2-D analysis of a building frame under gravity load and fire 

Dat Duthinh1

Summary 

A two-dimensional finite-element model is developed that provides some 

insight into the behavior and collapse of high-rise steel buildings with open web 

floor systems. For one prescribed temperature distribution that corresponds to a 

two-story, quarter-span fire, the diagonals of the heated trusses buckle 

inelastically, causing considerable sag in the fire floors. This behavior puts a high 

tension demand on the truss connections to the perimeter column, which remains 

at moderate temperatures in this model and does not experience buckling. Our 

analysis is based on temperatures and material properties that were selected for 

illustrative purposes. Therefore no claim is made as to its applicability to any 

specific structure. 

Introduction 

The objective of this paper is to present an approach to the analysis of a tall 

steel building frame similar to the analysis by Usmani et al. [1]. For the purpose 

of comparison with other published work, and to use one specific example to 

illustrate the analysis method, the data used in this paper are based on 

information from Reference 2. The choice of a 2-D model is justified by the need 

to perform simple but informative calculations covering a wide variety of 

conditions. We modeled the structural system independently of connection 

details. Connections are outside the scope of the analysis presented in this paper. 

Structural model 

The vertical plane considered in the model includes a column and five 

longitudinal floor trusses and slabs at about 4/5 of the height of the building 

being considered [2]. The column extends 22 m (72 ft) to a height of six floors, 

and both its upper and lower ends are pinned, with the upper end free to translate 

vertically. The upper chords of the floor trusses are simply supported at the 

(right) internal end, and connected to the perimeter column by hinges. In the 

actual structure, a double floor truss carries a tributary floor slab 2 m (80 in) wide 

and is in turn supported by two perimeter columns, whereas in our model, a 

single truss supported by a single column carries a 1 m (40 in) wide slab. The 

floor spans 18.3 m (60 ft) from the perimeter column to the core. 

 

1

 National Institute of Standards and Technology, Gaithersburg, MD. 20899, USA 
The trusses, slabs, and column that supports them are simulated by threenode beam finite elements, capable of modeling a wide variety of cross sections,

with a mesh density and number of integration points specified by the user. One

particularly attractive feature of this element is its capability of supporting linear 

temperature gradients across its section and along its length. 

Material properties 

The various steels range in nominal yield strength from 248 MPa (36 ksi) in 

the floor trusses to 448 MPa (65 ksi) in the column [2]. They are all modeled by 

bilinear stress-strain curves, with a tangent modulus about 0.5 % of the elastic 

modulus. Steel properties above 1100°K are relatively scarce, but indicate a 

stabilization of yield and ultimate strengths between 1100°K and 1300°K. 

The lightweight concrete slab is also modeled as a bilinear material, with

compressive strength of 21 MPa or 3000 psi [2]. For simplicity, the top chord of 

the floor truss is assumed to act in a perfect composite way with the slab and 

allow the tensile strength at the bottom of the slab to be equal in magnitude to the

compressive strength at the top. The simplification inherent in transforming the

steel top chord into an equivalent concrete section disregards the differential 

thermal expansion between steel and concrete. 

Loading 

The floor slabs are acted upon by dead and live loads consistent with [2]. The 

column end loads are determined by a separate analysis of the damaged structure 

estimated from [2]. In addition, the column dead load is applied along its length.

The behavior of the structure is greatly influenced by thermal loads. This 

paper uses a strictly conventional fire. For comparison with [1], a single 

temperature distribution T represented by an exponential function of time t, with 

a reference temperature T0 = 300°K, is used. The time rate of change of the

temperature, represented by coefficient a = 0.005, depends, among other factors, 

on the location and intensity of the fire, and the quality of the insulation. 

( ) ( ) (1 ) 0 max 0

at T t T T T e − = + − − (1) 

A two-floor fire, with maximum temperature Tmax = 1273°K, heats the

structure over the quarter-span closest to the perimeter column. Over that span,

the slab of the second floor is uniformly heated, whereas the slabs of the floors

below and above it have linear temperature gradients across their thickness, with 
the bottom of the lower slab and the top of the upper slab remaining at 300°K at 

all times. In the three-quarters of the span not directly under fire, the temperature 

decreases linearly from the maximum at quarter-span to normal room 

temperature at the core. Between the first and the third floors, the column 

temperature is also described by Eq. (1), with Tmax = 400°K, whereas the rest of 

the column remains at 300°K at all times. 

Results 

Nonlinear, static, large deformation analysis accounting for P-delta effects 

and buckling was performed. The analysis proceeded in a number of load steps, 

the first corresponding to gravity loads at the start of the fire (normal room 

temperature). Subsequent steps occurred at 200 s intervals, with the maximum 

temperature attained, to within 1°K, at 1400 s. Results are shown in Figs. 1-3. 

At room temperature, even under the severe load redistribution due to initial 

damage to the building, non-linear effects are negligible, i.e., the structure still 

behaves linearly. The maximum floor sag is 35 mm (1.4 in), causing the 

horizontal span to decrease and the column to pull in slightly. Approaching 200 s 

and a temperature of 915°K, the heated trusses begin to show distress, especially 

in the compressed diagonal and vertical members, which buckle inelastically. 

(The temperatures referred to in these results are the hottest temperatures in the 

structure at any given time.) At 200 s the maximum floor sag increases to 

335 mm (13.2 in), and the column is pushed out (peak of 38 mm or 1.5 in) by the 

thermal expansion of the second floor. At that time the connection of the second 

floor to the perimeter column experiences its maximum compression of 125 kN 

(28 kips). Because the third floor slab has a thermal gradient with its top surface 

at room temperature, its lateral expansion is much smaller than for the second 

floor slab, and its sag is larger. The connection between the third floor slab and 

the column is always in tension (Fig. 2). As expected, the bottom floor slab, 

heated at the top and cool at the bottom, bows upward. As the temperature 

continues to rise, more of the second floor truss fails, and the increasing sag 

begins to pull the column in. The horizontal deflection of the column becomes 

positive (inward), and the connection force between the column and the second 

floor turns to tension. This inward movement of the column relieves the tension 

in the connection between the column and the third floor. Further temperature 

rise causes further weakening in the third floor truss, which eventually becomes 

active in pulling the column in. At the peak temperature of 1273°K, the 

maximum lateral deflection in the column (183 mm or 3.3 in) occurs at the third 

floor, inward, and the connection between the column and the third floor 

experiences a tension of 185 kN (41.6 kips). The connection forces supplied by 

the interior column to the floors are similar to that of the external column. 
NIST analysis of connections is ongoing and will indicate whether this or 

other connections can supply the calculated demand, and if not, at what 

temperatures connection failures will occur. In this regard, the composite 

behavior of the concrete slab with the steel truss may become questionable at a 

temperature and strain level yet to be determined. 

For comparison with Quintiere et al. [3], our results show that the truss 

diagonals buckle inelastically, and there is considerable reserve strength after the 

first diagonal buckles. At the highest temperatures analyzed, seven diagonals 

and vertical had buckled in each of the heated floors. This conclusion assumes 

that the various structural connections maintain their integrity throughout the fire. 

Conclusions 

A model is developed that provides some insight into the behavior and collapse 

of high-rise steel buildings with open web floor systems. For the particular 

temperature distribution we selected from among those assumed by Usmani et al. 

[1], the diagonals of the heated trusses buckle inelastically, causing considerable 

sag in the fire floors. This behavior puts a high tension demand on the truss 

connections to the perimeter column, which remains at moderate temperatures in 

this model and does not experience buckling. This is the major difference 

between our results and Usmani’s, even though the heated trusses in our model 

are exposed to a much higher temperature and the column to a more severe load 

that reflects load redistribution in the damaged structure. One possible 

explanation for the difference is that failure modes may be very sensitive to 

material properties. Similar high tension demands are also imposed on the floor 

connections to the internal column 

References 

1 Usmani, A.S., Chung, Y.C. and Torero, J.L. (2003): “How did the WTC 

towers collapse: a new theory,” Fire Safety Journal, vol. 38, no. 6, Oct. pp. 501-

533. 

2 FEMA (2002): “World Trade Center Building Performance Study”, Federal 

Emergency Management Agency Report 403, Sep. 

3 Quintiere, J.G., di Marzo, M. and Becker, R. (2002): “A suggested cause of 

the fire-induced collapse of the World Trade Towers,” Fire Safety Journal, vol. 

37, no. 7, Oct. pp. 707-716. 
Acknowledgment 

Discussion with Michael Riley produced useful insight on FEA. The guidance 

provided by project leader Emil Simiu is also gratefully acknowledged. 

Fig. 1 Deformed shape at 1273°K 
Fig. 2 Horizontal force (kN) between column and 

second and third floors vs. temperature (°K) 

Fig. 3 Vertical deflections (mm) of second 

and third floors vs. temperature (°K)