NONLINEAR STRESS-STRAIN BEHAVIOUR OF RC ELEMENTS AND RC FRAMES EXPOSED TO FIRE Meri Cvetkovska 1 ABSTRACT Structure exposed to fire should not undergo a significant damage or collapse as long as it takes to evacuate people or to extinguish the fire. For a given specific loading, element geometries and support conditions, the fire resistance is defined as a time when plastic hinges are formed and the structure has lost its bearing capacity. The objective of this study is to develop a nonlinear finite element program for predicting the structural behaviour of planer reinforced concrete frame structures, exposed to fire. Two numerical moduls are incorporated in this program. The modul FIRE-T carries out the nonlinear transient heat flow analysis associated with fire and the second modul FIRE-S predicts the response of reinforced concrete elements subjected to the thermal histories predicted by FIRE-T. The modul FIRE-S is based on a tangent stiffness formulation and an iterative analysis approach. The complex features of structural behavior in fire conditions, such as thermal expansion, shrinkage, creep, cracking or crushing, and material properties changing with temperature are considered. The use of the program FIRE for defining the fire resistance of three-bay, two-story reinforced concrete frame is presented. For a given specific loading and element geometry, different fire scenarios are analyzed. Key Words: heat transfer, fire resistance, structural response, material nonlinearity 1 Assistant Mr.., Univerzitet Sv. Kiril i Metodij, Gradezen fakultet, Partizanski Odredi bb, 1 Skopje, Republika Macedonija, tel: 116-66 (ext. 118), e-mail: cvetkovska@gf.ukim.edu.mk 1
1. Introduction The fire resistance of reinforced concrete structural elements is determined by subjecting a specimen to a standard fire test. The fire resistance of the element is then expressed as the time duration of acceptable behavior, given either in terms of critical temperatures at specified locations, or in terms of the capacity to sustain load during testing. Standard tests do not, however, reliably predict behavior because actual fires and structural restraints cannot be adequately simulated. Fire tests provide reliable information on local behaviour of the elements, but the question about the formation of the possible mechanisms which influence global behaviour of the structure as a whole, remains open. The response of complex structures subjected to a real fire loading is estimated by methods of computational modelling of thermodinamic and thermomechanical processes. However, the development of various numerical methods for the analysis of the nonlinear response of structures under fire does not reduce the necessity, nor the importance of the afore mentioned experimental tests. They are necessary to determine the mechanical properties of materials at elevated temperatures, as well as to check the adequacy of the developed computational methods for the analysis of the response of structures. This paper presents a computational procedure for the nonlinear analysis of a reinforced concrete plane frame structure subjected to fire loading. The program FIRE carries out the nonlinear transient heat flow analysis (modul FIRE-T) and nonlinear stressstrain response associated with fire (modul FIRE-S). The solution technique used in FIRE is a finite element method coupled with time step integration. A nonlinear heat flow analysis is used to account for the temperature dependence of material properties and the thermal boundary conditions. This nonlinearity requires the use of an iterative procedure within any given time step. While cracks appear, or same parts of the element crush, the heat penetrates in the cross section easier, but in this study it is neglected. In modeling the fire response of reinforced concrete structures, it has been assumed that the heat flow is separable from the structural analysis. While the temperature field is defined in a small time interval, the response of the structural element can be determinate too. A direct stiffness method approach of nonlinear structural analysis is implemented in this program. An iterative procedure within a given time step is required to solve the problem because the structural stiffness depends on the deformed shape, on the current temperature distribution and the previous response history of the structure. The used analysis procedure does not account for the effects of large displacements on equilibrium equations. 2. Heat conduction The governing differential equation of heat transfer in conduction [1] is:?? T?? T?? T? T (? x )? (? y )? (? z )?? c (1)? x? x? y? y? z? z? t 2
where:? x,y, z is a thermal conductivity (temperature dependent);? is a density of the material (temperature dependant); c is a specific heat (temperature dependent). The fire boundary conditions can be modeled in terms of both convective and radiative heat transfer mechanisms. The heat flow cosed by convection is : q? h (T T ) (2) c c z? f where: hc is coefficient of convection (for the surface directly exposed to fire it s value is h c =25 W/m 2 C, and h c =9 W/m 2 C for the unexposed surface); Tz is the temperature on the boundary of the element; T f is the temperature of the fluid around the element. The heat flow cosed by radiation is : r c 4 z,a 4 f,a r? T T? q? V?? ( T? T )? h? (3) 2? T? T?? T T? z f 2 h r? V?? c z, a f,a z,a? f,a (4) where: h r is coefficient of radiation (temperature dependant); V is a radiation view factor (recommended V? 1. );? is a resultant coefficient of emission??? f? z ;? f?. 8 is the coefficient of emission for the fire compartment;? z?. 7 is the coefficient of emission for? 8 the surface of the element;? c? 5.67? 1 is Stefan-Boltzmann constant; T z, a is the absolute temperature of the surface; T f, a is the absolute temperature of the fluid. The solution of the differential equation (1), which involves the boundary conditions defined by equations (2) and (3), in FEM is given by a system of algebraic equations over the unknown nodal temperatures:?? T?? K??? K?? T?? R?? T P C? 1 2? (5) R -radiation C -capacity matrix (temp. dependent); P -vector of temperature loads (convection and radiation included); T -vector of unknown nodal temperatures; T - where:? K1?-conductivity matrix (temp. dependent);? K2?-convection matrix;?? matrix (temp. dependent);?? vector of temperature derivatives over time. If the heat capacity of the material is taken under consideration and if thermal load is time dependent, problem becomes transient and for solving the Equation (5) an iterative procedure has to be used [3]. In a small time interval we assumed that the time derivative of the temperature is constant: Tt?? t? Tt Tt? Tt?? t? (6)? t By summarizing Equation (5) for time t and t?? t and assuming that the capacity matrix in small time interval is constant ( C t? C t?? t ), the heat flow equation becomes:? 2?? 2? K? t t?????? t?? t? Ct Tt?? t?? Kt? Ct Tt? Pt?? t Pt? K t??? K1t??? K 2??? Rt? (7) 3
Equation (7) together with the initial and boundary conditions completely solve the problem. Taking the radiation into account makes the problem nonlinear. This problem is solved by involving new iterative procedure in every time step. Problem becomes nonlinear too, when temperature dependent physical properties of the materials are assumed. In that case the conductivity and capacity matrix are defined at the beginning of each time step based on the temperature from the previous time step. 3. Nonlinear stress-strain analysis of plane frame structures The response of a reinforced concrete plane frame structure exposed to fire is predicted by modul FIRE-S. This modul accounts for: dimensional changes caused by temperature differentials, changes in mechanical properties of materials with changes in temperature, degradation of sections by cracking and/or crushing and acceleration of shrinkage and creep with an increase of temperature. Determining the fire response of reinforced concrete structure is thus a complex nonlinear analysis problem in which the strength and stiffness of a structure as well as internal forces continually change due to restraints imposed by the structural system on free thermal expansion, shrinkage, or creep to maintain compatibility among all structural elements. Because frames are modeled as an assemblage of members connected to joints, the basic analytical problem is to find the deformation history of the joints U (t ) when external loading at the joints R ( t ) and temperature history within the members T (t ) are specified. Since only two dimensional frames are considered, each joint has three degrees of freedom, two translations and one rotation. Likewise, there are two forces and a moment at each joint. The overall system stiffness matrix of a structure is assembled by incorporating the stiffness contribution of each member. Each member is treated as a linear beam element modeled by simple beam theory and is composed of a linear elastic material. In a reinforced concrete, the first condition holds if members are of usual properties, but the second condition is violated for virtually all loading conditions. The materials in a reinforced concrete structure are nonlinear and detailed knowledge of the strain states existing within members is necessary in order to obtain the member stiffness matrix. Hence, the member must be further discretized. The basic discretization sheme used in the program to determine stiffness properties and internal forces is shown in Fig.1. Structure members are subdivided into a number of segments such that by calculating segment properties, it is possible to determine overall member properties. Each segment is treated as a standard beam element in which axial force is assumed to be constant and bending moment to vary linearly along the length of a segment. However, in order to calculate section properties and internal forces and moments within a segment, it is necessary to discretize cross sections further into subslices. a) b) 4
Fig.1: Geometric idealization of a structure: a) a structure discretizated to members and segments, b) cross-section of a segment discretized to subslices The subslices associated with each segment can be envisioned as uniaxially loaded prisms. Therefore, only uniaxial stress states are considered, equivalent to the assumption that the effect of multiaxial stress components is negligible. This type of prismatic model allows only the effects of axial and flexural stiffness to be considered in modeling structural behaviour, neglecting the effect of shear in member idealization. Local effects near member ends are not considered. However, in most linear structures these effects are of secondary importance and principal structural action is due to flexure and axial deformation. Internal axial force and bending moment at each end of a segment are found by summing the force and moment contributions of subslices discretizing the cross section. The mechanical strain in each subslice can be found directly from the extension and curvature of a cross section:? y DCG? r, j? i?? a??? (8) r, j where:? i is the total mechanical strain in the subslice r;? a is the total strain in the reference plane; y is the distance from subslice to reference plane;? is the curvature; DCG is distance from the reference plane to centroidal plane. In order to find current stress in each subslice, that part of the current total mechanical strain which gives rise to stress, must be determined: j r,j f? i?? i?? i (9) j f where:? i is stress related strain for time step i and iteration j;? i is free strain in subslices for time step i. Free strains are determined for each subslice at the beginning of a time step and do not vary during iteration within a time step. For the concrete subslice they are calculated according to the following equations: f,c f,c cr,c tr,c th,c? i?? i? 1??? i??? i??? i (1) f,s f,s cr,s th,s? i?? i? 1??? i??? i (11) f cr where:? i is free strain for concrete, or steel subslice, for time step i;?? i is free creep tr,c strain accumulated over current time step i;?? i is transient strain accumulated only in th concrete subslice over current time step i;?? i is free thermal expansion accumulated over current time step i. The??? relations, recommended by EC2, part 1.2 [2], take into account the creep of concrete and steel at elevated temperatures. It is done by moving the maxima in the stress-strain curves to higher strains with higher temperatures. When compressed concrete is heated for the first time, the total strain is different from the total strain measured in constant temperature creep tests and an additional irrecoverable transient strain must be taken into account. This transient strain is a function of the level of stress and the thermal expansion (Anderberg and Thelandersson, 1978) [6]. The increment of transient strain at any given time step can be computed as: 5
? tr,c c th,c?? i?? 2.35?? ' i (12) fc tr,c th c, where:?? i is transient strain accumulated over current time step i;?? i is free thermal expansion accumulated over current time step i;? c is applied compressive stress; f c ' is compressive strength at ambient conditions. When the transient strain is not included in the structural analysis, the result is stiffer structural model at elevated temperatures, in which the computed thermal stresses become very large and failure is predicted to occur much earlier than experimentaly observed (Ellingwood and Lin, 1991) [5]. 4. Fire resistance of plane frame structure Thermal and structural response of three-bay, two-story reinforced concrete frame exposed to different fire scenarios (fire appears in different zones), have been investigated analytically using the method outlined in this paper. The frame is assumed to be fixed at the base, the uniform load is q=45kn/m (the same for the two stories), all columns and beams are designed with the same cross section (4? 4cm), columns are symmetrically reinforced with 8?16mm and details of beam reinforcement are shown in Fig.6 and Fig.7. The concrete cover for all elements is a=2.5cm. ISO 834 standard fire is considered. EC2, part 1.2 [2] recommendations for thermal and mechanical properties of concrete and steel at high temperatures are assumed. The yield strength of reinforcement at ambient temperature is f y? 4Mpa, and the compressive strength of concrete is f c? 3Mpa. symmetry 64.3 46.4 64.3 94.6 48.2 5.1 4.1 114.7 99.2 4.5 46.7 93. 16.3 14.6 11.6 6.4 3.4 15.5 17.7 33.6 67.4 38.1 9.2 13.2 141.5 3 1 2 57.3 46. 35.9 45.2 28.5 66. 28.5 44.7 11.7 3m t=. h t=3.9 h 3m 24.5 2. 5m 5m 5m 6.4 18.6 Fig.2: Bending moment redistribution for the frame exposed to fire in the first story middle bay ( fire resistance of the frame is t=3.9h)? y=-.21? y=-.568? y=-.42? y=-.373? y=-.432? x=+.62? y=+.191? x=+.129? y=-.471? x=+.22? y=-.13? y=-.18 6? y=-.465? y=-.28? y=-.429? y=-3.481? x=+.745? y=+.21? x=+.781? y=-.1? x=+.83? y=-.11 +? y(cm) +? x(cm) t=. h t=3.9 h symmetry
Fig.3: Deformation history of the frame exposed to fire in the first story middle bay ( fire resistance of the frame is t=3.9h) Two different fire scenarios are analysed: fire in the first story middle bay (Fig.2 and Fig.3) and fire in the first story exterior bay (Fig.4 and Fig.5). In both cases the bottom of the elements exposed to fire becomes hotter than the top and tends to expand more than the top. This differential heating causes the ends of the elements to tend to lift from the supports thus increasing the reactions. This action results in a redistribution of moments. The negative moments increase while the positive moment decreases and tends to become negative. After time t=.5h, the negative moments begin to decrease again, but slowly. The first.5 hour of the fire leads to large thermal expansion in the floor beam above the fire compartment, pushing the giders in adjacent bays outward. When the fire is in the middle bay the lateral displacements of the joints are restrained by beam and column frame elements away from the fire zone, so significant compression force developes in the beam above the fire. This axial force acts as a prestressing force and delays the moment when yielding of the top reinforcement will occur, so it has a positive effect on the fire resistance of the frame. This is not a case when fire is in the exterior bay and free expansion is partly allowed. Structural behavior is illustrated in Fig 3 and Fig.5, where the deformation of the frame model due to each fire scenario is shown. 16.8 85.4 1 131.8 16.8 3 46.4 69.4 3.6 4 135.1 117.4 39.9 82. 115.6 134.1 111.9 2 17.7 33.6 5. 41.2 85.2 15.6 27.2 2.6 Fig.4: Bending moment redistribution for the frame exposed to fire in the first story exterior bay ( fire resistance of the frame is t=3.45h)? x= -2.678? y=+.728? x= -.322? y=+.721? x= -1.433? y= -4.578 39.9 5.5? x= -.222? y= -.517? x= -.111? y=+.21? x= -.158? y=+.185? x= -.67? y= -.312? x= -.114? y= -.263? y= -.28? y= -.42? y= -.458? y= -.571? y= -.13? y= -.21 62.7 93.8 46.2 62.7 47.6 3m 3m. 31.2 5m 5m 5m. 24.3 +? y(cm) +? x(cm) 7
Fig.5: Deformation history of the frame exposed to fire in the first story exterior bay ( fire resistance of the frame is t=3.45h) The concrete directly exposed to fire develops large compression stresses due to thermal gradients. At the ends of the beam fire has the same effect as the uniform load does, so concrete at the bottom of the cross section crushes, while concrete at the top of the cross section cracks (Fig.6). At the middle of the span the effect is opposite (Fig.7). Cracked 2? 14+4? 16 26 cm 14 cm 26 cm 14 cm 14 cm 26 cm Crushed 4? 14 t?. h t? 2. h t? 3.45 h Fig.6: Degradation profiles for beam at the interior support, when fire is in the first story exterior bay 2?14 26 cm 14 cm 26 cm 14 cm 14 cm 26 cm Cracked Crushed 4?14 t?. h t? 2. h t? 3.45 h Fig.7: Degradation profiles for beam mid-span, when fire is in the first story exterior bay The temperature in the reinforcement directly influences the fire endurance of reinforced concrete elements. The concrete protects reinforcement from high temperatures, so bars which are close to the fire are hotter, and opposite. Thus the increase in negative moment at the ends of the beam can be accommodated, but the redistribution that occurs is sufficient to cause yielding of the top reinforcement (el.3, 4 and 5, Fig.8a). In the same cross section the reinforcing bars which are on the side of the fire (el.1 and el.2, Fig.8a) are all the time in compression by the action of the fire and the negative bending moment. The yield strength is reduced because of the high temperatures, so they start to yield very soon (el.2 is hotter and start to yield first). The resulting decrease in positive moment at the mid-span means that the bottom reinforcement can be heated to higher temperature before failure would occur (Fig.8b). At the beginning the bottom reinforcement is in tension and the top reinforcement is in compression. After some time, as e result of fire action and moment 8
redistribution, they change the sign. When temperature penetrates deeper in the cross section the bending moment tends to become positive again (Fig.2 and Fig.4) and the influence of the uniform load is dominant, so the stresses have the same sign as at the beginning of fire action. The temperature in el.1 (Fig.8b) is lower then in el.2, the compression effect caused by fire is less then for el.2, so this reinforcement doesn t change the sign.? s / fy (T) in % 1 el.3-el.5 7 4 1-2.5 1 1.5 2 2.5 3 3.5 el.1, el.2-5 -8-11 time (h) el.1 el.2 el.3, 4, 5? s / fy (T) in % 1 75 el.3 5 25.5 1 1.5 2 2.5 3 3.5-25 el.1, el.2-5 el.1 el.2-75 el.3-1 time (h) a) b) Fig.8:Time redistribution of stresses in the reinforcement bars at: a) interior support, b) midspan of the beam, when fire is in the first story exterior bay A typical??? relationship for steel at high temperatures is presented on Fig.9a. Although stresses, as an absolute value, continually increase (el.2, Fig.8a), a descending branch appears which is result of a reduction in the yield strength of the steel. El.5 (Fig.9b) is in the upper part of the cross section, the temperature is low and the yield strength of the steel has the initial value, so the??? relationship is typical for ambient temperature.? s (Mpa) -5-4 -3-2 -1 stresses in el.2 temperature in el.2 el.2 1 8 6 4 2 Temperature (C o )? s(mpa) 4 3 2 1 el.5 stresses in el.5 temperature in el.5 4 3 2 1 Temperature (C o ) -5-1 -15? s*1-3 -2-25 -3 5 1 15 2 25? s*1-3 a) b) Fig.9:??? relationships for the reinforcement bars at the interior support of the beam, when fire is in the first story exterior bay The fire resistance of the frame, when fire is in the first story middle bay, is 3 hours and 55 minutes. First plastic hinges are formed after 35 minutes, at the ends of the beam exposed to fire (section No.1 at Fig.2). Much latter, as a result of moment redistribution, plastic hinges are formed in section No.2. The development of plastic zone in the whole first and last segment of the beam exposed to fire (the segment stiffness becomes zero) is the reason for numerical instability. This moment is treated as a moment of failure. Really, failure occurs few minutes later, when plastic hinge will be formed in section No.3. The fire resistance of the frame, when fire is in the first story exterior bay, is 3 hours and 25 minutes. First plastic hinge is formed after 25 minutes, at the upper end of the exterior column exposed to fire (section No.1 at Fig.4). After 4 minutes plastic hinge is formed in section No.2, and after 45 minutes in section No.3. The development of plastic zone in the whole last segment of the beam (section No.2) is the reason for numerical instability. This 9
moment is treated as a moment of failure. Really, failure occurs few minutes later, when plastic hinge will be formed in section No.4 and the beam will fail in flexure. 5. Conclusion The model proposed in this study is capable of predicting the fire resistance of planar reinforced concrete structural members with a satisfactory accuracy. The computer program FIRE have been developed as analytical tool to study the fire response of reinforced concrete frame structures. Histories of: displacements, internal forces and moments, stresses and strains in concrete and steel reinforcement, as well as current states of concrete (cracking and crashing) and steel reinforcement (yielding) are calculated subject to temperature field development in the thermal time history of the structure. Since a physical testing program for investigating the response of a large variety of structural elements under differing restraint, loading, and fire conditions is impractical and expensive, analytical studies supported by the results of physical experiments could efficiently provide the data needed to resolve questions related to the design of structures for fire safety. Parametric studies, helping to identify important design considerations, could be easily achieved throughout implementation of this program. The time response capability of FIRE can also be used to assess potential modes of failure more realistically and to define the residual capacity of structure after attack of fire. LITERATURE [1] D.R.Pitts and L.E.Sissom "Theory and Problems of Heat Transfer" McGraw-Hill Book Company, (1977) [2] EUROCODE 2: Design of Concrete Structures -Part 1-2: 'General rules-structural Fire Design" European Committee for standardization [3] M.Cvetkovska "Heat Transfer in Structures and Stress-Strain Analysis Using Finite Element Method" M.sc. Thesis, University Sv.Kiril and Metodi, Macedonia, (1992) [4] R.Iding, B.Bresler and Z.Nizamuddin, "FIRES-RC II,, A Computer Program for the Fire Response of Structures -Reinforced Concrete Frames", Report No. UCG FRG 77-8, University of California -Berkeley, (1977) [5] Ellingwood B. and Lin T.D., Flexure and Shear Behavior of Concrete Beams during Fires, Journal of the Structural Engineering, Vol.117, No.2,pp.44-458, February 1991 [6] Huang Z. and Platten A., Nonlinear Finite Element Analysis of Planer re inforced Concrete Members Subjected to Fire, ACI Structural Journal, Vol.94, No.3, pp.272-282, May-June1997 [7] Saje F., Saje B.M., Planinc I., Turk G. and Bratina S., Analysis of Fire Resistance of Reinforced Concrete Planar Frames, Proceedings Third International Conference on Concrete Under Severe Conditions, Vancouver, Canada, 21 [8] Cvetkovska M. and Lazarov L., "Nonlinear Stress Strain Behavior of RC Elements Exposed to Fire", Proceedings of the 1 th Yugoslav Congress of Civil Engineers, Yugoslavia, 1998 [9] Cvetkovska M. and Lazarov L., "Nonlinear Stress Strain Behavior of RC Beams in Nonstacionary Temperature field", The First International Conference on Engineering Computational Technology, Edinburgh, Scotland, August 1998 1
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