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2643 results in Civil and geotechnical engineering

Page 30 of 133

  • 14 - Introduction to prestressed concrete

  • from Part 2 - Prestressed concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 403-413

  • 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.

  • Appendix A - Elastic neutral axis

  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 483-484

  • 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.

  • 2 - Design properties of materials

  • from Part 1 - Reinforced concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 13-18

  • 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).

  • Preface to the third edition

  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp xxiii-xxiii

  • 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.

  • 12 - Footings, pile caps and retaining walls

  • from Part 1 - Reinforced concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 311-376

  • 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:

  • • the entire footing sustains pressure from the subsoil – see Figures 12.2(2)b(i) and b(ii)

  • • only part of the footing sustains the subsoil pressure – see Figure 12.2(2)b(iii).In the above discussion, the self-weight of the footing is included in the loading.

  • Preface to the second edition

  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp xxi-xxii

  • 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:

  • • updated tables and figures to reflect the amendments and addenda to AS 3600–2009 promulgated by Standards Australia International since its first publication

  • • additional information on fire design, detailing and cover, long-term deflection, as well as aspects of partially prestressed concrete design; and

  • • an expanded Appendix on strut-and-tie modelling, encompassing the latest publications on the topic plus a numerical example.

    • Just as significant, another 37 tutorial problems have been added to the various chapters of the book. This makes a total of 108.

  • 9 - Slabs

  • from Part 1 - Reinforced concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 185-254

  • 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.

  • 11 - Walls

  • from Part 1 - Reinforced concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 296-310

  • 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.

  • 10 - Columns

  • from Part 1 - Reinforced concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 255-295

  • Summary

    Introduction

    Columns exist in all conventional building structures. Whereas beams, slabs or even trusses may be used to span the floors, columns carry loads vertically, floor by floor, down to the foundations. Even in specialised systems such as shear-wall, shear-core and framed-tube structures, columns are used to support parts of the floor areas.

    Figure 10.1(1)a shows a portion of a three-dimensional building frame. For the purposes of discussion on the role of columns, the frame may be taken as representative of other popular building systems, such as multistorey flat slabs, as well as beam/slab and column structures. At each level, the floor spans in both the x and z directions. As a result, bending occurs in both the x–y and y–z planes. Thus, for a typical column, AB, the forces acting at the top end, or joint A, include:

  • • N, the axial force equal to the portion of the vertical load (from the floor immediately above) to be carried by column AB plus the axial load transmitted by the column above (i.e. column CA)

  • • Mx, the bending moment about the z-axis

  • • Mz, the bending moment about the x-axis.

    • These are illustrated in Figure 10.1(1)b. Note that a similar set of end forces also exists at the bottom end (joint B).

    These three-dimensional forces are statically indeterminate, and computer-based structural analysis procedures are normally relied on to determine their values. Because of the interaction between axial force and bending moments, the analysis and design of reinforced concrete columns, subjected to either uniaxial or biaxial bending, are considerably more complicated than the treatments for beams. In addition, the stability of slender columns and problems associated with column side-sway need to be considered. Consequently, analysis and design formulas for columns are nonlinear and the process is cumbersome. It is impossible to carry out a direct rigorous design of a reinforced concrete column. In practice, with the use of column interaction diagrams, a trial-and-error approach based on a preliminary column section can be used. A computer may also be programmed to do this rather tedious work.

    In this chapter, the basic strength equations for centrally loaded columns and columns subjected to uniaxial bending are given in detail, together with the procedure for constructing the column interaction diagram. As advanced topics, two methods of analysis for arbitrary sections are included: the iterative approach and the semi-graphical procedure.

  • Preface to the first edition

  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp xix-xx

  • Summary

    Introduction

    of the contents of this book were originally developed in the late 1980s at the University of Wollongong, New South Wales. The contents were targeted towards third-year courses in reinforced and prestressed concrete structures. The book was believed useful for both students learning the subjects and practising engineers wishing to apply with confidence the then newly published Australian Standard AS 3600–1988. In 1995 and following the publication of AS 3600–1994, the contents were updated at Griffith University (Gold Coast campus) and used as the learning and teaching material for the third-year course, ‘Concrete structures’ (which also covers prestressed concrete). In 2002, further revisions were made to include the technical advances of AS 3600–2001. Some of the book's more advanced topics were used for part of the Griffith University postgraduate course, ‘Advanced reinforced concrete’.

    In anticipation of the publication of the current version of AS 3600, which was scheduled for 2007, a major rewrite began early that year to expand on the contents and present them in two parts. The effort continued into 2009, introducing in Part 1 ‘Reinforced concrete’, inter alia, the new chapters on walls, as well as on footings, pile caps and retaining walls, plus an appendix on strut-and-tie modelling. In addition, a new Part 2 had been written, which covered five new chapters on prestressed concrete. The entire manuscript was then thoroughly reviewed and revised as appropriate following the publication of AS 3600–2009 in late December 2009.

    In line with the original aims, the book contains extensive fundamental materials for learning and teaching purposes. It is also useful for practising engineers, especially those wishing to have a full grasp of the new AS 3600–2009. This is important, as the 2009 contents have been updated and expanded significantly and, for the first time, provisions for concrete compressive strength up to 100 MPa are included. The increase in concrete strength has resulted in major changes to many of the analysis and design equations.

  • 3 - Analysis and design of rectangular beams for bending

  • from Part 1 - Reinforced concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 19-60

  • Summary

    Introduction

    The ultimate strength theory underpins the analysis and design of reinforced and prestressed concrete structures, and has been since the promulgation of the Concrete Structures Standard, AS 3600–1988. The fundamentals of the theory are described in this chapter and its application is demonstrated in detail using beams with rectangular cross sections. The discussion covers singly and doubly reinforced beams.

    Definitions

    Analysis

    Given the details of a reinforced concrete section, the ultimate resistance (Ru) against a specific action effect or combination of loads may be determined analytically using a method adopted by a code of practice or some other method. The process of determining Ru is referred to as analysis.

    Design

    Given the action effect (S*) for moment, shear or torsion, and following a computational process, we can arrive at a reinforced concrete section that would satisfy AS 3600–2009 (the Standard) requirement as given in Equation 1.1(1): The process of obtaining the required section of a member is called design.

    Ultimate strength method

    If a given action effect at any section of a reinforced concrete member under load is severe enough to cause failure, then the ultimate strength has been reached. The primary responsibility of a designer is to ensure that this will not occur.

    The ultimate strength method incorporates the actual stress–strain relations of concrete and steel, up to the point of failure. As a result, such an analysis can predict quite accurately the ultimate strength of a section against a given action effect. Because the design process can be seen as the analysis in reverse, a design should fail when the ultimate load is reached. The capacity reduction factor (ϕ) is used in Equation 1.1(1) to cover unavoidable uncertainties resulting from some theoretical and practical aspects of reinforced concrete, including variation in material qualities and construction tolerances.

    Ultimate strength theory

    Basic assumptions

    The following is a summary of the basic assumptions of ultimate strength theory:

  • • 1.Plane sections normal to the beam axis remain plane after bending; that is, the strain distribution over the depth of the section is linear (this requires that bond failure does not occur prematurely – see Chapter 8).

  • • 2.The stress–strain curve of steel is bilinear with an ultimate strain approaching infinity (see Figure 3.3(1)a).

  • 8 - Bond and stress development

  • from Part 1 - Reinforced concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 171-184

  • Summary

    General remarks

    An assumption is made in the analysis and design of beams in bending that plane sections normal to the beam axis remain plane after bending (see Section 3.3.1). This enables the establishment of the compatibility condition, which is fundamental to the derivation of bending analysis and design equations. The compatibility condition can be maintained only if there is a complete bond between the concrete and the longitudinal steel bars.

    Inadequate bond strength will lead to premature failure of the beam in bending or in shear; it will invalidate the analysis and design equations for reinforced concrete.

    Anchorage bond and development length

    The concept of anchorage bond can be readily explained using the free-body diagram of a beam segment illustrated in Figure 8.1(1).

    If at section a-a (Figure 8.1(1)), the steel bar of an area Ab is to be able to develop the yield strength (fsy), the resultant of the ultimate bond stress (u) around and along the embedded length must be at least equal to the tensile force (T). Otherwise, the bar would be ‘pulled out’ of the concrete, resulting in bond failure.

    By considering the equilibrium of the horizontal forces, we have

    from which

    where lb is the basic development length, db is the diameter of the bar and u is the apparent bond strength.

    Mechanism of bond resistance

    Amongst other topics, the book by Park and Paulay (1975) contains a wealth of information from published literature on bond and the reader is referred to that all-encompassing book for an in-depth study. Only a brief discussion on the nature of bond is given here, so that the reader can better understand the design equations recommended in AS 3600–2009 (the Standard).

    For plain reinforcing bars of Class N or L, bond resistance is provided by:

  • • cohesion or chemical bond between mortar and the bar surface

  • • the wedging action of dislodged sand particles at the interface between steel and concrete

  • • friction between steel and concrete.

  • 6 - Ultimate strength design for shear

  • from Part 1 - Reinforced concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 123-154

  • Summary

    Introduction

    This chapter deals with the ultimate strength design of reinforced concrete beams for shear. The derivation of relevant formulas and application of the Standard recommendations are presented in some detail. The design examples cover both the transverse and longitudinal shear.

    Transverse shear stress and shear failure

    Principal stresses

    For a beam in bending as depicted in Figure 6.2(1)a, the bending stress is given as

    and the shear stress as

    where M and V are the moment and shear force at the beam section; y is the distance from the neutral axis (positive downwards); Q′ is the first moment of the area beyond the level where shear stress (v) is being computed, with respect to the neutral axis; b′ is the width of the section at the level in question; and I is the moment of inertia.

    In Figure 6.2(1)a, Element 1 is located at the neutral axis level and Element 4 is at the extremity of the tensile region, whereas Elements 2 and 3, respectively, are somewhere within the tensile and compressive zones of the beam. All of the elements are, at a given beam section, a distance x from the left support. The bending and shear stresses acting on each of the elements are readily computed using Equations 6.2(1) and 6.2(2), respectively. Figure 6.2(1)b shows the free-body diagrams of these elements of the beam. For each of the free-bodies, a Mohr circle can be constructed (see Figure 6.2(1)c), which yields the magnitudes and orientations of the principal stresses (fmax) and (fmin). Figure 6.2(1)d shows the four free-body diagrams acted upon by the principal stresses.

    It is obvious in Figure 6.2(1)d that, as concrete is weak in tension, the principal tensile stress (fmax) will cause cracking in each element in a direction 90° to itself. In Element 1, the direction of the crack is 45° anticlockwise from the direction of the neutral axis (plane) as indicated by the letters NA. This crack is caused by shear stress alone. On the other hand, only bending stress exists in Element 4 (i.e. fmax = f1).

  • 7 - Ultimate strength design for torsion

  • from Part 1 - Reinforced concrete
  • Yew-Chaye Loo, Griffith University, Queensland, Sanaul Huq Chowdhury, Griffith University, Queensland
  • Book:
    Reinforced and Prestressed Concrete
    Published online:
    12 December 2018
    Print publication:
    26 July 2018, pp 155-170

  • Summary

    Origin and nature of torsion

    Torsion is a three-dimensional action; it is the moment about the longitudinal axis of the structural member. Occasionally, torsional moment is also referred to as twisting moment or torque.

    In a three-dimensional structure, there are numerous situations in which torsion occurs. Figure 7.1(1) shows two typical cases.

    For the case of the cantilever bent beam or bow girder in Figure 7.l(1)a, the torsional moment (T) is produced by the transverse load (P) acting eccentrically with respect to the axis of the beam. As it is a statically determinate structure, adequate design for torsion is vitally important – collapse of the system will result if failure in torsion occurs.

    The grillage system shown in Figure 7.1(1)b is often used for beam-and-slab floor structures. The system is statically indeterminate. Torsion of the girder is a result of the unbalanced end moments at C of the two cross beams spanning in the z-direction. Note that, for convenience, torsional moments may be indicated in the x–y plane by double-headed arrows following the right-hand screw rule. As the system is statically indeterminate, failure of beam AB in torsion would not automatically mean collapse of the grillage. However, serious serviceability problems of the beam (torsional cracking) can be expected as well as the redistribution of bending moments in the two cross beams (DC and CE).

    Torsional reinforcement

    Figure 7.1(2) depicts a rectangular beam in pure torsion. On each face of the beam, shear stresses are induced by the torsional moment. At the instance when the principal tensile stress exceeds the tensile strength of concrete, cracking occurs at 90° to the principal stress direction.

    Typical failure patterns of beams in pure torsion are shown in Figure 7.1(3). These are typical results of the tests carried out by Lei (1983). It is obvious that, on each face of the beam, the torsional shear cracks are inclined with respect to the axis of the beam. Thus, the use of closed ties only is effective in resisting torsional moment. Longitudinal steel placed behind the four faces of the beam, through dowel and other actions, is also helpful in resisting torsion.

Page 30 of 133