Abrish Rc i (Rearranged)_2

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1 AAiT Department of Civil Engineering

CHAPTER I

Introduction

1.1. General Introduction

Concrete is a conglomerate, artificial stone like material obtained by hardening and curing a

mixture of mainly cement, water and aggregates and sometimes admixtures. The main

cementing constituents of portland cement are tri-calcium silicate (3CaO.SiO2 abbreviated

C3S) and di-calcium silicate (2CaO.SiO2 abbreviated C2S) which when combined with water

undergo chemical reactions that result in cementing gels called tobermorite gels. They are

so named due to their similarity in composition and structure to a natural mineral

discovered in Tobermory, Scotland. The hydration reaction is according to the following

equations

→

→

C2SH and CSH are di- and mono-calcium silicate gels respectively, and CH is Ca(OH)2

There are hypotheses trying to explain as to how these gels impart strength to concrete.

The first hypothesis says it is due to force of adsorption. According to this hypothesis these

gels are about 1/1000 of the size of portland cement grains (≈10μm) and have enormous

surface area (about 3*106cm2/g) which results in immense attractive forces between

particles as atoms on each surface are attempting to complete their unsaturated bonds by

adsorption. These forces cause particles of the gel to adhere to each other and to every

other particle in the cement paste. The second hypothesis says that the gel attracts one

another and everything around them due to adhesive Vander Waals forces.

Whichever way, tobermorite gels form the heart of hardened concrete in that it cements

everything together. The finished product, plain concrete has a high compressive strength

and low resistance to tension, such that its tensile strength is approximately one-tenth of its

compressive strength. Consequently, tensile and shear reinforcement has to be provided to

resist tension to compensate for the weak tension regions in reinforced concrete.

It is this deviation in the composition of a reinforced concrete section from the homogeneity

of steel or wood that requires a modified approach of structural design, as will be explained

in subsequent chapters. The two component of the heterogeneous reinforced concrete

section are to be so arranged and proportioned that optimal use is made of the two

materials involved.

Design of concrete sections involves determining the cross sectional dimensions of concrete

structural members and the required quantity of reinforcement. A large number of

parameters have to be dealt with in design of concrete sections such as geometrical width,

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depth, area of reinforcement, steel strain, concrete strain and steel stress. Consequently,

trial and adjustment are necessary in the choice of concrete sections, with assumptions

based on conditions at site, availability of the constituent materials, particular demands of

the owners, architectural and headroom requirements, applicable codes and environmental

conditions.

1.2. Concrete and Reinforced Concrete

Concrete is a stone like material obtained by permitting a carefully proportioned mixture of

cement, sand, and gravel or other aggregate, and water to harden informs of the shape and

dimensions of the desired structure. The bulk of the material consists of fine and coarse

aggregates. Cement and water interact chemically to bind the aggregate particles in to a

solid mass. Additional water above that needed for this chemical reaction is necessary for

workability of the mixture that enables it to fill the forms and surround the embedded

reinforcing steel prior to hardening.

Concrete has high compressive strength which makes it suitable for members primarily

subjected to compression such as columns and arches. However its tensile strength is very

small compared with its compressive strength (10-15%). This prevents its economical use in

members subjected to tension either entirely (such as in tie rods) or over part of their cross

sections (such as beams or other flexural members).

To offset this limitation, it was found possible to use steel with its high tensile strength to

reinforce concrete, chiefly in those places where its low tensile strength would limit the

carrying capacity of the member. Such a member, where steel bars are embedded in the

concrete in such a way that the tension forces needed for moment equilibrium after the

concrete cracks can be developed in the bars is called a reinforced concrete member.

The resulting combination, known as reinforced concrete, combines many of the advantages

of each material:

Relatively low cost, Good weather and fire resistance, Good compressive

strength, Excellent formability of concrete

High tensile strength, much greater ductility and toughness of steel

It is this combination that allows the almost unlimited range of uses and possibilities of

reinforced concrete in the construction of buildings, bridges, dams, tanks, reservoirs and a

host of other structures. Reinforced concrete is a dominant structural material throughout

the world because of the wide availability of constitutions of concrete and reinforcing steel

bars, the relatively simple skills required for its construction and the economy of reinforced

concrete compared to other forms of construction.

Advantages of Reinforce Concrete:

The following advantages are observed in using RC member compared with other existing

structural member made form steel or timber:

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1. Reinforced concrete is mouldable into any desired shape ,and this variability allows

the shape of the structures to be adapted to its function in an economical manner

and furnishes the architect with a wide range of possibilities or aesthetically

satisfying structural solutio0ns,

2. Unlike others i.e. steel or timber, RC does not deteriorate with time,

3. It is fire, weather and corrosion resistant,

4. It is monolithic, i.e. can be assumed that it is made from one piece.

5. Most of the constituent materials, with the exception of cement and additives, are

usually available at low cost locally or at small distance from construction sites.

6. Concrete has high compressive strength.

7. Low maintenance

Disadvantages of Reinforced Concrete

The followings are disadvantages of using Reinforce concrete as construction material:

1. It is difficult for quality control as it is usually produced on site.

2. It is very difficult to dismantle for repair or change,

3. The form work costs whether steel or timber are rather expensive compared with

those of concrete and reinforcing steel,

4. It is difficult to supervise after pouring,

5. In ordinary Reinforced Concrete structures, large portion of the section are not

utilized, but this disadvantage is absent in pre-stressed concrete members.

6. Its tensile strength is very low

Despite its advantages, RC has found exceptionally extensive applications in construction

industry owing to its durability, strength, stiffness and availability of cheap local materials.

1.3. Material Aspect of Reinforced Concrete

To understand and interpret the total behavior of a composite element requires knowledge

of the characteristics of its components. Concrete is produced by the collective mechanical

and chemical interaction of a large number of constituent materials. Hence a discussion of

the function of each of these components is vital prior to studying concrete as a finished

product. In this manner, the designer and the materials engineer can develop skills for the

choice of the proper ingredients and so proportion them as to obtain an efficient and

desirable concrete satisfying the designers strength and serviceability requirements.

1.3.1. Concrete

1.3.1.1. Ingredients of Concrete

a) Cement

Cement is the most important ingredient of concrete because it is the hydration reaction

that gives strength to concrete. This ingredient is also the most expensive in plain concrete

production.

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Portland cement is produced from a mixture of ground clay (contains Si02 and Al2O3) and

lime (CaO) and other minor ingredients such as MgO and Fe2O3 by heating to the point of

incipient fusion (clinkering temperature). The clinker is then ground to different degrees of

fineness to get cement.

Table 1.3.1.1-1 shows the main chemicals in Portland cement and the relative contribution

of each component towards the rate of gain in strength. The early strength of Portland

cement is higher with higher percentages of C2S. If moist curing is continuous, later strength

levels will be greater, with higher percentages of C2S. C3A contributes to the strength

developed during the first day after placing the concrete because it is the earliest to

hydrate.

Component Rate of

Reaction Heat

Liberated Ultimate

Cementing Value

Tricalcium silicate, C3S Dicalcium silicate,C2S Tricalcium aluminate, C3A Tetracalcium aluminoferrate, C4AF

Medium Slow Fast Slow

Medium Small Large Small

Good Good Poor Poor

Table 1.3.1.1-1Properties of Cements

Type of Cement Component (%) General

Characteristics C3S C2S C3A C4AF CaSO4 CaO MgO

Normal: I 49.0 15.0 12.0 8.0 2.9 0.8 2.4 All-purpose cement

Modified: II 45.0 29.0 6.0 12.0 2.8 0.6 3.0 Comparative low heat liberation; used in large structures

High early strength: III 56.0 15.0 12.0 8.0 3.9 1.4 2.6 High strength in 3 days

Low heat: IV 30.0 46.0 5.0 13.0 2.9 0.3 2.7 Used in mass concrete dams

Sulphate resisting: V 43.0 36.0 4.0 12.0 2.7 0.4 1.6 Used on sewers and structures exposed to sulphate

Table 1.3.1.1-2 Percentage Composition of Portland Cements

When portland cement combines with water during setting and hardening, lime is liberated

from some of the compounds. The amount of lime liberated is approximately 20% by weight

of the cement. Under unfavorable conditions, this might cause disintegration of a structure

owning to leaching of the lime from the cement. Such a situation should be prevented by

adding a siliceous mineral such as pozzolan to the cement. The added mineral reacts with

the lime in the presence of moisture to produce strong calcium silicate.

The size of the cement particles strongly influences the rate of reaction of cement with

water. For a given weight of finely ground cement, the surface area of the particles is

greater than that of the coarsely ground cement. This results in a greater rate of reaction

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with water and a more rapid hardening process for larger surface areas. This is one reason

for the high early strength type-III cement.

Type of cement affects durability of concrete also. Disintegration of concrete due to cycles

of wetting, freezing, thawing, and drying and propagation of resulting cracks is a matter of

great importance. The presence of minute air voids throughout the cement paste increases

the resistance of concrete to disintegration. This can be achieved by the addition of air-

entraining admixtures to the concrete while mixing.

Disintegration due to chemicals in contact with the structure, such as in the case of port

structure and sub-structure can also be slowed down or prevented. Since the concrete in

such cases is exposed to chlorides and sometimes sulphates of magnesium and sodium, it is

sometimes necessary to specify sulphate-resisting cement. Usually, type II cement will be

adequate for use in seawater structures.

Since the different types of cement generate different degrees of heat at different rates, the

type of structure governs the type of cement to be used. The bulkier and heavier in cross

section the structure is the less the generation of heat of hydration that is desired. In

massive structures such as dams, piers, and caissons, type IV cement are advantageous to

use. From this discussion it is seen that the type of structures, the weather, and other

conditions under which it is built and will be used are the governing factors in the choice of

the type of cement that should be used.

b) Water and Air

Water

Water is required in the production of concrete in order to precipitate chemical reaction

with the cement, to wet the aggregates and to lubricate the mixture for easy workability.

Normally, drinking water can be used in mixing. Water having harmful ingredients such as

silt, oil, sugar or chemicals is destructive to the strength and setting properties of cement. It

can disrupt the affinity between the aggregate and the cement paste and can adversely

affect workability of a mixture.

Excessive water leaves uneven honeycombed skeleton in the finished product after

hydration has taken place while too little water prevents complete chemical reaction with

the cement. The product in both cases is a concrete that is weaker than and inferior to

normal concrete.

Entrained Air

With the gradual evaporation of excess water from the mix, pores are produced in the

hardened concrete. If evenly distributed, these could give improved characteristics to the

product. Very even distribution of pores by artificial introduction of finely divided uniformly

distributed air bubbles throughout the product is possible by adding air-entraining agents

such as vinsol resin. Air entrainment increases workability, decreases density, increases

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durability, reduces bleeding and segregation, and reduces the required sand content in the

mix. For these reasons, the percentage of entrained air should be kept at the required

optimum value for the desired quality of the concrete. The optimum air content is 9% of the

mortar fraction of the concrete. Air entraining in excess of 5-6% of the total mix

proportionally reduces the concrete strength.

c) Aggregates

Aggregates are those parts of the concrete that constitute the bulk of the finished product.

They comprise 60 to 80% of the volume of the concrete and have to be so graded that the

whole mass of concrete acts as a relatively solid homogeneous, dense combination, with the

smaller sizes acting as an inert filler of the voids that exist between the larger particles.

Since the aggregates constitute the major part of the mixture, the more aggregate is used in

the mix the cheaper is the cost of the concrete, provided that the mixture is of reasonable

workability for the specific job for which it is used.

Aggregates are of two types: coarse aggregates and fine aggregates. Coarse aggregates are

usually manufactured by crushing stone and fine aggregates are natural sand obtained by

the natural disintegration of rock or artificial sand obtained by artificially crushing stones.

Coarse Aggregate

Properties of the coarse aggregates affect the strength of hardened concrete and its

resistance to disintegration, weathering, and other destructive effect. The coarse aggregate

must be clean of organic impurities and must bond well with the cement gel. Table 1.3.1.1-3

gives grading or particles size distribution requirements of coarse aggregates by Ethiopian

Standard for Concrete and Concrete Products, ES C.D3.201.

Nominal size of graded aggregate

Percentage passing through test sieves having square openings

75mm 63mm 37.5mm 19mm 13.2mm 9.5mm 4.75mm

38-5 100 - 95 - 100 30 – 70 - 10 - 35 0 - 5

19-5 - - 100 95 – 100 2.8 25 - 55 0 - 10

13-5 - - 12 100 90 - 100 40 - 85 0 - 10 Table 1.3.1.1-3 Grading requirements for coarse aggregates [ES C.D3.201]

Coarse aggregate shall be free of injurious amounts of organic impurities. The amount of

deleterious substance in coarse aggregate shall not exceed the limits specified in Table

1.3.1.1-4.

Deleterious substance Maximum percentage by mass

Friable soft fragments 3.00

Coal and lignite 1.00

Clay lumps 0.25

Materials passing 63μm sieve including crushed dust 1.50

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Table 1.3.1.1-4 Permissible limits for deleterious substances in coarse aggregates [ES C.D3.201]

Other requirements are soundness and resistance to abrasion. Concerning soundness,

coarse aggregate shall not show loss in mass exceeding 12 percent when subjected to five

cycles of wetting and drying with sodium sulphate solution or 18 percent when magnesium

sulphate solution is used. The maximum loss in mass when coarse aggregate is subjected to

abrasion test shall not exceed 50 percent.

Fine Aggregates

Fine aggregate is smaller filler made of sand. It ranges in size from No.4 to No. 100(4.75 mm

to 150μm). A good fine aggregate should always be free of organic impurities, clay, or any

deleterious material or excessive filler of size smaller than No. 100 sieve. It should

preferably have a well-graded combination. The following requirements are given by

Ethiopian Standards [ES D3.201].

The grading requirement of fine aggregate shall be within the limit specified in table1.3.1.1-

5

The fine aggregate shall not also have more than 45 percent retained between any two

consecutive sieves. The fineness modulus shall not be less than 2.0 or more than 3.5 with a

tolerance of ± 0.2.

Sieve Percentage passing

9.50mm 4.75mm 2.36mm 1.18mm 600μm 300μm 150μm

100 95 - 100 80 - 100 50 - 85 25 - 60 10 - 30 2 - 10

Table 1.3.1.1-5 Grading requirements for fine aggregates [ES D3.201]

Table 1.3.1.1-6 gives limits of deleterious substances for fine aggregates.

Deleterious Substance Maximum percentage by mass

Friable particles Clay or fine silt (materials passing 63μm sieve) in fine aggregates used for

- Concrete subject to abrasion - All other concrete

Coal Ignite

1.0

3.0 5.0 1.0

Table 1.3.1.1-6 Permissible limits for deleterious substance in fine aggregates [ES C.D3.201]

Fine aggregates, when subjected to five cycles of soundness test, shall not show loss in

mass exceeding 10 percent when sodium sulphate solution is used or 15 percent

magnesium sulphate solution is used.

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Characteristics of the finished product, concrete can be varied considerably by varying the

proportion of its ingredients. Thus, for a specific structure it is economical to use concrete

with the desired characteristics though it may be weak in others. For example, concrete for

building should have high compressive strength whereas for water tanks, water tightness is

of prime importance.

Performance of concrete in service depends on properties both in the plastic and hardened

states.

1.3.1.2. Properties in the Plastic State

a) Workability - is an important property and concerns the ease with which the mix can be

mixed, handled, transported and placed with little loss of homogeneity so that after

compaction it surrounds all reinforcements completely, fills the form work and results in

concrete with the least voids.

b) Temperature - Care should be taken to minimize the temperature due to evolving heat

of hydration if cement is greater than or equal to400kg/m3 and the least dimension of

concrete to be placed at a single time is 600mm or more.

1.3.1.3. Properties in the Hardened State

a) Compressive strength

The main measure of the structural quality of concrete is its compression strength. Tests for

this property are made on cylindrical specimen of height equal to twice the diameter

(usually 6x12 inches, i.e. 150x300mm) originally as specified by American society for Testing

and materials (ASTM). According, the cylinder specimens are moist cured at about 70±50F,

generally for 28 days and then tested in the laboratory at a specified rate of loading usually

to reach the maximum stress in 2 to 3 minutes. The compression strength obtained from

such test is known as the cylinder strength fc or fck and this is the main property specified for

design purpose.

Depending up on the mix (especially the water cement ratio) and the time and quality of

curing, compressive strength of concrete can be obtained up to 100 MPa . For most practical

and ordinary use(fck) available ranges between 20 to 50 MPa.

The compressive strength is calculated from failure load divided by cross-sectional area

resisting the load and reported in units of force per square area. In EBCS 2-1995, concrete is

graded based on tests of 150 mm cubes at the age of 28 days which may be considered as

the characteristic cube compression strength in MPa and graded as C5, C15, C20, C30, C40,

C50 and C60 the numbers being characteristic compressive strength in MPa.This may be

converted to equivalent cylinder compressive strength fck as

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The 28 day compressive strength may be obtained from 7 days compressive strength using

experimentally developed empirical relations. One formal is

√

S7 and S28 are7 and 28 day strengths in psi (W.A. Slater)

Strength can be increased by

Decreasing W/C ratio

Using high strength aggregates because that makes 65-75% of the volume of

concrete.

Grading the aggregates to produce a small percentage of voids in the concrete

Moist curing the concrete after it has set

Vibrating the concrete in the forms while plastic

Concrete strength is chiefly influenced by W/C ratio, it can be estimated by,

A and B are empirical constants that depend on age, curing condition, type of cement

properties of aggregates and testing method. W/C is water cement ratio.

Figure 1.3.1.3-1Effect W/C ration on strength

Other factor affecting concrete strength is degree of compaction.

Figure 1.3.1.3-2Effect of degree of compaction on strength

b) Tensile strength – It is used to design for shear, torsion and crack width. This is much

lower than compressive strength and generally falls between 8 and 15 percent of

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compressive strength. It is difficult to determine from tension test due to problem

with gripping and is indirectly determined from split-cylinder test or flexure test

(modulus of rupture) or from empirical formulae.

In a split-cylinder test, a 150mm*300mm compression test cylinder is placed on its side

and loaded in compression along the diameter as shown in figure 1.3.1.3-3. The splitting

tensile strength, fct is determined as,

(

)

Figure 1.3.1.3-3 Split-cylinder test procedure

In flexure test a plain concrete beam, generally 150mm*150mm*750mm long is loaded in

flexure at the third points of a 600mm span until it fails due to cracking on the tension face

as shown in figure 1.3.1.3-4. It can be estimated by,

Figure 1.3.1.3-4 Flexure test

EBSC 2-1995 uses the following empirical formula,

Where fctk – tensile strength of concrete in MPa

fck– characteristic cylinder strength in MPa

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c) Creep

It is strain that occurs under constant sustained compressive load. It is also defined as

deformation of a member under sustained load. It results in stress redistribution and

additional deformation and should be considered. For example, in the design of RC

beams for allowable stress, the effects of creep are taken into account by reducing the

modulus of elasticity of concrete usually by 50%.

Creep is

Proportional to stress

Increases with increase in W/C ratio

Decrease with relative humidity of atmosphere

d) Volume change

Shrinkage is the shortening of concrete during hardening and drying under constant

temperature. The prime cause of shrinkage is due to loss of a layer of adsorbed water from

the surface of the gel particles. It depends on relative humidity (but recoverable on wetting

and of composition of the concrete.

Essentially, Shrinkage occurs as the moister diffuses out of the concrete which result the

exterior to shrink more rapidly than the interior. This leads to tensile stresses in the outer

skin of the concrete and compression stresses in the interior. The effect of shrinkage can be

reduced by using less cement and by adequate moist curing.

e) Density

Increase in density results in increase in strength. Density can be increased by using

denser aggregate, graded aggregates, vibrating and reducing w/c ratio.

f) Durability

Concrete durability has been defined by the American Concrete Institute as its resistance

to weathering action, chemical attack, abrasion and other degradation processes.

Concrete should be capable of withstanding

Weathering such as corrosion and mainly freezing and thawing. This can be

improved by increasing water tightness.

Chemical reaction

Wear

1.3.1.4. Proportioning and Mixing Concrete

Component of a mix should be selected to produce concrete with desired characteristics at

lowest cost possible. For economy the amount of cement should be kept to a minimum.

Because of the larger number of variables involved, it is usually advisable to proportion

concrete mixes by making and trail batches.

The selection of the relative proportion of cement, water and aggregate is called mix design.

The important requirements in mix design are the following, which can sum up as

workability, strength, durability and economy.

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a) The fresh concrete must be workable or placeable

b) The hardened concrete must be strong enough to carry the loads for which it has

been designed

c) The hardened concrete must be able to withstand the condition to which it will be

exposed to in service life

d) It must be capable of being produced economically.

A start is made with selection of W/C ratio, then largest size of aggregate (dictated by

sectional dimension of structural members and spacing of reinforcements). Then several

trial batches are made with varying ratio of aggregates to obtain the desired workability

with the least cement. Test should be made to evaluate compressive strength and other

desired characteristics. Observations should be made of the slump and appearance of

concrete. After a mix has been selected, some changes may have to be made after some

field experience with it.

If this is expensive or not justified the mix proportions which are appropriate for grades C5

to C30 may be taken from EBCS 2-1995 “Structural use of concrete” page 90.

Minimum mixing time measured from the time the ingredients are put together is given in

table 1.3.1.4-1. Over mixing can remove entrained air and increase fines requiring more

water for workability. The maximum mixing time may be taken 3 times the minimum mixing

time as a guide.

Capacity of mixer (m3) Time of mixing (minutes)

1.5 1.5

2.3 2.2

3.0 2.5

4.5 3.0 Table 1.3.1.4-1 Minimum mixing time for production of Portland cement concrete

After mixing the concrete, the chemical reaction of cement and water in the mix is relatively

slow and requires time and favorable temperature for its completion. This setting time is

divided in to three distinct phases as:

1. First phase: time of initial set, requires from 30 to 60 minutes for completion, at

which the mixed concrete decreases its plasticity and develops pronounced

resistance to follow,

2. Second phase: time of final set requires from 5 to 6 hours after mixing operation,

where the concrete appears to be relatively soft solid without surface hardening,

3. Third phase: time of progressive hardening, may take about one month after mixing

where the concrete almost attains the major portions of its potential hardness and

strength.

1.3.2. Reinforcing Steel

It is a high-strength and high cost steel bar used in concrete construction (e.g., in a beam or

wall) to provide additional strength. When reinforcing steel is used with concrete, the

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concrete is made to resist compression stress and the steel is made to resist tensile stress

with or without additional compressive stress.

When RC elements are used, sufficient bond between the two materials must be developed

to ensure that there is no relative movement between the steel bars and the surrounding

concrete. This bond may be developed by,

chemical adhesion

natural roughness

closely spaced rib-shaped surface deformation of reinforcement bars as shown in

figure

Reinforcing bars varying 6 to 35 mm in size are available in which all are surface deformed

except φ6.

Figure 1.3.2-1 Type of reinforcement bars

Some bar size and areas for design purpose available in Ethiopia are given in table

Diameter φ (mm)

6 8 10 12 14 16 20 24….

Area (mm2) 28 50 78.5 113 154 200 314 450

Weight (Kg/m) 0.222 0.395 0.619 0.888 1.210 1.570 2.470 3.500 Table 1.3.2-1Reinforcement bar properties that are available in Ethiopia

Characteristic properties of reinforcing bars are expressed using its yield strength, fy (fyk) and

modulus of elasticity Es. Fy ranges between 220 to 500MPa, with 300MPa common in our

country. Es ranges between 200 to 210GPa.

1.4. Concrete Placement and Curing

1.4.1. Concrete Placement

When concrete is discharged from the mixer, precaution should be exercised to prevent

segregation. Vibration is desirable after pouring the fresh concrete because it eliminates

voids and brings particles into close contact. The resulting consolidation also ensures close

contact of the concrete with the forms, with reinforcement and other embedded items.

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For consolidation of structural concrete, immersion vibration are recommended. Oscillation

should be at least 7000 vibration per minute when the vibrator head is immersed in

concrete. Each yd3 (0.765m3) of concrete should be vibrated at least 1 minute.

Formwork retains concrete until it has set and produced the desired shape and sometimes

the desired surface finish. Formwork must be supported on false work of adequate strength

and rigidity. Forms must also be tight, yet they must be of low cost and often easily

demountable to permit reuse.

Early striking forms is generally desirable to permit quick reuse, start curing as soon as

possible and allow repairs and surface treatment while the concrete is still green and

condition are favorable for good bond.

The time between casting of concrete and removal of the formwork depends mainly on the

strength development of the concrete and on the function of the formwork. Provided the

concrete strength is confirmed by test on cubes stored under the same condition, formwork

can be removed when the cube strength is 50% of the nominal strength or twice the stress

to which it will then be subjected whichever is greater, provided such earlier removal will

not result in unacceptable deflection such as due to shrinkage and creep [EBCS 2-1995].

In the absences of more accurate data the following minimum periods are recommended by

EBCS 2-1995.

1. For non-load bearing parts of formwork like vertical forms for beams, columns and

walls …………………………………………………………………………………………………………… 18 hours

2. For soffit formwork to slabs ……………………………..……………………………………………. 7 days

3. For props to slabs …………………………………………..…………………………………………….. 14 days

4. For soffits formwork to beams ……………..………………………………………………………. 14 days

5. For props to beams ………………………………..…………………………………………………….. 21 days

1.4.2. Curing Concrete

While more than enough water for hydration is incorporated into normal concrete mixes,

the loss due to evaporation from the time the concrete is placed is usually so rapid that

complete hydration may be delayed or prevented. Rapid drying causes also drying shrinkage

surface cracks. Therefore it is important to keep fresh concrete moist for several days after

placing either by sprinkling, ponding or by surface sealing. This operation is called curing. If

curing is properly done for a sufficiently long period, curing produces stronger and more

watertight concrete. The most common field practice is curing by sprinkling. Portland

cement concrete should be cured this way for 7-14 days. Curing is especially important in

hot climate to replenish water lost due to rapid evaporation.

1.5. Stress-Strain Relation for Concrete and Reinforcements

Strength and deformation of reinforced concrete members can be calculated from stress-

strain relations of concrete and reinforcement steel and the dimensions of the members.

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1.5.1. Concrete

a) Uniaxial Stress Behavior

Under practical conditions concrete is seldom stressed in one condition only (Uniaxial

stress). Nevertheless an assumed uniaxial stress conditions can be justified in many cases.

Compressive Stress Behavior

The compressive strength and deformation characteristics (σ-ε) of concrete is usually

obtained from cylinders with h/d = 2, normally h =300mm and d=150mm. Loaded

longitudinally at a slow strain rate to reach maximum stress in 2 or 3 minutes. Smaller size

cylinders or cubes are also used particularly for production control and the compressive

strength of these units is higher. These can be converted with appropriate conversion

factors obtained from tests to standard cylinder or cube strengths.

Figure1.5.1-1 presents typical stress-strain curves obtained from concrete cylinders loaded

in uniaxial compression.

Figure 1.5.1-1Stress-strain curves for concrete cylinders loaded in uniaxial compression

The curves are almost linear up to about half of the compressive strength. The peak of the

curve for high strength concrete is relatively sharp but for low strength concrete the curve

has a flat top. The strain at maximum stress is approximately 0.002. At higher strains, after

the maximum stress is reached, stress can still be carried even though cracks parallel to the

directions of loading become visible in the concrete. Tests by Rusch have indicated that the

shape of stress-strain curve before maximum stress depends on the strength of the

concrete with more curvature for weaker concrete. A widely used approximation for the

shape of stress-strain curve before maximum stress is reached is a second-degree parabola.

The extent of falling branch behavior adopted depends on the limit of concrete strain

assumed useful (0.0035 for LSD and 0.003 for USD).

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Tangent and Secant Moduli of Elasticity

Three ways of defining the modulus of elasticity are illustrated in figure 1.5.1-2. The slope of

the line that is tangent to a point on the stress-strain curve, such as A, is called the tangent

modulus of elasticity, ET, at the stress corresponding to point A. The slope of the stress-

strain curve at the origin is initial tangent modulus of elasticity. The secant modulus of

elasticity at a given stress is the slope of the line through the origin and through the point on

the curve representing that stress for example point B. Frequently, the secant modulus is

defined by using the point corresponding to 0.4 - 0.5 of the compressive strength (fck),

representing the service-load stress. Whenever Ecm is used it usually means the secant

modulus in MPa.

Figure 1.5.1-2 Initial, tangent and secant modulus of elasticity

In the absence of more accurate data, in case accuracy is not required, an estimate of the

mean secant modulus Ecm can be obtained from table 1.5.1-1 for given concrete grades as

given by EBCS 2-1995.

Grades of Concrete C15 C20 C25 C30 C40 C50 C60

fck (N/mm2) 12 16 20 24 32 40 48

fctm (N/mm2) 1.6 1.9 2.2 2.5 3.0 3.5 4.0

fctk (N/mm2) 1.1 1.3 1.5 1.7 2.1 2.5 2.8

Ecm(GPa) 26 27 29 32 35 37 39

For concrete cubes of size 200 mm, the grade of the concrete is obtained by multiplying the cube strength by 1.05 (EBCS2 – 2.3).

Table 1.5.1-1Grades of Concrete and their strength characteristics

The following empirical formula is also given by EBCS 2-1995, in which Ecm is in GPa and fck is

in MPa.

⁄

The stress-strain curve in figure 1.5.1-3 is simplified for design to a parabolic rectangular

stress block as given by EBSC 2-1995.

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Figure 1.5.1-3Idealized and design stress-strain diagram for concrete

When the load is applied at a fast strain rate, both the strength and modulus of elasticity of

concrete increase, for example it is reported that for a strain rate of 0.01/sec the concrete

strength may increase by as much as 17%.

Rusch, conducting long term loading tests on confined concrete found that the sustained

load compressive strength is 0.8 of in short-term strength, where short term strength is

determined from an identically old and identically cast specimen that is loaded to failure

over a 10-minute period when the specimen under sustained load has collapsed. In practice

concrete strength considered in design of structures is short-term strength at 28 days. The

strength reduction due to long term will be partly offset by higher strength attained by

concrete at greater ages.

Creep strains due to long-term loading cause modification in the shape of the stress-strain

curve. Some curves obtained by Rusch for various rates of loading are given in figure 1.5.1-4.

It can be seen that for various rates of loading, the maximum stress reached gradually

decreases but the descending branch falls less quickly, the strain at which maximum stress is

reached increases with a decreasing rate of loading (strain).

Figure 1.5.1-4 Stress-strain curves for concrete with various rates of axial compressive loadings

Tensile Stress Behavior

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It is difficult to get tensile strength of concrete from direct tension test due to difficulties of

holding specimens to achieve axial tension and the uncertainties of secondary stresses

induced by the grips of testing devices. Therefore, it is indirectly determined from split-

cylinder test or from flexure test on plain concrete beams of 150mm square cross-section.

The split-cylinder strength σ from theory of elasticity is,

(

)

The split-cylinder strength ranges from 0.5 to 0.75 of the modulus of rupture. The difference

is mainly due to non-linear stress distribution near failure in flexural members when failure

is imminent.

Because of the low tensile strength of concrete, tensile strength of concrete is usually

ignored for flexure in strength calculations of reinforced concrete members. When it is

taken in to account like for shear or torsion the stress-strain curve in tension may be

idealized as a straight line up to the tensile strength. Within this range the modulus of

elasticity in tension may be assumed to be the same as in compression.

Poison's Ratio

Poison’s ratio for concrete is usually in the range 0.15 to 0.2; however values between 0.1

and 0.3 have been determined. Poisons ratio is generally lower for high strength concrete.

At high compressive stresses the transverse strains increase rapidly owing to internal

cracking parallel to the direction of loading.

b) Combined Stress behavior

In many structural situations concrete is subjected to direct and shear stresses. By

transformation of stresses, stress at a point can be represented by three mutually

perpendicular principal stresses.

In spite of extensive research, no reliable theory has been developed for determining the

failure strength of concrete under a general three dimensional state of stress.

Biaxial Stress Behavior

Some investigators reported that the strength of concrete subjected to biaxial compression

may be as much as 27% higher than uniaxial strength.

For equal biaxial compressive stresses, the strength increase is approximately 16%. The

strength in biaxial tension is approximately equal to the uniaxial tensile strength.

On other planes than the principal, normal and shear stresses act. Mohr's failure theory is

used to obtain strength for this combined case. Figure 1.5.1-5 shows how a family of Mohr's

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circle for failures in tension, compression and other combinations is enclosed in an envelope

curve.

Figure 1.5.1-5 Strength of concrete under general two-dimensional stress system

A failure curve for elements with direct (normal) stress in one direction combined with

shear stress shown in figure1.5.1-6.

Figure 1.5.1-6 Combinations of normal stress and shear stress causing failure of concrete

The curve shows that the compressive strength of concrete is reduced in the presence of

shear stress.

c) Creep

Figure 1.5.1-7 shows that the stress-strain relationship of concrete is a function of time. The

final creep strain may be several times as large as the initial elastic strain. Generally creep

has little effect on the strength of a structure but it results in increase in service load

deflections.

The creep deformation due to constant axial compressive stress is shown in figure 1.5.1-7.

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Figure 1.5.1-7 Typical creep curve for concrete with constant axial compressive stress

The creep proceeds at a decreasing rate with time. The magnitude of creep strain depends

on the composition of the concrete (aggregate type and proportions, cement type and

content and W/C ratio), the environment and the stress-time history.

d) Shrinkage in Concrete

When concrete loses moisture by evaporation, it shrinks. Shrinkage strains are independent

of the stress in the concrete. If restrained, shrinkage strains can cause cracking of concrete

and generally results in increase in deflection of structural members with time.

A curve showing the increase in shrinkage strain with time appears in figure1.5.1-8. The

shrinkage occurs at a decreasing rate with time. The final shrinkage strains vary greatly

being generally in the range 0.0002 to 0.0006 but sometimes as much as 0.0010.

Figure 1.5.1-8Shrinkage strain of concrete

1.5.2. Reinforcement Steel

Bar Shape and Size

Reinforcement steel bars are round in cross-section. To restrict longitudinal movement of

the bars relative to the surrounding concrete and for force transfer from the bars to the

concrete, deformations are rolled on to the bar surfaces. Minimum requirements for

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deformations such as spacing, height and circumferential coverage have been established

by experimental research. ASTM specifications require the deformations to have average

spacing not exceeding 0.7 of the nominal bar diameter and a height at least 0.04 to 0.05 of

the bar diameter. The deformations must cover 75% of the bar circumference. The angle

that these deformations make with the axis of the bar should not be less than 45°. Generally

longitudinal ribs are also present.

Figure 1.5.2-1 Deformed Bar

Deformed steel bars are produced in sizes ranging from 8mm to 35mm in Ethiopia. Ø6mm is

plain bar and is used for stirrups.

Monotonic Stress Behavior

Typical stress-strain Curves for reinforcement steel figure 1.5.2-2 are obtained from

monotonic tension test. The curve exhibits an initial linear elastic portion, a yield plateau a

strain hardening range and finally a range in which the stress drops of until fracture occurs.

The slope of the linear elastic portion gives modulus of elasticity, which ranges from 200 to

210 GPa.

The yield strength fy is a very important property of reinforcement steel and is used as

design stress in ultimate strength design (USD) and design stress obtained from σy in limit

state design (LSD).

Figure 1.5.2-2 Typical stress-strain curves for reinforcement steel

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σy can easily be read for ductile steel. It is taken as stress at 0.2% offset for steel without

well-defined yield plateau.

The minimum strain in the steel at fracture is essential for the safety of the structure that

the steel be ductile enough to undergo large deformation before fracture. This should

usually be 4.5 to 12%.

Generally the stress-strain curves for steel in tension and compression are assumed to be

identical. Tests have shown that this is a reasonable assumption.

The effect of fast rate of loading is to increase the yield strength of steel. For example, it has

been reported that for strain rate of 0.01/sec the lower yield strength may be increased by

14%.

In design it is necessary to idealize the shape of the stress-strain curve. Generally the curve

is simplified idealizing it as two straight lines. EBCS 2 gives the simplified stress-strain curve

shown in figure 1.5.2-3 for LSD.

Figure 1.5.2-3 Idealized and design stress-strain diagram for reinforcing steel

Reversed Stress Behavior

If reversed (tension-compression) type loading is applied to a steel specimen in the yield

range, a stress-strain curve of the type shown infigure1.5.2-4 (a) is obtained. This figure

shows that under reversed loading the stress-strain curve becomes non-linear at a stress

much lower than the initial yield strength. This effect is called Bauschinger effect.

Figure1.5.2-4 (b) gives an elastic perfectly plastic idealization for reversed loading which is

only an approximation. Reversed loading curves are important when considering the effect

of high intensity seismic loading on members.

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Figure 1.5.2-4a) Bauschinger effect for steel under reversed loading, b)Elastic-perfectly plastic idealization for steel under reversed loading

1.6. Overview of Design Philosophies

Design is a process used in engineering to specify how to create or do something. A design

must satisfy such requirements like functional, performance and resource usage. It is also

expected to meet restrictions on the design process, time of completion, cost, or the

available tools for doing the design.

“Structural design can be defined as a mixture of art and science, combining the engineer’s

feeling for the behavior of a structure with a sound knowledge of the principles of statics,

dynamics, mechanics of materials, and structural analysis, to produce a safe economical

structure that will serve its intended purpose” (Salmon and Johnson 1990). It is the process

of determining the dimensions and layout of the load resisting (structural) components of a

structure to satisfy the purpose of use, to possess safety and durability, and to be

economical. In civil works, buildings, bridges, dams, retaining walls, highway pavements,

aircraft landing strips are typical with individual specialized design procedure.

Structural Analysis is the assessment of the performance of a given structure under given

loads and other effects, such as support movements or temperature change.

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Figure 1.6-1 Reinforced Concrete Building Components

This course provides the first encounter on the analysis and design of the individual

structural elements of reinforced concrete structures, with emphasis on:

• Beams – horizontal members carrying lateral loads and subjected to flexural

stress,

• Slabs – horizontal plate elements carrying gravity loads and subjected to

flexural stress, and

• Columns – vertical members carrying primarily axial load but generally

subjected to axial compressive force with or without bending moment.

In (reinforced concrete) buildings, architectural planning and design is carried out to

determine the arrangement and layout of the building to meet the client’s requirements.

The structural engineer then determines the best structural system or forms to realize the

architect’s concept. The structural analysis versus design cycle is represented by the

flowchart in figure 1.6-2.

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Figure 1.6-2The Structural Design Process

Once the form and structural arrangement have been finalized the structural design

procedure consists of the following:

a. idealization of the structure into component parts

b. load estimation on the various structural components

c. analysis to determine the maximum internal stresses and strains

d. design of sections and reinforcement arrangements

e. detail drawings and bar schedule preparation

Serviceability, Strength and Structural Safety

To serve its purpose, a structure must be safe against collapse and serviceable in use.

Serviceability requires that deflections be adequately small; that cracks, if any, be kept to

tolerable limits; that vibrations be minimized; etc. Durability requirements are concerned

with the deterioration and decay of materials with age and environmental impact.

Safety requires that the strength of a structure, built as designed, could be predicted

accurately, safety could be ensured by providing a carrying capacity just barely in excess of

the known loads. However there are a number of sources of uncertainty in the analysis,

design and construction of RC structures. These sources of uncertainty, which require a

definite margin of safety, may be listed as follows:

1. Actual loads may differ from those assumed.

2. Actual loads may be distributed in a manner different from that assumed.

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3. The assumptions and simplifications inherent in any analysis may result in calculated

load effects – moments, shears, etc. different from those that, in fact, act in the

structure.

4. The actual structural behavior may differ from that assumed, owing to imperfect

knowledge.

5. Actual member dimensions may differ from those specified.

6. Reinforcement may not be in its proper position.

7. Actual material strength may be different from that specified.

The purpose of structural design is to provide a structure with least possible construction

and maintenance costs, provision of necessary space and of all guaranteeing satisfactory

performance during the lifetime of the structure. Satisfactory performance in this context

implies that under all unfavorable action of load combination imposed on the structure:

- The existence of adequate safety against collapse must be ensured,

- Limited deformation showing structure function shall not be impaired, and

- Adequate safety against any hazardous events to enable escape of the occupants

must be possible.

Hence, design involves selection of structural forms, assessment of the dimension of the

various members for the selected structural forms to satisfy the stated performances,

maintaining a proper balance between safety and economy.

Therefore, a structure must be designed on the basis of strength, serviceability and

durability requirements.

There are three methods of concrete design. These are

1. The Working Stress Design (WSD) method

2. The Ultimate Strength Design (USD) method (also called Load Factor Method (LFD))

3. The Limit State Design (LSD) method

1.6.1. The Working Stress Design (WSD) method

In this method the section of reinforced concrete members are designed assuming straight-

line stress-strain relationships, i.e., the response and stresses are elastic. The stresses in the

concrete and steel at service load are kept below a stress called allowable or permissible

stress, which is obtained dividing the ultimate strength of the materials by safety factor. For

instance, the allowable compressive stress in extreme fiber of concrete should not exceed

0.425 fck and that of tensile stress in steel 0.52 fyk, for class-I works.

The internal bending moments and forces for a structure are calculated assuming linear

elastic behavior. Because of elastic stress distribution is assumed in design, it is not really

applicable to a semi-plastic (elasto-plastic) material such as concrete, nor is it suitable when

deformations are not proportional to the load, as in slender columns. It has also been found

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to be unsafe when dealing with the stability of structures subject to overturning forces. This

method was used from 1900-1950 for the design of reinforced concrete members.

1.6.2. The Ultimate Strength Design (USD) method

After about half a century of practical experience and laboratory tests the knowledge of the

behavior of structural concrete under load has vastly increased and the deficiencies of

elastic theory (working stress design method) have become evident. The deficiencies of

WSD are,

i. Reinforced concrete sections behave in-elastically at high loads; hence elastic

theory cannot give a reliable prediction of the strength of the members because

inelastic strains are not taken into account.

ii. Because reserve of strength in the inelastic range of stress-strain of concrete is

not utilized, the Working Stress Design Method is conservative and hence results

in uneconomical design.

iii. The stress-strain curve for concrete is nonlinear and is time dependent. Creep

strains can be several times elastic strains. Therefore, modular ratio used in WSD

is a crude approximation. Creep Strains can cause a substantial redistribution of

stresses and actual stresses in structures are far from allowable stress used in

design.

In the ultimate strength method, sections are designed taking the actual inelastic strains

into account. The design stresses used are the ultimate strengths of materials and for safety

the loads are magnified or scaled up by load factors. Typical load factors used are 1.4 for

dead load and 1.7 for live load. Structural analysis is carried out either assuming linear

elastic behavior of the structure up to ultimate load or by taking some account of the

redistribution of actions due to the non-linear behavior at high loads.

As this method does not apply factors of safety to material stresses, it cannot directly take

account of variability of the materials, and also it cannot be used to calculate the deflections

or cracking at working loads.

USD method became accepted as an alternative design method in building codes of ACI in

1956 and of UK in 1957. This method was popular from 1950 up to 1960s.

1.6.3. The Limit State Design (LSD) method

More recently, it has been recognized that the design approach for reinforced concrete

should ideally combine the best features of ultimate strength and working stress design.

This is desirable because if sections are proportioned by ultimate strength requirements

alone there is a danger that although the load factors are adequate to ensure safety against

strength failure, the cracking and deflections at service loads may be excessive. Cracking

may be excessive if the steel stresses are high or if the bars are badly distributed.

Deflections may be critical if the shallow section, which are possible in USD, are used and

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the stress are high. Thus, to ensure a satisfactory design, the deflections and crack widths

must be checked for service loads to make sure that they lie within reasonable limiting

values dictated by functional requirements of the structure. This check requires the use of

elastic theory. Therefore, in the LSD method structures will be designed for strength at

ultimate loads (ULS), and deflection and crack width checked at service loads (SLS).

This design philosophy is gaining acceptance in many countries throughout the world

including Ethiopia. EBCS2-1995 is based on the LSD method.

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Assignment I

1. Enumerate the advantages of concrete and major weaknesses of concrete and give ways

to overcome the weaknesses.

2. What factors make concrete a dominant structural material?

3. Discuss the different hypotheses on strength development of concrete and give your

position with your rationale.

4. Enumerate the factors that affect the strength and performance of concrete and

describe how these affect strength and performance of concrete.

5. How are characteristic strengths and moduli of elasticity of concrete and steel obtained

for design?

6. What are the two most important factors affecting strength of concrete?

7. What requirements should form-work and false-work meet in order to produce quality

structural concrete?

8. Discuss the care that should be exercised in concrete placement and curing.

9. What is the importance and use of codes in reinforced concrete design?

10. What is the weakness of WSD method that led to the development of USD method?

Describe the weaknesses of USD that led to the development of the LSD method?

11. Compare and contrast structural concrete and structural steel. Give the advantages and

disadvantages of each.

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CHAPTER II

LIMIT STATE DESIGN FOR FLEXURE

2.1. Introduction

The WSD method discussed in chapter I have some shortcomings that led to the

development of USD and LSD. The Limit State Design (LSD) method combines the best

features of WSD and USD and has gained acceptance in many countries throughout the

world including Ethiopia. Ethiopian Building Code Standards (EBCS) are based on the LSD

method.

The Limit State Design Method is based on the limit state design philosophy. This design

philosophy considers that any structure that has exceeded a limit state for which it was

designed is unfit for the intended function or use. The limit state may be reached because

the structure is in danger of collapse (ultimate limit state) or because excessive deflection

has resulted in the structure's being unable to carry out its design functions (serviceability

limit state). Other limit states may be reached due to vibration, cracking, durability, fire or

various other factors, which mean that the structure can no longer fulfill the purpose for

which it was designed. These limit states are classified into three as ultimate, serviceability

and special limit states.

1. Ultimate Limit State (ULS) concerns:

failure by rupture, loss or stability, transformation into a mechanism

loss of equilibrium

failure caused by fatigue

To satisfy the design requirements of the ULS,

Appropriate safety factors are used

The most critical combinations of loads are considered.

Brittle failure is avoided (Ductility is ensured).

Accuracy of concrete works checked.

2. Serviceability Limit State (SLS) concerns not failure of structures but:

deformation

vibrations which cause discomfort to people

damage (cracking) - appearance, durability or function

To satisfy the design requirements of the SLS,

Minimum depth for defection requirements is provided

Adequate cover is provided and

Necessary detailing of reinforcement.

3. Special Limit States concerns:

extreme earthquakes, fires, explosion or vehicular collisions

A special feature of this philosophy is that it uses statistics to assess the variation in the

contributions of the factors influencing the limit states of a structure. These are material

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strength and loads, which affect resistance (capacity) of structural members and action

effects (internal actions) respectively.

The distributions of material strength and variation in structural loads follow normal or

Gaussian distribution. Section capacity and internal actions follow a similar distribution.

The number of specimens with extremely low strength or extremely high strength, though

small is never zero. It is, therefore, possible to have the situation in which two extremes are

reached simultaneously and if this is the extreme of high load together with low strength,

then a limit state of collapse may be reached. The probability of the collapse limit state

being reached will not be zero, but it will be kept sufficiently low by selecting suitable design

stresses and design loads that the probability may practically be taken as zero.

The use of statistical procedures has resulted in what are called characteristic strength and

characteristic loads as reference values. Characteristic strength of a material is that value

below which some percent of the test results fall (5% according to EBCS 2-1995 for concrete

and steel).

Where fk = characteristics strength fm = mean strength, δ =standard deviation, K1 = a factor that ensures the probability of the characteristics strength is not being exceeded is small. (K1 =1.64)

Figure 2.1-1Characteristic strength definition

Table 2.1-1 gives different grades of concrete and characteristic cylinder compressive

strength in MPa. These values are obtained for standard cubes and cylinders at a slow rate

of loading to reach maximum stresses with in 2 or 3 minutes.

Grades of Concrete C15 C20 C25 C30 C40 C50 C60

fck 12 16 20 24 32 40 48 Table 2.1-1 Grades of concrete and characteristic compressive strength fck

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Where fcu = characteristic standard cube strength (obtained from 150mm cubes), fck= characteristic standard cylinder strength (for 150mm diameter and

300mm high cylinder).

The characteristic tensile strength of concrete is calculated using,

⁄

The characteristic strength of reinforcement steel, fyk is defined as the fractile of the proof

stress fy or the 0.2% offset strength.

The same basic procedure as for strength may be used for the calculation of characteristic

loads but the practically insufficient statistical information reduce the effectiveness of the

approach for loads. Hence these are defined in and given by codes. Characteristic load is

that value of the load, which has an acceptable probability of not being exceeded during the

service life of the structure. EBCS 1-1995 gives values of characteristic permanent loads Gk

and characteristic imposed loads Qk and EBCS8-1995 gives characteristic seismic loads AEd.

The LSD method is a design method that involves identification of all possible modes of

failure and determining acceptable factors of safety against exceedence of each limit state.

These factors of safety are those which take care of material variability γm and load

variability γF.

Suitable design stress fd is obtained as,

γm allows for differences that may occur between the strength of the material as

determined from laboratory tests and that achieved in the structure. The difference may

occur due to a number of reasons including method of manufacture, duration of loading,

corrosion and other factors.

Table 2.1-2 gives partial safety factors of materials at ULS according to EBCS 2-1995

Material and workmanship

Concrete γc Steel γs

Class I Class II Class I Class II

Design Situation

Persistent and transient

1.50 1.65 1.15 1.20

Accidental 1.30 1.45 1.00 1.10 Table 2.1-2Partial safe factors of materials at ULS according to EBCS 2 -1995

Persistent design situations refer to conditions of normal use. Transient design situations

refer to temporary conditions such as during construction or repair. Accidental design

situations refer to exceptional conditions such as during fire, explosion or impact.

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The difference in values for the two materials is indicative of the comparative lack of control

over the production of concrete the strength of which is affected by such factors as

water/cement ratio, degree of compaction, rate of drying, etc., which frequently cannot be

accurately controlled on site to conditions in factory.

Design stress of concrete in compression is,

Design stress of concrete in tension is,

⁄

Design stress of steel for both in tension and compression,

The characteristics load is given by EBCS 2-1995 as,

Where Fk = characteristics load, Fm = mean load, δ =standard deviation, K2 = a factor that ensures the probability of the characteristics load being exceeded is small. (K2 =1.64)

Figure 2.1-2Characteristics load definition

Suitable design loads are obtained from characteristic loads by applying partial safety factor

for loads or load factors γF.

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γF accounts for possible increase of loads above those considered in design, relative

accuracy in determining the loads, inaccuracy in the analysis and design stage, difference

between dimensions shown on structural drawings and as built due to inaccuracy,

construction and the importance of the limit state that is considered.

Load combination for ULS

1. Permanent action (Gk) and only one variable action (Qk)

2. Permanent action (Gk) and two or more variable action (Qk)

∑

3. Permanent action, variable action and accidental (seismic) action

Load combination for SLS

1. Permanent action (Gk) and only one variable action (Qk)

2. Permanent action (Gk) and two or more variable action (Qk)

∑

With the design loads for ULS on the structure, the structure is assumed to be on the verge

of collapse and ultimate moments and forces are determined by structural analysis. Analysis

can be carried out assuming linear elastic response (with or without plastic redistribution of

moments), non-linear response, or plastic response.

Finally serviceability requirements will be checked for the structure under service loads.

Elastic methods of analysis may be applied for analysis in the Serviceability Limit States.

Statics of beam action

A beam is a structural member that supports applied loads and its own weight primarily by

internal moments and shears. Figure 2.1-3a) shows a beam that supports its own dead

weight w, plus some applied load P. If the axial applied load, N, is equal to zero as shown,

the member is referred to as a beam. If N is a compressive force, the member is called a

beam-column. If it were tensile, the member would be a tension tie. These cause bending

moments, distributed as shown in figure 2.1-3 b).The bending moments are obtained

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directly from the loads using the laws of statics and for a given span and combination of

loads w and P. The moment diagram is independent of the composition or size of the beam.

The bending moment is referred to as a load effect. Other load effects include shear force,

axial force, torque, deflection and vibration.

At any section within the beam, the internal resisting moment, M, shown in figure 2.1-3 c)is

necessary to equilibrate the bending moment. An internal resisting shear, V, is also required

as shown. The internal resisting moment, when the cross section fails, is referred to as the

moment capacity or moment resistance. The word "resistance" can also be used to describe

shear resistance or axial load resistance.

The beam shown in figure 2.1-3will safely support the loads if at every section the resistance

of the member exceeds the effects of the loads.

Figure 2.1-3 Internal force in a beam

The internal resisting moment, M, results from an internal compressive force, C, and an

internal tensile force, T, separated by a lever arm, jd, as shown in figure 2.1-3 (d).

The conventional elastic beam theory results in the equation σ = My/I, which for an

uncracked, homogeneous rectangular beam without reinforcement gives the distribution of

stresses shown in figure 2.1-4.The stress diagram shown in figure 2.1-4 (c) and (d) may be

visualized as having a "volume," and hence one frequently refers to the compressive stress

block and the tensile stress block. This is equal to the volume of the compressive stress

block shown in figure 2.1-4 (d). In a similar manner one could compute the force T from the

tensile stress block. The forces, C and T, act through the centroids of the volumes of the

respective stress blocks. In the elastic case these forces act at h/3 above or below the

neutral axis, so that jd = 2h /3. From above equations we can write,

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Figure 2.1-4Elastic beam stresses and stress blocks

Stress-strain distribution for beams

In RC structures such as beams, the tension caused by bending moment is chiefly resisted by

the steel reinforcement while the concrete alone is usually capable of resisting the

corresponding compression. Such joint action of the two type of materials is assured if the

relative slip is prevented which is achieved by using deformed bars with high bond strength

at the steel-concrete interface. Figure 2.1-5 shows a simple test beam installed with gauges

to measure strains at different levels. The measured strains are seen to be linear as shown

in figure 2.1-5 (b). Corresponding stress are computed from strains at each level using

Hook’s Law i.e. E = σ/ε. The results are plotted in figure 2.1-5 (c) and are found to be

parabolic in nature.

Figure 2.1-5 Side view of test beam with gauges

To illustrate the stress-strain development for increased loading, consider the following,

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Figure 2.1-6 Loading on a simply supported beam

Figure 2.1-7 Stress and strain distribution

Major points to notice are,

At low loads where tensile stress is less than or equal to the characteristics tensile

strength of concrete (fctk), the stress & strain relation shown in figure 2.1-7 (a)

results.

At increased loading, tensile stress larger than fctk in figure 2.1-7 (b) cause cracks

below neutral axis (NA) and the steel alone carry all tensile force. If the compressive

stress at extreme fiber is less than fc'/2, stresses and strains continue to be closely

proportional (linear stress distribution) otherwise non-linear.

For further increment of load, the stress distribution is no longer linear as shown in

figure 2.1-7 (c).

If the structure say the beam has reached its maximum carrying capacity one may conclude

the following on the cause of failure.

1. When the amount of steel is small at some value of the load the steel reaches its

yield point. In such circumstances:

The steel stretches a large amount.

Tension cracks in the concrete widen, visible and significant deflection of the

beam occurs.

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Compression zone of concrete increase resulting in crushing of concrete

(secondary compression failure).

Such failure is gradual and is preceded by visible sign, widening and lengthening of

cracks, marked increase in deflection.

2. When a large amount of steel is used, compressive strength of concrete would be

exhausted before the steel starts yielding. Thus concrete fails by crushing.

Compression failure through crushing of concrete is sudden and occurs without

warning.

3. When the amount of tensile strength of steel and compressive strength of concrete

are exhausted simultaneously then the type of failure that occurs is called balanced

failure.

Therefore it is a good practice to dimension sections in such a way that, should there be

overloading, failure would be initiated by yielding of the steel rather than crushing of

concrete.

2.2. ULS of Singly Reinforced Rectangular Beams

In the ULS the materials are used to their maximum capacity, i.e., the concrete is strained to

its maximum usable strain of 0.0035 and steel to its design stress fyd (εs<0.01 also) as given

by EBSC 2-1995.

Design of reinforced concrete sections may be carried out using equations or charts and

tables. You may have to design irregular compressed areas like a triangle, trapezium, or

composite areas. The bases for all these are strain compatibility and equilibrium equations.

Therefore, we have to begin with stress-strain diagrams to derive expressions for flexural

strength of reinforced concrete members.

2.2.1. Basic assumptions at ULS

Three basic assumptions are made when deriving the expression for flexural strength of

reinforced concrete sections.

Sections perpendicular to the axis of bending that are plane before bending remain

plane after bending.

The strain in the reinforcement is equal to the strain in the concrete at the same

level.

The stress in the concrete and reinforcement can be computed from the strains by

using stress-strain curves for concrete and steel.

The three assumptions already made are sufficient to allow calculation of the strength and

behavior of reinforced concrete elements. For design however, several additional

assumption are introduced to simplify the problem with little loss of accuracy.

The tensile strength of concrete is neglected in flexural strength calculations.

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Concrete is assumed to fail when maximum compressive strain reaches a limiting

value of 0.0035 in bending and 0.002 in axial compression according to EBCS 2-1995.

The compressive stress-strain relationship for concrete may be based on stress-

strain curves or may be assumed to be rectangular, trapezoidal, parabolic or any

other shape as long as it is in agreement with comprehensive tests.

The maximum tensile strain in the reinforcements is taken to be 0.01 according to

EBCS 2-1995.

Stress-strain curve for steel is known.

The strain diagram shall be assumed to pass through one of the three points A, B or

C as shown in figure 2.2.1-1 as given by EBSC 2-1995.

Figure 2.2.1-1 Strain diagram in the ULS

Figure 2.2.1-2 Idealized stress-strain diagrams

2.2.2. Analysis of Singly Reinforced Concrete Beams

Stress and Strain Compatibility and Equilibrium

Two requirements are satisfied through the analysis and design of reinforced concrete

beams and columns.

1. Stress and strain compatibility. The stress at any point in a member must correspond

to the strain at that point.

2. Equilibrium. The internal forces must balance the external load effects.

Consider the stress and strain distribution at ULS for a rectangular cross section of singly

reinforced concrete beam subjected to bending as shown in the figure below.

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Figure 2.2.2-1 Singly reinforced rectangular beam

1. The triangular stress distribution applies when the stresses are very nearly

proportional to the strains, which generally occur at the loading level encountered

under working condition and it is, therefore, used at the serviceability limit state.

2. The rectangular-parabolic stress block represents the distribution at failure when the

compressive strains are within the plastic range and it is associated with the design

for the ultimate limit state.

3. The equivalent rectangular stress block us a simplified alternative to the rectangular-

parabolic distribution.

1. If one wants to use the idealized parabolic-rectangular stress block given in EBCS 2-

1995, as shown in figure 2.2.1-2

Figure 2.2.2-2 Derivation

and

Moment about the T,

Moment about the C,

αc and βc are values calculated by integrating the stress-strain diagram for the different

location of the N.A. depth. i.e.

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∫

∫

i. εcm≤2‰and N.A. within the section

ii. εcm≥2‰ and N.A. within the section

iii. εcm≥2‰ and N.A. outside the section

To understand the mechanics behind the derivation of the above equations, referring to

figure 2.2.2-2, the capacity of section when the εcm ≥ 0.002 and N.A. within the section,

⇒

⇒

(

) ( (

))

C is the compression stress resultant,

∫

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∫ (

) ( (

))

Taking moment about the top fiber,

∫

(

)

∫ (

) ( (

))

(

)

2. If one wants to use the rectangular stress block given in EBCS 2-1995,

Tensile force in the reinforcement bars become,

Compressive force in the concrete,

The moment resistance of the cross-section is,

One should note that, the rectangular stress block approximation is only valid if the

concrete stain is 0.0035 and that if the steel is below fracture stain (0.01). The justification

for reducing the depth of N.A by 80% is shown below.

Figure 2.2.2-3 Derivation

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From similarity of triangle,

⇒

To find the compressive force for the parabolic rectangular stress block,

[ ] [ ]

To find the moment arm βc, taking moment about the top fiber,

[ ]

[ ]

)

(

)

Thus to accommodate both αc and βc, the depth of the equivalent rectangular stress block is

reduced by 80%.

Example

Calculate the moment capacity of a beam with b = 250mm, h = 500mm and cover to

reinforcement of 25mm. The beam is reinforced with 3ф20 bars with fyk = 400Mpa and fck =

30MPa. Use both parabolic-rectangular stress block and rectangular stress block. Comment

on the accuracy of rectangular stress block approximation.

2.2.3. Types of flexural failures

There are three types of flexural failures of reinforced concrete sections: tension,

compression and balanced failures. These three types of failures may be discussed to

choose the desirable type of failure from the three, in case failure is imminent.

a) Tension Failure

If the steel content As of the section is small, the steel will reach fyd before the concert

reaches its maximum strain εcu of 0.0035. With further increase in loading, the steel force

remains constant at fyd As, but results a large plastic deformation in the steel, wide cracking

in the concrete and large increase in compressive strain in the extreme fiber of concrete.

With this increase in strain the stress distribution in the concrete becomes distinctly non-

linear resulting in increase of the mean stress. Because equilibrium of internal forces should

be maintained, the depth of the N.A decreases, which results in the increment of the lever

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arm z. The flexural strength is reached when concrete strain reaches 0.0035. With further

increase in strain, crushing failure occurs. εs may also be so large as to exceed 0.01. This

phenomenon is shown in figure 2.2.3-1. This type of failure is preferable and is used for

design.

Figure 2.2.3-1 Tension failure

b) Compression Failure

If the steel content As is large, the concrete may reach its capacity before steel yields. In

such a case the N.A depth increases considerably causing an increase in compressive force.

Again the flexural strength of the section is reached when εc= 0.0035.The section fails

suddenly in a brittle fashion. This phenomenon is shown in figure 2.2.3-2.

Figure 2.2.3-2Compression failure

c) Balanced Failure

At balanced failure the steel reaches fyd and the concrete reaches a strain of 0.0035

simultaneously. This phenomenon is shown in figure 3.2.3-3.

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Figure 2.2.3-3 Balanced failure

Figure 2.2.3-4Strain Diagrams for tension (1), Balanced (2) and compression (3) failures

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2.2.4. Design equations for singly reinforced rectangular beams

Compression failures are dangerous because they are brittle and occur suddenly giving little

visible warning. Tension failures, however, are preceded by large deflections and wide

cracking and have a ductile character. To ensure that all beams have the desirable

characteristic of visible warning if failure is imminent, as well as reasonable ductility at

failure, it is recommended that the depth of the N.A be limited or the steel ratio be limited

to a fraction of ρb. In our code of practice, EBCS 2-1995 limits the depth of the N.A to,

Where δ is percent plastic moment redistribution = (moment after redistribution)/original

moment.

In the case of no moment redistribution, δ = 1.0

⁄

Usually d is obtained from serviceability limit state. EBCS 2-1995 gives the following

minimum effective depth,

(

)

Where, fyk = characteristics strength of reinforcement (MPa) Le = effective span βa = constant from table

Member Simply Supported End Spans Interior Spans Cantilevers

Beams 20 24 28 10

Slabs a) Span ratio 2:1 b) Span ratio 1:1

25 35

30 40

35 45

12 10

Table 2.2.4-1 Values of βafrom EBCS 2-1995

The design of singly reinforced section can be carried out using chart or tables found in

EBCS-2-1995 Part 2 and are summarized below.

Referring to figure 2.2.2-2, the force cared by the compression and tensile zone can be

calculated using,

and

Moment capacity of the section in terms of tension force in the steel,

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Where ρ is defined as geometrical ratio of steel reinforcement and is given by,

In the general design tables No 1a and No 1b in EBCS 2-1995 part 2, the design of the

section is formulated using empirical parameters Km and Ks.

√

√

⁄

Moment capacity of the section in terms of the compression force C in the concrete is,

For the limiting case of x/d = 0.45, μsd,s = μ*sd,s = 0.295

Steps to be followed

a. Design using tables

1. Evaluate Km

2. Enter the general design table No 1.a using Km and concrete grade,

a. If Km ≤ Km*, the value of Km*show shaded in design Table No 1.a, then the

section is singly reinforced.

Enter the design table No 1.a using Km and concrete grade

Read Ks from the table corresponding to the steel grade and Km

Evaluate As

b. Design using general design chart

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1. Calculate

2. Enter the general design chart,

If , section is singly reinforced.

Evaluate Z from by reading value of from chart using

Evaluate

Minimum reinforcement

At Some sections of continuous beams, moment may be so small that require a small

amount of steel. If the moment is less than that which cracks the section and with any load

causing cracking moment, failure is sudden and brittle. To prevent this, it is recommended

that a minimum reinforcement, As,min required to resist Mcracking be provided. As,min is

obtained from the cracking moment. Empirical relations are given in codes and standards.

EBCS2-1995 gives for beams,

2.2.5. Design equation for singly reinforced one-way slabs

One-way slabs carrying predominantly uniform load are designed on the assumption that

they consist of a series of rectangular beams of 1 m width spanning between supporting

beams or walls.

Figure 2.2.5-1One-way Slab Panels

A rectangular slab panel is classified as one way slab if the ratio of the long span to that of

the short span is greater than two. If the long span/short span is less than 2, the slab is

classified as two-way slab; the load in this case is transmitted along two orthogonal

directions.

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One way slabs may be simply supported or continuous over a number of supports. The

bending moments, on which design is to be based, are calculated from elastic analysis in the

same way as for beams. Approximate analysis could also be used in the case of continuous

slabs as recommended in some code of practices.

The flexural design of one-wayslab sections are treated in the same manner as for singly

reinforced rectangular beam sections, considering the slab as strips of beams having a width

of 1m. The reinforcement bar obtained is distributed uniformly with spacing between bars

given as,

Where: as – An area of reinforcement bar to be used

As – Total area of steel required

2.1. ULS of doubly reinforced rectangular section

Occasionally, beams are built with both tension reinforcement and compression

reinforcements. The effect of the compression reinforcement on the behavior of the beam

and the reason it used are discussed in this section.

Figure 2.1-1 Effect of compression reinforcement

The effect of the compression on the design resistance capacity is very little. For normal

steel tension reinforcement ration (ρ ≤ 0.015), the increase in the moment is generally < 5%.

The notable difference between section with or without compression reinforcements is that

the NA depth of the section with compression reinforcement is less than the later, and the

effectiveness of compression steel decrease as it moves away the compression face.

Reasons for providing compression reinforcement

1. Reduce sustained-load deflection

First and most important, the addition of compression reinforcement reduces the

long term deflection of a beam subjected to sustained loads.

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Figure 2.1-2 Effect of compression reinforcements on deflection under sustained loading

Creep of the concrete in the compression zone transfers load from the concrete to

the compression steel, reducing the stress in the concrete.

2. Increased ductility

The addition of the compression reinforcements causes reduction in the depth of the

compression block.

Figure 2.1-3 Effect of compression reinforcements on strength and ductility.

3. Change of mode of failure from compression to tension

When ρ > ρb, beam fails in a brittle manner, through crushing of the compression

zone before the reinforcement yields.

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Figure 2.1-4 Moment curvature diagram for beams with or without compression reinforcement

4. Fabrication ease

When assembling the reinforcement cage for the beam, it is customary to provide

bars in the corners of stirrups to hold the stirrups in place in the form and also to

help anchor the stirrups. If developed properly, these bars in effect are compression

reinforcements, although they are generally disregarded in design.

Analysis of Beams with Double Reinforcements

If the depth of an RC beam is limited due to architectural or other reasons the section may

not have sufficient compressive area of concrete to resist the moment induced in it. In such

cases the capacity of the section can be increased by placing steel in the compression zone.

This additional steel carries the additional compressive force that is required to resist

moment ΔM over and above the maximum capacity of the section as singly reinforced

section as shown in figure 2.3-5.

Figure 2.1-5 Doubly reinforced section

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For analysis the beam is hypostatically divided into two beams. Beam 1 consisting of the

concrete web and sufficient steel at the bottom so that Ts1 = Cc and having a maximum

capacity of a singly reinforced section. Beam 2 consisting of the compression reinforcement

at the top and the remaining tension reinforcement to carry additional moment.

Beam 1

Beam 2

In the derivation of the above formula, the stress in the compression reinforcement has

been shown as fs2.

If the steel has yielded (εs2 ≥ εyd)

If the steel has yielded (εs2 < εyd)

In the case of analysis type of problem, the steel may not have yielded. The analysis of such

a section a best carried but by assuming first that all the steel has yielded, the calculation

can be modified later if it is found that some or all of the steel have not yielded.

The step to be followed is, calculate first αc assuming all steel yielded and reading values of

εs1 and εs2 from the chart and compare with εyd.

The design resistance or capacity of the section can be calculated by

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For design problems the following procedures can be followed,

a. Design using tables

1. Evaluate Km

2. Enter the general design table No 1.a using Km and concrete grade,

If Km> Km*, the value of Km*show shaded in design Table No 1.a, then the

section is doubly reinforced.

- Evaluate Km/ Km* and d2/d

- Read Ks, Ks’, ρ and ρ’ from the same table corresponding to Km/ Km*, d2/d

and concrete grade

- Evaluate

b. Design using general design chart

1. Calculate

2. Enter the general design chart,

If , section is doubly reinforced.

- Evaluate Z from chart using

- Evaluate

- Calculate

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2.2. ULS of T- and L- Sections

Reinforced concrete floors or roofs are monolithic and hence, a part of the slab will act with

the upper part of the beam to resist longitudinal compression. The resulting beam cross-

section is, then, T-shaped (inverted L), rather than rectangular with the slab forming the

beam flange where as part of the beam projecting below the slab forms the web or stem.

Figure 2.2-1 Slab and Beam floor System

The T -sections provide a large concrete cross-sectional area of the flange to resist the

compressive force. Hence, T-sections are very advantageous in simply supported spans to

resist large positive bending moment, whereas the inverted T-sections have the added

advantage in cantilever beam to resist negative moment.

As the longitudinal compressive stress varies across the flange width of same level, it is

convenient in design to make use of an effective flange width (may be smaller than the

actual width) which is considered to be uniformly stressed.

Effective flange width (according to EBCS 2, 1995)

Figure 2.2-2 Typical slab

The part of the slab that is acting together with the beam, called effective flange width be is

provided in codes of practices. The EBCS recommends that the effective flange width for T-

sections and L- sections must not exceed:

For in interior beams (T-sections)

{

⁄

For in edge beams (L-sections)

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{

Le is the effective span length and bw is width of the beam.

The neutral axis of a T-beam may be either in the flange or in the web, depending upon the

proportion of the cross-section, the amount of tensile steel and the strength of the

materials. If the calculated depth to the neutral axis is less or equal to the slabs thickness, hf

the beam can be analyzed as if it were a rectangular beam of width equal to be. If the NA is

in the web x>hf, a method is developed which account for the actual T -shaped compression

zone. The compression block shall be divided into two parts; one is for the compression in

the flange (Beam F) and the other is for the compression in the web (Beam W). T-beams

with compression flanges rarely require compression reinforcement, but if this is

unavoidable, the same principles apply as for doubly reinforced sections for the

compression in the web.

When designing T- and L- sections, since the compression blocks are irregular in shape, it is

one of the special cases where the equivalent rectangular stress block approximation are

used instead of the parabolic rectangular one. Referring to figure 2.4-3,

Assume b = be,

Usually,

We solve for x from the above quadratic equation,

i. If , section is T- or L-, thus it is convenient to consider two

hypothetical beams: Beam F and Beam W

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Figure 2.2-3ULS T-section

Beam F

(

) or (

)

The force in the remaining steel area Asw is balanced by compression in the rectangular

portion of the beam. (i.e. Asw = As - Asf)

Beam W

or

The total moment capacity of the section now becomes,

ii. If 0.8x ≤ hf, then the beam is considered to be a “rectangular beam” for the

calculation purpose. The effect of small area of the web under compression is

insignificant.

Note:- In the derivation of the design resistance capacity of the section, it was assumed that

fs = fyd. This has to be verified by determining the NA and checking the strain profile.

Beam W

Beam F

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Assignments Problems on Chapter 2

1. Determine the imposed (uniformly distributed) load and the tensile steel of the

singly reinforced rectangular beam of L = 8.0m simply supported, thickness of

supporting brick wall = 300 mm, width b = 300 mm, effective depth d = 550 mm,

total depth D = 600 mm, grade of concrete = C25 and characteristic strength of the

steel = 415MPa. (Height of brick wall=?)

2. A singly reinforced beam has a width of 300mm and an effective depth of 600 mm.

The concrete is C25 and the steel is Grade 300. Determine

a. the maximum design moment of resistance of the section and the required

reinforcement, if the maximum aggregate size is 20mm and the cover to the

main reinforcement is 40 mm, and

b. the area of reinforcement required to resist a moment of 350kNm.

3. Determine the moment of resistance of a section whose width is 200mm, effective

depth 450mm and is reinforced with 2Φ20 and 2Φ16 bars. The concrete is C30 and

steel is????.

4. A singly reinforced beam constructed from C25 concrete has a width of 200mm and

an effective depth of 450mm. If the reinforcement has characteristic yield strength

of 420MPa, determine the maximum capacity of the section and the reinforcement.

5. A doubly reinforced beam constructed from C30 concrete has a width of 250mm and

an effective depth of 450mm. The reinforcement has a characteristic strength of

300MPa, and the axis depth of the compression reinforcement, if required, is 50mm.

If the moment applied to the section is 400kNm, determine the reinforcement

requirements.

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6. Determine the ultimate moment capacity of the doubly reinforced beam of b = 350

mm, d' = 60 mm, d = 600 mm, As = 2945 mm2, A’s = 1256 mm2, using C30 and S415.

7. A T-beam has an effective flange width of 1370mm, a flange thickness of 150mm, a

web width of 250mm and an overall depth of 500mm. The concrete is C25 and the

steel 300MPa. The axis distance from the reinforcement to any face is 50mm. Design

suitable reinforcement for the following moments:

a. 400kNm (sagging), and

b. 300kNm (hogging).

8. A beam with a flange width of 1500 mm, flange thickness 150 mm, effective depth

800mm and web width 350mm carries a moment of 2000kNm. If the concrete is C25

and the steel strength is 300MPa, design suitable reinforcement.

9. Design the main reinforcement(s) of a rectangular RC beam section having a width of

300mm, a total depth of 500mm and carries a positive moment of 400kN-m. Use C30

concrete, fyk = 420MPa and d' = 50mm.

10. Determine the maximum permissible span (in meters) of a simply supported RC one-

way slab having a thickness of 15cm and reinforced with Φ10c/c100mm. The

working live-load on the slab is 5kPa; C25 concrete and steel fyk = 300MPa are used.

Consider both ultimate and serviceability limit states to select the appropriate value.

11. For the reinforced concrete cantilever beam shown in the figure below, design the

total depth (not be higher than 600mm), as well as the flexural reinforcements. Use

C30 concrete, fyk = 300Mpa, d’ = 60mm. (The given loads are factored loads.)

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12. Determine (a) the theoretical and (b)

practical curtailment point for 2 diam

20 bars for the cantilever beam

shown below. Assume b = 300mm, d

= 700mm, C25 concrete, fyk =

300MPa and a total uniformly

distributed design load of 20 kN/m.

RCS – I Serviceability limit state Chapter V


CHAPTER III

LIMIT STATE DESIGN FOR SHEAR

3.1. Introduction

Beams resist loads primarily by means of internal moment M and shear V. In the design of

reinforced concrete members flexure is usually considered first, (i.e. sections are

proportioned and areas of longitudinal reinforcement determined for the moment M),

because flexural failure is ductile. The beams are then designed for shear. Because shear

failure is frequently sudden and brittle, the design for shear should ensure that shear

strength equals or exceeds the flexural strength at all points in the beam. Fig 3.1 shows

internal forces of a simple beam.

Figure 3.1-1 Internal force in beams

3.2. Basic Theory

Stresses in an Uncracked Beam

From the FBD in Fig. 3.1-1 c, it can be seen that dM/dx = V. Thus shear forces exist in those

parts of a beam where the moment changes from section to section. The shear stresses, V

on elements 1 and 2 cut out of a beam (Fig. 3-2-1 a) is calculated from the equation,

Where V = shear force on the cross section

I = moment of inertia of the section

Q = first moment of part of the cross-sectional area about the centroid

b = width of the member at which the stresses are calculated



e ) Photograph of half of a cracked reinforced concrete beam

Figure 3.2-1 Normal, shear and principal stress in a homogenous un-cracked beam

For uncracked rectangular beam Fig. 3-2-1b gives the distribution of shear stresses on a

section. In regions where we have M and V we have biaxial states of stress and the principal

stresses are

√(

)

√(

)



The principal stresses on the elements are shown in Fig. 3-2-1c. The surfaces on which

principal stresses act in an uncracked beam are plotted by curved lines as in Fig 3-2-1d and

are known as stress trajectories. Since concrete cracks when the principal stresses exceed

the tensile strength of the concrete, the initial cracking pattern resembles the family of

curves (stress trajectories) shown in Fig 3-2-1d.

Two types of cracks can be seen. The vertical cracks occurred first, due to flexural stresses.

These start at the bottom of the beam where the flexural stresses are the largest. The

inclined cracks at the ends of the beam are due to combined shear flexure. These are

commonly referred to as inclined cracks, shear cracks or diagonal tension cracks. Such

cracks must exist before a beam can fail in shear.

Although there is similarity between the planes of maximum principal tensile stress and the

cracking pattern, it is by no means perfect, because in RC flexural cracks generally occur

before the principal tensile stress at mid height become critical. Once the flexural cracks

have occurred, the tensile stress perpendicular to the cracks drops to zero. To maintain

equilibrium, a major redistribution is necessary. As a result, the onset of inclined cracking in

a beam cannot be predicted from the principal stresses unless shear cracks precedes

flexural cracking. This very rarely happens in RC, but it does occur in some pre-stressed

beams.

Shear transfer of reinforced concrete beams heavily relies on the tensile and compressive

stresses of the concrete. Most of the time the problem of concrete in shear design is not

shear stress exceeding the shear strength of the concrete; rather, it is the major principal

stress exceeding the tensile strength of concrete due to the low tensile strength. When the

tensile stress exceeds the tensile strength then cracks will form. With the formation of

cracks ensues a complex pattern of stresses.

3.3. Mechanism of shear resistance in concrete beams without shear

reinforcements

After formation of cracks we will have a different stress distribution. Fig 4-3-1a shows a

cracked beam.



Figure 3.3-1 Calculation of average shear stress between cracks

The concrete below the neutral axis in a cracked reinforced concrete beam is in a state of

pure shear because tensile stress is zero.

From fig 3-3-1b

⇒

From the moment equilibrium of the element

⇒

If the shaded portion of Fig 3-3-1b is isolated, the force ΔT must be transferred by horizontal

shear stress on the top of the element. The average value of these stresses below the top of

the crack is

The distribution of the average horizontal shear stress is shown in Fig 3-3-1d. Since the

vertical shear stresses on an element are equal to the horizontal shear stresses on the same

element, the vertical shear stress distribution will be as shown in Fig. 3-3-1d.

Fig 3.3-2 shows mechanism of shear resistance across an inclined crack in a beam without

shear reinforcement (stirrups). Observe that a typical vertical plane cuts (passes): the

compression zone, the crack and the flexural reinforcement, unlike the entire section of the

un-cracked homogenous beam. Shear resistance along A, B, C is provided by the sum of



shear in the compression zone Vcz, the vertical component of force due to aggregate

interlock Vay and force due to dowel action of the longitudinal reinforcement Vd.

Figure 3.3-2 Internal force in a cracked beam without stirrups

Immediately after inclined cracking it is found that 40-60% of the shear is resisted by Vay and

Vd,

Considering portions D, E, F below the crack and summing up moments about E we see that

Va and Vd will have moment about E in the clockwise direction which should be balanced by

moment due to compression force C’1.

From horizontal force equilibrium on vertical face A, B, D, E we see that T1=C1+C’1 and finally

T1 and C1+C’1 must equilibrate the external moment at the section.

As the crack widens Va decreases and much of the resistance is provided by Vcz and Vd. As Vd

gets larger it leads to splitting crack in the concrete along the reinforcement. When this

crack occurs Vd drops to zero. When Va and Vd disappear so do V’cz and C’1 with the result

that all shearing is transmitted in the width AB above the crack. This may cause crushing of

concrete in region AB.

It is important to note also that, if C’1 =0, T2 = T1 and T2=C1. In other words, the inclined crack

has made the tensile force at C to be a function of the moment on the vertical section A, B,

D, E. This shift in tensile force must be considered when determining bar cutoff points and

when anchoring bars.

For a typical RC beam, the approximate proportions are

It has been found that the dowel action is generally the first to reach its capacity followed

by failure of the aggregate interlock, which is followed by shear failure of the concrete in



compression (abruptly and explosively). However, the precise proportion is difficult to

establish and the shear strength is represented by a single expression accounting for all

mechanisms.

The shear resistance of the concrete depends on the tensile strength of concrete, shear

span to depth ratio, av/d, size of the member, aggregate interlock and the amount of

longitudinal reinforcement. Empirical relations are given in codes which may consider all of

these factors or only some.

EBCS 2 gives empirical relations as a function of the tensile strength of the concrete fctd, area

of longitudinal reinforcement, effective depth d, and breadth of web, bw.

⁄

As the area of tensile reinforcement anchored beyond the intersection of the steel and the

line of possible 450 cracking starting from the edge of the section.

3.4. Design of shear reinforcement

In subsection 3.3 we saw that formation of diagonal cracks is followed by widening of cracks

and brittle compression failure. This type of failure can be suppressed and development of

full flexural capacity can be ensured using shear reinforcement. Inclined stirrups, bent up

longitudinal bars or vertical stirrups can be used.

Figure 3.4-1 Inclined cracks and shear reinforcements



Figure 3.4-2 Vertical and Inclined Shear reinforcements

The most commonly used type is vertical stirrup. The use of bent bars has almost

disappeared. Inclined stirrups cannot be used for beams resisting shear reversal such as

building resisting seismic loads.

Stirrups restrain the cracks from opening wide and so not only maintain the shear resistance

due to aggregate interlock and dowel action but also contribute to shear resistance. The

shear resistance at section of the beam is categorized into two contributions as the part

resisted by concrete and as shear resisted by stirrups.

Where: Vc = Vcz+Vay+Vd, shear resisted by the concrete Vs = shear resisted by the stirrups

The stirrups are required to carry shear over and above the capacity of the concrete.

Figure 3.4-3 Distribution of internal shears in a beam with web reinforcements



Figure 3.4-4 Internal force in a cracked beam with stirrups

The amount of shear reinforcement or the spacing S of the stirrups having cross sectional

area Av (of the two vertical bars) is obtained from a mathematical model called “Truss-

Analogy”. This model was proposed by Professor Mörsch in 1902 for the design of beam for

shear. The stirrups are modeled as vertical tension members, the longitudinal flexure

reinforcement as horizontal tension members the concrete diagonals between cracks as

diagonal compression members and the concrete in flexural compression as top horizontal

compression members as shown in fig 3.4-5.

The shear reinforcement spacing ‘S’ can be calculated as follows because it has to carry ‘Vs’,

The horizontal projection of cross section A-A is and

No. of stirrups in this width is;

Therefore,

EBCS 2 gives



Figure 3.4-5 Truss analogy

Before determining the spacing of reinforcement S, whether diagonal compression failure of

concrete occurs or not should be checked. The average compression stress in the concrete

diagonal in concrete diagonals in fig.3.4-6 b is



Figure 3.4-6Forces in stirrups and compression diagonals

The shear V on B-B has been replaced by diagonal compression force D and axial tension

force Nv as shown in Fig 3.4-6c.

(

)

⇒

V is the shear at which diagonal compression failure occurs. Internal shear induced by loads

should be less than V. If internal shear is greater than or equal to 0.5fcdbwZ then the section

has to be increased.

EBCS 2 gives even a smaller limit on Vsd to avoid diagonal compression failure,

If Vsd>VRd, then the beam section has to be increased.

Assuming uniform shear stress distribution in the concrete the resultant of V and D act at

mid height of the section as a result, Nv acts through mid-height which means that Nv/2 acts

in each of the top and bottom chord members.

At the bottom total tension, ⁄

⁄ ,

At the top net compression, ⁄

⁄ .

In design the value of θ should be 25º ≤ θ ≤ 65º. The choice of small value of θ reduces the

number of stirrups required but increases the compression stress in the web and increases

Nv, and hence the shift of moment diagrams. The opposite is true for large angles.

Because the shear within a distance of D from face of support is resisted by the support for

a 450 crack, the maximum design shear force is taken as the one at a distance d from face of

support.

Minimum shear reinforcement according to EBCS 2

Because shear failure of RC beams without shear reinforcement is brittle and sudden,

minimum shear reinforcement should be provided in regions of beam where we need no

shear reinforcement theoretically.



The practical design procedure recommended by EBCS 2 is essentially empirical and may be

summarized as follows:

• Calculate the design shear force at the section to be designed, Vsd

• If Vsd ≥ VRd, Diagonal compression failure, increase the cross sectional dimension

• If Vsd ≤ VRd and Vsd ≥ Vc, provide shear reinforcement , S, as

• If Vsd ≤ Vc , provide minimum reinforcement, ρmin

Maximum spacing according to EBCS 2,

S ≤ Smax =0.5d ≤ 300 mm if V ≤ 2/3VRd,

S ≤ Smax =0.3d ≤ 200 mm if VRd>V > 2/3VRd

The first limit is given so that a 450 crack will be intercepted by at least one stirrup.

Commonly used stirrup bars have diameters ranging between 6 mm and 10 mm.

3.5. Bond and development length

3.5.1. Bond

In reinforced concrete, the concrete carries compression and the steel carries tension. In the

tension zone there is no slip between the concrete and the steel transfers its tension to the

surrounding concrete by shear stresses at the bar-concrete interface. This interface shear

stress is called bond stress.

This bond when fully developed enables the two materials to form a composite structure. If

this bond could not be developed then the bars pull out of the concrete and the tension

drops to zero.

The bond strength varies along the length of the bar Fig 3.5.1-1e and usually average bond

stress is used.



Figure 3.5.1-1 Steel, concrete and bond stresses in a cracked beam

The mechanism by which smooth plane bars develop this bond is by adhesion between the

concrete and the bar surface and by a small amount of bar friction.

For a bar loaded in tension both of these will be lost quickly because of reduction in

diameter due to Poisson’s ratio and the bar pulls out. For this reason, smooth plane bars are

not used as reinforcement. For cases where smooth bars are embedded in concrete (anchor

bolts, stirrups made of small diameter bars, etc) mechanical anchorage in the form of hooks,

nuts and washers on the embedded end, or similar devices are used.

In deformed bars although adhesion and friction are present at first loading, this will be lost

quickly leaving the load to be transferred by bearing on the ribs (Fig 3.5.1-2). If these

bearing forces are too big, the radial component will cause splitting along the reinforcement

which propagate out to the surface along the shortest distance (fig 3.5.1-3)



Figure 3.5.1-2 Bond transfer mechanism

The load at which splitting failure develops is a function of:

1. The minimum distance from the bar to the surface of the concrete or to the next bar.

The smaller this distance, the smaller the splitting load.

2. The tensile strength of the concrete

3. The average bond stress. As this increase the wedging force increase leading to a

splitting failure.

Figure 3.5.1-3 Typical splitting failure surfaces



Bond stresses arise from two situations; from anchorage of bars and from change of bar

force along the length of the bar such as due to change in bending moment (fig 3.5.1-4).

Average bond stress in a beam

⇒

If there are more than one bars

⇒

Where ∑ p is the sum of perimeters of all

bars

Alternatively

3.5.2. Development Length

The full design tensile strength of a deformed straight bar can be developed at a given

section provided the bar extends into the concrete a sufficient length beyond the section.

The length of the bar required to develop the length of the bar is known as development

length, lb or anchorage length. Since bond stress varies along the length of the bar, the

concept of development length is used instead of bond stress in codes.

This length lb is a function of

a) Whether we have tension or compression, longer lb is required for tension. b) The quality of concrete surrounding the bar which in turn is affected by depth of the

section and position of the bar. c) The diameter and grade of steel.

EBCS 2 gives the basic anchorage length lb for a bar of diameter as

Figure 3.5.1-4 Average flexural bond stress



Where: fbd is the design bond strength defined below.

The design bond strength fbd depends on the type of reinforcement, the concrete strength

and the position of the bar during concreting. The bond conditions are considered to be

good for:

a) all bars which are in the lower half of an element b) all bars in elements whose depth does not exceed 300mm. c) all bars which are at least 300mm from top of an element in which they are placed d) all bars with an inclination of 45-900 with the horizontal during concreting

For good bond conditions the design bond strength of may be obtained from

For other bond conditions the design bond strength may be taken as 0.7*the value for good

bond conditions. Local bond should be checked at sections where there are high shear

combined with rapid changes in bending moments such as: simply supported ends of a

member, points of contra flexure, supports of a cantilever, and points where tension bars

are terminated.

The required anchorage length lbnet depends on the type of anchorage and on the stress in

the reinforcement.

[ ]

[ ]

Where: a = 1.0 for straight bar in tension or in compression a = 0.7 for hook anchorage with standard hooks in fig 3.5.2-1 As[calculated] and As [provided] are area of reinforcement calculated, and provided respectively. lb,min = minimum anchorage length:

for bars in tension lb,min = 0.3lb ≥ 10ϕ or lb,min ≥ 200 mm

for bars in compression lb,min = 0.6lb ≥ 10ϕ or lb,min ≥ 200 mm

Figure 3.5.2-1 Standard hooks



3.5.3. Lapped splices

When the available standard length of bars (which is 12m) is less than the required length

we extend reinforcement bars by lap splices. In lapped splices, the force in one bar is

transferred to the surrounding concrete which in turn transfers the force to the adjacent

bars. Due to the stress influence of the two bars in the surrounding concrete a large

development length is required for lapped splices than for anchorage.

The requirement in lapped splices is to locate in regions of small bending moment and avoid

splicing in critical zones (large tension zones).

EBCS 2 gives the lap length `lo’ to be at least,

Where:

lb net and a are given in section 4.5.2

a1 is obtained from the following table; it is a function of percentage of rebars

lapped at one section. Lapped joints are considered to be at the same section

if the distance between their centers does not exceed the required lap

length.

Distance between

two adjacent laps

Distance to the

nearest surface

Percentage of reinforcement

lapped with in required lap length

a b 20% 25% 33% 50% 100%

a < 10 and /or b < 5 1.2 1.4 1.6 1.8 2.0

a > 10 and b >5 1.0 1.1 1.2 1.3 1.4

Table 3.5.3-1 values for a1

3.5.4. Bar cutoff

For economy some bars are cut off where they are no longer needed where the remaining

bars are adequate to carry the tension. The location of points where bars are cutoff is a

function of the tension due to moment and shear.

The flexure envelope tension diagram will be displaced horizontally by a1 as shown in Fig.

3.5.4-1 to take care of additional tension resulting from shear force.



Figure 3.5.4-2 Tensile force or M/Z diagram

The displacement a1 depends on the spacing of potential shear cracks and may be taken as:

a) members without shear reinforcement (slabs) a1 = 1.0d b) members with Vsd < 2Vc, a1= 0.75d c) members with Vsd ≥ 2Vc, a1= 0.5d

Where Vsd is the applied design shear force Near points of zero moment a1 ≥ d shall be taken for both positive and negative moments. The anchorage length of reinforcement is as follows 1) Reinforcement shall extend beyond the point at which it is no longer required to resist

tension for a length given by lb or lb.net ≥ d provided that in this case the continuing bars are capable of resisting twice the applied moment at the section

2) The anchorage length of bars that are bent up as shear reinforcement shall be at least equal to 1.3lb.net in zones subject to tension and to 0.7lb.net in zones subject to compression

When considering anchorage of bottom reinforcement at supports, the following must be

applied

1. At least one-quarter of the positive moment reinforcement in simple beams and one-half of the positive moment reinforcement in slabs shall be extended along the same face of the member in to the support

2. The anchorage of this reinforcement shall be capable of developing the following tensile force

3. The anchorage length is measured from a) The face of the support for a direct support

4. A plane inside the support located at a distance of 1/3 the width of the support from the face of the support for an indirect support

5. The anchorage length of the bottom reinforcement at intermediate supports shall be at least 10Ф.



CHAPTER IV

SERVICEABILITY LIMIT STATE

4.1. Introduction

In design of structures, the chief items of behavior of which are of practical significance are

1) The strength of the structure. i.e., the magnitude of loads which will cause

the structure to fail and

2) The deformations, such as deflections and extent of cracking, which the

structure will undergo when loaded under service conditions.

In the previous chapter we mainly deal with the strength design of RC beams. It is also

important that member performance in normal service be satisfactory, when loads are

those actually expected to act, (i.e. when load factor is 1.0), which is not guaranteed simply

by providing adequate strength. Serviceability studies are carried out based on elastic

theory, with stress in both concrete and steel assumed to be proportional to strain. The

concrete on the tension side of the neutral axis may be assumed un-cracked, partially

cracked, or fully cracked, depending on the loads and material strengths.

The concept of Serviceability limit states has been introduced in chapter 2 and for RC

structures these states are often satisfied by observing empirical rules which affect the

detailing only. In some circumstances, however, it may be desired to estimate the behavior

of a member under working conditions, and mathematical methods of estimating

deformations and cracking must be used.

The major SLS for reinforced concrete structures are: excessive crack widths, excessive

deflections, and undesirable vibrations.

Historically, deflections and crack widths have not been a problem for RC building

structures. With the advent high strength steel (fyk ≥ 400 MPa), the reinforcement stresses

at service loads have increased by about 50%. Since crack widths, deflections and fatigue

are all related to steel stress, each of these has become more critical.

4.2. Elastic analysis of beam sections

4.2.1. Section Un-cracked

As long as the tensile stress in the concrete is smaller than the tensile strength of concrete

(fctk) the strain and stress is the same as in an elastic, homogeneous beam. The only

difference is the presence of another material, i.e. the steel reinforcement. As it can be

shown, in the elastic range, for any given value of strain, the stress in the steel is 'n' times

that of the concrete, where n =Es/Ec is the modular ratio. In calculation the actual steel and

concrete cross-section could be replaced by a fictitious section (transformed section)

thought of as consisting of concrete only. In this section the actual steel area is replaced

with an equivalent concrete area (nAs) located at the level of the steel. Once the



transformed section has been obtained, the beam is analyzed like an elastic homogeneous

beam.

Figure 4.2.1-1 Un-cracked Transformed Section

4.2.2. Section Cracked

When the tension stresses fct exceeds fctk, cracks form in the tension zone of the section. If

the concrete compressive stress is smaller than approximately 0.5fck and the steel has not

reached the yield strength, both materials continue to behave elastically.

At this stage, it is assumed that tension cracks have progressed all the way to the neutral

axis and that sections that are plane before bending remain plane in the bent member. This

situation of the section, strain and stress distribution is shown in the figure 4.2.2-1 below.

Figure 4.2.2-1 Cracked Transformed Section

To determine the location of the NA (a), the moment of the tension area about the NA is set

equal to the moment of the compression area, which gives,



Having obtained 'a' by solving this equation, the moment of inertia and other properties of

the transformed section can be determined as in the preceding case. Alternatively, one can

proceed from basic principle by accounting directly for the forces which act in the cross-

section as shown in figure 4.2.2-1.

From the strain distribution,

Applying Hooke’s Law and using modular ration the above equation becomes,

For horizontal force equilibrium,

But equation can be simplified to which reduces to

Equating equation and and solving for practical value of k yields,

√

Note that satisfies the stress -strain relation as well as the equilibrium of horizontal

forces and hence is a useful relation for analysis.



4.3. Serviceability Limit States of Cracking

4.3.1. General

The occurrence of cracks in reinforced concrete is inevitable because of the low tensile

strength of concrete. Structures designed with low steel stresses at service load serve their

intended function with very limited cracking. Crack widths are of concern for three main

reasons: aesthetic appearance, leakage and corrosion.

Aesthetic appearance:- The limits on aesthetic acceptability are difficult to set because of

the variability of personal opinion. The maximum crack width that will neither impair a

structure’s appearance nor create public alarm is probably in the range of 0.25 to 0.38 mm.

Leakage:- Crack control is important in the design of liquid-retaining structures. Leakage is

basically a function of the crack width.

Corrosion:- Concrete made from portland cement usually provide good protection for

reinforcement steel due its high alkalinity. Corrosion of the reinforcement happens when an

electrolytic cell is formed due to the carbonization of the concrete or chlorides penetrate

through the concrete reaches the bar surface. The time taken for this to occur will depend

on whether or not the concrete is cracked, the environment, the thickness of the cover, and

the permeability of the concrete. If the concrete is cracked, the time required for a

corrosion cell to be established is the function of the crack width.

At present cracking is controlled by specifying maximum allowable crack widths at the

surface of the concrete for given type of environment.

4.3.2. Causes of Cracks

1. Load induced cracks: Tensile stress induce by loads, moments, and shear cause

distinctive crack patterns as shown in Fig. 4.3.2-1



Figure 4.3.2-1 Load-induced cracks

2. Heat of hydration cracking: a frequent cause of cracking in structures is restrained

contraction resulting from the cooling down to ambient temperatures of very young

members which have expanded due to heat of hydration which developed as the

concrete was setting. A typical heat of hydration cracking pattern of a wall cast on

foundation concrete is shown in Fig. 4.3.2-2. Such cracking can be controlled by

controlling the heat rise due to the heat of hydration and the rate of cooling, or both; by

placing the wall in short lengths; or by reinforcements.



Figure 4.3.2-2 Heat of hydration cracking

3. Plastic slumping cracks: plastic shrinkage and slumping of the concrete occurs as newly

placed concrete bleeds and surface dries, results in settlement cracks along the

reinforcement as shown in Fig 4.3.2-3a, or a random cracking pattern, referred to as

map cracking shown in Fig. 4.3.2-3b. These types of cracks can be avoided by proper mix

design and by preventing rapid drying of the surface during the first hour or so after

placing. Map cracking can also occur due to alkali-aggregate reaction.

Figure 4.3.2-3 Other types of cracks

4. Cracks caused by corrosion: rust occupies two to three times the volume of the metal

from which it is formed. As a result, if rusting occurs, a bursting force is generated at the

bar location which leads to splitting cracks and eventual loss of cover (4.3.2-3b). Such

cracking looks similar to bond cracking (4.3.2-1e) and may accompany bond cracking.

4.3.3. EBCS’s Provisions of Cracking

(1) For reinforced concrete, two limit states of cracking: the limit state of crack formation

and the limit state of crack widths are of interest.

(2) The particular limit state to be checked is chosen on the basis of the requirements for

durability and appearance. The requirements for durability depend on the conditions of

exposure and the sensitivity of the reinforcement to corrosion.



4.3.3.1. Minimum Reinforcement Areas

(1) In assessing the minimum area of reinforcement required to ensure controlled cracking

in a member or part of a member which may be subjected to tensile stress due to the

restraint of imposed deformations, it is necessary to distinguish between two possible

mechanisms by which such stress may arise. The two mechanisms are:

(a) Restraint of intrinsic imposed deformations - where stresses are generated in

a member due to dimensional changes of the member considered being

restrained (for example stress induced in a member due to restraint to

shrinkage of the member).

(b) Restraint of extrinsic imposed deformations - where the stresses are

generated in the member considered by its resistance to externally applied

deformations (for example where a member is stressed due to settlement of

a support).

(2) It is also necessary to distinguish between two basic types of stress distribution within

the member at the onset of cracking. These are:

(a) Bending - where the tensile stress distribution within the section is

triangular (i.e. some part of the section remains in compression).

(b) Tension - where the whole of the sections subject to tensile stress.

(3) Unless more rigorous calculation shows a lesser area to be adequate, the required

minimum areas of reinforcement may be calculated from the relation given

sctefctes AKfKA /.

Where sA = area of reinforcement

ctA = area of concrete within tensile zone. The tensile zone is that part of

the section which calculated to be in tension just before formation of the first crack.

S = maximum stress permitted in the reinforcement immediately after

formation the crack. This may be taken as 100% of the yield strength of the reinforcement, A lower value may, however, be needed to satisfy the crack with limits

efctf . = tensile strength of the concrete effective at the time when the cracks

may first be arises from dissipation of the heat of hydration, this may be within 3-5 days from casting depending on the environmental conditions, the shape of the member and the nature of the formwork. When the time of cracking cannot be established with confidence as being less than 28 days, it is suggested that a minimum tensile strength of 3 MPa be adopted.

eK = a coefficient which takes account of the nature of the stress

distribution within the section immediately prior to cracking. The stress distribution is that resulting from the combination of effects of loading and restrained imposed deformation.

= 1.0 for pure tension = 0.4 for bending without normal compressive force



K = a coefficient which allows for the effect of non-uniform self-equilibrating stresses Values of K for various situations are given below: a) tensile stresses due to restraint of intrinsic deformations generally

K = 0.8 for rectangular sections when h ≤ 300mm, K = 0.8 h ≥ 800mm, K =1.0

b) tensile stresses due to restraint of extrinsic deformations K = 1.0.

Parts of sections distant from the main tension reinforcement, such as outstanding parts of

a section or the webs of deep sections, may be considered to be subjected to imposed

deformations by the tension chord of the member. For such cases, a value in the range of

0.5 < K < 1.0 will be appropriate.

(4) The minimum reinforcement may be reduced or even be dispensed with altogether if

the imposed deformations sufficiently small that it is unlikely to cause cracking. In such

cases minimum reinforcement need only be provided to resist the tensions due to the

restraint.

4.3.3.2. Limit state of Crack Formation

(1) The maximum tensile stresses in the concrete are calculated under the action of design

loads appropriate to a serviceability limit state and on the basis of the geometrical

properties of the transformed un-cracked concrete cross section.

(2) The calculated stresses shall not exceed the following values:

a) Flexure ctkct f70.1

b) Direct tension ctkct f

(3) In addition to the above, minimum reinforcement in accordance with Chapter 7 shall be

provided for the control of cracking.

4.3.3.3. Limit state of Crack Widths

4.3.3.3.1. General

(1) Adequate protection against corrosion may be assumed provided that the minimum

concrete covers in section (EBCS-2, 1995 section 7.1.3) are complied with and

provided further that the characteristic crack widths kw do not exceed the limiting

values given in Table 4.3.3.3.1 1 appropriate to the different conditions of exposure.



Type of

Exposure

Dry

environment:

Interior of

buildings of

normal

habitation or

offices

(Mild)

Humid environment:

Interior components

(e.g. laundries); exterior

components;

components in non-

aggressive soil and/or

water

(Moderate)

Seawater and/ or aggressive

chemical environment:

Components completely or

partially submerged in

seawater; components in

saturated salt air; aggressive

industrial atmospheres

(Severe)

Characteristic

crack width, kw

0.4

0.2

0.1

Table 4.3.3.3.1-1 Characteristic Crack Width for Concrete Members

4.3.3.3.2. Cracks due to Flexure

(1) Checking of the limit state of flexural crack widths is generally not necessary for

reinforced concrete where

(a) at least the minimum reinforcement given by section 4.3.3.1 is provided

(b) the reinforcement consists of deformed bars, and

(c) their diameter does not exceed the maximum values in Table 4.3.3.3.2-1.

mmwk 4.0 mmwk 2.0

as MP( ) )(mm as MP( ) )(mm

160 40 160 25

200 32 200 16

240 25 240 12

280 20 320 6

320 16 400 4 Table 4.3.3.3.2-1 Maximum Bar Diameter for which Checking Flexural Crack width may be omitted

Note: Where necessary linear interpolation may be used

In Table 4.3.3.3.2 1

s is the steel stress under service condition

kw is the permitted characteristic crack width

(2) If crack widths have to be calculated, the following approximate equations may be

used in the absence of more accurate methods

kw = 1.7 mw

mw = smmS

Where kw = the characteristic crack width

mw = the mean crack width



mS = the average distance between cracks

sm = the mean strain of the reinforcement considering the contribution of

concrete in tension. (3) The average distance between cracks may be obtained from

r

m kkS

2125.050

Where: = diameter

1k a coefficient which characterizes the bond properties of the bars

= 0.8 for deformed bars = 1.6 for plain bars

2k a coefficient representing the influence of the form of the stress diagram.

= 0.50 for bending = 1.00 for pure tension =

121 2/)( for bending with tension

21, are the larger and the smaller concrete strains, respectively, below the

neutral axis of the cracked section given in Fig. 4.3.3.3.2-1

The coefficient r is defined as

efc

sr

A

A

,

(5.15)

Where sA is the area of the reinforcement contained in efcA ,

efcA , is the section of the Zone of the concrete (effective embedment zone)

where the reinforcing bars can effectively influence the crack widths shown by the shaded area in Fig 4.2.3.3.2-1.

Figure 4.3.3.3.2-1 Definition of efcA ,

(4) The mean strain of the reinforcement may be obtained as

2

211s

sr

s

s

smE

s

s

E

4.0



Where: s is the service stress in the steel and may be obtained by elastic theory

using modular ratio equal to 10.

rs is the steel stress at rupture of concrete section; i.e., stress for the

cracked section under the action of the theoretical moment crM

1 is a coefficient which characterizes the bond properties of the bars and

is equal to = 1.0 for high bond bars = 0.5 for plain bars

2 is a coefficient representing the influence of the duration of the

application or repetition of the loads. = 1.0 at the first loading = 0.5 for sustained loads or for a large number of lead cycles

4.3.3.3.3. Cracking due to Shear

(1) Checking of shear crack widths is not necessary in slabs and in the web of beams if the

spacing of the stirrups does not exceed the values given in Table 4.3.3.3.3-1.

kw (mm) 0.4 0.2

ayd MPf ( ) 220 400 360 500 220 400 360 500

Bond Properties (1) (2) (1) (2) (1) (2) (1) (2)

csd VV 300 250 200 150

csdc VVV 3 250 200 150 100

sdV > cV3 200 150 100 75

Table 4.3.3.3.3-1 Maximum spacing (mm) of Vertical Stirrups for which checking of shear crack width can be omitted

Where (1) Plain bars (2) high bond bars

In 4.3.3.3.3-1,

kw is the permitted characteristic crack width

sdV is the shear acting during the combination under consideration

cV is the shear resistance of concrete given in chapter 3;

i.e. Vc= 0.25 fctd k1 k2 bw d

(2) If more precise data are available, then the widths of the shear cracks in the webs of

beams can be calculated for sustained loads by means of Eq. mw = smmS

together with the following equations:



a

ww

csds

s

s

sd

c

s

ssm

r

m

mwk

MPdb

VV

EV

V

E

xdkks

wkw

40cossin

1.

4.01

sin25.050

7.1

2

21

Where mw = the mean crack width (see Eq. 5.12)

= the angle of inclination of the stirrup from the horizontal

wk = a correction coefficient to take account of the effect of slope of the

stirrups on the spacing of the cracks.

= 1.2 for vertical stirrups ( = 090 )

= 0.8 for inclined stirrups with = 045 to 060

w = the geometric percentage of web reinforcement

x = the height of the compression zone in the cracked section. (3) When several adjacent bars in the same layer are bent in the same zone (for example, at

the corners of a frame), the diameter of mandrel shall be chosen with a view to avoiding

crushing or splitting of the concrete under the effect of the pressure that occurs inside the

bend. (See Eqn. 7.7 in ECBS-2, 1995)

4.4. Serviceability Limit States of Deflection

4.4.1. General

In addition to limitation on cracking, described in the preceding sections, it is usually

necessary to impose certain controls on deflections of beams to ensure serviceability.

Excessive deflections can lead to cracking of supported walls and partitions, ill-fitting doors

and windows, poor roof drainage, misalignment of sensitive machinery and equipment, or

visually offensive sag. It is important, therefore, to maintain control of deflections, in one

way or another, so that members designed mainly for strength at prescribed overloads will

also perform well in normal service.

According to EBCS-2, 1995, the deflection of a structure or any part of the structure shall not

adversely affect the proper functioning or appearance of the structure. This may be ensured

either by keeping calculated deflections below the limiting values or compliance with the

requirements for minimum effective depth given in section 5.3.2.

4.4.2. Limits on Deflections

The final deflection including the effects of temperature, creep and shrinkage of all

horizontal members shall not in general exceed the value

200

eL



Where Le is the effective span

For roof or floor construction supporting or attached to non-structural elements such as

partitions and finishes likely to be damaged by large deflections, that part of the deflection

which occurs after the attachment of the non-structural element shall not exceed the value

mmLe 20

350

In any calculation of deflections, the design properties of the materials and the design loads

shall be those appropriate for a serviceability limit states.

4.4.3. Requirements for Effective Depth

The minimum effective depth given below shall be provided unless computation of

deflection indicates that smaller thickness may be used without exceeding the limits

stipulated in the above section.

a

eyk Lfd

4006.04.0

Where: fyk is the characteristic strength of the reinforcement in MPa

a is the appropriate constant from Table 5.6, and for slabs carrying

partition walls likely to crack, shall be taken as oa L150

Lo is the distance in meters between points of zero moments; and for a

cantilever, twice the length to the face of the support.

Member Simply Supported

End spans

Interior spans

Cantilevers

Beams 20 24 28 10

Slabs a)Span ratio 2:1 b)Span ratio 1:1

25 35

30 40

35 45

12 10

Table 4.4.3-1 Values of a

Note: For slabs with intermediate span ratio interpolate linearly

4.4.4. Calculation of Deflection

Before discussing about how to calculate deflections, it is important to understand the

deflection behavior of beams. Let us consider the load–deflection history of a reinforced

concrete fixed ended beam shown in Fig. 4.4.4-1.



Figure 4.4.4-1 Load-deflection behavior of a concrete beam

Initially the beam is uncracked and stiff (O-A). With further load, cracking occurs when the

moment at the ends exceed the cracking moment, Mcr. When a section cracks, its moment

of inertia decreases leading to a decrease in the stiffness of the beam (A-B). Cracking in the

mid-span region causes further reduction in stiffness (point B). Eventually, the

reinforcements would yield at the ends, or at mid-span, leading to increased deflection with

little change in load (points D and E). The service load is represented by point C. he beam is

essentially elastic at point C, the nonlinear load deflection being caused by a progressive

reduction of flexural stiffness due to increased cracking as the loads are increased.

With time, the service load deflection would increase from C to C’, due to creep and

shrinkage of concrete. The short-time, or instantaneous, deflection under service loads

(point C) and the long-time deflection under service loads (point C’) are both of interest in

design.

4.4.5. Immediate Deflections

Unless values are obtained by a more comprehensive analysis, deflections which occur

immediately on application of load shall be computed by the usual elastic method as sum of

the two parts i and ii , but nor more than max .



)(

)(75.0

2

max

2

2

xdzAE

ML

xdzAE

MML

IE

ML

ss

k

ss

crkii

icm

cri

Unless the theoretical cracking moment is obtained by a more comprehensive method, it

shall be computed by

Mcr = 1.70 fctk Z

Where: i = deflection due to the theoretical cracking moment on uncracked

transformed section

ii = deflection due to the balance of applied moment over and above

cracking value and acting on a section with an equivalent stiffness of 75% of the cracked value.

max = deflection of the fully cracked section

Ii = moment of inertia of the un-cracked transformed concrete section Mk = the maximum applied moment at mid-span due to sustained characteristic loads; for cantilevers Mk is the moment at the face of the support Z = section modulus d = effective depth of the section x = neutral axis depth at the section of maximum moment z = internal lever arm at the section of the maximum moment = deflection coefficient depending on the loading and support conditions

(e.g. = 5/48 for simply supported span subjected to UDL)

Note: The value of x and z may be determined for service load condition using a modular

ratio of 10, or for the ultimate load condition.

4.4.6. Long Term Deflections

The deflection of reinforced concrete beams increases with time due to creep and

shrinkage. Additional deflections two or three times as large as the immediate deflections

may result eventually.

The presence of compression reinforcement can reduce the additional deflection due to

shrinkage and creep significantly. In addition to the content of the compression steel, the

extent of the long-term deflection depends on humidity, temperature, curing conditions and

age of concrete at the time of loading, ratio of stress to strength and many other factors.

For this reason, only estimates can be made for long-term deflections.

The effect of creep is to increase the strain in the concrete with time, as illustrated in the

figure below.



Figure 4.4.6-1 Effect of creep

Deflection calculations can allow for this increase in stress by reducing the value of the

elastic modulus for concrete, Ec to an effective elastic modulus, Ec,eff

Where, ϕ is a creep coefficient taken from table 4.4.6-1 below.

Age of concrete at loading (days)

Notional size 2Ac/U (mm)

Dry Atmospheric Condition (Inside)

Humid Atmospheric Condition (Outside)

50 150 600 50 150 600

1 5.5 4.6 3.7 3.6 3.2 2.9

7 3.9 3.1 2.6 2.6 2.3 2.0

28 3.0 2.5 2.0 1.9 1.7 1.5

90 2.4 2.0 1.6 1.5 1.4 1.2

365 1.8 1.5 1.2 1.1 1.0 1.0 Table 4.4.6-1 Creep coefficient, ϕ, for normal weight concrete

According to EBCS-2, unless values are obtained by more comprehensive analysis, the

additional long-term deflection of flexural members shall be obtained by multiplying the

immediate deflection caused by sustained load considered, computed in accordance with

Section 4.4.5, by the factor

6.0]/2.12[ ' ss AA

Where: As’= area of compression reinforcement As= area of tension reinforcement



Assignment set III

1. A floor system is supported by beams spaced at 3m on the center lines which are

simply supported at one end and fixed at the other. The beam are 8m in span with

web width 250mm, overall depth limited to 500mm and slab thickness 100mm. The

floor is subjected to a super imposed service load of 4KN/m2. Design the typical

interior beam for flexure and shear reinforcement. Use concrete C30, steel S300 and

Class I works.

2. The beam reinforced for shear and flexure as shown below is made of concrete C25,

steel S420 and Class I works. It is supporting a factored dead load of 6KN/m and a

factored live load of 50KN/m. Calculate bar-cut off and development length for the

critical sections.

3. A rectangular reinforced concrete beam 250x700mm is reinforced with 8Ф20 in two

rows to sustain a maximum bending moment of 180KNm. Evaluate the stress in the

concrete and steel. Assume n = 8.

4. A rectangular reinforced concrete beam has depth of 600mm and width of 200mm

and is reinforced with 6Ф22 in two rows. If fck = 24MPa and steel fyk = 280 MPa and

Class I work is used. Determine

a. The moment capacity just before cracking

b. The maximum moment due to service load which the section may sustain if

the allowable stress for steel and concrete is 0.52fyk and 0.425fck respectively.

5. A simply supported rectangular reinforced concrete beam with b/D = 300/500, d’/d’’

= 50/50 is reinforced with As1 = 1473mm2 and As2 = 402mm2. If Ec = 30.5GPa and Es =

200GPa, fctk = 2.6MPa, DL = 15KN/m, LL = 8KN/m and L = 8m,

a. Calculate the immediate and long term deflection.

b. Crack width for moderate exposure

Abrish Rc i (Rearranged)_2

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