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:
gateIntroduction
This article presents the calculation method for determining the buckling resistance of a pinned column with intermediate restraints in accordance with Eurocode standards. The procedure is based on an example from the Access Steel design examples collection and is compared with the calculation process implemented in Consteel’s steel member design functions, specifically within the Member Checks module.
In the following sections, a step-by-step guide is provided to demonstrate how the member check functionality can be applied to simple cases, highlighting both methodology and practical usage.
Input Data for the Example
The example considers a pinned column in a multi-storey building, subjected to a design axial force of $N_{Ed}$ = 1000 kN. The column has a total length of 10.50 m and is laterally restrained about the y–y axis at intervals of 3.50 m.
The member is a rolled HEA 260 section made of S235 steel. The cross-section is classified as Class 1. The geometric properties of the section are: height h = 250 mm, width b = 260 mm, web thickness $t_w$ = 7.5 mm, flange thickness $t_f$ = 12.5 mm, and fillet radius r = 24 mm. The cross-sectional area is A = 86.8 cm², with moments of inertia $I_y$ = 10450 cm⁴ and $I_z$ = 3668 cm⁴.
The material properties are defined according to EN 1993-1-1. Since the maximum thickness is less than 40 mm, the yield strength is taken as $I_y$ = 235 N/mm². The partial safety factors are γM0 = 1.0 and γM1 = 1.0.
Determining Design Buckling Resistance of a Compression Member
The design buckling resistance of the column $N_{b,Rd}$ is evaluated by determining the reduction factor χ for both principal buckling directions. This requires the calculation of the elastic critical forces $N_{cr}$, which form the basis for identifying the governing buckling mode.
Elastic critical force for the relevant buckling mode $N_{cr}$
The Young’s modulus is taken as $E=210000 \frac{N}{mm^2}$. The buckling lengths in the respective planes are $L_{cr,y} = 10.50m$ for buckling about the y–y axis and $L_{cr,z} = 3.50m$ for buckling about the z–z axis. Observe that the buckling lengths for the strong and weak axes differ according to the support conditions, which must be determined by the engineer in manual calculations.
$$N_{cr,y}=\frac{π^2*E*I_{y}}{L_{cr,y^2}}=1964.5 kN$$
$$N_{cr,z}=\frac{π^2*E*I_{z}}{L_{cr,z^2}}=6206.0 kN$$
In Consteel, the elastic critical force for the relevant buckling mode can be determined using the Individual Member Design approach. This is accessible in the Member Checks tab under the Steel module, where selected members can be added and evaluated.
Once a member is selected, the analysis results are automatically loaded, provided that first- or second-order analysis results are available. Ensure that the analysis has been run in the Analysis tab and the cross section check on the Global ckecks tab before proceeding to the Member Checks section.
For the pinned column with intermediate restraints, the relevant buckling cases, strong and weak axis, are selected, and the dominant load combination is automatically indicated with a *. Consteel identifies the intermediate restraints separately for each direction and divides the member into segments accordingly to help determine the correct buckling lengths.
Design parameters for each segment are set with the three-dot icon:
At this step, users must verify the assigned values. By default, the first value is applied, and the correct buckling shape or effective length factor should be confirmed based on engineering judgment.
In order to use the critical load multiplier selection option, make sure to perform the calculation first:
In order to check whether the correct critical load multiplier was selected, you can examine the effective length factor, which is calculated based on it (in this case, it is 1 for both directions). In our example, the relevant buckling shapes for the y–y and z–z directions are as follows:
The elastic critical force $N_{cr}$ is calculated automatically, regardless of whether the effective length factor was entered manually or the critical load multiplier was selected.
| Access Steel – manual calculation | Consteel using the effective length factor | Consteel using the critical load multiplier | |
| $N_{cr,y}$ | 1964.5 kN | 1962.53 kN | 1973.76 kN |
| $N_{cr,z}$ | 6206.0 kN | 6189.01 kN | 6218.96 kN |
Once all parameters are defined, the design check is executed by clicking the Check button, and the results are displayed.
Results can be reviewed and filtered by member, load combination, and buckling case. Lateral-torsional buckling checks follow a similar procedure, with segment boundaries adjustable and critical moments calculated either analytically or using the critical load multiplier.
Non-dimensional slenderness
In order to determine the reduction factor, the non-dimensional slenderness λ must be calculated based on the elastic critical force corresponding to the relevant buckling mode.
$$\overline{\lambda_y} = \sqrt{\frac{A*f_y}{N_{cr,y}}}=\sqrt{\frac{86.8*23.5}{1965}}=1.016$$
$$\overline{\lambda_z} = \sqrt{\frac{A*f_z}{N_{cr,z}}}=\sqrt{\frac{86.8*23.5}{6206}}=0.573$$
In Consteel, the detailed calculations for strong and weak axis buckling can be reviewed separately on the Results tab:
Reduction factor
For axial compression, the value of χ corresponding to the relevant non-dimensional slenderness $\overline{\lambda}$ should be determined from the appropriate buckling curve in accordance with EN 1993-1-1 §6.3.1.2.
For $\frac{h}{b}= \frac{250mm}{260mm} = 0.96 < 1.2$ and $t_f = 12.5 mm< 100 mm$
- buckling about axis y-y, buckling curve b, imperfection factor $\alpha=0.34$
$$\varphi_y=0.5*[1+0.34(1.019-0.2)+1.019^2]=1.158$$
$$\chi_y=\frac{1}{1.158+\sqrt{1.158^2-1.019^2}}=0.585$$
- buckling about axis z-z, buckling curve c, imperfection factor $\alpha=0.49$
$$\varphi_y=0.5*[1+0.49(0.573-0.2)+0.573^2]=0.756$$
$$\chi_y=\frac{1}{0.756+\sqrt{0.756^2-0.573^2}}=0.801$$
$$\chi=min(\chi_y;\chi_z)$$
$$\chi=0.585<1.00$$
Design buckling resistance of a compression member
$$N_{b,Rd}=\chi*\frac{A*f_y}{\gamma_{M1}}=0.585*\frac{86.8*23.5}{1.0}=1193 kN$$
$$\frac{N_Ed}{N_{b,Rd}}=\frac{1000}{1193}=0.84<1.00$$
Conclusion
This example demonstrates the application of the isolated member approach for a simple compression member. For more complex cases or alternative stability verification methods, such as the imperfection approach or the general method, refer to the dedicated article on stability design methods, where their principles and applications are discussed in detail.
Download modelConsteel recommends to use the General Method from EN 1993-1-1 for the evaluation of out-of-plane strength of members and sturctures. In addition, the scaled imperfection based 2nd order approach is available.
Did you know, that when linear buckling eigenform affine imperfections are used, Consteel can scale automatically the selected eigenmodes to perform a Eurocode compatible design? And you can even combine several imperfections?
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Bending:
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Did you know that in addition the standard Type 1 and Type 2 response spectrums defined by Eurocode 8, you can use also user-defined spectrums with Consteel?
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Introduction
When a beam, bent in a plane, is allowed to move and twist freely between its two support points, in addition to bending, sudden perpendicular displacement and twisting may occur: causing the beam to deviate out of its original plane. This phenomenon is illustrated in Figure 1, showing a single supported beam with I-section bent around the strong axis. As the bending moment in the vertical plane increases, reaching a critical value, the beam undergoes abrupt lateral movement and twisting between the supports. This phenomenon is called lateral torsional buckling (LTB), which is a loss of stability mode that can apply to both perfect beams and real beams.
The design of the beam against LTB is fully analogous to the design of a compressed column against flexural buckling. The analogy is illustrated in Table 1, where the corresponding parameters are shown that affect the two buckling resistances:
| Flexural (column) buckling | Lateral torsional buckling |
|---|---|
| design force ($N_{Ed}$) | design moment ($M_{Ed}$) |
| critical force ($N_{cr}$) | critical moment ($M_{cr}$) |
| column slenderness ($\frac{}{\lambda}$) | beam slenderness ($\frac{}{\lambda}_{LT}$) |
| buckling reduction factor ($\chi$) | buckling reduction factor ($\chi_{LT}$) |
| buckling resistance ($N_{b,Rd}$) | buckling resistance ($M_{b,Rd}$) |
The critical moment of the perfect beam is determined at the location of the maximum value of the My,Ed design bending moment diagram. For a doubly symmetrical I cross-section:
$$M_{cr}=C_1\frac{\pi^2EI_z}{(k_z⋅L)^2}\left[\frac{I_\omega }{I_z}+ \frac{(k_zL)^2GI_t}{\pi^2EI_z}\right] ^{0.5} $$
where kz is the coefficient of restraint about the weak axis of the cross-section, G is the shear modulus, and It and Iω are the pure (St. Venant) and warping torsional moments of inertia of the cross-section. The value of the factor C1 depends on the shape of the bending moment diagram and its value can be found in appropriate tables and manuals. For a constant moment diagram, C1=1.0. The formula for the other design parameters, in particular the buckling reduction factor $\chi_{LT}$, depends on the design standard considered.
Lateral torsional buckling resistance by EN1993-1-1
The design of the beam against LTB (load capacity check) according to EC3-1-1 shall be carried out in the following steps:
gateDesigning a lattice girder
The design of the bars of a truss (lattice girder) structure does not require any special theoretical knowledge: normally, the truss bars are designed as compressed and/or tensioned bars, neglecting bending moments and shear forces. The dimensioning of compression bars is nowadays carried out using a model-based computer procedure. For details, see the knowledge base material Design of columns against buckling. Here, only the determination of the deflection length of the compressed bars is presented.
The most important parameter for the dimensioning of a compressed bar is the slenderness:
$$\overline{\lambda}=\sqrt\frac{Af_y}{N_{cr}}$$
where
$$N_{cr}=\frac{\pi^2El}{(kL)^2}$$
where the buckling length factor k is recommended by EN1993-1-1 to facilitate manual calculations:
| Type of the bar | Direction of buckling | k |
|---|---|---|
| chord | – in-plane – out-of-plane | 0.9 0.9 |
| bracing | – in-plane – out-of-plane | 0.9 1.0 |
Software using model-based computational methods (e.g. Consteel software) determines the elastic critical force Ncr directly by finite element numerical methods, taking into account the behaviour of the entire lattice girder, instead of the above conservative formula. The following example is intended to illustrate the relationship between the manual design procedure proposed by the standard and the results of the modern model-based numerical procedure.
- Let the structural model of the lattice girder under consideration be the Consteel model shown in Figure 1.
- Let the load shown correspond to the design load combination of the girder.
- Determine the deflection length of the most stressed compressed chord member using finite element numerical stability analysis.
(Consteel software)
Relationship between procedures
The steps of the calculation are:
Buckling stability analysis
The stability analysis of the elastic model shows the governing buckling mode of the lattice structure and the corresponding elastic critical load factor αcr (Figure 2).
We can see that the upper chord of the perfectly elastic model is deflected laterally under load. The load that causes this elastic buckling is the critical load, whose value is given by the product of the design load and the critical load factor αcr=5.99.
gateThe evolution of compressed bar (column) design
One of the characteristic features of steel structures made of bars (e.g. lattice girders) is the compressed bar. We speak of a compressed bar when the structural element, which usually has a straight axis, is loaded by a compressive force P applied centrally (Figure 1).
Figure 2 illustrates the evolution of compressed bar (column) design. In the beginning (in the old days), master builders determined the load-bearing capacity of compressed columns of different materials and sizes on the basis of the experience accumulated over the centuries, passed down from master to apprentice. A significant change was brought about by the application of classical mathematical differential analysis to engineering. The Swiss mathematician and physicist Euler (1707-1783) solved the problem of the deflection of a compressed elastic line, which could be applied to the solution of the elastic compressed bar (Euler’s force). In the following centuries, engineers recognised that Euler’s force only gave an acceptable approximation to the real load capacity of a compressed bar in certain cases (mainly for large slender bars). Many solutions for the bearing capacity of a compressed bar were developed that were more advanced than the Euler formula, but it was not until the huge structural engineering boom following World War II that significant changes were made. Compression bar experiments were carried out in every major structural laboratory in the world, and a database of over two thousand experiments was compiled from the results. The load capacity of the pressure bar was given by a formula based on the database, using the method of mathematical statistics.
This methodology is still dominant today: ‘the dimensioning of the compressed bar has become a political issue for the steel construction profession…’. Understanding the principle of compressed bar design is therefore essential for the structural engineer.
The right side of the Figure 2 also contains a hint for the future. At the level of scientific research, it is already present that the load capacity of a real compressed column can be determined by mathematical-mechanical simulation. Indeed, in the near future, databases that go beyond anything we know today can be created using supercomputers. On the basis of such a gigantic database, artificial intelligence could, at least in principle, supersede existing engineering knowledge and methodology. But the reality is that structural engineering is not one of the pull sectors (such as the defense or automotive industries), so this new shift in design theory is certainly a long way off.
In the following, the Euler force and the experimentally based standard design formula, which are of major importance to structural steel engineering today, are discussed in detail.
Buckling strength of the ideal columns: the Euler force
Assume that the hinged compressed column shown in the Figure 3 has the following properties:
- perfectly straight,
- its material is perfectly linearly elastic,
- centrally compressed.
Under the above conditions, perform the compressed column experiment using Consteel software: run the Linear Buckling Analysis (LBA) calculation. The result is illustrated in Figure 3.
gateIn Consteel, the calculation of cross sectional interaction resistance for Class 3 and 4 sections is executed with the modified Formula 6.2 of EN 1993-1-1 with the consideration of warping and altering signs of component resistances. Let’s see how…
Application of EN 1993-1-1 formula 6.2
For calculation of the resistance of a cross section subjected to combination of internal forces and bending moments, EN 1993-1-1 allows the usage -as a conservative approximation- a linear summation of the utilization ratios for each stress resultant, specified in formula 6.2.
$$\frac{N_{Ed}}{N_{Rd}}+\frac{M_{y,Ed}}{M_{y,Rd}}+\frac{M_{z,Ed}}{M_{z,Rd}}\leq 1$$
As Consteel uses the 7DOF finite element and so it is capable of calulcating bimoment, an extended form of the formula is used for interaction resistance calculation to consider the additional effect.
$$\frac{N_{Ed}}{N_{Rd}}+\frac{M_{y,Ed}}{M_{y,Rd}}+\frac{M_{z,Ed}}{M_{z,Rd}}+\frac{B_{Ed}}{B_{Rd}}\leq 1$$
Formula 6.2 ignores the fact that not every component results the highest stress at the same critical point of the cross-section.
In order to moderate this conservatism of the formula, Consteel applies a modified method for class 3 and 4 sections. Instead of calculating the maximal ratio for every force component using the minimal section moduli (W), Consteel finds the most critical point of the cross-section first (based on the sum of different normal stress components) and calculates the component ratios using the W values determined for this critical point. Summation is done with considering the sign of the stresses caused by the components corresponding to the sign of the dominant stress in the critical point.
(For class 1 and 2 sections, the complex plastic stress distribution cannot be determined by the software. The Formula 6.2 is used with the extension of bimoments to calculate interaction resistance, but no modification with altering signs is applied)
Example
Let’s see an example for better explanation.
GATEDid you know that you could use Consteel to determine the optimum number of shear connectors for composite beams?
In composite beam design, the required number of shear studs is not only a detailing issue but a direct part of the structural resistance mechanism. The modelling and design environment in Consteel allows this number to be determined in a rational and automated way, consistent with EN 1994-1-1:2010.
The process begins with the definition of a composite beam cross-section (Macro section). Two standard types are available: a solid slab composite beam and a profiled steel sheeting composite beam. The effective width is defined at input level, but the actual effective width used in analysis is calculated automatically based on span, geometry, and stud spacing assumptions. This ensures that the structural response is captured realistically, while maintaining simple input control for the user.
After defining the cross-section, the member is created and design parameters are assigned through object properties. These parameters govern key aspects of composite behaviour, including stud spacing, support conditions, and whether shear stud design is performed manually or automatically.
For shear connector design, Consteel applies a plastic distribution model. The governing section of the beam, typically corresponding to the maximum bending moment, is identified automatically. From this section, shear studs are distributed symmetrically along the beam length.
When automatic optimisation is selected, the software determines the minimum number of shear studs required to satisfy bending resistance. It then increases the number iteratively until the composite section achieves sufficient capacity. The resulting value represents the optimal number of shear connector positions for the given loading and geometry.
The key parameters used in this evaluation include:
- nopt: optimal number of shear stud positions in the governing region
- nact: actual number of stud positions applied
- sL, sR: spacing of studs on the left and right side of the critical section
- nmin: minimum required number of stud positions based on resistance
The ratio nact / nopt is used as a direct measure of utilisation of the shear connector system.
There is also the option to define the number of studs manually. In that case, Consteel distributes them uniformly along the member and checks compliance with detailing rules such as minimum and maximum spacing. This is typically useful when construction constraints or predefined detailing standards take priority.
Composite beam design itself is carried out according to EN 1994-1-1:2010. Plastic bending resistance is evaluated for Class 1 and 2 cross-sections, while shear, concrete flange crushing, and longitudinal shear are checked at critical sections. Shear buckling is treated according to EN 1993-1-5. Lateral-torsional buckling is not included in the current design scope.
For profiled sheeting, shear stud resistance is reduced depending on rib orientation, in line with Eurocode 4 rules.
Overall, the key point is that the number of shear connectors is no longer an assumed input. It is derived from structural demand through an automated and transparent process, which makes it easier to reach an efficient and consistent composite design without repeated manual iteration.
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Try Consteel for freeDid you know that you could use Consteel to determine automatically the second order moment effects for slender reinforced concrete columns?
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