The estimation of the air temperature distribution in the area of fire is performed. The temperature distribution in the structure is evaluated by solving the non-linear transient heat conduction problem which is governed by the diffusion equation with corresponding boundary and initial conditions. The problem is non-linear because both, thermal material parameters and radiation boundary condition, are temperature dependent. The finite element method is used to solve the problem. The time dependent mechanical response of the structure due to simultaneous action of mechanical and temperature loads is the main topic of our work. The numerical procedure stems from the Reissner theory of planar beams in which the membrane and flexural deformations are accounted for. The strains, displacements, and rotations may all take arbitrarily large values. At this stage of research, the effect of shear deformations is neglected. Besides the well-known Bernoulli hypothesis of a straight cross-section, the compatibility of reinforcement and concrete is assumed. A new element based on pseudocurvature interpolation as a sole unknown function of the problem is employed in mechanical analysis. The main advantages of this element are that it is locking-free and that the constitutive equations are consistently taken into account. The accuracy of non-linear fire resistance analysis depends on the choice of constitutive equations for concrete and steel reinforcement, and also by the adequate choice of models describing thermal and creep deformations. In the present formulation, the constitutive equations for concrete and reinforcement proposed by the European prestandard ENV 1992-1-2 are included in our models. The beneficiary effect of the tensional strength of concrete is neglected. The effect of temperature change in concrete and reinforcement is also defined according to European standards. An assumption that the total strain increment may be considered as a sum of mechanical strain increment and thermal strain increment is introduced. A part of the total strain results from the viscous creep strain of the reinforcement, which is introduced into our calculation by the model, presented by Williams-Leir[].

The stress and strain distributions in the structure due to an arbitrary conservative mechanical and thermal loading are estimated by the incremental–iterative procedure. The time interval of the duration of the fire is divided into time steps. For each time step, Newton's method is used to evaluate the unknown values of deformation, kinematic and static quantities at the end of the time step from the values of the quantities at the beginning of the time step. The system of non-linear equilibrium equations is linearised and solved for each time step separately. The length of the time step depends on the convergence of Newton's method and on the rate of temperature change. The accuracy of the method is analysed on a relatively simple example of approximately 12 m long simply supported beam of T-shaped cross-section. The beam is loaded by the uniform load and is exposed to fire. The results of experiments were published by Gustaferro et al[1]. The web of the beam was exposed to a fire in which the ambient temperature increased according to the ASTM standard fire curve. The lower and the vertical sides of the flange were insulated. Since the report does not give the thermal material properties, the conductivity, the specific heat, and the thermal expansion coefficient were taken according to the standard ENV 1992-1-2, and are all taken to be temperature dependent. The temperature distribution was evaluated for a typical cross-section. It was assumed that the temperature distribution is equal in all cross-sections. The simply supported beam is modelled by finite elements in which the pseudocurvature of the centroid axis is interpolated by the fourth degree Lagrange polynomials. The 5-point Lobatto integration scheme is used for the numerical integration along the axis of the element. The comparison between numerical and experimental values of the vertical displacement at the midspan of the beam shows that the chosen values of thermal material properties are not adequate since the evaluated displacements were too large. The time to numerical failure of the structure is also underestimated (250 minutes compared to 370 minutes observed in the experiment). The same conclusion may be drawn also by the comparison of experimental and numerical values of temperature in the reinforcement in which the numerically estimated temperature was much too high. Only if the radiation emissivity er of the surface is reduced from 0.56 to 0.3, numerical values of reinforcement temperature become similar to measured temperatures in the reinforcement. Then the numerical failure of the structure occurs after 325 minutes, which is closer to the actual time of failure. It is evident that an additional source of heat must have influenced the structure during the first 25 minutes of the experiment which was not accounted for in the previous analysis. If this event is to be simulated by the numerical analysis the experimental values of the reinforcement temperature instead of numerical values have to be employed in the mechanical analysis. In this case the displacements virtually coincide with the measured values for up to 180 minutes. Afterwords the viscous creep becomes an important factor. Therefore, in the last analysis, the viscous creep is additionally introduced to the material model in which the values of creep constants are taken according to Williams- Leir (1983) for steel X-60. In this case the calculated values of displacements are virtually equal to the measured ones for up to 320 minutes when the numerical procedure fails. Then the estimated displacement is 46.68 cm compared to the measured one which is approximately 46 cm.

1

A.H. Gustaferro, M.S. Abrams, E.A.B. Salse: Fire-Resistance of Prestressed concrete Beams, Study C: Structural Behaviour During Fire Tests (RD009.01B), PCA, Research and Development Bulletin, 1�29, 1971.

2

G. Williams-Leir: Creep of Structural Steel in Fire: Analytical Expressions, Fire and Materials, 7, 73�78, 1983. doi:10.1002/fam.810070205

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £122 +P&P)