Samir Shehada_ ACI 05_RC Design 1

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Design of Reinforced Concrete Structures (I) ECIV 3316

Instructors Name: Prof. Samir M. Shihada

E-Mail: [email protected] or, [email protected]

Office: Administration Building, Room B243

Office Phone: +970 8 2644400 - Ext. No. 2814

Office Hours: 11:00-12:00, Saturdays through Wednesdays. Course Description:

Strength of reinforced concrete; design of short columns; beam in flexure and shear; one-way slabs; development and anchorage of reinforcement and isolated footings.

Objectives: 1. Students will gain a basic understanding of the integration of analysis and

design. 2. Students will learn how to design reinforced concrete members, including

short columns, beams, one-way slabs, and isolated footings for applicable strength and serviceability limit states according to ACI 318-2008.

3. Students will ultimately learn how to design a reinforced concrete building frame system.

Instructional Methods:

1. Three lecture hours per week covering theoretical background in addition to solving numerical examples.

2. One discussion hour per week focusing on a comprehensive design project. This design project will be completed throughout the semester. Design groups will be assigned by the Teaching Assistant.

Textbook: Reinforced Concrete Design, draft of third edition (available on my webpage).

References: 1. Building Code Requirements for Reinforced Concrete, ACI 318-08, Farmington

Hills, MI, USA. 2. Reinforced Concrete, Mechanics and Design, by J.K. Wight and J.G.

MacGregor 5th Ed., Prentice Hal, 2005. 3. Design of Reinforced Concrete , by J.C. McCormac and J. Nelson, 7th Ed.,

Wiley, 2006.

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Course Outline: 1- Introduction: 2- Materials and Properties:

Concrete Steel reinforcement

3- Design Requirements: 4- Design of Columns:

Axially Loaded Short Columns 5- Design for Flexure:

Singly Reinforced Rectangular Sections T-Shaped Sections Irregular Sections

6- Design for Shear: 7- Design of One-way Slabs:

Solid Ribbed

8- Development of Reinforcement: Development lengths Lap Splices Bar Cutoffs

9- Design of Isolated Footings (Concentrically Loaded): Square Rectangular

10- Applications Comprehensive Design Project

Attendance:

Regular attendance is strongly recommended for maintaining pace with the lectures and the progress of the class.

Grading Policy: The students will be evaluated by a mid-term exam, a final exam and assigned comprehensive project. The final grades for this course will be based on the following percentages: Midterm Exam. 25 % Final Comprehensive Exam. 60 %

Design Project 15% Total 100 %

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CHAPTER 1 : INTRODUCTION

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1.1 Reinforced Concrete Hardened plain concrete is characterized by its high

compressive strength and its relatively low tensile strength. The

addition of steel reinforcement, which is characterized by its

high tensile strength in the tension regions, helps improve the

resistance in these regions. The final product is known as

reinforced concrete.

Steel is used as a reinforcement element due to some factors,

which are,

Steel improves the resistance of concrete in the tension

regions due to its high tensile and compressive strengths.

Steel and concrete have similar thermal expansion

coefficients; 0.000010 to 0.000013 for concrete and

0.000012 for steel per degree Celsius, thus causing

negligible internal stresses resulting from temperature

changes, which in turn, means a good bond between the two

materials.

Steel adds ductility which is required in the design process.

The construction of reinforced concrete structures requires the

use of a form to take the shape of the built member. The

reinforcement is held in place in the form during the casting

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operation. Once the concrete has hardened to the required

strength, only then the forms are removed.

1.2 Advantages of Reinforced Concrete

Reinforced concrete is used as a prime construction material

universally. It is used in constructing bridges, buildings,

underground structures, hydraulic structures, and so many other

uses. The remarkable success of reinforced concrete may be

attributed to its numerous advantages. These include the

following:

Durability, especially fire resistance.

Ability to be formed in different shapes.

Rigidity, which means comfort for the occupants.

Low-maintenance cost.

Economy due to availability of inexpensive local materials.

1.3 Disadvantages of Reinforced Concrete

Concrete has a low tensile strength, thus requiring the use of

steel reinforcement.

Low strength per unit weight compared to other structural

materials such as structural steel.

Requirements of forms and shoring, which involves labor,

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cost and time.

The properties of concrete are variant due to proportioning

and mixing.

Quality control needs much more attention, compared to

other materials such as structural steel.

Time-dependent volume changes that may cause deflections

or cracking, if restrained.

1.4 Historical Development

In 1824, Portland cement was patented by the English Joseph

Aspdin. The name Portland was used due to the resemblance of

cement to the building stone quarried on the Isle of Portland of

the English coast.

The advantages of using steel reinforced concrete in building

construction was first discovered in 1850 by a Frenchman

called Joseph Lambot. He found that by adding thin steel bars

or steel fibers to concrete, he could greatly increase the strength

of the concrete, making it better for use in a variety of

applications. In the early years, reinforced concrete was used

for making a number of items, such as reinforced garden tubs,

road guardrails, and reinforced concrete beams. Buildings that

used reinforced concrete in their construction were constructed

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all over the world, especially in the United States, Canada and

Europe. In 1878, the first reinforced concrete system was

patented in the United States by an American by the name of

Thaddeus Hyatt. Mathias Koenen, in Berlin, Germany, was the

first experimenter to deduce methods of computation for load

tests, publishing his analysis of tests conducted in Germany in

1886. Koenen based his theory of flexure on the following

premises: (1) plane sections perpendicular to the neutral axis

prior to bending remain so following bending; (2) stress is

proportional to strain; (3) there is perfect bond between

concrete and steel; and (4) tension stresses in the concrete are

not considered. Koenens basic procedure is still generally

accepted and used today.

In the United States, the American Concrete Institute became an

influential organization in the development of reinforced

concrete standards. Chartered in 1906 as the National

Association of Cement Users, it changed its name to ACI in

1913.

1.5 Vertical Load Structural Systems

Building Frame: it is a complete frame system that provides

support for gravity loads, shown in Figure 1.1.

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Moment-Resisting frame: it is a frame in which members

and joints are capable of resisting forces, including lateral

loads, primarily by flexure, shown in Figure 1.2.

Bearing Wall: it is a complete vertical load-carrying space

frame.

1.6 Structural Elements A concrete building may contain some or all of the following

main structural elements, which are to be dealt with in detail in

the following chapters of this book. Figures 1.1 show the main

structural elements in a building frame system. These are

discussed, in short, below.

Slab is a horizontal plate element which is usually designed

to resist gravity loads. The depth of the slab is usually very

small compared to its length or width. It is usually designed

to resist shear forces and bending moments.

Beam is long, horizontal or inclined member with limited

width and depth. Its main function is to support slab loads. It

is designed to resist shearing forces, bending moments and

torques, if any.

Column is a member that supports beam or slab loads. It is

designed to support axial loads and moments.

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Wall is a vertical plate element that resists gravity as well as

lateral loads as in the case of retaining wall. It is usually

designed to resist shear forces and bending moments.

Stair is a structural member that provides means of

movement from one floor to another in a structure. It is

designed to resist shear forces and bending moments.

Footing, or foundation, is a member that supports column or

wall loads and transmit them directly to the soil. The footing

is designed to resist shearing forces and bending moments.

Figure 1.1: Building Frame System

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Figures 1.2 show the main structural elements in a moment-resisting frame system. The frame consists of main beams (girders) and columns. The load on the slab is transferred to the girders then to the columns and footings.

Figure 1.2: Moment-Resisting Frame System

CHAPTER TWO MATERIALS 6

2 CHAPTER 2: MATERIALS

2.1 Introduction

Concrete is used in constructing buildings, harbors, runways, water

structures, power plants, pressure vessels. In order to use concrete

satisfactorily, the designer, the site engineer, and the contractor need to be

familiar with construction materials and their technologies.

There are two common structural materials used in construction: concrete

and steel. They sometimes complement one another, and sometimes

compete with one another.

The man on the site needs to know more about concrete than about steel.

Steel is manufactured under controlled conditions in a sophisticated plant.

On a concrete building site, the situation is totally different. While the

quality of cement is guaranteed, transporting, placing, compacting and

curing of concrete greatly influence the final product. It is the competence

of the contractor and the supplier which controls the actual quality of

concrete in the finished structure.


2.2 Concrete

Concrete for reinforced concrete consists of aggregate particles bound

together by a paste made from Portland cement and water. The paste fills

the voids between the aggregate particles, and after the fresh concrete is

placed, it hardens as a result of exothermic chemical reactions between

cement and water to form a solid and durable structural material.

Although there are several types of ordinary Portland cements, most

concrete for buildings is made from Type I ordinary cement.

In recent years, there has been a substantial increase in the use of other

chemical additives for cement dispersion, acceleration or retardation of

initial set, and improvement of workability. The so called Super-plasticizers

are being used in many applications where high strength concrete with

substantial slump is desired.

Aggregates are particles that form about three-fourths of the volume of

finished concrete. According to their particle size, aggregates are classified

as fine or coarse. Coarse aggregates consist of gravel or crushed rock

particles not less than 5 mm in size. Fine aggregates consist of sand or

pulverized rock particles usually less than 5 mm in size. Aggregates alone

exhibit a linear stress-strain relationship and so does the hydrated cement

paste. On the other hand, concrete exhibits a non-linear relationship due to

presence of interfaces and the development of micro cracking at the

interfaces under load, as indicated in Figure 2.1.


Figure 2.1: Stress-strain relations for cement paste, aggregate, and concrete

Mixing water should be clean and free of organic materials that react with the cement or the reinforcing bars in case of reinforced concrete. The quantity of water relative to that of the cement, called water-cement ratio, is the most important item in determining concrete strength. An increase in this ratio leads to a reduction in the compressive strength of concrete as shown in Figure 2.2.

It is important that concrete has workability adequate to assure its consolidation in the forms without excessive voids. This property is usually indirectly measured in the field by the slump test. The necessary slump may be small when vibrators are used to consolidate the concrete.


Proper curing of concrete requires that the water in the mix is not to be allowed to evaporate from the concrete until the concrete has gained the desired strength.

Figure 2.2: Relation between compressive strength and water-cement ratio of concrete

2.3 Mechanical Properties of Concrete

In this section, mechanical properties of concrete, including compressive strength, tensile strength, modulus of elasticity, shrinkage and creep will be briefly covered.

2.3.1 Concrete Compressive Strength

The compressive strength of concrete cf is mainly affected by the water-

cement ratio, degree of compaction, age, and temperature. It is determined through testing standard cylinders 15 cm in diameter and 30 cm in length in uniaxial compression at 28 days (ASTM C470). Test cubes 10 cm 10 cm


10 cm are also tested in uniaxial compression at 28 days (BS 1881), shown in Figure 2.3.

The ACI Code is based on the concrete compressive strength as measured by a standard test cylinder. Designers have to pay attention to make the necessary transformation when concrete compressive strength is based on the cube test. The concrete compressive strength as measured by the standard cylinder test can be approximated by Eq.(2.1)

cf Cylinder = 0.80 cf Cube ( 2.1)

According to ACI code 1.1.1 for structural concert cf shall not be less than

17 MPa, and no maximum value of cf shall apply unless restricted by a specific code provision.

Figure 2.3: Cube and cylinder test specimens

2.3.2 Concrete Tensile Strength

The strength of concrete in tension is low compared to its compressive strength. Tests have shown that the tensile strength ranges from 8 to 15 percent of the compressive strength cf .

Generally, the ratio of tensile strength to compressive strength is lower, the higher is the compressive strength as indicated in Figure 2.4. There are several factors which affect the relationship between the two strengths, the main one is being the method of testing the concrete in tension, the size of the specimen, the shape, surface texture of coarse aggregate, and the moisture condition of the concrete. The large difference between the tensile


and compressive strengths of concrete is attributed to the formation of fine cracks throughout the concrete.

Figure 2.4: Relation between tensile and compressive strength of concrete

Tensile strength of concrete is important in structures where cracks are not permitted, such as in the design of liquid containers.

2.3.2.1 Standard Tension Tests:

Three main tests are used to measure the tensile strength of concrete.

A. The Direct Tension Test Direct tension tests are rarely used because of the problem of gripping the specimen and due to the secondary stresses developing at the ends of the specimens.

B. The Split Cylinder Test (ASTM 496) In this test, a standard compression test cylinder 15 cm in diameter and 30 cm in length is placed on its side and loaded in compression along a diameter until splitting occurs along the vertical diameter as shown in Figure 2.5.

The splitting tensile strength ctf is given by


dlPfct p

2= ( 2.2)

where

P = maximum applied load

l = length of the cylinder

d = diameter of the cylinder

Figure 2.5: Split cylinder test

ACI 8.6.1 gives an approximate value for ctf

= cct ff l78.1 ( 2.3)

where

cf is given in kg/cm2.

= modification factor reflecting the reduced mechanical properties of lightweight concrete, all relative to normal weight concrete of the same compressive strength.

According to ACI 8.6.1, = 0.85 for sand-lightweight concrete and = 0.75 for all-lightweight concrete, Linear interpolation between 0.75 and


0.85 shall be permitted, on the basis of volumetric fractions, when a portion of the lightweight fine aggregate is replaced with normal weight fine aggregate.

Linear interpolation between 0.85 and 1.0 shall be permitted, on the basis of volumetric fractions, for concrete containing normal weight fine aggregate and a blend of lightweight and normal weight coarse aggregates. For normal weight concrete, = 1.0.

C. The Modulus of Rupture Test (ASTM C78) In this test, a plain concrete beam 15 cm by 15 cm in cross section and 75 cm in length is loaded to failure in bending at the third points of a 60 cm span as shown in Figure 2.6.

The modulus of rupture is given by

2

6hbMfr = ( 2.4)

where

M = maximum bending moment

b = width of specimen

h = depth of specimen.


Figure 2.6: Modulus of rupture test

ACI 9.5.2.3 gives an approximate value for rf

cr ff = l2 ( 2.5)

where cf is given in kg/cm2.

2.3.3 Modulus of Elasticity

The modulus of elasticity of concrete cE is defined as the ratio of normal

stress to corresponding strain for compression stresses below proportional limit. Since the stress-strain diagram for concrete is nonlinear as evident in Figure 2.7, the slope of the curve is variable, making the determination of such modulus a tough task. The secant method is usually used to determine cE , being the slope of the line drawn from a compressive stress

of zero to a compressive stress of 0.45 cf .

According to ACI 8.5.1 and for normal weight concrete

cc fE = 15100 ( 2.6)

where cf is given in kg/cm2.


Figure 2.7: Stress-strain diagram

2.3.4 Creep

Creep is defined as the long-term deformation caused by the application of loads for long periods of time, usually years. The total deformation is divided into two parts; the first is called instantaneous deformation occurring right after the application of loads, and the second which is time dependent is called creep. Long-term deformation increases at a slowing rate for a period of two to three years with maximum value recorded at a period of five years.

ACI 9.5.2.5 states that additional long-term deflection resulting from creep and shrinkage of members under bending is determined by multiplying the immediate deflection caused by the sustained load by a factor Dl given by

rx

l+

=D 501 ( 2.7)

where

dbAs=r is ratio of 'sA to db


sA = cross sectional area of compression reinforcement

b = width of cross section

d = effective depth of cross section

x = time-dependent factor for sustained load

Dl = multiplier for additional deflection due to long-term effects

The time dependent factors x are given at different periods of time

3 months ---------------1.00

6 months ---------------1.20

12 months--------------1.40

60 months--------------2.00

Creep can be reduced through using high-strength concrete, good curing of concrete, and using reinforcement on the compression side of the cross section as evident in Eq. (2.7). Multipliers for long-term deflection are also given in Figure 2.8.

Figure 2.8: Long-term deflection multipliers


2.3.5 Shrinkage

Shrinkage of concrete is defined as the reduction in volume of concrete due to loss of moisture. If the concrete member is not restrained, no stresses will be produced. On the other hand, stresses will be developed in case of restraining the concrete member in any form. Once the allowable tensile stresses are exceeded, tension cracking will take place.

Shrinkage can be reduced through using a low water-cement ratio, good curing of concrete, nonporous aggregates, shrinkage reinforcement, and expansion joints.

ACI 7.12.2.1 specifies that a minimum shrinkage and temperature reinforcement ratio of 0.0018 is to be used in one-way slabs perpendicular to the main reinforcement (for yf =4200 kg/cm

2), 0.002 for yf =2800

kg/cm2, but not less than 0.0014.

2.4 Ultra-High Performance Concrete (UHPC)

Concrete with target strengths greater than 2000 kg/cm2 has been developed for outstanding mechanical performance and shows a very promising future in construction applications. With the addition of fibers, UHPC shows very high compressive and tensile strength as well as high ductility when compared with conventional concrete and called Ultra High Performance Fiber Reinforced Concrete (UHPFRC).

In recent years; UHPFRC has been successfully applied to dam repair, bridge deck overlay, coupling beams in high-rise buildings and other specialized structures.

2.5 Light-Weight Concrete

Concrete lighter in weight than normal-weight concrete is used for thermal insulation and to reduce dead load. It is made using light-weight aggregates or foaming agents added to the mix.


2.6 Reinforcement

Steel and steel alloys are widely used as construction materials throughout the world. Steel is an iron-carbon alloy with the carbon content less than 2 %. Structural steel is an alloy with carbon content ranges from 0.80 % to 2 %. Cast iron contains more than 2 % of carbon, thus characterized by its hardness and brittleness. Steel has three main uses in construction; structural steel, forms, pans, and steel reinforcement.

The low tensile strength of plain concrete, a brittle material, results in limited structural applications since most structural elements carry loads that create tensile stresses of significant magnitude. The addition of high-strength ductile reinforcement that bonds strongly to concrete produces a tough ductile material capable of transmitting tension and suitable for constructing many types of structural elements, e.g., slabs, beams, and columns.

Most concrete members are reinforced with steel in the form of bars. Steel reinforcement imparts a great strength and toughness to concrete. Moreover, reinforcement reduces creep and minimizes the width of cracks.

2.6.1 Properties of Steel Reinforcement

Steel reinforcement in the form of longitudinal bars is used to resist tensile forces resulting from direct tension and/or flexure. Reinforcement is also used to resist stresses resulting from shear and/or torsion. In some cases, reinforcement is used to resist compressive stresses.

Reinforcement comes in two forms; round steel bars or welded wire fabric WWF. When bars have smooth surfaces, they are called plain, and when they have projections on their surfaces, they are called deformed. Round bars come in diameters ranging from 6 mm to 50 mm.

Reinforcement is to be free from mud, oil, or other nonmetallic coatings that decrease bond. Welded wire fabric is a prefabricated reinforcement


welded together to form rectangular or square mesh, usually used in slab or wall reinforcement.

Epoxy-coated reinforcing bars are used to minimize corrosion of reinforcement under severe environmental conditions, such as brifge decks and parking garages.

The most important mechanical property of steel reinforcement is the yield stress yf . Mild steel has a well-defined yield stress on the stress-strain

curve, while high strength steel does not have a well-defined yield stress as shown in Figure 2.9 For high strength steels, ACI 3.5.3.2 defines the yield stress as the stress corresponding to a strain of 0.0035. Grade 4200 kg/cm2 (60 ksi) and 2800 kg/cm2 (40 ksi) are the commonly used steel grades in our region. Currently Grade 5200 kg/cm2 (75 ksi) is particularly used in high-rise buildings where it is used in combination with high strength concretes. Grade 4200 kg/cm2 is the most commonly used main reinforcement since Grade 5200 kg/cm2 is rather expensive. When crack widths are to be minimized through reducing reinforcement stresses, grade 2800 kg/cm2 is to be preferred over grade 4200 kg/cm2 and 5200 kg/cm2.

The modulus of elasticity sE is defined as the slope of the stress-strain

curve.

ACI 8.5.2 gives the modulus of elasticity for mild and high strength steels as

6100.2 =sE ( 2.8)

where sE is given in kg/cm2.

Table 2.1shows weights and cross sectional areas of different bar sizes while Table 2.2 shows US bar sizes and their metric equivalence.


Figure 2.9: Stress-strain relation for steel reinforcement

Table 2.1: Properties of reinforcing bars

Cross sectional area (cm2 ) f

Number of bars

mm

Weight Kg/m

1 2 3 4 5 6 7 8 9 10

6 0.222 0.28 0.57 0.85 1.13 1.41 1.7 1.98 2.26 2.54 2.83

8 0.395 0.50 1.01 1.51 2.01 2.51 3.02 3.52 4.02 4.52 5.03

10 0.617 0.79 1.57 2.36 3.14 3.93 4.71 5.5 6.28 7.07 7.85

12 0.888 1.13 2.26 3.39 4.52 5.65 6.79 7.92 9.05 10.2 11.3

14 1.208 1.54 3.08 4.62 6.16 7.7 9.24 10.8 12.3 13.9 15.4

16 1.578 2.01 4.02 6.03 8.04 10.1 12.1 14.1 16.1 18.1 20.1

18 1.998 2.54 5.09 7.63 10.2 12.7 15.3 17.8 20.4 22.9 25.4

20 2.466 3.14 6.28 9.42 12.6 15.7 18.8 22.0 25.1 28.3 31.4

22 2.984 3.80 7.60 11.4 15.2 19.0 22.8 26.6 30.4 34.2 38.0

25 3.853 4.91 9.82 14.7 19.6 24.5 29.5 34.4 39.3 44.2 49.1


Table 2.2: US bars and their metric equivalence

US Units Mteric Units

Size No. Diameter (inch)

Size No. Diameter (mm)

3

4

5

6

7

8

9

10

11

14

18

0.375

0.500

0.625

0.750

0.875

1.000

1.128

1.270

1.410

1.693

2.257

10

13

16

19

22

25

29

32

36

43

57

9.50

12.7

15.9

19.1

22.2

25.4

28.7

32.3

35.8

43.0

57.3

CHAPTER THREE DESIGN REQUIREMENTS 21

1 CHAPTER 3: DESIGN REQUIREMENTS

3.1 Structural Concrete Design At first, the general planning is carried out by the architect to set out the

layout of the building floors based on customer's needs. Only then, the

structural engineer determines the most appropriate structural system to

ensure strength, serviceability and economy of the building. This is done

through the following steps.

1. Setting out the building structural system/systems.

2. Evaluating the external loads on the members. These loads include own

weights of the members, which are estimated at the start, in addition to other

loads that the members are intended to support. Own weights of the members

are to be checked later once the design process is done.

3. Carrying out the structural analysis using computer or manual calculations

to determine the internal forces. The analysis is done using manually or using

computer software.

4. Determination of member dimensions and required reinforcement.


5. Preparation of structural drawings.

3.2 Types of Concrete Design

Concrete design can be classified into three main categories; plain concrete design, reinforced concrete design, and prestressed concrete design.

3.2.1 Plain Concrete Design

With the advent of reinforced concrete, plain concrete is hardly used as a structural material. It is mainly used for nonstructural members. This is due to the low strength of concrete in tension which results in large sections, especially, when required to resist tensile stresses resulting from direct tension or bending.

3.2.2 Reinforced Concrete Design

The compressive strength of concrete is high while its tensile strength is low. To alleviate the situation, high tensile strength reinforcement in the form of steel bars is added in the tension regions to enhance the capacity of concrete members as shown in Figure 1.1. The reinforcement is usually placed in the forms before casting the concrete. Once hardened, the resulting composite material is called reinforced concrete.


Figure 1.1: Mechanics of reinforced concrete: (a) beam and loads; (b) a plain concrete beam; (c) a reinforced concrete beam

1.1.1 Prestressed Concrete Design

Since the strength of reinforced concrete can be enhanced by the elimination of cracking, prestressing is used to produce compressive stresses in tension regions. Prestress is applied to a concrete member by high-strength steel tendons in the forms of bars, wires, or cables that are first tensioned and then anchored to the member. When the tendons are tensioned before the concrete is cast around them, the concrete member is called pre-tensioned. When the tendons are passed through ducts and tensioned after the concrete has hardened and gained enough strength, the concrete member is called post-tensioned.


When compared to classical reinforced concrete design, prestressed concrete design produces lighter sections, thus allowing the economic use of much longer spans.

1.2 Design Versus Analysis

It involves the determination of the type of structural system to be used, the cross sectional dimensions, and the required reinforcement. The designed structure should be able to resist all forces expected to act during the life span of the structure safely and without excessive deformation or cracking.

1.2.1 Analysis

It involves the determination of the capacity of a section of known dimensions, material properties and steel reinforcement, if any to external forces and moments.

1.3 Limit States of Reinforced Concrete Design

When a structural element becomes unfit for its intended use, it is said to have reached a limit state. The limit states are classified into three groups:

1.3.1 Ultimate Limit States

These involve structural collapse of some structural elements or the structure altogether. These limit states should be prevented as they tend to cause loss of life and property. Elastic instability, rupture, progressive collapse, and fatigue are forms of these limit states.

1.3.2 Service Limit States

These involve the disruption of the functional use of the structure, not its collapse. A higher probability of occurrence can be tolerated than in case of an ultimate limit state since there is less danger of loss of life. Excessive deflections, immoderate crack widths, and annoying vibrations are forms of these limit states.


1.3.3 Special Limit States

These involve damage or failure due to abnormal conditions such as collapse in severe earthquakes, damage due to explosions, fires, or deterioration of the structure and its main structural elements.

Generally, for buildings, a limit state design is carried out first in order to proportion the elements, and second a serviceability limit state is conducted to check whether these elements satisfy those serviceability limit states.

1.4 Design and Building Codes

A code is a set of technical specifications that control the design and construction of a certain type of structures. Theoretical research, experiments, and past experience help in the process of setting these specifications. The purpose of such code is to set minimum requirements necessary for designing safe and sound structures. It also helps to provide protection for the public from dangers resulting from the use of inadequate design and construction techniques.

There are two types of codes; the first is called structural code, and the second is called building code. A structural code is a code that involves the design of a certain type of structures (reinforced concrete, structural steel, etc.). The structural code that will be used extensively throughout this textbook is The American Concrete Institute (ACI 318-08), which is one of the most solid codes due to its continuing modification, improvement, and revision to incorporate the latest advancements in the field of reinforced concrete design and construction. Supplements containing such revisions are made on yearly basis. Every three or six years, a comprehensive code edition is made, combining all revisions made since the last comprehensive edition.

A building code, on the other side, is a code that reflects local conditions such as earthquakes, winds, snow, and tornadoes in the specifications. Usually the building code which describes the prevailing conditions in a


certain city or state, is used in addition to the main structural or national code. Prior to the year 2000, there were three model codes: the Uniform Building Code (UBC), the Standard Building Code (SBC) and the Basic Building Code (BBC). In 2000, these three codes were replaced by the International Building Code (IBC) , which is updated every three years.

1.5 Design Methods

Two methods of design for reinforced concrete have been dominant. The Working Stress method was the principal method used from the early 1900s until the early 1960s. Since the publication of the 1963 edition of the ACI Code, there has been a rapid transition to Ultimate Strength Design. Ultimate Strength Design is identified in the code as the Strength Design Method. The 1956 ACI Code (ACI 318-56) was the first code edition which officially recognized and permitted the Ultimate Strength Design method and included it in an appendix. The 1963 ACI Code (ACI 318-63) dealt with both methods equally. The 1971 ACI Code (ACI 318-71) was based fully on the strength approach for proportioning reinforced concrete members, except for a small section dedicated to what is called the Alternate Design Method. In the 1977 ACI Code (ACI 318-77) the Alternate Design Method was demoted to Appendix B. It has been preserved in all editions of the code since 1977, including the 1999 edition mentioned in Appendix A. In the 2002 code edition, the so called Alternate Design Method was taken out.

1.5.1 The Strength Design Method

At the present time, the strength design method is the method adopted by most prestigious design codes; including the 2008 version of the ACI building code (ACI 318-08). In this method, elements are designed so that the internal forces produced by factored loads do not exceed the corresponding strength capacities and allow for some capacity reduction. The factored loads are obtained by multiplying the working loads (service


loads) by factors usually greater than unity. The favored mode of failure is the one that ensures a controlled local failure of members in a ductile rather than brittle manner.

1.5.1.1 Shortcomings:

1. The use of elastic methods of analysis to determine the internal forces in the members, which are associated with the factored loads, is inconsistent. This is due to the fact that when the ultimate load is approached, steel and concrete are no longer behaving elastically, a basic requirement of the validity of the elastic methods of design.

2. Regardless of the method of design used, structures are expected to behave elastically or nearly under normal working loads. Under this condition, the strength method can not be used and the working stress analysis should be made to determine the deformations and crack widths.

1.5.2 The Working-Stress Design Method

Before the introduction of the strength-design method in the ACI building code in 1956, the working stress design method was used in design. This method is based on the condition that the stresses caused by service loads without load factors are not to exceed the allowable stresses which are taken as a fraction of the ultimate stresses of the materials, cf for concrete

and yf for steel. In this method, linear elastic relationship between stress

and strain is assumed for both concrete and steel reinforcement. The working stress-design method will generally result in designs that are more conservative than those based on the strength design method. Now only the design of sanitary structures holding fluids is based on the working-stress design method since keeping stresses low is a logical way to limit cracking and prevent leakage.


1.5.2.1 Shortcomings

1. No way to account for degrees of uncertainty of various types of loads. Dead loads, for example, can be predicted more accurately than live loads which are usually variant and harder to predict.

2. Experimental investigations showed that analysis according to the working-stress design method does not predict actual behavior, especially, at high stresses.

3. The elastic theory does not allow for prediction of the ductility of a structural member. Consideration of ductility, however, is of a vital importance in the field of design for most dynamic effects.

4. The working stress design method does not make allowances for varying quality control, standard of construction and variations indicating the magnitude of damage that may be caused by possible failure of a particular element.

5. It has been confirmed by tests that the working stress design method does not give correct information with respect to the actual factor of safety against failure of reinforced concrete members. The factor of safety is defined as the ratio between the load that would cause the total collapse to that used as the service or working load. It has been found that the value of this factor is far different from the ratio of the strength to the so-called working stress.

1.6 Loads on Structures

All structural elements must be designed for all loads anticipated to act during the life span of such elements. These loads should not cause the structural elements to fail or deflect excessively under working conditions. Therefore, the designer must use the available codes to estimate these loads if such estimates are available. If not, the designer must use his own judgment to make these estimates which are needed for the analysis process


before embarking on the design process. The most important load types are listed below.

1.6.1 Dead Load (D.L)

The dead load is usually a load of permanent status, such as the own weight of the structure, its partitions, flooring and roofing. The exact value of the dead load is not known until the structural members have been proportioned. Once this is done, this load is calculated and used with other loads to design these members. Only then, the assumed loads are compared with the actual ones, if the difference is substantial such as in long spans, modifications of the assumed values are necessary to guarantee economy on one extreme and adequacy on the other.

1.6.2 Live Load (L.L)

The live load is a moving or movable type of load such as occupants, furniture, etc. Live loads used in designing buildings are usually specified by local building codes. Live loads depend on the intended use of the structure and the number of occupants at a particular time. The structural engineer must use a good judgment if the expected live load is not specified by the local code, or if he expects a larger value than the one specified by the code. Live loads are arranged in such a way to give maximum values for the internal forces. Table 1.1 shows typical live load values used by the ASCE 7-05.

Table 1.1: Typical live loads specified in ASCE 7-05 Apartment Buildings: Residential areas and corridors 200 kg/m2 Public rooms and corridors 480 kg/m2

Office Buildings: Lobbies and first-floor corridors 480 kg/m2 Offices 240 kg/m2 Corridors above first floor 380 kg/m2 File and computer rooms 400 kg/m2

Schools: Classrooms 195 kg/m2


Corridors above first floor 385 kg/m2 First-floor corridors 480 kg/m2

Stairs and Exit Ways: 480 kg/m2 Storage Warehouses: Light 600 kg/m2 Heavy 1200 kg/m2

Garages (cars): 200 kg/m2 Retail Stores: Firest floor 480 kg/m2 Upper floors 360 kg/m2

Wholesale, all Floors 600 kg/m2

1.6.3 Wind Load (W.L)

The wind load is a lateral load produced by wind pressure and gusts. It is a type of dynamic load that is considered static to simplify analysis. The magnitude of this force depends on the shape of the building, its height, the velocity of the wind and the type of terrain in which the building exists. Usually this load is considered to act in combination with dead and live loads.

1.6.4 Earthquake Load (E.L)

The earthquake load, which is also called seismic load, is a lateral load caused by ground motions resulting from earthquakes. The magnitude of such a load depends on the mass of the structure and the acceleration caused by the earthquake.

The provisions of the ACI Code provide enough ductility to allow concrete structures to stand earthquakes in low seismic risk regions. In moderate to high-risk regions, special arrangements and detailing are needed to guarantee ductility.

1.7 Safety Provisions

Safety is required to insure that the structure can sustain all expected loads during its construction stage and its life span with an appropriate factor of


safety. The factor of safety is used to account for the following uncertainties:

Real Loads may differ from assumed design loads, or distributed differently.

Material strengths could be smaller than those used in the design.

Executed dimensions or reinforcement are less than those specified by the designer.

Assumptions and simplifications are made during analysis or design.

The factor of safety should account for the expected type of failure and its consequences and for the importance of the member in terms of structural integrity.

The ACI strength design method, , involves a two-way safety measure. The first of which involves using load factors, usually greater than unity to increase the service loads. The magnitude of such a load factor depends on the accuracy of determining the type of load under consideration. The second safety measure specified by the ACI Code involves a strength reduction factor multiplied by the nominal (theoretical) strength to obtain design strength. The magnitude of such a reduction factor is usually smaller than unity. The load factors and the strength reduction factors will be discussed in detail in the following section.

1.7.1 Load Factors

These load factors are required for possible overloading resulting from;

Magnitudes of loads may vary from those assumed in design. Uncertainties involved in determination of internal force.

In the ACI 318-2002 Code, the load combination and strength reduction factors of the 1999 code were revised and moved to Appendix C, and remains in the ACI 318 08 code edition.


According to ACI 9.2.1, required strength U shall be at least equal to the effects of factored loads in Eqs. (1.1) through (1.7).

The effect of one or more loads not acting simultaneously is to be investigated.

a- Dead load and fluid load Combination:

( )FDU += 4.1 (1.1) b- Dead load, fluid load, temperature load, live load, soil load, roof load,

snow load, and rain load combination:

( ) ( ) rLHLTFDU 5.06.12.1 +++++= ( ) ( ) SHLTFDU 5.06.12.1 +++++= (1.2) ( ) ( ) RHLTFDU 5.06.12.1 +++++=

c- Dead load, roof live load, live load, rain load, wind load, and snow load

combination:

LLDU r 0.16.12.1 ++= LSDU 0.16.12.1 ++= LRDU 0.16.12.1 ++= WLDU r 8.06.12.1 ++= (1.3)

WSDU 8.06.12.1 ++= WRDU 8.06.12.1 ++=

d- Dead Load, wind load, live load, roof live load, snow load, and rain

load combination:

rLLWDU 5.00.16.12.1 +++= SLWDU 5.00.16.12.1 +++= (1.4)

RLWDU 5.00.16.12.1 +++=


e- Dead Load, earthquake load, live load, and snow load combination:

SLEDU 2.00.10.12.1 +++= (1.5)

f- Dead Load, wind load, and soil load combination:

HWDU 6.16.19.0 ++= (1.6)

g- Dead Load, earthquake load, and soil load combination:

HEDU 6.10.19.0 ++= (1.7)

Where

U = Required strength to resist factored loads, or internal forces

D = Dead loads, or related internal forces

F = Fluid loads, or related internal forces

T = Cumulative effects of temperature, creep, shrinkage, and differential settlement

L = Live loads, or related internal forces

H = Soil pressure, or related internal forces

rL = Roof live loads, or related internal forces

S = Snow loads, or related internal forces

R = Rain loads, or related internal forces

W = Wind loads, or related internal forces

E = Earthquake loads, or related internal forces

Regarding the above given equations, the following important notes are also given in ACI 9.2.1 and 9.2.2

a- The live load factor on L on Eqs. (1.3), (1.4) and (1.5) is permitted to be reduced to 0.5 except for garages, areas of public assembly, and all

areas where the live load is greater than 485 2/ mkg .


b- Where the wind load W has not been reduced by a directionality factor, it is permitted to use 1.3W instead of 1.6W in Eqs. (1.4) and (1.6).

c- Where earthquake load E is based on service level forces, 1.4 E is to be used in place of 1.0 E in Eqs. (1.5) and (1.7).

d- The load factor on H shall be set equal to zero in Eqs. (1.6) and (1.7) if the structural action due to H counteracts that due to W or E . Where lateral earth pressure provides resistance to actions from other forces it shall not be included in H but shall be included in the design resistance.

e- If the live load is applied rapidly, as may be the case for parking structures, loading docks, warehouse floors, elevator shafts, etc., impact effects should be considered. In all equations, substitute (L + impact) for L when impact should be considered.

For many members, the loads considered are dead, live, wind and earthquake.

Where the F, H, R, S , Lr and T loads are not considered equations (1.1) through (1.7) simplify to those given in Table (1.2) below.

Table 1.2: Required Strength for simplified load combinations

Loads Required Strength Equation NO.

Dead (D) and Live (L) D4.1

LD 6.12.1 +

(1.1)

(1.2)

Dead (D), Live (L) and wind (W)

LD 0.12.1 +

WD 8.02.1 +

LWD 0.16.12.1 ++

WD 6.19.0 +

(1.3)

(1.3)

(1.4)

(1.6)

Dead (D), Live (L) and ELD 0.10.12.1 ++ (1.5)


Earthquake (E) ED 0.19.0 + (1.7)

1.7.2 Strength Reduction Factors

According to the ACI Code 9.3.1, the nominal (theoretical) strength is multiplied by a strength reduction factor to obtain the design strength.

Design strength Required strength

The reasons for using the strength reduction factors include:

Allow for the probability of under-strength due to variations in material strengths and dimensions.

Allow for inaccuracies in the design equations. Reflect the degree of ductility and required reliability of the member

under the load effects being considered. Reflect the importance of the member in the structure.

In the ACI 318-2002 Code, the strength reduction factors were adjusted to be compatible with model building code.

According to ACI 9.3.2 strength reduction factors F are given as follows:

a- For tension-controlled sections ... F = 0.90

b- For compression-controlled sections,

Members with spiral reinforcement .. F = 0.75

Other reinforced members .... F = 0.65

c- For shear and torsion .. F = 0.75

d- For bearing on concrete ... F = 0.65

e- Post-tensioned anchorage zones .. F = 0.85

f- Strut and tie models . F = 0.75

An example of showing the importance of a member in a structure is that columns have smaller strength reduction factors, thus larger safety


measures than beams. This is due to the importance of columns when paying attention to their extensive type of failure which differs from the localized type of failure encountered in beams. Moreover, columns are less ductile than beams, thus requiring a larger factor of safety.

In ACI 10.3.4, sections are called tension-controlled when the net tensile strain in the extreme tension steel is equal to or greater than 0.005 when the concrete in compression reaches its crushing strain of 0.003, as shown in Fig. 1.2.a.

In ACI 10.3.3, sections are called compression-controlled when the net tensile strain in the extreme tension steel is equal to or less than ye

(permitted to be taken as 0.002 for reinforcement with 2/4200 cmkgfy = )

when the concrete in compression reaches its crushing strain of 0.003, as shown in Fig. 1.2.c.

There is a transition region between tension-controlled and compression-controlled sections, shown in Fig. 1.2.b.

(a) (b) (c)

Figure 1.2: Classification of sections for 2/4200 cmkgfy = ; (a) Tension-controlled section; (b) Section in transition between tension and compression; (c) compression-controlled section.


Example (1.1): For frame ABCD shown in Figure 1.2.a, determine the axial forces for which member AB should be designed for the following service loads are applied:

- a dead load of 1 t/m on member BC; - a live load 2.5 t/m on member BC; - a horizontal wind load of 5 tons at joint C, which may act to the

right or left on member AB and CD, respectively.

Figure 1.2.a: Frame

Solution:

The frame is analyzed using SAP 2000 structural analysis and design software for the following load combinations.

Combination (1): D + L, based on Eq. (1.2) ton/m5.2 (2.5) 1.6 (1) 1.2 60.120.1 =+=+= LDwu

.)(0.26)2/10(2.5 comptonsFAB == , as shown in Figure 1.2.b.

Combination (2): D + W, based on Eq. (1.3)

WDU 8.02.1 +=

Wind acts to the right

.)(0.4 comptonsFAB = , as shown in Figure 1.2.c.


Wind acts to the left

.)(0.8 comptonsFAB = , as shown in Figure 1.2.d.

Combination (3): D + L + W, based on Eq. (1.4)

LWDU 0.16.12.1 ++=


.)(5.14 comptonsFAB = , as shown in Figure 1.2.e.


.)(5.22 comptonsFAB = , as shown in Figure 1.2.f.

Combination (4): D + W, based on Eq. (1.6)

WDU 60.19.0 +=


.)(50.0 comptonsFAB = , as shown in Figure 1.2.g.


.)(50.8 comptonsFAB = , as shown in Figure 1.2.h.

Studying the four combinations, member AB should be designed for an axial compression load of 26.0 tons.


Figure 1.2: (continued); Frame and loading combinations, (b) D+L combination; (c) D+W combination (Right); (d) D+W combination (Left); (e) D+L+W combination (Right); (f) D+L+W combination (Left); (g) D+W combination (Right); (h) D+W combination (Left);


Example (1.2): The beam shown in Figure 1.3.a carries a uniformly distributed service dead load of 3 t/m, and a service live load of 1.5 t/m. Determine the maximum positive and negative bending moments for which beam ABC .

Solution:

Figure 1.3.a: Beam ABC

The Beam is analyzed using SAP 2000 structural analysis and design software for the following loading cases.

Maximum negative moment:

This case is evaluated by fully loading the two spans by dead and live loads.

mtonLDwu /0.6)5.1(6.1)3(2.160.120.1 =+=+=

The maximum negative moment is given as.

( ) mtM ve .75.18.max =- , as shown in Figure 1.3.c.

Maximum positive moment:

This case is evaluated by fully loading one of the two spans by dead and live loads while loading the other span by dead load only.

For the span loaded with dead and live loads,

LDwu 60.120.1 += = 1.2 (3) + 1.6 (1.5) = 6.0 t/m

For the other span, Dwu 20.1= = 1.2 (3) = 3.6 t/m

The maximum positive moment is given as.

mtM ve .0.12(max) =+ , as shown in Figure 1.3.e.


(b)

(c)

(d)

(d)

Figure 1.3 : (continued); (b) loads causing maximum negative moment at point B; (c) corresponding bending moment diagram; (d) loads causing maximum positive moment in span BC; (e) corresponding bending moment diagram.


Example (1.3): For frame ABCD shown in Figure 1.4, determine maximum positive and negative bending moments for which member BC should be designed for when the following service loads are acting:

- a dead load of 4 t/m on member BC; - a live load 3 t/m on member BC; - a horizontal wind load of 1 tons at joint C, which may act to the

right or left on member AB and CD, respectively.

Figure 1.4.a: Frame

Solution

The frame is analyzed using SAP 2000 structural analysis and design software for the following load combinations. Combination (1): D + L load

ton/m9.6 (3) 1.6 (4) 1.2 60.120.1 =+=+= LDwu


Figure 1.4.b: Dead and live loads

Combination (2): D + W load (on members CD and AB)

WDU 8.02.1 +=

ton/m4.8 (4) 1.2 2.1)( === Dverticalwu ton/m0.8 (1) 8.08.0)( === Whorizontalwu

Figure 1.4.c: Dead and wind loads (on member CD)


Figure 1.4.d: Dead and wind loads (on member AB)

Combination (3): D + L + W load (on members CD and AB)

LWDU 0.16.12.1 ++=

ton/m7.8 1(3)(4) 1.2 0.12.1)( =+=+= LDverticalwu

ton/m0.8 ) (1) 8.08.0)( === Whorizontalwu

Figure 1.4.e: Dead, Live and wind loads (on member CD)


Figure 1.4.f: Dead, Live and wind loads (on member AB)

Combination (4): D + W Load (on members CD and AB)

WDU 60.19.0 +=

ton/m3.6 (4) 0.9 9.0)( === Dverticalwu

ton/m1.6 ) (1) 6.16.1)( === Whorizontalwu

Figure 1.4.g: Dead and wind loads (on member CD)


Figure 1.4.h: Dead and wind loads

Based on the results obtained from the four combinations, member BC should be designed for a maximum negative bending moment of 252.57 ton.m and a maximum positive bending moment of 227.43 ton.m


1.8 Problems

P1.10.1 The beam shown in Figure P1.10.1 carries a uniformly distributed service dead load of 2.5 t/m, and a service live load of 2.0 t/m. Determine the maximum positive and negative bending moments for which beam ABC should be designed for.

Figure P1.10.1

P1.10.2 For frame ABCD shown in Figure P1.10.2, determine the axial forces for which the member CD should be designed for when the following service loads are applied:

- a dead load of 1.5 t/m on member BC; - a live load 2.0 t/m on member BC; - a horizontal wind load of 8 tons at joint C, which may act either to

the right or left.

Figure P1.10.2


P1.10.3 The multi-story frame shown in Figure P1.10.3 carries the following loads:

- two 5-ton service concentrated live loads applied at points G and H;

- a service dead load of 3 t/m and a service live load of 2 t/m on member CD;

- a service dead load of 5 t/m and a service live load of 1.5 t/m on member BE.

Determine the maximum positive and negative bending moments for which member BE should be designed for.

Figure P1.10.3

1

Design of Columns

Introduction

According to ACI Code 2.2, a structural element with a ratio of height-to-least lateral dimension exceeding three used primarily to support compressive loads is defined as column. Columns support vertical loads from the floor and roof slabs and transfer these loads to the footings.

Columns usually support compressive loads with or without bending. Depending on the magnitude of the bending moment and the axial force, column behavior will vary from pure beam action to pure column action.

Columns are classified as short or long depending on their slenderness ratios. Short columns usually fail when their materials are overstressed and long columns usually fail due to buckling which produces secondary moments resulting from the D-P effect.

Columns are classified according to the way they are reinforced into tied and spirally reinforced columns. Columns are usually reinforced with longitudinal and transverse reinforcement. When this transverse reinforcement is in the form of ties, the column is called tied. If the transverse reinforcement is in the form of helical hoops, the column is called spirally reinforced.

Since failure of columns often cause extensive damage, they are designed with a higher factor of safety than beams.

Types of Columns

Columns are divided into three types according to the way they are reinforced.

Tied Columns

A tied column, shown in Figure 1, is a column in which the longitudinal reinforcement bars are tied together with separate smaller diameter transverse bars (ties) spaced at some interval along the column height. These ties help to hold the longitudinal reinforcement bars in place during construction and ensure stability of these bars against local buckling. The cross sections of such columns are usually square, rectangular, or circular in shape. A minimum of four bars is used in rectangular and circular cross sections.

2

Figure 1: Tied column

Spirally-Reinforced Columns

They are columns in which the longitudinal bars are arranged in a circle surrounded by a closely spaced continuous spiral, shown in Figure 2. These columns are usually circular or square in shape. A minimum of six bars is used for longitudinal reinforcement.

Figure 2: Spirally-reinforced column

Composite Columns

A composite column is a column made of structural steel shapes or pipes surrounded by or filled by concrete with or without longitudinal reinforcement, shown in Figure 3.

3

Figure 3:Composite column

Behavior of Tied and Spirally-Reinforced Columns

Axial loading tests have proven that tied and spirally reinforced columns having the same cross-sectional areas of concrete and steel reinforcement behave in the same manner up to the ultimate load, as shown in Figure 4.a. At that load tied columns fail suddenly due to excessive cracking in the concrete section followed by buckling of the longitudinal reinforcement between ties within the failure region, as shown in Figure 4.b.

(a)

4

(b)

Figure 4: Failure of columns; (a) behavior of tied and spirally-reinforced columns; (b) failure of columns

For spirally reinforced columns, once the ultimate load is reached, the concrete shell covering the spiral starts to peel off. Only then, the spiral comes to action by providing a confining force to the concrete core, thus enabling the column to sustain large deformations before final collapse occurs.

Factored Loads and Strength Reduction Factors

Factored Loads

Load factors for dead, live, wind or earthquake live loads combinations are shown in Table 1.

Table 1: Required Strength for simplified load combinations

Loads Required Strength Equation NO.

Dead (D) and Live (L) D4.1

LD 6.12.1 +

(1.1)

(1.2)

Dead (D), Live (L) and wind (W)

LD 0.12.1 +

WD 8.02.1 +

LWD 0.16.12.1 ++

WD 6.19.0 +

(1.3)

(1.3)

(1.4)

(1.6)

Dead (D), Live (L) and Earthquake (E)

ELD 0.10.12.1 ++

ED 0.19.0 +

(1.5)

(1.7)

Strength Reduction Factors

According to ACI 9.3.2 strength reduction factors F for compression-controlled sections are given as follows:

Members with spiral reinforcement F = 0.75

5 Other reinforced members F = 0.65

The basic equation is given by

nu PP F (1)

where

uP = factored axial load

F = strength reduction factor

nP = nominal axial load

Short Axially Loaded Columns

Figure 5: Uniaxial stress-strain curves for steel and concrete

When axial compressive loads are applied through the centroid of the cross section of a short column, concrete and steel reinforcement are shortened by the same amount due to their composite action. The ultimate load is attained when the reinforcement reaches its yield stress and the concrete reaches its 28-day compressive strength simultaneously, shown in Figure 5.

From equilibrium of forces in the vertical direction,

nsncno PPP += ( 2)

or,

( ) yssgcno fAAAfP +-= ( 3) Where

noP = nominal axial capacity of section at zero eccentricity

ncP = nominal axial load carried by concrete

6

nsP = nominal axial load carried by steel reinforcement

gA = gross sectional area of column

sA = cross sectional area of reinforcement

cf = concrete compressive strength at 28-days

Equation (3) yields larger values than those obtained from laboratory testing due to the better quality of the tested concrete cylinders. Reducing the compressive strength in Equation (3) by 15 % gives results in close agreement with those obtained through testing schemes.

( ) yssgcno fAAAfP +-= 85.0 (4) The above equation is appropriate for determining axial load capacities of already designed columns. Equation (4) could be modified to suit the process of designing columns through the following substitution

ggs AA r=

where gr is the reinforcement ratio

( ) ygggggcno fAAAfP rr +-= 85.0

[ ])85.0(85.0 cygcgno fffAP -+= r (5)

To account for accidental eccentricity resulting from misalignment of reinforcement, voids in the concrete section, unbalanced moments in the beam, or misalignment of columns from one floor to another, ACI Code R10.3.6 and R10.3.7 reduce the strength of tied columns by 20 % and spirally reinforced columns by 15 %.

For capacity calculation of tied columns, the following equation is to be used;

( )[ ]yssgcu fAAA'f85.0)8.0(65.0P +-= , or

( )[ ]yssgcu fAAA'f85.052.0P +-= ] (6) For capacity calculation of spirally reinforced columns, the following equation is to be used;

( ) ( )[ ( ) yssgcu fAAAfP +-= 85.085.075.0 ], or

( )[ yssgcu fAAAfP +-= 85.06375.0 ] (7) For design purposes of tied and spirally reinforced columns respectively,

( )[ ]cygcgu 'f85.0f'f85.0A52.0P -+= r (8)

[ ( )cygcgu fffAP -+= 85.085.06375.0 r ] (9)

7

Design of Spiral

Laboratory tests have proved that compressive strength of the concrete confined within a spiral is increased due to the lateral pressure exerted on the concrete core by the spiral hoops, as shown in Figure 6.

(b) (c) Figure 6: (a) Influence of lateral pressure 2f on the ultimate compressive strength; (b) lateral pressure on core; (c) lateral pressure on spiral

The ultimate compressive strength of laterally pressured cylinders is given by

21 10.4 fff c += (10)

where

1f = compressive strength of test cylinders in biaxial compression at 28-days.

cf = compressive strength of test cylinders in uniaxial compression at 28-days.

2f = applied horizontal pressure.

The spiral is proportioned so that additional compressive strength provided by the confining action of the spiral is equal to the strength provided by the spalled concrete shell covering the spiral when the spiral is stressed to its yield. This is given by

( ) ( )ccgc AfAAf 210.485.0 =- or,

( )( )

-

=

-= 1

10.485.0

10.485.0

2c

gc

c

cgc

AAf

AAAf

f (11)

where

(a)

8

gA = columns gross sectional area

cA = area of concrete core based on a diameter measured out-to-out of spiral

Consider a concrete cylinder equal in depth to the pitch of the spiral S and neglect the slope of the spiral. Cutting the cylinder vertically along a diameter gives the following equilibrium equation in the horizontal direction as shown in Figure 7.

(a) (b) Figure 7: (a) Free body diagram of core and spiral cut-along a diameter; (b) one turn of spiral

22 fSDfa csys =

SDfa

fc

sys22 = (12)

where

sa = cross-sectional area of spiral

syf = yield stress of spiral

cD = core diameter = diameter minus twice the concrete cover

S = spirals pitch

Substituting Equation (12) into Equation (11)

( ) ( ) cc

syscgc ASD

faAAf

210.485.0 =-

SDfa

AAf

c

sys

c

gc =

-

1

20.885.0 (13)

letting sr be the ratio of volume of spiral reinforcement in one turn to volume of core inside it ,

or

SDa

SDDa

c

s

c

css

4)4/( 2

==p

pr

and 4

SDa css

r= (14)

9Substituting Equation (14) into Equation (13) gives

441

20.885.0 sys

c

sycs

c

gc fSD

fSDAAf rr

==

-

or,

-

= 1

41.0

c

g

sy

cs A

Af

fr (15)

The constant in the previous equation is replaced by 0.45 to get the equation given in ACI 9.10.3.

And

-

= 1

45.0

c

g

sy

cs A

Af

fr (16)

Combining equations (14) and (16), the pitch of the spiral S is given as

-

=

sy

c

c

gc

s

ff

AA

D

aS145.0

4 (17)

Columns Subjected To Pure Axial Tension

The strength under pure axial tension is computed assuming that the section is completely cracked and subjected to a uniform strain equal to, or less than ye . The axial capacity of the

concrete is ignored and the axial strength in tension is given by the following equation.

ysu fAP F= (18)

where F is the strength reduction factor for axial tension = 0.90, and sA is the area of column

reinforcement.

Design Considerations

Maximum and Minimum Reinforcement Ratios

ACI Code 10.9.1 specifies that a minimum reinforcement ratio of 1 % is to be used in tied or spirally reinforced columns. This minimum reinforcement is needed to safeguard against any bending, reduce the effect of shrinkage and creep and enhance ductility of columns. Maximum reinforcement ratio is limited to 8 % for columns in general to avoid honeycombing of concrete.

For compression member with a cross section larger than required by consideration of loading, ACI Code 10.8.4 permits the minimum area of steel reinforcement to be based on the gross sectional area required by analysis. The reduced sectional area is not to be less than one half the actual cross sectional dimensions. In regions of high seismic risk, ACI Code 10.8.4 is not applicable.

10Minimum Number of Reinforcing Bars

ACI Code 10.9.2 specifies a minimum of four bars within rectangular or circular sections; or one bar in each corner of the cross section for other shapes and a minimum of six bars in spirally reinforced columns.

Clear Distance between Reinforcing Bars

ACI Code 7.6.3 and 7.6.4 specify that for tied or spirally reinforced columns, clear distance between bars, shown in Figure 8, is not to be less than the larger of 1.50 times bar diameter or 4 cm. This is done to ensure free flow of concrete among reinforcing bars. The clear distance limitations also apply to the clear distance between lap spliced bars and adjacent lap splices since the maximum number of bars occurs at the splices.

Figure 8: Clear distance between bars

Concrete Protection Cover

ACI Code 7.7.1 specifies that for reinforced columns, the clear concrete cover is not to be taken less than 4 cm for columns not exposed to weather or in contact with ground. It is essential for protecting the reinforcement from corrosion or fire hazards.

Minimum Cross Sectional Dimensions

With the 1971 Code, minimum sizes for compression members were eliminated to allow wider utilization of reinforced concrete compression members in smaller size and lightly loaded structures, such as low-rise residential and light office buildings. When small sections are used, there is a greater need for careful workmanship. For practical considerations, column dimensions are taken as multiples of 5 cm.

Lateral Reinforcement

Ties are effective in restraining the longitudinal bars from buckling out through the surface of the column, holding the reinforcement cage together during the construction process, confining the concrete core and when columns are subjected to horizontal forces, they serve as shear reinforcement. Spirals, on the other hand, serve in addition to these benefits in compensating for the strength loss due to spalling of the outside concrete shell at ultimate column strength.

11Ties

According to ACI Code 7.10.5.1, for longitudinal bars 32 mm or smaller, lateral ties 10 mm in diameter are used. In our country and in some neighboring countries, ties 8 mm in diameter are used in column construction.

Tests have proven that spacing between ties has no significant effect on ultimate strength of columns.

ACI Code 7.10.5.2 specifies that vertical spacing of ties is not to exceed the smallest of:

16 times longitudinal bar diameter.

48 times tie diameter.

Least cross sectional dimension.

ACI Code 7.10.5.3 specifies that ties are arranged in such a way that every corner and alternate longitudinal bar is to have lateral support provided by the corner of a tie with an included angle of not more than 135 degrees. Besides, no longitudinal bar is to be farther than 15 cm clear on each side along the tie from such a laterally supported bar. When longitudinal bars are located around the perimeter of a circle, circular ties are used. Figure 9.a shows a number of tie and spiral arrangements.

12

Figure 9.a: Tie and spiral arrangements

13Spirals

According to ACI Code 7.10.4.2 spirals not less than 10 mm in diameter are to be used in cast-in-place construction. The clear pitch of the spiral is not to be less than 2.5 cm and not more than 7.5 cm as dictated by ACI Code 7.10.4.3. The smaller limit is set to ensure flow of concrete between spiral hoops while the larger limit is set to ensure effective confinement of concrete core. The diameter of the spiral could be changed to ensure that the spacing lies within the specified limits.

Bundled Bars

For isolated situations requiring heavy concentration of reinforcement, bundles of standard bar sizes can save space and reduce congestion for placement and compaction of concrete. Bundling of parallel reinforcing bars in contact is permitted but only if ties enclose such bundles. According to ACI Code 7.6.6, groups of parallel reinforcing bars bundled in contact to act as one unit are limited to four in any one bundle, as shown in Figure 9.b.

Figure 9.b: Bundled bars

Column Reinforcement Details

When column offset are necessary, longitudinal bars may be bent subject to the following limitations.

1. Slope of the inclined portion of an offset bar with axis of column must not exceed 1 in 6, shown in Figure 10.

14

Figure 10: Offset Bars

2. Portion of bar above and below the offset must be parallel to axis of column.

3. Horizontal support at offset bends must be provided by lateral ties, spirals, or parts of the floor construction. Ties or spirals, if used, shall be placed not more than 15 cm from points of bend. Horizontal support provided must be designed to resist 1.5 times the horizontal component of the computed force in the inclined portion of an offset bar.

4. Offset bars must be bent before placement in the forms.

5. When a column face is offset 7.5 cm ,or more, longitudinal column bars parallel to and near the face must not be offset bent. Separate dowels, lap spliced with the longitudinal bars adjacent to the offset column faces, must be provided as shown in Figure 11. In some cases, a column might be offset 7.5 cm or more on some faces, and less than 7.5 cm on the remaining faces, which could possibly result in some offset bent longitudinal column bars and some separate dowels being used in the same column.

Figure 11: Separated Dowels

15Column Lateral Reinforcement

Ties

In tied reinforced concrete columns, ties must be located at no more than half a tie spacing above the floor or footing and at no more than half a tie spacing below the lowest horizontal reinforcement in the slab or drop panel above. If beams or brackets frame from four directions into a column, ties may be terminated not more than 7.5 cm below the lowest horizontal reinforcement in the shallowest of such beams or brackets, shown in Figure 12.

(a) (b)

Figure 12: Beams on all column faces

Spirals

Spiral reinforcement must extend from the top of footing or slab in any story to the level of the lowest horizontal reinforcement in slabs, drop panels, or beams above. If beams or brackets do not frame into all sides of the column, ties must extend above the top of the spiral to the bottom of the slab or drop panel, shown in Figure 13.

(a) (b)

Figure 13: Beams on all column faces

16

Design Procedure for Short Axially Loaded Columns

1. Evaluate the factored axial load uP acting on the column.

2. Decide on a reinforcement ratio gr that satisfies ACI Code limits. Usually a 1 % ratio is

chosen for economic considerations.

3. From equations (8) or (9) for tied and spirally reinforced columns respectively, determine the gross sectional area gA of the concrete section.

4. Choose the dimensions of the cross section based on its shape. For rectangular sections, the ratio of the longer to shorter side is recommended to not exceed 3.

5. Readjust the reinforcement ratio by substituting the actual cross sectional area in Equations (8) or (9). This ratio has to fall within the specified code limits.

6. Calculate the needed area of longitudinal reinforcement ratio based on the adjusted reinforced ratio and the chosen concrete dimensions.

7. From reinforcement tables, choose the number and diameters of needed reinforcing bars. For rectangular sections, a minimum of four bars is needed, while a minimum of six bars is used for circular columns.

8. Design the lateral reinforcement according to the type of column, either ties or spirals, as explained in the previous sections of this chapter.

9. Check whether the spacing between longitudinal reinforcing bars satisfies ACI Code requirements.

10. Draw the designed section showing concrete dimensions and with required longitudinal and lateral reinforcement.

Example (1):

The cross section of a short axially loaded tied column is shown in Figure 14. It is reinforced with mm166f bars.

Calculate the design load capacity of the cross section. Use cf = 280 kg/cm2 and yf = 4200

kg/cm2.

Figure 14:Column cross section

17Solution:

%21.1)25()40()100(10.12

==gr

Clear distance between bars = ( ) ( ) ( )

cm80.122

60.1380.024240=

---

Since the clear distance between bars is less than 15 cm, only one tie is required for the cross section.

The spacing between ties is not to exceed the smallest of

16 (1.6) = 25.60 cm

48 (0.80) = 38.40 cm

25 cm

f 8 mm ties spaced @ 25 cm (one set, since only corner bars are used).

Thus, ACI requirements regarding reinforcement ratio, clear distance between bars and tie spacing are all satisfied.

Substituting in Equation (6)

( )[ ]ysSgcu fAAA'f85.052.0P +-= ( ) ( )( ) ( )[ ]420010.1210.12254028085.01000/52.0Pu +-=

ton69.148Pu =

Example (2):

The cross section of a short axially loaded tied column is shown in Figure 14. It is reinforced with mm166f bars.

Calculate the design tensile load capacity of the cross section. Use cf = 280 kg/cm2 and yf =

4200 kg/cm2.

Solution:

( ) tonsfAP ysu 74.451000420010.1290.0 =

== f

Example (3): Design a short tied column to support a factored concentric load of 80 tons, with one side of the cross section equals to 25 cm.

Use cf = 280 kg/cm2, yf = 4200 kg/cm

2 and =gr 1%.

Solution: 1- The factored load uP is given as 80 tons.

182- The reinforcement ratio gr is given as 1%.

3- From Equation (8), determine the gross sectional area gA of the concrete section:

( )[ ]cygcgu 'f85.0f'f85.0A52.0P -+= r ( ) ( )[ ]28085.0420001.028085.0A52.0000,80 g -+=

2g cm16.554A =

4- Choose the dimensions of the cross section based on its shape:

h = 554.16/25 = 22.17 cm, use a 25 25 cm cross section.

5- Evaluate the adjusted gr by substituting the newly designed gross sectional area gA in

Equation (8)

Minimum reinforcement of 1 % will be used.

6- Calculate the needed area of longitudinal reinforcement ratio:

gas AA r= = 0.01 (554.16) = 5.54 cm2

7- From reinforcement tables, choose the number and diameters of needed reinforcement: use ( )216.6144 cmAmm s =f

8- Design the needed ties: The spacing between ties is not to exceed the smallest of

16 (1.4) = 22.40 cm

48 (0.80) = 38.40 cm

20 cm

Use ties f 8 mm spaced @ 20 cm (one set, since only corner bars are used).

9- Check whether the spacing between longitudinal reinforcing bars satisfies ACI Code requirements:

( ) ( ) ( ) ( ) cm0.4cm40.15.1cm6.1240.1280.024225Sclear >>=---=

10- The designed cross section together with needed longitudinal and lateral reinforcement is shown in Figure 15.

19

Figure 15: Designed cross section

Example (4): Design a short, spirally reinforced column to support a service dead load of 45 tons and a service live load of 60 tons.

Use cf = 280 kg/cm2, yf = 4200 kg/cm

2.and =gr 1%.

Solution: 1- LDu P60.1P20.1P += = 1.20 (45) + 1.6 (60) = 150 tons

2- The reinforcement ratio gr is given as 1%.

3- From equation (10.19.b), determine the gross sectional area gA of the concrete section:

[ ])85.0(85.06375.0 cygcgu fffAP -+= r ( ) ( )[ ]28085.0420001.028085.0A6375.0000,150 g -+=

2g cm54.847A =

4- Choose the dimensions of the cross section based on its shape: D = 32.85 cm, taken as 35 cm in diameter.

5- gr is taken as 1 % since smaller values are not allowed by the code.

6- Calculate the needed area of longitudinal reinforcement ratio:

== ggs AA r ( )( ) 22 cm62.9354/01.0 =p

7- From reinforcement tables, choose the number and diameters of needed reinforcement: use )cm78.10A(mm147 2s =f

8- Design the needed spiral: From Equation (16) and trying f 8 mm spiral,

cmDc 274435 =--=

20

-

=

sy

c

c

gc

s

ff

AA

D

aS

145.0

4

( )

( ) ( )( )( )( )

cmS 63.3

42002801

274/354/

2745.0

50.04

2

2=

-

=

p

p

taken as 3.50 cm (center-to-center)

cmSc 7.280.05.3 =-= , i.e within ACI Code limits.

Use f 8 mm spiral with a pitch of 3.5 cm, center-to-center.

9- Check whether the spacing between longitudinal reinforcing bars satisfies ACI Code requirements:

D = 35 2 (4) 2 (0.80) 1.40 = 24 cm

S = 2 (12)

2

43.51sino

= 10.41 cm

( ) cmcmcmSclear 0.440.15.101.940.141.10 >>=-=

10- The designed cross section together with needed longitudinal and lateral reinforcement is shown in Figure 16.

Figure 16: Designed cross section

1

Design of Beams for Flexure

Introduction

Beams are structural members carrying transverse loads that can cause bending moments, shear forces, and in some cases torsion.

This chapter deals only with shallow beams that are defined by ACI 10.7.1 as beams with depth to clear span ratio of less than 0.25.

Design of a beam starts with proportioning its sections to resist bending moments and choosing the required reinforcement. Once this is done, the chosen sections are checked and designed for shear and torsion. The last step in the design process is to check the bond between the reinforcement and concrete.

In this chapter, two problems are dealt with; analysis and design. In the first problem, the dimensions of the cross section together with the amount of reinforcement are given; it is only required to evaluate the bending capacity of the beam cross section. The second problem deals with evaluating the dimensions of the cross section and the required reinforcement, provided that either the bending moment or the loading causing it is given. In both cases, the properties of concrete and steel reinforcement need to be known.

In order to limit deflections, the depth of the cross section is chosen to fulfill the ACI Code serviceability requirements.

Design assumptions

The following assumptions provided by ACI 10.2 are helpful in deriving the basic equations that are used throughout this chapter.

The strain in both reinforcement and in concrete is assumed to be directly proportional to the distance from the neutral axis, even near ultimate strength.

Tensile strength of concrete is to be neglected in axial and flexural calculations of reinforced concrete.

Maximum usable strain at extreme concrete compression fiber is to be equal to 0.003.

Equilibrium of forces acting on the cross section and compatibility of strains between the concrete and the reinforcement are to be satisfied.

Stress in reinforcement below yf is taken as sE times steel strain. For strains

greater than that corresponding to yf , stress in reinforcement is considered to

2

be independent of strain and equal to yf . Thus, for ys ee < , sss Ef e= and

when ys ee , ys ff = .

The concrete compressive stress distribution may be assumed to be rectangular, trapezoidal, parabolic or any other shape that results in prediction of strength in substantial agreement with results of compressive tests.

Stress distribution for different stages of loading

Stage I (Un-cracked linear stage):

If a reinforced concrete beam is loaded in bending in such away that compressive stresses develop at the top fibers of the section while tensile stresses develop at the bottom fibers, the stress distribution according to the bending theory is given by

IyM

f =

(1)

where

f = normal stress

M = bending moment

y = distance from the neutral axis to the point under consideration, measured perpendicular to the beam axis.

I = moment of inertia of the concrete section in addition to that of the reinforcement.

(a) (b) (c) (d) (e) (f)

Figure 1: (a) Cross section; (b) strains; (c),(d),(e) and (f) stress distribution

In this stage, shown in Figure 1.b and Figure 1.c, strains and stresses are distributed linearly and satisfy the following:

cc ff <

rct ff <

where

cf = concrete compressive stress

3

cf = concrete compressive strength at 28 days

ctf = tensile stress of concrete

rf = modulus of rupture of concrete

This case is exclusively used in the design of un-cracked sections associated with water structures, when water is on the tension side of the section, using the working stress design method.

Stage II: Cracked linear stage

With higher loads applied to the beam, stresses are still distributed linearly and the tensile stress in concrete exceeds the modulus of rupture, as shown in Figure 1.d. Thus, the first of tensile cracks at the bottom surface starts to develop with the tensile and compressive strains satisfying the following:

cc ff <

rct ff >

The tensile strength of concrete in the area below the neutral axis is to be neglected. This stage is considered the basis for design of sections subjected to bending using the working stress design method.

Stage III: Cracked nonlinear stage

With further increase of the load, the compressive stresses in the concrete becomes nonlinear, while the strain is still proportional to the distance from the neutral axis, as shown in Figure 1.e.

cc ff <

rct ff >

This case is not used in design, as it is a transitory case between working stress and ultimate strength design methods.

Stage IV: Ultimate Strength Stage

With further increase in the load, the cracks push upward moving the neutral axis in that direction until failure takes place. The stress distribution is shown in 1.f. Depending on the properties of a beam, flexural mode of failure may be ductile or brittle as will be explained in the nex

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