Did you know that you can use Consteel to design a pre-engineered Metal Building with all its unique characteristics, including web-tapered welded members, the interaction of primary and secondary structural elements, flange braces, shear and rotational stabilization effect provided by wall and roof sheeting? 

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Did you know that in addition the standard Type 1 and Type 2 response spectrums defined by Eurocode 8, you can use also user-defined spectrums with Consteel? 

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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 bar	Direction of buckling	k
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 N_cr 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.

• Let the structural model of the lattice girder under consideration be the Consteel model shown in Figure 1.
• Let the load shown correspond to the design load combination of the girder.
• Determine the deflection length of the most stressed compressed chord member using finite element numerical stability analysis.

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 α_cr (Figure 2).

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 α_cr=5.99.

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Did you know that you can use Consteel to design simple supported, continuous and over-lapped purlins systems in Consteel, considering shear and rotational stiffness of attached roof sheeting? 

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If you haven’t tried Consteel yet, request a trial for free!

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Introduction

Reinforced concrete columns are essential structural elements in the construction industry. They are used, for example, in frame buildings, halls, family houses and bridges. They are used in both monolithic and prefabricated versions.

The designer aims to design safe and economical structures. As technology evolves, so do our building materials, and higher strength concrete can be produced at lower cost. As a result, the use of smaller cross-sections of columns can be advantageous.

As the columns become slenderer, stability issues and the calculation of second-order effects become more important. The ConSteel finite element program specializes in steel structures and therefore has fast and well automated solutions to stability problems.

Taking advantage of the existing features of the software, a new method for designing reinforced concrete columns, improved by ConSteel, has been made available in ConSteel version 16. It is based on the Nominal Curvature method described in Eurocode 5.8.8 [1].

To apply the Nominal Curvature Method, a lot of information is required, various material and geometric parameters. The purpose of this article is to show that the Nominal Curvature Method, as extended in ConSteel 16, answers all the questions that arise during design and is free of many of the shortcomings of the original method.

Overview of Eurocode 2 – desinging reinforced concrete columns

In this chapter, the design of reinforced concrete columns based on Eurocode 2, nominal curvature method, is presented in outline, focusing on the most important aspects. 

Material parameters

Partial safety factors:

• Modulus of elasticity of concrete
  𝛾_cE = 1.20
  
• Concrete
  𝛾_c = 1.50
  
• Steel reinforcement
  𝛾_s = 1.15

The material properties of concrete are dealt with in Eurocode 1992-1-1, Chapter 3.1.

Modulus of elasticity:

• Design value
  
  • 𝐸_cd = 𝐸_cm/γ_cE
    
  • Reducing the mean value with γ_cE partial safety factor
    
  • Applicable in ULS
    
  • In the case where creep is not considered, or considered elsewhere
    

Creep

The calculation of the creep coefficient is discussed in EN 1992-1-1, chapter 3.1.4. Here, various factors are used to determine the final value of the creep coefficient as a function of concrete strength, using diagrams. The values can also be determined according to Annex B of EN 1992-1-1. The two calculations give almost identical results.

Imperfections

The imperfections of concrete buildings are discussed in Eurocode 1992-1-1, Chapter 5.2. It divides the imperfections into two parts. One is the global inclination, which is shown in Figure 1(b). The other part is when the network points are not displaced but the elements in between are curved. This is the initial curvature (also known as a shape error), illustrated in part c) of Figure 1.

Inclination

The effect of imperfection due to inclination can be taken into account by calculating fictitious transverse forces (nominal loads). To do this, the value of the applied inclination is calculated as follows:

• Base value of inclination
  θ₀ = 1/200
  
• Height-dependent reduction factor
  α_h = 2/√𝐻
  
  where 𝐻 is the height
  
• reduction factor depending on the number of structural elements
  α_m = √0.5(1 + 1/𝑚)
  
  𝑚: number of vertical structural elements bearing the total load
  
• applied inclination
  θ_i = θ₀α_hα_m                   

Then, as shown in Figure 2, the 𝑁 the normal forces can be used to calculate the notional loads in the unbraced case: 𝐻_i = θ_i𝑁.

In braced case, for example a hinged-hinged column, 𝐻_i  force is not defined at the top of the column, but at the center, with the value: 𝐻_i = 2θ_i𝑁.

Second order effects

The method described in EN 1992-1-1, chapter 5.8.8, is applicable by default to isolated columns with constant cross-section and normal forces.

The design method uses the maximum second-order moment (𝑀₂). Its distribution along the length is not directly determined. For the sake of simplicity and to be conservative, it is usual to assume this second order bending moment to be uniform along the length, but the standard also permits a sinusoidal or parabolic distribution.

If we can realistically determine the distribution of curvature, the Eurocode allows the use of the method for global structures (EN 1992-1-1 5.8.5 (3)), but this is not usually possible for manual methods due to the interactions between the elements.

To use this method, it is essential to specify the buckling length, the value of the second order bending moment depends on it. For this purpose, the standard allows the use of the factors used in the elastic theory (for cantilever 𝑙₀ = 2𝐻, fix bottom – top hinged case 𝑙₀ = 0.7𝐻, etc.).

Calculation of design bending moment:

𝑀_Ed = 𝑀_0Ed + 𝑀₂

where

𝑀_0Ed is the 1^st order moment, including the effect of imperfections

𝑀₂ is the nominal 2^nd order moment (including the effect of any curvature)

Calculation of second order bending moment from curvature

Determine the nominal curvature first:

1/𝑟 = 𝐾_r𝐾_φ1/𝑟₀

where

• 𝐾_r is a correction factor depending on axial load
  
• 𝐾φ is a factor for taking account of creep
  
• 1/𝑟₀ is the theoretical (physical) curvature associated with failure

1/𝑟₀ = ε𝑦𝑑/0,45𝑑

The curvature belongs to the point where the concrete reaches its ultimate compressive strength ( ) and the tensioned reinforcement is starting to yield, i.e. the so-called “balanced” case.

The position on the design line is taken into account by the correction factor depending on axial load:

𝐾_r = (𝑛_u − 𝑛)/(𝑛_u − 𝑛_bal)

where

• 𝑛 = 𝑁_Ed / A_c𝑓_cd
  relative axial force
  
  • 𝑁_Ed
    design value of axial force
    
• 𝑛_u=1+ω
  • ω = A_s𝑓_yd / A_c𝑓_cd
    mechanical reinforcement ratio
    
  • A_s
    is the total area of reinforcement
    
  • Ac
    is the area of concrete cross-section
    
• 𝑛_bal =0.4
  value of n at maximum bending
  
• (0.4 applicable in the absence of further information)

Factor for taking account of creep:

𝐾_φ = 1 + βφ_ef ≥ 1

where

• φ_ef = φ(∞,0) 𝑀_0Eqp / 𝑀_0Ed
  effective creep
  
  • 𝑀_0Eqp first order quasi-permanent bending moment (SLS)
    
  • 𝑀_0Ed first order bending moment (ULS) – design combination
    
• β=0,35 + 𝑓_ck/200−λ/150
  
  • λ = 𝑙₀ / 𝑖
    slenderness
    
  • 𝑙₀
    buckling length
    
  • i = √𝐼_c/𝐴_c
    inertia radius of uncracked concrete

Second order bending moment

𝑀₂=𝑁_Ed𝑒₂

where

Second order eccentricity, where c is the factor depending on the curvature distribution. For constant cross-section 𝑐=𝜋² applicable (sinusoidal distribution). In case of constant distribution 𝑐=8 is applicable.

Design

Column Interaction Curve

According to the Interaction Curve, the failure of the cross-section always occurs, when the concrete reaches its ultimate strain (usually 𝜀_cd = 0,35%).

Depending on where we are on the interaction curve, the reinforcement on the other side:

• Tensioned, and yielding,
  
• Tensioned, and just started yielding,
  
• Tensioned, but elastic,
  
• Compressed and elastic.

Theoretical background of the development

The design procedure is an extension of the standard procedure described in Chapter 2. It automates the manual entry of the buckling lengths and defines the distribution of the second order bending moments.

The initial value of the curvature distribution on which the calculation is based is performed on the global structure and not on an isolated column. The curvature distribution is determined from the elastic buckling shapes calculated on the whole structure (Linear Buckling Analysis – LBA).

This can be considered a realistic curvature distribution for the concrete column because we calculate the buckling shape for the entire structure, taking into account the interaction of the structural elements.

This allows the method to be extended, so that the column can now be considered not only as an isolated element, but also as part of the whole structure, in accordance with the Eurocode (EN 1992-1-1 5.8.5 (3)).

The final step, the calculation of second order bending moments (𝑀₂), is performed on an isolated model in the spirit of the standard, but for this curvature it uses the values of the corresponding buckling shape along the column calculated on the full model.

The maximum value of the buckling shape for the curvature prescribed by the standard (1/𝑟) and the other values are varied in proportion.

The appropriate eigenvalue assignment is done by a procedure called buckling sensitivity analysis. Magnification is performed at the point of maximum curvature found along the length of the column.

With this method, we could theoretically calculate second-order moments for any structural element and the entire structure. However, at this stage of development, it is only considered for straight-axis beam elements defined as reinforced concrete columns.

Later on, if the need arises, the procedure can be developed into a general procedure after proper testing and verification. This could be a very useful feature for example for reinforced concrete arches or reinforced concrete frame columns with moment restraints.

Buckling sensitivity analysis

The main difficulty of the method is to find the right buckling shapes for the corresponding RC columns ins both direction (x and y). The program calculates a number of buckling shapes defined by the user.

Assuming each shape as a displaced shape, the summed deformation energy per bar element is calculated along each bar element of the structure.

The element with the highest deformation energy value calculated on the basis of the buckling length just tested is assigned a value of 100%, the other elements a proportional value. The buckling shape currently under consideration is assigned to the corresponding bar element.

Since a column can generally bend in both perpendicular directions, the test is performed in both local directions and 2 eigenvalues are assigned to each column (1-1 per direction).

Calculation of second order bending moments

According to Eurocode:

𝑀₂ = 𝑁_Ed𝑒₂

where 𝑐 = π₂

ConSteel calculates in a similar manner. Second order moments are calculated for each finite element of the reinforced concrete column. Three values are used. The first is the normal load at the finite element point (𝑁_Ed). The second is the second order eccentricity as defined in the Eurocode (𝑒₂).

After that, the third value is the ordinate of the buckling shape at the given finite element point, with the maximum of the buckling shape normalized to the unit value. Simply put, multiplying the first two values by this third value gives the second order moment at the given finite element point of the reinforced concrete column. This results in an improved moment distribution.

Differences compared to the standard procedure

Simplification of the calculation of the effective creep:

φ_ef = φ(∞,𝑡₀)

conservatively, we equate the effective creep with the final value of the creep factor, without reduction. This avoids errors such as, for example, if there is no bending moment in a quasi-permanent load combination, then the value of the effective creep factor is by definition zero.

Creep factor value in ConSteel

The values given in ConSteel are taken from Table 1 of the Eurocode guide for Reinforced Concrete Structures [6].

This is based on EN 1992-1-1, chapter 3.1.4, where various factors can be used to determine the final value of the creep factor as a function of concrete strength, using diagrams. 

Calculation of slenderness based on buckling analysis

Critical strength of the Euler beam. The formula rearranged:

where

There is no need to manually enter the buckling length, the slenderness calculation is fully automated.

Second order bending moment distribution

The distribution of the second order bending moment is the same as the buckling shape, taking into account the interaction between columns.

Demonstration of the method using a console example

You can see how to make the example model in our Reinforced Concrete Column – overview article.

Download the starter model at the end of this article and open the “separate_circle_column_cantilever.csm” file.

First order analysis

Displacement shape: realistic values:

• Slight vertical displacement
• in the direction where a greater horizontal load is applied, greater displacement
• slight displacement in the other direction due to the imperfection
• the displacement shape is curved in the direction of the horizontal load as expected

Check the internal forces:

N

• same as the vertical load, with constant distribution (no self-weight applied in the model)

negative sign -> compression

Vy+Mz

• we calculate the imperfection from the normal force
• 675*0,005 = 3,375 kN
• 3,38*3 = 10,14 kNm
• Distributions and valuer are as expected
• No Vy+Mz from applied loads, only from imperfection

Vz +My

• Imperfection calculated form the normal force
• 675*0,005 = 3,375 kN
• 3,38*3 = 10,14 kNm
• These internal forces are calculated from imperfection
• Additional 20*1,5 = 30 kN load
• 30+3,375 = 33,375 kNm
• 33,375*3 = 100,125 kNm
• Correct internal forces
• first order internal loads from applied loads + inclination

Buckling analysis & buckling sensitivity analysis

We can see from the coordinates, that this is indeed a planar buckling shape.

For a single column, it is rather easy, to check whether we have buckling shape in both directions. Here we only found one, so we need to find the other one as well.

Parameters of the buckling analysis need to be adjusted. We should increase the upper limit of the calculated number of buckling shapes.

Download the adjusted model at the end of this article and open the “separate_circle_column_cantilever_MoreBucklingShape.csm” file.

Now we have buckling shape in both x and y directions.

ULS design – EN 1992 condition

Everything is symmetrical.

GATE

When applying design rules in load combination filter, the most frequently used utilization type is Steel – Dominant results. What results are exactly considered by this option and what do corresponding limitations mean?

Introduction

There are four ways to apply load combination filter: based on limit states and load cases, manually, and by rules. Unlike the other three methods, filter by rules is only possible based on analysis and/or design results.

The most effective way to reduce the number of load combinations is definitely the use of design rules.

With design rules, load combinations can be selected based on utility ratios. Utilizations are available from several design checks, including dominant results and detailed verifications for steel elements, such as general elastic cross-section check, pure resistances, interactions, and global stability.

The meaning of the dominant check

The dominant check is not always the check which gives the maximal ratio but the one with the maximum RELEVANT ratio. Typical example: if plastic interaction formulas are valid, those results will be dominant over general elastic cross-section check results, although the latter are higher.

Steel – Dominant results

Steel – Dominant results option contains the utility ratio of the dominant check at every finite element node, in all load combinations. Meaning that there are as many utilization values as the number of load combinations calculated, in every FE node.

It is also important to understand the difference between the utilizations of Maximum of dominant results and Steel -Dominant results. Maximum of dominant results option contains the dominant utility ratio of the dominant load combination at every node, like an envelope of Steel-Dominant results. Meaning there is only one utilization value in every FE node. Also, it is the same as the dominant result table on Global checks tab.

When a rule is applied, the utilizations of the chosen utilization type are compared against the limitation. The load combinations which give the results that correspond to the limitation, are selected by the rule. Every FE node of the selected model portion is examined.

Limitations in case of Steel – Dominant results

• Maximum: to select the combinations which cause the maximum utilization at any node. It can be the same as Maximum of dominant results, except if there are combinations where the utilization is the same and it is maximal. In this case, here all the combinations are selected, while with Maximum of dominant results, there is always one maximum.
• More than % of maximum: to select the combinations as in ’Maximum’ plus those which cause utilization that is more than the given percentage of the maximum. E.g. at a certain point max utility ratio is 80%, Limitation= More than 90% of maximum. This rule will select all the load combinations which cause utility ratios between 0,9*80=72% and 80%.
• More than: to select the combinations which cause utilization more than the defined value at any point.

Let’s see an example of a simple 2D frame for better explanation. Right-side beam is in the portion for which three design rules were applied. Five points are selected for representation but of course all the nodes of the portion are examined against the rules’ limitations.

The utilizations of the five dedicated FE node in all 11 load combinations are shown on the below diagram. (To find all of these utilizations in the attached model, global checks must be calculated for the load combinations one-by-one.)

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Introduction

In ConSteel, there are three options for designing reinforced concrete columns: the Manual Nominal Curvature Method, the Automatic Nominal Curvature Method, and the Nominal Stiffness Method. 

Each method has its advantages and disadvantages and should be used in different situations. We will now briefly review these methods and show how they can be used. Example models and a flowchart guide is also available at the end of the overview.

You can find the related chapters within the Online Manual about how to access these features in the Structural design and Structural modeling chapters.

Summary table

The following table summarizes the most important information about the three methods. Click on the table to see it in full screen.

We will now illustrate the application of these methods with a few short examples.

Examples

Manual Nominal Curvature Method

Create section

Define structure without imperfections 

Define reinforcement

Define design parameters

First order analysis

Design

Automatic Nominal Curvature Method

Only the steps presented, which are different from the Manual Nominal Curvature Method.

Imperfections

Design parameters

First order analysis – with imperfections

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Consteel offers a range of load combination filtering options, which can be applied based on limit states, load cases, and analysis and design results. By applying different series of filters, designers can streamline their workflow and reduce calculation time.

Filtering options

Filtering is realized through the Load combination set definition window.

Filtering by limit states and by load cases are handled together with the checkboxes under the Limit states and Load cases buttons.

The 3-state checkboxes affect each other as they are not only used for selection but also for indication of the content. They can be manually set only to checked or unchecked. The middle state only appears when other filters are applied.

Filtering by limit states or load cases does not require any calculation results.

Filter by rules, on the other hand,is based on the actual analysis and/or design results. Different types of rules can be applied one by one or at the same time to select the desired load combinations.

When a rule is applied, all the load combinations that are selected on the Load combination set definition dialog- either with filtering by limit states/load cases or checked in manually- are examined at every position the rule indicates. Load combinations that meet the rule’s criteria are selected (remain checked in), while those that do not, become unchecked.

• With analysis rules, load combinations can be selected based on deformations or internal forces at either every finite element node or only at the member ends. This last one is included specifically for connection design. Deformations are checked in SLS combinations, internal forces are checked in ULS combinations only.
• With buckling rules, those ULS load combinations can be selected which have the elastic critical load factor (first buckling eigenvalue) less than the given limit.
• With design rules, load combinations can be selected based on utility ratios checked in every finite element node of the chosen portion. Utilizations are available from several design checks: dominant results and detailed verifications for steel elements such as general elastic cross-section check, pure resistances, interactions and global stability. Only ULS combinations can be filtered with design rules.

Interaction of the different filter types

Let’s see an example.

It is a simple 2D frame model, with 27 load combinations of various limit states generated. Analysis and design results are calculated for all load combinations.

If applying design rule to select only those load combinations which result dominant utilization over 50%,

4 load combinations will be selected (Load combination set 1):

But if ULS Accidental limit state is turned off before applying the same 50% filter,  

only one load combination is selected (see Load combination set 2).

Application of multiple rules

Applying multiple rules together results in the sum of the lists that would have been created separately.

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In Consteel 16, we introduced the function of load combination filter. Filtering is possible based on the load combinations’ limit state, load cases, and corresponding analysis and design results. The goal is to create different sets for the different steps of the optimization and reduce calculation time while making sure that all the relevant load combinations are considered. Let’s see what a conscious design workflow looks like in practice!

Description

It is a significant problem in almost all structural design projects that the standards define many possible load cases and combinations to evaluate. Although most of these load combinations are never relevant or provide decisive design situations, it is usually not evident which ones might be neglected safely, especially when considering, that different load combinations can be relevant for different parts of the structure, like primary or secondary structure, connections, etc. Accordingly, the optimization process is overloaded by a large amount of unnecessary calculations.

With the load combination filter function, a reduced list of load combinations aka a load combination set can be created and saved for the different steps of the optimization.

Detailed process

Modeling

The base of all optimization processes is a correctly built structural model. So, the first step is geometrical and structural modeling and load definition. It is advisable to run a first-order analysis for only one or two load cases and diagnostics to find possible modeling errors. Load combinations can be created after that. Every limit state that will be used during the whole design of the structure, should be defined. Consteel’s automatic load combination generation function is an efficient tool to do it.

Calculation and filter

On the Load combination set definition dialog, it is possible to create load combination sets by selecting the combinations based on their limit state and/or the load cases they contain. But usually, filtering on specific analysis or design results will likely be more effective in reducing the number of combinations. Using the above-described general workflow, the steps are as follows:

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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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