ADDIS ABABA INSTITUTE OF TECHNOLOGY ADDIS ABABA UNIVERSITY

ADDIS ABABA INSTITUTE OF TECHNOLOGY ADDIS ABABA UNIVERSITY DEPARTMENT OF CIVIL ENGINEERING CENG 3502- REINFORCED CONCRETE STRUCTURES I Academic Year: 2010/2011 COURSE OUTLINE 1. Introduction Mark Distribution:* 1.1. General Introduction Assignments ……. 10% 1.2. History of Development of concrete Mid-Semester Examination ……. 40% 1.3. Materials Aspect of Reinforced Concrete Final Examination ……. 50% 1.4. Concrete Placement and Curing Total ……. 100% 1.5. Stress-Strain Relation for Concrete and Reinforcement Steel References: 1.6. Overview of design Philosophies 1. Reinforced Concrete: Mechanics and Design, -------- Assignment I -------by James G MacGregor and James K Wight. 2. Design of Concrete Structures, by Arthur H. 2. Limit State Design for Flexure Nilson, David Darwin and Charles W. Dolan. 2.1. Introduction 3. Reinforced Concrete: A fundamental 2.2. ULS of Singly Reinforced Rectangular Approach, by Edward Nawy Sections (Beams, One-way Slabs) 4. Ethiopian Building Code Standards 2 (EBSC22.3. ULS of Doubly Reinforced Rectangular 1995) – Structural Use of Concrete Sections 2.4. ULS of T- and L- Sections 2.5. Bond, Anchorage and Development Instructors: Length Ato Misgun Samuel -------- Assignment II --------------- Mid-Semester Exam -------3. Limit State Design for Shear 3.1. Introduction 3.2. Mechanism of shear resistance in concrete beams without shear reinforcement 3.3. Design of shear reinforcement -------- Assignment III -------4. Serviceability Limit State 4.1. Introduction 4.2. Elastic Analysis of Beam Sections 4.3. Elastic Deflection and Crack Width 4.4. Moment-Curvature Relationship -------- Assignment IV --------------- Final Exam --------

* Mark distribution may be changed as the instructors feel otherwise.

RCS – I

CONTENTS

Contents CHAPTER I ...................................................................................................................................... 1 Introduction ................................................................................................................................... 1 1.1.

General Introduction........................................................................................................... 1

1.2.

History of Development of Concrete .................................................................................. 2

1.2.1.

Cement and Concrete ..................................................................................................... 2

1.2.2.

Reinforced Concrete ....................................................................................................... 3

1.2.3.

Design Specifications for Reinforced Concrete ............................................................... 5

1.3.

Material Aspect of Reinforced Concrete............................................................................. 5

1.3.1.

Concrete .......................................................................................................................... 6

1.3.1.1.

Ingredients of Concrete .............................................................................................. 6

1.3.1.2.

Properties in the Plastic State ................................................................................... 10

1.3.1.3.

Properties in the Hardened State ............................................................................. 11

1.3.1.4.

Proportioning and Mixing Concrete.......................................................................... 14

1.3.2. 1.4.

Reinforcing Steel ........................................................................................................... 15 Concrete Placement and Curing ....................................................................................... 16

1.4.1.

Concrete Placement ...................................................................................................... 16

1.4.2.

Curing Concrete ............................................................................................................ 17

1.5.

Stress-Strain Relation for Concrete and Reinforcements ................................................. 17

1.5.1.

Concrete ........................................................................................................................ 17

1.5.2.

Reinforcement Steel ..................................................................................................... 24

1.6.

Overview of Design Philosophies ...................................................................................... 27

CHAPTER II ................................................................................................................................... 31 LIMIT STATE DESIGN FOR FLEXURE .............................................................................................. 31 2.1.

Introduction ...................................................................................................................... 31

2.2.

ULS of Singly Reinforced Rectangular Beams ................................................................... 39

2.2.1.

Basic assumptions at ULS .............................................................................................. 39

2.2.2.

Analysis of Singly Reinforced Concrete Beams ............................................................. 40

2.2.3.

Types of flexural failures ............................................................................................... 45

2.2.4.

Design equation for singly reinforced rectangular beams ............................................ 48

2.2.5.

Design equation for singly reinforced one-way slabs ................................................... 50

2.3.

ULS of doubly reinforced rectangular section .................................................................. 51

2.4.

ULS of T- and L- Sections ................................................................................................... 53

AAiT Department of Civil Engineering

i Instructor Misgun S.

RCS – I

CONTENTS

CHAPTER III .................................................................................................................................. 56 LIMIT STATE DESIGN FOR SHEAR ................................................................................................. 56 3.1.

Introduction ...................................................................................................................... 56

3.2.

Basic Theory ...................................................................................................................... 56

3.3.

Mechanism of shear resistance in concrete beams without shear reinforcements ........ 58

3.4.

Design of shear reinforcement ......................................................................................... 61

3.5.

Bond and development length ......................................................................................... 66

3.5.1.

Bond .............................................................................................................................. 66

3.5.2.

Development Length..................................................................................................... 69

3.5.3.

Lapped splices ............................................................................................................... 71

3.5.4.

Bar cutoff....................................................................................................................... 71

CHAPTER IV .................................................................................................................................. 73 SERVICEABILITY LIMIT STATE........................................................................................................ 73 4.1.

Introduction ...................................................................................................................... 73

4.2.

Elastic analysis of beam sections ...................................................................................... 73

4.2.1.

Section Un-cracked ....................................................................................................... 73

4.2.2.

Section Cracked............................................................................................................. 74

4.3.

Serviceability Limit States of Cracking .............................................................................. 76

4.3.1.

General .......................................................................................................................... 76

4.3.2.

Causes of Cracks ............................................................................................................ 76

4.3.3.

EBCS’s Provisions of Cracking........................................................................................ 79

4.3.3.1.

Minimum Reinforcement Areas................................................................................ 79

4.3.3.2.

Limit state of Crack Formation.................................................................................. 80

4.3.3.3.

Limit state of Crack Widths ....................................................................................... 80

4.3.3.3.1.

General .................................................................................................................. 80

4.3.3.3.2.

Cracks due to Flexure ............................................................................................ 81

4.3.3.3.3.

Cracking due to Shear ........................................................................................... 83

4.4.

Serviceability Limit States of Deflection ........................................................................... 84

4.4.1.

General .......................................................................................................................... 84

4.4.2.

Limits on Deflections ..................................................................................................... 84

4.4.3.

Requirements for Effective Depth ................................................................................ 85

4.4.4.

Calculation of Deflection ............................................................................................... 85

4.4.5.

Immediate Deflections .................................................................................................. 86

4.4.6.

Long Term Deflections .................................................................................................. 87


ii Instructor Misgun S.

RCS – I

INTRODUCTION

Chapter I

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

+ 2 +

+ 2 + C2SH and CSH are di- and mono-calcium silicate gels respectively, and CH is Ca(OH)2 There are hypothesis 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. Concrete and reinforced concrete are used as construction materials in countries around the world for the construction of buildings, bridges, underground structures, water tanks,


1 Instructor Misgun S.

RCS – I INTRODUCTION Chapter I television towers, offshore oil exploration and production structures, highway and airfield pavements and many more 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. Other advantages of reinforced concrete are: • • • • • •

Mouldability to any desired shape in forms while plastic, Economy Fire resistance Suitability of material for architectural & other functions Rigidity Low maintenance

But concrete has also disadvantages; an important one is that quality control when manufactured in the field sometimes is not as good as for other construction materials that are made in factory. Another disadvantage is that concrete is a relatively brittle material i.e., it easily breaks in tension while it is very strong in compression. This disadvantage can however be offset by reinforcing concrete with steel in the tension zone. Other disadvantage, concrete is difficult to dismantle after hardening and large portion of the xsection is not effectively used due to cracks. Concrete requires formwork which most of the time is expensive and supervision after pouring is difficult. Concrete has relatively low strength per unit volume (or weight) and undergoes time dependent volume changes such as drying shrinkage, which if restrained, may cause cracking and deflections. Furthermore, deflections tend to increase with time due to creep of concrete under sustained loadings. 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, 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. History of Development of Concrete 1.2.1. Cement and Concrete Lime mortar was first used in structures in Minoan civilization in Crete about 2000BC. This type of mortar had the disadvantage of gradually dissolving when immersed in water and hence could not be used for exposed joints or underwater joints. About the third century AAiT Department of Civil Engineering


RCS – I INTRODUCTION Chapter I D.C the Rotilans discovered a volcanic ash called pozzolana (named after the village of pozzoli where the ash was discovered) which when mixed with limestone, burnt and ground had an excellent hydraulic cementing quality with far better strength which could also be used under water. They used light volcanic rock as aggregates. The most remarkable concrete structure built by Romans was the dome of Pantheon in Rome completed in A.D. 126. This dome had a span of 44 m, a span not exceeded until the 19th century. The lowest part of the dome was concrete with aggregate consists of broken bricks. As the builder approached the top of the dome they used lighter and lighter aggregate, using pumice at the top to reduce the dead-load moments. Although the outside of the dome was and still is covered with decorations, the marks of the forms are still visible on the inside. While designing the Eddy stone Lighthouse off the south coast of England just before A.D. 1800, the English engineer John Smeaton discovered that a mixture of burned limestone and clay could be used to make cement that would set under water and be water resistant. Owing to the exposed nature of this lighthouse, however, Smeaton reverted to the triedand-true Roman cement and mortised stonework. In the ensuing years a number of people used Smeaton's material but the difficulty of finding limestone and clay in the same quarry greatly restricted its use. In 1824, Joseph Aspdin mixed ground limestone and clay from different quarries and heated them in a kilnto make cement. Aspdin named his product Portland cement because concrete made from it resembled Portland stone, a high-grade limestone from the Isle of Portland in the south of England. This cement was used by Brunei in 1828 for the mortar used in the masonry liner of a tunnel under the Thames River and in 1835 for mass concrete piers for a bridge. Occasionally in the production of cement the mixture would be overheated, forming a hard clinker which was considered to be spoiled and was discarded. In 1845, L C. Johnson found that the best cement resulted from grinding this clinker. This is the material now known as Portland cement. Portland cement was produced in Pennsylvania in 1871 by D. O. Saylorand about the same time in Indiana by T. Millen of South Bend, but it was not until the early1880s that significant amounts were produced in the United States.

1.2.2. Reinforced Concrete W, B, Wilkinson of Newcastle-upon-Tyne obtained a patent in 1854 for a reinforced concrete floor system that used hollow plaster domes as forms. The ribs between the forms were filled with concrete and were reinforced with discarded steel mine-hoist ropes in the center of the ribs. In France, Lambot built a rowboat of concrete reinforced with wire in 1848 and patented it in 1855. His patent included drawings of a reinforced concrete beam and a column reinforced with four round iron bars. In 1861, another Frenchman, Coignet, published a book illustrating uses of reinforced concrete.



RCS – I INTRODUCTION Chapter I The American lawyer and engineer Thaddeus Hyatt experimented with reinforced concrete beams in the 1850s. His beams had longitudinal bars in the tension zone and vertical stirrups for shear. Unfortunately, Hyatt's work was not known until he privately published a book describing his tests and building system in 1877. Perhaps the greatest incentive to the early development of the scientific knowledge of reinforced concrete came from the work of Joseph Monier, owner of a French nursery garden. Monier began experimenting about 1850 with concrete tubs reinforced with iron for planting trees. He patented his idea in 1867; this patent was rapidly followed by patents for reinforced pipes and tanks (1986), flat plates (1869), bridges (1873) and stairs (1875). In1880-1881, Monier received German patents for many of the same applications. These were licensed to the construction firm Wayss and Freitag, which commissioned Professors Morsch and Bach of the University of Stuttgart to test the strength of reinforced concrete and commissioned Mr. Koenen, chief building inspector for Prussia, to develop a method of computing the strength of reinforced concrete. Koenen's book, published in 1886, presented an analysis which assumed that the neutral axis was at the mid-height of the member. The first reinforced concrete building in the United States was a house built on Long Island in 1875 by W. E, Ward, a mechanical engineer. E. L Ransome of California experimented on reinforced concrete in the 1870s and patented twisted steel reinforcing bar in1884. In the same year, Ransome independently developed his own set of design procedures. In 1888 he constructed a building having cast-iron columns and a reinforced concrete floor system consisting of beams and a slab made from flat metal arches covered with concrete. In1890, Ransome built the Leland Stanford, Jr. Museum in San Francisco. This two-story building used discarded cable car rope as beam reinforcement. In 1903 in Pennsylvania, hebuilt the first building in the United States completely framed with reinforced concrete. In the period from 1875 to 1900, the science of reinforced concrete developed through a series of patents. An English textbook published in 1904 listed 43 patented systems, 15 in France, 14 in Germany or Austria-Hungary, 8 in the United States, 3 in the United Kingdom and 3 elsewhere. Most of these differed in the shape of the bars and the manner in which the bars were bent. From 1890 to 1920, practicing engineers gradually gained knowledge of the mechanics of reinforced concrete, as books, technical articles, and codes presented the theories. In an 1894 paper to the French Society of Civil Engineers, Coignet (son of the earlier Coignet) and de Tedeskko extended Koenen's theories to develop the working stress design method for flexure, which was used universally from 1900 to 1950. During the past seven decades extensive research has been carried out on various aspects of reinforced concrete behavior, resulting in the current design procedures. Pre-stressed concrete was pioneered by E. Freyssinet who in 1928 concluded that it was necessary to use high-strength steel wire for pre-stressing because the creep of concrete AAiT Department of Civil Engineering


RCS – I INTRODUCTION Chapter I dissipated most of the pre-stress force if normal reinforcing bars were used to develop the pre-stressing force. Freyssinet developed anchorages for the tendons and designed and built a number of pioneering bridges and structures.

1.2.3. Design Specifications for Reinforced Concrete The first set of building regulations for reinforced concrete was drafted under the leadership of Professor Morsch of the University of Stuttgart and was issued in Prussia in 1904.Design regulations were issued in Britain, France, Austria and Switzerland between 1907and 1909. The American Railway Engineering Association appointed a Committee on Masonry in l890. In 1903 this committee presented specifications for Portland cement concrete. Between 1908 and 1910 a series of committee reports led to the Standard Building Regulations for the Use of Reinforced Concrete published in 1910 by the National Association of Cement Users, which subsequently became the American Concrete Institute. A Joint Committee on Concrete and Reinforced Concrete was established in 1904 by the American Society of Civil Engineers, the American Society for Testing and Materials, the American Railway Engineering Association and the Association of American Portland Cement Manufacturers. This group was later joined by the American Concrete Institute. Between 1904 and 1910 the Joint Committee carried out research. A preliminary report issued in 1913 lists the more important papers and books on reinforced concrete published between 1898 and 1911. The final report of this committee was published in 1916. The history of reinforced concrete building codes in the United States was reviewed in1954 by Kerekes and Reid. The first Ethiopian code was developed in 1972. This was revised in 1983 and had 3volumes (Ethiopian Standard Code of Practice: ESCP: 1983). The second revision was made in 1995 and this consists of 13 volumes which are currently used for design.

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.



RCS – I

INTRODUCTION

Chapter I

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

Rate of Reaction Medium Slow Fast Slow

Heat Liberated Medium Small Large Small

Ultimate Cementing Value Good Good Poor Poor

Table 1.3.1.1-1Properties of Cements

Type of Cement

C3S

C2S

Component (%) C3A C4AF CaSO4

CaO

MgO

Normal: I

49.0 15.0

12.0

8.0

2.9

0.8

2.4

Modified: II

45.0 29.0

6.0

12.0

2.8

0.6

3.0

High early strength: III

56.0 15.0

12.0

8.0

3.9

1.4

2.6

Low heat: IV

30.0 46.0

5.0

13.0

2.9

0.3

2.7

Sulphate resisting: V

43.0 36.0

4.0

12.0

2.7

0.4

1.6

General Characteristics All-purpose cement Comparative low heat liberation; used in large structures High strength in 3 days Used in mass concrete dams Used on sewers and structures exposed to sulphate

Table 1.3.1.1-2 Percentage Composition of Portland Cements



RCS – I INTRODUCTION Chapter I 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 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.



RCS – I INTRODUCTION Chapter I 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 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.



RCS – I

INTRODUCTION

Nominal size of graded aggregate

Percentage passing through test sieves having square openings 75mm 100 -

38-5 19-5 13-5

Chapter I

63mm -

37.5mm 95 - 100 100 12

19mm 30 – 70 95 – 100 100

13.2mm 2.8 90 - 100

9.5mm 10 - 35 25 - 55 40 - 85

4.75mm 0-5 0 - 10 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

Friable soft fragments Coal and lignite Clay lumps Materials passing 63μm sieve including crushed dust

Maximum percentage by mass 3.00 1.00 0.25 1.50

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 table1.3.1.15 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.



RCS – I

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

Chapter I

Percentage passing 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 1.0 Clay or fine silt (materials passing 63μm sieve) in fine aggregates used for - Concrete subject to abrasion 3.0 5.0 - All other concrete Coal Ignite 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. 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.



RCS – I

INTRODUCTION

Chapter I

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 = 0.80 The 28 day compressive strength may be obtained from 7 days compressive strength using experimentally developed empirical relations. One formal is = + 30 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. AAiT Department of Civil Engineering


RCS – I

INTRODUCTION

Chapter I

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 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, 2( -$ ' = , . , 01*23 4 56*27+89 3803 )*+ %

Figure 1.3.1.3-3 Split-cylinder test procedure



RCS – I INTRODUCTION Chapter I 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, ; 6; : = = , *8@A98 3803 < >?

Figure 1.3.1.3-4 Flexure test

EBSC 2-1995 uses the following empirical formula, /

' = 0.21 Where fctk – tensile strength of concrete in MPa fck– characteristic cylinder strength in MPa 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 relatively 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. AAiT Department of Civil Engineering


RCS – I

INTRODUCTION

Chapter I

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. 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 AAiT Department of Civil Engineering


RCS – I INTRODUCTION Chapter I water for workability. The maximum mixing time may be taken 3 times the minimum mixing time as a guide. Capacity of mixer (m3) 1.5 2.3 3.0 4.5

Time of mixing (minutes) 1.5 2.2 2.5 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 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.



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INTRODUCTION

Chapter I

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) Area (mm2) Weight (Kg/m)

6

8

10

12

14

16

20

24….

28 0.222

50 0.395

78.5 0.619

113 0.888

154 1.210

200 1.570

314 2.470

450 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. 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.



RCS – I INTRODUCTION Chapter I 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 stressstrain relations of concrete and reinforcement steel and the dimensions of the members.

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 AAiT Department of Civil Engineering


RCS – I INTRODUCTION Chapter I 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).

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 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 0.4 - 0.5 of the compressive strength (fck),



RCS – I INTRODUCTION Chapter I 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 2 fck (N/mm ) 12 16 20 24 32 40 48 2 fctm (N/mm ) 1.6 1.9 2.2 2.5 3.0 3.5 4.0 2 fctk (N/mm ) 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. CD = 9.5# 8&

GH

The stress-strain curve in figure1.5.1-3 is simplified for design to a parabolic rectangular stress block as given by EBSC 2-1995.



RCS – I

INTRODUCTION

Chapter I

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).



RCS – I

INTRODUCTION

Chapter I

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

Tensile Stress Behavior 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, σ=

2( -$ , . )*+ %

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 general lower for high strength concrete. At high compressive stresses the transverse strains increase rapidly owing to internal cracking parallel to the direction of loading.



RCS – I

INTRODUCTION

Chapter I

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 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.



RCS – I

INTRODUCTION

Chapter I

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.

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.



RCS – I

INTRODUCTION

Chapter I

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 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



RCS – I INTRODUCTION Chapter I 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

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%.



RCS – I INTRODUCTION Chapter I 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 1.5.2-3Idealized 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.

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



RCS – I

INTRODUCTION

Chapter I

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.

Figure 1.6-1 Reinforced Concrete Building Components

This material provides the first encounter on the analysis and design of the individual structural elements of reinforced concrete structures, with emphasis on:



RCS – I • • •

INTRODUCTION Chapter I 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.

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. b. c. d. e.

idealization of the structure into component parts load estimation on the various structural components analysis to determine the maximum internal stresses and strains design of sections and reinforcement arrangements detail drawings and bar schedule preparation

There are three methods of concrete design. These are AAiT Department of Civil Engineering


RCS – I INTRODUCTION Chapter I 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 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 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 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.

ii.

iii.

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. 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. 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



RCS – I INTRODUCTION Chapter I 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 is 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 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.



RCS – I

LIMIT STATE DESIGN FOR FLEXURE

Chapter II

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 31 AAiT Department of Civil Engineering Instructor Misgun S.

RCS – I LIMIT STATE DESIGN FOR FLEXURE Chapter II 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). = D 4 JG K 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 fck

C15 12

C20 16

C25 20

C30 24

C40 32

C50 40

C60 48

Table 2.1-1 Grades of concrete and characteristic compressive strength fck



RCS – I

LIMIT STATE DESIGN FOR FLEXURE =

Chapter II

1.25

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, ⁄

' = 0.21

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, =

MD

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 Design Situation Persistent and transient Accidental

Concrete γc

Steel γs

Class I

Class II

Class I

Class II

1.50

1.65

1.15

1.20

1.30

1.45

1.00

1.10

Table 2.1-2Partial safe factors of materials at ULS according to EBCS 2 -1995



RCS – I LIMIT STATE DESIGN FOR FLEXURE Chapter II 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. 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, =

0.85 *O7P 4 389% *OQ+27P , N?898 0.85 0?O93 4 389% *OQ+27P M

Design stress of concrete in tension is, '

' , M ⁄

N?898 ' 0.21 , Q55O9+27P 3O C 2 4 1995

Design stress of steel for both in tension and compression,

, M

The characteristics load is given by EBCS 2-1995 as, R RD 4 J K 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



RCS – I LIMIT STATE DESIGN FOR FLEXURE Chapter II Suitable design loads are obtained from characteristic loads by applying partial safety factor for loads or load factors γF. R = γT RU ,

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) R = 1.3V + 1.6W

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

R = 1.3V + 1.35 X W ZG

3. Permanent action, variable action and accidental (seismic) action R = V + WG + [ = 0.75(1.3V + 1.6W ) + [

Load combination for SLS 1. Permanent action (Gk) and only one variable action (Qk) R = V + W

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

R = V + 0.9 X W ZG

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 AAiT Department of Civil Engineering


RCS – I LIMIT STATE DESIGN FOR FLEXURE Chapter II 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 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. 980203Q58 ] *OQ+ 88530

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



RCS – I LIMIT STATE DESIGN FOR FLEXURE Chapter II 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,

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



RCS – I LIMIT STATE DESIGN FOR FLEXURE Chapter II To illustrate the stress-strain development for increased loading, consider the following,

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.



RCS – I

LIMIT STATE DESIGN FOR FLEXURE • •

Chapter II

Tension cracks in the concrete widens, visible and significant deflection of the beam occurs. 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.



RCS – I LIMIT STATE DESIGN FOR FLEXURE Chapter II 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. 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 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 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.



RCS – I LIMIT STATE DESIGN FOR FLEXURE Chapter II 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.

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 rectangularparabolic distribution. 1. If one wants to use the idealized parabolic-rectangular stress block given in EBCS 21995, as shown in figure 2.2.1-2

Figure 2.2.2-2 Derivation

^ = and = _ >+ Moment about the T,



RCS – I

LIMIT STATE DESIGN FOR FLEXURE ; = < = _ >+ (+ − ` +) = _ >+ (1 − ` )

Chapter II

Moment about the C, ; = ^ < = (+ − ` +) = + (1 − ` ) αc and βc are values calculated by integrating the stress-strain diagram for the different location of the N.A. depth. i.e. a 1000b (1 − 250b ) > +6 _ = d >@ c

a 1000b (1 − 250b ) > 6 +6 ` = 1 − d @ c

i.

εcm≤2‰and N.A. within the section _ =

` = ii.

8 − bD 4(6 − bD ) c

εcm≥2‰ and N.A. within the section _ = ` =

iii.

bD (6 − bD ) -c 12

3bD − 2 -c 3bD

bD (3bD − 4) + 2 2bD (3bD − 2 ) c

εcm≥2‰ and N.A. outside the section _ =

1 (125 + 64bD − 16bD ) 189

` = 0.5 −

(bD − 2) 40 7 125 + 64bD − 16bD

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, @f 2 2 = ⇒ @f = @ @ bD bD

bD 6 2 = ⇒ bD = 6 2 @f @f



RCS – I


Chapter II

26 26 = 1000 , . h1 − 250 , .i @f @f

C is the compression stress resultant, ck

= j + + >(@ − @f ) d

ck

= j 1000 , _ =

d

26 26 . h1 − 250 , .i >+6 + >(@ − @f ) @f @f

3bD − 2 = -c , >+ 3bD

Nℎ898 -c =

Taking moment about the top fiber, ck

` + = j +(@ − 6) + >(@ − @f ) l d

@ +

@ − @f m 2

26 26 @ − @f ` + = j 1000 , . h1 − 250 , .i >+6 (@ − 6) + > l m @f @f 2 d ck

` =

bD (3bD − 4) + 2 - , 2bD (3bD − 2) c

Nℎ898 -c =

@ +

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, = 0.8@> The moment resistance of the cross-section is, ; = ^n = n

; = (+ − 0.4@) = 0.8@> (+ − 0.4@)

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.



RCS – I


Chapter II

Figure 2.2.2-3 Derivation

From similarity of triangle,

@f 0.002 4 = ⇒ @f @ 0.5714@ 0.0035 7 @

To find the compressive force for the parabolic rectangular stress block, 98Q #Q98Q O oQ>5+8p 4 Q98Q O o5+8p& 1 4 > @ 4 > @ 0.8095 > @ 3 7

_ 0.8095

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

c

` ∑ 98Q @ #Q98Q O oQ>5+8p 4 Q98Q O o5+8p #@ 4 sr &) @ 1 4 @ ` > 4 > , @. #@ 4 & 2 3 7 7 ` 0.3367 >@ ` 0.416@ 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.



RCS – I


Chapter II

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 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 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.



RCS – I


Chapter II

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.

Figure 2.2.3-3 Balanced failure



RCS – I


Chapter II

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



RCS – I


Chapter II

2.2.4. Design equation 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, @ ≤ 0.8(K − 0.44) +

Where δ is percent plastic moment redistribution = (moment after redistribution)/original moment. In the case of no moment redistribution, δ = 1.0 @H ≤ 0.448 +

Usually d is obtained from serviceability limit state. EBCS 2-1995 gives the following minimum effective depth, + = u0.4 + 0.6

w

v 400 `x

Where, fyk = characteristics strength of reinforcement (MPa) Le = effective span βa = constant from table Member Beams Slabs a) Span ratio 2:1 b) Span ratio 1:1

Simply Supported 20

End Spans 24

Interior Spans 28

Cantilevers 10

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, ;y = (+ − ` +) = z>+ (+ − ` +) = z>+ (1 − ` )



RCS – I

LIMIT STATE DESIGN FOR FLEXURE Chapter II ;y (1 ) = z − ` = _ (1 − ` ) = _ (1 − ` ) = JD >+

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

>+

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. ; JD = { = >+ =

+

∗ , JD = JD O9

; ; 1 = (+ − ` +) + (1 − ` )

*83 J = =

; { H>

@ = 0.45 +

1 (1 − ` )

; J +

Moment capacity of the section in terms of the compression force C in the concrete is, ; = (+ − ` +) = _ >+(+ − ` +) = _ >+ (1 − ` ) ; = _ (1 − ` ) = } , 27 P879Q* +802P7 5ℎQ93 $O. 1 >+

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 =

J ; +



RCS – I


Chapter II

b. Design using general design chart 1. Calculate } =

~

2. Enter the general design chart,

• • •

If} } ∗ , section is singly reinforced. Evaluate Z from - =
~

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. d.

EBCS2-1995 gives for beams, zD Y =

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



RCS – I LIMIT STATE DESIGN FOR FLEXURE Chapter II 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. 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:

>Q

as – An area of reinforcement bar to be used As – Total area of steel required

2.3. ULS of doubly reinforced rectangular section 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 steelcarries 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 3.3-1.

Figure 2.3-1Doubly reinforced section



RCS – I


Chapter II

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, • 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/dand concrete grade - Evaluate = ′ =

J ; z ^8702*8 9827O95%873 +

J′ ; z′ O%198002O7 9827O95%873 +

b. Design using general design chart 1. Calculate } , =

~

2. Enter the general design chart,

• -

If} > } ∗ , section is doubly reinforced. Evaluate Z from chart using - = + Calculate G =

∗ ∗ ;, ; , − ;, + (38702*8 9827O958%873) < (+ − + )

∗ ; , − ;, = (5O%198002O7 9827O958%873) (+ − + )

fs in the above expression depends on the yielding of the compressive steel and it can be checked by reading the value of b from the chart, If it is found that the steel has yielded, = ,

2 b ]

O9, C

If it is found that the steel has not yielded, = C b ,

2 b


C 52 Instructor Misgun S.

RCS – I


Chapter II

2.4. 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 2.4-1 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 2.4-2 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 Tsections and L- sections must not exceed: For in interior beams (T-sections)

w

> ≤ 5 5 ⁄5 >8Q% 01Q527P >! +

For in edge beams (L-sections)



RCS – I


w

> ≤ 10 >! + ℎQ* 3ℎ8 5*8Q9 +20Q3758 3O Q+Q5873 >8Q% >! +

Chapter II

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, =

= ^

0.8@ > =

=

0.8@ >

;y = 0.8@ > (+ − 0.4@)

We solve for x from the above quadratic equation, i.

If 6 = 0.8@ > ℎ , section is T- or L-, thus it is convenient to consider two hypothetical beams: Beam F and Beam W



RCS – I


Chapter II

Beam W

Beam F

Figure 2.4-3ULS T-section

Beam F = (> − >! )ℎ

=

(> − >! )ℎ

;y , = l+ 4

m or ;y , = #> 4 >! &? l+ 4

m

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 ! = 0.8@ >! @

! 0.8 >!

;y ,! ! #+ 4 0.4@& or ;y ,! >! #0.8@&#+ 4 0.4@& The total moment capacity of the section now becomes, ;y ;y , ;y ,! 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.



RCS – I

LIMIT STATE DESIGN FOR SHEAR

Chapter III

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 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,



RCS – I

Where

=

W >


Chapter III

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



RCS – I LIMIT STATE DESIGN FOR SHEAR Chapter III 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 ~xf: = ~ Yf: =

c c + {l m + c 2 2

c c − {l m + c 2 2

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 has 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.



RCS – I


Chapter III

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 ^=

; ; + ; ; Q7+ ^ + ^ = ⇒ ^ = < < <

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 =

^ O9 = >! @ >! <

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 59 AAiT Department of Civil Engineering Instructor Misgun S.

RCS – I LIMIT STATE DESIGN FOR SHEAR Chapter III 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, = + x +

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 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 = 20 – 40% 15 – 25% x 35 – 50%



RCS – I LIMIT STATE DESIGN FOR SHEAR Chapter III It has 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. = 0.25 ' JG J >! +

JG #1 50z& t 2.0 , N?898 z

H> + !

J #1.6 4 +& ] 1.0 , + 20 27 %83890 1.0 2 50% O 3?8 >O33O% 9827O958%873 20 389%27Q38+. 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



RCS – I


Chapter III

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 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 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



RCS – I


Chapter III

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 “TrussAnalogy”. 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 diagonals 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’, = ($O. O 03299A10& @ #RO958 5Q9928+ >6 8Q5? 03299A1&

RO958 5Q9928+ >6 8Q5? 03299A1 ,

The horizontal projection of cross section A-A is

Therefore,

<
EBCS 2 gives

+



RCS – I


Chapter III

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



RCS – I


Chapter III

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. = =

, Nℎ898, 98Q

=

; Q7+ 98Q = >! < cos sin

1 = ,tan + . sin >! < cos <>! tan

RO9 = 45d ⇒ =

≤ 0.5 >! <,

2 ≤ <>!

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, y = 0.25 >! +

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,^ = ;H< +

$H 2,

At the bottom net compression, = ;H< −

$H 2.

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. AAiT Department of Civil Engineering


RCS – I

zD Y =


0.4 = >!

Chapter III

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.



RCS – I


Chapter III

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)



RCS – I


Chapter III

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



RCS – I LIMIT STATE DESIGN FOR SHEAR Chapter III 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 ^=

; ; ; ^ = , ; = @; < <

^ =

@ = }x¦ )+ @ <

^ = = }x¦ )+ @ < ⇒ }x¦ =

)+ <

If there are more than one bars ⇒ }x¦ =

∑1<

Where ∑ p is the sum of perimeters of all bars Alternatively }x¦ =

^ + = = )+ @ )+ @ 4@

Figure 3.5.1-4 Average flexural bond stress

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 φ as



RCS – I

* =

§ 4


Chapter III

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) b) c) d)

all bars which are in the lower half of an element all bars in elements whose depth does not exceed 300mm. all bars which are at least 300mm from top of an element in which they are placed 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 = ' O9 1*Q27 >Q90 = 2' O9 +8O9%8+ >Q90 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. *,Y ' = Q* Where:

oxx' p ] *,D Y o¨:f p

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



RCS – I


Chapter III

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, *f = QG ∗ *,Y ' > *f,D Y

Where:

*f,D Y = 0.3 ∗ Q ∗ QG ∗ * > 15 φ O9 > 200%% 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

a

Percentage of reinforcement lapped with in required lap length

b

20%

25%

33%

50%

100%

a < 10 φ and /or

b < 5φ

1.2

1.4

1.6

1.8

2.0

a > 10φ and

b >5φ

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. ^=

; $ + < 2

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.



RCS – I


Chapter III

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) b) members with Vsd < 2Vc, c) members with Vsd ≥ 2Vc,

a1 = 1.0d a1= 0.75d 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 onehalf 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. AAiT Department of Civil Engineering


RCS – I

SERVICEABILITY LIMIT STATE

Chapter IV

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 73 AAiT Department of Civil Engineering Instructor Misgun S.

RCS – I SERVICEABILITY LIMIT STATE Chapter IV 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, >

@ 4 7 #+ 4 @& 0 2



RCS – I SERVICEABILITY LIMIT STATE Chapter IV 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 crosssection as shown in figure 4.2.2-1. From the strain distribution, b b = @ +−@

Applying Hooke’s Law and using modular ration the above equation becomes, = , Nℎ25ℎ 02%1*2280 3O @ 7(+ − @) @ 7 = = - O9 @ = -+ (2) + + 7

For horizontal force equilibrium, =^

@> = (22) 2 +82727P z =

388* 9Q32O, 8©AQ32O7 (22)>85O%80 >+

->+ = z>+ 2

=

(222) 2z

But equation (2) can be simplified to - + -7 = 7 which reduces to 7 = (1 − -) (2) -

Equating equation (222) and (2) and solving for practical value of k yields,

- = −z7 + (z7) + 2z7 (88 +892Q32O7 O7 3ℎ8 *853A98 7O38)

Note that () satisfies the stress -strain relation as well as the equilibrium of horizontal forces and hence is a useful relation for analysis.



RCS – I


Chapter IV

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



RCS – I


Chapter IV

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



RCS – I SERVICEABILITY LIMIT STATE Chapter IV 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.



RCS – I


Chapter IV

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: Restraint of intrinsic imposed deformations - where stresses are generated in (a) 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 As = K e Kf ct.ef Act / σ s Where

As = area of reinforcement Act = 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. σ 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 f ct.ef = 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



RCS – I

Ke =

K=

SERVICEABILITY LIMIT STATE Chapter IV being less than 28 days, it is suggested that a minimum tensile strength of 3 MPa be adopted. 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 a coefficient which allows for the effect of non-uniform selfequilibrating 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 σ ct = 1.70 f ctk b) Direct tension

σ ct = f ctk

(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 wk do not exceed the limiting values given in Table 4.3.3.3.1 1 appropriate to the different conditions of exposure.



RCS – I


Type of Exposure

Chapter IV

Dry environment: Interior of buildings of normal habitation or offices

Humid environment: Interior components (e.g. laundries); exterior components; components in nonaggressive soil and/or water

Seawater and/ or aggressive chemical environment: Components completely or partially submerged in seawater; components in saturated salt air; aggressive industrial atmospheres

(Mild)

(Moderate)

(Severe)

0.4

0.2

0.1

Characteristic crack width, wk

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. wk = 0.4mm σ s ( MPa ) φ (mm) 160 200 240 280 320

40 32 25 20 16

wk = 0.2mm σ s ( MPa ) φ (mm) 160 200 240 320 400

25 16 12 6 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

σ s is the steel stress under service condition wk 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 wk = 1.7 wm wm = S m ε sm



RCS – I


Chapter IV

Where wk = the characteristic crack width wm = the mean crack width S m = the average distance between cracks

ε sm = the mean strain of the reinforcement considering the contribution of concrete in tension. (3)

The average distance between cracks may be obtained from

S m = 50 + 0.25k1 k 2 Where:

φ ρr

φ = diameter k1 a coefficient which characterizes the bond properties of the bars = 0.8 for deformed bars = 1.6 for plain bars k 2 a coefficient representing the influence of the form of the stress diagram. = 0.50 for bending = 1.00 for pure tension = (ε 1 + ε 2 ) / 2ε 1 for bending with tension ε 1 , ε 2 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 ρ r is defined as

ρr = Where

As Ac ,ef

(5.15)

As is the area of the reinforcement contained in Ac , ef

Ac , ef 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 Ac , ef



RCS – I SERVICEABILITY LIMIT STATE The mean strain of the reinforcement may be obtained as (4)

ε sm

σs 

σ 1 − β 1 β 2  sr = Es   σs 

  

2

Chapter IV

 σ  ≥ 0.4 s Es 

σ s is the service stress in the steel and may be obtained by elastic theory

Where:

using modular ratio equal to 10. σ s r is the steel stress at rupture of concrete section; i.e., stress for the cracked section under the action of the theoretical moment M cr β1 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 β 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. wk (mm)

f yd ( MPa ) Bond Properties Vsd ≤ Vc

0.4 220 (1)

400 (2) 300

0.2 360 500 (1) (2) 250

220 400 (1) (2) 200

360 500 (1) (2) 150

Vc ≤ Vsd ≤ 3Vc

250

200

150

100

Vsd > 3Vc

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, wk is the permitted characteristic crack width Vsd is the shear acting during the combination under consideration Vc 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. wm = S m ε sm together with the following equations:



RCS – I


Chapter IV

wk = 1.7 k w wm s m = 50 + 0.25k1 k 2

 σ  ≥ 0 .4 s Es  V − Vc 1 σ s = sd . ≥ 40MPa bw d ρ w (sin α + cos α )

ε sm

σs 

φ d−x ≤ ρ r sin α

V 1 −  c = E s   V sd 

  

2

wm = the mean crack width (see Eq. 5.12) α = the angle of inclination of the stirrup from the horizontal k w = 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 ( α = 900 ) = 0.8 for inclined stirrups with α = 450 to 600 ρ 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)

Where

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

δ=

Le 200



RCS – I Where Le is the effective span


Chapter IV

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

δ =

Le ≤ 20 mm 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. f yk  Le   d =  0.4 + 0.6 400  β a 

Where:

fyk is the characteristic strength of the reinforcement in MPa

β a is the appropriate constant from Table 5.6, and for slabs carrying partition walls likely to crack, shall be taken as β a ≤ 150 Lo 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 Beams Slabs a)Span ratio 2:1 b)Span ratio 1:1

Simply Supported 20

End spans 24

Interior spans 28

Cantilevers

25 35

30 40

35 45

12 10

10

Table 4.4.3-1 Values of β 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.



RCS – I


Chapter IV

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 δ i and δ ii , but nor more than δ max .



RCS – I


δ i = βL2

M cr E cm I i

δ ii = β L2

M k − M cr 0.75 E s As z (d − x)

δ max = β L2

Chapter IV

Mk E s As z (d − x)

Unless the theoretical cracking moment is obtained by a more comprehensive method, it shall be computed by Mcr = 1.70 fctk Z

δ i = deflection due to the theoretical cracking moment on uncracked transformed section δ 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. δ 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 β = deflection coefficient depending on the loading and support conditions (e.g. β = 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. Where:

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.



RCS – I


Chapter IV

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 C, =

C 1ª

Where, ϕ is a creep coefficient taken from table 4.4.6-1 below. Age of concrete at loading (days) 1 7 28 90 365

Notional size 2Ac/U (mm) Dry Atmospheric Humid Atmospheric Condition (Inside) Condition (Outside) 50 150 600 50 150 600 5.5 4.6 3.7 3.6 3.2 2.9 3.9 3.1 2.6 2.6 2.3 2.0 3.0 2.5 2.0 1.9 1.7 1.5 2.4 2.0 1.6 1.5 1.4 1.2 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

[2 − 1.2 As' / As ] ≥ 0.6 Where:

As’= area of compression reinforcement As= area of tension reinforcement



ADDIS ABABA INSTITUTE OF TECHNOLOGY ADDIS ABABA UNIVERSITY

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