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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.

Fig. 1  Lateral torsional buckling (LTB) mode of beams under bending
Fig. 1  Lateral torsional buckling (LTB) of beams under bending

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) bucklingLateral 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 ($M_{b,Rd}$)
Table 1: Analogy between flexural and LT buckling modes

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:

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Designing 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 barDirection of bucklingk
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.

Fig. 1  Structural model and design load combination of the examined lattice girder
(ConSteel software)
Fig. 1  Structural model and design load combination of the examined lattice girder
(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).

Fig. 2 Buckling mode and critical load factor given by numerical analysis
Fig. 2 Buckling mode and critical load factor given by numerical analysis

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.

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This overview delves into Consteel’s solution, offering an alternative approach to calculating effective cross-sectional properties and reshaping conventional methodologies in structural analysis and design.

Determine the strength utilization of a double symmetric welded I cross-section with 384×4 web and 300×8 flanges, if the internal forces on the cross-section are NEd=500kN compressive force and My,Ed=100kNm bending moment. The material grade of the cross section is S235.

Calculation of cross-sectional properties

First, take the cross-section data (symmetric welded I-section), from which the Consteel software generates the EPS (and GSS) (see Online Manual/10.1.1 The EPS model) cross-section model (Fig. 1).

Fig. 1 Cross-section data model
Fig. 1 Cross-section data model

If the cross-section is class 4, the effective model is determined by the assumed normal stress distribution. According to EC3-1-1, the EPS model of a class 4 cross-section can be defined in two ways :

– method A: based on pure stress conditions,

– method B: based on complex stress condition.

In order to compare the results, the cross-section properties will be calculated by method A at first and then by method B.

Cross-sectional properties by method A

Aeff=4720mm2   

Fig. 2 Effective properties due to pure compressive force N
Fig. 2 Effective properties due to pure compressive force N

Weff,y,min=864080mm3

ez=14.5mm 

Fig. 3 Effective properties due to pure bending moment My
Fig. 3 Effective properties due to pure bending moment My

Cross-sectional properties by method B

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Introduction

Are you wondering how a web opening would influence the lateral-torsional buckling resistance of your beam? Check it precisely with a Consteel Superbeam based analysis

It is often required to let services pass through the web of beams. In such cases the common solution is to provide the required number of opening in the webplate. Such an opening can have a circular or rectangular shape, depending on the amount, size and shape of pipes or ventilation or cable trays.

Beams must be designed to have the required against lateral-torsional buckling. The design procedure defined in Eurocode 3 is based on the evaluation of the critical bending moment value which provides the slenderness value, needed to calculate the reduction factor used for the design verification.

There is no analytical formula provided in the code for beams with web openings. Would the neglection of such cutouts cause a miscalculated and unsafe estimation of the critical moment value?

The following demonstration will be made with a 6 meters long simple supported floor beam with a welded section.

6 meters long simple supported floor beam with a welded section

Exposed to a linear load of 10 kN/m, the critical bending moment value of the solid web beam can be obtained by performing a Linear Buckling Analysis (LBA) with Consteel.

(LBA) with Consteel
Download model file

The obtained critical multiplier for the first buckling mode is 3.00 which means that the actually applied load intensity can be multiplied by 3.00 to reach the critical load level. The corresponding critical moment will have the value of Mcr = 3.0 * 47.18 = 141.54 kNm yielding a slenderness of 1.286 (Mpl,Rd = 234.20 kNm) and a lateral-torsional buckling resistance of 0.394 * 234.20 = 92.27 kNm. With this value the actual utilization ratio is at 51%.

How would this value change if a rectangular opening needs to be cut into the web of this beam?

Analytical formula for critical bending moment

By looking to the analytical formula (ENV 1993-1-1 F.4) to calculate the critical moment of double symmetric sections loaded at eccentric load application point it becomes obvious that the section properties having effect on the moment value are Iz, Iw and It.

ENV 1993-1-1 F.4 analytical formula

An opening in the web has no effect on the first two values and has very little effect on the last one. As it has been already shown in previous article, the presence of such an opening can have effect on the vertical deflection, but as long as the lateral stiffness of a beam is much lower than it’s strong axis stiffness, the vertical deflections can be neglected when the lateral-torsional buckling resistance is calculated. The usual linear buckling analysis (LBA) performed also by Consteel neglects the pre-buckling deformations.

Therefore one can expect that in general web openings can be disregarded when the critical moment value is calculated.

Analysis with Consteel Superbeam

Beam finite elements cannot natively consider the presence of web openings. In order to obtain the precise analysis result, it is possible to use shell finite elements. The new Superbeam functionality comes as a solution in such cases. Instead of using beam finite elements, let’s use shell elements!

Opening can be positioned easily along the web, either as an individual opening or as a group of openings placed equidistantly. The opening can be rectangular, circular or even hexagonal. Circular openings can be completed with an additional circular ring stiffener.

The rectangular opening for this example can be easily defined with this tool. As there is no need to provide any additional opening on the remaining part of the beam, only the first part which includes the opening will be modelled with shell elements and the rest can still be modelled with beam finite elements. Using this technique, the total degrees of freedom of the model can be kept as low as possible. When using Superbeam, the designer has the choice whether to use beam or shell finite elements, as appropriate.

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Modeling of tapered elements

Stability calculation of tapered members is always a difficult problem despite its popularity in steel hall construction.

Generally in analysis software for the stability analysis a segmented but uniform beam element method is used where a member with I or H cross section and with variable web depth is divided into n segments and the depth of each segment is taken equal to the real depth measured at the middle of the segment. The lengths of the segments were taken equal, except at both ends where additional shorter segments are added in order the better approximate the real depth of the elements to be modeled. Such model captures correctly the in-plane displacements, but cannot consider accurately the additional torsion coming from the axial stresses due to warping in the flanges which are not parallel with the reference line in case of tapered elements.

This simplification may cause incorrect results in calculating buckling modes involving torsional displacements like flexural-torsional buckling of columns or lateral-torsional buckling of beams especially in cases where the beam flanges have longer unbraced lengths.

Modelling web tapered elements with a sequence of segments of constant depth

Consteel analysis model for tapered members

In order to improve the accuracy of the stability analysis of structural models including tapered members Consteel uses a special tapered beam finite element. A basics of this unique finite element have already been published by other researchers however up to now Consteel is the only commercial software which has implemented this finite element for the buckling analysis.

The mentioned problems arising from the non-parallel flanges can be fixed by considering appropriate additional terms in the element stiffness matrix. The final stiffness matrix can be written as a sum of original stiffness matrix and the additional terms:  

Where KS is for the original stiffness matrix with uniform cross section and KT contains the additional terms valid for doublesymmetric and monosymmetric I and H cross sections. 

The additional terms in KT use the following special cross section parameters:

Where IflzT and IflzB are the intertias of the flange related to z axis (parallel to the web), for upper (T) and bottom (B) flanges, respectively,  aand aB is the distance between the centerline of upper and lower flange and the line parallel with the reference axis of the element and passing through the shear center of the middle cross section, as seen on the picture below in case of double symmetric I and H cross section.

Definition of aand ain case of double symmetric I and H sections

Additionally daT/dand daB/dmeans the angle between the upper and lower flanges and the line parallel with the reference line of the element and passing through the shear center of the middle cross section, respectively. As an approximation these can be expressed as: 

where aflT and aflB are the angles between the flanges and the element reference line, ẟshear is the angle between the lines passing through the centers of gravity and shear centers of the extreme cross sections of the elements.

Definition of aflT and aflB in case of double symmetric I and H sections

Comparison of results

This part shows some validation examples for the accuracy of the implemented new finite element compared to published numerical results and analysis by shell elements. The examples show the very high accuracy of this element even in the most challenging buckling cases where the segmented uniform beam element method yields some extent of inaccuracy.

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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.

Picture0: Studied Intermediate frame

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.

o   they have a fundamental period of vibration smaller than the followings

  4*Tc 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:

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 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:

Picture1: Bending moment using ELF method

This way Consteel can perform an automatic strength and stability verification for the seismic 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

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.

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.

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):

Picture4: Envelop bending moment diagram of maximum values, obtained with MRSA and CQC combination

The results of the strength verification are the following:

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 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:

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 (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 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.

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It is known that the lateral torsional buckling resistance of a simple supported beam can be significantly increased if its compressed flange is connected adequately to a trapezoidal decking.

Eurocode allows to consider the compressed flange as fully restrained laterally, if the following condition is met (EN 1993-1-1 BB2.):

If the condition is not met, Eurocode doesn’t provide too much guidance how the contribution of a less rigid sheeting can be considered.

The following graph shows the effect of shear stiffness on the ultimate load capacity of a 10 m long simple supported IPE purlin loaded at the level of the top flange. It is assumed that several of these purlins are uses with 3 meters of distance between their centers.

The blue line in the graph shows the ultimate load carried by this purlin, in function of the actual shear stiffness. Shear stiffness of the sheeting measured by “S” is shown on the horizonal axis and a reduction factor applied on the plastic moment resistances is shown on the vertical axis.

The graph has been built with the help of Consteel software, using 7DOF beam finite elements. Such beam elements are adequate to consider the presence of any shear restraint.

In case of S=0, the value is calculated without any contribution. By increasing the shear stiffness the line approaches the maximum value of 1, when no reduction of the plastic moment resistance due to lateral torsional buckling is required. In other words when lateral buckling checks can be disregarded.

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