Etabs Concrete

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Concrete Frame

Design Manual


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ETABS

IntegratedThree-Dimensional

Static and Dynamic Analysis and Design

ofBuilding Systems

CONCRETE FRAME DESIGN MANUAL

COMPUTERS &

STRUCTURES

INC.

R

Computers and Structures, Inc.Berkeley, California, USA

Version 7.1

December 2000


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COPYRIGHT

The computer program ETABS and all associated documentation are

proprietary and copyrighted products. Worldwide rights of ownership

rest with Computers and Structures, Inc. Unlicensed use of the program

or reproduction of the documentation in any form, without prior written

authorization from Computers and Structures, Inc., is explicitly prohib-

ited.

Further information and copies of this documentation may be obtained

from:

Computers and Structures, Inc.

1995 University Avenue

Berkeley, California 94704 USA

Tel: (510) 845-2177

Fax: (510) 845-4096

E-mail:[email protected]

Web:www.csiberkeley.com

Copyright Computers and Structures, Inc., 19782000.

The CSI Logo is a registered trademark of Computers and Structures, Inc.

ETABS is a registered trademark of Computers and Structures, Inc.


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DISCLAIMER

CONSIDERABLE TIME, EFFORT AND EXPENSE HAVE GONE

INTO THE DEVELOPMENT AND DOCUMENTATION OF ETABS.

THE PROGRAM HAS BEEN THOROUGHLY TESTED AND USED.

IN USING THE PROGRAM, HOWEVER, THE USER ACCEPTS

AND UNDERSTANDS THAT NO WARRANTY IS EXPRESSED OR

IMPLIED BY THE DEVELOPERS OR THE DISTRIBUTORS ON

THE ACCURACY OR THE RELIABILITY OF THE PROGRAM.

THIS PROGRAM IS A VERY PRACTICAL TOOL FOR THE DE-

SIGN OF REINFORCED CONCRETE STRUCTURES. HOWEVER,

THE USER MUST THOROUGHLY READ THE MANUAL AND

CLEARLY RECOGNIZE THE ASPECTS OF REINFORCED CON-

CRETE DESIGN THAT THE PROGRAM ALGORITHMS DO NOT

ADDRESS.

THE USER MUST EXPLICITLY UNDERSTAND THE ASSUMP-

TIONS OF THE PROGRAM AND MUST INDEPENDENTLY VER-

IFY THE RESULTS.


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Table of Contents

CHAPTER I Introduction 1O v e r v i e w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . 3

CHAPTER II Design Algorithms 5

Design Load Combinations . . . . . . . . . . . . . . . . . . . . . . . . 6

Design and Check Stations . . . . . . . . . . . . . . . . . . . . . . . . 7

Identifying Beams and Columns . . . . . . . . . . . . . . . . . . . . . 8

Design of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Design of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Design of Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Beam/Column Flexural Capacity Ratios . . . . . . . . . . . . . . . . 18

P- Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Element Unsupported Lengths . . . . . . . . . . . . . . . . . . . . . 20

Special Considerations for Seismic Loads . . . . . . . . . . . . . . . 21

Choice of Input Units . . . . . . . . . . . . . . . . . . . . . . . . . . 22

CHAPTER III Design for ACI 318-99 23

Design Load Combinations . . . . . . . . . . . . . . . . . . . . . . . 23

Strength Reduction Factors . . . . . . . . . . . . . . . . . . . . . . . 26

Column Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Generation of Biaxial Interaction Surfaces . . . . . . . . . . . . . 27

Check Column Capacity . . . . . . . . . . . . . . . . . . . . . . 29Determine Factored Moments and Forces. . . . . . . . . . . 29Determine Moment Magnification Factors . . . . . . . . . . 29

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Determine Capacity Ratio . . . . . . . . . . . . . . . . . . . 31

Design Column Shear Reinforcement . . . . . . . . . . . . . . . 32Determine Section Forces . . . . . . . . . . . . . . . . . . . 33Determine Concrete Shear Capacity . . . . . . . . . . . . . 34Determine Required Shear Reinforcement . . . . . . . . . . 36

Beam Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Design Beam Flexural Reinforcement . . . . . . . . . . . . . . . 37

Determine Factored Moments . . . . . . . . . . . . . . . . . 37Determine Required Flexural Reinforcement . . . . . . . . . 37

Design Beam Shear Reinforcement. . . . . . . . . . . . . . . . . 44Determine Shear Force and Moment . . . . . . . . . . . . . 44Determine Concrete Shear Capacity . . . . . . . . . . . . . 46Determine Required Shear Reinforcement . . . . . . . . . . 46

Design of Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Determine the Panel Zone Shear Force. . . . . . . . . . . . . . . 47

Determine the Effective Area of Joint . . . . . . . . . . . . . . . 48

Determine the Effective Area of Joint . . . . . . . . . . . . . . . 49Check Panel Zone Shear Stress. . . . . . . . . . . . . . . . . . . 49


CHAPTER IV Design for UBC 97 53



Column Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58


Check Column Capacity . . . . . . . . . . . . . . . . . . . . . . 60Determine Factored Moments and Forces. . . . . . . . . . . 60Determine Moment Magnification Factors . . . . . . . . . . 60Determine Capacity Ratio . . . . . . . . . . . . . . . . . . . 62


Beam Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Design Beam Flexural Reinforcement . . . . . . . . . . . . . . . 68Determine Factored Moments . . . . . . . . . . . . . . . . . 68

Determine Required Flexural Reinforcement . . . . . . . . . 69Design Beam Shear Reinforcement. . . . . . . . . . . . . . . . . 75

Determine Shear Force and Moment . . . . . . . . . . . . . 76Determine Concrete Shear Capacity . . . . . . . . . . . . . 77Determine Required Shear Reinforcement . . . . . . . . . . 78

Design of Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Determine the Panel Zone Shear Force. . . . . . . . . . . . . . . 79

Determine the Effective Area of Joint . . . . . . . . . . . . . . . 80

Check Panel Zone Shear Stress. . . . . . . . . . . . . . . . . . . 80

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ETABS Concrete Design Manual


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CHAPTER V Design for CSA-A23.3-94 85



Column Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89


Check Column Capacity . . . . . . . . . . . . . . . . . . . . . . 91Determine Factored Moments and Forces. . . . . . . . . . . 91Determine Moment Magnification Factors . . . . . . . . . . 91Determine Capacity Ratio . . . . . . . . . . . . . . . . . . . 94


Beam Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Design Beam Flexural Reinforcement . . . . . . . . . . . . . . 101

Determine Factored Moments . . . . . . . . . . . . . . . . 101Determine Required Flexural Reinforcement . . . . . . . . 102

Design Beam Shear Reinforcement . . . . . . . . . . . . . . . . 109Determine Shear Force and Moment. . . . . . . . . . . . . 110Determine Concrete Shear Capacity . . . . . . . . . . . . . 111Determine Required Shear Reinforcement. . . . . . . . . . 112

CHAPTER VI Design for BS 8110-85 R1989 115

Design Load Combinations . . . . . . . . . . . . . . . . . . . . . . 115

Design Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 18

Column Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 18

Generation of Biaxial Interaction Surfaces . . . . . . . . . . . . 119

Check Column Capacity. . . . . . . . . . . . . . . . . . . . . . 120Determine Factored Moments and Forces . . . . . . . . . . 121Determine Additional Moments . . . . . . . . . . . . . . . 121Determine Capacity Ratio . . . . . . . . . . . . . . . . . . 123

Design Column Shear Reinforcement. . . . . . . . . . . . . . . 124

Beam Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Design Beam Flexural Reinforcement . . . . . . . . . . . . . . 125Determine Factored Moments . . . . . . . . . . . . . . . . 126Determine Required Flexural Reinforcement . . . . . . . . 126

Design Beam Shear Reinforcement . . . . . . . . . . . . . . . . 131

CHAPTER VII Design for Eurocode 2 133


Design Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 36

Column Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 37

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Generation of Biaxial Interaction Surfaces . . . . . . . . . . . . 137

Check Column Capacity. . . . . . . . . . . . . . . . . . . . . . 139Determine Factored Moments and Forces . . . . . . . . . . 139Determine Code Total Moments . . . . . . . . . . . . . . 139Determine Capacity Ratio . . . . . . . . . . . . . . . . . . 141

Design Column Shear Reinforcement. . . . . . . . . . . . . . . 142

Beam Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 46


Design Beam Shear Reinforcement . . . . . . . . . . . . . . . . 153

CHAPTER VIII Design for NZS 3101-95 157


Strength Reduction Factors. . . . . . . . . . . . . . . . . . . . . . . 160

Column Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 61Generation of Biaxial Interaction Surfaces . . . . . . . . . . . . 161

Check Column Capacity. . . . . . . . . . . . . . . . . . . . . . 163Determine Factored Moments and Forces . . . . . . . . . . 163Determine Moment Magnification Factors . . . . . . . . . 164Dynamic Moment Magnification . . . . . . . . . . . . . . 166Determine Capacity Ratio . . . . . . . . . . . . . . . . . . 167

Design Column Shear Reinforcement. . . . . . . . . . . . . . . 167Determine Section Forces . . . . . . . . . . . . . . . . . . 168Determine Concrete Shear Capacity . . . . . . . . . . . . . 169Determine Required Shear Reinforcement. . . . . . . . . . 171

Beam Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 74


Design Beam Shear Reinforcement . . . . . . . . . . . . . . . . 182Determine Shear Force and Moment. . . . . . . . . . . . . 182Determine Concrete Shear Capacity . . . . . . . . . . . . . 183Determine Required Shear Reinforcement. . . . . . . . . . 184

CHAPTER IX Design Output 189

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189Graphical Display of Design Input and Output . . . . . . . . . . . . 190

Tabular Display of Design Input and Output . . . . . . . . . . . . . 191

Member Specific Information . . . . . . . . . . . . . . . . . . . . . 194

References 197

Index 199

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C h a p t e r I

Introduction

Overview

ETABS features powerful and completely integrated modules for design of both

steel and reinforced concrete structures (CSI 1999, 2000). The program provides

the user with options to create, modify, analyze and design structural models, all

from within the same user interface.

The program provides an interactive environment in which the user can study the

stress conditions, make appropriate changes, such as revising member properties,

and re-examine the results without the need to re-run the analysis. A single mouse

click on an element brings up detailed design information. Members can be

grouped together for design purposes. The output in both graphical and tabulated

formats can be readily printed.

The program is structured to support a wide variety of the latest national and inter-

national building design codes for the automated design and check of concrete and

steel frame members. The program currently supports the following concrete frame

design codes:

U.S. ACI (ACI 1999),

U.S. UBC (UBC 1997),

Canadian (CSA 1994),

Overview 1


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British (BSI 1989),

European (CEN 1992), and

New Zealand (NZS 3101-95).

The design is based upon a set of user-specified loading combinations. However,

the program provides a set of default load combinations for each design code sup-

ported in ETABS. If the default load combinations are acceptable, no definition of

additional load combinations are required.

In the design of the columns, the program calculates the required longitudinal and

shear reinforcement. However the user may specify the longitudinal steel, in which

case a column capacity ratio is reported. The column capacity ratio gives an indica-

tion of the stress condition with respect to the capacity of the column.

Every beam member is designed for flexure and shear at a user defined number of

stations along the beam span.

The presentation of the output is clear and concise. The information is in a form that

allows the engineer to take appropriate remedial measures in the event of member

overstress. Backup design information produced by the program is also provided

for convenient verification of the results.

English as well as SI and MKS metric units can be used to define the model geome-

try and to specify design parameters.

Organization

This manual is organized in the following way:

Chapter II outlines various aspects of the concrete design procedures of the ETABS

program. This chapter describes the common terminology of concrete design as im-

plemented in ETABS.

Each of six subsequent chapters gives a detailed description of a specific code of

practice as interpreted by and implemented in ETABS. Each chapter describes the

design loading combination, column and beam design procedures, and other spe-

cial consideration required by the code. In addition Chapter IV describes the joint

design according to the UBC code.

Chapter III gives a detailed description of the ACI code (ACI 1999) as imple-

mented in ETABS.

2 Organization



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Chapter IV gives a detailed description of the UBC concrete code (UBC 1997)

as implemented in ETABS.

Chapter V gives a detailed description of the Canadian code (CSA 1994) as im-

plemented in ETABS.

Chapter VI gives a detailed description of the British code (BSI 1989) as imple-mented in ETABS.

Chapter VII gives a detailed description of the Eurocode 2 (CEN 1992) as im-

plemented in ETABS.

Chapter VIII gives a detailed description of the New Zealand code (NZS 1997)

as implemented in ETABS.

Chapter IX outlines various aspects of the tabular and graphical output from

ETABS related to concrete design.

Recommended Reading

It is recommended that the user read Chapter II Design Algorithms and one of six

subsequent chapters corresponding to the code of interest to the user. Finally the

user should read Design Output in Chapter IX for understanding and interpreting

ETABS output related to concrete design. If the users interest is in the UBC con-

crete design code, it is recommended that the user should also read the chapter re-

lated to ACI code.

Recommended Reading 3

Chapter I Introduction


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C h a p t e r II

Design Algorithms

This chapter outlines various aspects of the concrete design and design-check pro-

cedures that are used by the ETABS program. The concrete design and check may

be performed in ETABS according to one of the following design codes:

The 1995 American Concrete Institute Building Code Requirements for Struc-

tural Concrete,ACI 318-99(ACI 1999).

International Conference of Building Officials1997 Uniform Building Code:

Volume 2: Structural Engineering Design Provisions,Chapter 19 Concrete,

UBC 1997(ICBO 1997).

The 1994 Canadian Standards Association Design of Concrete Structures for

Buildings, CSA-A23.3-94(CSA 1994).

The 1989 British Standards Institution Structural Use of Concrete,BS 8110-85

R1989(BSI 1989).

The 1992 European Committee for Standardization, Design of Concrete Struc-tures,EUROCODE 2(CEN 1992).

The 1995 Standards New Zealand Concrete Structures Standard, NZS 3101-95

(NZS 1995).

Details of the algorithms associated with each of these codes as implemented in

ETABS are described in the subsequent chapters. However, this chapter provides a

background which is common to all the design codes.

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For referring to pertinent sections of the corresponding code, a unique prefix is as-

signed for each code.

References to the ACI 318-99 code has the prefix of ACI

References to the UBC 1997 code has the prefix of UBC

References to the Canadian code carry the prefix of CSA

References to the British code carry the prefix of BS

References to the Eurocode 2 carry the prefix of EC2

References to the New Zealand code carry the prefix of NZS

In writing this manual it has been assumed that the user has an engineering back-

ground in the general area of structural reinforced concrete design and familiarity

with at least one of the above mentioned design codes.

Design Load Combinations

The design load combinations are used for determining the various combinations of

the load cases for which the structure needs to be designed/checked. The load com-

bination factors to be used vary with the selected design code. The load combina-

tion factors are applied to the forces and moments obtained from the associated load

cases and the results are then summed to obtain the factored design forces and mo-

ments for the load combination.

For multi-valued load combinations involving response spectrum, time history, and

multi-valued combinations (of type enveloping, square-root of the sum of the

squares or absolute) where any correspondence between interacting quantities is

lost, the program automatically produces multiple sub combinations using max-

ima/minima permutations of interacting quantities. Separate combinations with

negative factors for response spectrum cases are not required because the program

automatically takes the minima to be the negative of the maxima for response spec-

trum cases and the above described permutations generate the required sub combi-

nations.

When a design combination involves only a single multi-valued case of time his-

tory or moving load, further options are available. The program has an option to re-

quest that time history combinations produce sub combinations for each time step

of the time history.

For normal loading conditions involving static dead load, live load, wind load, and

earthquake load, and/or dynamic response spectrum earthquake load, the program

has built-in default loading combinations for each design code. These are based on

6 Design Load Combinations



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the code recommendations and are documented for each code in the corresponding

chapters.

For other loading conditions involving time history, pattern live loads, separate

consideration of roof live load, snow load, etc., the user must define design loading

combinations either in lieu of or in addition to the default design loading combina-tions.

The default load combinations assume all static load cases declared as dead load to

be additive. Similarly, all cases declared as live load are assumed additive. How-

ever, each static load case declared as wind or earthquake, or response spectrum

cases, is assumed to be non additive with each other and produces multiple lateral

load combinations. Also wind and static earthquake cases produce separate loading

combinations with the sense (positive or negative) reversed. If these conditions are

not correct, the user must provide the appropriate design combinations.

The default load combinations are included in design if the user requests them to be

included or if no other user defined combination is available for concrete design. If

any default combination is included in design, then all default combinations will

automatically be updated by the program any time the user changes to a different

design code or if static or response spectrum load cases are modified.

Live load reduction factors can be applied to the member forces of the live load case

on an element-by-element basis to reduce the contribution of the live load to the

factored loading.

The user is cautioned that if time history results are not requested to be recovered in

the analysis for some or all the frame members, then the effects of these loads will

be assumed to be zero in any combination that includes them.

Design and Check Stations

For each load combination, each beam, column, or brace element is designed or

checked at a number of locations along the length of the element. The locations are

based on equally spaced segments along the clear length of the element. By default

there will be at least 3 stations in a column or brace element and the stations in a

beam will be at most 2 feet (0.5m if model is created in SI unit) apart. The number

of segments in an element can be overwritten by the user before the analysis is

made. The user can refine the design along the length of an element by requesting

more segments. See the section Frame Output Stations Assigned to Line Objects

in theETABS Users Manual Volume 1 (CSI 1999) for details.

Design and Check Stations 7

Chapter II Design Algorithms


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When using 1997 UBC design codes, requirements for joint design at the beam to

column connections are evaluated at the topmost station of each column. The pro-

gram also performs a joint shear analysis at the same station to determine if special

considerations are required in any of the joint panel zones. The ratio of the beam

flexural capacities with respect to the column flexural capacities considering axial

force effect associated with the weak beam-strong column aspect of any beam/col-umn intersection are reported.

Identifying Beams and Columns

In ETABS all beams and columns are represented as frame elements. But design of

beams and columns requires separate treatment. Identification for a concrete ele-

ment is done by specifying the frame section assigned to the element to be of type

beam or column. If there is any brace element in the frame, the brace element would

also be identified as either a beam or a column element based on the assigned sec-tion to the brace element.

Design of Beams

In the design of concrete beams, in general, ETABS calculates and reports the re-

quired areas of steel for flexure and shear based upon the beam moments, shears,

load combination factors, and other criteria which are described in detail in the code

specific chapters. The reinforcement requirements are calculated at a user-defined

number of stations along the beam span.

All the beams are only designed for major direction flexure and shear. Effects due

to any axial forces, minor direction bending, and torsion that may exist in the beams

must be investigated independently by the user.

In designing the flexural reinforcement for the major moment at a particular section

of a particular beam, the steps involve the determination of the maximum factored

moments and the determination of the reinforcing steel. The beam section is de-

signed for the maximum positiveMu+ and maximum negativeMu

- factored moment

envelopes obtained from all of the load combinations. Negative beam momentsproduce top steel. In such cases the beam is always designed as a rectangular sec-

tion. Positive beam moments produce bottom steel. In such cases the beam may be

designed as a rectangular- or a T-beam. For the design of flexural reinforcement,

the beam is first designed as a singly reinforced beam. If the beam section is not

adequate, then the required compression reinforcement is calculated.

8 Identifying Beams and Columns



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In designing the shear reinforcement for a particular beam for a particular set of

loading combinations at a particular station due to the beam major shear, the steps

involve the determination of the factored shear force, the determination of the shear

force that can be resisted by concrete, and the determination of the reinforcement

steel required to carry the balance.

Special considerations for seismic design are incorporated in ETABS for ACI,

UBC, Canadian, and New Zealand codes.

Design of Columns

In the design of the columns, the program calculates the required longitudinal steel,

or if the longitudinal steel is specified, the column stress condition is reported in

terms of a column capacity ratio, which is a factor that gives an indication of the

stress condition of the column with respect to the capacity of the column. The de-sign procedure for the reinforced concrete columns of the structure involves the fol-

lowing steps:

Generate axial force-biaxial moment interaction surfaces for all of the different

concrete section types of the model. A typical interaction surface is shown in

Figure II-2.

Check the capacity of each column for the factored axial force and bending mo-

ments obtained from each loading combination at each end of the column. This

step is also used to calculate the required reinforcement (if none was specified)

that will produce a capacity ratio of 1.0.

Design the column shear reinforcement.

The generation of the interaction surface is based on the assumed strain and stress

distributions and some other simplifying assumptions. These stress and strain dis-

tributions and the assumptions vary from code to code. A typical assumed strain

distribution is described inFigure II-1.

Here maximum compression strain is limited to c . For most of the design codes,

this assumed distribution remains valid. However, the value of c varies from code

to code. For example, c 0.003 for ACI, UBC and New Zealand codes, and

c 0.0035for Canadian, British and European codes. The details of the generation

of interaction surfaces differ from code to code. These are described in the chapters

specific to the code.

Design of Columns 9



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A typical interaction surface is shown inFigure II-2. The column capacity interac-

tion volume is numerically described by a series of discrete points that are gener-

ated on the three-dimensional interaction failure surface. The coordinates of these

points are determined by rotating a plane of linear strain in three dimensions on the

section of the column as described inFigure II-1.

The area associated with each rebar is placed at the actual location of the center of

the bar and the algorithm does not assume any simplifications in the manner in

which the area of steel is distributed over the cross section of the column. The inter-

action algorithm provides corrections to account for the concrete area that is dis-

placed by the reinforcing in the compression zone.

10 Design of Columns


DIRECTION 1

DIRECTION 2

3 2

1a a

DIRECTION 3

+

+

+

Neutral AxisDirection



c

c

c

c

c

c

0

0

0

ReinforcementBars

Reinforcement

Bars

Reinforcement

Bars

Varying LinearStrain Plane



Figure II-1

Idealized Strain Distribution for Generation of Interaction Surfaces


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The effects of code specified strength reduction factors and maximum limit on the

axial capacity are incorporated in the interaction surfaces. The formulation is based

consistently upon the general principles of ultimate strength design, and allows for

rectangular, square or circular, doubly symmetric column sections. In addition to

axial compression and biaxial bending, the formulation allows for axial tension andbiaxial bending considerations as shown inFigure II-2.

Design of Columns 11


M x

M y

Axial tension

C u r v e # 1

Axial compression

M by

C ur ve #N R C VPbx

Pma x

Mbx

C ur ve #2

12

3

Pby

-P0

+P0

Figure II-2

A Typical Column Interaction Surface


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The capacity check is based on whether the design load points lie inside the interac-

tion volume in a force space, as shown in Figure II-3. If the point lies inside the vol-

ume, the column capacity is adequate, and vice versa. The point in the interaction

volume (P, Mx , and My ) which is represented by point L is placed in the interac-

tion space as shown inFigure II-3. If the point lies within the interaction volume,

the column capacity is adequate; however, if the point lies outside the interaction

volume, the column is overstressed. As a measure of the stress condition of the col-

umn, a capacity ratio is calculated. This ratio is achieved by plotting the point L, de-

fined by P, Mxand My, and determining the location of point C. The point C is de-

fined as the point where the line OL (if extended outwards) will intersect the failure

surface. This point is determined by three-dimensional linear interpolation between

the points that define the failure surface. The capacity ratio, CR, is given by the ra-

tio OL OC .

12 Design of Columns


Axial Compression

Axial Tension

MX MY

My

MxP

oL

C

Lines DefiningFailure Surface

Figure II-3Geometric Representation of Column Capacity Ratio


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If OL = OC (or CR=1) the point lies on the interaction surface and the column is

stressed to capacity.

If OL < OC (or CR OC (or CR>1) the point lies outside the interaction volume and the col-

umn is overstressed.

The capacity ratio is basically a factor that gives an indication of the stress condi-

tion of the column with respect to the capacity of the column. In other words, if the

axial force and biaxial moment set for which the column is being checked is ampli-

fied by dividing it by the reported capacity ratio, the point defined by the resulting

axial force and biaxial moment set will lie on the failure (or interaction volume) sur-

face.

Design of Columns 13


Figure II-4Moment CapacityMu at a Given Axial LoadPu


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The shear reinforcement design procedure for columns is very similar to that for

beams, except that the effect of the axial force on the concrete shear capacity needs

to be considered.

For certain special seismic cases, the design of column for shear is based on the ca-

pacity-shear. The capacity-shear force in a particular direction is calculated fromthe moment capacities of the column associated with the factored axial force acting

on the column. For each load combination, the factored axial load is calculated, us-

ing the ETABS analysis load cases and the corresponding load combination factors.

Then, the moment capacity of the column in a particular direction under the influ-

ence of the axial force is calculated, using the uniaxial interaction diagram in the

corresponding direction as shown inFigure II-4.

Design of Joints

To ensure that the beam-column joint of special moment resisting frames possesses

adequate shear strength, the program performs a rational analysis of the beam-

column panel zone to determine the shear forces that are generated in the joint. The

program then checks this against design shear strength.

Only joints having a column below the joint are designed. The material properties

of the joint are assumed to be the same as those of the column below the joint.

The joint analysis is done in the major and the minor directions of the column. The

joint design procedure involves the following steps:

Determine the panel zone design shear force,Vuh

Determine the effective area of the joint

Check panel zone shear stress

The following three sections describe in detail the algorithms associated with the

above mentioned steps.

Determine the Panel Zone Shear Force

For a particular column direction, major or minor, the free body stress condition of

a typical beam-column intersection is shown inFigure II-5.

The force Vuh is the horizontal panel zone shear force that is to be calculated. The

forces that act on the joint are Pu ,Vu ,MuL

and MuR

. The forces Pu andVu are axial

force and shear force, respectively, from the column framing into the top of the

joint. The momentsMuL and Mu

R are obtained from the beams framing into the

14 Design of Joints



22/207

joint. The joint shear force Vuh is calculated by resolving the moments into Cand T

forces. Noting that T CL L

and T CR R

,

V = T + T - V uh

L R u

The location ofCor Tforces is determined by the direction of the moment. The

magnitude of Cor Tforces is conservatively determined using basic principles of

ultimate strength theory ignoring compression reinforcement as follows. The maxi-

Design of Joints 15


Figure II-5

Beam-Column Joint Analysis


23/207

mum compression, Cmax , and the maximum moment, Mmax , that can be carried by

the beam is calculated first.

C =max 0.85f bdc

M = Cmax maxd

2

Then the Cand Tforces are conservatively determined as follows:

C = T = C maxmax

1 1

abs M

M

The moments and the Cand Tforces from beams that frame into the joint in a direc-

tion that is not parallel to the major or minor directions of the column are resolved

along the direction that is being investigated, thereby contributing force compo-nents to the analysis. Also Cand Tare calculated for the positive and negative mo-

ments considering the fact that the concrete cover may be different for the direction

of moment.

In the design of special moment resisting concrete frames, the evaluation of the de-

sign shear force is based upon the moment capacities (with reinforcing steel

overstrength factor, , and no factors) of the beams framing into the joint,

(ACI 21.5.1.1, UBC 1921.5.1.1). The Cand Tforce are based upon these moment

capacities. The column shear forceVu is calculated from the beam moment capaci-

ties as follows:

V = M + M

Hu

u

L

u

R

See Figure II-6. It should be noted that the points of inflection shown on Figure II-6

are taken as midway between actual lateral support points for the columns. If there

is no column at the top of the joint, the shear force from the top of the column is

taken as zero.

The effects of load reversals, as illustrated in Case 1 and Case 2 ofFigure II-5areinvestigated and the design is based upon the maximum of the joint shears obtained

from the two cases.

Determine the Effective Area of Joint

The joint area that resists the shear forces is assumed always to be rectangular in

plan view. The dimensions of the rectangle correspond to the major and minor di-

mensions of the column below the joint, except if the beam framing into the joint is

16 Design of Joints



24/207

very narrow. The effective width of the joint area to be used in the calculation is limited to the

width of the beam plus the depth of the column. The area of the joint is assumed not to exceed

the area of the column below. The joint area for joint shear along the major and minor directions

is calculated separately (ACI R21.5.3).

It should be noted that if the beam frames into the joint eccentrically, the above assumptionsmay be unconservative and the user should investigate the acceptability of the particular joint.

Design of Joints 17


POINT OF

INFLECTION Vu

TOP OF BEAM

COLUMN

ABOVE

COLUMN

HEIGHT

(H) PANEL

ZONE

CR

CL RT

LT h

uVu

LM

u

RM

Vu

COLUMN

BELOW

ELEVATION

POINT OFINFLECTION

Figure II-6

Column Shear Force,Vu


25/207

Check Panel Zone Shear Stress

The panel zone shear stress is evaluated by dividing the shear forceVuh by the effec-

tive area of the joint and comparing it with the following design shear strengths

(ACI 21.5.3, UBC 1921.5.3) :

v

fc

20 for joints confined on all four sides,

15

,

fc , for joints confined on three faces or on two opposite faces,

12 for all other joints, fc

,

where = 0.85 (by default). (ACI 9.3.2.3, UBC 1909.3.2.3, 1909.3.4.1)

A beam that frames into a face of a column at the joint is considered in ETABS to

provide confinement to the joint if at least three-quarters of the face f the joint is

covered by the framing member (ACI 21.5.3.1, UBC 1921.5.3.1).

For light weight aggregate concrete, the design shear strength of the joint is reduced

in ETABS to at least three-quarters of that of the normal weight concrete by replac-

ing the fc with

min ,,

f f fcs factor c c

3 4 (ACI 21.5.3.2, UBC 1921.5.3.2)

For joint design, the program reports the joint shear, the joint shear stress, the al-

lowable joint shear stress and a capacity ratio.

Beam/Column Flexural Capacity Ratios

At a particular joint for a particular column direction, major or minor, the program

will calculate the ratio of the sum of the beam moment capacities to the sum of the

column moment capacities, (ACI 21.4.2.2, UBC 1921.4.2.2).

M Me g6

5(ACI 21.4.2.2, UBC 1921.4.2.2)

The capacities are calculated with no reinforcing overstrength factor, , and in-

cluding factors. The beam capacities are calculated for reversed situations (Cases

1 and 2) as illustrated in Figure II-5 and the maximum summation obtained is used.

The moment capacities of beams that frame into the joint in a direction that is not

parallel to the major or minor direction of the column are resolved along the direc-

18 Beam/Column Flexural Capacity Ratios



26/207

tion that is being investigated and the resolved components are added to the sum-

mation.

The column capacity summation includes the column above and the column below

the joint. For each load combination the axial force,Pu , in each of the columns is

calculated from the ETABS analysis load combinations. For each load combina-tion, the moment capacity of each column under the influence of the corresponding

axial load Pu is then determined separately for the major and minor directions of the

column, using the uniaxial column interaction diagram, seeFigure II-4. The mo-

ment capacities of the two columns are added to give the capacity summation for

the corresponding load combination. The maximum capacity summations obtained

from all of the load combinations is used for the beam/column capacity ratio.

The beam/column flexural capacity ratios are only reported for Special Mo-

ment-Resisting Frames involving seismic design load combinations.

P- Effects

The ETABS design algorithms require that the analysis results include the P- ef-

fects. The P- effects are considered differently for braced or nonsway and

unbraced or sway components of moments in frames. For the braced moments

in frames, the effect of P-

is limited to individual member stability. For un-

braced components, lateral drift effects should be considered in addition to indi-

vidual member stability effect. In ETABS, it is assumed that braced or

nonsway moments are contributed from the dead or live loads. Whereas,unbraced or sway moments are contributed from all other types of loads.

For the individual member stability effects, the moments are magnified with mo-

ment magnification factors as in the ACI, UBC, Canadian, and New Zealand codes

or with additional moments as in the British and European codes.

For lateral drift effects, ETABS assumes that the P- analysis is performed and

that the amplification is already included in the results. The moments and forces ob-

tained from P- analysis are further amplified for individual column stability effect

if required by the governing code as in the ACI, UBC, Canadian, and New Zealandcodes.

The users of ETABS should be aware that the default analysis option in ETABS for

P-

effect is turned OFF. The default number of iterations for P-

analysis is 1.

The user should turn the P- analysis ON and set the maximum number of itera-

tions for the analysis. For further reference, the user is referred toETABS Users

Manual Volume 2(CSI 1999).

P-

Effects 19



27/207

The user is also cautioned that ETABS currently considers P- effects due to axial

loads in frame members only. Forces in other types of elements do not contribute to

this effect. If significant forces are present in other types of elements, for example,

large axial loads in shear walls modeled as shell elements, then the additional forces

computed for P- will be inaccurate.

Element Unsupported Lengths

To account for column slenderness effects the column unsupported lengths are re-

quired. The two unsupported lengths arel33

andl22

. These are the lengths between

support points of the element in the corresponding directions. The lengthl33

corre-

sponds to instability about the 3-3 axis (major axis), and l22

corresponds to instabil-

ity about the 2-2 axis (minor axis).

Normally, the unsupported element length is equal to the length of the element, i.e.,the distance between END-I and END-J of the element. SeeFigure II-7. The pro-

gram, however, allows users to assign several elements to be treated as a single

member for design. This can be done differently for major and minor bending.

Therefore, extraneous joints, as shown inFigure II-8, that affect the unsupported

length of an element are automatically taken into consideration.

20 Element Unsupported Lengths


Figure II-7

Axes of Bending and Unsupported Length


28/207

In determining the values for l22

and l33

of the elements, the program recognizes

various aspects of the structure that have an effect on these lengths, such as member

connectivity, diaphragm constraints and support points. The program automati-

cally locates the element support points and evaluates the corresponding unsup-

ported element length.

Therefore, the unsupported length of a column may actually be evaluated as being

greater than the corresponding element length. If the beam frames into only one di-

rection of the column, the beam is assumed to give lateral support only in that direc-

tion.

The user has options to specify the unsupported lengths of the elements on an ele-

ment-by-element basis.

Special Considerations for Seismic LoadsThe ACI code imposes a special ductility requirement for frames in seismic regions

by specifying frames either as Ordinary, Intermediate, or Special moment resisting

frames. The Special moment resisting frame can provide the required ductility and

energy dissipation in the nonlinear range of cyclic deformation. The UBC code re-

quires that the concrete frame must be designed for a specific Seismic Zone which

is either Zone 0, Zone 1, Zone 2, Zone 3, or Zone 4, where Zone 4 is designated as

Special Considerations for Seismic Loads21


Figure II-8

Unsupported Lengths and Interior Nodes


29/207

the zone of severe earthquake. The Canadian code requires that the concrete frame

must be designed as either an Ordinary, Nominal, or Ductile moment resisting

frame. The New Zealand code also requires that the concrete frame must be de-

signed as either an Ordinary, Elastically responding, frames with Limited ductility,

or Ductile moment resisting frame.

Unlike the ACI, UBC, Canadian, and New Zealand codes, the current implementa-

tion of the British code and the Eurocode 2 in ETABS does not account for any spe-

cial requirements for seismic design.

Choice of Input Units

English as well as SI and MKS metric units can be used for input. But the codes are

based on a specific system of units. All equations and descriptions presented in the

subsequent chapters correspond to that specific system of units unless otherwisenoted. For example, the ACI code is published in inch-pound-second units. By de-

fault, all equations and descriptions presented in the chapter Design for ACI

318-99 correspond to inch-pound-second units. However, any system of units can

be used to define and design the structure in ETABS.

22 Choice of Input Units



30/207

C h a p t e r III

Design for ACI 318-99

This chapter describes in detail the various aspects of the concrete design procedure

that is used by ETABS when the user selects the ACI 318-99 Design Code (ACI

1999). Various notations used in this chapter are listed inTable III-1.

The design is based on user-specified loading combinations. But the program pro-

vides a set of default load combinations that should satisfy requirements for the de-

sign of most building type structures.

ETABS provides options to design or check Ordinary, Intermediate (moderate seis-

mic risk areas), and Special (high seismic risk areas) moment resisting frames as re-

quired for seismic design provisions. The details of the design criteria used for the

different framing systems are described in the following sections.

English as well as SI and MKS metric units can be used for input. But the code is

based on Inch-Pound-Second units. For simplicity, all equations and descriptions

presented in this chapter correspond to Inch-Pound-Second units unless otherwisenoted.

Design Load Combinations

The design load combinations are the various combinations of the prescribed load

cases for which the structure needs to be checked. For the ACI 318-99 code, if a

Design Load Combinations 23


31/20724 Design Load Combinations


Acv Area of concrete used to determine shear stress, sq-in

Ag Gross area of concrete, sq-in

As Area of tension reinforcement, sq-in

As

Area of compression reinforcement, sq-inA

s required( ) Area of steel required for tension reinforcement, sq-in

Ast Total area of column longitudinal reinforcement, sq-in

Av Area of shear reinforcement, sq-in

a Depth of compression block, in

ab

Depth of compression block at balanced condition, in

b Width of member, in

bf

Effective width of flange (T-Beam section), in

bw Width of web (T-Beam section), inCm Coefficient, dependent upon column curvature, used to calculate mo-

ment magnification factor

c Depth to neutral axis, in

cb

Depth to neutral axis at balanced conditions, in

d Distance from compression face to tension reinforcement, in

d

Concrete cover to center of reinforcing, in

ds Thickness of slab (T-Beam section), in

Ec Modulus of elasticity of concrete, psi

Es Modulus of elasticity of reinforcement, assumed as 29,000,000 psi

(ACI 8.5.2)

fc Specified compressive strength of concrete, psi

fy Specified yield strength of flexural reinforcement, psi

fy 80000, psi (ACI 9.4)

fys Specified yield strength of shear reinforcement, psi

h Dimension of column, in

Ig Moment of inertia of gross concrete section about centroidal axis,

neglecting reinforcement, in4

Ise Moment of inertia of reinforcement about centroidal axis of

member cross section, in4

Table III-1

List of Symbols Used in the ACI code


32/207Design Load Combinations 25

Chapter III Design for ACI 318-99

k Effective length factor

L Clear unsupported length, in

M1

Smaller factored end moment in a column, lb-in

M2 Larger factored end moment in a column, lb-inMc Factored moment to be used in design, lb-in

Mns Nonsway component of factored end moment, lb-in

Ms Sway component of factored end moment, lb-in

Mu Factored moment at section, lb-in

Mux Factored moment at section about X-axis, lb-in

Muy Factored moment at section about Y-axis, lb-in

Pb

Axial load capacity at balanced strain conditions, lb

Pc Critical buckling strength of column, lbPmax Maximum axial load strength allowed, lb

P0

Axial load capacity at zero eccentricity, lb

Pu Factored axial load at section, lb

r Radius of gyration of column section, in

Vc Shear resisted by concrete, lb

VE

Shear force caused by earthquake loads, lb

VD L

Shear force from span loading, lb

Vu

Factored shear force at a section, lb

Vp Shear force computed from probable moment capacity, lb

Reinforcing steel overstrength factor

1 Factor for obtaining depth of compression block in concrete

d Absolute value of ratio of maximum factored axial dead load to maxi-

mum factored axial total load

s Moment magnification factor for sway moments

ns Moment magnification factor for nonsway moments

c Strain in concrete

s Strain in reinforcing steel

Strength reduction factor

Table III-1

List of Symbols Used in the ACI code (continued)


33/207

structure is subjected to dead load (DL) and live load (LL) only, the stress check

may need only one load combination, namely 1.4 DL + 1.7 LL (ACI 9.2.1). How-

ever, in addition to the dead and live loads, if the structure is subjected to wind

(WL) and earthquake (EL) loads, and considering that wind and earthquake forces

are reversible, then the following load combinations have to be considered (ACI

9.2).

1.4 DL

1.4 DL + 1.7 LL (ACI 9.2.1)

0.9 DL 1.3 WL

0.75 (1.4 DL + 1.7 LL 1.7 WL) (ACI 9.2.2)

0.9 DL 1.3 * 1.1 EL

0.75 (1.4 DL + 1.7 LL 1.7 * 1.1 EL) (ACI 9.2.3)

These are also the default design load combinations in ETABS whenever the ACI

318-99 code is used. The user is warned that the above load combinations involving

seismic loads consider service-level seismic forces. Different load factors may ap-

ply with strength-level seismic forces (ACI R9.2.3).

Live load reduction factors can be applied to the member forces of the live load

condition on an element-by-element basis to reduce the contribution of the live load

to the factored loading.

Strength Reduction Factors

The strength reduction factors, , are applied on the nominal strength to obtain the

design strength provided by a member. The factors for flexure, axial force, shear,

and torsion are as follows:

= 0.90 for flexure, (ACI 9.3.2.1)

= 0.90 for axial tension, (ACI 9.3.2.2)

= 0.90 for axial tension and flexure, (ACI 9.3.2.2)

for axial compression, and axial compression

and flexure (spirally reinforced column), (ACI 9.3.2.2)

for axial compression, and axial compression

and flexure (tied column), and (ACI 9.3.2.2)

= 0.85 for shear and torsion. (ACI 9.3.2.3)

26 Strength Reduction Factors



34/207

Column Design

The user may define the geometry of the reinforcing bar configuration of each con-

crete column section. If the area of reinforcing is provided by the user, the program

checks the column capacity. However, if the area of reinforcing is not provided by

the user, the program calculates the amount of reinforcing required for the column.

The design procedure for the reinforced concrete columns of the structure involves

the following steps:

Generate axial force/biaxial moment interaction surfaces for all of the different

concrete section types of the model. A typical biaxial interaction surface is

shown inFigure II-2. When the steel is undefined, the program generates the

interaction surfaces for the range of allowable reinforcement 1 to 8 percent

for Ordinary and Intermediate moment resisting frames (ACI 10.9.1) and 1 to 6

percent for Special moment resisting frames (ACI 21.4.3.1). Calculate the capacity ratio or the required reinforcing area for the factored ax-

ial force and biaxial (or uniaxial) bending moments obtained from each loading

combination at each station of the column. The target capacity ratio is taken as

one when calculating the required reinforcing area.

Design the column shear reinforcement.

The following three subsections describe in detail the algorithms associated with

the above-mentioned steps.

Generation of Biaxial Interaction Surfaces

The column capacity interaction volume is numerically described by a series of dis-

crete points that are generated on the three-dimensional interaction failure surface.

In addition to axial compression and biaxial bending, the formulation allows for ax-

ial tension and biaxial bending considerations. A typical interaction diagram is

shown inFigure II-2.

The coordinates of these points are determined by rotating a plane of linear strain in

three dimensions on the section of the column. See Figure II-1.The linear straindiagram limits the maximum concrete strain, c , at the extremity of the section

to 0.003 (ACI 10.2.3).

The formulation is based consistently upon the general principles of ultimate

strength design (ACI 10.3), and allows for any doubly symmetric rectangular,

square, or circular column section.

Column Design 27



35/207

The stress in the steel is given by the product of the steel strain and the steel modu-

lus of elasticity, s sE , and is limited to the yield stress of the steel, fy (ACI 10.2.4).

The area associated with each reinforcing bar is assumed to be placed at the actual

location of the center of the bar and the algorithm does not assume any further sim-

plifications in the manner in which the area of steel is distributed over the cross sec-

tion of the column, such as an equivalent steel tube or cylinder. SeeFigure III-1.

The concrete compression stress block is assumed to be rectangular, with a stress

value of 0.85fc (ACI 10.2.7.1). SeeFigure III-1. The interaction algorithm pro-

vides correction to account for the concrete area that is displaced by the reinforce-

ment in the compression zone.

The effects of the strength reduction factor, , are included in the generation of the

interaction surfaces. The maximum compressive axial load is limited to Pn(max)

,

where

P = f A - A + f Ac g st y st n(max) 0.85 [ 0.85 ( ) ]

spiral column, (ACI 10.3.5.1)

P = f A - A f Ac g st y st n(max) 0.80 [ 0.85 ( ) + ]

tied column, (ACI 10.3.5.2)

= 0.70 for tied columns, and (ACI 9.3.2.2)

= 0.75 for spirally reinforced columns. (ACI 9.3.2.2)

28 Column Design


c

d'

C a = c1

2sC

1sC

0.85 f'cc = 0.003

s4

s3

s2

s1

Ts4

Ts3

(i) Concrete Section (ii) Strain Diagram (iii) Stress Diagram

Figure III-1

Idealization of Stress and Strain Distribution in a Column Section


36/207

The value of used in the interaction diagram varies from

(compression) to

(flexure) based on the axial load. For low values of axial load, is increased lin-

early from (compression) to (flexure) as the Pn decreases from the smaller of

Pb

or 0.1f Ac g to zero, where P

b is the axial force at the balanced condition. The

factor used in calculating Pn and Pb is the (compression). In cases involving

axial tension, is always (flexure) which is 0.9 by default (ACI 9.3.2.2).

Check Column Capacity

The column capacity is checked for each loading combination at each check station

of each column. In checking a particular column for a particular loading combina-

tion at a particular station, the following steps are involved:

Determine the factored moments and forces from the analysis load cases and

the specified load combination factors to giveP M Mu ux uy, ,and . Determine the moment magnification factors for the column moments.

Apply the moment magnification factors to the factored moments. Determine

whether the point, defined by the resulting axial load and biaxial moment set,

lies within the interaction volume.

The factored moments and corresponding magnification factors depend on the

identification of the individual column as either sway or non-sway.

The following three sections describe in detail the algorithms associated with the

above-mentioned steps.

Determine Factored Moments and Forces

The factored loads for a particular load combination are obtained by applying the

corresponding load factors to all the load cases, givingP M Mu ux uy, ,and . The fac-

tored moments are further increased for non-sway columns, if required, to obtain

minimum eccentricities of (0.6 0.03 h) inches, where h is the dimension of the

column in the corresponding direction (ACI 10.12.3.2).

Determine Moment Magnification Factors

The moment magnification factors are calculated separately for sway (overall sta-

bility effect), s and for non-sway (individual column stability effect), ns . Also the

moment magnification factors in the major and minor directions are in general dif-

ferent (ACI 10.0, R10.13).

Column Design 29



37/207

The moment obtained from analysis is separated into two components: the sway

( )Ms and the non-sway (Mns ) components. The non-sway components which are

identified by ns subscripts are predominantly caused by gravity load. The sway

components are identified by the s subscripts. The sway moments are predomi-

nantly caused by lateral loads, and are related to the cause of side sway.

For individual columns or column-members in a floor, the magnified moments

about two axes at any station of a column can be obtained as

M M Mns s s . (ACI 10.13.3)

The factor s is the moment magnification factor for moments causing side sway.

The moment magnification factors for sway moments, s , is taken as 1 because the

component moments Ms and Mns are obtained from a second order elastic (P- )

analysis (ACI R10.10, 10.10.1, R10.13, 10.13.4.1).

The program assumes that a P- analysis has been performed in ETABS and, there-

fore, moment magnification factor s for moments causing sidesway is taken as

unity (ACI 10.10.2). For the P- analysis the load should correspond to a load com-

bination of 1.4 dead load + 1.7 live load (ACI 10.13.6). See also White and Hajjar

(1991). The user should use reduction factors for the moment of inertias in ETABS

as specified in ACI 10.11. The moment of intertia reduction for sustained lateral

load involves a factor d

(ACI 10.11). This d

for sway frame in second-order anal-

ysis is different from the one that is defined later for non-sway moment magnifica-

tion (ACI 10.0, R10.12.3, R10.13.4.1). The default moment of inertia factor in

ETABS is 1.

The computed moments are further amplified for individual column stability effect

(ACI 10.12.3, 10.13.5) by the nonsway moment magnification factor, ns , as fol-

lows:

M Mc ns , where (ACI 10.12.3)

Mc is the factored moment to be used in design.

The non-sway moment magnification factor, ns , associated with the major or mi-

nor direction of the column is given by (ACI 10.12.3)

nsm

u

c

C

P

P

=

0.75

1.0

1

, where (ACI 10.12.3)

C = + M

Mm

a

b

0.6 0.4 0.4 , (ACI 10.12.3.1)

30 Column Design



38/207

Ma andMb are the moments at the ends of the column, andMb is numerically

larger thanMa .M Ma b is positive for single curvature bending and negative

for double curvature bending. The above expression ofCm is valid if there is no

transverse load applied between the supports. If transverse load is present on

the span, or the length is overwritten,Cm 1. Cm can be overwritten by the user

on an element by element basis.

P = EI

klc

u

2

2( )

, where (ACI 10.12.3)

kis conservatively taken as 1, however ETABS allows the user to over-

ride this value (ACI 10.12.1),

lu is the unsupported length of the column for the direction of bending

considered. The two unsupported lengths arel22

andl33

corresponding to

instability in the minor and major directions of the element, respectively.

SeeFigure II-7.These are the lengths between the support points of the

element in the corresponding directions.

EI is associated with a particular column direction:

EI =E I

+

c g

d

0.4

1 , where (ACI 10.12.3)

d

maximum factored axial sustained (dead) load

maximum factored axial total load.(ACI 10.0,R10.12.3)

The magnification factor, ns , must be a positive number and greater than one.

ThereforePu must be less than 0.75Pc . IfPu is found to be greater than or equal to

0.75Pc , a failure condition is declared.

The above calculations are done for major and minor directions separately. That

means that s , ns , Cm , k, lu ,EI, and Pc assume different values for major and minor

directions of bending.

If the program assumptions are not satisfactory for a particular member, the usercan explicitly specify values of s nsand .

Determine Capacity Ratio

As a measure of the stress condition of the column, a capacity ratio is calculated.

The capacity ratio is basically a factor that gives an indication of the stress condi-

tion of the column with respect to the capacity of the column.

Column Design 31



39/207

Before entering the interaction diagram to check the column capacity, the moment

magnification factors are applied to the factored loads to obtainP M Mu ux uy, ,and .

The point (P M Mu ux uy, , ) is then placed in the interaction space shown as point L in

Figure II-3. If the point lies within the interaction volume, the column capacity is

adequate; however, if the point lies outside the interaction volume, the column is

overstressed.

This capacity ratio is achieved by plotting the point L and determining the location

of point C. The point C is defined as the point where the line OL (if extended out-

wards) will intersect the failure surface. This point is determined by three-

dimensional linear interpolation between the points that define the failure surface.

SeeFigure II-3.The capacity ratio, CR, is given by the ratioOL

OC.

If OL = OC (or CR=1) the point lies on the interaction surface and the column is

stressed to capacity.

If OL < OC (or CR OC (or CR>1) the point lies outside the interaction volume and the col-

umn is overstressed.

The maximum of all the values of CR calculated from each load combination is re-

ported for each check station of the column along with the controlling

P M Mu ux uy, ,and set and associated load combination number.

If the reinforcing area is not defined, ETABS computes the reinforcement that will

give an interaction ratio of unity.

Design Column Shear Reinforcement

The shear reinforcement is designed for each loading combination in the major and

minor directions of the column. In designing the shear reinforcing for a particular

column for a particular loading combination due to shear forces in a particular di-

rection, the following steps are involved:

Determine the factored forces acting on the section,Pu andVu . Note thatPu is

needed for the calculation of Vc .

Determine the shear force, Vc , that can be resisted by concrete alone.

Calculate the reinforcement steel required to carry the balance.

32 Column Design



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For Special and Intermediate moment resisting frames (ductile frames), the shear

design of the columns is also based upon the probable and nominal moment capaci-

ties of the members, respectively, in addition to the factored moments. Effects of

the axial forces on the column moment capacities are included in the formulation.

The following three sections describe in detail the algorithms associated with theabove-mentioned steps.

Determine Section Forces

In the design of the column shear reinforcement of anOrdinary moment re-

sisting concrete frame, the forces for a particular load combination, namely,

the column axial force, Pu , and the column shear force,Vu , in a particular direc-

tion are obtained by factoring the ETABS analysis load cases with the corre-

sponding load combination factors.

In the shear design ofSpecial moment resisting frames (seismic design) the

column is checked for capacity-shear in addition to the requirement for the Or-

dinary moment resisting frames. The capacity-shear force in a column,Vp , i n a

particular direction is calculated from the probable moment capacities of the

column associated with the factored axial force acting on the column.

For each load combination, the factored axial load,Pu , is calculated. Then, the

positive and negative moment capacities,Mu andMu

, of the column in a par-

ticular direction under the influence of the axial forcePu is calculated using the

uniaxial interaction diagram in the corresponding direction. The design shear

force, Vu , is then given by (ACI 21.4.5.1)

V V + V u p D+ L (ACI 21.4.5.1)

where, Vp is the capacity-shear force obtained by applying the calculated prob-

able ultimate moment capacities at the two ends of the column acting in two op-

posite directions. Therefore, Vp is the maximum ofVP1 and VP2 , where

V = M + M

LP

I

-

J

+

1, and

V = M + M

LP

I

+

J

-

2, where

MI

, MI

= Positive and negative moment capacities at end I of

the column using a steel yield stress value of fyand no factors ( 1.0),

Column Design 33



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MJ

, MJ

= Positive and negative moment capacities at end J of

the column using a steel yield stress value of fyand no factors ( 1.0),and

L = Clear span of column.

For Special moment resisting frames is taken as 1.25 (ACI 10.0, R21.4.5.1).

VD L

is the contribution of shear force from the in-span distribution of gravity

loads. For most of the columns, it is zero.

ForIntermediate moment resisting frames, the shear capacity of the column

is also checked for the capacity-shear based on the nominal moment capacities

at the ends and the factored gravity loads, in addition to the check required for

Ordinary moment resisting frames. The design shear force is taken to be the

minimum of that based on the nominal ( 1.0) moment capacity and modified

factored shear force. The procedure for calculating nominal moment capacityis the same as that for computing the probable moment capacity for special mo-

ment resisting frames, except that is taken equal to 1 rather than 1.25 (ACI

21.10.3.a, R21.10). The modified factored shear forces are based on the speci-

fied load factors except the earthquake load factors are doubled (ACI

21.10.3.b).

Determine Concrete Shear Capacity

Given the design force setPu andVu , the shear force carried by the concrete, Vc , is

calculated as follows:

If the column is subjected to axial compression, i.e.Pu is positive,

V = f + P

AAc c

u

g

cv2 12000

, where (ACI 11.3.1.2)

fc

100 psi, and (ACI 11.1.2)

V f + PA

Ac c u

g

cv

3.5 1500

. (ACI 11.3.2.2)

The termP Au g must have psi units.Acv is the effective shear area which is shown

shaded inFigure III-2. For circular columns Acv is taken to be equal to the gross

area of the section (ACI 11.3.3, R11.3.3).

34 Column Design



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If the column is subjected to axial tension,Pu is negative,

V = f + P

AAc c

u

g

cv2 1500

0

(ACI 11.3.2.3)

ForSpecial moment resisting concrete frame design, Vc is set to zero if the

factored axial compressive force, Pu , including the earthquake effect is small

( )P f A / u c g 20 and if the shear force contribution from earthquake, V

E, is

Column Design 35


Figure III-2

Shear Stress Area, Acv


43/207

more than half of the total factored maximum shear force over the length of the

member Vu (V VE u 0.5 ) (ACI 21.4.5.2).

Determine Required Shear Reinforcement

Given Vu and Vc , the required shear reinforcement in the form of stirrups or tieswithin a spacing,s, is given for rectangular and circular columns by

A = V / V s

f dv

u c

ys

( ) , for rectangular columns and (ACI 11.5.6.1, 11.5.6.2)

A = V / V s

f Dv

u c

ys

( )

0.8

, for circular columns. (ACI 11.5.6.3, 11.3.3)

Vu

is limited by the following relationship.

( / )V V f Au c c cv 8 (ACI 11.5.6.9)

Otherwise redimensioning of the concrete section is required. Here , the strength

reduction factor, is 0.85 (ACI 9.3.2.3). The maximum of all the calculatedAv val-

ues obtained from each load combination are reported for the major and minor di-

rections of the column along with the controlling shear force and associated load

combination label.

The column shear reinforcement requirements reported by the program are based

purely upon shear strength consideration. Any minimum stirrup requirements to

satisfy spacing considerations or transverse reinforcement volumetric considera-

tions must be investigated independently of the program by the user.

Beam Design

In the design of concrete beams, ETABS calculates and reports the required areas

of steel for flexure and shear based upon the beam moments, shears, load combina-

tion factors, and other criteria described below. The reinforcement requirements

are calculated at a user defined number of check/design stations along the beam

span.

All the beams are only designed for major direction flexure and shear. Effects

due to any axial forces, minor direction bending, and torsion that may exist in the

beams must be investigated independently by the user.

The beam design procedure involves the following steps:

36 Beam Design



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Design beam flexural reinforcement

Design beam shear reinforcement

Design Beam Flexural Reinforcement

The beam top and bottom flexural steel is designed at check/design stations along

the beam span. In designing the flexural reinforcement for the major moment for a

particular beam for a particular section, the following steps are involved:

Determine the maximum factored moments

Determine the reinforcing steel

Determine Factored Moments

In the design of flexural reinforcement of Special, Intermediate, or Ordinary mo-ment resisting concrete frame beams, the factored moments for each load combina-

tion at a particular beam section are obtained by factoring the corresponding mo-

ments for different load cases with the corresponding load factors.

The beam section is then designed for the maximum positive Mu+ and maximum

negative Mu- factored moments obtained from all of the load combinations .

Negative beam moments produce top steel. In such cases the beam is always de-

signed as a rectangular section. Positive beam moments produce bottom steel. In

such cases the beam may be designed as a Rectangular- or a T-beam.

Determine Required Flexural Reinforcement

In the flexural reinforcement design process, the program calculates both the ten-

sion and compression reinforcement. Compression reinforcement is added when

the applied design moment exceeds the maximum moment capacity of a singly re-

inforced section. The user has the option of avoiding the compression reinforce-

ment by increasing the effective depth, the width, or the grade of concrete.

The design procedure is based on the simplified rectangular stress block as showninFigure III-3(ACI 10.2). Furthermore it is assumed that the compression carried

by concrete is less than 0.75 times that which can be carried at the balanced condi-

tion (ACI 10.3.3). When the applied moment exceeds the moment capacity at this

designed balanced condition, the area of compression reinforcement is calculated

on the assumption that the additional moment will be carried by compression and

additional tension reinforcement.

Beam Design 37



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The design procedure used by ETABS, for both rectangular and flanged sections

(L- and T-beams) is summarized below. It is assumed that the design ultimate axial

force does not exceed 0.1f Ac g (ACI 10.3.3), hence all the beams are designed for

major direction flexure and shear only.

Design for Rectangular Beam

In designing for a factored negative or positive moment, Mu , (i.e. designing top or

bottom steel) the depth of the compression block is given by a (seeFigure III-3),

where,

a d dM

f b

u

c

2 2

0.85 , (ACI 10.2.7.1)

where, the value of is 0.90 (ACI 9.3.2.1) in the above and the following equa-

tions. Also 1

andcb

are calculated as follows:

1 = 0.85 0.05

fc 4000

1000, 0.65 0.85

1 , (ACI 10.2.7.3)

38 Beam Design


0.85f'c

c

b

d

As

(i) BEAMSECTION

(ii)STRAINDIAGRAM

(iii) STRESSDIAGRAM

a = c1

=0.003

s

A'sd'

Cs

TsTc

Figure III-3

Design of Rectangular Beam Section


46/207

c E

E + fd =

+ fd

b

c s

c s y y

87000

87000. (ACI 10.2.3, 10.2.4)

The maximum allowed depth of the compression block is given by

a cbmax 0.75 1 . (ACI 10.2.7.1, 10.3.3)

If a a max , the area of tensile steel reinforcement is then given by

A M

f d a

su

y

2

.

This steel is to be placed at the bottom ifMu is positive, or at the top ifMu is

negative.

If a a max , compression reinforcement is required (ACI 10.3.3) and is calcu-

lated as follows:

The compressive force developed in concrete alone is given by

C f bac0.85 max , and (ACI 10.2.7.1)

the moment resisted by concrete compression and tensile steel is

M C d a

uc

max

2

.

Therefore the moment resisted by compression steel and tensile steel is

M M Mus u uc .

So the required compression steel is given by

A M

f d ds

us

s

( ) , where

f E c dc

s s

0.003 . (ACI 10.2.4)

The required tensile steel for balancing the compression in concrete is

A M

f d a

s

uc

y

1

2

max

, and

Beam Design 39



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the tensile steel for balancing the compression in steel is given by

A M

f d ds

us

y

2

( ) .

Therefore, the total tensile reinforcement, A A As s s 1 2 , and total com-pression reinforcement is As

.As is to be placed at bottom and As is to be

placed at top ifMu is positive, and vice versa ifMu is negative.

Design for T-Beam

In designing for a factored negative moment,Mu , (i.e. designing top steel), the cal-

culation of the steel area is exactly the same as above, i.e., no T-Beam data is to be

used. SeeFigure III-4. IfMu 0 , the depth of the compression block is given by

a d d Mf b

u

c f

2 20.85

.

The maximum allowed depth of compression block is given by

a cbmax

0.75 1

. (ACI 10.2.7.1, 10.3.3)

40 Beam Design


c

bf

d

As

(i)BEAMSECTION

(ii)STRAINDIAGRAM

(iii)STRESSDIAGRAM

=0.003

s

ds0.85f'c

Cf

Tf

0.85f'c

Cw

Tw

bw

As'

C s

Ts

d' fs'

Figure III-4

Design of a T-Beam Section


48/207

If a ds , the subsequent calculations forAs are exactly the same as previously

defined for the rectangular section design. However, in this case the width of

the compression flange is taken as the width of the beam for analysis. Whether

compression reinforcement is required depends on whethera a max .

If a ds

, calculation forAs is done in two parts. The first part is for balancingthe compressive force from the flange, Cf, and the second part is for balancing

the compressive force from the web, Cw , as shown inFigure III-4. Cf is given

by

C f b b d f c f w s

0.85 ( ) .

Therefore, A =C

fs

f

y

1 and the portion of Mu that is resisted by the flange is

given by

M = C d d

uf f

s

2 .

Again, the value for is (flexure) which is 0.90 by default. Therefore, the

balance of the moment, Mu to be carried by the web is given by

M = M Muw u uf .

The web is a rectangular section of dimensions bw andd, for which the design

depth of the compression block is recalculated as

a d d M

f b

uw

c w

1

2 2

0.85 .

If a a1

max , the area of tensile steel reinforcement is then given by

A M

f d a

s

uw

y

2

1

2

, and

A A As s s 1 2 .

This steel is to be placed at the bottom of the T-beam.

Beam Design 41



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If a a1

max , compression reinforcement is required (ACI 10.3.3) and is

calculated as follows:

The compressive force in web concrete alone is given by

C f bac0.85 max . (ACI 10.2.7.1)

Therefore the moment resisted by concrete web and tensile steel is

M C d a

uc

max

2 , and

the moment resisted by compression steel and tensile steel is

M M Mus uw uc .

Therefore, the compression steel is computed as

A M

f d ds

us

s

( ) , where

f E c d

cs s

0.003 . (ACI 10.2.4)

The tensile steel for balancing compression in web concrete is

A

M

f d as

uc

y

2

2

max

, and

the tensile steel for balancing compression in steel is

A M

f d ds

us

y

3

( ) .

The total tensile reinforcement, A A A As s s s 1 2 3 , and total com-

pression reinforcement isA

s

.A

s is to be placed at bottom andA

s

istobeplaced at top.

Minimum Tensile Reinforcement

The minimum flexural tensile steel provided in a rectangular section in an Ordinary

moment resisting frame is given by the minimum of the two following limits:

42 Beam Design



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Af

fb d

fb ds

c

y

w

y

w

max and3 200

or (ACI 10.5.1)

A As s required 4 3 ( ). (ACI 10.5.3)

Special Consideration for Seismic Design

For Special moment resisting concrete frames (seismic design), the beam design

satisfies the following additional conditions (see alsoTable III-2):

The minimum longitudinal reinforcement shall be provided at both at the top

and bottom. Any of the top and bottom reinforcement shall not be less than

As min( )

(ACI 21.3.2.1).

A ff

b df

b dc

y

w

y

ws(min)

max and3 200 or (ACI 10.5.1)

A As requireds(min)

4

3 ( )

. (ACI 10.5.3)

The beam flexural steel is limited to a maximum given by

A b ds w 0.025 . (ACI 21.3.2.1)

At any end (support) of the beam, the beam positive moment capacity (i.e. as-sociated with the bottom steel) would not be less than 1/2 of the beam negative

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