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38520 results in Cambridge Textbooks

Page 1170 of 1926

  • 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 1170 of 1926