Comparison of chosen methods for estimation of critical lateral torsional buckling bending moment of web-tapered I-beams. In this article, the elastic critical bending moments of the web-tapered I-beams calculated by the analytical and numerical solutions developed last years by researchers involved in the topic were compared with own calculations carried out with available common tools. The main goal was to verify the accuracy and convergence of the results provided by different modern methods and different finite bar elements 1D with 7 degrees od freedom at the node (7DOF).

Click the button bellow to download and read the full article. (PL)

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Critical temperature calculation in Consteel

The calculation of the critical temperature is available in Consteel since the release of version 14. As an introduction of this feature, we prepared a video that gives some theoretical background on the topic, and demonstrates its usage in Consteel. It is shown how to prepare the model, how to execute the analysis and design, and how to create documentation about the critical temperature results.

Check out our user guide to learn more!

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Consteel 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

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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).

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This article aims to cover the theoretical background of the shear field stiffness determination methods implemented in Consteel. Modeling with the shear field stiffness based method will also be compared with shell modeling of trapezoidal deckings in Consteel.

Theoretical background

Modeling the shear stiffness of trapezoidal deckings is used to utilize their contribution to stabilizing the main structure. The possibility to consider the shear stiffness of sheetings is implemented at finite element level in Consteel and ensures easy modeling through its application onto beam elements.

Shear panel definitions

For the discussion let’s establish some basic definitions regarding shear panels.

• Dimensions:
  • L [m]: width of the shear field, also the span of the stabilized beam
  • L_s [m]: complete length of the shear field parallel to the ribs
  • a [m]: effective shear field length for only one connecting beam element
• Stiffnesses:
  • G_s [kN/m]: specific shear stiffness considering a 1 m long strip of an “L_s“ long shear field
  • S [kN]: shear stiffness of the complete shear field

Determaination of the shear field stiffness

The general formula to calculate the shear stiffness in Consteel is the following:

There are 4 methods implemented in Consteel to determine the shear field stiffness:

1. Schardt/Strehl method: (K1, K2), DIN 18807-1:1987-06 [1]

2. improved Schardt/Strehl method: (K1, K2, K1*, K2* and e_L)

3. Bryan/Davies method: (K1, K2, K1*, K2*, e_L, α₁, α₂, α₃, α₄)

4. Eurocode 3: (1993-1-3 10.1.1 (10)) [2]

See in more detail here: Determination of shear field stiffness and application in Consteel

The first 3 methods are based on the 1. one, the Schardt/Strehl method. These methods operate on the same principle by calculating the shear stiffness from values (K₁, K₂, etc.) provided by the manufacturer of the sheetings. The 2. and 3. methods are more developed versions of the 1. one, trying to more accurately calculate the shear stiffness by introducing additional parameters to account for more sources of the overall shear stiffness of the sheeting.

The 4. method that is found in Eurocode 3 can be more generally applied since that doesn’t require such product specific values.

A basic assumption in case of these methods is that the sheeting is connected at every rib to the beams that it stabilizes (purlins in most cases). An additional modifying factor in case of all these methods is if the trapezoidal sheeting is not fixed at every rib, but every second rib, then the final “S” shear stiffness should be substituted by 0,2*S.

Theoretical background of the Schardt/Strehl method

This approach is based on a model assuming a fully linear elastic behaviour of the diaphragm. The ultimate limit state is therefore defined by yielding in the corner radius at the flange-web transition. The mechanical model also assumes that the sheeting is fixed to the substructure at all 4 edges. Shear forces R_Q and R_L are acting on the sheeting at the individual fixed points on the lower flanges where the sheeting is screwed to the substructure. The number of waves in the sheeting is assumed to be large enough so that the individual forces acting at the transverse edges in the middle are assumed to be constant (n>10). The length “L_s” of the shear field can be arbitrary, but should be in reasonable proportion to the width “L” of the shear field (<4).

Based on these assumptions the mechanical analysis can be isolated to a half of one wave of the sheeting as shown on the left-hand side of the following figure:

On the right-hand side the considered internal forces are shown for one slice of the sheeting.

Assumptions for the mechanical model:

• The M_s and M_z moments are neglected (shown in brackets on the right-hand side figure).
• Transverse bending moments “m_i” at the level of the plate are considered. These moments have a 0 value in the center of the lower and upper flanges.
• The longitudinal stresses σ_z are constant over the thickness “t_i” of the plates and linearly distributed over the height “h_i”.
• The flexibility of connections is neglected.

The method accounts for the following effects:

• Shear deformation: corresponding value: K₁ [m/kN] shows the sheetings compliance coming from shear deformation. The lower this value is, the more stiffness the sheeting has.
• Warping deformation: corresponding value: K₂ [m²/kN] shows the sheetings compliance coming from warping deformation. The lower this value is, the more stiffness the sheeting has.

Component considering shear deformation

The value K₁ can be calculated from the following formula based on the properties of the trapezoidal sheeting:

where

• ∑l [mm]: Summed up length of all the plates within one full wave
• br [mm]: length of one complete wave
• G [N/mm²]: shear modulus
• t_core [mm]: structural thickness. (generally: t_core = t_nominal – 0,04 mm)

The formula of the K₁ value is similar to how it should be calculated in case of a planar plate, but its thickness corrected with the ∑l/br ratio, or in other words the ratio of the complete length of the plates to the length of one wave. The K₁ shear deformation compliance value is directly proportional to the ∑l/br ratio, therefore if a certain trapezoidal sheetings height is increased with everything else left the same, the corresponding K₁ value would increase, and the stiffness coming from shear deformation would decrease. On the other hand the K₁ value is inversely proportional to “G” shear modulus and “t_core“ structural thickness, so if either of these values would increase, K₁ would decrease, and the stiffness coming from shear deformation would increase.

Component considering warping deformation

The K₂ parameter further softens the structure taking into account the warping deformations. The detailed calculation of the K₂ parameter will not be shown here due to its extensiveness and complexity. The details of this calculation can be found in the literature [5]. The calculation is based on the “Folded Plate” theory [8]. To obtain the K₂ parameter the warping displacements and warping coordinates have to be calculated for the sheeting. The following figure shows an example for the normalized warping displacements and deformed shapes for k=1 and k=2:

k=1 and k=2 are connected to individual solutions for the differential equation system describing the mechanical behavior based on the “Folded Plate” theory.

Calculating shear stiffness

The behavior of the two components in the formula of the shear stiffness is different. The part that considers the shear deformation only depends on the effective width “a”, but independent from the total length of the shear field “L_s”. On the other hand the part that considers the warping deformation is also dependent on the total length of the shear field “L_s”. The K₂ parameter in the denominator is divided by L_s, which means that the longer the total shear field is, the larger the specific shear stiffness is going to get because of the contribution of the warping deformations.

Also if the total length of the shear field “L_s“ would get really low, then K₂/L_s would approach infinity, which means that the stiffness approaches zero. For this reason a minimal length for sheetings “L_s,min” is also provided by the manufacturers, which gives the specific shear stiffness “G_s” a minimum value.

Comparison against shell models

The effect of the shear stiffness of a trapezoidal sheeting can be modeled in multiple ways. In Consteel additionally to built-in shear field field object applicable on beam elements, the sheeting can be modeled directly with shell elements. This latter approach is more complicated and time consuming to set up, but should provide similar results. Such a comparison was prepared in Consteel.

Examined structure

Stabilized beam: IPE300 S235

Span: L = 4140 mm

Type of trapezoidal sheeting: Hoesch T 35.1

• Examined thicknesses: 0,75 mm; 1 mm; 1,25 mm
• Examined sheeting lengths: 2 m, 3 m

Horizontal line load: q_y = 10 kN/m

The load and the trapezoidal sheeting are both acting on the centerline of the stabilized beam.

Consteel shear field model

In this modeling version the stabilizing effect of the trapezoidal sheeting is modeled by the shear field object implemented into Consteel.

Example figure: L_s = 2 m, a = 2 m, G_s = 3293 kN/m, S = 6586 kN

Consteel shell model

In this modeling version, the trapezoidal sheeting is modeled by shell elements. The thicknesses of the shell elements are equal to the structural thickness t_core. The model of the trapezoidal sheeting is included in a frame made from beam elements. The sheeting is connected to the frame by link elements at the lower flanges where the sheeting is screwed to the substructure. The frame is included in the model in order to connect the shell elements to the main beam. The beams of the frame have a cross-section that has relatively insignificant weaker axis inertia compared to the shear stiffness of the sheeting. The shell elements are connected through the frame to the main beam by link elements that only transfer force in their axial direction.

Example figure: L_s = 2 m, a = 2 m

The shell elements are also supported in the vertical direction along the middle lines of its top and bottom flanges in order to eliminate the bending deformation resulting from the eccentric compression load on the sheeting, since the shear field model also does not take this effect into account. The edges on both sides of the sheeting are supported against “x” and “z” directional displacements in order to account for the sheeting being fixed to the substructure at all 4 of its edges. The line supports on the plate elements are shown on the following picture viewing the structure from below.

Horizontal displacement examination

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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.

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

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

Where I_flzT and I_flzB are the intertias of the flange related to z axis (parallel to the web), for upper (_T) and bottom (_B) flanges, respectively,  a_T and a_B 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.

Additionally d_aT/d_x and d_aB/d_x means 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 a_flT and a_flB are the angles between the flanges and the element reference line, ẟ_shear is the angle between the lines passing through the centers of gravity and shear centers of the extreme cross sections of the elements.

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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Introduction

As you may already know, you can check the max, min and min-max envelope diagrams for (first and second order) analysis results in Consteel. But you can also create your own envelope figures…

How it works

By default the envelope figures can be requested for ULS and SLS combinations. These default options will use the results of all of the calculated ULS or SLS load combinations:

To create your own envelope figures:

If you choose the Define envelope figure… option from the load combinations dropdown menu of the analysis tab, you will get the User defined envelopedialog:

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Part 1: Buckling utilization differences

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.

This article reviews how these two methods relate to each other in terms of the final member utilization. For the sake of simplicity, we consider only members subjected to pure compression, undergoing flexural-buckling. For case (1) the chapter 6.3.1 is used while for case (2) the imperfections are considered to take the shape of a buckling mode and the corresponding chapter is the 5.3.2 (11).

It is an obvious expectation that these two standard procedures should yield the same utilization for the same problem. However, this is by far not the case in general.

Let’s see the following simple example of 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 standard amplitude for the buckling mode based imperfection is calculated by Eq. 5.9-5.11, that is equal to v₀ = 13.4 mm. The next figure shows the model, the buckling mode shape – which is a classic flexural buckling about the weak axis – and the second order weak axis bending moment distribution.

• Important to note that the verification based on equivalent geometrical imperfection should be calculated from the results of a second order analysis using the linear cross section check defined by Eq. 6.2 (or Eq. 6.1 for elastic cases).

The most utilized cross-section will be the middle one, where the second order bending moment value depends naturally on the level of compressive force according to the well-known amplification relationship:

$$M_{zII}=N_{c,Ed}*v_0*\frac1{1-\frac{N_{cr}}{N_{c,Ed}}}$$

where is the applied compressive force, N_c,Ed is the amplitude of the equivalent geometrical imperfection and 1/(1-(N_cr/N_c,Ed)) is the amplification factor depending on the elastic critical load N_cr = 347,6 kN (Euler load). The utilization of this critical cross-section can be calculated according to Eq. (6.2):

$$U^{IMP}_{max}=\frac{N_{c,Ed}}{N_{Rd}} +\frac{M_{z,II}}{M_{z,Rd}} =N_{c,Ed} \Bigg[\frac{1}{N_{Rd}} +\frac{1}{M_{z,Rd}} *v_0*\frac1{1-\frac{N_{cr}}{N_{c,Ed}}}\Bigg]$$

Figure 2. shows the relationship between the applied compressive force and the second order bending moment and utilization of the middle cross-section where N_Rd = 1264,6 kN and M_z,Rd = 29,28 kNm. It can be clearly seen that the utilization is nonlinearly depends on the compressive force level due to the nonlinearity of the second order bending moment. The utilization corresponding to the 100% value gives the buckling resistance of the column:

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Introduction

In case of a model with a lot of load cases (wind in different directions, with and without internal pressure, snow,seismic etc.), hundreds of load combinations can be generated acc. to EC but many of these combinations are irrelevant. Running an Analysis on all of the load combinations can take a lot of time, that could be saved, if the relevant load combinations are calculated only.

How to do that

1. To find the relevant load combinations and save calculation time: At first perform only elastic first order analysis and global checks based on that. Even first order calculation can take much time if there are non-linear elements (practically tension-only rods) in the model. It is possible to temporarily change the non-linear elements to linear ones in order to let Consteel calculate the load combination analysis results with superposition of the load cases which is much quicker than calculation with non-linear elements.

2. Under the Global checks tab, on the results table, selection of the load combinations with significant utilization can be done. (As on all tables in Consteel, multiple selection of different load combinations can be performed easily using Ctrl+select, or Shift+select features of Windows) With a right click on the selected combinations, use Select only these combination for the Analysis function to apply your choice to the next run.

GATE

The main objective of the research paper is to present the technical and economical results obtained for standardized structures with small and medium spans. The obtained results can represent a starting point for designers as well as for investors, who intend to build single storey steel structures. According to the research, there have been identified a series of interrelated factors which may represent sources of savings in the optimization process. The authors highlight the work strategy in creating standardized structural systems that will improve product performances. The paper analyzes three standardized configurations including the following structural solutions: frame structures made of hot rolled beams and columns, thin-walled frame structures and structures using trusses and rectangular hollow sections columns. All configurations have been analyzed using Consteel 7.0 design software, for Bucharest region, loads evaluation being performed according to the current standards. Free height of the building varies between 4.00m and 6.00m, bay between
4.00m and 5.00m, and the span between 8.00m and 12.00m. The article presents the principle of the structural configurations and gives reference charts in order to estimate the steel consumption per square meter, aiming structural performance in what concerns price per square meter and execution time.

Click the button bellow to download and read the full article. (ROU)

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