Introduction
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 modelIntroduction
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:
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.
gateDid you know that you could use Consteel to Consider the shear stiffness of a steel deck as stabilization for steel members?
In many practical steel structures, trapezoidal decking is treated only as a load-bearing surface. In reality, when properly connected to the supporting members, it behaves as a shear diaphragm and contributes to the overall stability of the structure. This effect can be directly taken into account in Consteel by applying shear field stiffness to beam elements.
The stabilizing effect comes from the in-plane shear stiffness of the deck. Under horizontal loading, the sheeting deforms and transfers forces between structural members. This behavior can be described by a single parameter, the shear stiffness (S), which represents the resistance of the diaphragm against deformation.
The overall stiffness is influenced by several components, including the shear deformation of the sheet, profile geometry, fastener slip, and connection flexibility. These contributions together define how effectively the deck can restrain phenomena such as lateral-torsional buckling.
A key requirement for this behavior is proper fastening. Typically, the sheeting must be connected along its edges and fixed to supporting members at each rib to ensure reliable diaphragm action.
In engineering practice, shear stiffness is determined using standardized or manufacturer-based methods rather than detailed analytical models. Consteel supports several established approaches:
- Schardt/Strehl method (DIN 18807), based on parameters describing shear and warping deformation
- Improved Schardt/Strehl method, including the effect of fastener spacing
- Bryan/Davies method, considering additional structural parameters
- Eurocode-based method, using general geometric properties of the sheeting
These methods differ in complexity and required input data, but all aim to provide a realistic stiffness value for use in global analysis. If the sheeting is not fixed at every rib, the calculated stiffness must be reduced accordingly.
The shear field object in Consteel allows engineers to include the diaphragm effect without detailed shell modeling. The calculated shear stiffness can be assigned directly to beam elements, providing additional lateral restraint.
The process involves selecting a trapezoidal sheet profile, choosing the appropriate calculation method, and defining the relevant geometric and connection parameters. The software then determines the stiffness and incorporates it into the structural model.
Including shear stiffness in the analysis can lead to higher critical load factors and reduced displacements, resulting in more efficient structural designs. However, it also means that the decking becomes part of the stabilizing system.
Any later modifications to the sheeting, such as openings or changes in fastening, may reduce this effect and should therefore be carefully assessed.
The shear stiffness of trapezoidal steel decking provides a measurable and often significant contribution to structural stability. By incorporating this effect in Consteel, engineers can achieve more realistic analysis results and optimize their designs while maintaining structural safety.
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Try Consteel for freeIntroduction
This verification example studies a simple fork supported beam member with welded section (flanges: 200-12 and 100-12; web: 400-8) subjected to bending about major axis. Constant bending moment due to concentrated end moments and triangular moment distribution from concentrated transverse force is examined for both orientations of the I-section. Critical moment and force of the member is calculated by hand and by the Consteel software using both 7 DOF beam finite element model and Superbeam function.
Geometry
Normal orientation – wide flange in compression
Constant bending moment distribution
Triangular bending moment distribution – load on upper flange
Triangular bending moment distribution – load on bottom flange
Reverse orientation – narrow flange in compression
Constant bending moment distribution
Triangular bending moment distribution – load on upper flange
Triangular bending moment distribution – load on bottom flange
Calculation by hand
Factors to be used for analitical approximation formulae of elastic critical moment are taken from G. Sedlacek, J. Naumes: Excerpt from the Background Document to EN 1993-1-1 Flexural buckling and lateral buckling on a common basis: Stability assessments according to Eurocode 3 CEN / TC250 / SC3 / N1639E – rev2
Normal orientation – wide flange in compression
Constant bending moment distribution
Reverse orientation – narrow flange in compression
Computation by Consteel
Version nr: Consteel 15 build 1722
Normal orientation – wide flange in compression
Constant bending moment distribution
- 7 DOF beam element
First buckling eigenvalue of the member which was computed by the Consteel software using the 7 DOF beam finite element model (n=25). The eigenshape shows lateral torsional buckling.
Superbeam
First buckling eigenvalue of the member which was computed by the Consteel software using the Superbeam function (δ=25).
Introduction
This verification example studies a simple fork supported beam member with welded section (flanges: 200-12; web: 400-8) subjected to bending about major axis. Constant bending moment due to concentrated end moments and triangular moment dsitribution from concentrated transverse force is examined. Critical moment and force of the member is calculated by hand and by the Consteel software using both 7 DOF beam finite element model and Superbeam function.
Geometry
Constant bending moment distribution
Triangular bending moment distribution – load on upper flange
Triangular bending moment distribution – load on bottom flange
Calculation by hand
Constant bending moment distribution
Triangular bending moment distribution
Computation by Consteel
Version nr: Consteel 15 build 1722
Constant bending moment distribution
7 DOF beam element
First buckling eigenvalue of the member which was computed by the Consteel software using the 7 DOF beam finite element model (n=16). The eigenshape shows lateral torsional buckling.
Superbeam
First buckling eigenvalue of the member which was computed by the Consteel software using the Superbeam function (δ=25).
Triangular bending moment distribution – load on upper flange
7 DOF beam element
First buckling eigenvalue of the member which was computed by the Consteel software using the 7 DOF beam finite element model (n=16).
Superbeam
First buckling eigenvalue of the member which was computed by the Consteel software using the Superbeam function (δ=25).
Triangular bending moment distribution – load on bottom flange
(more…)Perfect the understanding of your structure with advanced buckling sensitivity results illustrated on proper mode shape and colored internal force diagrams.
gateConsteel 14 is a powerful analysis and design software for structural engineers. Watch our video how to get started with Consteel.
Contents
- Set analysis parameters
- Perform first and second order analysis
- Perform buckling analysis
- Analysis results in graphics and in tables
- Results: deformation, internal forces, reactions
Part 2 – Imperfection factors
The Eurocode EN 1993-1-1 offers basically two methods for the buckling verification of members:
(1) based on buckling reduction factors (buckling curves) and
(2) based on equivalent geometrical imperfections.
In the first part of this article, we reviewed the utilization difference and showed the relationship between the two methods. It was concluded that the method of chapters 6.3.1 (reduction factor) and 5.3.2 (11) (buckling mode based equivalent imperfection) are consistent at the load level equal to the buckling resistance of the member, so when the member utilization is 100%. The basic result of the procedure in 5.3.2 (11) is the amplitude (largest deflection value) of the equivalent geometrical imperfection. However, the Eurocode gives another simpler alternative for the calculation of this amplitude for compressed members in section 5.3.2(3) b) in Table 5.1, where the amplitude of an initial bow is defined as a portion of the member length for each buckling curves (Fig. 1.). We use the first column (“elastic analysis”) including smaller amplitude values.
It is an obvious expectation that these two standard procedures should yield at least similar results for the same problem. However, this is by far not the case in general.
In order to show the significance of the imperfection amplitudes this part is dealing with these two calculation methods, the variation of their values and the effect on the buckling utilization.
Let’s see again the simple example of Part 1: a simply supported, compressed column with a Class 2 cross-section (plastic resistance calculation allowed). The column is 6 meters high and has an IPE300 cross-section made of S235 steel. The two methods are implemented into Consteel and on Figure 2. it can be seen, that the two values for the amplitude of the geometrical imperfection is very different – e0 = 24 mm by the 5.3.2(3) b) Table 5.1 (L/250) and e0 = 13,4 mm by the 5.3.2 (11) (same as in Part 1).
Click the button bellow to download and read the full article at page 187-195.
In this paper a numerical study is presented which examines a steel frame with two different finite element programs. Stability failure is more frequent in a lot of cases than strength failure hence it is important to focus on these failure modes: global, in-plane-, out-of-plane -, lateral-torsional- and local buckling. Three models were used with different elements such as shell elements and 7 DOF beam elements. 7 DOF beam elements were used in the first model, shell elements were used in the other two. The first of the shell models gave too much local buckling shapes therefore it was improved with local constraints and that is the third model where global buckling shapes can be examined. There are three different procedures to calculate the resistance: (i) the general method, (ii) the method of the reduction factors, and (iii) the simulation. The analysis results of the different programs and design methods were compared to each other and to the manual calculation based on the Eurocode 3 standards.
gate