Design theories

3.1.1 Development of design

The specific aim of structural design is, for a given framing arrangement, to determine the member sizes to support the structure’s loads. The historical basis of design was trial and error. Then with development of mathematics and science the design theories –elastic, plastic and limit state –were devel-oped, which permit accurate and economic designs to be made. The design theories are discussed; design methods given in BS 5950: Part 1 are set out briefly. Reference is also made to Eurocode 3 (EC3). The complete codes should be consulted.

3.1.2 Design from experience

Safe proportions for members such as depth/thickness, height/width, span/depth etc. were determined from experience and formulated into rules.

In this way, structural forms and methods of construction such as beam–

column, arch–barrel vault and domes in stone, masonry and timber were developed, as well as cable structures using natural fibres. Very remark-able structures from the ancient civilizations of Egypt, Greece, Rome and the cathedrals of the middle ages survive as a tribute to the ingenuity and prowess of architects using this design basis. The results of the trial-and-error method still survive in our building practices for brick houses. An experimental design method is still included in the steel code.

3.1.3 Elastic theory

Elastic theory was the first theoretical design method to be developed. The behaviour of steel when loaded below the yield point is much closer to true elastic behaviour than that of other structural materials (Figure 3.1). All sections and the complete structure are assumed to obey Hooke’s law and

Structural steel design 33

Stress (N/mm2) Stress (N/mm2)

(a)

(b)

Figure 3.1 Stress–strain diagrams: (a) structural steels – BS 5950 and EC3; (b) plastic design.

recover to their original state on removal of load if not loaded past yield.

Design to elastic theory was carried out in accordance with BS 449, The Use of Structural Steel in Building.

For design the structure is loaded with the working loads, that is the maxi-mum loads to which it will be subjected during its life. Statically determinate structures are analysed using simple theory of statics. For statically indeter-minate structures, linear or first-order elastic theory is traditionally used for analysis. The various load cases can be combined by superposition to give the worst cases for design. In modern practice, second-order analysis taking account of deflections in the structure can be performed, for which com-puter programs and code methods are available. In addition, analysis can be performed to determine the load factor which will cause elastic instability where the influence of axial load on bending stiffness is considered. Dynamic analyses can also be carried out. Elastic analysis continues to form the main means of structural analysis.

In design to elastic theory, sections are sized to ensure permissible stresses are not exceeded at any point in the structure. Stresses are reduced where instability due to buckling such as in slender compression members, unsup-ported compression flanges of slender beams, deep webs etc. can occur.

34 Structural steel design

0.25P P 0.153 0.247

0.178

Bending Bending and axial load

Partially

Figure 3.2 Loading, deflection, bending and stress distributions: (a) elastic analysis; (b) plastic analysis.

Deflections under working loads can be calculated as part of the analysis and checked against code limits. The loading, deflection and elastic bending moment diagram and elastic stress distribution for a fixed base portal are shown in Figure 3.2(a).

Structural steel design 35 The permissible stresses are obtained by dividing the yield stress or elastic critical buckling stress where stability is a problem by a factor of safety.

The one factor of safety takes account variations in strengths of materi-als, inaccuracies in fabrication, possible overloads etc. to ensure a safe design.

3.1.4 Plastic theory

Plastic theory was the next major development in design. This resulted from work at Cambridge University by the late Lord Baker, Professors Horne, Heyman etc. The design theory is outlined.

When a steel specimen is loaded beyond the elastic limit the stress remains constant while the strain increases, as shown in Figure 3.1(b).

For a beam section subjected to increasing moment this behaviour results in the formation of a plastic hinge where a section rotates at the plastic moment capacity.

Plastic analysis is based on determining the least load that causes the struc-ture to collapse. Collapse occurs when sufficient plastic hinges have formed to convert the structure to a mechanism. The safe load is the collapse load divided by a load factor.

In design the structure is loaded with the collapse or factored loads, obtained by multiplying the working loads by the load factor, and analy-sed plastically. Methods of rigid plastic analysis have been developed for single-storey and multistorey frames where all deformation is assumed to occur in the hinges. Portals are designed almost exclusively using plastic design. Software is also available to carry out elastic–plastic analysis where the frame first acts elastically and, as the load increases, hinges form succes-sively until the frame is converted to a mechanism. More accurate analyses take the frame deflections into account. These secondary effects are only of importance in some slender sway frames. The plastic design methods for multistorey rigid non-sway and sway frames are given in BS 5950.

The loading, collapse mechanism and plastic bending moment diagram for a fixed-base portal are shown in Figure 3.2(b). Sections are designed using plastic theory and the stress distributions for sections subjected to bending only and bending and axial load are also shown in the figure. Sections require checking to ensure that local buckling does not occur before a hinge can form. Bracing is required at the hinge and adjacent to it to prevent overall buckling.

3.1.5 Limit state theory and design codes

Limit state theory was developed by the Comitée Européen Du Béton for design of structural concrete and has now been widely accepted as the best design method for all materials. It includes principles from the elastic and

36 Structural steel design

plastic theories and incorporates other relevant factors to give as realistic a basis for design as possible. The following concepts are central to limit state theory:

1 Account is taken in design of all separate conditions that could cause failure or make the structure unfit for its intended use. These are the various limit states and are listed in the next section.

2 The design is based on the actual behaviour of materials in structures and performance of real structures established by tests and long-term observations. Good practice embodied in clauses in codes and speci-fications must be followed in order that some limit states cannot be reached.

3 The overall intention is that design is to be based on statistical methods and probability theory. It is recognized that no design can be made completely safe; only a low probability that the structure will not reach a limit state can be achieved. However, full probabilistic design is not possible at present and the basis is mainly deterministic.

4 Separate partial factors of safety for loads and materials are specified.

This permits a better assessment to be made of uncertainties in loading, variations in material strengths and the effects of initial imperfections and errors in fabrication and erection. Most importantly, the factors give a reserve of strength against failure.

The limit state codes for design of structural steel now in use are BS 5950:

Part 1 (2000) and EC3 (1993). All design examples in the book are to BS 5950 and many of these design examples are to EC3 for comparison purpose. EC3 is not discussed. However, references are made in some cases.

In limit state philosophy, the steel codes are Level 1 safety codes. This means that safety or reliability is provided on a structural element basis by specifying partial factors of safety for loads and materials. All relevant separate limit states must be checked. Level 2 is partly based on proba-bilistic concepts and gives a greater reliability than a Level 1 design code.

A Level 3 code would entail a fully probabilistic design for the complete structure.

3.2 Limit states and design basis