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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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.

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This paper discusses a combination of best practices and procedures from recent work in Europe and the US, providing rational and economical calculations addressing the complexities associated with frame design using nonprismatic members. Recommendations are provided in the context of US design practice. A primary objective is to achieve maximum simplicity, transparency, and design speed while facilitating rigor of the underlying calculations. The paper provides several focused examples illustrating the recommended design verification procedures.

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Contents

• Set design parameters
• Perform cross section check
• Examine the design results in the Section Module
• Stability design according to the general method
• Member check – stability design for members
• Serviceability check

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Introduction

There are different ways to evaluate the stability of a structure. It is important to know the differences between those methods and the limits of applicability but it is also important to recognize the equalities in pure cases.

Methods of stability design

In Eurocode 1993-1-1, and so in Consteel, there are 3 methods to verify the stability of a model:

• Imperfection approach (described in Section 5.2 and 5.3)

The structural model is subjected to appropriate geometrical imperfections and after completing a second order analysis, only the cross section resistances need to be checked

• Isolated member approach ( described in section 6.3.1, 6.3.2 and 6.3.3)

       The method is based on two essential simplifications:

1. Structural member isolation: The relevant member is isolated from the global structural model by applying special boundary conditions (supports, restraints or loads) at the connection points which are taken into account in the calculation of the buckling resistance
2. Buckling mode separation: The buckling of the member is calculated separately for the pure modes: flexural buckling for pure compression and lateral-torsional buckling for pure bending. The two effects are connected by applying special interaction factors.

• General method (described in section 6.3.4)

The basic idea behind the general method is that it no longer isolates members and separates the pure buckling modes, but considers the complex system of forces in the member and evaluates the appropriate compound buckling modes. The method offers the possibility to provide solutions where the isolated member approach is not entirely appropriate:

-The general method is applicable not only for single, isolated members, but also for sub frames or complete structural models where the governing buckling mode involves the complete frame.

-The general method can examine irregular structural members such as tapered members, haunched members, and built up members.

-The general method is applicable for any irregular load and support system where separation into the pure buckling modes is not possible.

Implementation of different approaches in Consteel

Isolated member approach is basically the reduction factor method and it can be performed in Member checks function in Consteel where it is possible to define the design parameters (e.g. effective length) by hand.

In Global checksfunction, cross-section and global buckling checks can be executed automatically according to the General method which does not require the direct introduction of effective lengths and other parameters depending on the distribution and combination of stresses along the member.

However, in pure cases (pure compression, or pure bending), buckling length calculated from general method can be equated with isolated member approach. In the following, a „how to“ example will be shown on a pure compression column:

How can I get the buckling length of a certain member?

Both the Isolated member approach and the General method require to calculate the slenderness value of the member. In case of the first method this is done through the use of buckling lengths. By using the General Method, the slenderness is calculated as the ratio of two amplifiying factors.

For the calculation of the equivalent buckling length we will need the first amplifier only, called critical elastic factor or alpha critical factor, obtained through a numerical analysis called linear buckling analysis.

Determination of alpha critical factor with buckling analysis:

Alpha critical factor: Minimum amplifier for the design loads to reach the elastic critical resistance of the structural component with regards to lateral buckling.

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After introducing the Eurocode standards several theses have been published on the now much-discussed phenomenon of lateral-torsional buckling of steel structural elements under pure bending. According to that, researchers are working on the development of such new design methods which can solve the problems of the design formulae given by the EN 1993-1-1. This paper gives a detailed review of the proposals for novel hand calculation procedures for the prediction of LT buckling resistance of beams. Nowadays, the application of structural design softwares in practical engineering becomes more common and widespread. Recognizing this growing interest, the main objective of our research work is the development of a novel, computer-aided design method. In this paper, the details of a general type stability design procedure for the determination of the LT buckling resistance of members under pure bending are introduced. Here, the theoretical basis of the proposed method is clarified, the calculation procedure is detailed and some results for the evaluation of the appropriateness of the method are also presented. Based on the evaluations it can be stated that the new, general type design method is properly accurate and has several advantages on the stability check of beams under bending

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In the second article of this series, Dr József Szalai of ConSteel Solutions demonstrates practical examples where the “General Method” of EN 1993-1-1 shows advantages compared to the conventional approaches.

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Clause 6.3.4 of EN 1993-1-1 describes a “General Method” for lateral and lateral torsional buckling of structural components, ideally suited to software applications. Although the UK National Annex places some limitations on the use of this method, it is possible that the approach will become more widely used.

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The new versions of the EN 1993-1-1 (EC3-1-1) and the EN 1993-1-5 (EC3-1-5) standards have introduced the general method designing beam-column structures; see [1] and [2]. The design method requires 3D geometric model and finite element analysis. In a series of papers we present this general design approach. The parts of the series are the following:

• Part 1: 3D model based analysis using general beam-column FEM

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