## 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:

$$\frac{}{\lambda}=\sqrt\frac{Af_y}{N_{cr}}$$

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 Ξ±_{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 *Ξ± _{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

Letβs 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

πΎ_{cE}= 1.20 - Concrete

πΎ_{c}= 1.50 - Steel reinforcement

πΎ_{s}= 1.15

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

**Modulus of elasticity:**

- Design value

- πΈ
_{cd}= πΈ_{cm}/Ξ³_{cE} - Reducing the mean value with Ξ³
_{cE}partial safety factor - Applicable in ULS
- In the case where creep is not considered, or considered elsewhere

- πΈ

**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

ΞΈ_{0}= 1/200 - Height-dependent reduction factor

Ξ±_{h}= 2/βπ»

where π» is the height - reduction factor depending on the number of structural elements

Ξ±_{m}= β0.5(1 + 1/π)*π*: number of vertical structural elements bearing the total load - applied inclination

ΞΈ_{i}= ΞΈ_{0}Ξ±_{h}Ξ±_{m}

Then, as shown in Figure 2, the π the normal forces can be used to calculate the notional loads in the unbraced case: π»_{i} = ΞΈ_{i}π.

In braced case, for example a hinged-hinged column, π»_{i} force is not defined at the top of the column, but at the center, with the value: π»_{i} = 2ΞΈ_{i}π.

### 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 (π_{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 π_{0} = 2π», fix bottom – top hinged case π_{0} = 0.7π», etc.).

Calculation of design bending moment:

π_{Ed} = π_{0Ed} + π_{2}

where

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

π_{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/π = πΎ_{r}πΎ_{Ο}1/π_{0}

where

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

1/π_{0} = Ξ΅π¦π/0,45π

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 βbalancedβ case.

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

πΎ_{r} = (π_{u} β π)/(π_{u} β π_{bal})

where

- π = π
_{Ed}/ A_{c}π_{cd}

relative axial force

- π
_{Ed}

design value of axial force

- π
- π
_{u}=1+Ο- Ο = A
_{s}π_{yd}/ A_{c}π_{cd}

mechanical reinforcement ratio - A
_{s}

is the total area of reinforcement - Ac

is the area of concrete cross-section

- Ο = A
- π
_{bal}=0.4

value of n at maximum bending - (0.4 applicable in the absence of further information)

Factor for taking account of creep:

πΎ_{Ο} = 1 + Ξ²Ο_{ef} β₯ 1

where

- Ο
_{ef}= Ο(β,0) π_{0Eqp}/ π_{0Ed}

effective creep

- π
_{0Eqp}first order quasi-permanent bending moment (SLS) - π
_{0Ed}first order bending moment (ULS) β design combination

- π
- Ξ²=0,35 + π
_{ck}/200βΞ»/150

- Ξ» = π
_{0}/ π

slenderness - π
_{0}

buckling length - i = βπΌ
_{c}/π΄_{c}

inertia radius of uncracked concrete

- Ξ» = π

Second order bending moment

π_{2}=π_{Ed}π_{2}

where

Second order eccentricity, where c is the factor depending on the curvature distribution. For constant cross-section π=π^{2} applicable (sinusoidal distribution). In case of constant distribution π=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 π_{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 (π_{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/π) 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:

π_{2} = π_{Ed}π_{2}

where π = Ο_{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 (π_{Ed}). The second is the second order eccentricity as defined in the Eurocode (π_{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:

Ο_{ef} = Ο(β,π‘_{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 β 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 β Dominant results*

*Steel β 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 β*More than % of maximum*β 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*

Letβs 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 rulesβ 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**Did you know that you could use Consteel to include in your model a wide range of cold-formed macro sections?**

Download the example model and try it!

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