CHAPTER I. Introduction

Transcription

1 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.SiO 2 abbreviated C 3 S) and di-calcium silicate (2CaO.SiO 2 abbreviated C 2 S) 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 C 2 SH 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*10 6 cm 2 /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, 1

2 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 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: 2

3 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 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 Concrete 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. 3

4 Portland cement is produced from a mixture of ground clay (contains Si0 2 and Al 2 O 3 ) and lime (CaO) and other minor ingredients such as MgO and Fe 2 O 3 by heating to the point of incipient fusion (clinkering temperature). The clinker is then ground to different degrees of fineness to get cement. Table 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 C 2 S. If moist curing is continuous, later strength levels will be greater, with higher percentages of C 2 S. C 3 A 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 Tricalcium silicate, C 3 S Medium Medium Dicalcium silicate,c 2 S Slow Small Tricalcium aluminate, C 3 A Fast Large Tetracalcium aluminoferrate, C 4 AF Slow Small Table Properties of Cements Ultimate Cementing Value Good Good Poor Poor Type of Cement Component (%) General C 3 S C 2 S C 3 A C 4 AF CaSO 4 CaO MgO Characteristics Normal: I All-purpose cement Modified: II Comparative low heat liberation; used in large structures High early strength: III High strength in 3 days Low heat: IV Used in mass concrete dams Sulphate resisting: V Used on sewers and structures exposed to sulphate Table 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 4

5 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 airentraining 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 5

6 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 gives grading or particles size distribution requirements of coarse aggregates by Ethiopian Standard for Concrete and Concrete Products, ES C.D Nominal size of graded aggregate Percentage passing through test sieves having square openings 75mm 63mm 37.5mm 19mm 13.2mm 9.5mm 4.75mm Table 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 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

7 Table 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 table 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 9.50mm 4.75mm 2.36mm 1.18mm 600μm 300μm 150μm Percentage passing Table Grading requirements for fine aggregates [ES D3.201] Table gives limits of deleterious substances for fine aggregates. Deleterious Substance Maximum percentage by mass Friable particles 1.0 Clay or fine silt (materials passing 63μm sieve) in fine aggregates used for - Concrete subject to abrasion All other concrete 5.0 Coal Ignite 1.0 Table 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. 7

8 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 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/m 3 and the least dimension of concrete to be placed at a single time is 600mm or more 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 f c or f ck 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(f ck ) 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 , 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 f ck as 8

9 The 28 day compressive strength may be obtained from 7 days compressive strength using experimentally developed empirical relations. One formal is S 7 and S 28 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 Effect W/C ration on strength Other factor affecting concrete strength is degree of compaction. Figure Effect 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 9

10 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 The splitting tensile strength, f ct is determined as, ( ) Figure 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 It can be estimated by, Figure Flexure test EBSC uses the following empirical formula, Where f ctk tensile strength of concrete in MPa f ck characteristic cylinder strength in MPa 10

11 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 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. 11

12 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 Structural use of concrete page 90. Minimum mixing time measured from the time the ingredients are put together is given in table 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 (m 3 ) Time of mixing (minutes) Table 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 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 12

13 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 Type of reinforcement bars Some bar size and areas for design purpose available in Ethiopia are given in table Diameter φ (mm) Area (mm 2 ) Weight (Kg/m) Table Reinforcement bar properties that are available in Ethiopia Characteristic properties of reinforcing bars are expressed using its yield strength, f y (f yk ) and modulus of elasticity E s. F y ranges between 220 to 500MPa, with 300MPa common in our country. E s ranges between 200 to 210GPa Concrete Placement and Curing 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. 13

14 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 yd 3 (0.765m 3 ) 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 ]. In the absences of more accurate data the following minimum periods are recommended by EBCS 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 days 4. For soffits formwork to beams days 5. For props to beams days 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 Stress-Strain Relation for Concrete and Reinforcements Strength and deformation of reinforced concrete members can be calculated from stressstrain relations of concrete and reinforcement steel and the dimensions of the members. 14

15 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. Figure presents typical stress-strain curves obtained from concrete cylinders loaded in uniaxial compression. Figure Stress-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 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 ( for LSD and for USD). 15

16 Tangent and Secant Moduli of Elasticity Three ways of defining the modulus of elasticity are illustrated in figure 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, E T, at the stress corresponding to point A. The slope of the stressstrain 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 of the compressive strength (f ck ), representing the service-load stress. Whenever E cm is used it usually means the secant modulus in MPa. Figure 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 E cm can be obtained from table for given concrete grades as given by EBCS Grades of Concrete C15 C20 C25 C30 C40 C50 C60 f ck (N/mm 2 ) f ctm (N/mm 2 ) f ctk (N/mm 2 ) E cm (GPa) 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 Grades of Concrete and their strength characteristics The following empirical formula is also given by EBCS , in which E cm is in GPa and f ck is in MPa. The stress-strain curve in figure is simplified for design to a parabolic rectangular stress block as given by EBSC

17 Figure Idealized 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 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 Stress-strain curves for concrete with various rates of axial compressive loadings Tensile Stress Behavior 17

18 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 splitcylinder 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 shows how a family of Mohr's 18

19 circle for failures in tension, compression and other combinations is enclosed in an envelope curve. Figure 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 figure Figure 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 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

20 Figure 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 figure The shrinkage occurs at a decreasing rate with time. The final shrinkage strains vary greatly being generally in the range to but sometimes as much as Reinforcement Steel Bar Shape and Size Figure Shrinkage strain of concrete 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 20

21 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 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 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 f y is a very important property of reinforcement steel and is used as design stress in ultimate strength design (USD) and design stress obtained from σ y in limit state design (LSD). Figure Typical stress-strain curves for reinforcement steel 21

22 σ y 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 for LSD. Figure 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 infigure (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. Figure (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. 22

23 Figure a) 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. 23

24 Figure 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

25 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. 25

26 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 durability requirements. on the basis of strength, serviceability and 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 The Working Stress Design (WSD) method In this method the section of reinforced concrete members are designed assuming straightline 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 f ck and that of tensile stress in steel 0.52 f yk, 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 26

27 to be unsafe when dealing with the stability of structures subject to overturning forces. This method was used from for the design of reinforced concrete members 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 This method was popular from 1950 up to 1960s 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 27

28 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. EBCS is based on the LSD method. 28

29 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. 29

30 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 30

31 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 for concrete and steel). Where f k = characteristics strength f m = mean strength, δ =standard deviation, K 1 = a factor that ensures the probability of the characteristics strength is not being exceeded is small. (K 1 =1.64) Figure 2.1-1Characteristic strength definition Table 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 f ck Table Grades of concrete and characteristic compressive strength f ck 31

32 Where f cu = characteristic standard cube strength (obtained from 150mm cubes), f ck = 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, f yk is defined as the fractile of the proof stress f y 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 gives values of characteristic permanent loads G k and characteristic imposed loads Q k and EBCS gives characteristic seismic loads A Ed. 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 γ m and load variability γ F. Suitable design stress f d is obtained as, γ m 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 gives partial safety factors of materials at ULS according to EBCS Material and Concrete γ c Steel γ s workmanship Class I Class II Class I Class II Design Situation Persistent and transient Accidental Table 2.1-2Partial safe factors of materials at ULS according to EBCS 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. 32

33 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 as, Where F k = characteristics load, F m = mean load, δ =standard deviation, K 2 = a factor that ensures the probability of the characteristics load being exceeded is small. (K 2 =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 γ F. 33

34 γ F 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 (G k ) and only one variable action (Q k ) 2. Permanent action (G k ) and two or more variable action (Q k ) 3. Permanent action, variable action and accidental (seismic) action Load combination for SLS 1. Permanent action (G k ) and only one variable action (Q k ) 2. Permanent action (G k ) and two or more variable action (Q k ) 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 b).the bending moments are obtained 34

35 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 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 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 (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 The stress diagram shown in figure (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 (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, 35

36 Stress-strain distribution for beams Figure 2.1-4Elastic beam stresses and stress blocks 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 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 (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 (c) and are found to be parabolic in nature. Figure Side view of test beam with gauges To illustrate the stress-strain development for increased loading, consider the following, 36

37 Figure Loading on a simply supported beam Major points to notice are, Figure Stress and strain distribution At low loads where tensile stress is less than or equal to the characteristics tensile strength of concrete (f ctk ), the stress & strain relation shown in figure (a) results. At increased loading, tensile stress larger than f ctk in figure (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 (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. 37

38 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 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 and steel to its design stress f yd (ε s <0.01 also) as given by EBSC 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 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. 38

39 Concrete is assumed to fail when maximum compressive strain reaches a limiting value of in bending and in axial compression according to EBCS The compressive stress-strain relationship for concrete may be based on stressstrain 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 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 as given by EBSC Figure Strain diagram in the ULS Figure Idealized stress-strain diagrams 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. 39

40 Figure 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 rectangularparabolic distribution. 1. If one wants to use the idealized parabolic-rectangular stress block given in EBCS , as shown in figure Moment about the T, and Figure Derivation Moment about the C, α c and β c are values calculated by integrating the stress-strain diagram for the different location of the N.A. depth. i.e. 40

41 i. ε cm 2 and N.A. within the section ii. ε cm 2 and N.A. within the section iii. ε cm 2 and N.A. outside the section To understand the mechanics behind the derivation of the above equations, referring to figure , the capacity of section when the ε cm and N.A. within the section, ( ) ( ( )) C is the compression stress resultant, 41

42 ( ) ( ( )) Taking moment about the top fiber, ( ) ( ) ( ( )) ( ) 2. If one wants to use the rectangular stress block given in EBCS , 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 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 Derivation 42

43 From similarity of triangle, To find the compressive force for the parabolic rectangular stress block, [ ] [ ] To find the moment arm β c, taking moment about the top fiber, [ ] [ ] ) ( ) Thus to accommodate both α c and β c, 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 f yk = 400Mpa and f ck = 30MPa. Use both parabolic-rectangular stress block and rectangular stress block. Comment on the accuracy of rectangular stress block approximation 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 A s of the section is small, the steel will reach f yd before the concert reaches its maximum strain ε cu of With further increase in loading, the steel force remains constant at f yd A s, 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 nonlinear 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 43

44 arm z. The flexural strength is reached when concrete strain reaches With further increase in strain, crushing failure occurs. ε s may also be so large as to exceed This phenomenon is shown in figure This type of failure is preferable and is used for design. b) Compression Failure Figure Tension failure If the steel content A s 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 ε c = The section fails suddenly in a brittle fashion. This phenomenon is shown in figure c) Balanced Failure Figure Compression failure At balanced failure the steel reaches f yd and the concrete reaches a strain of simultaneously. This phenomenon is shown in figure

45 Figure Balanced failure Figure Strain Diagrams for tension (1), Balanced (2) and compression (3) failures 45

46 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 ρ b. In our code of practice, EBCS 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 gives the following minimum effective depth, ( ) Where, f yk = characteristics strength of reinforcement (MPa) L e = effective span β a = constant from table Member Simply Supported End Spans Interior Spans Cantilevers Beams Slabs a) Span ratio 2:1 b) Span ratio 1: Table Values of β a from EBCS The design of singly reinforced section can be carried out using chart or tables found in EBCS Part 2 and are summarized below. Referring to figure , 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,

47 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 part 2, the design of the section is formulated using empirical parameters K m and K s. 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, μ sd,s = μ* sd,s = Steps to be followed a. Design using tables 1. Evaluate K m 2. Enter the general design table No 1.a using K m and concrete grade, a. If K m K m *, the value of K m *show shaded in design Table No 1.a, then the section is singly reinforced. Enter the design table No 1.a using K m and concrete grade Read K s from the table corresponding to the steel grade and K m Evaluate A s b. Design using general design chart 47

48 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, A s,min required to resist M cracking be provided. A s,min is obtained from the cracking moment. Empirical relations are given in codes and standards. EBCS gives for beams, 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 One-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. 48

49 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: a s An area of reinforcement bar to be used A s 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 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. 49

50 Figure 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 Effect of compression reinforcements on strength and ductility. 3. Change of mode of failure from compression to tension When ρ > ρ b, beam fails in a brittle manner, through crushing of the compression zone before the reinforcement yields. 50

51 Figure 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 Figure Doubly reinforced section 51

52 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 T s1 = C c 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 f s2. If the steel has yielded (ε s2 ε yd ) If the steel has yielded (ε s2 < ε yd ) 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 α c assuming all steel yielded and reading values of ε s1 and ε s2 from the chart and compare with ε yd. The design resistance or capacity of the section can be calculated by 52

53 For design problems the following procedures can be followed, a. Design using tables 1. Evaluate K m 2. Enter the general design table No 1.a using K m and concrete grade, If K m > K m *, the value of K m *show shaded in design Table No 1.a, then the section is doubly reinforced. - Evaluate K m / K m * and d 2 /d - Read K s, K s, ρ and ρ from the same table corresponding to K m / K m *, d 2 /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 53

54 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 crosssection 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 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 Typical slab The part of the slab that is acting together with the beam, called effective flange width b e 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) 54

55 { L e is the effective span length and b w 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, h f the beam can be analyzed as if it were a rectangular beam of width equal to b e. If the NA is in the web x>h f, 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 = b e, 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 55

56 Beam W Beam F Figure 2.2-3ULS T-section Beam F ( ) or ( ) The force in the remaining steel area A sw is balanced by compression in the rectangular portion of the beam. (i.e. A sw = A s - A sf ) Beam W The total moment capacity of the section now becomes, or ii. If 0.8x h f, 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 f s = f yd. This has to be verified by determining the NA and checking the strain profile. 56

57 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. 57

58 6. Determine the ultimate moment capacity of the doubly reinforced beam of b = 350 mm, d' = 60 mm, d = 600 mm, A s = 2945 mm2, A s = 1256 mm 2, using C30 and S 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, f yk = 420MPa and d ' = 50mm. 10. Determine the maximum permissible span (in meters) of a simply supported RC oneway slab having a thickness of 15cm and reinforced with Φ10c/c100mm. The working live-load on the slab is 5kPa; C25 concrete and steel f yk = 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, f yk = 300Mpa, d = 60mm. (The given loads are factored loads.) 58

59 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. 59

60 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 Basic Theory Stresses in an Uncracked Beam Figure Internal force in beams From the FBD in Fig 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 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 60

61 Serviceability limit state Chapter V e ) Photograph of half of a cracked reinforced concrete beam Figure Normal, shear and principal stress in a homogenous un-cracked beam For uncracked rectangular beam Fig b 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 ( ) ( ) 61