Structural systems: behaviour and design

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Structural systems: behaviour

and design

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First published 2010

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Structural systems: behaviour and design. Volume 1: Plane structural systems. Stavridis L.T. ISBN: 978-0-7277-4106-6 Structural systems: behaviour and design. Volume 2: Spatial structural systems, foundations and dynamics. Stavridis L.T. ISBN: 978-0-7277-4107-3

Structural analysis with ﬁnite elements. Rugarli P. ISBN: 978-0-7277-4093-9

Steel—concrete composite buildings: Designing with eurocodes. Collings D. ISBN: 978-0-7277-4089-2

Designers guide to eurocode 1: Actions on bridges. EN 1991-2, EN 1991-1-1, -1-3 to -1-7 and EN 1990 Annex A2. Calgaro J.-A., Tschumi M., Gulvanessian H. ISBN: 978-0-7277-3158-6

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epi tosout! to akribÝ& epizhtein o´son h tou pr_Ügmato_{& ’}_ýsi_{& epid}_Ýcetai’’

AristotÝlh& ‘‘Because it is the essence of education to seek as much accuracy as the nature of things allows’’ Aristoteles

Preface

What a technically educated person (engineer, architect or constructor) understands today by the term ‘structural design’ is practically the same as what his fellow man meant 500 — or even more — years ago, namely a procedure based on the application of a particular item of knowledge, and because of which a structure will ‘stand up’ and not ‘fall down’, under whatever actions it may be exposed to during its life span.

However, what changed over the years and took the so-called ‘structural theory’ out of the realm of empiricism and intuition was the introduction of analysis in the assessment of structural behaviour and its elevation to an applied scientiﬁc ﬁeld. Of course, the evolution of computational methods together with the wide availability of computer facilities played decisive roles in this.

Structural mechanics is now a highly demanding subject, not only from the point of view of its analytical treatment regarding structural behaviour but also with regard to its evaluation and practical application in structural design. Both of these directions have quite distinct characteristics.

The analytical approach always poses the question:Given a structural conﬁguration and a

certain loading, what is the response and its corresponding deformations? This is a problem governed by strict analytical conditions and requirements, establishing in this way the scien-tiﬁc character of the subject matter, but sometimes creating the illusion that the process of analysis is an end in itself. Of course, the solution to this problem is nowadays ensured, due to powerful numerical methods and the wide availability of personal computers.

On the other hand, the practical aspects focus on the application of load-carrying behaviour in the conceptual and working design of structures, by posing the essential

question: Given the physical environment and the prevailing service requirements, what

structural system made of the appropriate materials will meet the necessary load-bearing requirements in an economical and aesthetically satisfactory way? This problem does indeed constitute an end in itself.

It is the first approach which has become established, over the years, in education for many reasons, all of them hinging on the fact that it is this approach which ensures the appropriate ‘scientific’ profile. The second approach, although more realistic, seldom attracts the attention it deserves in the educational civil engineering curricula, a fact that the student unfortunately becomes aware of at a rather later stage, to his or her disappointment.

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Indeed, the almost exclusive concentration on the computational aspect of structural mechanics dramatically deprives the young and inexperienced engineer from the physical perception of the load-carrying characteristics of a structural system, something which is not due solely to his or her lack of experience. The student concerned, being used to the study of complicated computational scenarios, tends to lack the mental clarity that permits the direct structural perception which is so often required either by the collaborating architect or the constructor on the building site.

In this book, an approach to the understanding of load-carrying mechanisms and the behaviour of a wide range of structural systems is presented, with subsequent application to relevant design decisions, which, relying in principle on physical comprehension, is carried out through simple analytical reasoning. However, despite the prevailing non-computer-oriented philosophy, it is recognised throughout this book that the computational procedure in the design ofﬁce practice needs in every case to be under-pinned by the appropriate computer software. A necessary condition, though, for the successful use of such software is the ability of the user to derive, even approximately, results based on a well-cultivated structural perception regarding the load-carrying action and the subsequent preliminary design, which should in any case precede any computer modelling and the corresponding computation: this is exactly what the present book aims at.

Thus, through the study of basic load-bearing actions and the behaviour of typical systems, such as simply supported and continuous beams, frames, arches, cable structures of any type, grids, plates, shells, rectilinear and curved thin-walled beams, and multi-storey systems, an attempt has been made to bring insights into these load-carrying characteristics that are necessary for their design and are usually overshadowed by a strictly analytical examination. In this respect, particular attention has been given to the use of reinforced and prestressed concrete, as well as to composite structures, in addition to the ‘traditional’ consideration of steel as the most ‘convenient’ material. Due attention has also been given to the plastic analysis and design of skeletal structures as well as to second-order theory and stability effects, with an emphasis on their practical and efﬁcient use.

However, it has been deemed an absolute necessity to consider in advance and in some detail the handling of plane skeletal statically determinate and indeterminate structures in two respective chapters. Moreover, the dynamic behaviour of discrete-mass structures has been examined in the penultimate chapter, where not only the response of multi-storey systems to earthquake but also that of beams and plates to human-induced dynamic actions as well as to machine operation is discussed.

Of course, complete consideration of structural design must also include the founda-tion problem. Thus, the ﬁnal chapter is dedicated to this issue, dealing primarily with the behaviour and design of shallow foundations, but also with soil—structure interaction and, to a lesser extent, the pile foundations.

It is hoped that, through the systematic examination of the above subjects, the basic elements of structural perception are emphasised which permit the safe preliminary design and dimensioning of any structure, such as a bridge, building, rooﬁng of a large space etc. These elements constitute the basis of not only appropriate computer

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modelling but also of the deliberate acceptance — or not — of the numerical results produced by software, which is a very important issue.

This book follows a strictly progressive path in the presentation of various subjects. Thus, later chapter build only on previously examined concepts, and some basic knowledge of elementary mechanics is considered a prerequisite.

As is well known — since the ﬁrst centuryBC,from the Roman architect Vitruvius — a

successful structural concept requires the satisfaction of four characteristic properties,

namely technical safety, functionality, economy and aesthetic quality. Technical safety

means that the available strength should be greater than the resulting response; functionality means, structurally, the control of different deformational characteristics, annoying vibrations included; economy means the successful selection of the structural and foundation system, as well as of the appropriate construction method; and aesthetic quality means the achievement of structural elegance.

While the ﬁrst two criteria are the subject of a knowledgeable technical analysis, successfully meeting the last two requires, on the part of the engineer, ingenuity, creativity and aesthetic judgment — properties not acquired by studying the prevailing design characteristics of the structural systems but are nevertheless prerequisites. Thus, although the successful design of a structure is based on the technical and aesthetic talent of the engineer, the knowledge of the structural principles put forward in this book are an essential basis for any such endeavour.

This book is aimed at everyone engaged in the study of structural analysis and design, either as a student, a practising engineer, or even as an architect seeking for a more profound structural understanding of bridge or building engineering. I hope that this books proves useful and achieves its goals.

Finally, I would like to express my gratitude to Thomas Telford Ltd for undertaking the publication of this edition of this book and the excellent collaboration with the editorial team, whose authoritative and knowledgeable direction under Matthew Lane has led to an entirely satisfactory result.

L.T. Stavridis Athens, June 2010

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1 Introductory concepts

1.1 Loads

Loads constitute theraison d’eˆtre of structural systems, and therefore their examination

precedes anything else. Structures are built and designed in such a way as to carry, safely and in a functionally satisfactory way, only certain loads, and it is the responsibility of the designer to prescribe the loads which the structure is logically expected to be exposed to during its life.

The determination of loads raises a difﬁcult problem. Because of the obvious need to design structures by loading them in a manner agreed in advance for each category, structural loads are subject to regulations which typically differ from country to country. Thus, for example, the same bridge would be designed for different loads in the USA than in the UK or in Japan.

Before examining the causes of the existence of loads, a distinction betweenstatic and

dynamic loads should be made.

A loadP is considered to act statically when the time t1needed for its full

develop-ment is deﬁnitely longer than thefundamental period T of the structure. This corresponds

practically to the time it takes for a structure to perform a complete oscillation, when an arbitrary deﬂection from its equilibrium situation is imposed on the structure and then left free to oscillate (Figure 1.1).

Thus a wind gust increasing from zero to its maximum in 3 s represents a static force for a short, stiff building having a fundamental period of 0.5 s, whereas for a tall, ﬂexible building having a fundamental period of 6 s the wind gust must be considered as a dynamic loading. It is clear that the way in which a dynamic loading is handled differs radically from that of a static one, for the simple reason that, because of the induced permanent motion, inertia forces are developed, which depend — at each moment — on the varying corresponding displacement of the structure. It is because of this complexity that it is always preferable, when feasible, to replace a dynamic loading with an equivalent static one.

Loads, from consideration of their natural origin, can be categorised as below.

(1) Gravity loads (g¼ 9.81 m/s2)

Loads which consist of the weight of the permanent components of a structure are

characterised as dead loads. They are determined according to the speciﬁc weight of

each material (e.g. reinforced concrete¼ 25.0 kN/m3, steel¼ 78.5 kN/m3). These

loads are clearly static ones.

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Loads due to the occupancy or the use, in general, of mainly horizontal surfaces are

characterised aslive loads. This use may arise from human activities (rooms in residences

or ofﬁces with equipment, people on pedestrian bridges or standing in stadiums, etc.) or from equipment installed in industrial spaces, or it may represent trafﬁc loads due to moving vehicles on road bridges, as well as trains on railway bridges.

These loads are expressed either as surface loads (kN/m2), or as concentrated loads

(kN). They are termed ‘live’ because they may change their position in the structure permanently. This fact always poses the question of the conﬁguration of these live loads, which then leads to consideration of the most unfavourable response in a structure.

Live loads — especially moving trafﬁc — are usually considered to act in a static manner, even if, from their very nature, they act dynamically. This is accounted for

by an additional percentage of the acting vehicle weight, known as theimpact factor.

However, these trafﬁc loads also always induce horizontal braking forces, which usually constitute about 5% of their corresponding weight.

Normal human activities on speciﬁcally formed surfaces, as occur in rooms used for gymnastics and in dance halls, as well as on pedestrian bridges, also have a dynamic inﬂuence.

(2) Loads arising from the soil that surrounds and/or supports the structure . Earth pressure is exerted on (mainly) vertical surfaces of the structure that come into

contact with the surrounding soil. Earth pressure basically arises from the fact that a soil volume resting freely on a horizontal level usually shows a more or less formed slope, the free sides of which are standing in a state of equilibrium (Figure 1.2).

This physical slope angle generally depends on the nature of the soil; for example,

for sands this angle is roughly equal to 308. Thus, when this sloped surface is

bound, through a structural surface, to be retained at a different angle to the physical one — usually vertically — the soil exerts on that surface a force that is stronger the greater the divergence from the physical slope angle.

. This situation is in fact similar to that of the hydrostatic pressure, where the corre-sponding ‘physical slope angle’ obviously equals zero. This is why the earth pressure

P

t t1

t1 > T Static loading t1 < T Dynamic loading

When the system is left free it will vibrate with the fundamental period T

P

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(kN/m2) is determined as the product of the height of soil mass, times its speciﬁc weight, multiplied by a factor less than that applied for water, i.e. less than unity. Such additional pressures also act on a retaining wall surface, when live loads have to be considered on the free horizontal soil surface (see Chapter 16).

. The soil, on which the structure is supported, may exhibit settlement due to causes which are either unrelated to the structure (e.g. groundwater lowering) or related to the deformability of the structure under loads acting on it. In the latter case, there is a substantial interaction between the foundation soil and the structure. Whatever the case, soil settlement deﬁnitely constitutes a loading on the structure, causing internal forces within it, unless the structure is statically determinate (as will be examined in Chapters 2 and 3).

. During an earthquake the foundation soil exhibits horizontal and/or vertical vibra-tions, which are imposed on the foundations of the structure. As a result of this dynamic loading, the whole structure is subjected to strong internal forces. As will be examined in Chapter 16, it is the imposed accelerations on the foundation that actually cause this state of stress in the structure; the ensuing soil settlement may be an additional factor, as explained above.

(3) Loads arising from the aquatic environment

The acting water may be either free, as in the case of a marine structure or a dam, or contained in the surrounding soil. The main effect of water is the hydrostatic pressure, which acts on every surface embedded within a water-containing environment. In the case of strong water motion relative to the structure, additional hydrodynamic pressures come into play.

If the structure is conﬁned with its lower part in soil containing groundwater, buoyancy forces develop, which are merely the upward hydrostatic pressure on the lower horizontal surface of the structure (Figure 1.3). Although the buoyancy force decreases the weight of the structure acting on the soil, the lower surface of the structure is acted on by the unreduced weight in the upward direction.

(4) Loads arising from climatic conditions

. Wind forces represent basically a horizontal loading that is dynamic in character, as explained above. The wind force experienced by a structure is directly related to

Retaining wall

Free slope surf ace

Underlying soil

Earth pressure

Slope angle

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the form of the structure and its height above the earth’s surface. A thorough determination of wind forces is particularly difﬁcult, as the forces may interact with the ensuing structural deformations (e.g. in the case of cable-suspended bridges), leading to complex aerodynamic calculations. However, in most cases wind pressures can be determined from the appropriate structural codes.

. Snow — in fact a gravity loading — is taken into account by means of a prescribed pressure on a roof, according to the appropriate structural codes. The prescribed pressure values take into account the geographic region as well as the altitude of the speciﬁc site.

. Temperature variation may be uniform (an increase or reduction) or linearly distributed across the thickness of a structural element. It should be noted that the unavoidable shrinkage of concrete during its hardening may be simulated by a

uniform temperature fall of 208C. This causes a state of stress in the structure,

which may not be negligible (see Chapter 15). A non-uniform temperature change always appears in structural elements exposed to the atmosphere (space covering roofs, bridges, etc.). It should be noted that, as in the case of soil settlement, a temperature variation in a statically determinate structure does not give rise to any internal forces at all (see Chapter 2).

(5) Special impact loads

Impact loads are dynamic in nature and are due to events such as an explosion, a collision of a body with the structure (e.g. a ship colliding with a bridge pier), etc.

The slab is acted on by the full (unreduced) weight of the structure The soil is acted on by a reduced pressure because of buoyancy

Pressure on the slab Pressure on the soil

Water table Water table

Water pressure Water pressure

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1.2 The structural behaviour of basic materials

The structural strength of a material is determined on the basis of the relationship

between the stress and the strain " experienced by an appropriate bar specimen

subjected to pure tension or pure compression.

Consider a specimen with a lengthL and a section A, subjected to a force F acting

along its longitudinal axis. Assuming a uniform distribution of the force on the

section, the concept ofstress is introduced as

¼ F/A (kN/m2₎

In this way the stress is uniformly distributed over the whole section A of the specimen

(Figure 1.4). The force F concerns the whole specimen externally, while the stress

characterises the internal state of stress of its longitudinal ﬁbres. It may be considered

that the application of the force F and the development of the stress over the

whole sectionA represent equivalent factors, given that F¼ A.

The result of the application of the forceF is an elongation or shortening of the

specimen, depending on whether F is applied as a tensile or as a compressive force.

Thus the concept ofstrain" may be introduced as

" ¼ /L (non-dimensional number)

The relationship between with " is obtained experimentally, by loading the

specimen progressively and at a constant rate, starting from a null loading level and

continuously recording the pairs of magnitudes and " on a diagram over two

ortho-gonal axes, as they are calculated on the basis of the measured values ofF and.

1.2.1 Steel

Figure 1.5(a) shows the experimentally obtained diagram for the behaviour of steel. The diagram represents a test done on steel in tension, but applies equally to steel in

compression. It is clear that, for each pair of values (, "), the force F and the

Deformed specimen L L δ σ F F A A σ = F/A ε = δ/L

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corresponding deformation can be determined as

F¼ A and ¼ " L (a)

It should be noted that the test is made by imposing on the specimen a speciﬁc defor-mation " rather than a stress . The rate of application d"/dt of " is approximately 5 105/s. It can be seen from Figure 1.5(a) that during the test four distinct behaviours are observed.

In region (1) the stress is proportional to the strain ", which means — as will be

shown later — that the corresponding force F is also proportional to the deformation

. This fundamental statement, widely known as Hooke’s law, can be written as ¼ " E

The magnitude ofE is derived geometrically as the slope of the straight segment (1). It

represents the most basic quantitative characteristic of the load-carrying behaviour of

the material and is called the modulus of elasticity. It can be thought of as a measure

of the resistance (stiffness) offered by the material against its unit elongation or short-ening. In the case of steel, the value ofE is approximately 2.1 108kN/m2.

Substituting the above two equations (a) into the preceding equation gives the

relationship between the external forceF and the deformation of the specimen:

F¼ (E A/L)

The expression (E A/L) is called the axial stiffness of the specimen (kN/m). It gives the

force required (kN) to deform (either elongate or shorten) the specimen by 1 m; in other fy fy σ εy εy εy ε: ‰ εpl εtot σ εtot= εy+ εpl

Idealised_{σ–ε diagram for steel} (b) 2.0 5.0 8.0 E Unloading (a) (1) (3) (4) (2) 0.10 ε

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words, it gives the resistance (stiffness) (kN) that the specimen offers when subjected to an axial deformation of 1 m.

The last equation may also be written in the form ¼ F (L/E A)

which shows how the axial deformation of the specimen can be calculated from the

externally applied force F. The expression (L/E A) is called the axial ﬂexibility (or

deformability) (m/kN). It gives the axial deformation of the specimen (m) when an axial force of 1 kN is applied. It is clear that these two concepts (and values) of ‘stiffness’ and ‘ﬂexibility’ are inversely related.

The behaviour of the steel in region (1) is calledlinear behaviour. This is a very

con-venient material property because it is governed by an absolute proportionality between the forceF (or stress) and the deformation (or strain "). Thus if the force F (or the stress) is multiplied by a factor k, the deformation (or the strain ") is also multiplied by the same factor. However, this property is valid only up to a certain stress levelfy, at

which point region (2) of the behaviour of the steel begins (see Figure 1.5(a)).

Up to the point where the stress reaches the value fy, if the load is removed from

the specimen the deformation (as well the strain ") will return to zero. In other

words, up to this point the deformation is reversible, and for this reason the region up tofyis called theelastic region. For bars used in concrete reinforcement the value of fy

may be considered equal to 420 N/mm2, and the corresponding "y, i.e. (fy/Es), is

about 2%. Representative values of fy for the two basic quality types of structural

steel are 240 N/mm2and 360 N/mm2.

Once the stress has reached the value fy, the specimen can be deformed further

without an increase in the applied force (i.e. stress). The steel exhibits zero stiffness (the slope of the graph is equal to zero), which means it ‘yields’, and so the stress fy is

called the yield stress. This state prevails up to a value of" of roughly 10%, where the steel acquires a little stiffness again, as can be seen from the small slope at the beginning of region (3). To obtain a further increase in the stress", a small increase in (i.e. in the force F) is required, and this property of the steel is called hardening.

After region (1), the steel no longer exhibits ‘elastic behaviour’. Removing the loading from the specimen at any time will lead to a course parallel to the ‘elastic region’, and so will not lead to zero deformation but to a ‘remaining’ deformation, represented by the strain "pl. Thus the specimen cannot revert to its initial length and

will be either a bit longer or a bit shorter, according to whether the stress imposed was a tensile or compressive one. Therefore, it is said that the material has undergone a plastic deformation, i.e. a plastic strain "pl. It is clear that the total deformation of the

specimen is represented by the total strain"tot(see Figure 1.5(a)), which is given by

"tot¼ "yþ "pl

The material property of yielding, i.e. of deformability under the constant stressfy, is a

very important structural property of steel, called theductility, which confers beneﬁcial

effects on the load-carrying capacity of structures. This holds not only for steel structures but also for reinforced or prestressed concrete structures, as will be explained later with

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respect to the plastic behaviour of these materials. Now, regarding the very small slope of region (3), this minimal stiffness is considered to be practically zero, exactly as in the

yield state, and consequently the very small increase in stress fy in this region is

simply ignored. So, the state of yielding can be considered to extend up to values of"

of about 10%, but for practical design purposes the value is taken as 8%. However,

for the detailed control requirements in an investigational study, the aforementioned hardening property of steel must be taken into account.

It can be see from region (4) in Figure 1.5(a) that to increase the deformation (i.e. of strain") further requires increasingly less force F (i.e. stress ), until the point where the specimen fails. This region is of no practical interest for the design of structures.

From the foregoing it can be concluded that, for practical purposes, the maximum

level of stress on the steel must not exceed the valuefy. Thus in practice the behaviour

of steel can be taken as the behaviour of a homogeneous, ideal, elastic—plastic material (see Figure 1.5(b)). Such behaviour characteristics are used in the development of the classic methods of structural analysis.

This idealised material behaviour is generally used in this book, but special attention will be given to those peculiarities of concrete structures which do not necessarily conform to this ideal behaviour.

The stress—strain diagrams in Figure 1.6(a) are for naturally hard steel, the strength of which is due entirely to its chemical composition. This quality of steel is used for both structural elements and reinforcing bars. However, when prestressed concrete is used, a prestressing steel of very high strength is needed (for reasons that will be explained later), and this is obtained through a cold forming procedure.

The stress—strain diagram for high-strength steel is shown in Figure 1.6(c). Comparing this with the graph for natural steel, it is immediately observed that there is no ‘yield plateau’, which means that there is not a clearly recognisable yield stress in the way

that has been previously established. However, a yield stress fPy may be considered,

which corresponds to the point where removing the load from the specimen will lead to

a remaining plastic strain "pl of 2%. This yield stress may be considered as constant

after the value "Py 8% has been reached, as shown in the idealised diagram in

Figure 1.6(b). For practical purposes the actual ultimate tensile strength of the prestressing steel beyondfPycan thus be ignored.

1.2.2 Concrete

Steel exhibits practically the same stress—strain diagram whether the force applied is a tension or compression force. However, the behaviour of concrete under tension is totally different from its behaviour under compression, as the compressive strength of concrete is much higher than its tensile strength. The stress—strain diagram for a specimen of concrete under compression is shown in Figure 1.7.

It is seen that up to a loading level roughly equal to one-third of the maximum

compressive strength fc the concrete exhibits linear elastic behaviour. The constant

slope in this region represents the value of the modulus of elasticity of concrete Ec,

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Reinforcing steel Structural steel E ~E ~240 ~400 σ: N/mm2 σ: N/mm2 σ: N/mm2 ~1600 ~1600 2‰ 2‰ 8‰ ε 8.0 10.0 _{ε: ‰} ε εPy

Idealisedσ–ε diagram for prestressing steel (b) (a) (c) fPy Prestressing steel Unloading

Figure 1.6 Characteristics of the behaviour of reinforcing and prestressing steel

1.5–2.5‰ σ: N/mm2 Linear behaviour ~fc/3 ~20 to ~45 fc ε Figure 1.7 Stress—strain diagram for concrete under compression

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After the point fc/3 and up to the highest sustainable stress fc the relationship

between and " is no longer linear, and consequently Hooke’s law does not apply.

The strain corresponding to the highest stress levelfc lies roughly between 1.5% and

2.5%. The stress fc represents the compressive strength of the specimen. Any further

deformation of the concrete requires an applied stress of less thanfc. This fact makes

concrete in this last region of the stress—strain diagram, from a design point of view, effectively useless.

An interesting fact observed experimentally is that increasing the speed of

imposing the deformation increases the maximum sustainable stress (i.e. strength) fc.

It is found that if a stress" is imposed at a speed of 102to 101/s — which corresponds

to the rate during an earthquake — there is an increase in the compressive strengthfc

of 50%. It should be noted that no such increase in strength is observed in the case of steel.

As has been already pointed out, the tensile strength of concretefct is much lower

than its compressive strength. The tensile strength can generally be estimated using the empirical relationship

f_ct¼ 0:45 pffiffiffif_c ðN=mm2Þ

and is of the order of 2 N/mm2.

Obviously, the concept of a yield stress for concrete simply does not exist. Exceeding the tensile strength of concrete leads automatically to cracking. In most concrete struc-tures the load under which cracking occurs is only a part of the loading that the structure

has to support underservice conditions, i.e. under conditions of everyday use. Thus it is

necessary to place reinforcing bars in the concrete in order to overcome the tensile stresses within the structure. Despite this measure, the concrete can still crack, and this poses different requirements on the design from those for a steel-only structure.

The placement of reinforcing bars actually creates a new composite material,reinforced

concrete, which is able to carry not only compressive but also tensile internal forces. The particularities of the behaviour of reinforced concrete under axial tension are dealt with in Section 1.3.

1.2.2.1 Creep

If a compressive stress0within the elastic range is applied to a concrete specimen and is

maintained at a constant level for a long time, it will be seen that, after an initial elastic shortening, the specimen will continue to shorten as time progresses. This phenomenon

is calledcreep and is an unavoidable property of concrete (Figure 1.8).

This deformation will continue for a period of more than 1 year, and will ﬁnally take a value that is larger than the initial elastic deformation. This extra deformation due to creep depends on the initial elastic deformation "eland thecreep factor ’. The value

of ’ depends on the age of the specimen at the time when the stress 0 was applied

(accounted for by a factor k), the relative humidity of the environment (a dry

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The ﬁnal creep strain"cr, i.e. the creep strain after an inﬁnite time, can be written as (see Figure 1.8) "cr ¼ ’1 "el¼ ’1 0 E_c where ’1¼ ’ k

Depending on the age of the concrete at the time when the stress0is applied, the factor

k takes different values, being 1.8, 1 and 0.02 for ages of 1 day, 28 days and 5 years,

respectively.

It must be pointed out that this deformation is not reversible because, after the unloading of the specimen, i.e. after the removal of the stress 0 that caused "el, a

remaining deformation is observed.

Thus, the total strain"totof the specimen at an inﬁnite time is

"tot¼ "elþ "cr¼ (1 þ ’1) "el

The value of’₁normally lies between 2.0 and 3.0.g

The basic problem with regard to creep is to determine the ﬁnal strain"₁when an initial stress0is applied at time0(the ‘age’ of the concrete) and then varies (say, increases)

gradually over time until its ﬁnal value ₁ (Figure 1.9). Systematic research over the

years has led to a practical and very functional conclusion known as Trost’s proposal

(Menn, 1990) which, assuming that the principle of superposition of creep deformations is generally valid, can be expressed as (see Figure 1.9)

"1¼_E0

c ð1 þ ’1Þ þ

1 0

E_c ð1 þ ’1Þ

where the value of the coefﬁcient (which depends on 0) is of the order of 0.85.

The last relationship may be written in the form "1¼_E0

initþ

1 0

E_dif

The compressive stress of concrete remains permanently constant Immediate deformation Gradual (slow) deformation σ0 σ0 σ0 εel = σ0/E εcr = φ∞ · εel ε tot = εel + εcr εtot = σ0 · (1 + φ∞)/E = σ0/Einit

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where

E_init¼ Ec

1þ ’₁ and Edif ¼

E_c

1þ ’₁

This expression of"₁is of particular importance. First, it shows that the effect of creep is

generally equivalent to a reduction in the modulus of elasticityEcof concrete. For the

initially applied stress0a value of the modulus equal toEinitis considered, whereas for

any further gradual increment in the stress — assumed as (₁ — 0) — the value Edif

applies. In other words, the concrete responds instantaneously to an initially applied

stress through the value Ec, but over the course of time responds with the valueEinit,

on the condition that the applied stress remains constant. For any further response

due to an additional gradual variation in the initially imposed stress, the value Edif

must be applied.

In this way, an instantaneous elastic deformationw due to an initial stress results in

the valuew₁¼ w (1 þ ’₁) by applying the modulusEinit, whereas the gradual

appli-cation of an additional stress corresponding to an ‘instantaneously produced’ elastic

deformationwresults in the valuew_,1¼ w (1 þ ’₁) by applying the modulus

Edif. Of course the total deformation that actually occurs isw1þ w,1.g

When looking at the deformation properties of concrete, shrinkage must also be

considered. During its curing and drying there is a shrinking of the concrete mass, and this is expressed as the corresponding shortening strain "s. For practical purposes

the value of "s is taken as 20 105. This is equivalent to the effect of a uniform

temperature fall of 208C, assuming a thermal coefﬁcient of 105/8C, a value that is

also valid for steel.

1.2.2.2 Relaxation

The property of concrete relaxation has characteristics opposite to those of creep. More

precisely, if a shortening strain 0/Ec is imposed on a concrete specimen through an

applied compressive stress 0 and care is taken such that this deformation is kept

Additional deformation ε = ∆σ/Edif

Influence of time duration +

ε = σ0/Einit

Einit = E/(1 + ϕ∞) Edif = E/(1 + µ · ϕ∞)

ε∞

Variation of the initially applied stress (Trost’s proposal)

Gradual (slow) additional application of stress Instantaneous application of stress σ σ∞ σ0 σ0 ∆σ t τ0 t∞

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constant (i.e. is blocked), then as time progresses there is a decrease in the concrete stress (Figure 1.10). In other words, the stress required for the initially imposed specimen shortening to remain constant decreases continuously over time.

This reduction in stress under constant strain is called relaxation and is due to the

creep property of concrete. It can be explained as follows: if the initial compressive

stress 0 is held constant, the specimen should shorten further over time, according

to its creep behaviour. It is obvious that, in this hypothetical conﬁguration, in order to keep the initial shortening intact, an additional tensile stress has to be applied, which of course leads to a reduction in the initial compressive stress (see Figure 1.10). The amount of relaxation is easily determined on the basis of Trost’s relation (see Section 1.2.2.1). It must simply hold that

"1¼_E0 c ð1 þ ’1Þ þ1_E 0 c ð1 þ ’1Þ ¼_E0 c

from which it is obtained that

1¼ 0 1

’1

1þ ’₁

In this way, for’₁¼ 2.0 and with ¼ 0.85, the ﬁnal value ₁amounts to only 26% of

the initially applied stress0. This last equation plays a particular role in evaluating the

stress in statically indeterminate concrete structures under an imposed deformation, as

will be explained in Section 5.5.1. g

It should be noted that the phenomenon of relaxation occurs also in prestressing steel, wherein there is a percentage loss of the initially applied stress under constant strain. This loss depends on the bar diameter and is roughly in the region of 5%.

1.3 Behaviour of a reinforced concrete member under tension

The peculiarity of concrete as a structural material, as compared with steel, does not lie so much in the fact that the—" diagram is not linear after a certain stress level, but in

Immediate deformation σ0 Blocked deformation σ∞ < σ0 σ0 Free deformation Why the stress decreases εel

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its inability to afford a signiﬁcant tensile resistance, a fact that makes the placement of reinforcing steel bars necessary, as already noted above. Thus, an understanding of the behaviour of a simple reinforced concrete member under axial tension is of basic importance.

Consider a straight concrete member (Figure 1.11) with a reinforcing bar placed at the centre of its square section and sticking out a little at both ends of the member so that a self-equilibrating pair of tensile forcesZ can be applied at its extremities. Let AcandAs

be the cross-sectional areas of the concrete and the steel member, respectively, andEc

andEs the corresponding moduli of elasticity.

Under the action of the forces Z the steel bar becomes elongated. It is of absolutely

basic importance that this steel elongation is imposed on the concrete too. This is

achieved through the adhesion or bond forces between the steel and the concrete,

which are more efﬁcient if the surface of the steel bar is roughly formed (e.g. ribbed). The bond between the two materials ensures that all along the reinforcing bar"s¼ "c.

Obviously, if there is a total absence of a bond then "c¼ 0, which means that the

concrete would be completely uninvolved in carrying the tensile load — something that is undesirable, as will become clear.

Before the material stresses near the ends of the member are examined, a section near the middle of the specimen is considered, in order to determine the tensile stress

of the concrete c and the steel s at this point. If Zc and Zs are the tensile forces

carried by the concrete and the steel, respectively, then considering the equilibrium of, say, the left-hand part as a free body (see Figure 1.11):

Z¼ Zcþ Zs¼ Ac cþ As s¼ Ac ("c Ec)þ As ("s Es)

Bond stresses Bond stresses Bond stresses

Steel stress

Concrete tensile steel

Equivalent section Ac,i, Ec Ac, Ec Z Z Z ZS τS ZS ZC ZC Z Z_τ σc (fct) σs σ c Ib Ib Ib As, Es Z = Zs + Zc Z = Zs + Zτ Z_τ = Zc Zs < Z

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Given the identical strains of the two materials ("s¼ "c), thens¼ c (Es/Ec) and, more

speciﬁcally, on the basis of the last equation: s¼ "s Es¼ Z A_sþ A_c Ec E_s (a) and c¼ "c Ec¼ Z Acþ As E_s E_c ¼_AZ c;i (b)

From the above equation it may be deduced that the concrete behaves as if its section were homogeneous and equal to

A_c_;i¼ A_cþ A_s Es E_c

This cross-sectionAb,i, which is somewhat larger thanAc, is called theequivalent

cross-section (see Figure 1.11). The concept of the idealised cross-cross-section, which will also be used in other cases in this book, allows the material — in this case the concrete — to be considered as ‘homogenous’, by considering a cross-section that is larger than its real dimensions.

If the initial magnitude of the forceZ is small enough, producing a stress c in the

concrete that is lower than its tensile strengthfct— thus leaving the concrete uncracked

— it is immediately concluded from equation (a) that s<Z/As and consequently,

according to Figure 1.11, the tensile force on the reinforcing bar Zs¼ As s at the

considered point is always smaller than the external force Z. However, in order to

explain the longitudinal equilibrium of the reinforcing bar considered as a free body, it must be concluded, and indeed should be intuitively clear, that friction (or adhesion) stresses are developed along the bar on its lateral surface. These are better

known asbond stressess, and contribute to the overall equilibrium (see Figure 1.11).

The distribution of these bond stresses along the reinforcing bar is shown qualitatively in Figure 1.11 (Menn, 1990). It can be seen that these stresses develop over a restricted lengthlband then disappear. This lengthlb, known as thebond or anchorage length, allows

the introduction of the external forceZ into the concrete in order to produce a uniformly

distributed state of stress. The anchorage length depends on the layout of the reinfor-cement and the quality of the concrete.

The bond stresses act on the concrete in the opposite sense to the way they act on the steel reinforcement, and they explain its equilibrium as a free body, given that no force is acting at its left-hand end, while in the location of the reinforcing section the tensile force Zc¼ Ac c is acting. The integral of the bond stresses acting over the lengthlb

gives the bond force Z which, on the basis of equilibrium of the reinforcing bar, is

Z¼ Z — As s, while from the longitudinal equilibrium of the concrete part Z¼ Zc

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Because of the distribution of the bond stresses, the steel stress initially has a high value, which falls rapidly to a lower value towards the end of the anchorage length, and thereafter remains constant. Conversely, the concrete stress is initially zero and

increases gradually up to a constant value c. Obviously, this pattern of stress

distri-bution applies equally, in a symmetrical manner, to the right-hand end of the

member.g

The case is now considered wherein the applied forceZ increases gradually up to such a

value that the concrete stresscbecomes equal to the tensile strengthfct. This value of

the external forceZr, thecracking force, results from the condition

f_ct¼ Zr

A_c;i (c)

The concrete is now cracked in some place — not necessarily predeterminable — over the whole height of its cross-section (Figure 1.12) and, as the steel bar has practically no contact with the surrounding concrete at the crack, it develops the full cracking forceZr.

The steel stress in the region of the crack is increased from the value given by

expression (a) up to the valuesr¼ Zr/As. As shown in Figure 1.12, the introduction

of the force Zr at the ends of the specimen through the developed bond stresses is

applied again, in an inverse manner, i.e. from ‘inside’ towards the ‘outside’, at both

sides of the crack. A minimal increase in the external force Zr will create further

cracks, as the concrete has already reached its tensile strength limit. The same stress picture is repeated between two consecutive cracks, as if they were both ‘free’ edges with regard to the bond stresses and the development of the steel and concrete stresses. It is clear that a new crack will occur each time that the maximum concrete stress (maxc¼ fct) is reached in a region.

Ib Ib Zr Zr Zr Zr Ib Ib Ib Ib <2Ib

Next crack location

Crack Crack Crack Crack

Bond stresses Crack Zr Zr σs Steel stress Concrete tensile stress σc (fct) σc (fct)

fct

σc < fct

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If two cracks develop at a distance 2 lbapart, they will inevitably lead to a third crack

at the middle of the distance between them (see Figure 1.12). As the maximum concrete stress at the middle of the two new ‘halves’ is lower than the tensile strengthfct, it can be

concluded that the occurrence of further cracking in these two ‘halves’ is impossible. For

the same reason, if two consecutive cracks are closer together than 2 lba new crack

cannot form between them. Thuslbis the minimum possible distance between cracks,

and there is therefore a maximum number of cracks that cannot be exceeded. This

crack distribution with a crack spacing of the order of lb is known as the stabilised

crack pattern (Menn, 1990).

It should be noted that the cracking of concrete is essential for the economical use of

reinforcement. Otherwise, the steel would be limited to a working stress s0which, as

previously determined, cannot exceed the values0¼ fct (Es/Ec).

However, this process of completion of the crack pattern is feasible only when at the ﬁrst crack the steel stresssris smaller than the yield stress, i.e. if

sr¼ Zr/As< fsy

If this is not the case there will be an undesirable yielding of the reinforcement wherever the ﬁrst crack appears, together with an uncontrolled crack width. Thus maintaining the steel stress at ﬁrst cracking below the yield stress is an absolutely necessary condition for the progressive formation of multiple cracks, thus avoiding the danger of the formation of a few excessively wide cracks. This condition leads to a minimum amount of reinforcement required in a given section, independent of the tensile force acting on it. In practice, many, not always controllable, factors lead the concrete to experience tensile stress, so that the minimum reinforcement should always be used in order to protect the structure against such an undesirable situation.

This minimum reinforcementAs,minfor a concrete sectionAccan be directly

deter-mined from the last equation if the cracking force Zr is estimated ﬁrst from equation

(c) as:Zr¼ fct Ac. Then, practically:

As,min¼ ( fct/fsy) Ac

It should be emphasised that in the event of a change in the cross-section of the concrete

Acalong the examined element, the minimum reinforcement must be determined based

on the larger cross-section, as shown in Figure 1.13 for the case of a tensioned element with a hole in its interior. If region B is reinforced with a minimum reinforcement based

C B C

AB AC

Minimum reinforcement according to cross-sectional area at C

Z Z

Figure 1.13 The minimum reinforcement required in the case of a change in the cross-section of the concrete

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on the sectionAB, then the cracking force of region C, which is equal tofct AC, would

automatically lead the reinforcement in region B to fail (i.e. yield), leading to the

occurrence of uncontrollably wide cracks.g

Something that is of particular interest in the design of a reinforced concrete structure is, of course, the crack width in the regular crack pattern conﬁguration. It is known that the concrete that surrounds the steel protects it from corrosion because of the prevailing pH of the local micro-environment. The crack width that complies with maintaining

this situation is of the order of 0.2—0.3 mm. If s is the spacing between the cracks in

the stabilised crack pattern (approximately equal to lb), and "sm represents the

average strain on the reinforcing steel, then the crack widthw, ignoring the elongation

of the concrete, is w¼ s "sm

Assuming that, approximately,"sm¼ (0.8 s)/Es(Menn, 1990):

w¼ 0.80 s lb/Es

It is clear that, in order to restrict the width of the cracks, the steel stress must be limited

accordingly.g

Once the above described crack pattern has been attained, the cracked specimen can be loaded further. However, the cracked specimen has a greatly reduced stiffness compared with the uncracked specimen, as it is only the reinforcing bar that offers any resistance to the external tensile force.

It is evident from the foregoing that the specimen can be loaded until the valueZy,

when the steel stress Zy/As becomes equivalent to the yield stress (Figure 1.14).

Beyond this level the specimen cannot be loaded further. Thus the ultimate tensile load of the specimen is

Zy¼ As fsy

1.4 Behaviour of a prestressed concrete member under tension

The case of a concrete member under tension may be used as an elementary introduction to the concept of prestressing.

Consider a concrete member with a duct embedded along its central line, and assume that the duct has sufﬁcient bond with the concrete and contains a loose steel cable wire

that has no bond with the concrete. The cable has a cross-sectional area AP and is

The ultimate tensile load is independent of the concrete section

As Zy = As · fsy

Zy

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anchored at the left-hand end of the concrete member, while its other end hangs out freely (Figure 1.15).

Using a hydraulic jack a tensile force P is applied at the free end of the cable;

simultaneously, an equal and opposite force is exerted on the concrete. The cable is

in equilibrium, with the applied force P at its free end and the equal and opposite

force exerted on it by the concrete at its anchored end. The concrete member, on the other hand, is in equilibrium due to the compressive anchor force acting at its

left-hand end and the compressive forceP acting through the jack at its other end.

The two cooperating systems, i.e. the concrete and the steel, at ﬁrst work totally independently of each other, given that no bond forces exist between them. The steel

cable undergoes a tension and an elongation under the jacking force P, while the

concrete undergoes a compression and a shortening under the action of the equal and

opposite forceDP(DP¼ P). When the externally applied jacking force reaches a desired

value P0, the cable is anchored to the right-hand free end of the member and the

hydraulic jack is removed. A cement grout is then poured into the steel duct, giving a complete bond between the steel and the cement. The concrete is now subjected to a

compressive forceP0, while an equal tensile force acts on the cable.

After this technical intervention, the composite member can carry any tension force Z, leaving a compressive stress on the concrete — i.e. without cracks — provided the force Z is smaller than P0(see Figure 1.15). As will be seen below, a value ofZ slightly greater

thanP0is needed so that the concrete will be under no stress at all, as would be the case

in a reinforced concrete member in its unloaded state. The concrete member may carry a further tensile load until acracking load ZPris reached by exploiting the tensile strength

of the concrete, exactly as described in Section 1.3 (Figure 1.16).

The establishment of a bond between the cable wire and the concrete allows the equations given in Section 1.3 to be applied directly:

P₀ A_cþ fct¼ Z_Pr A_c;i¼ Z_Pr A_cþ A_P Es Ec Z Z Without bond

For Z < P0 the member continues to be compressed

The concrete is compressed with a stress P0/Ac DP = P0

Restoration of bond P0 P0 DP DP Ac Ap

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This equation yields the value ofZPr, which is clearly greater thanP0. The forceZ which

leads the concrete to experience null stress may be determined by omitting from the left-hand side of the equation the stress fct. It is clear that even this force is slightly

greater thanP0.

Consider now the cable stress, which has the initial value P0¼ P0/AP at the

prestressing level. On reaching the cracking force ZPr it shows an increase P

which, according to equation (a) in Section 1.3, is given by

P ¼

Z_Pr A_Pþ A_c Ec

E_s

This increase is small in comparison to the initial stress P0. However, while the

prevailing cable stress immediately before cracking is (P0þ P), immediately after

cracking it is ZPr/AP¼ (P0þ P)þ ( fct Ab)/AP. This abrupt change corresponds to

the previously existing concrete tensile force which, after the crack has occurred, is transferred to the cable itself. From this point on and for every further increase in the

tensile force, only the cable section AP offers any resistance, and thus the member

has a much lower stiffness when cracked than when uncracked. The member remains functional regarding the maximum allowable crack width, provided that the further

increase in the steel stress does not exceed about 200 N/mm2.

It should be noted here that, in order to confront the cracking danger better, reinforcing steel should additionally be used. Up to the moment the external tensile

force ZPr is reached, the reinforcing steel develops only a small tensile stress, as it

begins to be tensioned only after the vanishing of the compressive stress of the concrete. After cracking occurs, the reinforcing steel, together with the prestressed

cable, can be stressed further only up to a level of 200 N/mm2.g

Z ZPr Z ZPr/Ac,i = P0/Ac + fct (Conclusion: ZPr > P0) Without bond Restoration of bond Equivalent section P0 P0 σc = P0/Ac Ac Ac,i

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Exactly as in the case of the reinforced concrete member, the ultimate tensile forceZyof

the member with additional reinforcing steel corresponds to the yield stress of both the reinforcing and prestressing steel (Figure 1.17):

Zy¼ AP fPyþ As fsy g

When designing a tensioned concrete member it must be kept in mind that reinforcing and prestressing steel have practically the same modulus of elasticity, but the yield stress of prestressing steel is about four times greater than that of reinforcing steel. Thus, for the hypothetical case of a concrete tension member, using either only prestressed steel or only reinforcing steel, and assuming that the concrete and the steel have the same cross-sections in both cases, it can be concluded that (see Section 1.5):

. The cracking force is much greater for a prestressed member than for a simply rein-forced member. The elongation of the prestressed member is far less than if the force were taken up by the prestressing steel alone.

. The ultimate tension load resulting from the yield stress of the steel is about four times greater for the prestressed member than for the reinforced one.

It is interesting to note that the cost ratio of prestressed to reinforcing steel for the same cross-sectional area is less than 4.

1.4.1 Loss of prestress

At this point it should be pointed out that, because of the creep and shrinkage of concrete, there will be some shortening of the member, and consequently of the tensioned cable itself, leading to a reduction in the prestressing force. This loss of prestressing force can be determined on the basis of Trost’s relation, as described in Section 1.2.2.1.

Due to the initial prestressing force P0, a uniform compressive stress 0¼ P0/Ac is

applied to the member, whereas the developing prestressing loss P0 induces a

gradual stress variation 0¼ —P0/Acin the member.

The total shortening strain of the concrete"totis due to the superposed effects of0

and0on one side, and the shrinkage strain on the other. According to Trost’s

‘consti-tutive relation’ (see Section 1.2.2.1), "tot ¼ 0 E_c=ð1 þ ’Þþ 0 E_c=ð1 þ ’Þþ "s As · fsy AP · fPy AP · fPy As · fsy As AP

The ultimate tensile load is independent of the prestressing force P0

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Now, the shortening" of the member due to creep and shrinkage will be " ¼ "tot "0

where"0¼ 0/Ecrepresents the instantaneous member shortening due to0. As" is

identical to the shortening strain of the cable, the loss of prestressP0will be

P0¼ " EP AP

From the above equations, if the elastic shortening due to P0 is neglected, the

pre-stressing loss is obtained as P0¼ n

0Ac þ "s EcAc

1þ n ð1 þ ’Þ

while the ratio of the loss to the initial prestressing force is P0=P0¼ n

’ þ "s ðEb=0Þ

1þ n ð1 þ ’Þ where

n¼ EP/Ecand ¼ AP/Ac

It is clear that the force (P0 P0) should be taken as the effective prestressing

force for the member, and it is this force, rather than P0, which should not be

exceeded by an external tensile load if the member is to be kept uncracked.

Moreover, it is clear that the ultimate tension loadZyof the member remains unaffected

by the loss of prestressP0.

1.5 Numerical examples

1.5.1 Reinforced concrete

Consider a reinforced concrete bar, 3 m long and having a cross-section 30 30 cm

(Ab¼ 0.09 m2). The reinforcement area is As¼ 8.00 cm2, and Ec¼ 3 107kN/m2 and

Es¼ 2 108kN/m2.

The cracking load Zr of the bar can be determined according to equation (c) in

Section 1.3, assuming a tensile strength of the concretefctof 200 kN/m2:

200¼ Zr

0:302_{þ 8:00 10}4_6:67

ThusZr¼ 19.07 kN.

This force causes a steel stress according to equation (a) of s0¼

19:07

8:00 104þ 0:302_0:15¼ 1333:6 kN=m

2

with a corresponding strain "s¼ s/Es¼ 1333.0/(2 108)¼ 6.66 106. The bar

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Up to this point the concrete remains uncracked, but immediately afterwards the bar has to be considered as cracked, and thus only the reinforcing area can offer a resistance to the external force. This causes an abrupt change in the steel stress and an abrupt change in the axial stiffness of the bar. The steel stress becomes

sr¼

19:07

8:00 104¼ 23 837:0 kN=m

2 _{(abrupt change)}

The member can be further loaded and still remain functionally usable with an accepted

crack width until the steel stresssreaches 200 000 kN/m2. That means that it can be

loaded up to the value

Zs¼ 19.07 þ (200 000.0 23 837.0) (8.00 104)¼ 160.00 kN

The member elongation is

¼200 000:0 1333:6

ð2 108_Þ 3:0 ¼ 2:98 10

3_m _{(acceptable value)}

The ultimate load in tensionZycorresponds to the steel yield stress (fsy¼ 420 000 kN/m2),

and is obtained as

Zy¼ As fsy¼ (8.00 104) 420 000.0 ¼ 336.0 kN

The variation in the reinforcement stress with the increasing external forceZ is shown in

Figure 1.18. At a very early stage, i.e. at the cracking load, the abrupt change in stiffness can be observed, which means that the use of the bar as a tensioned element in a functionally satisfactory way (small crack width, acceptable deformations) is very restricted. σs: kN/m2 420 000 200 000 23 837 19.1 160.0 336.0

Cracking Functionally adequate Ultimate tensile load

Z: kN

8.0 cm2

0.30 m 0.30 m

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1.5.2 Prestressed concrete

The same member as in Section 1.5.1 is considered, but this time the reinforcement (of the same sectional area) is prestressed steel with a yield stressfPyof 1 600 000 kN/m2.

The reinforcement is prestressed with an initial stress equal to 70% of its yield stress,

i.e. with a force approximately equal to 900 kN ( 8.00 104 0.70 1 600 000). The

same force is also applied to the concrete:

c¼ 900:0 0:302 ¼ 10 000:0 kN=m 2 _{(acceptable value)} P0¼ 900:0 8:00 104¼ 1 125 000:0 kN=m 2

The member can be loaded, remaining at ﬁrst in an uncracked condition, after the restoration of the bond, as set out in Section 1.3. In order to reach the cracking force

ZPr, the null concrete stress must ﬁrst be attained, and then the maximum tensile

stress of the concrete. This means that the cracking force has to produce in the

homoge-neous (uncracked) concrete member a ‘tensile stress’ equal to cþ fct, i.e. equal to

(10 000þ 200) kN/m2. According to Section 1.4,

10 200:0 ¼ ZPr

0:302_{þ 8:00 10}4_6:67

Thus ZPr¼ 972.4 kN.

The load that will overcome the concrete stress is then 972.4 10 000/102 000 ¼

953.3 kN. This value is somewhat higher than the value of 900 kN, as was previously noted would be the case.

Due toZPr, the initial stressP0of the reinforcement increases byP:

P ¼

972:4

8:00 104þ 0:302_0:15¼ 68 000.0 kN/m

2 _{(small in comparison to}

P0Þ

From this point on it is the reinforcement alone that carries the increasing external tension force.

It is understandable that, after cracking, there is an abrupt increase in the reinforcement stress, because the tensile concrete force is transferred to the steel. This increase is equal to

200.0 0.302/8.00 104¼ 22 500.0 kN/m2

The member can be loaded further and remain functionally usable with an accepted crack width up to the point where the resulting increase in the steel stressP, beyond the

one that prevailed immediately before cracking, reaches 200 000 kN/m2. This maximum

usable loadZscan be obtained as

Zs¼ 972.4 þ (200 000.0 22 500.0) (8.00 104)¼ 979.2 þ 151.4 ¼ 1114.4 kN

Now consider the case when a mild steel reinforcement, say with the same

cross-section As¼ 8.00 cm2, is also present. Initially this reinforcement is compressed, but

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develops a stress, according to equation (a) in Section 1.3, equal to sr¼

972:4 953:3

2 8:00 104þ 0:302 0:15¼ 1284:8 kN=m

2 _{ðnegligibleÞ}

After cracking has occurred, the effective section area consists of both the prestressed

and the reinforcing steel. Further loading of the member can occur up to the point Zs

only, where the stress increase in both reinforcements of 200 000.0 kN/m2 will not

have been surpassed:

Z_s¼ 972:4þ ð200 000:0 22 500:0Þð8:00104Þ þ ð200 000:0 1284:8Þð8:00104Þ

¼ 1273:4 kN

The ultimate loadZycorresponds to the steel yielding stress.

In the case when there is only prestressing reinforcement: Zy¼ AP fPy¼ (8.00 104) 1 600 000.0 ¼ 1280.0 kN

In the case when there is both prestressing and reinforcing steel, it can be concluded that it is the mild steel that will yield ﬁrst. Therefore, ﬁnally:

Zy¼ AP fPyþ As fsy¼ 1280.0 þ (8.00 104) 420 000.0 ¼ 1280.0 þ 336.0

¼ 1616.0 kN

The stress in the prestressing steel during the gradual increase in the external forceZ

is shown in Figure 1.19. The following conclusions may be established:

Prestressing Cracking Ultimate load Ultimate load Functionally adequate Functionally adequate σs: kN/m2 1 600 000 1 393 000 1 193 000 1 125 000 0.30 m 0.30 m 0.30 m 0.30 m 8.0 cm2 8.0 cm2 8.0 cm2 Z: kN 900 972 1114 1280 1273 1616

Structural systems: behaviour and design