shown in Figs. 8 and 10, the reduction rate of the strength decreases with the increase in concrete cover thickness during fire exposure. After 120 min fire exposure, the strengths of the beam were 23%, 35%, 52%, and 69% of the strength at room temperature for C = 20, 30, 40, and 50 mm (0.78, 1.17, 1.56, and 1.95 in.), respectively. By properly increasing the concrete cover thickness, the fire resistance of RC flexural members can be improved significantly. This increase in the concrete cover thickness not only reduces the load capacity of a beam of fixed depth and reinforcement ratio at room temperature, but also possibly increases the width of surface cracks, and hence decreases the serviceability. Therefore, it is not practical to excessively increase the concrete cover thickness for improving the behavior of RC flexural members under fire conditions.
800 600 ASTM E119 fire
400
C=20mm C=30mm
200
C=40mm C=50mm
0
0
30
60
90
120
150
180
Time, min Fig. 8. Temperatures of corner rebar (1 mm = 0.039 in.).
3.4. Validation The validity of the computer model was established by comparing predicted results from the model with the measured values from fire tests for columns, Lie and Woollerton [17]. The geometric and material properties of the tested columns used in the analysis are given in Table 5. Taking Column I-2 as an example, the
Fig. 9. Details and properties of the studied beam with different concrete cover thickness (1 mm = 0.039 in.; 1 MPa = 145 psi; 1 kN m = 738 lb ft).
180
160 150 140
Fire endurance, min
Fire endurance, min
160 140 120 100 C=30mm
80
C=40mm
200
250
120 110 100 90
C=30mm
80
C=40mm
70
C=50mm
60
130
300
60
C=50mm
200
250
Cross-section width, mm
Cross-section width, mm
(a)
(b)
300
Fig. 7. Effects of beam cross-section width and concrete cover thickness (C) on fire endurance of RC beams (1 mm = 0.039 in.). (a) Based on the critical temperature of tensile steel. (b) Based on the strength of the beam.
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180
1
160 0.75
Relative strength
Strength, kN.m
140 120 100 80 C=20mm
60
C=30mm
40
C=40mm
20
C=50mm
0
0.5
0 0
30
60
90
120
150
C=20mm C=30mm C=40mm C=50mm
0.25
180
0
30
60
Time, min
90
120
150
180
Time, min
Fig. 10. Load capacity with different concrete cover thicknesses (1 kN m = 738 lb ft; 1 mm = 0.039 in.).
Table 5 Summary of relevant parameters and results. Column no.
Size (square) (mm)
Reinf. Qty-Size
Comp. strength MPa
Applied load (kN)
Design load (kN)
Load level*
I-2 I-3 I-4 I-7 I-8 I-9 II-1 II-2 II-3 II-4 II-5 II-6 II-9 III-1 III-2 III-14
305 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305
4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M 4-25M
36.9 34.2 35.1 36.1 34.8 38.3 41.6 43.6 35.4 52.9 49.5 46.6 37.1 39.6 39.3 37.9
1333 800 711 1067 1778 1333 342 1044 916 1178 1067 1076 1333 800 1000 1178
3726 3517 3587 3665 3564 3835 4090 4245 3610 4965 4702 4477 4550 3935 3912 3804
0.358 0.227 0.198 0.291 0.499 0.348 0.084 0.246 0.254 0.237 0.227 0.240 0.293 0.203 0.256 0.310
Fire endurance (min) Measured
Predicted
170 218 220 208 146 187 340 201 210 227 234 188 225 242 220 183
167 221 234 193 117 171 304 213 209 217 221 215 192 233 208 185
1 kN = 225 lb; 1 mm = 0.039 in.; 1 MPa = 145 psi. Aggregate type: Siliceous; Strength of steel = 444 MPa; The clear cover of the main steel = 38 mm; relative humidity: 65–75%; M = mm. 0 a load level = applied load=ð0:85f c Ac þ f y As Þ, where fc = concrete strength at room temperature, Ac = area of concrete, fy = main steel strength at room temperature, As = area of steel.
Fig. 11. Temperature of concrete at various depths along centerline as function of exposure time (1 mm = 0.039 in.).
predicted values of the results (temperatures and deformations) from the analysis are compared with the measured values from fire tests in Figs. 11 and 12. The predicted load level effect on fire endurance is shown in Fig. 13, together with the measured values.
Fig. 12. Axial deformation of columns as a function of exposure time (1 mm = 0.039 in.).
It can be seen from Fig. 11 that, with the exception of the temperatures measured at 64 mm (2.5 in.) depth from 70 to 140 min, there is good agreement between the predicted and measured values. The temperatures measured at the center of the column show
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360 330
Fire Rsistance, min
300 270 240 210 180 150 120 90 60 30 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Load Level Fig. 13. Effect of load level on the fire endurance of columns.
initially a relatively slow rise before 100 °C. This temperature behavior may be the result of thermal-induced migration of the moisture toward the center of the column, where as shown in the figures, the influence of the migration is most pronounced Lie and Irwin [18]. Fig. 12 shows the calculated and measured axial deformations of the columns during exposure to fire. The numerical analysis predicts reasonably well, the trend in the progression of the deformation with time. The fire endurance decreased with load level, Fig. 13, until the column could no longer support the applied load, Table 5. It is easy to determine the fire resistance of column for given applied load level, and vice versa.
Acknowledgements The authors gratefully acknowledge the National Science Foundation (NSF), Grant No. 0425699 (Contract Monitor: Dr. O. Shinaishin), and the United States Department of Agriculture (USDA), Agreement No. 58-5148-7-175 for the interest, encouragement, and financial support.
Appendix A. Properties of concrete (siliceous aggregate) A.1. stress–strain relationships [14]
4. Conclusions Based on the current study, the following conclusions can be drawn: The numerical model used in this paper is capable of predicting the fire resistance of square or rectangular concrete columns, with an accuracy that is adequate for practical purposes. Using the curve developed in this study for the effect of load level on the fire endurance of columns, the fire resistance of square RC columns with the same similitude conditions can be evaluated easily for any given applied load, and vice versa. For axially loaded square columns, the fire endurance increases with the cross-section size at the same load level. Two relationships for axially load column have been developed: (i) fire endurance with cross-section size and, (ii) load capacity in the fire condition with cross-section size. The fire resistance of RC columns can be determined based on these two relationships, and the test data or numerical results of the reference column. Because the fire resistance of RC beams mainly depends on the load ratio and the temperature of the reinforced bars, the cross-section depth has almost no effect on the fire endurance of RC beams. The fire endurance increases only slightly with the increase of cross-section width. By properly increasing the concrete cover thickness, the fire resistance of RC beams can be improved significantly. However, it is not practical to excessively increase the concrete cover thickness for improving the fire resistance of RC beams. Although the cover thickness of 50 mm was observed to be the most suitable for the 2-h ASTM fire exposure time, further studies are needed to establish an optimal cover thickness related to effective depth.
y¼
8 > < 2x x2
06x61
x > 2ðx1Þ2 þx
x>1
:
0
0
where y ¼ r=f cT ; x ¼ e=e0T , and f cT , concrete strength at elevated temperature, is given by Hertz [11]:
" , 0
0
f cT ¼ f c 1
1þ
2 8 64 !# T T T T þ þ þ T1 T1 T1 T1
For siliceous aggregate, the proposed values for T1, T2, T8, T64 are: Siliceous aggregate: T1 = 15,000, T2 = 800, T8 = 570, and T64 = 100,000. A.2. Specific heat (J/kg °C) [23]
C c ðTÞ ¼ 900
for 20 C 6 T 6 100 C
C c ðTÞ ¼ 900 þ ðT 100Þ for 100 C < T 6 200 C C c ðTÞ ¼ 1000 þ ðT 200Þ=2 for 200 C < T 6 400 C C c ðTÞ ¼ 1100
for 400 C < T 6 1200 C
A.3. Thermal conductivity (W/m °C) [8] Lower limit:
T
T 2 kðTÞ ¼ 1:36 0:136 100 þ 0:0057 100
for 20 C 6 T 6 1200 C
Upper limit:
T
T 2 kðTÞ ¼ 2 0:2451 100 þ 0:0107 100
for 20 C 6 T 6 1200 C
478
D.V. Reddy et al. / Engineering Structures 99 (2015) 468–478
Appendix B. Properties of steel reinforcement B.1. Stress–strain relationship [23]
r¼
8 e EsT > > < f yT þ ðf uT > > :
0:62 e e eeyT 3 f yT Þ 1:5 euT yT 0:5 eyT euT eyT
jej 6 eyT
euT P jej > eyT
h i h i f yT ¼ f y = 1 þ 24ðT=1000Þ4:5 ; f uT ¼ f u = 1 þ 36ðT=1000Þ6:2 EsT ¼ f yT =eyT ; EsT ¼ f yT =eyT
euT ¼ 0:18 0:23ðT=1000Þ P 0:04; eyT ¼ 0:18 0:23ðT=1000Þ P 0:04 in which r, e = the stress and mechanical strain of steel reinforcement, respectively; fy, fu = the yield strength and ultimate strength of steel reinforcement at room temperature, respectively; f yT , f uT = the yield strength and ultimate strength of steel reinforcement at temperature T, respectively; eyT = the yield mechanical strain of steel reinforcement at temperature T, and taken as 0.001874 [9]; euT = the ultimate mechanical strain of steel reinforcement at temperature T. B.2. Thermal strain
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