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Summary

Introduction

The ultimate strength analysis and design of T-beams and other flanged sections as well as nonstandard or irregular-shaped sections are dealt with in this chapter. The procedures involved together with derivation of relevant formulas and the application of relevant Standard recommendations are presented, with illustrative examples. The treatment of continuous beams and other statically indeterminate structures is briefly described.

T-beams and other flanged sections

Definitions

Beams with a cross-sectional shape of T, L, I or a box are referred to collectively as flanged beams. Flanged sections are a popular choice for major structures, such as road and railway bridges, due to their structural efficiency. A beam-and-slab building floor system may be considered as an assembly of T-beams. An example is shown in Figure 4.2(1)a.

The analysis and design of flanged beams may seem to be comparatively more involved than the analysis and design of rectangular beams, due to more complicated geometry. However, using simple assumptions and some approximations, the formulas developed for singly and doubly reinforced rectangular sections can be adapted for use in the analysis and design of flanged beams.

Effective flange width

Figure 4.2(1) shows a typical T-beam. A three-dimensional stress analysis would indicate that, depending on the dimensions of the beam, the bending stress distribution over the width of the flange (b) is not uniform as it would be for a rectangular section. Instead, the stress varies from a maximum (f1) near the top of the web to a minimum (f2) at the two edges of the flange, as in Figure 4.2(1)b. However, the stresses f1 and f2 are statically indeterminate.

To simplify the problem, the actual section in Figure 4.2(1)b may be replaced by an ‘effective’ one, as shown in Figure 4.2(1)c, in which the effective flange width is taken as

where A is the area of the stress diagram E-A-B-C-F. In the effective section, the maximum stress (f1) is uniformly distributed over the effective width (bef) and can be computed in the usual way.

Summary

Introduction

Prestressing may be seen as an elaborate and active way to reinforce concrete when its tensile capacity is insufficient. Whereas the traditional reinforcement becomes active mainly after the concrete has exceeded its cracking strength, the purpose of prestressing is to prevent cracking from occurring. This is done by introducing compressive stress in the concrete to neutralise the anticipated tensile stress developed under load.

In a traditional reinforced concrete design, the safety margin can always be increased by providing more reinforcement. The same may not be true in prestressed concrete, as over-prestressing can cause cracking or perhaps failure before even any external loading is applied. As a result, prestressed concrete analysis and design are more complicated and mechanics-based than for reinforced concrete, which relies more on empirical formulas. In practice, prestressed concrete also requires a higher level of technology in its construction.

By nature, prestressing is more efficient than the traditional reinforcement in that the stress in concrete, either tensile or compressive (caused by self-weight or other forms of dead load), can be neutralised before any additional (live) loading is applied. Consequently, for a given design, the maximum permissible prestressed concrete span can be considerably larger than a reinforced one. Following some fundamentals given in this chapter, Chapter 15 presents the bending theory of fully prestressed concrete beams based on the critical stress state criteria (which ensures that no cracking or overstressing in tension or compression would ever occur throughout the life of the beam under service load). The design of beams in bending using the critical stress state approach is given in Chapter 16. To comply with the Australian Standard (AS) 3600–2009 (the Standard), in practice, all prestressed beams must have the required strength. The bending strength analysis of fully and partially prestressed concrete beams is discussed in Chapter 17. The last chapter, Chapter 18, reviews the analysis and design of end blocks for prestress anchorage.

Non-engineering examples of prestressing

Wooden barrel

Figure 14.2(1) shows an assembly of curved barrel staves with tapered ends that are held together by the metal hoops forced from both ends towards the centre. The forced expansion of the hoops produces the compressive stress between the staves. To enhance efficiency, the hoops may be heated up before installation. It is well known that a well-made wooden barrel cannot only hold itself together, but is also water tight.

Summary

The elastic, or working stress method, has been dropped from the various design codes and Standards for many years. However, some of the analysis equations remain useful in computing deflections. In particular, neutral axis (NA) positions are needed in the assessments of the effective moments of inertia.

The procedure for determining the position of the NA of a fully cracked rectangular section is summarised in this appendix, accompanied by some useful formulas.

Summary

Introduction

Selection of materials and a decision on the associated properties constitute an essential first step in any structural design. In this chapter, the relevant information and data required for the design of reinforced and prestressed concrete structures are covered in some detail.

Concrete

Characteristic strengths

The characteristic compressive strength at 28 days is defined as the strength value of a given concrete that is exceeded by 95% of concrete cylinders tested (made from the same concrete). The test method for determining is described in AS 1012.9–2014. The value of is used in all Standard-based design calculations. Thus, in mix design, the strength to aim for (or the 28-day mean in situ compressive strength) is

where s is the standard deviation of the test data and k varies from 1.25 to 2.5 depending on the grade of concrete and the manufacturing process (see AS 1379–2007 for all aspects of the testing and assessment of concrete or AS 1012.9–2014 for determining the compressive strength of concrete specimens).

As per Clause 3.1.1.2 of AS 3600–2009 (the Standard), fcm at a given age may be taken as 90% of the mean cylinder strength of the same age or fcmi. Refer to Clause 3.1.1.1 for more recommendations. Various design procedures, such as those described by Ryan and Samarin (1992) and by Murdock and Brook (1982), may be used to determine the proportions of the constituent materials (water, cement, fine and course aggregates and admixtures as appropriate).

As a function of, the characteristic flexural tensile strength as per Clause 3.1.1.3 of the Standard, may be taken as

or it may be determined for a given concrete by the laboratory test described in AS 1012.11–2000 (R2014). This value is used mainly in cracking moment and deflection calculations (see Clauses 8.1.6.1 and 8.5.3.1, respectively, in the Standard), and in determining the minimum slab and pad-footing reinforcement (Clause 9.1.1).

A second tensile strength, the characteristic principal (or splitting) tensile strength, is required mainly in the design for longitudinal shear (Clause 8.4.3 in the Standard).

Summary

This latest edition of the book is written in response to the domestic and international market needs as well as on the request of the publisher, Cambridge University Press. It retains the features of the original book but to enhance readability, every chapter now begins with an introduction and concludes with a summary, followed if applicable by an enlarged set of tutorial problems. In all 20 new problems are added making a grand total of 128.

The original Chapter 3 on bending analysis and design was judged by some peer-reviewers as being unduly lengthy. It is now split into two consecutive chapters. Based on a previous appendix, a new Chapter 13 on strut-and-tie modelling is added incorporating the latest material on the subject. Since the last edition, many significant amendments have been made and addenda added to the Australian Standard AS 3600–2009. These are incorporated in the current edition in the form of new and updated tables and figures. Also included are the latest information and recommendations on fire resistance, detailing including cover, long-term deflection, as well as torsion design.

Summary

Introduction

For a traditional building or bridge structure, the vertical forces in the walls, columns and piers are carried to the subsoil through a foundation system. The most common system consists of footings. In soft soil, because the bearing capacity is low, piles are needed to transfer the forces from the superstructure to deeper grounds where stiffer clay, sand layers or bed rock exist. The wall or column forces are each distributed to the piles or group of piles through a footing-like cap – a pile cap.

In civil and structural engineering, slopes often need be to cut to provide level grounds for construction. To ensure stability at and around the cuts or to meet similar requirements, the use of retaining walls for the disturbed soil and backfill is sometimes necessary.

The design of reinforced concrete footings and pile caps is generally governed by shear or transverse shear for wall footings, and transverse or punching shear for column footings and pile caps. Retaining walls behave like a cantilever system, resisting the horizontal pressures exerted by the disturbed soil or backfill (or both) by bending action.

The analysis of the forces acting above and below typical wall footings and their design are presented in Section 12.2. The treatments for footings supporting single and multiple columns are given in Section 12.3 whereas Section 12.4 deals with pile caps. Illustrative and design examples are given to highlight the application of the analysis and design procedures.

The horizontal soil pressure analysis and the design of retaining walls are discussed in Section 12.5, which also includes illustrative and design examples.

Wall footings

General remarks

There are two main types of wall loading to be carried by the footing – the concentric loading as illustrated in Figure 12.2(1) and the eccentric loading as shown in Figure 12.2(2).

Concentric loading leads to uniform pressure on the subsoil, which normally requires a symmetrical footing – see Figure 12.2(1). On the other hand, eccentric loading, depending on the ratio of M* and N*, produces two possible pressure distributions:

Summary

The second edition retains all of the features of the original book on the explicit and implicit advice of our peers via the mandatory Cambridge University Press review process. To limit the volume size, the old Appendix C, ‘Development of an integrated package for design of reinforced concrete flat plates on personal computer’, has been removed, being of diminishing practical importance. To enhance the contents, new and important materials are added, some of which were also on the advice of the reviewers:

Summary

One-way slabs

Theoretically, all slabs are two-dimensional systems with bending in two orthogonal directions or two-way bending. In practice, however, some slabs are supported on parallel and continuous (line) supports in one direction only. Figure 9.1(1)a shows a typical slab of this type, in which the line supports take the form of a continuous wall or a girder, which is supported on discrete columns. Note that the coordinate system adopted here is different from that used in the earlier chapters. This is to conform to AS 3600–2009 (the Standard) convention.

If such a slab is subjected to a uniformly distributed load only, then bending mainly occurs in the direction perpendicular to the support – the x-direction in Figure 9.l(1)a. The bending in the y-direction may be ignored. Such a slab is referred to as a one-way slab.

Since bending occurs only in the x-direction, the entire slab structure can be seen as a very wide beam running continuously in the x-direction. For analysis and design, it is convenient to consider a strip of a unit width, for example, 1 metre. This is illustrated in Figure 9.1(1)b. The strip can be further assumed to be continuous over standard (beam) supports. However, such an assumption is valid only if the column-supported girders are rigid enough to prevent significant bending in the y-direction.

The 1-metre wide continuous strip can be treated as a beam of the same width. Consequently, the forces in this one-way strip include only the bending moment (Mx) – or simply M – and transverse shear (V) as depicted in Figure 9.l(1)c. A simplified method for computing these forces in a multispan strip is provided in Clause 6.10.2 of the Standard.

The use of a 1-metre wide strip is convenient for design purposes because the computed steel reinforcement area is given in mm2/m. Then the required spacing of a given size bar can be obtained directly from Table 2.3(2) in Section 2.3.

Summary

Australian Standard precast prestressed concrete bridge girder sections

Summary

In the past, when design provisions were lacking, wall panels were usually taken as nonload-bearing. Due to the increased acceptance of precast techniques in building construction and a trend towards reinforced concrete core structures, walls have now become popular as load-bearing elements.

A wall is a vertical planar continuum with a thickness much smaller than its height or length. If a wall is short, with a length of the same order as the thickness, it can be treated as a column. In fact, AS 3600–2009 (the Standard) defines a wall as an element wider than three times its thickness. Otherwise, the element is considered to be a column.

Walls are normally supported at the bottom end by a floor system and at the top end by a roof structure or another floor. Or a wall may be freestanding. Depending on the chosen structural system, walls may be supported on either or both sides by interconnecting walls or other structural elements. Consequently, a wall may act like a column, a beam cantilevered at one end or a slab standing vertically. Figure 11.1(1)a depicts a wall under vertical in-plane and lateral bending loads; Figure 11.l(1)b shows a wall that is under combined in-plane vertical (axial) and horizontal (shear) forces. At times a wall may be subjected to simultaneous axial, bending and shear forces.

Walls in tilt-up construction and low-rise structures are relatively easy to analyse, based on forces acting statically on the element. In multistorey buildings, shear walls are present in elevator shafts, central cores or stairwells; they may also exist as load-bearing planar continuums. In addition to resisting horizontal forces from wind and earthquake loading, shear walls also carry in-plane vertical forces due to dead and live loading. With the axial, bending and shear forces in a wall computed using an appropriate structural analysis procedure, the provisions in Section 11 of the Standard may be used to design a wall for strength and serviceability.

The Standard's strength design provisions for walls are highlighted in this chapter. Also summarised are relevant recommendations given in ACI 318–2014. Doh and Fragomeni (2004) have proposed a strength design formula covering walls in both one-way and two-way actions. This new formula is included in the discussion here and illustrative examples are provided to demonstrate the application of the three design procedures.

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4 - T-beams and irregular-shaped sections

Summary

9 - Two-Variable Regression Models

Contents

Part 2 - Prestressed concrete

14 - Introduction to prestressed concrete

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Appendix A - Elastic neutral axis

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2 - Design properties of materials

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8 - Bivariate Hypothesis Testing

Preface to the third edition

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12 - Footings, pile caps and retaining walls

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Frontmatter

Preface to the second edition

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Appendix A - Critical Values of Chi-Squared

Deidcation

9 - Slabs

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Appendix C - Australian Standard precast prestressed concrete bridge girder sections

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Appendix B - Critical Values of t

Acknowledgments to the Third Edition

List of Figures

11 - Walls

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