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 ELF-based 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 second-order 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:
1st mode f=0.90 Hz, T=1.109 sec
Picture6: 1st vibration mode
2nd mode f=3.00 Hz, T=0.334 sec
Picture7: 2nd
vibration mode
4th
mode f=4.265 Hz, T=0.234 sec
Picture8: 4th
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 2nd or 4th
is dominant for the mezzanine structure.
The
corresponding bending moment diagrams are, respectively:
Picture9: Bending
moment diagram calculated from the 1st vibration mode

Picture10: Bending
moment diagram calculated from the 2nd vibration mode
Picture11: Bending
moment diagram calculated from the 4th vibration mode
These bending
moments also justify the assumption made based on the colors, the 2nd
mode creates significant bending moments additionally to the first mode and the
4th mode creates significant bending moments additionally to the 1st
mode. But it seems that also the 2nd mode created significant
bending moments at this region.
It is
interesting to note, that the bending moment diagram from the 1st
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 2nd mode should be used together with the 1st
mode to correctly describe the platform region, as this region is not fully
dominated by the 1st mode only, the 2nd has a significant
contribution.
Similarly to the
mezzanine region, additionally to the 1st mode, here the 4th
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 fine-tuned 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.