Analysis of a portal steel frame subject to fire by use of a truss model

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1 Analysis of a portal steel frame subject to fire by use of a truss model P. G. Papadopoulos & A. Mathiopoulou Department of Civil Engineering, Aristotle University of Thessaloniki, Greece Abstract A plane steel frame is simulated by a truss model. The bars obey uniaxial elastoplastic stress-strain laws. Yield stress and modulus of elasticity are assumed constant up to 300 C and then linearly decreasing up to zero for 900 C. Semi-rigid beam-column connection is represented by a hinge and a bar connecting column and beam; this bar has a small length and properly selected cross-section area and elasticity modulus. Three fire scenarios are considered. Numerical experiments show that heat conduction time in steel is very short and can be ignored; it is enough to assume simplified temperature distributions over the cross-sections. A short computer program, with only about 200 Fortran instructions, is documented, for the step-by-step nonlinear structural analysis of plane truss models of steel frames subjected to fire. Geometrical nonlinearities are taken into account by writing, within each time step of the algorithm, the equilibrium conditions with respect to the deformed truss. The program is applied on a typical steel portal frame. In the output of the application, it is observed that, for temperatures lower than 300 C, influence of fire appears by thermal expansion and additional stresses, as well as by thermal bowing of beams due to temperature gradient over the cross-sections. Whereas, for temperatures higher than 300 C, plastic hinges are gradually formed due to yield stress reduction with temperature, and columns may buckle due to elasticity modulus reduction with temperature. Each of the above two phenomena may lead to a collapse of the portal frame, which is shown, in the analysis, by very large displacements. Keywords: portal steel frame, semi-rigid beam-column connection, truss model, fire, heat conduction, step-by-step algorithm, thermal expansion, thermal bowing, plastic hinges, buckling. é Heat Transfer VIII, B. Sund

2 496 Heat Transfer VIII 1 Introduction In a multi-bay multi-storey frame, a fire is usually restricted in one compartment only, thus a local collapse is possible. On the other hand, in a portal frame, consisting of one compartment only, a global collapse may occur during a fire. This is a reason for which structural analysis of portal frames subject to fire attracts a particular interest (Liew et al [5], Wong [9]). The structural analysis of a frame can be performed by the use of a truss model, which has a global stiffness matrix much simpler than those of the usual finite element simulations. And it is recognized that a simple truss model gives equivalent results with the finite element simulations (Absi [1]). A truss model (discontinuum) is used to simulate the mechanical behavior of a 2D or 3D solid (continuum), in a very simple way. The members of the truss obey uniaxial stress-strain laws, and special care must be taken for the determination of bar sections, so that the stress-strain response of the model approximates that of the continuum under consideration. In the nonlinear problems, the usual finite elements present some difficulties [4]. On the other hand, a bar of a truss model is the simplest possible finite element. And a truss can simply take into account physical nonlinearities by the uniaxial stress-strain laws of the bars [6], and geometrical nonlinearities by writing the equilibrium conditions with respect to the deformed truss [7]. Thus, in a frame simulated by a truss, the formation of plastic hinges, the buckling of columns and the possible collapse due to each of the above two reasons, shown in the analysis by very large displacements, can be taken into account. In the present work, the simulation of a plane steel frame by a truss model is described, particularly the simulation of the semi-rigid beam-column connection. And assumptions are presented for fire scenarios, temperature distribution over beam sections, and dependence of mechanical properties of steel on temperature. A short computer program, with only about 200 Fortran instructions, is documented, for the step-by-step nonlinear structural analysis of plane truss models of steel frame subjected to fire. This program is applied on a typical steel portal frame. Figure 1: a. Simulation of a steel beam element by a truss model. b. Simulation of a web element by a truss model.

3 Heat Transfer VIII Determination of bar sections of model A plane truss model, as shown in fig. 1a, can simulate a steel beam element. The flanges are obviously simulated by single bars. On the other hand, for a web element (fig. 1b), based on the biaxial linear elastic stress-strain relations: σ σ x y E = 1 ν 2 1 ν ν ε 1 ε x y, (1) with a value of Poisson ratio ν = 1/3, we can determine the bar sections of the corresponding truss model (fig. 1b), by the following simple formulas: A 2 3 cos ϑ sinϑ 1 = A3 ( 9 /8) ( bd / 2) A A 3 3 cos + A1 = A1 3, 9 bd ϑ, (2) bh ϑ + A2 = 2, sin A where the angle ϑ is subject to the following limitations: so that positive values for bar sections result. 21 < ϑ 45, (3) 3 Stress-strain-temperature laws of the bars Every bar of the truss model obeys a uniaxial elasto-plastic stress-strain law σ ε, as shown in fig. 2a, which is a function of the constitutive variable of plastic strain ε pl. In fig. 2b, the dependence of yield stress σ y and initial elasticity modulus E o of steel on the temperature T is shown. It is assumed that σ y, E o remain constant up to 300 C and then decrease linearly up to zero for 900 C. The above assumptions approximate the recommendations of various Codes (Wong [9], Zhao [10]). Figure 2: a. Uniaxial elastoplastic stress-strain law of a bar of the truss model. b. Dependence of yield stress σ y and initial elasticity modulus E o of steel on temperature T, where σ y20, E o20 values of σ y, E o for T = 20 C.

4 498 Heat Transfer VIII Figure 3: a. Semi-rigid beam-column connection of flush end-plate type. b. Given moment-rotation curve of the beam-column connection. c. Simulation of the beam-column connection in the truss model. 4 Beam-column connection In a steel frame, the columns are usually connected to the beams, by a semi-rigid connection of flush end-plate type (Broderick and Thomson [2]) as shown in fig. 3a, where the end-plate is connected to the beam by welding and to the column by bolts. The moment-rotation relation of such a semi-rigid connection, described by an elastoplastic diagram like that of fig. 3b, is defined by two parameters, the ultimate moment M u and the yield rotation φ y. These parameters are usually experimentally determined and are given with the input data of the structural analysis of a steel frame. In the truss model, the above beam-column connection is represented by a hinge and a bar connecting beam and column as shown in fig. 3c. This bar has a small length l, e.g. l = 1.0 cm, so that not to distort the geometry of the frame. The cross-section area A and the initial elasticity modulus E o of this bar are determined by the following formulas: M A = u, σ h y E o σy l =, (4) ϕ h y so that to reproduce the given moment-rotation relation M φ of the beamcolumn connection. 5 Fire scenarios Three different histories T t, of the gases temperature T with respect to time t during a fire, are considered, which are trilinear approximations of those presented in Liew et al. [5]: The standard ISO-834 fire which is unfavourable as not having a descending branch and two natural fires with high and low ventilation, as shown in fig. 4.

5 Heat Transfer VIII 499 Figure 4: The three fire scenarios under consideration. Figure 5: a. Heat conduction through a steel flange. b. Initial and final temperature distribution. c. Assumptions for temperature distribution over a steel beam cross-section. 6 Heat transfer The heat is mainly transferred by convection that is by the movement of hot gases (Drysdale [3]). Numerical experiments show that heat conduction through steel is very fast, but it is counterparted by the cool surroundings of the structure. The differential equation of heat conduction within a material, in the one-dimensional case, is written: 2 d T 1 dt =, (5) 2 d x α d t where α is the coefficient of thermal diffusivity for the material under consideration, which, for the steel, has the value α = m 2 /sec.

6 500 Heat Transfer VIII The above differential equation can be transformed to a difference equation, as follows: [T(x+ x,t) 2T(x,t)+ T(x x,t)]/ x 2 =[T(x,t + t) T(x,t)]/α t (6) This one-dimensional equation can be easily calculated by hand. Let apply it in the case of fig. 5a, where in a steel flange with a thickness of 15mm, the above side is cool with a room temperature of 20 C, whereas the convected hot gases of a fire heat the lower side and it is unfavourably assumed that this lower side immediately attains a temperature of 700 C. In fig. 5b, at left the initial step distribution of temperature perpendicularly to the flange is shown, and at right the final temperature distribution, which has been obtained after a calculation by hand. The heat conduction time, through this steel flange, proves very short, about 5sec, which is much shorter than the times of fire scenarios under consideration, e.g. 5sec<<500sec (see fig. 4). Thus, the heat conduction time through steel, as being very short, can be ignored. And it is enough to consider simplified assumptions for temperature distribution, over the steel beam cross-sections, as shown in fig. 5c (Wang et al. [8]). 7 Computer program Figure 6: Flow-chart of the main program. A short computer program, with only about 200 Fortran instructions, has been written for the step-by-step nonlinear structural analysis of a plane truss model of a steel frame exposed to fire. The flow-chart of the main program, shown in fig. 6, is described below:

7 Heat Transfer VIII 501 First, the input data are read: General data, that is yield stress σ y20 and initial elasticity modulus E o20 of steel for room temperature 20 C, as well as its coefficient of thermal expansion α T. For every node are read the support codes I x, I y, the initial coordinates x, y and the external loads P x, P y. For every bar are read the nodes K l, K r that it connects, left and right, and its cross-section area A. Also, the fire scenario T t is read. Within each time step of the algorithm, first the gases temperature T is updated, as well as the temperature over the cross-sections of the beams, according to the assumed temperature distributions. From the known temperatures, the present values for the yield stress σ y and the initial elasticity modulus E o, for every bar, are determined. Then, the subroutine NONL is called, which, by using the nonlinear equations of the problem, finds first from the present nodal coordinates x, y, the strains ε of the bars. Then, the NONL calls the subroutine SET (stress-strain-temperature law σ-ε-τ), for every bar, to find the new plastic strains ε pl, stresses σ and tangential elasticity moduli E of the bars. From the axial forces N of the bars, the NONL finally finds the nodal forces F x, F y which are nonzero because of the thermal expansion and bowing of the beams. Then, the subroutine STIF is called to form the present global stiffness matrix K of the truss, by use of the tangential elasticity moduli E of the bars (physical nonlinearity) as well as the present direction cosines of the bars (geometrical nonlinearity). The subroutine GAUS is called to solve the global equilibrium algebraic system K u = f, where f, u supervectors of nodal forces F x, F y and nodal displacement increments u x, u y, respectively. Thus, the total nodal displacements u x, u y, and the new nodal coordinates x, y are found. Again, the subroutine NONL is called and now the nodal forces F x, F y result nonzero only in the supports. At the end of each time-step of the algorithm, output is printed, that is, for every node, total displacements u x, u y, new coordinates x, y and nodal forces F x, F y. And for every bar, plastic strain ε pl, strain ε, stress σ and tangential elasticity modulus E. When a very large nodal displacement u max appears or a specific time t max is exceeded, the algorithm is interrupted. 8 Application The previously described computer program is applied on a typical semi-rigid plane steel portal frame exposed to fire (Liew et al. [5]). 8.1 Given data In fig. 7 are shown the given data of the application: In fig. 7a, the dimensions and loading of the plane steel portal frame and the support conditions, in fig. 7b detailed dimensions for column and steel cross-sections. Whereas in fig. 7c is presented the given moment-rotation curve M φ of the semi-rigid beam-column

8 502 Heat Transfer VIII Figure 7: Given data of the application. a. Supports, dimensions, loading of the portal frame. b. Detailed dimensions of column and beam crosssections. c. Moment-rotation curve of the beam-column connection. connection of flush end-plate type, which is elastoplastic and is defined by two parameters M u, φ y, that is ultimate moment and yield rotation, respectively. The mechanical properties of steel are: yield stress σ y20 = 275 N/mm 2 and initial elasticity modulus E o20 = N/mm 2 both for room temperature 20 C, whereas the coefficient of thermal expansion of steel is α T = 10-5 / C. For this application, the three fire scenarios of fig. 4 are considered. 8.2 Preparation of input for the program In fig. 8, it is demonstrated how the input data for the computer program are prepared for the present application. First, by exploit of symmetry, only half structure is considered by adding proper supports on the symmetry axis. Then column, beam and joint are simulated by truss elements. A system of reference axes is selected, the external loads on the nodes are noted, the cross-section areas of the bars are determined by use of formulas (2) and the cross-section area and initial elasticity modulus, of the small bar connecting column and beam are determined by use of formulas (4).

9 Heat Transfer VIII 503 Figure 8: Preparation of input data of the program for the application under consideration. 8.3 Results In the output of the computer program, for the application under consideration, the following are observed: For temperatures lower than 300 C, the influence of fire appears by thermal expansion and additional stresses, as well as by thermal bowing of beams due to temperature gradient over cross-sections (fig. 9a). Whereas, for temperatures higher than 300 C, plastic hinges are gradually formed due to reduction of yield stress with temperature. And columns may buckle due to reduction of elasticity modulus with temperature. Each of the above two phenomena may lead to a global collapse of the portal frame, which is shown, in the analysis, by very large displacements (fig. 9b).

10 504 Heat Transfer VIII Figure 9: Output of the application. a. For T<300 C, thermal expansion, bowing and additional stresses, b. For T>300 C, formation of plastic hinges, possible buckling of column and global collapse mechanism. 9 Conclusions 1. A truss model has a very simple global stiffness matrix and can easily describe physical nonlinearities by the uniaxial stress-strain laws of bars and geometrical nonlinearities by writing the equilibrium conditions with respect to the deformed truss. Thus, in a frame simulated by a truss model, the formation of plastic hinges and the buckling of columns can be easily taken into account, as well as the collapse due to each of the above two phenomena, which is shown, in the analysis, by very large displacements. 2. A plane steel frame can be easily simulated by a truss model. The flanges of steel beams are obviously simulated by single bars. Whereas, for the webs, simple formulae based on biaxial linear elasticity, give the corresponding bar sections of the truss model. 3. A semi-rigid beam-column connection of flush end-plate type, with given moment-rotation relation, is simulated in the truss model by a hinge and a bar connecting column and beam. This bar has a small length e.g. 1.0cm, so that not to distort the geometry of the frame and properly selected crosssection area and elasticity modulus, so that to reproduce the momentrotation relation of the beam-column connection. 4. The heat is mainly transferred by convection that is by the movement of hot gases. Numerical experiments show that heat conduction in steel is very fast, but it is counterparted by the cool surroundings of the structure. The heat conduction time in steel can be ignored, because it is much shorter than the times of fire scenarios. It is enough to make simplified assumptions for the temperature distribution on the steel beam cross-sections.

11 Heat Transfer VIII A short computer program, consisting of main program and four subroutines, with totally only about 200 Fortran instructions, for the step-bystep nonlinear structural analysis of plane truss models of steel frames exposed to fire, is documented by description of the flow-chart and an application. 6. The above computer program is applied on a typical steel portal frame. The moment-rotation curve of the semi-rigid beam-column connection is given. Three fire scenarios are tried. It is demonstrated how the input data of the above computer program are prepared by discretization and determination of bar sections of truss model. 7. The output of the application shows that, for temperatures lower than 300 C, influence of fire appears by thermal expansion and additional stresses, as well as by thermal bowing of beams due to temperature gradient over the cross-sections. For temperatures higher than 300 C, plastic hinges are gradually formed due to yield stress reduction with temperature, and columns may buckle due to elasticity modulus reduction with temperature. References [1] Absi E., Méthodes des Calculs Numérique en Élasticité, Eyrolles, Paris, [2] Broderick B.M. & Thomson A.W., The response of flush end-plate joints under earthquake loading. Journal of Constructional Steel Research, 58, pp , [3] Drysdale D., An Introduction to Fire Dynamics, Wiley, [4] F.E.No.Mech. (Finite Elements in Nonlinear Mechanics), International Conferences, Institute für Statik und Dynamik, I. 1978, II. 1981, III. 1984, Stuttgart, Germany. [5] Liew Richard J.Y., Tang L.K., Holmaas Tore, Choo Y.S., Advanced analysis for the assessment of steel frames in fire, Journal of Constructional Steel Research, 47, pp , [6] Papadopoulos P.G., Karayannis C.G., Seismic analysis of RC frames by network models, Computers and Structures, 28, pp , [7] Papadopoulos P.G., Xenidis H.C., A truss model with structural instability for the confinement of concrete columns, European Earthquake Engineering, pp , 1999 II. [8] Wang Y.C., Lennon T., Moore D.B., The behaviour of steel frames subject to fire, Journal of Constructional Steel Research, 35, pp , [9] Wong M.B., Plastic frame analysis under fire conditions, Journal of Structural Engineering ASCE, 127, pp , March [10] Zhao J.-C., Application of the direct iteration method for non-linear analysis of steel frames in fire, Fire Safety Journal, 35, pp , 2000.

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