Is a single dominant vibration mode sufficient, or should multiple vibration modes be considered in seismic analysis?

Steel portal frames are frequently used in industrial and logistics buildings as primary load-bearing structures. Their seismic behavior is strongly influenced by the stiffness of the roof diaphragm and by the interaction between the main portal frames and secondary structural subsystems such as endwalls.

In seismic design, engineers often assume that the global response of such buildings can be represented by a single dominant vibration mode. This assumption is valid when the roof diaphragm is sufficiently rigid and the first transverse mode mobilizes most of the structural mass. However, when the diaphragm is flexible or when different structural parts participate in different vibration modes, higher modes may also contribute to the seismic response.

This article investigates how the choice between a single-mode and a multi-modal approach affects the seismic design of steel halls modeled in Consteel. Through a comparative example, the study demonstrates the implications of different modal combination techniques and discusses how reliable internal forces can be obtained while maintaining compatibility with stability verification procedures according to EN 1993-1-1.

Case with a Rigid Roof Diaphragm

Single dominant mode

If a building is designed with a sufficiently rigid roof diaphragm, a single transverse vibration mode is typically able to mobilize close to 90% of the total participating mass. In such cases, the Single dominant mode method is an efficient and preferred design method.

A rigid roof diaphragm can be achieved by:

• Using an adequate trapezoidal steel deck, modeled in Consteel either
  • as a Shear Field with a high shear stiffness parameter (“S” value), or
  • by introducing equivalent dummy roof bracing diagonals with rod diameters calibrated to reproduce the diaphragm shear stiffness.
• Alternatively, real bracing elements may be added along the sidewall columns or from the eaves to the ridge along the building length.

Case without a Rigid Roof Diaphragm

If a rigid diaphragm is intentionally not assumed, a single vibration mode will generally not represent the full seismic response in the transverse direction.

Single dominant mode

A dynamic eigenvalue analysis is first performed to determine the natural vibration modes of the structure. In Consteel, this analysis calculates the eigenfrequencies and corresponding mode shapes based on the structural stiffness and mass distribution, considering both the elastic stiffness and second-order geometric stiffness of the structure. The first three vibration modes are then evaluated for their mass participation in the transverse direction.

After the calculation, the mass participation for each principal direction (X, Y, and Z) can be viewed in the Analysis tab under the Analysis report, in the Mass section. In the examined case:

• The first vibration mode is dominant in the transverse direction.
• It mobilizes only 62% of the total participating mass.
• This is below the 90% minimum requirement specified by EN 1998.

As a result, additional vibration modes must be considered to accurately capture the seismic response of the structure.

If only the first mode were used:

• The seismic action in the middle portal frames would be reasonably captured.
• More rigid structural parts (e.g. endwalls), insufficiently mobilized by the first mode, would not be properly represented.

When the Single dominant mode method is applied, Consteel automatically scales the first mode to represent 100% mass participation, regardless of the dynamic analysis result, as a measure the predict the effect of uncalculated and therefore missing modes. In this case this results an overestimation of 1.00/0.62=1.61 of the internal forces for the middle frames.

See bending moment diagrams obtained from 62% modal mass (Fig. 7) and from overestimated 100% mass (Fig. 8) considered in single dominant mode of Consteel.

Download the example model using the single dominant mode method:

Download model

All modal shapes, CQC summation

Since 90% of the participating mass is not achieved with the first modes, additional modes must be calculated. After increasing the number of computed modes to 25, the total participating mass in the transverse direction exceeds 90%.

In Consteel, modal loads are calculated for all dynamic shapes in all directions, and first-order analysis is used to determine displacements and internal forces for each mode. The Complete Quadratic Combination (CQC) method then summarizes these results. This Eurocode-recommended statistical method produces envelope diagrams of deformations and internal forces (without signs), representing the highest expected values.

Key aspects of the Complete Quadratic Combination (CQC) method:

• Modal internal forces are combined non-linearly.
• All resulting modal combinations are positive by definition.
• Final bending moment and axial force envelopes are obtained.

Limitations:

• The envelope results from a non-linear modal superposition.
• An equivalent linear load case producing identical results cannot be created.
• Therefore, linear buckling analysis with critical load multipliers corresponding exactly to the CQC envelope is not possible.
• Consequently, the advanced stability design method in Consteel (based on elastic buckling modes and the General Method of EN 1993-1-1) cannot be directly applied.

Download the example model using the all modal shapes, CQC summation method:

Download model

Selected modes, linear summation

Using the “Selected modes, linear summation” method in Consteel, selected vibration modes can be combined linearly. Relevant modal shapes are defined together with combination factors for each mode, and the software generates equivalent modal loads from the resulting combined vibration shapes. This is a technique used to approximate non-linear modal combinations by adequate linearized modal combinations.

From modal evaluation, the following modes are identified as globally relevant:

Mode	Mass participation
1	0.62
4	0.08
34	0.17

These three modes are largely independent.

Mode 4 produces bending moments with opposite signs in certain members; therefore, two combinations are defined:

• Combination 1: +1 · Mode 1, +1 · Mode 4, +1 · Mode 34
• Combination 2: +1 · Mode 1, −1 · Mode 4, +1 · Mode 34

The resulting combined modal shapes generate linearized equivalent seismic load cases that can be used in the subsequent structural analysis.

After the dynamic analysis and the first-order linear static analysis of the generated equivalent modal forces, the envelope of the resulting linearized seismic load cases provides the bending moments in the portal frames and the axial forces in the endwall members.

This approach avoids the non-linear CQC modal combination and instead uses a linear combination of selected modal results. As a result, the seismic effects are represented by equivalent linear load cases, which allows the application of the standard stability verification procedures in Consteel, including linear buckling analysis and the General Method according to EN 1993-1-1.

Download the example model using the selected modes, linear summation method:

Download model

Summary

The following table shows the bending moments from the horizontal seismic action (without the effect of other permanent loads included in the seismic combination) obtained in portal frames using the various approaches by Consteel

Bending moment [kNm]
	Single dominant mode	CQC	Selected modes
			combination 1	combination 2
Frame 1	14.77	10.40	12.55	4.19
Frame 2	24.29	15.10	12.71	17.78
Frame 3	24.14	15.02	12.66	17.68
Frame 4	14.75	10.39	12.65	4.18

Maximum utilization ratios considering global buckling

Highest ratio	Single dominant mode [kNm]	CQC	Selected modes
Frame 1	0.34	–	0.33
Frame 2	0.39	–	0.36
Frame 3	0.39	–	0.36
Frame 4	0.34	–	0.33

Similar comparison for the axial tension forces in the diagonal members of the endwalls

Bending moment	Single dominant mode [kNm]	CQC	Selected modes
			combination 1	combination 2
Endwall	2.82	7.38	9.64	6.45

Conclusion

The Single dominant mode method provides:

• A fast solution
• Conservative internal forces for the most dynamically sensitive members (e.g., intermediate portal frames)

Limitations:

• For structurally stiffer subsystems with different dynamic characteristics (e.g., endwalls), results may be inaccurate and potentially unsafe.

The Selected modes, linear summation method:

• Approximates the CQC multi-modal response with linearized equivalent modal load cases.
• Enables stability verification using:
  • Linear buckling analysis
  • General Method (EN 1993-1-1)
• Provides results close to the CQC reference while maintaining compatibility with standard design workflows.

This makes it a technically consistent solution when multiple transverse vibration modes must be considered but linear stability design procedures are required.

The use of 1-2-1 scale factors represents a simplified approach that is usually adequate for industrial halls. For a more complex Consteel example, see the Seismic design of frames of single-story industrial building with built-in mezzanine floors according to Eurocode 8 with Consteel article.

In everyday practice frames of pre-engineered 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 non-seismic 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 built-in partial mezzanine floor and with a platform placed above the roof.

Equivalent Lateral Force method

The most straightforward design approach is the Equivalent Lateral Force (ELF) method (EN 1998-1 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:

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 for a moment this second criteria and accept the ELF method.

When the ELF method is applied, only the first (fundamental) mode is used, with the total seismic mass of the building carried by this frame. As the seismic effect is described with one single fundamental vibration mode only, the representation of the seismic effect is a simple equivalent load case, called as dominant load. Using this regular load case all the common first and second-order 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 combinations. The results are visible here, respectively:

As it can be seen the structure is generally OK for strength, but there are some local overstresses at the platform and the utilization ratio is very high at the left corner. Regarding stability verifications the section seem to be weak. So – as expected – it is a key importance to be able to perform the stability verifications.

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 structure be more precisely calculated? The general approach proposed by EN 1998 is the Modal Response Spectrum Analysis (MRSA) (EN 1998-1 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 frame of the seismic action. Plus, the internal forces produced at ends of members connected to a given node are not even in static equilibrium…. And even the sign of the internal forces or deformations 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 or by a linear combination of multiple load cases, no linear buckling analysis can be performed and therefore the automatic buckling feature of CosSteel 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 (out-of-plane and vertical), but these are not important for our example.

EN 1998 requires to consider enough vibration modes in each direction to reach a minimum of 90% of the seismic mass.

With Consteel the first 7 vibration modes have been calcualated and the results are shown in the table. Direction ‘1′ means horizontal in-plane direction while ‘2’ means horizontal out-of-plane direction and ‘z’ means the vertical direction. We are concentrating now on the vibrations which happen in the plane of this frame.

As visible, the fundamental mode has high contribution (77%) but does not reach the required level. The difference may justify the initial doubts about only using this single mode and disregard all the others. To fulfill the 90% minimum criteria, the second mode (17%) must be also considered, but visible even the 4th and the 6th have non-zero (although less then 5%) contributions in this direction.

As said before Consteel can perform only strength verifications but no stability verifications based on results obtained from an 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):

The results of the strength verification are the following:

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?

Consteel approach

Seismic modal analysis with “selected modes”

Luckily Consteel provides a very flexible approach, called as „selected modes” method. This allows the user to pick the vibration modes 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 together already 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

2^nd mode f=3.00 Hz, T=0.334 sec

4^th mode f=4.265 Hz, T=0.234 sec

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:

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 (of 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.

gate