## Introduction

During the lifetime of a steel structure changes often happen. These changes usually result an increase of loads acting on some of its elements which therefore may need to be strengthened.

Strengthening is usually done by welding additional steel plates to the existing members. In the case of I sections, usually, the flanges are reinforced to increase the bending moment capacity or the web is stiffened to avoid local buckling or crippling at support regions.

This paper will focus on the increase of bending moment capacity.

## Lateral-torsional buckling resistance

The usual practice is to either increase the compression flange thickness by adding additional plates to it, or by widening it with the help of angles, as can be seen in the pictures below.

Although these can be very efficient ways to increase the bending moment capacity of a beam, welding on site is a complex process and might require the temporary removal of structural or non-structural elements connected to the flange of the beam. Welding especially “above the head” is difficult, the quality of weld seam needs to be properly checked.

Bending moment capacity of a beam might be limited by lateral-torsional buckling. If the section is not sufficiently restrained laterally against torsion, its actual load-bearing capacity will be lower than the value which depends purely on its section resistance.

In such cases, if the LTB behaviour could be directly improved, there would be no need to strengthen its cross-section along its full length. Here comes the **Superbeam** as a possible help.

Additional lateral restraining elements are often difficult to be added, therefore this is often not an option.

If we look at what LTB resistance of an I section depends on, we can see, that if we don’t want to change its cross section along its full length, it depends on the value of the reduction factor responsible to consider lateral-torsional buckling χ_{LT}.

This reduction factor is calculated from the slenderness value of the beam, which needs to be improved (reduced) to result a lower, more favourable reduction factor.

Without changing the cross section, the only way to do this is by improving the critical moment value. Increasing this value can be made not only by changing the cross-section but also by changing the boundary conditions.

The value of parameters ‘k’ and ‘kw’ depend on the boundary conditions, where ‘k’ means a factor which depends on how the section is fixed against weak axis bending at its ends and ‘kw’ means a factor which depends on how the section is fixed against warping. Warping is the phenomenon when the upper and lower flange of an I section twist in opposite directions.

To change the end conditions is typically difficult, but a certain limitation of the twist of flanges relative to each other ie. preventing or limiting warping might be possible. Limitation of this twist can be obtained by connecting the flanges by an additional element which has non-zero torsional stiffness. This torsional stiffness will prevent the counter-rotation of the flanges and therefore the warping and allowing to consider a ‘kw’ value different than 1.0 in this formula.

Consteel supports several such strengthening profiles and can determine the torsional stiffness to be considered in preventing or limiting warping.

## Analysis with Consteel Superbeam

Let’s take the following case. We have a simple supported 5 m long beam loaded by a uniform load of 20 kN/m acting at the top flange, on top of its self weight, without any intermediate lateral support. Its section is a welded I profile, made of S235, 10 mm thick plates, flange width of 200 mm and total section depth of 320 mm.

As we can expect, in the case of such a large unbraced length, the bending moment resistance would be strongly limited by lateral-torsional buckling, and therefore we can expect that strengthening by the proposed method is viable.

The critical moment of this beam is obtained in Consteel using linear buckling analysis with 7DOF beam elements option of the Superbeam, which has found the critical multiplier of 2.88.

This results M_{cr} = 2.88*64.18=184,84 kNm and a slenderness λ of 1,036 and reduction factor of 0,519.

The final bending moment resistance is 103 kNm.

Let’s further assume that this resistance needs to be increased by 30% due to new requirements. Let’s see whether a successful strengthening without modifications of the cross-section would be possible.

gate## Introduction

Beam with welded I sections are often executed with slender webs. This is mainly due to the recognition that the main contributors to bending stiffness of a beam are the flanges. The web plate’s main role is to safely keep these flanges away from each other and carry the shear stresses which might be present. Significant weight saving can be achieved with the use of slender webs, but there are some aspects to take care about.

When slender web plates are exposed to longitudinal, uniform normal stresses, above a certain stress level its distribution will no longer remain uniform. A compressed region of a plate distant from its lateral supports may buckle in a direction perpendicular to the acting external normal stresses, causing a subsequent transfer of stresses from the affected region to other neighbouring regions remaining in their unbuckled position.

This buckling remains limited to a part of the plate keeping other parts intact and therefore is called as local buckling. Local buckling usually does not result an immediate collapse of the structure, due to possibility of the stresses to redistribute and often even a substantial amount of further load increases are possible.

The tendency of a compressed plate to suffer local buckling is characterized by its slenderness value defined by the following formula

where σ_{cr} is the critical stress level above of which the stress redistribution and local buckling starts to appear. A higher critical stress will result in lower slenderness value which indicates that the plate can carry higher compressive stresses without the initiation of local buckling.

## Analysis of cross-sections with beam finite elements

The well-known beam finite elements used by usual structural design software do not “see” the internal composition of the cross-section. During structural analysis the sections are represented by certain integrated cross-sections properties assuming the validity of several assumptions including the Bernoulli-Navier Hypothesis and the non-deformability of the cross-section. A local buckling of any of its internal plates would violate these assumptions making hard to create the equivalent cross-sections properties.

In the modern design practice followed by Eurocode the phenomenon of local buckling is handled by the use of effective section properties. Regions subject to possible local buckling of compressed plates of a cross-sections are “eliminated” and the section properties are calculated based on the remaining parts of the cross-sections.

Design verifications use these effective cross-section properties to calculate the resistance of cross-sections exposed compressive forces. When required by Eurocode, the effect of appearance of local buckling can also be reflected in a structural analysis using beam finite elements with the use of effective cross-section properties, instead of the original gross section properties. This is mainly required to prove serviceability criteria.

## Analysis of cross-sections with Consteel Superbeam

The Consteel Superbeam function makes possible to confirm directly the presence of local buckling using the same beam element based model, but using a mixed beam and shell finite element modelling and analysis approach. Using the Superbeam tool, complete structural members or parts of them can be alternatively 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.

Contrary to beam finite elements, modelling with shell finite elements doesn’t have the previously mentioned limitations. This approach can fully consider the shape and location of the cross-section’s internal components instead of the use of an integrated overall section property. When a linear buckling analysis (LBA) is performed, the critical stress multipliers corresponding to the actual stress distribution can be obtained. Additionally to the load multipliers, the corresponding buckling shapes are also available, giving direct indication on the location, shape and appearance of local buckling within the compressed parts of the cross-section.

The use of effective cross-section concept is very convenient but there might be cases when more insight view is desired. The following example gives an idea where the Superbeam function can be helpful.

## Demonstrative example

Let’s consider a 12 m long simple supported welded beam with the following parameters

The beam is laterally restrained at third points at the level of its upper flange. The beam is loaded with its self-weight plus a uniformly distributed load of 10 kN/m acting at the level of upper flange.

When the beam is analysed with 7DOF beam finite elements, one can obtain the critical load multiplier of 5.2 of the global buckling mode, which is lateral-torsional buckling (LTB) in this case.

The beam finite element cannot give any visible indication about possible local buckling in compressed plates of the cross-sections.

As the maximum bending moment occurs in the middle third of this beam, it seems enough to analyse this part mode deeply with the Superbeam function. An LBA with the mixed beam and shell model gives comparable critical multiplier of 5.22 with some numeric perturbances in the part modelled with shell elements.

gate**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.

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.

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 M_{cr} = 3.0 * 47.18 = 141.54 kNm yielding a slenderness of 1.286 (M_{pl,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 I_{z}, I_{w} and I_{t}.

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.

gate## Web openings and their deflection effect on beams

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 openings in the web plate. Such an opening can have a circular or rectangular shape, depending on the amount, size and shape of pipes or ventilation or cable trays.

If the structural engineer has the freedom to position these openings along the beam, where to place them? What would be its effect on the deflection of the beam?

The effect of such openings on the deflection is more important when the length of the opening along the beam is increased. As circular openings are made with equal length and depth, they are usually less critical than rectangular openings.

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

Exposed to a linear load of 10 kN/m, the deflection at mid-span of the solid web beam is 4.6 mm.

Let’s assume that a 250 mm deep rectangular opening with a length of 400 mm needs to be provided on the web, at a distance of 300 mm from the left support.

**Traditional analysis with beam finite elements**

Consteel 7DOF beam finite elements are very powerful, but cannot consider natively such opening. The usual approach is to build a Vierendeel-type of model, by using additional beam elements with a T shape section „above” and „below” the opening. These additional beam elements are defined eccentrically to the reference line of the solid-web beam.

Eccentricities can be easily defined in Consteel using both smart and traditional link elements.

The deflection with this refined model will be equal to 4.8 mm.

**Analysis with Consteel Superbeam** – use shell elements for more precise analysis results

gate
In our series we have shown in Part 1 and Part 2 how the spring stiffness „K” is determined for and edge and for an intermediate stiffener. In this part we will show how to proceed further.

The goal is to determine how efficiently can this stiffener support the connected compressed plate. To consider local buckling of the compressed plate the effective widths will be calculated. These widths can be either calculated for a plate supported at both ends or for a plate supported at one end only, using Table 4.1 and Table 4.2, respectively.

If a stiffener fulfills the minimum constructive requirements, we will first assume that it is rigid enough the act as a support. Based on this we work out the effective lengths for the connected compressed plates, assuming also that they are fully loaded up to the yield strength. Once we know the effective widths of the plate parts connected to the stiffener, we also know the load this stiffener is supposed to be able to carry without buckling, to justify the first assumption. Therefore, next we calculate the flexural buckling resistance of the stiffener. If we find out that it is lower than what was implicitly assumed previously, we go back to the first step, lower the stress to be compatible with the obtained buckling resistance of the stiffener (distortional buckling). This lower stress level may of course result different effective widths. An iterative procedure is started until the satisfactory result is found. After completion the final stiffener buckling resistance value will be incorporated into an equivalent effective thickness applicable for the stiffener and the connected effective plate parts.

In the above described iterative process the „K” value of the stiffener will be needed to determine the flexural buckling resistance of the stiffener. This resistance will be calculated using the help of a simple beam model, where the stabilization provided the remaining parts of the section is used as a continuous bedding of „K”. The cross section of the beam corresponds to the stiffener and connected plate parts and will be exposed to the actual stress level of the iteration, assumed as a simplification to be constant along the length. The length of the beam in this model is unknown at the moment, it should be equal to the half wavelength of the buckling shape which is expected to be around 1.5-3 times the height of the studied cold-formed section.

Distortional buckling will appear only if the length of the stiffener is much longer than this typical length range. As a further simplification Eurocode assumes infinite length as the possible worst case, because for this condition an analytic solution exists in the literature:

In this formula „K” is the spring value – shown in Part 1 and Part 2 – represents the spring stiffness value what the remaining section can provide to the stiffener. I_{s}and A_{s}are section properties of the corresponding stiffener together with connected effective widths of supported plates, namely area and inertia around an axis perpendicular to the expected direction of buckling of the stiffener. E is the Young modulus of the steel material.

The formula gives the critical stress level where elastic buckling would appear. Once this has been found, reduced slenderness is calculated, and the distortional buckling resistance can be obtained by using a special buckling curve given by (5.12a-c) formulas of EN 1993-1-3.

The validity of formula (5.15) can be easily demonstrated with a simple Consteel model.

Let’s assume the Z section from our 1st blog exposed to compression force and let’s assume that we have already calculated effective width values. Yield strength of the material is 235 MPa and coating thickness is 0.04 mm. We found that the effective width of the lip and flange are c_{eff}=13 mm and b_{eff}=29 mm, respectively.

The section properties can be calculated with Consteel

Distortional buckling – Further secret formulas of EN 1993-1-3. Check our article for the first secret formula if you hadn’t read it yet.

In our series we continue with the second „secret” formula of EN 1993-1-3.

This formula (5.11) is used when the ability of an intermediate stiffener to stabilize a compressed web plate is studied.

When the intermediate stiffener is rigid enough, it divides efficiently the longer web plate into shorter plates. During buckling a rigid stiffener will not displace and the plate will buckle between the ends (stabilized by flanges) and this stiffener:

If the stiffener is not rigid enough, already at a much lower stress level the distortional buckling will appear which is characterized by a displacement of the stiffener and by the web plate buckling with a half-wave equal to the plate length:

In such a case Eurocode still accepts that web plate buckling will occur between ends and intermediate stiffener, but reduces the thickness of the intermediate stiffener zone, to compensate for the less than ideal rigidity. This compensation will depend on the buckling resistance of the stiffener when subject to compression.

The simplified model used by Eurocode in the procedure can be seen below. As a simplification it is assumed that the web plate is connected to the rest of the section with hinges. It means that it’s buckling shape is not influenced by the stiffness of connected parts, they do not encaster its ends, its ends can freely rotate. The ends of the plate are supported and there the plate can provide an elastic bedding to the stiffener, which will be considered as a „help” when its buckling resistance will be determined. This bedding is represented by the K spring stiffness:

The spring value which represents in this model the bedding provided by the plate is calculated as the ratio of the displacement obtained from the application of a unit load at the center of gravity of the stiffneer. In this case the unit load is applied perpendicularly to the web plate. Formula (5.11) gives the deflection (δ) from such unit load (*u*).

## 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 – *e _{0} = 24 mm* by the 5.3.2(3) b) Table 5.1 (L/250) and

*e*by the 5.3.2 (11) (same as in Part 1).

_{0}= 13,4 mm## 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

*contains the additional terms valid for doublesymmetric and monosymmetric I and H cross sections.*

**K**_{T}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 (

*) flanges, respectively*

_{B}*, a*and a

_{T }*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.*

_{B}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.

gateEN 1993-1-3 contains 3 „secret” formulas. The first two are used to determine the effective cross section due to distortional buckling when edge or intermediate stiffeners are involved. The third is used to calculate the distortion of the whole cross section when analyzed with a connected sheeting.

The physical meaning of all three formulas can be easily shown with simple ConSteel models which helps designer to understand the underlaying mechanial model.

The first formula (5.10) is used when the ability of an edge stiffener to stabilize a compressed flange of Z or C section is studied. During distortional buckling the intersection point of flange with the lip (it is called as edge stiffener) is expected to move in a direction perpendicular to the flange. This formula gives the stiffness value provided by the Z or C section, when is assumed that during deformations the point of intersection of the web with the flange doesn’t move. This assumption corresponds to attaching supports to these nodes as seen on picture 5.6 of EN 1993-1-3.

When a compressed edge stiffener would buckle, it will be partially restrained by the section with these attached supports. Depending on the distribution of normal stresses on the section, one or two edge stiffeners might be under compression. If both stiffeners are under compression and tend to buckle, the restraining capacity of the section will have to be shared between them. This sharing requirement is reflected by the coefficent k_{f}. The spring stiffness value will be used as a distributed spring support when the buckling resistance of the edge stiffener is calculated.

Stiffness values are typically calculated as the ratio of a displacements obtained from the application of a unit load. In this case the unit loads are applied parallelly to the expected displacement of the compressed edge stiffeners.

The ConSteel model shown below reproduces the stiffness calculation for a 1 m long Zee section (as a simplification the unit loads are placed at the intersection of the flange with the lip and not at the center of the gravity of the edge zone):

Z purlin, nominal thickness = 1.30 mm, 200 mm deep, 72 mm wide symmetric flanges, 15.5 mm deep lips

in case of M_{y} bending: k_{f} = 0, b_{1} = b_{2} = 72 mm, h_{w} = 198.7 mm, t = 1.26 mm

K_{1} = 210000*1.26^3/4*(1-0.3^2)*1/(72^2*198.7+72^3+0.5*72*72*198.7*1.0) = 0.08 N/mm^{2}

The resulting vertical displacement from the point load is 12.9 mm.

gateIn 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.

## 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:

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:

This way Consteel can perform an automatic strength and stability verification for the seismic combinations. The results are visible here, respectively:

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.

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):

The results of the strength verification are the following:

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:

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

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

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

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:

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 (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 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.