CFD-NZS-3101-2006

Concrete Frame Design Manual NZS 3101-06

Concrete Frame Design Manual

NZS 3101-06 For ETABS 2015

ISO ETA082914M29 Rev. 0 Proudly developed in the United States of America December 2014

Copyright

Copyright Computers & Structures, Inc., 1978-2014 All rights reserved. The CSI Logo, SAP2000, ETABS, and SAFE are registered trademarks of Computers & Structures, Inc. Watch & LearnTM is a trademark of Computers & Structures, Inc. The computer programs SAP2000 and ETABS and all associated documentation are proprietary and copyrighted products. Worldwide rights of ownership rest with Computers & Structures, Inc. Unlicensed use of these programs or reproduction of documentation in any form, without prior written authorization from Computers & Structures, Inc., is ex-plicitly prohibited. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior explicit written permission of the publisher. Further information and copies of this documentation may be obtained from: Computers & Structures, Inc. www.csiamerica.com [email protected] (for general information) [email protected] (for technical support)

http://www.csiamerica.com/mailto:[email protected]:[email protected]

DISCLAIMER

CONSIDERABLE TIME, EFFORT AND EXPENSE HAVE GONE INTO THE DEVELOPMENT AND TESTING OF THIS SOFTWARE. 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 THIS PRODUCT.

THIS PRODUCT IS A PRACTICAL AND POWERFUL TOOL FOR STRUCTURAL DESIGN. HOWEVER, THE USER MUST EXPLICITLY UNDERSTAND THE BASIC ASSUMPTIONS OF THE SOFTWARE MODELING, ANALYSIS, AND DESIGN ALGORITHMS AND COMPENSATE FOR THE ASPECTS THAT ARE NOT ADDRESSED.

THE INFORMATION PRODUCED BY THE SOFTWARE MUST BE CHECKED BY A QUALIFIED AND EXPERIENCED ENGINEER. THE ENGINEER MUST INDEPENDENTLY VERIFY THE RESULTS AND TAKE PROFESSIONAL RESPONSIBILITY FOR THE INFORMATION THAT IS USED.

Contents

Chapter 1 Introduction

1.1 Organization 1-2

1.2 Recommended Reading/Practice 1-2

Chapter 2 Design Prerequisites

2.1 Design Load Combinations 2-1

2.2 Design and Check Stations 2-3

2.3 Identifying Beams and Columns 2-3

2.4 Design of Beams 2-3

2.5 Design of Columns 2-4

2.6 Design of Joints 2-5

2.7 P-Delta Effects 2-6

i

Concrete Frame Design NZS 3101-06

2.8 Element Unsupported Lengths 2-6

2.9 Choice of Input Units 2-7

Chapter 3 Design Process

3.1 Notation 3-1

3.2 Design Load Combinations 3-4

3.2.1 Limits on Material Strength 3-5 3.2.2 Strength Resistance Factors 3-6

3.3 Column Design 3-6

3.3.1 Generation of Biaxial Interaction Surface 3-8 3.3.2 Calculate Column Capacity Ratio 3-11 3.3.3 Determine Capacity Ratio 3-16 3.3.4 Required Reinforcing Area 3-18 3.3.5 Design Column Shear Reinforcement 3-18

3.4 Beam Design 3-28

3.4.1 Design Beam Flexural Reinforcement 3-28 3.4.2 Design Beam Shear Reinforcement 3-38 3.4.3 Design Beam Torsion Reinforcement 3-42

3.5 Joint Design 3-46

3.5.1 Determine the Panel Zone Shear Force 3-47 3.5.2 Determine the Effective Area of Joint 3-49 3.5.3 Check Panel Zone Shear Stress 3-49 3.5.4 Beam-Column Flexural Capacity Ratios 3-50

Appendix A Second Order P-Delta Effects Appendix B Member Unsupported Lengths and Computation of

K-Factors Appendix C Concrete Frame Design Preferences Appendix D Concrete Frame Overwrites

ii

Contents

Appendix E Error Messages and Warnings

Bibliography

Chapter 1 Introduction

The design of concrete frames is seamlessly integrated within the program. Initiation of the design process, along with control of various design parameters, is accomplished using the Design menu.

Automated design at the object level is available for any one of a number of user-selected design codes, as long as the structures have first been modeled and analyzed by the program. Model and analysis data, such as material properties and member forces, are recovered directly from the model database, and no additional user input is required if the design defaults are acceptable.

The design is based on a set of user-specified loading combinations. However, the program provides default load combinations for each supported design code. If the default load combinations are acceptable, no definition of additional load combinations is required.

In the design of 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 indication of the stress condition with respect to the capacity of the column.

The biaxial column capacity check is based on the generation of consistent three-dimensional interaction surfaces. It does not use any empirical formula-tions that extrapolate uniaxial interaction curves to approximate biaxial action.

1 - 1


Interaction surfaces are generated for user-specified column reinforcing con-figurations. The column configurations may be rectangular, square or circular, with similar reinforcing patterns. The calculation of moment magnification factors, unsupported lengths, and strength reduction factors is automated in the algorithm.

Every beam member is designed for flexure, shear, and torsion at output stations along the beam span.

All beam-column joints are investigated for existing shear conditions.

For Ductile and Limited Ductile frames, the shear design of the columns, beams, and joints is based on the probable moment capacities of the members. Also, the program will produce ratios of the beam moment capacities with respect to the column moment capacities, to investigate weak beam/strong column aspects, including the effects of axial force.

Output data can be presented graphically on the model, in tables for both input and output data, or on the calculation sheet prepared for each member. For each presentation method, the output is in a format that allows the engineer to quickly study the stress conditions that exist in the structure and, in the event the member reinforcing is not adequate, aids the engineer in taking appropriate remedial measures, including altering the design member without rerunning the entire analysis.

1.1 Organization This manual is designed to help you quickly become productive with the con-crete frame design options of NZS 3101-06. Chapter 2 provides detailed de-scriptions of the Design Prerequisites used for NZS 3101-06. Chapter 3 provides detailed descriptions of the code-specific process used for NZS 3101-06. The appendices provide details on certain topics referenced in this manual.

1.2 Recommended Reading/Practice It is strongly recommended that you read this manual and review any applicable Watch & Learn Series tutorials, which are found on our web site,

1 - 2 Organization

Chapter 1 - Introduction

http://www.csiamerica.com, before attempting to design a concrete frame. Ad-ditional information can be found in the on-line Help facility available from within the programs main menu.

Recommended Reading/Practice 1 - 3

Chapter 2 Design Prerequisites

This chapter provides an overview of the basic assumptions, design precondi-tions, and some of the design parameters that affect the design of concrete frames. In writing this manual it has been assumed that the user has an engi-neering background in the general area of structural reinforced concrete design and familiarity with NZS 3101-06 codes.

2.1 Design Load Combinations The design load combinations are used for determining the various combina-tions of the load cases for which the structure needs to be designed/checked. The load combination factors to be used vary with the selected design code. The load combination factors are applied to the forces and moments obtained from the associated load cases and are then summed to obtain the factored design forces and moments for the load combination.

For multi-valued load combinations involving response spectrum, time history, moving loads (only applicable for SAP2000) 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 maxima/minima permutations of in-teracting quantities. Separate combinations with negative factors for response

2 - 1


spectrum cases are not required because the program automatically takes the minima to be the negative of the maxima for response spectrum cases and the above described permutations generate the required sub combinations.

When a design combination involves only a single multi-valued case of time history or moving load, further options are available. The program has an option to request that time history combinations produce sub combinations for each time step of the time history. Also an option is available to request that moving load combinations produce sub combinations using maxima and minima of each design quantity but with corresponding values of interacting quantities.

For normal loading conditions involving static dead load, live load, wind load, and earthquake load, or dynamic response spectrum earthquake load, the pro-gram has built-in default loading combinations for each design code. These are based on the code recommendations and are documented for each code in the corresponding manuals.

For other loading conditions involving moving load, time history, pattern live loads, separate consideration of roof live load, snow load, and so on, the user must define design loading combinations either in lieu of or in addition to the default design loading combinations.

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. However, 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, all default combinations will automatically be updated by the program any time the design code is changed 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.

2 - 2 Design Load Combinations

Chapter 2 - Design Prerequisites

The user is cautioned that if moving load or time history results are not requested to be recovered in the analysis for some or all of the frame members, the effects of those loads will be assumed to be zero in any combination that includes them.

2.2 Design and Check Stations For each load combination, each 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. The number of segments in an element is requested by the user before the analysis is performed. The user can refine the design along the length of an element by requesting more segments.

When using the NZS 3101-06 design code, requirements for joint design at the beam-to-column connections are evaluated at the top most station of each column. The program 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/column intersection are reported.

2.3 Identifying Beams and Columns In the program, all beams and columns are represented as frame elements, but design of beams and columns requires separate treatment. Identification for a concrete element is accomplished by specifying the frame section assigned to the element to be of type beam or column. If any brace element exists in the frame, the brace element also would be identified as a beam or a column element, depending on the section assigned to the brace element.

2.4 Design of Beams In the design of concrete beams, in general, the program calculates and reports the required areas of steel for flexure and shear based on the beam moments, shears, load combination factors, and other criteria, which are described in detail

Design and Check Stations 2 - 3


in the code-specific manuals. The reinforcement requirements are calculated at a user-defined number of stations along the beam span.

All the beams are designed for major direction flexure, shear and torsion only. Effects due to any axial forces and minor direction bending 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 designed for the maximum positive and maximum negative factored moment envelopes obtained from all of the load combinations. Negative beam moments produce top steel. In such cases, the beam is always designed as a Rectangular section. Positive beam moments produce bottom steel. In such cases, the beam may be designed as a Rectangular beam 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, the required compression reinforce-ment is calculated.

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 into the program for the NZS 3101-06 code.

2.5 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 design procedure for the reinforced concrete columns of the structure involves the following steps:

2 - 4 Design of Columns


Generate axial force-biaxial moment interaction surfaces for all of the dif-ferent concrete section types in the model.

Check the capacity of each column for the factored axial force and bending moments obtained from each loading combination at each end of the col-umn. This step is also used to calculate the required reinforcement (if none was specified) that will produce a capacity ratio of 1.0.

The generation of the interaction surface is based on the assumed strain and stress distributions and some other simplifying assumptions. These stress and strain distributions and the assumptions are documented in Chapter 3.

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 must be considered.

For certain special seismic cases, the design of columns for shear is based on the capacity shear. The capacity shear force in a particular direction is calculated from the 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 using load cases and the corresponding load combination factors. Then, the moment capacity of the column in a particular direction under the in-fluence of the axial force is calculated, using the uniaxial interaction diagram in the corresponding direction as documented in Chapter 3.

2.6 Design of Joints To ensure that the beam-column joint of Ductile and Moderately Ductile 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 that have 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 performed 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

Determine the effective area of the joint

Design of Joints 2 - 5


Check panel zone shear stress

The joint design details are documented in Chapter 3.

2.7 P-Delta Effects The program design process requires that the analysis results include P-delta effects. The P-delta effects are considered differently for braced or non-sway and unbraced or sway components of moments in columns or frames. For the braced moments in columns, the effect of P-delta is limited to individual member stability. For unbraced components, lateral drift effects should be considered in addition to individual member stability effects. The program assumes 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 by moment magnification factors, as documented in Chapter 3 of this manual.

For lateral drift effects, the program assumes that the P-delta analysis is performed and that the amplification is already included in the results. The moments and forces obtained from P-delta analysis are further amplified for individual column stability effect if required by the governing code, as in the NZS 3101-06 codes.

Users should be aware that the default analysis option in the program is that P-delta effects are not considered. The user can include P-delta analysis and set the maximum number of iterations for the analysis. The default number of iteration for P-delta analysis is 1. Further details about P-delta analysis are provided in Appendix A of this design manual.

2.8 Element Unsupported Lengths To account for column slenderness effects, the column unsupported lengths are required. The two unsupported lengths are l33 and l22. These are the lengths between support points of the element in the corresponding directions. The

2 - 6 P-Delta Effects


length l33 corresponds to instability about the 3-3 axis (major axis), and l22 cor-responds to instability 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. The program, however, allows users to assign several elements to be treated as a single member for design. This can be accomplished differently for major and minor bending, as documented in Appendix B of this design manual.

The user has options to specify the unsupported lengths of the elements on an element-by-element basis.

2.9 Choice of Input Units English as well as SI and MKS metric units can be used for input. 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 oth-erwise noted. For example, the NZS 3101-06 code is published in New-ton-millimeter-second units. By default, all equations and descriptions presented in the Design Process chapter correspond to Newton-millimeter -second units. However, any system of units can be used to define and design a structure in the program.

Choice of Input Units 2 - 7

Chapter 3 Design Process

This chapter provides a detailed description of the code-specific algorithms used in the design of concrete frames when the NZS 3101-06 (NZS) codes have been selected. For simplicity, all equations and descriptions presented in this chapter correspond to Newton-millimeter-second units unless otherwise noted.

3.1 Notation The various notations used in this chapter are described herein:

Acv Area of concrete used to determine shear stress, mm2

Ag Gross area of concrete, mm2

As Area of tension reinforcement, mm2

sA Area of compression reinforcement, mm2

As(required) Area of steel required for tension reinforcement, mm2

Ast Total area of column longitudinal reinforcement, mm2

Asv Area of shear reinforcement, mm2

3 - 1


Av /s Area of shear reinforcement per unit length of the member, mm2/mm

a Depth of compression block, mm

ab Depth of compression block at balanced condition, mm

amax Maximum allowed depth of compression block, mm

b Width of member, mm

bf Effective width of flange (T-beam section), mm

bw Width of web (T-beam section), mm

Cm Coefficient, dependent upon column curvature, used to calculate moment magnification factor

c Depth to neutral axis, mm

cb Depth to neutral axis at balanced conditions, mm

d Distance from compression face to centroid of tension reinforce-ment, mm

d Concrete cover to center of reinforcing, mm

Ec Modulus of elasticity of concrete, MPa

Es Modulus of elasticity of reinforcement, assumed as 200,000 MPa (NZS 5.3.4)

fc Specified compressive strength of concrete, MPa

c,of Concrete overstrength capacity (for columns) = (fc +15) MPa

fy Specified yield strength of flexural reinforcement, MPa. The value of fy used in design calculation is limited to a torsional longitudinal reinforcement of 500 MPa for both tension and compression (NZS 5.3.3).

fyt Specified yield strength of shear reinforcement, MPa

h Dimension of beam or column, mm

hf Thickness of slab (T-beam section), mm

3 - 2 Notation

Chapter 3 - Design Process

Ig Moment of inertia of gross concrete section about centroidal axis, neglecting reinforcement, mm4

k Effective length factor

L Clear unsupported length, mm

Ma Smaller factored end moment in a column, N-mm Mb Larger factored end moment in a column, N-mm Mc Factored moment to be used in design, N-mm Mns Non-sway component of factored end moment, N-mm Ms Sway component of factored end moment, N-mm M* Factored moment at a section, N-mm M*2 Factored moment at a section about 2-axis, N-mm M*3 Factored moment at a section about 3-axis, N-mm Nb Axial load capacity at balanced strain conditions, N

Nc Critical buckling strength of column, N

Nmax Maximum axial load strength allowed, N

0*N Design axial load derived from overstrength considerations, N

N* Factored axial load at a section, N

Vc Shear force resisted by concrete, N

VE Shear force caused by earthquake loads, N

VD+L Shear force from span loading, N

Vp Shear force computed from probable moment capacity, N *oV Shear force determined from flexural overstrength of element, N

Vs Shear force resisted by steel, N

V* Factored shear force at a section, N

1 Average stress factor in equivalent stress block

1 Factor for obtaining depth of compression block in concrete

Notation 3 - 3


d Absolute value of ratio of maximum factored axial dead load to maximum factored axial total load

b Moment magnification factor for nonsway frames

s Moment magnification factor for sway frames

c Strain in concrete

s Strain in reinforcing steel

Strength reduction factor

o, fy Reinforcing steel overstrength factor

3.2 Design Load Combinations The design load combinations are the various combinations of the prescribed load cases for which the structure is to be checked. The program creates a number of default design load combinations for a concrete frame design. Users can add their own design load combinations as well as modify or delete the program default design load combinations. An unlimited number of design load combinations can be specified.

To define a design load combination, simply specify one or more response cases, each with its own scale factor. The scale factors are applied to the forces and moments from the load cases to form the factored design forces and moments for each design load combination. There is one exception to the preceding. For spectral analysis modal combinations, any correspondence between the signs of the moments and axial loads is lost. The program uses eight design load com-binations for each such loading combination specified, reversing the sign of axial loads and moments in major and minor directions.

As an example, if a structure is subjected to dead load, G, and live load, Q, only, the NZS 3101-06 design check may need one design load combination only, namely, 1.2G +1.5Q. However, if the structure is subjected to wind, earthquake, or other loads, numerous additional design load combinations may be required.

The program allows live load reduction factors to be applied to the member forces of the reducible live load case on a member-by-member basis to reduce the contribution of the live load to the factored responses.

3 - 4 Design Load Combinations


The design load combinations are the various combinations of the analysis cases for which the structure needs to be checked. For this code, if a structure is sub-jected to dead load (D), live load (L), wind (W), earthquake (E), and snow (S) loads, and considering that wind and earthquake forces are reversible, the fol-lowing load combinations may need to be defined (AS/NZS 1170.0, 4.2.2):

1.35D (AS/NZS 1170.0, 4.2.2(a))

1.2D + 1.5L (AS/NZS 1170.0, 4.2.2(b))

1.2D + 1.5(0.4 L) (AS/NZS 1170.0, 4.2.2(c))

0.9D 1.0W 1.2D + 0.4L 1.0W

(AS/NZS 1170.0, 4.2.2(e)) (AS/NZS 1170.0, 4.2.2(d))

1.0D + 0.3L 1.0E (AS/NZS 1170.0, 4.2.2(f))

These are also the default design load combinations in the program whenever the NZS 3101-06 code is used. The user should use other appropriate design load combinations if roof live load is separately treated, or if other types of loads are present.

Live load reduction factors can be applied to the member forces of the live load analysis on a member-by-member basis to reduce the contribution of the live load to the factored loading.

When using the NZS 3101-06 code, the program design assumes that an analysis for P-delta effects is not required. The user should verify that this assumption is correct and alter any earthquake loading combinations accordingly, if required.

3.2.1 Limits on Material Strength

The upper and lower limits of cf for Ductile and Limited Ductile frames should be 70 MPa and 25 MPa respectively (NZS 5.2.1).

25 70MPacf (NZS 5.2.1)

For remaining frame types, the upper and lower limits of cf shall be as follows:

Design Load Combinations 3 - 5


25 100MPacf (NZS 5.2.1)

The lower characteristic yield strength of longitudinal reinforcement, fy, should be equal to or less than 500 MPa for all frames (NZS 5.3.3). The lower charac-teristic yield strength of transverse (stirrup) reinforcement, fyt, should not be greater than 500 MPa for shear or 800 MPa for confinement (NZS 5.3.3).

When the compression strength of concrete used in design is beyond the given limits or when the yield strength of steel used in design exceeds the given limits, the code does not cover such cases. The code allows use of cf and fy beyond the given limits provided special study is conducted (NZS 5.2.1).

The program does not enforce any of these limits for column PMM interaction check or design and flexure design of beam. The specified strengths are used for design. The user is responsible to use the proper strength values while defining the materials.

3.2.2 Strength Resistance Factors

The strength reduction factor, , is defined as given below (NZS 2.3.2.2):

Type of action effect Strength reduction factor () b = Flexure with or without axial tension or compression 0.85 c = Axial compression 0.85 s = Shear 0.75 t = Torsion 0.75 For actions derived from overstrength of elements (2.6.5) 1.00

3.3 Column Design The program can be used to check column capacity or to design columns. If the geometry of the reinforcing bar configuration of each concrete column section has been defined, the program will check the column capacity. Alternatively, the program can calculate the amount of reinforcing required to design the column based on provided reinforcing bar configuration. The reinforcement requirements are calculated or checked at a user-defined number of check/design

3 - 6 Column Design


stations along the column span. 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 interacting diagram is shown in Figure 3-1. For reinforcement to be designed, the program generates the interaction surfaces for the range of allowable reinforcement: 0.8 to 8 percent for Ordinary and Intermediate moment resisting frames (NZS 10.3.8.1) and 0.8 percent to 18Ag /fy for Ductile, Limited, and Nominal Ductile moment resisting frames (NZS 10.4.6.2).

Figure 3-1 A typical column interaction surface

Calculate the capacity ratio or the required reinforcing area for the factored axial force and biaxial (or uniaxial) bending moments obtained from each

Column Design 3 - 7


loading combination at each station of the column. The target capacity ratio is taken as the Utilization Factor Limit when calculating the required rein-forcing area.

Design the column shear reinforcement.

The following sections describe in detail the algorithms associated with this process.

3.3.1 Generation of Biaxial Interaction Surfaces

The column capacity interaction volume is numerically described by a series of discrete points that are generated on the three-dimensional interaction failure surface. In addition to axial compression and biaxial bending, the formulation allows for axial tension and biaxial bending considerations. A typical interaction surface is shown in Figure 3-1.

The coordinates of these points are determined by rotating a plane of linear strain in three dimensions on the section of the column, as shown in Figure 3-2. The linear strain diagram limits the maximum concrete strain, c, at the extremity of the section, to 0.003 (NZS 7.4.2.3).

The formulation is based consistently upon the general principles of ultimate strength design (NZS 7.4), and allows for any doubly symmetric rectangular, square, or circular column section.

The stress in the steel is given by the product of the steel strain and the steel modulus of elasticity, sEs, and is limited to the yield stress of the steel, fy (NZS 5.3.3). 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 simplifications with respect to distributing the area of steel over the cross-section of the column, as shown in Figure 3-2.

The concrete compression stress block is assumed to be rectangular, with a stress value of 1 ca f (NZS 7.4.2.7), as shown in Figure 3-3. The interaction algorithm provides correction to account for the concrete area that is displaced by the re-inforcement in the compression zone. The depth of the equivalent rectangular block, 1c, where:

3 - 8 Column Design


Figure 3-2 Idealized strain distribution for generation of interaction surface

Column Design 3 - 9


Figure 3-3 Idealization of stress and strain distribution in a column section

1 = 0.85 0.004 ( )55 ,cf (NZS 7.4.2.7c)

1 = 0.85 0.008 ( )30 ,cf (NZS 7.4.2.7d)

10.75 0.85, (NZS 7.4.2.7)

10.65 0.85, (NZS 7.4.2.7)

The interaction algorithm provides correction to account for the concrete area that is displaced by the reinforcement in the compression zone.

Default values for for various actions are provided by the program but can be overwritten using the Preferences.

The effect of the strength reduction factor, , is included in the generation of the interaction surface. The maximum axial resistance is given by:

N* 0.85 [1 cf (Ag Ast) + fy Ast] (Ordinary, Nominal) (NZS 10.3.4.2)

No* 0.70 [1 cf (Ag Ast) + fy Ast] (Ductile, Limited Ductile) (NZS 10.4.4)

3 - 10 Column Design


3.3.2 Calculate Column Capacity Ratio

The column capacity ratio is calculated for each design load combination at each output station of each column. The following steps are involved in calculating the capacity ratio of a particular column for a particular design load combination at a particular location:

Determine the factored moments and forces from the load cases and the specified load combination factors to give No*, M*2, and M*3.

Determine the moment magnification factors for the column moments.

Apply the moment magnification factors to the factored moments. Deter-mine 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 nonsway.

The following three sections describe in detail the algorithms associated with that process.

3.3.2.1 Determine Factored Moments and Forces The loads for a particular design load combination are obtained by applying the corresponding factors to all of the analysis cases, giving No*, M*2, and M*3. The factored moments are further increased, if required, to obtain minimum eccen-tricities of (15 + 0.03h) mm, where h is the dimension of the column in the corresponding direction (NZS 10.3.2.3.5(c)). The computed moments are fur-ther amplified by using Moment Magnification Factors to allow for Lateral Drift Effect and Member Stability Effect.

3.3.2.2 Determine Moment Magnification Factors The moment magnification factors are applied in two stages. First the moments are separated into their sway and non-sway components. The non-sway components are amplified for lateral drift effect. Although this amplification may be avoided for braced frames according to the code, the program treats all

Column Design 3 - 11


frames uniformly to amplify non-sway components of moments. These ampli-fied moments are further amplified for individual member stability effect.

3.3.2.2.1 Lateral Drift Effect For all frames, the moment magnification factor for lateral drift effect is applied only to the sway moment in the program.

ns s sM M M= +

The moment magnification factors for moments causing sidesway in the major and minor directions, 2s and 3s , can be different. The moment magnification factors, 2s and 3s , can be taken as 1.0 if a P- analysis is carried out. The program assumes that the program analysis models P- effects; therefore, 2s and 3s are taken as 1.0.

It is suggested that the P- analysis be performed at the factored load level (White and Hajjar 1991). The necessary factors for a P- analysis for the NZS 3101-06 code should be (1.0 D + 0.4 L)/ c with the loading standard NZS/AS 1170.0, where c is the strength reduction factor for compression and is equal to 0.85.

The user is reminded of the special analysis requirements, especially those re-lated to the value of EI used in analysis (NZS 10.3.2.3.5). In the program, the EI values are computed based on gross cross-section areas. The user has the option to reduce the EI values for analysis purposes using a scale factor on a section-by-section basis. If the program assumptions are not satisfactory for a particular member, the user can explicitly specify values of s2 and s3.

3.3.2.2.2 Member Stability Effects All compression members are designed using the factored axial load, *oN , ob-tained from the analysis and a magnified factored moment, cM . The magnified moment is computed as,

2c bM M= , (NZS 10.3.2.3.5)



where 2M is the column maximum end moment obtained from elastic analysis after considering minimum eccentricity and lateral drift effect, and cM is the maximum moment associated with the major or minor direction of the column occurring at the end or at an interior point within the span of the column. The moment magnification factor, b , for moments not causing sidesway is given by

* 1.01

0.75

mb

c

CN

N

=

, where (NZS 10.3.2.3.5)

( )

2

2cEIN

kL

= ,

k is conservatively taken as 1, however the user can override the value,

EI is associated with a particular column direction given by

0.401

c g

d

E IEI =

+, and (NZS 10.3.2.3.5)

Maximum factored axial dead loadMaximum factored total axial loadd

= ,

0.6 0.4 0.4,amb

MCM

= + (NZS 10.3.2.3.5)

aM and bM are the moments at the ends of the column, and bM is numeri-cally larger than .aM a bM M is positive for single curvature bending and negative for double curvature bending. The preceding expression of mC 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, or for any other case, 1mC = (NZS 10.3.2.3.5). mC can be overwritten by the user on an element-by-element basis.



The magnification factor, b , must be a positive number and greater than one. Therefore N* must be less than 0.75 cN . If N* is found to be greater than or equal to 0.75 cN , a failure condition is declared.

The preceding calculations use the unsupported lengths of the column for major and minor directions separately. That means that n, ns, Cm, k, lu, EI, and N* assume different values for major and minor directions of bending.

If the program assumptions are not satisfactory for a particular member, the user can explicitly specify values of n and ns.

3.3.2.3 Dynamic Moment Magnification For seismic design of Ductile frames and frames with Limited ductility, the moment is further amplified for dynamic effects of higher modes as follows:

* *0 col,joint,elastic col0.3m bM R M h V = (NZS CD3.2.5)

where,

*M = The design moment for column.

col,joint,elasticM = The column moment at the center of the joint obtained from linear elastic analysis.

*colV = The design shear for the column.

mR = The moment reduction factor, which is taken as a function of axial force and the dynamic magnification factor, , for Ductile mo-ment resisting frames (NZS Table D.1). The user can overwrite this.

0 = An overstrength factor for a joint zone. This value is equal to the ratio of beam input overstrength moment, obM , at the intersec-tion point being considered, to the corresponding sum of the

seismic design moment ( )EbM . It is given by oboEb

MM

=

and



is computed for each direction (NZS CD3.2.2). At the base of columns, the value of 0 is taken as 1.2. The user can overwrite this factor in the major and minor directions.

= The dynamic magnification factor, which depends on the funda-mental period of the building and on the height of the beam column joint being considered in the structure (NZS CD3.2.3). The max-imum value of max is computed as follows:

max 10.6 0.85T = + , where max1.3 1.8. (NZS D3.2.3)

The value of dynamic magnification factor, , varies over the height of the building. At the base of the building and at the top of the upper story the value of is taken as 1.0. Between 30% of the height of the frame above the base and the third highest level in the frame, is taken as max . The value of at the 2nd level is the larger of 1.3 or that obtained by linear interpolation between the values at the base of the column and max at 30% of the height of the building. For the 2nd to top level the value is the larger of 1.3 or that obtained by linear interpolation between max at the third highest level and 1.0 at the highest level.

The user can overwrite this, and

= the modification which is given by the following equation (NZS D3.2.3(b)):

,

1.4 1.0.2.5

o

o fy n

MM

=

(NZS Eq. D-2)

The maximum value of is 1.0 at the base of the column and in the top story of the building,

where

oM = Sum of the bending moments acting on the beam at the faces of the column being considered when overstrength moments act on the beam.

nM = Corresponding sum of the moments when the beams are sustain-ing their nominal strength moments.



,o fy = As defined in NZS 2.6.5.6.

bh = the overall dimension of the beam at the beam-column joint.

For all levels, except the base of the columns and the top story of the building, the product of the dynamic magnification factor and the modification factor

1.2 (NZS D3.2.3(c)).

When columns are part of more than one frame biaxial action, the dynamic magnification and the modification factors ( ) for the moments in the column for the first frame shall be as given in the preceding sections. The corresponding dynamic magnification and the modification factors for the simultaneous actions from the second or subsequent frames is taken as 1.0. Where the enclosed angle between two frames is less than 45, the dynamic magnification for the two frames shall be the same dynamic magnification and modification factors.

In the current implementation of the program, the three parameters mR , 0 , and can be overwritten.

3.3.3 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 condition of the column with respect to the capacity of the column.

Before entering the interaction diagram to check the column capacity, the mo-ment magnification factors are applied to the factored loads to obtain No*, M*2, and M*3. The point (No*, M*2, M*3) is then placed in the interaction space shown as point L in Figure 3-4. 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. Point C is defined 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, as shown in Figure 3-4. The capacity ratio, CR, is given by the ratio OL/OC.



If OL = OC (or CR = 1), the point lies on the interaction surface and the column is stressed to capacity.

Figure 3-4 Geometric representation of column capacity ratio

If OL < OC (or CR < 1), the point lies within the interaction volume and the column capacity is adequate.

If OL > OC (or CR > 1), the point lies outside the interaction volume and the column is overstressed.

The maximum of all the values of CR calculated from each design load combination is reported for each check station of the column along with the controlling No*, M*2, and M*3 set and associated design load combination name.



3.3.4 Required Reinforcing Area

If the reinforcing area is not defined, the program computes the reinforcement that will give a column capacity ratio equal to the Utilization Factor Limit, which is set to 0.95 by default.

3.3.5 Design Column Shear Reinforcement

The shear reinforcement is designed for each design combination in the major and minor directions of the column. The following steps are involved in designing the shear reinforcing for a particular column for a particular design load combination resulting from shear forces in a particular direction:

Determine the factored forces acting on the section, N** and V*. Note that N* is needed for the calculation of vc.

Determine the shear stress, vc, which can be resisted by concrete alone.

Calculate the reinforcement steel required to carry the balance.

For Special Moment Resisting Ductile frames and frames with Limited Ductil-ity, the shear design of the columns is also based on the overstrength moment capacities of the column, in addition to the factored shear forces (NZS 10.4.2).

The following three sections describe in detail the algorithms associated with this process.

3.3.5.1 Determine Section Forces In the design of the column shear reinforcement of an Ordinary Moment

Resisting concrete frame, the forces for a particular design load combina-tion, namely, the column axial force, N*, and the column shear force, V*, in a particular direction are obtained by factoring the load cases with the corre-sponding design load combination factors.

In the shear design of Seismic Moment Resisting Ductile frames and frames of Limited Ductility (NOT Elastically responding frames), the shear capac-ity of the column is checked for capacity shear in addition to the requirement for the Ordinary Moment Resisting frames (NZS 10.4.2). The capacity shear



force in the column, V*, is determined from consideration of the maximum forces that can be generated at the column. Two different capacity shears are calculated for each direction (major and minor). The first is based on the probable moment strength of the column, while the second is computed from the probable moment strengths of the beams framing into the column. The design strength is taken as the minimum of these two values, but never less than the factored shear obtained from the design load combination.

V* = min{ ceV , beV } V*, factored (NZS 10.4.7.2.1)

where

ceV = Capacity shear force of the column based on the probable

maximum flexural strengths of the two ends of the column.

beV = Capacity shear force of the column based on the probable

moment strengths of the beams framing into the column.

In calculating the capacity shear of the column, ,ceV the flexural overstrength at the two ends of the column is calculated for the existing factored axial load. Clockwise rotation of the joint at one end and the associated counterclockwise rotation of the other joint produces one shear force. The reverse situation produces another capacity shear force, and both of these situations are checked, with the maximum of these two values taken as .ceV

For each design load combination, the factored axial load, N*, is calcu-lated. Then, the overstrength positive and negative moment capacities,

prM+ and ,prM of the column in a particular direction under the influence

of the axial force *oN is calculated using the uniaxial interaction diagram in the corresponding direction. Then the capacity shear force is obtained by applying the calculated overstrength moment capacities at the two ends of the column acting in two opposite directions. Therefore, ceV is the maximum of 1

ceV and 2 ,

ceV

{ }1 2max ,c c ce e eV V V= (NZS 10.4.7.2.1)



where,

1I Jc

e

M MV

L

++= ,

2c I J

eM MV

L

+ += ,

,I IM M+ = Positive and negative overstrength moment capacities

( ),p pM M + at end I of the column using a steel yield stress value of o,fyfy and no reduction factor ( = 1.1),

,J JM M+ = Positive and negative probable maximum moment ca-

pacities ( ),p pM M + at end J of the column using a steel yield stress value of o,fyfy and no reduction factor ( = 1.1) (NZS 2.6.3.2), and

L = Clear span of the column.

The overstrength moment capacities are determined using a strength reduction factor, , of 1.0 and the reinforcing steel stress equal to o,fyfy, where o,fy is set equal to 1.25 for Grade 300 and 1.35 for Grade 500 for both Ductile Moment Resisting frames and frames with Limited Ductility (NZS 2.6.5.6).

For Ductile Moment Resisting frames, the shear capacity of the column is also checked for additional factored loads, in addition to the checks required for capacity design and factored loads. The factored shear force is based on the specified load combinations, which are regular load combinations except the earthquake load factor is taken to be 1.7 (NZS 4.4.5.8).

If the column section was identified as a section to be checked, the us-er-specified reinforcing is used for the interaction curve. If the column section was identified as a section to be designed, the reinforcing area envelope is calculated after completing the flexural (PMM) design of the column. This envelope of reinforcing area is used for the interaction curve.



If the column section is a variable (non-prismatic) section, the cross-sections at the two ends are used, along with the user-specified re-inforcing or the envelope of reinforcing for check or design sections, as appropriate. If the user overwrites the length factor, the full span length is used. However, if the length factor is not overwritten by the user, the clear span length will be used. In the latter case, the maximum of the negative and positive moment capacities will be used for both the positive and negative moment capacities in determining the capacity shear.

In calculating the capacity shear of the column based on the flexural strength of the beams framing into it, beV , the program calculates the maximum probable positive and negative moment capacities of each beam framing into the top joint of the column. Then the sum of the beam mo-ments is calculated as a resistance to joint rotation. Both clockwise and counterclockwise rotations are considered separately, as well as the rota-tion of the joint in both the major and minor axis directions of the column. The shear force in the column is determined assuming that the point of inflection occurs at mid-span of the columns above and below the joint. The effects of load reversals are investigated and the design is based on the maximum of the joint shears obtained from the two cases.

{ }1 2max ,b b be e eV V V=

where,

=1eV Column capacity shear for clockwise joint rotation,

=2eV Column capacity shear for counterclockwise joint rotation,

HMV re 11 = .

It should be noted that the points of inflection shown in Figure 3-5 are taken at midway between actual lateral support points for the columns, and H is taken as the mean of the two column heights. If there is no column at the top of the joint, H is taken to be equal to one-half of the height of the column below the joint.



Figure 3-5 Column shear force Vu

HMV re 22 = ,

=1rM Sum of beam moment resistances with clockwise joint rotations,

=2rM Sum of beam moment resistances with counterclockwise joint rotations, and



=H Distance between the inflection points, which is equal to the mean height of the columns above and below the joint. If there is no column at the top of the joint, the distance is taken as one-half of the height of the column at the bottom of the joint.

For the case shown in Figure 3-5, 1eV can be calculated as follows:

1 .L Ru u

eM MV

H+

=

The expression for beV is applicable for the determination of both the major and minor direction shear forces. The calculated shear force is used for the design of the column below the joint. When beams are not oriented along the major and minor axes of the column, appropriate components of the flexural capacities are used. If the beam is oriented at an angle with the column major axis, the appropriate component, Mpr cos or Mpr sin, of the beam flux capacity is used in calculating Mr1 and Mr2. Also the positive and neg-ative moment capacities are used appropriately based on the orientation of the beam with respect to the column local axis.

3.3.5.2 Determine Concrete Shear Capacity The nominal shear strength provided by the concrete alone for normal density concrete, Vc, is calculated as follows:

The nominal shear strength for a section is computed as,

=c a n b cvV k k v A (NZS 10.3.10.3.1)

where, ka is equal to 1.0 for maximum aggregate size of 20 mm or more and equal to

0.85 for a maximum aggregate size of 10 mm. Interpolation may be used for intermediate sizes (NZS 10.3.10.3.1). The program default for ka is 1.0, which can be overwritten by the user in the design overwrites.

kn allows for the effect of axial loads, which is computed as follows:



**

*

**

1 12 if under axial tension, 0,

1 if under flexure only, 0,

1 3 if under compression, >0.

bc g

n

bc g

N v Nf A

k N

N v Nf A

+


Figure 3-6 Shear stress area, cvA

*oN is negative for tension. In any case, cv is not taken less than zero (NZS

10.4.7.2.6).

3.3.5.3 Determine Required Shear Reinforcement The average shear stress is computed for a rectangular section as,

** .

w

Vvb d

= (NZS 7.5.1)

For other types of sections, wb d is replaced by cvA , the effective shear area, which is shown in Figure 3-6 .

RECTANGULAR

dd'

b

dd'

b

cvA

SQUARE WITH CIRCULAR REBAR

cvA

cvA

CIRCULAR

dd'

DIRECTION OF SHEAR

FORCE

DIRECTION OF SHEAR

FORCE

DIRECTION OF SHEAR

FORCE

RECTANGULAR

dd'

b

dd'

b

cvA

SQUARE WITH CIRCULAR REBAR

cvAcvA

cvA

CIRCULAR

dd'

DIRECTION OF SHEAR

FORCE

DIRECTION OF SHEAR

FORCE

DIRECTION OF SHEAR

FORCE

DIRECTION OF SHEAR

FORCE

DIRECTION OF SHEAR

FORCE

DIRECTION OF SHEAR

FORCE



The average shear stress, *v , has a maximum limit, maxv , which is given as:

{ }max min 0.2 ,8MPacv f = (NZS 10.3.10.2.1)

The shear reinforcement per unit spacing is computed as follows:

*v s c

s yt

A v vs f d

=

for rectangular hoops or ties, (NZS 10.3.10.4.2)

( )*''

2 s cvs yt

v vAs f d

=

for circular hoops or ties, (NZS 10.3.10.4.2)

if * max ,v v>

a failure condition is declared. (NZS 10.3.10.2.1)

In calculating the design shear reinforcement, a limit is imposed on the fyt as

500MPa.ytf (NZS 7.5.8 and NZS 9.3.9.2)

The maximum of all the calculated vA s values, obtained from each load combination, is reported for the major and minor directions of the column, along with the controlling shear force and associated load combination number.

In designing the column shear reinforcement, the following limits are imposed on the concrete compressive strength:

100MPacf (Ordinary and Elastic) (NZS 5.2.1)

70MPacf (Ductile and Limited) (NZS 5.2.1)

For all columns and at any station, the minimum area of transverse stirrup and circular hoop reinforcement is imposed as follows:

( ) *core

10.0065

3.3gtv c

c yt c c g

AmA f N hs A f f A

(Stirrups) (NZS 10.3.10.6.1)



( ) * core1 0.00842.4 4

gtv c

c yt c c g

AmA f hNs A f f A

(Hoops) (NZS 10.3.10.5.1)

In potential plastic hinge locations, as described later, of Seismic Moment Re-sisting Ductile frames and frames with Limited ductility, the minimum area of transverse stirrup and circular hoops is imposed as follows (NZS 10.4.7.4.1, NZS 10.4.7.5.1):

( ) *core'

1.30.0060

3.3gtv c o

c yt c g

AmA f N hs A f f A

(Stirrups)

( ) * core1.3 0.0084

2.4 4gtv c o

c yt c g

AmA f N hs A f f A

(Hoops)

In the preceding four equations for calculating minimum shear reinforcement, the following limits are imposed:

1.5gc

AA

(NZS 10.3.10.5.1(a) and NZS 10.3.10.6.1)

0.4tm (NZS 10.3.10.5.1(a) and NZS 10.3.10.6.1)

800MPaytf (NZS 10.3.10.5.1(a) and NZS 10.3.10.6.1)

For the definition of the potential plastic hinge, it is assumed in the current version of the program that any beam and column segment near the joint is a potential plastic hinge. The length of the plastic hinge, hinge ,L in a column depends on the level of axial compression in it, and it is taken as follows:

( )( )( )

*

*hinge

*

if 0.25,

2 if 0.25 0.50,

3 if 0.50.

o c c g

o c c g

o c c g

h N f A

L h N f A

h N f A

< =


to satisfy spacing or volumetric requirements must be investigated inde-pendently of the program by the user.

3.4 Beam Design In the design of concrete beams, the program calculates and reports the required areas of steel for flexure and shear based on the beam moments, shear forces, torsions, design load combination factors, and other criteria described in the text that follows. The reinforcement requirements are calculated at a user-defined number of check/design stations along the beam span.

All beams are designed for major direction flexure and shear only. Effects resulting from any axial forces, torsion and minor direction bending that may exist in the beams must be investigated independently by the user.

The beam design procedure involves the following steps:

Design flexural reinforcement

Design shear reinforcement

3.4.1 Design Beam Flexural Reinforcement

The beam top and bottom flexural steel is designed at check/design stations along the beam span. The following steps are involved in designing the flexural reinforcement for the major moment for a particular beam for a particular section:

Determine the maximum factored moments

Determine the reinforcing steel

3.4.1.1 Determine Factored Moments In the design of flexural reinforcement of Special, Intermediate, or Ordinary Moment Resisting concrete frame beams, the factored moments for each design load combination at a particular beam section are obtained by factoring the

3 - 28 Beam Design


corresponding moments for different load cases with the corresponding design load combination factors.

The beam section is then designed for the factored moments obtained from all of the design load combinations. Positive moments produce bottom steel. In such cases, the beam may be designed as a Rectangular or a T-beam. Negative moments produce top steel. In such cases, the beam is always designed as a rectangular section.

3.4.1.2 Determine Required Flexural Reinforcement In the flexural reinforcement design process, the program calculates both the tension and compression reinforcement. Compression reinforcement is added when the applied design moment exceeds the maximum moment capacity of a singly reinforced section. The user has the option of avoiding the compression reinforcement by increasing the effective depth, the width, or the grade of concrete.

The design procedure is based on the simplified rectangular stress block, as shown in Figure 3-7 (NZS 7.4.2.7).

The design procedure used by the program for both rectangular and flanged sections (T-beams) is summarized in the following subsections.

3.4.1.2.1 Design for Rectangular Beam In designing for a factored negative or positive moment, M* (i.e., designing top or bottom steel), the depth of the compression block is given by a (see Figure 3-7), where,

*2

1

2,

c b

Ma d d

f b=

(AS 8.1.2.2)

where, the value b is taken as 0.85 by default (NZS 2.3.2.2) in the preceding and the following equations. The factor 1 is calculated as follows (NZS 7.4.2.7):

1 0.85 for 55MPacf =

1 10.85 0.004( 55) for 55MPa, 0.75 0.85c cf f =

Beam Design 3 - 29


Figure 3-7 Rectangular beam design

The value 1 and cb are calculated as follows:

1 0.85 for 30,cf = (NZS 7.4.2.7)

1 10.85 0.008( 30), 0.65 0.85cf = (NZS 7.4.2.7)

cb

c y sc d

f E

= +

(NZS 7.4.2.8)

The maximum allowed depth of the rectangular compression block, amax, is given by:

amax = 0.751cb. (NZS 7.4.2.7, 9.3.8.1)

If a amax (NZS 9.3.8.1), the area of tension reinforcement is then give by:

*

.

2

s

b y

MAaf d

=

3 - 30 Beam Design


The reinforcement is to be placed at the bottom if M* is positive, or at the top if M* is negative.

If a > amax (NZS 9.3.8.1), compression reinforcement is required (NZS 7.4.2.9) and is calculated as follows:

The compressive force developed in the concrete alone is given by:

C = 1 cf bamax (NZS 7.4.2.7)

and the moment resisted by concrete compression and tension reinforcement is:

M*c = C max2

ad

b.

Therefore the moment required to be resisted by compression reinforcement and tension reinforcement is:

M*s = M* M*c.

The required compression reinforcement is given by:

( )( )1

*s

ss c b

MAf f d d '

=

, where

maxs c , s yc d 'f E f .

c =

(NZS 7.4.2.2, 7.4.2.4)

The required tension reinforcement for balancing the compression in the con-crete is:

As1 = *

max.

2

c

y b

Maf d

and the tension reinforcement for balancing the compression reinforcement is given by:

Beam Design 3 - 31


As2 = ( )

*

's

y b

Mf d d

Therefore, the total tension reinforcement, As = As1 + As2, and the total com-pression reinforcement is A's. A s is to be placed at the bottom and A's is to be placed at the top if M* is positive, and vice versa if M* is negative.

3.4.1.2.2 Design for T-Beam In designing a T-beam, a simplified stress block, as shown in Figure 3-8, is as-sumed if the flange is under compression, i.e., if the moment is positive. If the moment is negative, the flange comes under tension, and the flange is ignored. In that case, a simplified stress block similar to that shown in Figure 3-8 is assumed in the compression side.

(I) BEAM SECTION

sA

(II) STRAIN DIAGRAM

(III) STRESS DIAGRAM

sA

wb

fb

d

d sf

0.003 =1 cf

wTsTs fT

fC

wC

sC

fh

c

1 cf

(I) BEAM SECTION

sA

(II) STRAIN DIAGRAM

(III) STRESS DIAGRAM

sA

wb

fb

d

d sf

0.003 =1 cf

wTsTs fT

fC

wC

sC

fh

c

1 cf

Figure 3-8 T-beam design

Flanged Beam Under Negative Moment

In designing for a factored negative moment, M* (i.e., designing top reinforce-ment), the calculation of the reinforcement area is exactly the same as described previously, i.e., no flanged beam data is used.

3 - 32 Beam Design


Flanged Beam Under Positive Moment

If M* > 0, the depth of the compression block is given by:

a = d *

2'

1

2.

c b f

Md

f b

(NZS 7.4.2)

The maximum allowable depth of the rectangular compression block, amax, is given by:

amax = 0.751cb. (NZS 7.4.2.7, 9.3.8.1)

If a hf, the subsequent calculations for As are exactly the same as previously defined for the rectangular beam design. However, in this case the width of the beam is taken as bf. Compression reinforcement is required when a > amax.

If a > hf, calculation for As has two parts. The first part is for balancing the compressive force from the flange, Cf, and the second part is for balancing the compressive force from the web, Cw, as shown in Figure 3-8.

Cf is given by:

( )1 .f c f w fC f b b h= (NZS 7.4.2.7)

Therefore, As1 = y

f

fC

and the portion of M* that is resisted by the flange is

given by:

* .2f

f f bh

M C d

=

Therefore, the balance of the moment, M* to be carried by the web is:

M*w = M* M*f.

The web is a rectangular section with dimensions bw and d, for which the depth of the compression block is recalculated as:

Beam Design 3 - 33


21

1

2 *wc b w

Ma d d .f b

=

(NZS 7.4.2 )

If a1 amax (NZS 9.3.8.1), the area of tension reinforcement is given by:

As2 = *

1

2

w

b y

Maf d

, and

As = As1 + As2.

This reinforcement is to be placed at the bottom of the flanged beam.

If a1 > amax (NZS 9.3.8.1), compression reinforcement is required and is cal-culated as follows:

The compressive force in the web concrete alone is given by:

1 maxw c wC f b a = (NZS 7.4.2.7)

and the moment resisted by the concrete web and tension reinforcement is:

max

2*c w b

aM C d . =

The moment resisted by compression and tension reinforcement is:

M*s = M*w M*c.

Therefore, the compression reinforcement is computed as:

( )( )1

*s

ss c b

MAf f d d '

=

, where

maxs c , s yc d 'f E f .

c =

(NZS 7.4.2.2, 7.4.2.4)

The tension reinforcement for balancing compression in the web concrete is:

3 - 34 Beam Design


As2 = *

max

2

c

y b

Maf d

and the tension reinforcement for balancing the compression reinforcement is:

As3 = ( )

*

.'

s

y b

Mf d d

Total tension reinforcement is As = As1 + As2 + As3, and the total compression reinforcement is A's. As is to be placed at the bottom, and A's is to be placed at the top.

3.4.1.2.3 Minimum and Maximum Tensile Reinforcement The minimum flexural tension reinforcement required in a beam section is given by the maximum of the two limits:

,4

cs w

y

fA b d

f

(NZS 9.3.8.2.1)

1.4 .wsy

b dAf

(NZS 9.3.8.2.1)

An upper limit of 0.04 times the gross web area on both the tension reinforce-ment and the compression reinforcement is imposed upon request as follows:

0.04 Rectangular beam,0.04 Flanged beam,

0.04 Rectangular beam,0.04 Flanged beam.

sw

sw

bdA

b d

bdA

b d

3.4.1.2.4 Special Consideration for Seismic Design For Seismic Moment Resisting concrete Ductile frames and frames with Limited Ductility (not Elastically responding structures), the following additional con-ditions are enforced for beam design (see also Table 3-1):

Beam Design 3 - 35


Table 3-1: Design Criteria

Type of Check/ Design

Ordinary Moment Resisting Frames (Non-Seismic)

Intermediate Moment Resisting Frames (Seismic)

Special Moment Resisting Frames (Seismic)

Column Check (interaction)

Specified Combinations



Column Design (interaction)


1% < < 8%


1% < < 8%


1% < < 6% = 1.0

Column Shears




Column shear capacity = 1.0 and fy 1.0 Vc = 0 (conditional)

Beam Design Flexure


0.04

( )2,min. 0.22 / /s cf syA D d f f bd


0.04

( )2,min. 0.22 / /s cf syA D d f f bd


0.025

(min)

3 200max andcs w wsy sy

fA b d b d

f f

Beam Min. Moment Override Check

No Requirement 1end end3

M Mu u+

{ }end1

maxspan 5M M ,Mu u u

+ +

{ }span max1

max5u

M M ,Mu u +

1end end2

M Mu u+

{ }end

1maxspan 4

M M ,Mu u u+ +

{ }1 maxspan end4M M ,Mu u u +

Beam Design Shear



Specified Combinations Beam Capacity Shear (Ve) with = 1.0 and = 1.25 plus VD+L Vc = 0 (conditional)

3 - 36 Beam Design


Table 3-1: Design Criteria

Type of Check/ Design

Ordinary Moment Resisting Frames (Non-Seismic)

Intermediate Moment Resisting Frames (Seismic)

Special Moment Resisting Frames (Seismic)

Joint Design No Requirement No Requirement Checked for shear

Beam/Column Capacity Ratio No Requirement No Requirement Checked

The minimum longitudinal reinforcement shall be provided at both the top and bottom. Any of the top and bottom reinforcement shall not be less than As(min) if tensile reinforcement is required.

'

(min) 4c

s wy

fA b d

f or (NZS 9.4.3.4(b))

(min) (required)4 .3s s

A A (NZS 9.3.8.2.3)

The beam flexural steel is limited to a maximum given by

0.025 .s wA b d (NZS 9.4.3.3)

106c

s wy

fA b df

+ (NZS 9.4.3.3)

At any section of beam within a potential plastic hinge region for Ductile Moment Resisting frames, the compression reinforcement areas, sA , shall not be less that 1/2 of the tension reinforcement area, sA , at the same section (NZS 9.4.3.4). At any section of a beam within a potential plastic hinge re-gion for moment resisting frames with Limited Ductility, the compression reinforcement areas, 'sA , shall not be less that 3/8 of the tension reinforce-ment area, sA , at the same section (NZS 9.4.3.4(a))

Beam Design 3 - 37


At least of the larger of the top reinforcement required at either end of the beam in a Ductile frame or in a frame with Limited Ductility shall be con-tinued throughout its length (NZS 9.4.3.4(c)).

3.4.2 Design Beam Shear Reinforcement

The shear reinforcement is designed for each load combination at each station along the length of the beam. In designing the shear reinforcement for a partic-ular beam, for a particular load combination, at a particular station due to the beam major shear, the following steps are involved:

Determine the factored shear force, V*.

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

Determine the shear reinforcement required to carry the balance.

For Seismic Moment Resisting frames, the shear design of the beam is also based on the overstrength moment capacities of the members.

The following three sections describe in detail the algorithms associated with these steps.

3.4.2.1 Determine Shear Force and Moment In the design of the beam shear reinforcement of an Ordinary Moment Re-

sisting concrete frame, the shear forces and moments for a particular design load combination at a particular beam section are obtained by factoring the associated shear forces and moments with the corresponding design load combination factors.

In the shear design of Seismic Moment Resisting Ductile frames and frames of Limited Ductility (NOT Elastically responding frames), the shear force, V*, is calculated from the overstrength moment capacities of each end of the beam, the gravity shear forces. The procedure for calculating the design shear force in a beam from the overstrenth moment capacity is the same as that described for a column earlier in this chapter. See Table 3-1 for a summary.

The design shear force is then given by (NZS 9.4.4.1.1)

3 - 38 Beam Design


{ }* 1 2max ,o e eV V V=

1 1e o D LV V V += +

2 2e o D LV V V += +

where Vo is the capacity shear force obtained by applying the calculated overstrength moment capacities at the two ends of the beams acting in two opposite directions. Therefore, Vo is the maximum of Vo1 and Vo2, where

, ,1

o I o Jo

M MV

L

++= , and

, ,2

o I o Jo

M MV

L

+ += , where

,o IM = Moment capacity at end I, with top steel in tension, using a

steel yield stress value of o,fy fy and no reduction factors ( = 1.0).

,o JM+ = Moment capacity at end J, with bottom steel in tension, using

a steel yield stress value of o,fy fy and no reduction factors ( = 1.0).

,o IM+ = Moment capacity at end I, with bottom steel in tension, using

a steel yield stress value of o,fy fy and no reduction factors ( = 1.0).

,o JM = Moment capacity at end J, with top steel in tension, using a

steel yield stress value of o,fy fy and no reduction factors ( = 1.0).

L = Clear span of beam.

The moment strengths are determined using a strength reduction factor of 1.0 and the reinforcing steel stress equal to o,fy fy, where o,fy fy is as defined in NZS 2.6.5.5. If the reinforcement area has not been overwritten for ductile beams, the value of the reinforcing area envelope is calculated after com-

Beam Design 3 - 39


pleting the flexural design of the beam for all the design load combinations. Then this enveloping reinforcing area is used in calculating the moment capacity of the beam. If the reinforcing area has been overwritten for ductile beams, this area is used in calculating the moment capacity of the beam. If the beam section is a variable cross-section, the cross-sections at the two ends are used along with the user-specified reinforcing or the envelope of reinforcing, as appropriate. If the user overwrites the major direction length factor, the full span length is used. However, if the length factor is not overwritten, the clear length will be used. In the latter case, the maximum of the negative and positive moment capacities will be used for both the nega-tive and positive moment capacities in determining the capacity shear.

VD+L is the contribution of shear force from the in-span distribution of gravity loads with the assumption that the ends are simply supported.

3.4.2.2 Determine Concrete Shear Capacity The shear force carried by the concrete, Vc, is calculated as:

Vc = vc Acv. (NZS 9.3.9.3.4)

The allowable shear stress capacity is given by:

c = kd ka b. (NZS 9.3.9.3.4)

The basic shear strength for rectangular section is computed as,

b =

+

dbA

w

s1007.0 cf , where (NZS 9.3.9.3.4)

cf 50 MPa, and (NZS 9.3.9.3.4)

0.08 cf b 0.2 cf (NZS 9.3.9.3.4)

The factor ka allows for the influence of maximum aggregate size on shear strength. For concrete with a maximum aggregate size of 20 mm or more, ka shall be taken as 1.0. For concrete where the maximum aggregate size is 10 mm or less, the value of ka shall be taken as 0.85. Interpolation may be used between these limits. The program default for ka is 1.0.

3 - 40 Beam Design


The factor kd allows for the influence of member depth on strength and it shall be calculated from the following conditions:

For members with shear reinforcement equal to or greater than the nominal shear reinforcement given in NZS 9.3.9.3.4, kd = 1.0

For members with an effective depth equal to or smaller than 400 mm, kd = 1.0

For members with an effective depth greater than 400,

( )0.25400 /dk d= where d is in mm

3.4.2.3 Determine Required Shear Reinforcement The average shear stress is computed for rectangular and flanged sections as:

* = *

.w

Vb d

(NZS 7.5.1)

The average shear stress is limited to a maximum limit of

vmax = min{ }0.2 , 8 MPa .cf (NZS 7.5.2, 9.3.9.3.3)

The shear reinforcement is computed as follows:

If * s ( )2cv or h max(300 mm, 0.5bw)

sAv = 0. (NZS 9.3.9.4.13)

If s ( )2cv < * sc,

sAv = 1 .

16w

cyt

bff

(NZS 9.3.9.4.15)

If sc < * smax, (NZS 9.3.9.4.2)

( )*.s cv

s yt

v vAs f d

=

Beam Design 3 - 41


If * > max, a failure condition is declared. (NZS 7.5.2, 9.3.9.3.3)

The maximum of all of the calculated Av/s values, obtained from each load combination, is reported along with the controlling shear force and associated load combination.

The beam shear reinforcement requirements considered by the program are based purely on shear strength considerations. Any minimum stirrup require-ments to satisfy spacing and volumetric considerations must be investigated independently of the program by the user.

3.4.3 Design Beam Torsion Reinforcement

The torsion reinforcement is designed for each design load combination at each station along the length of the beam. The following steps are involved in designing the longitudinal and shear reinforcement for a particular station due to the beam torsion:

Determine the factored torsion, T*.

Determine special section properties.

Determine critical torsion capacity.

Determine the torsion reinforcement required.

Note that the torsion design can be turned off by choosing not to consider torsion in the Design Preferences.

3.4.3.1 Determine Factored Torsion In the design of beam torsion reinforcement, the torsions for each load combi-nation at a particular beam station are obtained by factoring the corresponding torsions for different load cases with the corresponding load combination factors.

In a statically indeterminate structure where redistribution of the torsion in a member can occur due to redistribution of internal forces upon cracking, the design T* is permitted to be reduced in accordance with the code (NZS 7.6.1.3). However, the program does not automatically redistribute the internal forces and

3 - 42 Beam Design


reduce T*. If redistribution is desired, the user should release the torsional degree of freedom (DOF) in the structural model.

3.4.3.2 Determine Special Section Properties For torsion design, special section properties, such as Acp, Aoh, Ao, pcp, and ph, are calculated. These properties are described in the following (NZS 7.1).

Aco = Area enclosed by outside perimeter of concrete cross-section

Ao = Gross area enclosed by shear flow path

pc = Outside perimeter of concrete cross-section

po = Perimeter of area Ao

tc = Assumed wall thickness of an equivalent tube for the gross section

to = Assumed wall thickness of an equivalent tube for the area enclosed by the shear flow path

In calculating the section properties involving reinforcement, such as Ao, po, and to, it is assumed that the distance between the centerline of the outermost closed stirrup and the outermost concrete surface is 50 mm. This is equivalent to a 38-mm clear cover and a 12-mm stirrup. For torsion design of flanged beam sections, it is assumed that placing torsion reinforcement in the flange area is inefficient. With this assumption, the flange is ignored for torsion reinforcement calculation. However, the flange is considered during Tcr calculation. With this assumption, the special properties for a rectangular beam section are given as:

Aco = bh (NZS 7.1)

Ao = (b 2c)(h 2c) (NZS 7.1)

pc = 2b + 2h (NZS 7.1)

po = 2(b 2c) + 2(h 2c) (NZS 7.1)

tc = 0.75 Ao/po (NZS 7.1)

to = 0.75 Aco/pc (NZS 7.1)

Beam Design 3 - 43


where, the section dimensions b, h, and c are shown in Figure 9-3. Similarly, the special section properties for a flanged beam section are given as:

Aco = bwh + (bf bw)hf (NZS 7.1)

Ao = (bw 2c)(h 2c) (NZS 7.1)

pc = 2bf + 2h (NZS 7.1)

po = 2(h 2c) + 2(bw 2c) (NZS 7.1)

tc = 0.75 Ao/po (NZS 7.1)

to = 0.75 Aco/pc (NZS 7.1)

where the section dimensions bf, bw, h, hf, and c for a flanged beam are shown in Figure 3-9. Note that the flange width on either side of the beam web is limited to the smaller of 3hf (NZS 7.6.1.7).

cb 2

h

sd

Closed Stirrup in Rectangular Beam

Closed Stirrup in T-Beam Section

ch 2 h

b

ch 2

wb



ch 2 ch 2

fh

fb

hh

b

c

c

c c

c

c

b 2c

wb 2c

wb

cb 2

h

sd



ch 2 h

b

ch 2

wb



ch 2 ch 2

fh

fb

hh

b

c

c

c c

c

c

b 2c

wb 2c

wb

Figure 3-9 Closed stirrup and section dimensions for torsion design

3.4.3.3 Determine Critical Torsion Capacity The critical torsion capacity, Tcr, for which the torsion in the section can be ignored is calculated as:

3 - 44 Beam Design


0.1 'cr t co c cT A t f= (NZS 7.6.1.2)

where Aco and tc are as described in the previous section, and f 'c is the specified concrete compressive strength. The stress due to torsion should also be limited in order to ignore torsion, defined as:

*

0.08 ' .2 ct o o

T fA t

(NZS 7.6.1.3)

3.4.3.4 Determine Torsion Reinforcement If the factored torsion, T*, is less than the threshold limit, Tcr, and meets the torsion stress limit, torsion can be safely ignored (NZS 7.6.1). In that case, the program reports that no torsion reinforcement is required. However, if T* ex-ceeds the threshold limit, it is assumed that the torsional resistance is provided by closed stirrups and longitudinal bars (NZS 7.6.4.1).

If T* > Tcr and/or the torsion stress limit is not met, the required closed stirrup area per unit spacing, At /s, is calculated as:

,t tn oyt

A v ts f= (NZS 7.6.4.2)

and the required longitudinal reinforcement is calculated as:

,tn o oly

v t pAf

= (NZS 7.6.4.3)

where the torsional shear stress vtn is defined as:

*

.2tn t o o

TvA t

=

(NZS 7.6.1.6)

The minimum closed stirrups and longitudinal reinforcement shall be such that the following is satisfied, where At /s can be from any closed stirrups for shear and Al can include flexure reinforcement, provided it is fully developed.

1.5 .t l o co y o

A A A tsp f A

= (NZS 7.6.2)

Beam Design 3 - 45


The term AtAl /po shall not be taken greater than 7At/s.

An upper limit of the combination of V* and T* that can be carried by the section is also checked using the equation:

min(0.2 ' ,8 MPa)n tn cv v f+ < (NZS 7.6.1.8)

For rectangular sections, bw is replaced with b. If the combination of V* and T* exceeds this limit, a failure message is declared. In that case, the concrete section should be increased in size.

The maximum of all of the calculated Al and At /s values obtained from each load combination is reported along with the controlling combination.

The beam torsion reinforcement requirements reported by the program are based purely on strength considerations. Any minimum stirrup requirements or lon-gitudinal rebar requirements to satisfy spacing considerations must be investi-gated independently of the program by the user.

3.5 Joint Design 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 checked. The material properties of the joint are assumed to be the same as those of the column below the joint.

The joint analysis is completed 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, huV

Determine the effective area of the joint

Check panel zone shear stress

The algorithms associated with these three steps are described in detail in the following three sections.

3 - 46 Joint Design


3.5.1 Determine the Panel Zone Shear Force

Figure 3-9 illustrates the free body stre

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CFD-NZS-3101-2006