Software version: ConSteel 17 Build 3303

Keywords: Modeling; Analysis; Design; Lattice girder; Getting started;

Model examples

**Design objective, choice of design standard**

This design guide is intended for novice ConSteel 17 users and provides a step-by-step guide for designing a simply supported lattice girder. The geometry of the lattice girder to be designed is known from the architectural conceptual design, (**Figure 1**). According to the concept, the lattice girder chords are made of hot-rolled sections of HEA120, while the lattice bars are made of cold-formed SHS80x4 sections. The design of the connections is not included in this guide.

It is well known that structural design is always carried out according to a certain standard or its version. The selection of the standard can be made from the Design Standard menu when creating a new model in the Project Center, or it can be modified later in the Standards tab [S1] selection panel (Figure 2).

The desired design standard can be selected from the list on the left of the panel. In this case, we select the EN Recommended option [**S2**]. The parameters applied by the selected standard can be accessed by selecting the corresponding row of the middle table [**S3**] of contents in the right-hand table [**S4**]. In Figure 2, the combination factors corresponding to Table 1.1 of the EC0 standard have been selected, whose parameters are shown in the right-hand table.

**Setting the grid raster editor**

First, let’s set the size of the raster according to the span of the structure by using the corresponding button [**1**] of the tool group on the left, which will bring up the **Grid and Coordinate System** adjustment panel (**Figure 3**).

For example, for the 19.6m long support, the **Size** window can be set to 20000 millimeters [**2**]. To update the setting, press **Enter**. With the above setting, the raster will be 20m wide in X and Y directions, the raster line density will be 1000mm, and the step spacing will be 250mm. It is convenient to add the grid support model in the X-Z global coordinate plane, so the raster editor will be rotated to the X-Z plane. To do this, select the **XZ** plane option [**3**].

**Loading**** initial cross-sections**

One fundamental characteristic of general structural analysis programs is that they can only work with specifically defined cross-sections. Therefore, the first step is to choose the initial cross-sections for the task, according to the conceptual design. This may seem contradictory to the simple manual methods taught in basic statics courses, where the specific dimensions of the cross-section were often irrelevant information (e.g. calculation of internal forces). When using computer programs, however, we need to provide specific cross-sections even if their dimensions do not affect the static quantities to be calculated (e.g. in the present case, the normal forces of a truss beam). Nevertheless, we should aim to select cross-sections that match the geometrical size of the structure. In this case, the preliminary design served this purpose.

Initially, the section library for the current model is empty, so we first need to select the appropriate cross-sections. To do this, go to the **Structure Members** tab [**4**] and select the **Section Administration** option [**5**] on the left side of the horizontally positioned tool group, then select the “**From Library**…” button [**6**] in the panel that appears (**Figure 4**).

**Figure 5** shows the loading of the HEA120 section, compliant with the European standard, which will be assigned to the chords. We select the region of the cross-section standard (**European**) and then its type (**H profiles**). From the list that appears, we can select the type of section (**HEA**) [**7**] and then the height of the section (**120**) [**8**]. By pressing the **Load** button [**9**], the program learns the cross-section, and from then on it knows everything about the cross-section and can work with it. Repeat the procedure as many times as you need different sections. Finally, the window is closed by pressing the **Close** button [**10**].

In our case, also a **CF-SHS 80×4** closed section (from **Library/Hollow sections – cold formed/CF-SHS/CF-SHS 80×4**) was loaded for the bracing members (**Figure 6**).

Later on, you can obtain all the information about the cross-section using the **Section** **module**. To do this, select the cross-section in the table by clicking on the corresponding row and then click on **Properties…** to display all the properties of the cross-section, such as type data; cross-section characteristics, etc. The program works with two cross-section mechanical models in a dual manner. The **GSS** (General Solid Section) model [**11**] is used for static calculations and the **EPS** (Elastic Plate Segment) model [**12**] for standard design operations (**Figure 7**). The cross-sectional properties (surface area, moments of inertia, etc.) can be displayed by pressing the button [**13**].

**Building the structural model**

Building of a structural (geometric) model consists of two main steps:

– creating the member geometry

– determining of supports.

**Creating the member geometry**

First, switch the coordinate input/display type to *Absolute* value setting. This is done using the switch [**14**] located in the bottom row of the editor window (if the Δ sign is yellow, the switch is set to *Relative*) (**Figure 8**).

To begin creating the geometry, start by establishing the editing lines that define the structural shape. To do this, select the **Draw Line** function [**15**] under the **Geometry** tab (**Figure 9**).

It’s a good practice to establish the center of the lower chord of the truss at the origin of the global coordinate system, and from there, initiate the line segments using the drawing option [**16**]. The endpoints of these segments can be defined by specifying their X coordinate. The X (or any other) coordinate can be entered by typing the appropriate letter, entering the coordinate value in the corresponding coordinate window, and pressing **ENTER**. The result is shown in **Figure 10**. Of course, the network can be added in a different, possibly more practical order.

After creating the frame for the geometry, we proceed to add the bars that form the lattice girder. The frame serves as a good starting point for adding the bars, although it should be noted that one can also pick up the bars directly without the frame. The division of the chords into bars can be done in various ways: (i) each straight chord section consists of a single bar member, or (ii) both the upper and lower chords consist of two or more symmetrical bar members, or (iii) each section between nodes is a separate bar member. The choice will be determined by the design principle to be applied later: will the chord sections always consist of a single cross-section, or will the possibility of changing cross-sections within sections be retained? In the present case, we assume that we do not wish to change the cross-section along the chords, and therefore we prefer to choose solution (ii).

To add the members, we need to establish snapping points on the created reference lines. You can set the intercepts in the bottom right of the editor window. Since the bars are divided into eights along the lattice girder chords, we set n=8 [**17**] (**Figure 11**) and press **Enter**.

After the above adjustment, the red snapping points appear close to the reference line. You can create the bars by selecting two-two (start and end) snapping points (red dots) as follows. To add a bar, select the Edit Bar function [18] under the Structural Members tab. This will bring up the Member Editor dialog, where in this case it is sufficient to select the appropriate cross-section [19] (**Figure 12**).

The given bar can then be positioned using the snapping points. The operation is repeated until all the bars have been placed.

Based on the above steps, we create the structural model of the lattice girder: the lower and upper chords are modeled with two members (**Figure 13**), and the lattice members with a series of bar elements (**Figure 14**).

The style of the model display can be selected using the style buttons in the left-hand column of the editor window (**Figure 15**).

**Adding supports**

For general details about adding supports, see chapter **5.9 Supports in the Online Manual**. Here, only the support of the actual lattice girder is shown.

The grid of bars created in the previous section must be made structurally stable in 3D, i.e. properly supported. In the present case, we will add the following supports:

– XYZ fixed-pinned support at the top point on the left end of the truss;

– YZ-directional rolling-pinned support at the top point on the right-hand end of the truss;

– YZ-directional rolling-pinned supports at the end of the truss, in the nodes of the bottom chord.

– Y-directional pinned supports at the inner nodes of the upper chord.

As an example, place XYZ (spatial hinge) support at the top left end node of the truss. Select the **Point Support** function [**20**] under the **Structural Members** tab, which will display the adjustment dialog (**Figure 16**). The support type can be selected in the upper window of the dialog [**21**].

For the graphical display of the point supports, we recommend the hidden line or solid view (**Figure 16. c** or **16.d**). It is possible that the graphical symbol of the placed support may not be visible due to its relatively small size and/or being obscured. It is a common mistake that the user does not see the placed support and places it multiple times on the same node: this will result in a model error later! The size of the graphic symbol can be increased by using the adjustment slider at the bottom right of the editor window [**22**] (**Figure 17**). Use the slider or, if necessary, the arrow to increase the size of the graphic symbol of the support object until it is sufficiently visible. If the support is obscured, it becomes visible by rotating the model.

You can place the supports one after the other, using the left mouse button, at the corresponding nodes of the model. The complete support system, and thus the complete structural model, is shown in **Figure 18**.

**Load model creation**

**Load model structure**

First, let’s determine the structure of the load model. Select the **Load Cases and Load Groups** function [**23**] under the **Load** tab, which will bring up a panel defining load groups and load cases (*load case* = a group of loads that are inseparable in time and space; *load group* = a group of load cases where the loads can be mutually exclusive in load combinations) (**Figure 19**). When a new model is created, the load structure on the left side of the dialog box contains by default one load group with one load case. Start by selecting the appropriate load group or load case name on the left, then rename it on the right to a name that suits your needs.

For example, change the initial name of the load group to *‘Dead load’* [**24**] and the name of the load set to *‘Self-weight’* [**25**]. Confirm the action with the **Apply** button.

If you have more than one load case in the ’Dead Load’ load group, you can add them [**27**] by using the **New load case** button [**26**] in the bottom table of the dialog (**Figure 20**).

You can add a new load group by clicking on the **New load group** button [**28**] at the top of the dialog, but first select the appropriate category in the window [**29**] next to it. In this case, the new load group contains snow, so select the ’*Snow*’ option (**Figure 21**).

Based on the well-thought-out load model, all load groups and load cases are added by repeating the above operations (**Figure 22**).

As a final step, assign the automatically generated structural self-weight to a selected load case. To do this, open the drop-down list **Load case including self-weight** [**30**] at the bottom of the table and select the appropriate load case, in this case the load *‘Self-weight’* (**Figure 23**). Finally, close the window by clicking on **OK**. This defines the structure of the load model. In the next step, we will apply loads to the appropriate load cases.

**Loading load cases**

We start to add actual loads to the load cases recorded in the load model structure. This can be a lengthy process, so we will first summarise the most important technical elements of applying loads and then present the actual loads.

*Technical elements*

Let us open the window under the **Loads** tab [**31**], where we select the load case to which we will assign specific load components (**Figure 24**).

The loads added next are added to this load case until the selection above is overwritten by another selection. We will now describe the elements and technical aspects of load application. In the case of lattice girders, the loads are usually concentrated forces (the bars of a lattice girder are usually not directly loaded). Therefore, only the definition of the concentrated force is discussed below.

**Concentrated load in the global system**

Select the **Point Load** function [**32**] under the **Loads** tab, and then enter the components of the concentrated force as interpreted in the global system [**33**] in the corresponding input fields [**34**]. Clicking on the corresponding structural node will place the concentrated force in the structural model as part of the current load case (**Figure 25**).

**Concentrated load in the local system**

In certain types of loads, the direction of the concentrated force may not align with the direction of the global coordinate system. For example, in the case of wind loads, the load may act perpendicular to the roof surface, in our case the upper chord. In such cases, instead of calculating components in the global directions, we switch to the local system. To do this, we select the **Local coordinate system** option [**35**], after which the concentrated load [**36**] can be placed on the structural model in two steps: (1) selecting the appropriate structural node with the mouse, and then (2) selecting the relevant member whose axis forms the **X** coordinate axis of the local system (**Figure 26**).

*Applied load cases*

Using the above technical elements, the following load cases have been applied:

**Load combinations**

The next step is to create load combinations. To do this, select the **Load Combinations** button [**37**] under the **Loads** tab, which will bring up the **Load Combinations** table. Select the option **“ Automatic generation of load combinations“** [

**38**], which will display the symbolic formulas of the load combination types for the design situations (

**Figure 27**).

In our case, we need the options *‘Persistent and transient design situations – Eq. 6.10’* and *‘Accidental design situation – Eq. 6.11.b’*. The setting is confirmed with the **Apply** button, which causes the generated load combinations to appear in the table (**Figure 28**). In the table, if necessary, undesirable combinations can be deleted after selection [**39**], partial factors can be rewritten, and new combinations can be created if necessary [**40**].

**Verifying the structural model**

Before performing a final analysis, it is useful to ensure that the structural model is correct. Select the **Analysis Parameters** button [41] under the **Analysis** tab, which will bring up the analysis setup dialog, where you should first select only the “*First order analysis*” option [42] (Figure** 29).**

Switch to the load combinations table [**43**] and switch off all combinations (select all the checkboxes in the Load combination set field with a left-click, then right-click on any checkbox to clear the checkmarks), except one in which only symmetric loads are present (in our case, this is *load combination 3 (Lc-3)*, in which only the permanent load and the total snow load are present) (**Figure 30**).

After the setup, press the **Apply** and **Calculate **buttons, wait for the analysis to run, and then examine the results. Possible aspects of the analysis are summarised below.

After the analysis is performed, the deformed shape of the structure is displayed. The deformation diagram can be scaled by using the slider next to the adjustments fields [**44**]. The content of the graphical display can be adjusted by using four fields (**Figure 31**). The contents of the windows are as follows:

– selection of analysis types (first-order; second-order; stability; etc.) [**45**];

– selection of load combination or load set [**46**];

– selection of the type of data to be displayed (displacements; stresses; etc.) [**47**];

– select the type of drawing (diagram; color overlay; etc.) [**48**].

Let’s examine the displayed deformed shape. Aspects of the test are usually the following:

- Is the deformed shape symmetric for symmetric loads…?
- Does the deformed shape stay within the plane of the model for loads in the plane…?
- Is the maximum displacement value realistic…?
- Does the deformed shape adhere to engineering judgment…!?

In the next step, let us switch to the internal forces, in our case the “**N**” normal force plot [**49**] (**Figure 32**), and examine the plot from the following points of view:

- Does the axial force decrease in the top chords of the trusses toward the ends of the beams…?;
- Is the value of the maximum normal force in the chords realistic…?;
- Does the axial force decrease in the diagonal bracing towards the center of the beam…?;
- Does the shape of the diagram adhere to engineering judgment (e.g., are there tensioned bars along the top chord of a two-span beam or compressed bars along the bottom chord)…!?

**Analysis **

If the result of the model check in the previous step is successful, we can proceed to the final analysis. To do this, enable all relevant load combinations and then perform the first-order analysis. If the calculation finishes without an error message, the model is ready for design.

The first step is to determine the relevant load combination (perhaps the first two or three) for which you want to save the figures and use them in other documents. This is done by using the automatic cross-section check function of the program (**Figure 33**). Switch to the **Global checks** tab and press the **Design Settings** button [**50**], which will bring up the *Design…* control panel, where only the options *‘Cross section check’* [**51**] and *‘First-order’* analysis [**52**] are activated.

After the setting, the **Calculation** button causes the program to perform a cross-sectional resistance test for every finite element node. The program displays the final results of the check in a color-coded graph and a table (**Figure 34**). The cross-sectional utilization is shown in the last column [**53**] and the corresponding load combination is shown in the fourth column from the left [**54**]. Based on the contents of the last column, the relevant load combination (or at most two or three load combinations) can be selected to determine the design. In the present case, this is the load combination Lc-14.

** **

**Documenting the results**

The ConSteel software has advanced documentation features (*Online Manual, 13. Documentation*), the use of which is beyond the scope of this manual. Therefore, only the saving of the most important graphical diagrams and their use in other (e.g. Word or Mathcad) working documents is presented here.

For the internal forces, it is sufficient to document Figure N for now, as the effect of bending moment and shear force is negligible in the design of light lattice girders. In general, the procedure for documenting a figure is described below.

Select the “N” normal force figure (**Figure 35**) calculated from the first-order analysis for the load combination of interest (in our case Lc-14). Adjust the scale of the diagram’s largest ordinate using the upper slider [**55**]. A value of the figure can be marked by right-clicking on the corresponding point of the model and selecting the ’*Marker*’ option [**56**] in the window that appears. This attaches a label indicating the value of the force. We place as many labels as necessary to make the diagram meaningful from an engineering perspective. In the present case, the maximum compressive and tension forces are specified, to which the other values can be related.

To save a figure, click on the **Document **tab. When you select the “*Create a snapshot of the current state*” function [**57**], the image setting table appears, where you have to enter the name of the image [**58**] from which the file name is generated (use only the right characters), and then select the appropriate image size [**59**] and font size [**60**] (**Figure 36**).

Once you have set up the image in the image frame indicated by the dashed line using the camera/screen movement, press the **Create** or **Modify** button, which will add the image to the gallery. Then press the **Manage picture** button [**61**], which will display the contents of the gallery, where you can select the current image [**62**], and press the **Save selected picture to file button** [**63**] (**Figure 37**).

These operations will save the image in the selected format and with the specified filename in the selected folder. The resulting image can be inserted into any document (e.g. Word or Mathcad).

**Checking load-bearing capacity**

The program can automatically check the cross-sectional and stability resistance of the structure in a complex way. To do this, switch to the **Global checks** tab and press the **Design settings** button [**64**], which will bring up the **Design…** control panel, where you must select the *’Cross-section check*’ [**65**] and the *’Buckling check’* [**66**] option. According to the standard, the detailed calculation is performed based on the results of the *‘Second order’* elastic [**67**] analysis (**Figure 38**). The calculation of the elastic critical load factor is performed using the *Automatic* option [**68**], where the program determines the value of the critical factor per structural element, which results in a more economical design.

After setting, pressing the **Calculation** button will perform all the necessary load-bearing capacity checks. The program displays the final result in a color-coded graph and table. As already known, any cross-section can be labeled with a label showing the utilization in %. The details of the check can also be viewed by right-clicking on the last data [**69**] in the corresponding row of the table label (**Figure 39**).

At this point, the complete spectrum of checks for the specific cross-section is displayed (Figure 40), where you can find the analysis considered critical by the program [70], as well as its details [71].

**Assessment of results**

If we open the tables of **Compression** and **Bending about the major axis** within the tables of **Pure resistances****, **we can conclude the following (**Figure 41**):

- the dominant cross-section internal force is the normal force
**N**, which results in a utilization of 109.3% [**72**]; - the
**M**bending moment causes only 12.9% utilization of the cross-sectional resistance [_{y}**73**]; - the cross-sectional check by the conservative interaction formula leads to a 122.2% utilization [
**74**]; - however,
**Figure 40**shows that the authoritative check was given by the general global stability check formula according to EN 1993-1-1 6.3.4, which gave a utilization of 162.5% [**70**].

The above dominant utilization is higher than the result of the check assuming pure compression. This is due to the inclusion of the **M _{y}** bending moment. Neglecting the M

_{y}moment for “heavier” lattice girders (e.g. lattice bridge structures) can lead to serious safety issues.

**Design**

After the above quick and automatic check, we can immediately see that our lattice structure does not meet either the strength or the global buckling limit state. The program also allows us to look behind the results and understand the reasons. In the present case, we see that the lattice girder chords and the two-two tensioned lattice bars at the supports are significantly overloaded, with inadequate cross-sectional dimensions. The problem can be eliminated by increasing the indicated cross-sections: with a few minutes of work, we can find the right cross-section types and sizes. A possible solution is to replace the upper chord with an **HEB140** section and the lower flange with an **HEA140 **section**. **Additionally the supports, the last two members are modified to ** SHS100x4 **sections (Figure 37). After checking the new structural configuration, the result is shown in Figure 42. The modification was successful, and the structure now complies (the maximum overshoot of about 3% is still acceptable as it is within the margin of error).

## 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.

## The evolution of compressed bar (column) design

One of the characteristic features of steel structures made of bars (e.g. lattice girders) is the compressed bar. We speak of a compressed bar when the structural element, which usually has a straight axis, is loaded by a compressive force * P* applied centrally (Figure 1).

Figure 2 illustrates the evolution of compressed bar (column) design. In the beginning (in the old days), master builders determined the load-bearing capacity of compressed columns of different materials and sizes on the basis of the **experience accumulated over the centuries**, passed down from master to apprentice. A significant change was brought about by the application of classical mathematical differential analysis to engineering. The Swiss mathematician and physicist Euler (1707-1783) solved the problem of the deflection of a compressed elastic line, which could be applied to the solution of the elastic compressed bar (**Euler’s force**). In the following centuries, engineers recognised that Euler’s force only gave an acceptable approximation to the real load capacity of a compressed bar in certain cases (mainly for large slender bars). Many solutions for the bearing capacity of a compressed bar were developed that were more advanced than the Euler formula, but it was not until the huge structural engineering boom following World War II that significant changes were made. **Compression bar experiments** were carried out in every major structural laboratory in the world, and a database of over two thousand experiments was compiled from the results. The load capacity of the pressure bar was given by a formula based on the database, using the method of mathematical statistics.

This methodology is still dominant today:

‘the dimensioning of the compressed bar has become a political issue for the steel construction profession…’. Understanding the principle of compressed bar design is therefore essential for the structural engineer.

The right side of the Figure 2 also contains a hint for the future. At the level of scientific research, it is already present that the load capacity of a real compressed column can be determined by mathematical-mechanical simulation. Indeed, in the near future, databases that go beyond anything we know today can be created using supercomputers. On the basis of such a gigantic database, **artificial intelligence** could, at least in principle, supersede existing engineering knowledge and methodology. But the reality is that structural engineering is not one of the pull sectors (such as the defense or automotive industries), so this new shift in design theory is certainly a long way off.

In the following, the **Euler force** and the **experimentally based** standard design formula, which are of major importance to structural steel engineering today, are discussed in detail.

## Buckling strength of the ideal columns: the Euler force

Assume that the hinged compressed column shown in the Figure 3 has the following properties:

- perfectly
**straight**, - its material is perfectly linearly
**elastic**, **centrally**compressed.

Under the above conditions, perform the compressed column experiment using Consteel software: run the Linear Buckling Analysis (LBA) calculation. The result is illustrated in Figure 3.

gate**The latest version, Consteel 17 is officially out!** In 2023, our main focus for Consteel development is **improving usability**. New features prioritize efficient model manipulation, easy modification, and clear information presentation across Consteel, Descript, and our cloud-based platform, Steelspace. In this comprehensive video, we walk you through a step-by-step workflow guide, demonstrating how to leverage Consteel 17 to its full potential.

If you would like to delve deeper into the new features, **check out our detailed blog post** for an in-depth exploration of Consteel 17’s capabilities.

In Consteel, the calculation of cross sectional interaction resistance for Class 3 and 4 sections is executed with the modified Formula 6.2 of EN 1993-1-1 with the consideration of warping and altering signs of component resistances. Let’s see how…

## Application of EN 1993-1-1 formula 6.2

For calculation of the resistance of a cross section subjected to combination of internal forces and bending moments, EN 1993-1-1 allows the usage -as a conservative approximation- a linear summation of the utilization ratios for each stress resultant, specified in formula 6.2.

$$\frac{N_{Ed}}{N_{Rd}}+\frac{M_{y,Ed}}{M_{y,Rd}}+\frac{M_{z,Ed}}{M_{z,Rd}}\leq 1$$

As Consteel uses the 7DOF finite element and so it is capable of calulcating bimoment, an extended form of the formula is used for interaction resistance calculation to consider the additional effect.

$$\frac{N_{Ed}}{N_{Rd}}+\frac{M_{y,Ed}}{M_{y,Rd}}+\frac{M_{z,Ed}}{M_{z,Rd}}+\frac{B_{Ed}}{B_{Rd}}\leq 1$$

Formula 6.2 ignores the fact that not every component results the highest stress at the same critical point of the cross-section.

In order to moderate this conservatism of the formula, Consteel applies a modified method for class 3 and 4 sections. Instead of calculating the maximal ratio for every force component using the minimal section moduli (W), Consteel finds the most critical point of the cross-section first (based on the sum of different normal stress components) and calculates the component ratios using the W values determined for this critical point. **Summation is done with considering the sign of the stresses caused by the components corresponding to the sign of the dominant stress in the critical point.**

(For class 1 and 2 sections, the complex plastic stress distribution cannot be determined by the software. The Formula 6.2 is used with the extension of bimoments to calculate interaction resistance, but no modification with altering signs is applied)

## Example

Let’s see an example for better explanation.

GATE## Did you know that you could use Consteel to design web-tapered members?

Download the example model and try it!

Download modelIf you haven’t tried Consteel yet, **request a trial for** **free**!

**Did you know that you could use Consteel to** **determine automatically the second order moment effects for slender reinforced concrete columns?**

Download the example model and try it!

Download modelIf you haven’t tried Consteel yet, **request a trial for** **free**!

**Did you know that you could use Consteel to** **perform local and distortional buckling checks for cold-formed members?**

Download the example model and try it!

Download modelIf you haven’t tried Consteel yet, **request a trial for** **free**!

## 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

- 𝐸

**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

θ_{0}= 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}= θ_{0}α_{h}α_{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_{i}𝑁.

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_{i}𝑁.

### 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 (𝑀_{2}). 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 𝑙_{0} = 2𝐻, fix bottom – top hinged case 𝑙_{0} = 0.7𝐻, etc.).

Calculation of design bending moment:

𝑀_{Ed} = 𝑀_{0Ed} + 𝑀_{2}

where

𝑀_{0Ed} is the 1^{st} order moment, including the effect of imperfections

𝑀_{2} 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_{r}𝐾_{φ}1/𝑟_{0}

where

- 𝐾
_{r}is a correction factor depending on axial load - 𝐾φ is a factor for taking account of creep
- 1/𝑟
_{0}is the theoretical (physical) curvature associated with failure

1/𝑟_{0} = 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_{c}𝑓_{cd}

relative axial force

- 𝑁
_{Ed}

design value of axial force

- 𝑁
- 𝑛
_{u}=1+ω- ω = A
_{s}𝑓_{yd}/ A_{c}𝑓_{cd}

mechanical reinforcement ratio - A
_{s}

is the total area of reinforcement - Ac

is the area of concrete cross-section

- ω = A
- 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

- 𝑀
- β=0,35 + 𝑓
_{ck}/200−λ/150

- λ = 𝑙
_{0}/ 𝑖

slenderness - 𝑙
_{0}

buckling length - i = √𝐼
_{c}/𝐴_{c}

inertia radius of uncracked concrete

- λ = 𝑙

Second order bending moment

𝑀_{2}=𝑁_{Ed}𝑒_{2}

where

Second order eccentricity, where c is the factor depending on the curvature distribution. For constant cross-section 𝑐=𝜋^{2} 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 (𝑀_{2}), 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:

𝑀_{2} = 𝑁_{Ed}𝑒_{2}

where 𝑐 = π_{2}

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 (𝑒_{2}).

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(∞,𝑡_{0})

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.

GATEWhen 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*

** 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***option contains the dominant utility ratio of the dominant load combination at every node, like an envelope of**

*Maximum of dominant results***. Meaning there is only one utilization value in every FE node. Also, it is the same as the dominant result table on**

*Steel-Dominant results***tab.**

*Global checks*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*

*Steel – Dominant results*

: to select the combinations which cause the maximum utilization at any node. It can be the same as*Maximum*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.*Maximum of dominant results*: to select the combinations as in ’*More than % of 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%.*Maximum*: to select the combinations which cause utilization more than the defined value at any point.*More than*

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.)

gate