Calculation of structural responses using finite element analysis packages

The simple approach described in section 6.3.1 can be applied only to very simple structural members with uniaxial stress state. For frame structures or structural members in which stresses are not uniaxial, struc-tural analysis packages based on finite element approaches should be used. There are many structural analysis software packages commercially

t_k := ∆t, k := 1

Structural increment analysis Calculate increments of stresses, strains and displacements for temperature and time increments

Thermal analysis Calculate temperature field q(t,y,z)

Start

Pre-fire structural analysis

Calculate stresses, strains and displacements due to externally applied loads for ambient temperature

t_k< tend

Update stresses, strains and displacements Correct for non-linearities if necessary

End

k := k + 1 t_k := tk + ∆t no

yes

Figure 6.2 Overall calculation procedure for the structural behaviour of fire affected members.

available nowadays. However, since the fire involves high temperatures, the software packages to be used must be able to take into account the special characteristics of materials at high temperatures and the non-linearity both in geometry and material. Several computer packages were specially designed for modelling high-temperature phenomena, includ-ing FIRES-RC II (Idinclud-ing, Bresler and Nizamuddin, 1977b), FASBUS II (Iding and Bresler, 1987, 1990), SAFIR (Nwosu et al., 1999) and VULCAN (Huang, Burgess and Plank, 2003a, b). The VULCAN is capable of mod-elling the global three-dimensional behaviour of composite steel-framed buildings under fire conditions. The analysis considers the whole frame action and includes geometrical and material non-linearities within its beam–column and slab elements. It also includes the ability to repre-sent semi-rigid connections that degrade with temperature and partial

interaction between the steel section and slab. In addition to these specific software packages, other non-linear finite element structural analysis programs such as ABAQUS, ANSYS and DIANA can also be utilized for conducting the fire analysis of structures (Sanad et al., 1999).

Most finite element programs require data to be entered on the stress–

strain temperature behaviour of steel whether reinforcing, pre-stressing or structural and/or concrete as appropriate. Since the thermal strain is treated separately in most structural analysis programs, the strain used to define the stress–strain temperature behaviour required as an input in the program thus is the sum of all other strains. For steel this includes the clas-sical creep strain and the strain induced by the mechanical stress. In the case where the classical creep strain is negligible, this reduces to just the strain induced by the mechanical stress and therefore the stress–strain temperature behaviour can be simply represented by the temperature-dependent stress–strain equations as described in Chapter 5. For concrete, however, the transient strain is not negligible and the strain used to define the stress–strain temperature behaviour thus must include both the tran-sient strain and the strain induced by the mechanical stress. Therefore, the temperature-dependent stress–strain equations must be modified to include the transient strain before they can be as the input to the program (Li and Purkiss, 2005).

Although there are many computer simulation results published in literature, there have been limited comparative, or benchmark, tests com-missioned for both thermal response and, more importantly, structural response. The latter is very much affected by materials models and the exact formulation of the analysis techniques used. Sullivan, Terro and Morris (1993/4) gave some results following a survey of available com-puter software packages both for thermal and structural analyses. For the thermal analysis programs, particularly for steel structures, it seems that most packages gave comparable answers and that these answers were also in reasonable agreement with experimental data. However, for concrete structures, particularly at temperatures of around 100–200^◦C, agreement was less acceptable for the reason that the packages examined did not fully consider moisture transport. It was also noted that the fit between experimental and predicted temperatures was often improved by adjust-ing the values of the parameters definadjust-ing the thermal diffusivity and/or the thermal boundary conditions such as emissivity.

For the structural analysis packages investigated, a greater spread of acceptability was found. This spread was, in part, due to the fact that some of the packages investigated were developed for research (and thus did not have adequate documentation or were user unfriendly), in part due to the assumptions made within the analysis algorithms (such as no allowance for large displacements) and in part due to inadequate materials models (especially for concrete where either transient strain

or the effect of stress history was ignored). It appeared that many of the programs predicted correct trends but that the absolute results did not agree with experimental data. It was also noted that the effect of classical creep on the behaviour of steel was neglected.

6.4 EXAMPLES

The first example presented here is a circular steel tube filled with con-crete subjected to pure compression and its outside surface is exposed to a fire. The problem was solved using the simple approach described in section 6.3.1 by Yin, Zha and Li (2006). Because of the axial symmetry of the problem, the temperature and axial compressive stress are axial sym-metric. Figure 6.3 shows the temperature distributions along the radial direction at various different times when the composite column is exposed to the fire, the temperature of which is defined by the standard fire curve.

As is seen in Fig. 6.3, the variation of the temperature is much smaller in the steel tube than in the concrete core. This is because steel has a much greater thermal diffusivity than concrete. The high temperature in the steel tube together with the non-uniform temperature distribu-tion in the concrete core leads to a complicated distribudistribu-tion of the axial

0 0,05 0,1 0,15 0,2 0,25

Figure 6.3 Temperature distribution profiles of the circular steel tube filled with concrete (column diameter D= 500 mm, steel tube thickness hc= 20 mm, conductivity and specific heat are temperature dependent for both steel and concrete, the standard fire curve is used for the fire temperature, Yin, Zha and Li, 2006).

compressive stress, as demonstrated in Fig. 6.4. Note that the reduction in stress in the steel tube when fire-exposure time increases is due to the strength reduction caused by high temperature. Figure 6.5 shows the load–displacement curves of the composite column at various different fire-exposure times. The fire resistance of the column can be obtained by plotting the maximum loads of the load–displacement curves against the fire-exposure times.

The second example is a two-bay I-section steel frame with columns fixed at the base and beams uniformly loaded on the top. The analysis was

0 0,05 0,1 0,15 0,2

Figure 6.4 Stress distribution profiles along the radial direction at various times for different compressive strains (σuo is the concrete peak compres-sive stress at ambient temperature and σyois the steel yield stress at ambient temperature, Yin, Zha and Li, 2006).

0 0,5 1 1,5

Figure 6.5 Load–displacement curves with temperature effects (Yin, Zha and Li, 2006).

performed using the non-linear finite element analysis package ABAQUS by Ali, Senseny and Alpert (2004). Two main steps were followed in the analysis procedure. In the first step, the frame was analyzed under the applied load at room temperature to establish the pre-fire stress and deformation in the frame. In the second step, the history of fire tem-perature was calculated and was imposed on the deformed and loaded structure causing the steel to expand and the mechanical properties to degrade. Both geometric and material non-linearities were included in the simulations to account for the expected large displacements, plastic deformations and creep. Five different fire scenarios were investigated.

The highly non-linear problem was solved using iterative procedures with automatic time stepping.

Figure 6.6 shows the simulation results. It is seen that the frame expands slowly toward the wall until it reaches its maximum lateral displacement followed by rapid change in displacement direction and col-lapse away from the fire wall. The time to colcol-lapse is about 45 min. When the fire covers at least one bay of the frame, the time to collapse is largely unaffected by the fire scenario because the plastic hinges and the exces-sive deformations in the span close the wall very much control the failure.

The required space between the wall and steel depends on the extent of fire. The minimum clearance required between the frame and the wall increases with the length of the fuel burning. This behaviour is consistent with the simple case of uniformly heated steel members that are restrained

Lateral displacement

Dead load + self-weight = 12,5 kN/m

W24 × 104 W24 × 104

W27 × 146

L = 15 m L = 15 m

12 m

L_n (Extent of fire)

W27 × 146

W24 × 104

Uniform 800

600

400

200

0

0 15 30 45

Time (min) Lateral displacement (mm)Fire wall

60 75 90

−200

−400

L_h/L = 1⁺ Middle column is exposed to fire

L_h/L = 1⁻ Middle column is not exposed to fire

L_h/L = 2 L_h/L = 1,5

Figure 6.6 Lateral displacement histories for a two-bay steel frame. Lh/L= 1⁻ – fire is localized to the bay closest to the firewall excluding the middle column.

L_h/L= 1⁺– the bay closest to the wall is exposed to fire including the middle column. L_h/L= 1.5 – fire is extended beyond the first bay to cover half of the second bay. L_h/L= 2 – the two bays are heated except for the far column.

Uniform – the fire heats all columns and girders of the frame. L_k(0≤ L_h≤ 2L) is the length of the fuel burning measured from the firewall to anywhere within the two bay. (Copy with permission from Ali, Senseny and Apert, 2004).

only at one end in which longitudinal thermal expansion is proportional to the heated length of the member. A significant difference in lateral expansion is noticed between the two-bay fire scenario (Lh/L= 2) and the uniform fire case. The uniform fire scenario is mathematically equivalent to a fixed girder at the middle column, and results in lateral displacements similar to the one-bay fire case (Lh/L= 1).

Figure 6.7 is an example that shows how the moisture influences the heat transfer and thus the temperature distribution in the concrete struc-tural member. The problem shown here is a simple concrete wall of 400 mm thick. The wall has an initial porosity of 0.08 and the correspond-ing initial moisture content is 2% of concrete weight. The wall is subject to double-side fire exposure. The temperature results for the case where the moisture transfer is considered are taken from Tenchev, Li and Purkiss (2001b). The experimental data are taken from Ahmed and Hurst (1997).

The two temperature curves for the case where the moisture transfer is not considered are corresponding to different specific heat expressions, both of which are given in EN 1992-1-2 (one is recommended to use together with considering moisture transfer and the other is recommended to use without considering the moisture transfer). The comparisons of tempera-ture distribution profiles between different models demonstrate that the

0,1 0,15 0,2

x (m) (distance from centre)

T (°C)

x (m) (distance from centre)

(b) t = 60 min

Decoupled model-A Decoupled model-B Experiments Coupled model

Figure 6.7 Temperature distribution profiles of the concrete wall (400 mm) with moisture content 2% of the concrete weight at fire exposure times of 30 and 60 min, obtained from different models. Decoupled model-A uses the specific heat of dry concrete. Decoupled model-B uses the specific heat recom-mended in EN 1992-1-2 to take account the effects of moisture. Experimental data are taken from Ahmed and Hurst (1997). Coupled model considers the transfer of both heat and moisture and the results are taken from Tenchev, Li and Purkiss (2001b).