In everyday
practice frames of preengineered metal buildings are often designed as 2D
structures. Industrial buildings often have partial mezzanine floors, attached
to one of the main columns, to suit the technology. Additionally, such
buildings often have above the roof platforms for machineries.
When it comes to
seismic design, as long as seismicity is not deemed to be a strongly
controlling factor for final design, the mezzanines are just attached to the
same type of frames as used at other locations and are locally strengthened, if
necessary. Only the horizontal component of the seismic effect is considered in
most of the cases.
The following
picture shows a typical intermediate frame of a longer industrial hall, with
builtin partial mezzanine floor and with a platform placed above the roof.
Picture0: Studied
Intermediate frame
Equivalent
Lateral Force method
The most straightforward design approach is the Equivalent Lateral Force (ELF) method (EN 19981 4.3.2.2). There are certain conditions for the application of this method.
 (1)P. this method may be applied to buildings whose response is not significantly affected by contributions from modes of vibration higher than the fundamental mode in each principal direction
 (2) the requirement in (1)P is
deemed to be satisfied in buildings which fulfill both of the following
conditions
o
they have a fundamental period
of vibration smaller than the followings
§ 4*T_{c} or 2.0 sec
o
they meet the criteria for
regularity in elevation given in 4.2.3.3
When a dynamic
analysis is performed on this 2D frame, the following vibration modes are
obtained:
Mode 
Period of vibration (sec) 
1 
1.107 
2 
0.335 
3 
0.234 
4 
0.225 
5 
0.192 
6 
0.131 


Table1:
vibration modes
The first
condition is met, but the criteria for the regularity in elevation is difficult
to be judged. The first condition of 4.2.3.3(2) is met, but 4.2.3.3(3) is not
really, as the mass is not decreasing gradually from foundation to the top,
because of the heavily loaded above the roof platform.
Let us disregard
this second criteria and accept ELF method first.
When the ELF
method is applied, only the first (fundamental) mode is used, with the total
seismic mass of the building. As the seismic effect is described with one
single vibration mode only, the representation of the seismic effect is a
simple equivalent load case. Using this regular load case all the common first
and secondorder analysis can be performed, as also the linear buckling
analysis. For example, the bending moment diagram calculated from the dominant
mode (from left to right) is the following:
This way ConSteel can perform an automatic strength and stability verification for the seismic (EQU) combinations. The results are visible here, respectively:
Picture2: Utilization
ratios based on strength verifications using ELF method
Picture3: Utilization
ratios based on stability verifications using ELF method
Of course, the
platform column could be strengthened and close this exercise. But somebody can
still have some doubts about the applicability of this ELF method, due to the
criteria of vertical regularity.
Modal
Response Spectrum Analysis
How could this
be precisely calculated? The general approach proposed by EN 1998 is the Modal
Response Spectrum Analysis (MRSA) (EN 19981 4.3.3.3). This method is
applicable in all cases, where the fundamental mode of vibration alone does not
describe adequately the dynamic response of the structure. MRSA will take into
account all the calculated vibration modes, not only the fundamental and
therefore the precise seismic effect can be worked out on the structure. But
the main problem is that this will result an envelope of the maximum values of
internal forces and displacements, without any guarantee that these correspond
to the same time trace of the seismic action. Plus, they are not even in
equilibrium…. And even the sign is only positive due to the use of modal
combinations SRSS or CQC. And even worse, as the seismic action calculated this
way cannot be described by a single load case, no linear buckling analysis can
be done and therefore the automatic buckling feature of ConSteel cannot be
used.
Let us see what
MRSA with a CQC combination would give.
The first 7
vibration modes with the corresponding seismic mass participation values can be
seen in the next table. The first column shows the frequencies in Hz and the
second column shows the mass contribution factors in the horizontal direction.
The other columns mean the mass participation in the other directions
(outofplane and vertical), but these are not important for our example.
Table2: mass
participation factors
EN 1998 requires
to consider enough vibration modes in each direction to reach a minimum of 90%
of the seismic mass. As visible, the fundamental mode has a relatively low
contribution (77%) which justifies the initial doubts about only using this
single mode and disregard all the others. To fulfill the 90% minimum criteria,
the second mode must be also considered, but visible even the 4th and the 6th
have nonzero (although less then 5%) contribution.
As said before
ConSteel can perform only strength verifications but not stability
verifications based on results of MRSA combined with CQC modal combination rule.
The bending moment
diagram with the maximum possible values looks as shown below (all the bending
moment values from the multimodal result are without a sign, they must be
assumed as positive and negative values as well):
Picture5: Utilization
ratios based on strength verifications using MRSA method with CQC combination
As visible the
platform leg is still weak, it must be strengthened without a question. On
other hand the utilization ratio (without stability verification!!) at the left
corner is lower, therefore there is a chance the the ELFbased 97.9% strength
verification result could be still acceptable as safe, but the stability must
be checked somehow.
But it is also
visible, that generally the bending moments obtained by MRSA CQC are much lower
than those obtained with the ELF method. Why is this? And how can a stability
verification be performed?
Seismic modal
analysis with “selected modes” – ConSteel approach
Luckily ConSteel
provides a very flexible approach, called as „selected modes” method. This
allows the user to pick the vibration modes by himself/herself and create
linear combinations from them by specifying appropriate weighting factors. As a
result, a linear combination of the modal loads calculated from vibration modes
is obtained, instead of the quadratic SRSS or CQC combinations, which can be
considered already as a single equivalent load case and all the necessary first
and secondorder static and linear buckling analysis can be performed, as in
the case of ELF calculation.
The definition
of the „selected modes” and the specification of weighting factor is not an
automated process in ConSteel, it must be driven by the user. To be successful,
it is important to understand how the structure works.
Although the first
2 vibration modes fulfill the minimum 90% mass contribution requirement, let us
see the additionally also the 4th mode:
1^{st} mode f=0.90 Hz, T=1.109 sec
Picture6: 1^{st} vibration mode
2^{nd} mode f=3.00 Hz, T=0.334 sec
Picture7: 2^{nd}
vibration mode
4^{th}
mode f=4.265 Hz, T=0.234 sec
Picture8: 4^{th} vibration mode
The colors suggest
that the fundamental mode describes globally the structure, but the second
seems to affect additionally the platform region and the 2^{nd} or 4^{th}
is dominant for the mezzanine structure.
The
corresponding bending moment diagrams are, respectively:
Picture9: Bending moment diagram calculated from the 1^{st} vibration mode
Picture10: Bending moment diagram calculated from the 2^{nd} vibration mode
Picture11: Bending moment diagram calculated from the 4^{th} vibration mode
These bending
moments also justify the assumption made based on the colors, the 2^{nd}
mode creates significant bending moments additionally to the first mode and the
4^{th} mode creates significant bending moments additionally to the 1^{st}
mode. But it seems that also the 2^{nd} mode created significant
bending moments at this region.
It is
interesting to note, that the bending moment diagram from the 1^{st}
mode (picture 9) almost perfectly fits to the CQC summarized bending moment (or
course by assigning signs to the values based on the fundamental vibration
mode) (see picture 4), except in the regions of the platform and the mezzanine.
This means that in general the fundamental vibration modes describes quite well
the dynamic response of this frame. And because of this, the bending moments
could be calculated with the mass contribution factor corresponding to this
mode (77%). And this is the reason, why the ELF method gives higher bending
moment values, as there the same vibration mode was considered, but instead of
the corresponding mass (77%), with 100% of the seismic mass.
As we
discovered, the 2^{nd} mode should be used together with the 1^{st}
mode to correctly describe the platform region, as this region is not fully
dominated by the 1^{st} mode only, the 2^{nd} has a significant
contribution.
Similarly to the
mezzanine region, additionally to the 1^{st} mode, here the 4^{th}
mode must be used to better approach the correct result.
The definition
of the weighting factors could be done by the following – a bit arbitrary –
way. Let us take the reference the MRSA CQC values at selected points of the
structure and create corresponding rules for the linear combination to well
approach the value obtained with the CQC combination, considered as reference
value
Platform
region
CQC value 70.21 kNm
1st mode 61.38 kNm * 1.00 = 61.38 kNm
2nd mode 33.29 kNm * 0.265 = 8.82 kNm
Picture12: MRSA CQC Bending moment diagram considered as reference for the platform region
Mezzanine
region (internal column)
CQC value 26.79 kNm
1st mode 11.74 kNm * 1.00 = 11.74 kNm
4th mode 14.39 kNm * 1.045 = 15.037 kNm
or (sidewall
column)
CQC value 287.29 kNm
1st mode 272.87 kNm * 1.00 = 251.55 kNm
2nd mode 89.10 kNm * 0.16 = 35.74 kNm
Picture13: MRSA CQC Bending moment diagram considered as reference for the mezzanine region
As a summary the following 4 linear mode combinations could be set
For the frame in
general
1: Mode 1 * 1.00
For the platform
region
2: Mode 1 * 1.00
+ Mode 2 * 0.265
For the
mezzanine region
3: Mode 1 * 1.00 + Mode
4 * 1.045
4: Mode 1 * 1.00
+ Mode 2 * 0.16
Of course, other
weighting factors could be also set, as the condition we set was to meet the
target value. The more target values we define in the region, we can more
precisely set the factors. Usually it is recommended to keep the factor of the
fundamental mode as 1.00 (or close to 1.00) and adjust the other factors for
the modes appearing in the given mode linear combination as necessary.
With these 4
linear mode combinations we can already perform the automatic stability
verifications. And the answers the original questions.
Utilization
ratio at the left corner with stability verification included: strength 79.2%,
stability 102.1% compared to the ELF results of strength 97.9% and stability 128.2%.
So, the use of the ELF method was safe, the results can be accepted, the
structure works for strength verifications but shows a small overstress regarding
stability verification.
Leg of the
platform: There was already a strength problem based on MRSA CQC results,
therefore the post must be strengthened. The result of the stability
verification with the finetuned seismic force is 106%.
Conclusion
This post wanted
to call the attention of performing stability verifications for seismic
combinations as well, like for any other combinations. For structures, where
the ELF method is applicable, ConSteel can perform without problem these
stability verifications automatically. Unfortunately for irregular structures
the MRSA CQC method does not give directly a possibility. The special method
implemented in ConSteel called „selected modes” can be successfully used to
create loads, with the help of a linear combination with modes important for
parts of the structures and with the resulted loads the stability verifications
can already be executed.
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