General remarks

As discussed in Section 1.2, in addition to strength, durability and fire resistance, serviceability is a design requirement specified in AS 3600-2009 (the Standard). Practical recommendations are given in Clauses 8.5, 8.6 and 8.9 of the Standard, for the treatments of beam deflection, crack control and slenderness limits for beams, respectively. The Standard also touches very briefly on the vibration of beams, stating qualitatively in Clause 8.7 (and in Clause 9.5 for slabs), that ‘vibration of beams shall be considered and appropriate action taken where necessary to ensure that the vibrations induced by machinery, or vehicular or pedestrian traffic, will not adversely affect the serviceability of the structure’.

Although vibration topics are not dealt with in this book, the reader may refer to articles published by Chowdhury and Loo (2000, 2006), and by Salzmann et al (2003) for the damping characteristics of simple and continuous concrete beams.

In this chapter, details are provided on short and long-term deflection calculations, and on alternative design requirements of maximum span/effective depth ratio, in accordance with the Standard. The analysis of the total deflection of beams under repeated loading is also introduced as an advanced topic. For completeness, crack control of beams is discussed in some detail. All the deflection topics are supplemented by worked out examples. Recommendations for computing effective flange widths have been presented in Section 3.7.2. The reader is referred to the Standard for discussions on slenderness limits and vibrations.

Introduction

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 8.1(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 8.1(l)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, one metre. This is illustrated in Figure 8.1(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.

Introduction

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 6.1(1) shows two typical cases.

For the case of the cantilever bent beam or bow girder in Figure 6.1(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 6.1(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).

General remarks

It is recognised in Section 12.1 that prestressing tendons (either in the form of wire or strands of wire) reinforce the weaknesses of concrete in an active manner. Due to this, considerable concentrated forces are exerted at the extremities of a prestressed beam. At the end zones, these forces in pretensioned beams translate into intensive bond stresses in the steel concrete interface. In posttensioned beams, they induce acute lateral tensile stresses and the anchor heads (see Figure 12.4(3)a) create high bearing stresses on the concrete ends.

These stresses need to be fully considered and carefully designed for, to prevent cracking and even premature failure in the end zones. A properly reinforced end zone is referred to as an end block.

The nature and distribution of the bond stress in the end zones of a pretensioned beam are given in Section 16.2, which also includes the design method recommended in AS 3600-2009 (the Standard). Section 16.3 identifies the three types of stresses induced by a posttensioned anchorage system. These are the bursting stress and the spalling stress, both of which are tensile, and orthogonal or transverse to the axis of the posttensioned tendon. There is also the bearing (compressive) stress on the concrete behind the steel anchor head. The design for the bursting, spalling and bearing stresses is discussed in Section 16.4. Finally, the distribution and detailing of the end block reinforcement are presented in Section 16.5. Pretensioned beams For a pretensioning system, the transfer of the prestressing forces to concrete is achieved mainly by bond along the ‘transmission length’.

Introduction

In the past, when design provisions were lacking, wall panels were usually taken as non-load-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 10.1(1)a depicts a wall under vertical in-plane and lateral bending loads; Figure 10.1(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.

General remarks

Prestressing may be seen as an elaborate and active way to reinforce the weakness of concrete in tension. 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-prestress can cause cracking or perhaps failure before even any external loading is applied. As a result, prestressed concrete analysis and designing is 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 13 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).

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, it is often required to cut slopes 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 sometime 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 11.2. The treatments for footings supporting single and multiple columns are given in Section 11.3 whereas Section 11.4 deals with pile caps. Illustrative and design examples are given to highlight the application of the analysis and design procedures.