INTRODUCTION
Reports after major ground motions indicate that the so called softstorey
failure is one of the most typical types of damage concerning buildings. During
the Loma Prieta earthquake of 1989, for example, many woodframe buildings experienced
extensive damage concentrated at the level of one of the storeys (Maison
et al., 2011). The softstorey failure was also observed during the
HyougokenNanbu (Kobe) earthquake of 1995 causing damage to a large number of
steel and reinforced concrete buildings (Elkholy and Meguro,
2004). It has been observed during earthquakes that the failure of an upper
damaged storey induces large vertical impact loading acting on the lower part
of the building. In the case, when the loadbearing capacity of the structural
elements of lower storeys is not sufficiently large, the progressive collapse
of the whole structure can be initiated which is considered to be the worstcase
scenario (Talaat and Mosalam, 2009).
The research concerning impacts in buildings during earthquakes has been carried
out for more than two decades now. However, previous investigations were mainly
focused on structural collisions between buildings in the horizontal direction.
This phenomenon is known in the literature as the earthquakeinduced structural
pounding. The fundamental study on such collisions between insufficiently separated
buildings with different dynamic properties (modelled as singledegreeoffreedom
systems) was conducted by Anagnostopoulos (1988). Singledegreeoffreedom
structural systems were also used by other researchers (Chau
and Wei, 2001; Mahmoud et al., 2008; Jankowski,
2010). More advanced investigations were carried out on discrete multidegreeoffreedom
models with the mass of each storey lumped on the floor level (Maison
and Kasai, 1992; Anagnostopoulos and Spiliopoulos, 1992;
Karayannis and Favvata, 2005; Mahmoud
and Jankowski, 2009). Quite simple analysis concerning impacts between two
adjacent buildings, with the use of finite element method, was also conducted
by Papadrakakis et al. (1996). More detailed,
threedimensional, nonlinear finite element models of colliding buildings with
different dynamic characteristics were employed in the studies carried out by
Jankowski (2009, 2012).
On the contrary to the earthquakeinduced horizontal collisions in buildings,
the results concerning investigations on vertical impacts between the damaged
upper part of the building falling onto the lower storeys after the softstorey
collapse during ground motions are very limited. Migda and
Jankowski (2012) conducted the experimental investigation on the behaviour
of horizontally deformed steel columns (deformation as the result of earthquake
loading) that are additionally subjected to vertical impact load (Fig.
1). In that study, however, the introduced predeformation was a static
one and the influence of the dynamic effects of the ground motion was not considered
(limitations of the experimental setup). Therefore, the aim of the present paper
is to extend the study and investigate numerically the dynamic response of a
model of steel column under ground motion excitation that is additionally subjected
to vertical impact load. The numerical model of the specimen has been created
and its accuracy has been confirmed by comparing the results of the numerical
analysis with the experimental results. In the first stage of the numerical
study, a number of cases, concerning different predeformations of the column
as well as various impact load time histories (related to different dropheights)
have been considered.

Fig. 1: 
Schematic diagram of vertical impact load (F) acting on horizontally
deformed column with initial predeformation (D) 
Then, the detailed nonlinear analysis has been conducted by exciting horizontally
the base of the specimen using the harmonic ground motion excitation as well
as by applying impact vertical load at different times of the excitation. The
geometric nonlinearity (large strain analysis) as well as the elastoplastic
material behaviour with the strain rate effect have been considered in the numerical
analysis.
NUMERICAL MODEL AND ITS VALIDATION
For the purposes of numerical analysis, a finite element model of steel columns
which were previously tested experimentally (Migda and Jankowski,
2012), has been created using the commercial software MSC Marc (Fig.
2). The numerical model of the specimen, with the length of 800 mm and cross
section of 3x20 mm, has been constructed out of 640 fournode shell elements.

Fig. 2: 
Numerical model of the steel column with moving platform 
The moving platform which was mounted at the top of the column and used during
the experiments to drop a steel sphere onto it, has been considered to be important
element to be also included in the numerical analysis. Its model has been constructed
out of 240 fournode shell elements. All material properties as well as original
dimensions have been assured to be the same as in the experimental study. Taking
into account the boundary conditions valid during the experiment, the numerical
model of the specimen has been fixed at the bottom of the column and only vertical
movement at four corners of the platform (locations of linear bearings) has
been allowed.
The accuracy of the numerical model has been first verified by comparing the
results of the numerical analysis with the results obtained form the experimental
tests (Migda and Jankowski, 2012). For this purpose,
similarly as in the case of the experimental study, a static horizontal predeformation
has been introduced at the bottom of the numerical model of the column and the
specimen has been subjected to vertical impact load acting on its top (Fig.
3). The same impact load time histories as measured during the experiment,
by dropping a steel sphere onto the platform, have been used. The geometric
nonlinearity, by conducting the large strain numerical analysis, has been taken
into account. A number of cases, concerning different predeformations (relative
horizontal displacement between the top and the bottom of the column) as well
as various impact load time histories (related to different dropheights) have
been considered in the study. The difference between the results of the experiment
and the results of the numerical analysis has been assessed by calculating the
normalized Root Mean Square (RMS) error (Bendat and Piersol,
1971; Jankowski, 2003; Jankowski
and Walukiewicz, 1997):
where,
are the values from the time history record obtained from the experiment and
from the numerical analysis, respectively and NV denotes a number of values
in these history records.
The example of the comparison between the results of the numerical analysis
and the results obtained form the experimental tests, in the form of the horizontal
displacement time histories at the middle of the column (predeformation of
20 mm, drop height of 200 mm), is presented in Fig. 4. For
this case, the natural frequency of vibrations of a specimen as well as the
damping ratio is equal to 24.91 Hz and 0.68%, respectively.

Fig. 3: 
Numerical model of the predeformed steel column subjected
to vertical impact load 
Using Eq. 1, the RMS error for the time histories shown in
Fig. 4 has been calculated as equal to 7.90%. The average
RMS error for all cases considered has been found to be as small as 6.43% confirming
the accuracy of the numerical model created.
NUMERICAL ANALYSIS
Parametric analysis for statically predeformed columns: In the first
stage of the numerical analysis, the parametric investigation has been conducted
in order to assess the influence of the predeformation of the column as well
as the influence of the impact load on the response of the steel column. The
analysis has been conducted for the statically predeformed columns (Fig.
3) subjected to different impact load time histories. The horizontal predeformation
was increased from 0 mm by 10 mm up to 60 mm.

Fig. 4(ab): 
Horizontal displacement time histories at the middle of the
column (predeformation of 20 mm, drop height of 200 mm) (a) Experiment
and (b) Numerical analysis 
The impact load time histories, as measured during the experiment for different
drop heights (Migda and Jankowski, 2012), were applied
onto the platform. The above conditions allowed the numerical models of the
steel column to remain in the elastic range as well as to prevent from dynamic
stability loss.
The examples of the preliminary results in the form of the maps of mean normal
stress distributions at the bottom part of the predeformed specimen, for the
case of 60 mm predeformation, are presented in Fig. 5 and
6. In particular, Fig. 5 shows the mean
normal stresses before impact while Fig. 6 presents the mean
normal stresses at the stage of the peak impact force acting at the top of the
predeformed column. By comparing Fig. 5 with Fig.
6, it has been observed that the increase in the stress values at the bottom
part of the specimen due to impact load can be as large as 235.9%.
The results from the parametric analysis are presented in Fig.
7 and 8. Figure 7 shows the peak values
of the mean normal stress at the bottom of the specimen with respect to predeformation
(for different drop heights). It can be observed from the figure that the value
of the peak mean normal stress uniformly and substantially increases with the
increase in the predeformation. In the case of the drop height of 350 mm, for
example, the increase in the mean normal stress value for the deformed column
(predeformation of 60 mm) with relation to the undeformed specimen is as large
as 1321.4%.

Fig. 5: 
Map of mean normal stresses at the bottom part of the deformed
specimen (predeformation of 60 mm) before impact 
The peak values of the horizontal displacement of the column at its midheight
with respect to predeformation (for different drop heights) are presented in
Fig. 8. It can be seen from the figure that larger predeformation
results in larger amplitudes in horizontal vibrations of the column. An evident
trend of larger peak horizontal displacement for greater drop heights is also
visible. In the case of the drop height of 350 mm, for example, the increase
in the peak horizontal displacement of the column at its midheight for the
predeformed column (predeformation of 60 mm) with relation to the undeformed
specimen is as large as 469,0%.

Fig. 6: 
Map of mean normal stresses at the bottom part of the deformed
specimen (predeformation of 60 mm) at the time of the peak impact force
acting at the top of the column 
Nonlinear analysis under dynamic excitation: In the second stage of
the numerical analysis, the advanced nonlinear analysis has been conducted by
exciting horizontally the base the column (Fig. 2) using the
harmonic ground motion excitation. The sine wave having the excitation frequency
tuned with the natural frequency of the specimen has been used in the analysis.
Together with the geometric nonlinearity (large strain analysis), the nonlinear,
elastoplastic material behaviour has also been considered. Additionally, the
strain rate effect has been taken into account in the numerical analysis by
relating the yield strength of steel with the strain rate following the relation
obtained from the experimental study conducted by Ansell
(2006).
It has been assumed in the analysis that vertical impact takes place at different
times of the excitation. Figure 9 shows the horizontal displacement
time history of the column at its midheight (response at time rage 0.820.92
s) with marks indicating moments of impact for five different cases considered
in the study. The peak values of impact force has been applied at the following
times:
Case 1: t_{1} = 0.859 sec
Case 2: t_{2} = 0.862 sec
Case 3: t_{3} = 0.869 sec
Case 4: t_{4} = 0.877 sec
Case 5: t_{5} = 0.880 sec
Related to different stages of horizontal deformation of the specimen. Similarly
as in the parametric investigation, the impact load time histories measured
during the experiment (Migda and Jankowski, 2012) have
been used in the study. The examples of the results, in the form of the horizontal
displacement time histories of the column at its midheight for five different
cases of impact, as compared to the time history when impact is not applied,
are shown in Fig. 10.

Fig. 7: 
Peak mean normal stress vs. predeformation of the column
for different drop heights (H) 

Fig. 8: 
Peak horizontal displacement of the column at its midheight
vs. its predeformation for different drop heights (H) 

Fig. 9: 
Horizontal displacement time history of the column at its
midheight under harmonic ground motion (time rage 0.820.92 sec) with marks
indicating different moments of peak values of impact force 

Fig. 10: 
Horizontal displacement time histories of the column at its
midheight under harmonic ground motion for five different cases of impact
as compared to the time history when impact is not applied 
Additionally, the comparison between the displacement time histories for impact
with peak value of impact force at t_{3} = 0.869 sec (case 3 related
to the peak horizontal deformation of the column) with and without considering
the strain rate effect is also presented in Fig. 11.

Fig. 11: 
Comparison between the displacement time histories under harmonic
ground motion for impact case 3 with and without considering the strain
rate effect 
It can be seen from Fig. 10 that vertical impact has a significant
influence on the behaviour of the steel column under dynamic ground motion excitation
leading to the increase in its response. It can be seen that incorporation of
the nonlinearity of material behaviour has resulted in entering into the plastic
range in all the cases considered (see permanent displacement of the column
at Fig. 10). Moreover, the obtained results clearly indicate
that the time of impact plays a substantial role in the overall behaviour. It
can be seen from Fig. 10 that, for the case when impact takes
place at the time of peak horizontal deformation of the column (impact case
3), the increase in the peak structural response is as large as 1808,4% with
relation to the peak response without impact. On the other hand, when impact
takes place when the actual displacement of the horizontally excited column
is relatively small (impact cases 1 and 5), the influence of vertical impact
on the response is less significant. Figure 10 shows that,
in such situations, the increase in the peak horizontal displacement of the
column at its midheight is equal to 438.4%, as compared to the peak response
without impact.
It can also be seen from Fig. 11 that incorporation of the
strain rate effect in the numerical analysis is really important and the difference
between the responses with and without considering this effect can be substantial.
In the case when the strain rate effect is taken into account and vertical impact
takes place at the time of the peak horizontal deformation of the column (impact
case 3), the increase in the peak structural response is equal to 20.32%, as
compared to case when the effect in not considered.
DISCUSSION
The detailed numerical investigation concerning the behaviour of deformed steel
columns, that are additionally subjected to vertical impact load (the result
of softstorey failure during earthquake), has been presented in this paper.
In the first stage of the study, the analysis has been conducted for different
values of the static horizontal predeformation of the column. Then, the base
of the column has been dynamically excited by harmonic ground motion and vertical
impact load has been applied at different times of the excitation. The geometric
nonlinearity (large strain analysis) as well as the elastoplastic material
behaviour with the strain rate effect have been taken into account.
The results of the first stage of the numerical study show that with the increase
in the static predeformation of the column the peak mean normal stress values
induced at the bottom of the specimen as well as the peak horizontal displacement
at the middle of the column show a substantial increase trend for all height
drop values considered. The above conclusions indicate that the initial predeformation
of columns has a substantial negative influence, what is fully consistent with
the results of the experimental study concerning the behaviour of horizontally
deformed steel columns that are additionally subjected to vertical impact load
(Migda and Jankowski, 2012). It has been observed, however,
that even the deformed column is still capable to carry considerable vertical
impact load before its failure due to stability loss. This fact was also confirmed
during the experiments (Migda and Jankowski, 2012).
The results of the second stage of the numerical study show that vertical impact
may substantially influence the response of the column which is dynamically
excited in its horizontal direction. The results indicate that the time of impact
plays a substantial role in the overall behaviour under ground motion excitation.
It has been shown that the response may be increased significantly if impact
is initiated when the specimen is in the range of its peak horizontal deformation.
On the other hand, when impact takes place when the actual displacement of the
horizontally excited column is relatively small, the influence of vertical impact
on the response is less significant. Moreover, the results indicate that the
incorporation of the strain rate effect in the numerical analysis is really
important in order to increase its accuracy.
The study described in this paper has provided us valuable results concerning
the behaviour of deformed steel columns subjected to vertical impact load. However,
the response of buildings during earthquakes after a softstorey failure still
needs to be further investigated. This concerns in particular a need for the
detailed numerical simulations concerning the dynamic behaviour of the whole
building (not only chosen structural members) under different real earthquake
excitations.
ACKNOWLEDGMENT
Numerical calculations were carried out at the Academic Computer Centre (TASK)
in Gdansk.