Effect of shear restraint of the lateral torsional buckling resistance

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 the 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. 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. The line reaches the maximum value of 1, which is the ultimate load of this purlin, when lateral buckling checks are disregarded.

The required value of S, using the above formula is 16900 kN. If a typical roof sheeting of 35 mm depth is used and connecting screws are used at every rib, this condition can be fulfilled.

But if the screws are placed at every second rib only, the S value provided by the sheeting must be reduced by a factor of 5. The S values reduced by this factor of 5 would come to the range of 3000-7000 kN, being considerable below the required minimum S value, to disregard lateral torsional stability checks. These reduced S values are marked with vertical lines (yellow, orange and gray) on the following graph.

The very last vertical light blue line is placed at the required minimum S value of 16900 kN.

It is very well visible, that shear stiffnesses well below this minimum limit value can still provide almost 100% restraining effect, even if the screws are used at every second rib. In case of the thinner sheeting there appears although already a reduction of the ultimate load.

This picture shows the buckling shape of this simple supported purlin, under the presence of a shear restraint well below the minimum limit required by Eurocode.

There are not only screwed-down roofs used in the practice. On roofs on larger building often floating roofs are used. Such roofs can slide within certain limits independently from the sub-structure, allowing large areas to be built without interruption and dilatation which is very beneficial from water tightness point of view. In case of such roof systems, the lateral stiffness is much lower. Is there are test values available, they can be considered for design, but in such case evidently a full lateral restraint cannot be assumed and the lateral torsional buckling check must be performed by considering the actual values provided by the sheeting.

With the help of the mentioned 7DOF beam finite elements, these values can be considered in the analysis and in the forthcoming design step, built on the results of analysis. As visible from the graph, even a slightest actual shear stiffness can provide an increase of the ultimate load.

There might be also sheeting produced from aluminium, which can provide considerably lower stiffness than sheeting produced by steel. Or at parts there might be translucent panels made of plastic which is normally not considered as a load bearing component. But the analysis and design must be inline with the real structure, at such locations no restraint should be considered. This can also be correctly considered in the proposed analysis and design approach.

Seismic design of frames of single-story industrial building with built-in mezzanine floors according to Eurocode 8 with ConSteel

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 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.

• (1)P. this method may be applied to buildings whose response is not significantly affected by contributions from modes of vibration higher than the fundamental mode in each principal direction
• (2) the requirement in (1)P is deemed to be satisfied in buildings which fulfill both of the following conditions

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

§  4*T_c or 2.0 sec

o   they meet the criteria for regularity in elevation given in 4.2.3.3

When a dynamic analysis is performed on this 2D frame, the following vibration modes are obtained:

Mode	Period of vibration (sec)
1	1.107
2	0.335
3	0.234
4	0.225
5	0.192
6	0.131

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 this second criteria and accept ELF method first.

When the ELF method is applied, only the first (fundamental) mode is used, with the total seismic mass of the building. As the seismic effect is described with one single vibration mode only, the representation of the seismic effect is a simple equivalent load case. 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 (EQU) 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 be 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 trace of the seismic action. Plus, they are not even in equilibrium…. And even the sign 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, no linear buckling analysis can be done and therefore the automatic buckling feature of ConSteel 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. As visible, the fundamental mode has a relatively low contribution (77%) which justifies the initial doubts about only using this single mode and disregard all the others. To fulfill the 90% minimum criteria, the second mode must be also considered, but visible even the 4th and the 6th have non-zero (although less then 5%) contribution.

As said before ConSteel can perform only strength verifications but not stability verifications based on results of 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?

Seismic modal analysis with “selected modes” – ConSteel approach

Luckily ConSteel provides a very flexible approach, called as „selected modes” method. This allows the user to pick the vibration modes by himself/herself 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 fulfill the minimum 90% mass contribution requirement, let us see the additionally also the 4th mode:

1^st mode f=0.90 Hz, T=1.109 sec

Picture6: 1^st vibration mode

 

2^nd mode f=3.00 Hz, T=0.334 sec

Picture7: 2^nd vibration mode

4^th mode f=4.265 Hz, T=0.234 sec

Picture8: 4^th 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 2^nd or 4^th is dominant for the mezzanine structure.

The corresponding bending moment diagrams are, respectively:

Picture9: Bending moment diagram calculated from the 1^st vibration mode

Picture10: Bending moment diagram calculated from the 2^nd vibration mode

Picture11: Bending moment diagram calculated from the 4^th vibration mode

These bending moments also justify the assumption made based on the colors, the 2^nd mode creates significant bending moments additionally to the first mode and the 4^th mode creates significant bending moments additionally to the 1^st mode. But it seems that also the 2^nd mode created significant bending moments at this region.

It is interesting to note, that the bending moment diagram from the 1^st mode (picture 9) almost perfectly fits to the CQC summarized bending moment (or course by assigning signs to the values based on the fundamental vibration mode) (see picture 4), except in the regions of the platform and the mezzanine. This means that in general the fundamental vibration modes describes quite well the dynamic response of this frame. And because of this, the bending moments could be calculated with the mass contribution factor corresponding to this mode (77%). And this is the reason, why the ELF method gives higher bending moment values, as there the same vibration mode was considered, but instead of the corresponding mass (77%), with 100% of the seismic mass.

As we discovered, the 2^nd mode should be used together with the 1^st mode to correctly describe the platform region, as this region is not fully dominated by the 1^st mode only, the 2^nd has a significant contribution.

Similarly to the mezzanine region, additionally to the 1^st mode, here the 4^th mode must be used to better approach the correct result.

The definition of the weighting factors could be done by the following – a bit arbitrary – way. Let us take the reference the MRSA CQC values at selected points of the structure and create corresponding rules for the linear combination to well approach the value obtained with the CQC combination, considered as reference value

Platform region

CQC value           70.21 kNm

1st mode            61.38 kNm          * 1.00    = 61.38 kNm

2nd mode           -33.29 kNm        * -0.265 = 8.82 kNm

Picture12: MRSA CQC Bending moment diagram considered as reference for the platform region

Mezzanine region (internal column)

CQC value           26.79 kNm

1st mode            11.74 kNm          * 1.00   = 11.74 kNm

4th mode            14.39 kNm          * 1.045 = 15.037 kNm

or (sidewall column)

CQC value           287.29 kNm

1st mode            272.87 kNm       * 1.00  = 251.55 kNm

2nd mode           89.10 kNm          * 0.16 = 35.74 kNm

Picture13: MRSA CQC Bending moment diagram considered as reference for the mezzanine region

 

As a summary the following 4 linear mode combinations could be set

For the frame in general

1: Mode 1 * 1.00

For the platform region

2: Mode 1 * 1.00 + Mode 2 * -0.265

For the mezzanine region

3: Mode 1 * 1.00   + Mode 4 * 1.045

4: Mode 1 * 1.00 + Mode 2 * 0.16

Of course, other weighting factors could be also set, as the condition we set was to meet the target value. The more target values we define in the region, we can more precisely set the factors. Usually it is recommended to keep the factor of the fundamental mode as 1.00 (or close to 1.00) and adjust the other factors for the modes appearing in the given mode linear combination as necessary.

With these 4 linear mode combinations we can already perform the automatic stability verifications. And the answers the original questions.

Utilization ratio at the left corner with stability verification included: strength 79.2%, stability 102.1% compared to the ELF results of strength 97.9% and stability 128.2%. So, the use of the ELF method was safe, the results can be accepted, the structure works for strength verifications but shows a small overstress regarding stability verification.

Leg of the platform: There was already a strength problem based on MRSA CQC results, therefore the post must be strengthened. The result of the stability verification with the fine-tuned seismic force is 106%.

Conclusion

This post wanted to call the attention of performing stability verifications for seismic combinations as well, like for any other combinations. For structures, where the ELF method is applicable, ConSteel can perform without problem these stability verifications automatically. Unfortunately for irregular structures the MRSA CQC method does not give directly a possibility. The special method implemented in ConSteel called „selected modes” can be successfully used to create loads, with the help of a linear combination with modes important for parts of the structures and with the resulted loads the stability verifications can already be executed.

A sneak peek into getting ready for the Consteel 14 launch

These days everyone is getting prepared for something: the children are waiting for the summer break, university students are preparing for their last exams, the rest is waiting for re-openings.

We at Consteel, we are busy with preparing Consteel 14, along with Pangolin (Consteel’s new Grasshopper plugin) while still working from our homes.

We did a small interview with a few of our colleagues on the getting-ready phase.

How do you feel about Consteel 14 on a scale of 1-14, where 1= Meh  14= I can’t wait for the users to see it!   ?

The team’s answers average is 11, which means we are pretty close to the finish!  

Let's see what our colleagues, working in different areas said.

Software Engineer

What are you responsible for, regarding the new version?

• Pangolin, and the new Managed API underpinning it.

Pick something that you did (and is related to Consteel 14) and are especially proud of?

• The Managed API, which will make programming ConSteel models from the outside possible in the ubiquitous .Net framework.

What are you working on right now?

• Refinements of Pangolin.

Can you remember the hardest part during the development of something for Consteel 14 (if you want to / can name such)?

• Solving various conflicts between Consteel's and Grasshopper's design philosophy in a non-confusing way.

Can you remember the greatest moment during the development of something for Consteel 14 (if you can/would like to, you can explain it a bit)?

• Calculating the first 100% Pangolin created model.

How far away in time does the launch feel like?

• Minutes.

How do your workdays look like? (Are you getting busier and busier each day or is it constantly easy/hard)?

• Busier and busier as every change affects more and more.

Recommendation for the users to check out first after downloading Consteel 14?

• Pangolin.

Anything else, that you would like to share about getting prepared for the new version launch?

• Pangolin is immense but unrefined. The sooner and more you use it, the more suggestions and bug reports you to give us, the faster will it get better.

Development Engineer

What are you responsible for, regarding the new version?

• Engineering background and functional planning for critical temperature calculation and linearly distributed surface load. coordination of the development of the new features I have tested the new features of ConSteel and created verification examples related to the new functionalities.

Pick something that you did (and is related to Consteel 14) and are especially proud of?

• I think the intumescent paint design function is pretty thorough. It is unique that we calculate critical temperatures for every finite element. Also, I think it's quite reassuring that we automatically check back the critical temperature with the design utilization. It was a team effort, so I can't take all the credit, but I think it turned out great.

What are you working on right now?

• Testing of design calculation improvements.

How far away in time does the launch feel like?

• Feels like it's right around the corner compared to the months of development.

How do your workdays look like? (Are you getting busier and busier each day or is it constantly easy/hard)?

• Constantly busy.

Software Engineer

What are you responsible for, regarding the new version?

• Coordination of the development of the new features I have tested the new features of ConSteel and created verification examples related to the new functionalities.

Pick something that you did (and is related to Consteel 14) and are especially proud of?

• Load distribution of the linear surface load, calculation of the critical temperature.

What are you working on right now?

• On lots of bugfixes.

Can you remember the hardest part during the development of something for Consteel 14 (if you want to / can name such)?

• Correcting the wrong codes of someone else.

How far away in time does the launch feel like?

• Tomorrow.

Recommendation for the users to check out first after downloading Consteel 14?

• Surface load.

Development Engineer

What are you responsible for, regarding the new version?

• I have tested the new features of ConSteel and created verification examples related to the new functionalities.

Pick something that you did (and is related to Consteel 14) and are especially proud of?

• I have written the system design document of the Beam - Beam Link Element.

What are you working on right now?

• I am working on the verification of the effective section properties of the cold-formed sections.

How do your workdays look like? (Are you getting busier and busier each day or is it constantly easy/hard)?

• Sometimes it is very easy, but there are days when I am very busy.

Software Engineer

What are you responsible for, regarding the new version?

• Developing the smart link / beam-beam link feature, testing other new features, writing descript code.

Pick something that you did (and is related to Consteel 14) and are especially proud of?

• Smart link / beam-beam link.

What are you working on right now?

• Fixes and testing.

Can you remember the greatest moment during the development of something for Consteel 14 (if you can/would like to, you can explain it a bit)?

• When the engineering team is vocally satisfied.

How far away in time does the launch feel like?

• Too close :)

How do your workdays look like? (Are you getting busier and busier each day or is it constantly easy/hard)?

• It's definitely getting busier

Recommendation for the users to check out first after downloading Consteel 14?

• Smart link / beam-beam link :)

UX/UI designer

What are you responsible for, regarding the new version?

• New website's design and some other design related material.

Pick something that you did (and is related to Consteel 14) and are especially proud of?

• Pangolin logo

What are you working on right now?

• Refreshing our product brochures.

Can you remember the hardest part during the development of something for Consteel 14 (if you want to / can name such)?

• Designing the new website.

Can you remember the greatest moment during the development of something for Consteel 14 (if you can/would like to, you can explain it a bit)?

• When I finished the logo for the Pangolin plugin.

How far away in time does the launch feel like?

• It's around the corner.

How do your workdays look like? (Are you getting busier and busier each day or is it constantly easy/hard)?

• It's like a roller coaster from January. But with deadlines approaching, it becomes busier.

Recommendation for the users to check out first after downloading Consteel 14?

• Register on our website. :)

As you can see, Consteel 14 will be out soon, and its development happens with multiple dedicated experts' input.

We can't wait for the launch, stay tuned to get notified when you can access Consteel 14!

Utilizing Pangolin in a "Consteel first workflow"

Pangolin not only creates Consteel models for you but can also read and utilize your existing Consteel models. This is what we will look into more detail in this post.

Let's say you already have a model built in Consteel:

You can either save this model as a .smadsteel file from Consteel, and load it in Grasshopper, or as in this case, use Pangolin’s live connection to read it while both programs run:

The imported objects can be used in the further Grasshopper definition just like any other object created with Pangolin’s components.

For example when you get a huge spreadsheet with a list of all the loads and their positions on a floor from a partner, or there is just a particular load distribution that is easier to define algorithmically than placing them all manually:

Then you can just send over the newly added objects to the same Consteel model:

Of course, this was a really simple example.

Now let's take a more complex Consteel model:

Even in such more complex cases, you can use the various features of Pangolin to extend the model. In this example, we defined a new layer for the members to have this load (denoted by the green layer style in the above picture).

In Grasshopper, you can ask for all the layers, pick the one you wish to filter, and select the beams that belong to that layer using the default functionalities of Grasshopper, thanks to Pangolin's integration with implicit data conversions: