12th International Conference on
Shock & Impact Loads on Structures
15-16 June 2017, Singapore

RISK ESTIMATION FOR PROGRESSIVE COLLAPSE OF
BUILDINGS
Daigoro Isobe*, Kohei Oi† and Kota Azuma‡*University of Tsukuba
1-1-1 Tennodai Tsukuba-shi, Ibaraki 305-8573, Japane-mail: <isobe@kz.tsukuba.ac.jp> webpage: http://www.kz.tsukuba.ac.jp/~isobe/

Keywords: Progressive collapse, Key element index, Risk estimation, Collapse scale,
Structural strength, Steel framed building

Abstract. In this study, the collapse behaviours of steel-framed buildings were
simulated using the adaptively shifted integration (ASI) -Gauss code to investigate
the relation between a key element index, which indicates the contribution of astructural column to the vertical capacity of the structure, and the scale ofprogressive collapse. Collapses were initiated by removing specific columns frommodels designed based on different axial force ratios, and the collapse scales ofthe buildings were estimated using the total potential energy values of structuralmembers after collapse. The numerical results for various models with different
structural strengths confirmed that a larger integrated value for the key element
index resulted in a higher risk of progressive collapse. This index could be used to
predict and compare the progressive collapse risks for a building even whenvarious locations for the removed columns are assumed.
1

INTRODUCTION

In the case of an emergency such as an intense fire or the collision of an object with a building,there is a risk of progressive collapse. This was the phenomenon that occurred at the World Trade
Center (WTC) towers during the 9.11 terrorist attacks. The official statements released by theFederal Emergency Management Agency (FEMA) in 20021 and the National Institute of Standards
and Technology (NIST) in 20052 and 20083 concluded that the details of the failure process after
the decisive initial trigger that set the upper part in motion were very complicated and theirclarification would require large computer simulations. Among the various papers published after
the incident, the one published by Bazant et al.4 has become a milestone for numerous other works
concerning the progressive collapse of buildings. By defining an overload ratio, the ratio of theelastic strength of the lower structure to the total weight of the upper, falling structure, they
explained that the force applied to the lower structure by the upper overwhelmingly exceeded thedesign vertical load, and consequently, it would be practically impossible to design a high-rise tower
to avoid a progressive collapse such as that which occurred in the case of the twin towers. On theother hand, they also admitted that numerous computer simulations should be carried out to clarify
the collapse process, because their estimations were performed without considering the tilting of
the upper structure, member fracture, and other details.

†‡

Aisin Seiki Co., Ltd.
Graduate school, University of Tsukuba

1

Daigoro Isobe, Kohei Oi and Kota Azuma

Vertical load
Removed column(a) Initial model(b) Model with one column removed

Figure 1: Pushdown procedure for calculating key element index
In this study, the collapse behaviours of steel-framed buildings were simulated using theadaptively shifted integration (ASI) -Gauss code to investigate the relation between a key element
index that indicates the contribution of a structural column to the vertical capacity of the structure
and the scale of a progressive collapse. Collapses were initiated by removing specific columns
from models designed based on different axial force ratios. Various patterns of removed columns,of which the locations were restricted to a single floor, were investigated. The total potential energy
values of the structural members after the collapse were used to estimate the collapse scales ofthe buildings.
2

KEY ELEMENT INDEX

Because of differences in their span lengths and strengths, certain columns in a structure
support greater vertical loads than other columns and thus act as “key elements” in the structure.
Several indexes have been developed to evaluate the contribution of each column to the strengthof a structure, i.e., to determine the key elements, such as the redundancy index5 and sensitivity
index6. These indexes are effective at identifying highly sensitive columns that support the verticalload; however, they consistently identify those columns that would only cause a partial collapse at
the upper floors and thus they are not useful for estimating the scale of a progressive collapse.
Therefore, a new index called the key element index was proposed for estimating the contributionof base columns to the total collapse of a structure7.
The key element index is calculated as follows. First, a static pushdown analysis is performed,
as shown in Figure 1, by incrementally applying vertical loads at every structural joint in a modelled
structure. Even though several columns on the upper floors may yield in this process, the verticalloads continue to be applied in steps until one of the base columns yields. The total vertical load(including the floor loads) applied to a column at the step where one of the base columns yields isdefined as the ultimate yield strength PG of the structure. The ultimate yield strength of the initial,
undamaged structure is denoted as 0PG, and the ultimate yield strength of a structure with one basecolumn eliminated is denoted as 1PG. The key element index KI of column m, for example, can be
defined as the ratio of the ultimate yield strengths of the present and initial steps. The index canthus be written as follows:/(1)

The key element index at the nth step is then written as follows:/(2)

The index shown above indicates the contribution of the present column (or columns) to thestrength of the initial, undamaged structure and can be used for cases in which the columns are
removed simultaneously. However, the sensitivities of columns may change momentarily in cases
where the columns are removed sequentially within a certain interval (such as blast demolition
cases). The updated index, as shown below, is used for those cases with a sequential removal:

/

2

(3)

Daigoro Isobe, Kohei Oi and Kota Azuma

40 m
Z
Y
X

21 m

21 m

Figure 2: Numerical model
In this case, the index can be defined as the ratio of the ultimate yield strengths of the present andprevious steps. An example of the distribution of key element index values in a building model canbe found later in this paper.
3

NUMERICAL MODELS AND CONDITIONS

A ten-story, three-span steel framed building model, as shown in Figure 2, was constructed for the
progressive collapse analysis. Each span length of the model was 7 m, and the height between floors
was 4 m. The columns had box-type sections and were made of SM490 steel; the beams had H-type
sections and were made of SM400. The material properties are listed in Table 1. The columns were
subdivided with two linear Timoshenko beam elements per member, and the beams and floors were
subdivided with four elements. The floor elements were all modelled with elastic elements that did notyield. Several models with different strengths were prepared, as listed in Table 2, to determine theeffects of various structural strengths on the collapse scale of a building. A floor load of 800 kgf/m2 wasconsidered, and the sectional sizes of the columns and beams were based on the base shear
coefficients of buildings when designing the models. These models are referred to as models A, B, C,
D, and E, hereafter, from the strongest to the weakest.
The collapse modes of buildings were expected to change according to the number of removed
columns and their locations. Thus, the patterns of removed columns were selected as shown in Figure
Table 1: Material properties of steel members

SM400

206

245

0.3

7.9×10-6

SM490

206

325

0.3

7.9×10-6：Young’s modulus [GPa],：Yield stress [MPa]，：Poisson’s ratio, ：Density [kg/mm3]
Table 2: Structural strengths of each model
Maximum axial
ratioBase shear
coefficient

Model A

0.124

0.200

Model B

0.200

0.095

Model C

0.300

0.048

Model D

0.400

0.027

Model E

0.500

0.016

3

Daigoro Isobe, Kohei Oi and Kota Azuma(a) Pattern 1(b) Pattern 2(c) Pattern 3(d) Pattern 4

YX■: Removed column
Figure 3: Removed-column patterns
3, and the removal of columns was restricted to a single floor for simplicity. An analysis was carried out
for 10.0 s of simulation time with a time increment of 1.0 ms. The removal of columns was executed at
t = 1.0 s in the simulations.
A potential energy decrease ratio, as defined in the following equation, was used to evaluate thescale of the progressive collapse:
Potential energy decrease ratio =(4)where U0 and Uf are the potential energies of a numerical model at the initial and final states,
respectively. The potential energy U is defined as the sum of the potential energies of all the elementscomposing the model and is expressed as follows:∑(5)

where i, iM, m, g, and hi are the element number, number of elements besides fractured ones, mass ofthe element, gravitational acceleration, and height from the ground to the centre of element no. i,
respectively. The scale of the progressive collapse is overestimated. Thus, it can be underestimatedfrom a safety point of view by only applying Eq. (5) to non-fractured elements. The value calculated by
Eq. (4) indicates a larger scale of progressive collapse if the value is closer to 1.0.
4

PROGRESSIVE COLLAPSE BEHAVIOURS OF STEEL-FRAMED BUILDINGAs mentioned in the previous section, the patterns shown in Figure 3 were used to investigate
the collapse behaviours of the models. A total of 200 simulations were performed by varying thefloor of the removed columns from the 1st floor to the 10th floor.
The collapse sequences of the models are shown in Figure 4. The colours in the figures show
the yield function values, which are expressed as follows:(6)
where Mx, My, and N are the bending moments around the x axis and y axis and the axial force,
respectively. The terms with the subscript 0 are values that result in a fully plastic section in anelement if they act on the cross section independently. The red colour, for example, shows that theyield function of the element has reached to a value of 1.0 and has yielded. First, Figures 4(a) and4(b) show the results for models C and E when columns are removed with pattern 1 at the 4th floor,
respectively. Shock waves propagating throughout the building are observed after the collision
between the upper and lower structures occurs at t = 1.9 s. Then, model C, which has a relatively

4Daigoro Isobe, Kohei Oi and Kota Azuma

1.9 s

10.0s

(a) Model C, pattern 1 at 4th floor

1.9 s

3.0 s

4.0 s

10.0s

(b) Model E, pattern 1 at 4th floor

1.9 s

3.0 s

4.0 s

10.0s

(c) Model C, pattern 2 at 7th floor

1.9 s

3.0 s

4.0 s

10.0s

(d) Model E, pattern 2 at 7th floor
Figure 4: Collapse sequences under various conditions (continued)
5

Daigoro Isobe, Kohei Oi and Kota Azuma

1.9 s

3.0 s

4.0 s

10.0s

(e) Model C, pattern 2 at 4th floor

2.0 s

10.0 s(f) Model C, pattern 3 at 4th floor
Figure 4: Collapse sequences under various conditions
strong structural strength, withstands the collapse and stops it from progressing. On the other hand,model E, which has a relatively weak structural strength, cannot withstand the impact, and finallyadvances to a total collapse. Asymmetricity appears when columns are removed with pattern 2 atthe 7th floor for the same models (Figures 4(c) and 4(d)). Although the upper structure in model C
crashes after the impact, it does not advance to a progressive collapse. In contrast, the tilting of
the upper structure causes severe damage to the lower structure on impact in model E and triggers
a total collapse. Figures 4(e) and 4(f) show the cases where columns are removed with patterns 2
and 3 at the same 4th floor for model C. Here, it is confirmed that the collapse is stopped fromprogressing when the number of removed columns decreases from pattern 2 to pattern 3. However,
the risk of progression does not necessarily increase in accordance with the number of removed
columns; the collapse is stopped from progressing in Figure 4(a) when all 16 columns are removed.
The asymmetricity of the removed-column patterns might have relevance. Furthermore, it can beclearly confirmed by comparing Figures 4(c) and 4(e) that the risk of progression becomes higher
if the removed columns are located on lower floors.
The numerical results of 200 cases indicated that the collapse modes tended to vary dependingon the structural strength of the model, floor of the removed columns, and removed-column pattern.
Figure 5 shows the relation between the floor number where the columns are removed and thepotential energy decrease ratio for each pattern. The decrease ratio shows a clear dependency on
the location of the floor where the columns are removed in patterns 2, 3, and 4 (Figures 5(b), 5(c),and 5(d)), with a lower floor showing a higher ratio. This indicates a high risk of large-scale collapse
when the failure of columns occurs at lower floors. This is because the impact force of the upper
structure on the lower structure increases when the upper has a larger kinetic energy. It is also a
result of the tendency for a building to become more unstable if the columns at lower floors are
removed. The figure also shows that the decrease ratio becomes larger as the strength of the model
becomes weaker from model A to model E, and as the number of removed columns becomes larger.
However, there is a slight difference in this tendency in the case of pattern 1, as shown in Figure5(a). Although the number of removed columns is larger than in pattern 2, the removed-column
6Daigoro Isobe, Kohei Oi and Kota Azuma
1.0

Potential energy decrease ratio

Potential energy decrease ratio

1.0
0.8
0.6
0.4

A(n=0.124)
B(n=0.200)C(n=0.300)
D(n=0.400)E(n=0.500)

0.2
0.0
1

2

3

4

5

6

7

8

0.6
0.4

A(n=0.124)
B(n=0.200)C(n=0.300)
D(n=0.400)E(n=0.500)

0.2
0.0

9 10

1

2

3

4

5

6

7

8

9 10

Floor No.

Floor No.

(a) Pattern 1(b) Pattern 2
1.0

Potential energy decrease ratio

1.0

Potential energy decrease ratio

0.8

0.8
0.6
0.4

A(n=0.124)
B(n=0.200)C(n=0.300)
D(n=0.400)E(n=0.500)

0.2
0.0
1

2

3

4

5

6

7

8

0.8
0.6
0.4

A(n=0.124)
B(n=0.200)C(n=0.300)
D(n=0.400)E(n=0.500)

0.2
0.0
1

9 10

2

3

4

5

6

7

8

9 10

Floor No.

Floor No.

(c) Pattern 3(d) Pattern 4

Figure 5: Relation between floor number where columns are removed andpotential energy decrease ratio for each pattern(a) Pattern 2-1(b) Pattern 2-2(c) Pattern 2-3(d) Pattern 2-4(e) Pattern 2-5(f) Pattern 2-6(g) Pattern 2-7(h) Pattern 2-8(i) Pattern 2-9(j) Pattern 2-10

Y
X■: Removed column
Figure 6: Removed-column patterns for detailed investigation
pattern is fully symmetric and stable cases that do not lead to a large-scale collapse tend to appear
more often. The impact between the upper and lower structures occurs in a wider area with lesstilting in this case, and the impact force is distributed in a wider range compared to the otherpatterns, which should produce the stability.

7

Daigoro Isobe, Kohei Oi and Kota Azuma

5

RISK ESTIMATION FOR PROGRESSIVE COLLAPSE USING KEY ELEMENT INDEX

Next, by fixing the number of removed columns at 12, the removed-column patterns shown in
Figure 6 were selected to investigate the relation between the integrated values of KI and the riskof progressive collapse. Pattern 2-1 in the figure is the same as pattern 2 used in the previous
section, and it should be the most asymmetric pattern in Figure 6. Pattern 2-2, on the other hand,should be the most symmetric pattern of all. Other patterns were selected randomly. The
asymmetricity of each pattern was evaluated by calculating the moving distance of the centre ofgravity in the horizontal plane before and after the removal of columns, and the effect on thecollapse scale was also investigated.
Figure 7 shows the key element index values calculated for each column in model A, as an
example. The columns with larger KI values are illustrated with deeper grey in the figure. As awhole, the columns in the corner have smaller values and the ones at the lower floors have larger
values. Other models also had the same tendencies. The integrated values of KI are calculated bysumming up all the KI values for the removed columns.
Figure 8 shows the relation between the floor number where the columns are removed and thepotential energy decrease ratio for models A, C, and E. The tendencies of the plots are the sameas in the previous section, with a lower model strength and lower floor where the columns are
removed leading to a higher ratio or risk of large-scale collapse. If the results are surveyed for eachmodel, a variation in the ratio depending on the removed-column patterns can be observed. In
10F

5F

1.003

1.010

1.010

1.003

1.017

1.120

1.120

1.017

1.010

1.010

1.010

1.010

1.120

1.109

1.109

1.120

1.010

1.010

1.010

1.010

1.120

1.109

1.109

1.120

1.003

1.010

1.010

1.003

1.017

1.120

1.120

1.017

1.001

1.032

1.032

1.001

1.020

1.143

1.143

1.020

1.032

1.024

1.024

1.032

1.143

1.139

1.139

1.143

1.032

1.024

1.024

1.032

1.143

1.139

1.139

1.143

1.001

1.032

1.032

1.001

1.020

1.143

1.143

1.020

1.000

1.052

1.052

1.000

1.022

1.169

1.169

1.022

1.052

1.042

1.042

1.052

1.169

1.171

1.171

1.169

1.052

1.042

1.042

1.052

1.169

1.171

1.171

1.169

1.000

1.052

1.052

1.000

1.022

1.169

1.169

1.022

1.004

1.072

1.072

1.004

1.022

1.198

1.198

1.022

1.072

1.062

1.062

1.072

1.198

1.203

1.203

1.198

1.072

1.062

1.062

1.072

1.198

1.203

1.203

1.198

1.004

1.072

1.072

1.004

1.022

1.198

1.198

1.022

1.011

1.095

1.095

1.011

1.020

1.248

1.248

1.020

1.095

1.084

1.084

1.095

1.248

1.265

1.265

1.248

1.095

1.084

1.084

1.095

1.248

1.265

1.265

1.248

1.011

1.095

1.095

1.011

1.020

1.248

1.248

1.020

9F

4F

8F

3F

7F

2F

6F

1F

Figure 7: Key element index value of each column for model A

8

Daigoro Isobe, Kohei Oi and Kota Azuma

0.6
2-1
2-2
2-3
2-4
2-5
2-6
2-7
2-8
2-9
2-10

0.4
0.2
0.0
1

2

3

4

5

6

7

8

Potential energy decrease ratio

Potential energy decrease ratio

0.8

1.0

0.8
0.6
2-1
2-2
2-3
2-4
2-5
2-6
2-7
2-8
2-9
2-10

0.4
0.2
0.0

9 10

1

2

3

4

5

6

7

Floor No.

8

Potential energy decrease ratio

1.0

1.0

0.8
0.6
2-1
2-2
2-3
2-4
2-5
2-6
2-7
2-8
2-9
2-10

0.4
0.2
0.0

9 10

1

2

3

4

5

6

Floor No.

(a) Model A

7

8

9 10

Floor No.

(b) Model C(c) Model E

Figure 8: Relation between floor number where columns are removed andpotential energy decrease ratio for each model

0.6

1F
2F
3F
4F
5F
6F
7F
8F
9F
10F

0.4
0.2
0.0
12

13

14

15

16

0.8
0.6

1F
2F
3F
4F
5F
6F
7F
8F
9F
10F

0.4
0.2
0.0

17

12

13

14

15

16

Potential energy decrease ratio

0.8

Potential energy decrease ratio

Potential energy decrease ratio

1.0

1.0

1.0

0.8
0.6

1F
2F
3F
4F
5F
6F
7F
8F
9F
10F

0.4
0.2
0.0

17

12

13

14

15

Integrated value of KI

Integrated value of KI

Integrated value of KI(a) Model A(b) Model C(c) Model E

16

17

Figure 9: Relation between integrated values of KI and potential energy decrease
ratio for each model
1.0

0.6

1F
2F
3F
4F
5F
6F
7F
8F
9F
10F

0.4
0.2
0.0
0.000

0.005

0.010

0.015

0.020

0.025

1.0

0.8
0.6

1F
2F
3F
4F
5F
6F
7F
8F
9F
10F

0.4
0.2
0.0
0.000

0.002

0.004

0.006

0.008

Potential energy decrease ratio

0.8

Potential energy decrease ratio

Potential energy decrease ratio

1.0

0.8
0.6

1F
2F
3F
4F
5F
6F
7F
8F
9F
10F

0.4
0.2
0.0
0.000

0.001

0.002

0.003

0.004

0.005

Moving distance of the centre of gravity [m]Moving distance of the centre of gravity [m]Moving distance of the centre of gravity [m](a) Model A(b) Model C(c) Model E

Figure 10: Relation between moving distance of centre of gravity and potential energy
decrease ratio for each model
particular, a large-scale collapse occurred only in case of pattern 2-1, the most asymmetric patternof all, in model A, the strongest model of all. In contrast, the collapse scale became smaller in thecase of pattern 2-2, the most symmetric pattern of all, in models C and E, the weaker models.
Figure 9 shows the relation between the integrated values of KI and the potential energy
decrease ratio for models A, C, and E. It is evident that as the integrated value of KI becomeslarger, the ratio, and thus the collapse scale, becomes larger. Moreover, the critical integrated valueof KI where large-scale collapse begins becomes smaller as the strength of the model becomes
weaker from model A to model E. This means that there is some critical integrated value of KI tostart a large-scale collapse, which depends on the strength of the models. The conditions of pattern
2-1 on the 2nd floor of model A and pattern 2-2 in some cases of models C and E also show some
irregular tendencies, as observed in Figure 8.
To see the effect of the asymmetricity of the building before and after the removal of columns,
the relation between the moving distance of the centre of gravity in the horizontal plane and thepotential energy decrease ratio was investigated, as shown in Figure 10. The mass of the removed
columns is very small compared to that of the whole building, and the moving distance due to theremoval is trivial. However, it can be observed that the collapse scale becomes larger in accordance

9Daigoro Isobe, Kohei Oi and Kota Azuma

with the moving distance in the horizontal plane. It should also be noted that the moving distancesof the centre of gravity for patterns 2-1 and 2-2 are located at extreme positions in the figures,
which explains the results shown in Figures 8 and 9. The collapse scale of a building is greatly
affected by the asymmetricity or symmetricity of the structural failure.
6

CONCLUSIONS

By evaluating the numerical results using the key element index, it was found that a larger
integrated value for the key element index results in a higher risk of progressive collapse. However,some peculiar tendencies were observed in the cases of removed columns with extremely
symmetrical or asymmetrical locations. The critical integrated value of the key element index
needed to cause a large-scale progressive collapse tends to depend on the strength of the building,and the value cannot be generally determined. This is because the key element index is aparameter that is not related to the strength of the building itself, and the relative values for buildings
with different strengths cannot be compared. Therefore, the key element index may only be used
to predict and compare the risks of progressive collapse in the same building, but this may be doneeven when various locations for the removed columns are assumed.
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[2]
[3]
[4]
[5]
[6]
[7]

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