Shear Wall Design Manual

Transcription

1 Shear Wall Design Manual Turkish TS with Seismic Code 2007

2 Shear Wall Design Manual Turkish TS with Turkish Seismic Code 2007 For ETABS 2016 ISO ETA122815M51 Rev. 0 Proudly developed in the United States of America December 2015

3 Copyright Copyright Computers & Structures, Inc., All rights reserved. The CSI Logo, SAP2000, ETABS, and SAFE are registered trademarks of Computers & Structures, Inc. Watch & Learn TM 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 explicitly 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. info@csiamerica.com (for general information) support@csiamerica.com (for technical support)

4 DISCLAIMER CONSIDERABLE TIME, EFFORT AND EXPENSE HAVE GONE INTO THE DEVELOPMENT AND DOCUMENTATION 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.

5 Contents 1 Introduction 1.1 Notation Design Station Locations Default Design Load Combinations Dead Load Component Live Load Component Wind Load Component Earthquake Load Component Combinations that Include a Response Spectrum Combinations that Include Time History Results Combinations that Include Static Nonlinear Results Shear Wall Design Preferences Shear Wall Design Overwrites Choice of Units Pier Design 2.1 Wall Pier Shear Design Determine the Concrete Shear Capacity Determine the Require Shear Reinforcing Wall Pier End Zones Details of Check for Wall End Zone Requirements 2-3 i

6 Shear Wall Design TS Reinforcement for Wall End Zones Wall Pier Flexural Design Designing a Simplified Pier Section Checking a General or Uniform Reinforcing Pier Section Wall Pier Demand/Capacity Ratio Designing a General Reinforcing Pier Section Spandrel Design 3.1 Spandrel Flexural Design Determine the Maximum Factored Moments Determine the Required Flexural Reinforcing Spandrel Shear Design Determine the Concrete Shear Capacity Determine the Required Shear Reinforcing 3-11 Appendix A Shear Wall Design Preferences Appendix B Design Procedure Overwrites Appendix C Analysis Sections and Design Sections Bibliography ii

7 Chapter 1 Introduction This manual describes the details of the shear wall design and stress check algorithms used by the program when the user selects the TS design code. The various notations used in this manual are described in Section 1.1. The design is based on loading combinations specified by the user (Section 1.2). To facilitate the design process, the program provides a set of default load combinations that should satisfy requirements for the design of most building type structures. The program performs the following design, check, or analysis procedures in accordance with TS requirements: Design and check of concrete wall piers for flexural and axial loads (Chapter 2) Design of concrete wall piers for shear (Chapter 2) Design of concrete shear wall spandrels for flexure (Chapter 3) Design of concrete wall spandrels for shear (Chapter 3) Consideration of the wall end zones requirements for concrete wall piers using an approach based on the requirements of Turkish Earthquake code 2007 Section in TS (Chapter 3) 1-1

8 Shear Wall Design TS The program provides detailed output data for Simplified pier section design, Section Designer pier section design, Section Designer pier section check, and Spandrel design (Chapter 4) Notation Following is the notation used in this manual. A cv Area of concrete used to determine shear stress, mm 2 A g Gross area of concrete, mm 2 A h-min Minimum required area of distributed horizontal reinforcing steel required for shear in a wall spandrel, mm 2 / mm A s Area of tension reinforcing steel, mm 2 A sc A sc-max A sf A st A st-max A sw /s A swd A sw-min/s Area of reinforcing steel required for compression in a pier edge member, or the required area of tension steel required to balance the compression steel force in a wall spandrel, mm 2 Maximum area of compression reinforcing steel in a wall pier edge member, mm 2 The required area of tension reinforcing steel for balancing the concrete compression force in the extruding portion of the concrete flange of a T-beam, mm 2 Area of reinforcing steel required for tension in a pier edge member, mm 2 Maximum area of tension reinforcing steel in a wall pier edge member, mm 2 Area of reinforcing steel required for shear, mm 2 / mm Area of diagonal shear reinforcement in a coupling beam, mm 2 Minimum required area of distributed vertical reinforcing steel required for shear in a wall spandrel, mm 2 / mm 1-2 Notation

9 Chapter 1 Introduction A stw The required area of tension reinforcing steel for balancing the concrete compression force in a rectangular concrete beam, or for balancing the concrete compression force in the concrete web of a T-beam, mm 2 A' s Area of compression reinforcing steel in a spandrel, mm 2 B 1, B 2... C c Length of a concrete edge member in a wall with uniform thickness, mm Concrete compression force in a wall pier or spandrel, pounds C f Concrete compression force in the extruding portion of a T- beam flange, pounds C s C w D/C DB1 DB2 E s IP-max IP-min L BZ Compression force in wall pier or spandrel reinforcing steel, pounds Concrete compression force in the web of a T-beam, pounds Demand/Capacity ratio as measured on an interaction curve for a wall pier, unitless Length of a user-defined wall pier edge member, mm. This can be different on the left and right sides of the pier, and it also can be different at the top and the bottom of the pier. Width of a user-defined wall pier edge member, mm. This can be different on the left and right sides of the pier, and it also can be different at the top and the bottom of the pier. Modulus of elasticity of reinforcing steel, MPa The maximum ratio of reinforcing considered in the design of a pier with a Section Designer section, unitless The minimum ratio of reinforcing considered in the design of a pier with a Section Designer section, unitless Horizontal length of the wall end zone at each end of a wall pier, mm Notation 1-3

10 Shear Wall Design TS L w L s LL M r M d M dc M ds M dw N b NC max N d N left N max N max Factor Horizontal length of wall pier, mm. This can be different at the top and the bottom of the pier Horizontal length of wall spandrel, mm Live load Bending resistance, N-mm Designed bending moment at a design section, N-mm In a wall spandrel with compression reinforcing, the designed bending moment at a design section resisted by the couple between the concrete in compression and the tension steel, N-mm In a wall spandrel with compression reinforcing, the designed bending moment at a design section resisted by the couple between the compression steel and the tension steel, N-mm In a wall spandrel with a T-beam section and compression reinforcing, the designed bending moment at a design section resisted by the couple between the concrete in compression in the web and the tension steel, N-mm Axial load capacity at balanced strain conditions, N Maximum ratio of compression steel in an edge member of a wall pier, unitless Designed factored axial load at a section, N Equivalent axial force in the left edge member of a wall pier used for design, N. This may be different at the top and the bottom of the wall pier. Limit on the maximum compressive design strength specified by TS , N Factor used to reduce the allowable maximum compressive design strength, unitless. The TS specifies this factor to be 1.0. This factor can be revised in the preferences. 1-4 Notation

11 Chapter 1 Introduction N 0 N oc N ot N right NT max OC OL R LW RLL T s V cr V ds V d WL a a 1 Axial load capacity at zero eccentricity, N The maximum compression force a wall pier can carry with strength reduction factors set equal to one, N The maximum tension force a wall pier can carry with strength reduction factors set equal to one, N Equivalent axial force in the right edge member of a wall pier used for design, pounds. This may be different at the top and the bottom of the wall pier. Maximum ratio of tension steel in an edge member of a wall pier, unitless On a wall pier interaction curve the "distance" from the origin to the capacity associated with the point considered On a wall pier interaction curve the "distance" from the origin to the point considered Shear strength reduction factor as specified in the concrete material properties, unitless. This reduction factor applies to light-weight concrete. It is equal to 1 for normal weight concrete. Reduced live load Tension force in wall pier reinforcing steel, N Shear at inclined cracking, N The portion of the shear force in a spandrel carried by the shear reinforcing steel, N Designed shear force at a design section, N Wind load Depth of the wall pier or spandrel compression block, mm Depth of the compression block in the web of a T-beam, mm Notation 1-5

12 Shear Wall Design TS b s c d r-bot d r-top d s d spandrel f y f ys f' c f' cs Width of the compression flange in a T-beam, mm. This can be different on the left and right ends of the T-beam Distance from the extreme compression fiber of the wall pier or spandrel to the neutral axis, mm Distance from bottom of spandrel beam to centroid of the bottom reinforcing steel, mm. This can be different on the left and right ends of the beam. Distance from top of spandrel beam to centroid of the top reinforcing steel, mm. This can be different on the left and right ends of the beam. Depth of the compression flange in a T-beam, mm. This can be different on the left and right ends of the T-beam. Depth of spandrel beam minus cover to centroid of reinforcing, mm Yield strength of steel reinforcing, N/mm 2. This value is used for flexural and axial design calculations. Yield strength of steel reinforcing, N/mm 2. This value is used for shear design calculations. Concrete compressive strength, N/mm 2. This value is used for flexural and axial design calculations. Concrete compressive strength, N/mm 2. This value is used for shear design calculations. f' s Stress in compression steel of a wall spandrel, N/mm 2. h s p max p min Height of a wall spandrel, mm. This can be different on the left and right ends of the spandrel. Maximum ratio of reinforcing steel in a wall pier with a Section Designer section that is designed (not checked), unitless. Minimum ratio of reinforcing steel in a wall pier with a Section Designer section that is designed (not checked), unitless. 1-6 Notation

13 Chapter 1 Introduction t w t s ΣDL ΣLL ΣRLL α ε ε s ε' s γ m γ mc γ ms Thickness of a wall pier, mm. This can be different at the top and bottom of the pier. Thickness of a wall spandrel, mm. This can be different on the left and right ends of the spandrel. The sum of all dead load cases The sum of all live load cases The sum of all reduced live load cases The angle between the diagonal reinforcing and the longitudinal axis of a coupling beam Reinforcing steel strain, unitless Reinforcing steel strain in a wall pier, unitless Compression steel strain in a wall spandrel, unitless Material factor Material factor for concrete Material factor for steel 1.2. Design Station Locations The program designs wall piers at stations located at the top and bottom of the pier only. To design at the mid-height of a pier, break the pier into two separate "half-height" piers. The program designs wall spandrels at stations located at the left and right ends of the spandrel only. To design at the mid-length of a spandrel, break the spandrel into two separate "half-length" piers. Note that if a spandrel is into broken into pieces, the program will calculate the seismic diagonal shear reinforcing separately for each piece. The angle used to calculate the seismic diagonal shear reinforcing for each piece is based on the length of the piece, not the length of the entire spandrel. This can cause the required area of diagonal reinforcing to be significantly underestimated. Thus, if you break a Design Station Locations 1-7

14 Shear Wall Design TS spandrel into pieces, calculate the seismic diagonal shear reinforcing separately by hand Default Design Load Combinations The design load combinations are the various combinations of the load cases for which the structure is to be checked. For this code, if a structure is subjected to dead (G), live (Q), wind (W), and earthquake (E), and considering that wind and earthquake forces are reversible, the following load combinations may need to be defined (TS 6.2.6): 1.4G + 1.6Q (TS 6.3) 0.9G ± 1.3W (TS 6.6) 1.0G + 1.3Q ± 1.3W (TS 6.5) 0.9G ± 1.0E (TS 6.8a) 1.0G + 1.0Q ± 1.0E (TS 6.7a) 1.0G + 1.0Q 1.0E (TS 6.7b) These also are the default design load combinations in the program whenever the TS 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 case on a member-by-member basis to reduce the contribution of the live load to the factored loading Dead Load Component The dead load component of the default design load combinations consists of the sum of all dead loads multiplied by the specified factor. Individual dead load cases are not considered separately in the default design load combinations. See the description of the earthquake load component later in this chapter for additional information. 1-8 Default Design Load Combinations

15 Chapter 1 Introduction Live Load Component The live load component of the default design load combinations consists of the sum of all live loads, both reducible and unreducible, multiplied by the specified factor. Individual live load cases are not considered separately in the default design load combinations Wind Load Component The wind load component of the default design load combinations consists of the contribution from a single wind load case. Thus, if multiple wind load cases are defined in the program model, the preceding equations will contribute multiple design load combinations, one for each wind load case that is defined Earthquake Load Component The earthquake load component of the default design load combinations consists of the contribution from a single earthquake load case. Thus, if multiple earthquake load cases are defined in the program model, the preceding equations will contribute multiple design load combinations, one for each earthquake load case that is defined. The earthquake load cases considered when creating the default design load combinations include all static load cases that are defined as earthquake loads and all response spectrum cases. Default design load combinations are not created for time history cases or for static nonlinear cases Combinations that Include a Response Spectrum In the program all response spectrum cases are assumed to be earthquake load cases. Default design load combinations are created that include the response spectrum cases. The output from a response spectrum is all positive. Any program shear wall design load combination that includes a response spectrum load case is checked for all possible combinations of signs on the response spectrum values. Thus, when checking shear in a wall pier or a wall spandrel, the response spectrum contribution of shear to the design load combination is considered once as a positive shear and then a second time as a negative shear. Similarly, when checking moment in a wall spandrel, the response spectrum contri- Default Design Load Combinations 1-9

16 Shear Wall Design TS bution of moment to the design load combination is considered once as a positive moment and then a second time as a negative moment. When checking the flexural behavior of a two-dimensional wall pier or spandrel, four possible combinations are considered for the contribution of response spectrum load to the design load combination. They are: +N and +M +N and M N and +M n and M where N is the axial load in the pier and M is the moment in the pier. Similarly, eight possible combinations of N, M2 and M3 are considered for threedimensional wall piers. Note that based on the preceding, TS 6.8a and TS 6.8b are redundant for a load combination with a response spectrum, and similarly, TS 6.7a and TS 6.7b are redundant for a load combination with a response spectrum. For this reason, the program creates default design load combinations based on TS 6.8a and TS 6.7a only for response spectra. Default design load combinations using TS 6.8b and TS 6.7b are not created for response spectra Combinations that Include Time History Results The default shear wall design load combinations do not include any time history results. To include time history forces in a design load combination, the user must define the load combination. When a design load combination includes time history results, the design can be for the envelope of those results or for each step of the time history. The type of time history design can be specified in the shear wall design preferences (Appendix A). When envelopes are used, the design is for the maximum of each response quantity (axial load, moment, and the like) as if they occurred simultaneously. Typically, this is not the realistic case, and in some instances, it may be unconservative. Designing for each step of a time history gives the correct correspondence between different response quantities, but designing for each step can be very time consuming Default Design Load Combinations

17 Chapter 1 Introduction When the program gets the envelope results for a time history, it gets a maximum and a minimum value for each response quantity. Thus, for wall piers it gets maximum and minimum values of axial load, shear and moment; and for wall spandrels, it gets maximum and minimum values of shear and moment. For a design load combination in the program shear wall design module, any load combination that includes a time history load case in it is checked for all possible combinations of maximum and minimum time history design values. Thus, when checking shear in a wall pier or a wall spandrel, the time history contribution of shear to the design load combination is considered once as a maximum shear and then a second time as a minimum shear. Similarly, when checking moment in a wall spandrel, the time history contribution of moment to the design load combination is considered once as a maximum moment and then a second time as a minimum moment. When checking the flexural behavior of a wall pier, four possible combinations are considered for the contribution of time history load to the design load combination. They are: N max and M max N max and M min N min and M max N min and M min where N is the axial load in the pier and M is the moment in the pier. If a single design load combination has more than one time history case in it, that design load combination is designed for the envelopes of the time histories, regardless of what is specified for the Time History Design item in the preferences Combinations That Include Static Nonlinear Results The default shear wall design load combinations do not include any static nonlinear results. To include static nonlinear results in a design load combination, define the load combination yourself. If a design load combination includes a single static nonlinear case and nothing else, the design is performed for each step of the static nonlinear analysis. Otherwise, the design is performed for the last step of the static nonlinear analysis only. Default Design Load Combinations 1-11

18 Shear Wall Design TS Shear Wall Design Preferences The shear wall design preferences are basic properties that apply to all wall pier and spandrel elements. Appendix A identifies shear wall design preferences for TS Default values are provided for all shear wall design preference items. Thus, it is not required that preferences be specified. However, at least review the default values for the preference items to make sure they are acceptable. Please consult the program-specific on-line Help for information about reviewing and updating preferences Shear Wall Design Overwrites The shear wall design overwrites are basic assignments that apply only to those piers or spandrels to which they are assigned. The overwrites for piers and spandrels are separate. Appendix B identifies the shear wall overwrites for TS Note that the available overwrites change depending on the pier section type (Uniform Reinforcing, General Reinforcing, or Simplified T and C). Default values are provided for all pier and spandrel overwrite items. Thus, it is not necessary to specify or change any of the overwrites. However, at least review the default values for the overwrite items to make sure they are acceptable. When changes are made to overwrite items, the program applies the changes only to the elements to which they are specifically assigned; that is, to the elements that are selected when the overwrites are changed. Please consult the program-specific on-line Help for information about reviewing and updating preferences Choice of Units For shear wall design in this program, any set of consistent units can be used for input. Also, the system of units being used can be changed at any time. Typically, design codes are based on one specific set of units. The TS code is based on Newton-Millimeter-Second units. For simplicity, all equations and descriptions presented in this manual correspond to Newton-Millimeter-Second units unless otherwise noted. The shear wall design preferences allow the user to specify special units for concentrated and distributed areas of reinforcing. These units are then used for reinforcing in the model, regardless of the current model units displayed in the drop-down list on the status bar (or within a specific form). The special 1-12 Shear Wall Design Preferences

19 Chapter 1 Introduction units specified for concentrated and distributed areas of reinforcing can only be changed in the shear wall design preferences. The choices available in the shear wall design preferences for the units associated with an area of concentrated reinforcing are in 2, cm 2, mm 2, and current units. The choices available for the units associated with an area per unit length of distributed reinforcing are in 2 /ft, cm 2 /m. mm 2 /m, and current units. The current units option uses whatever units are currently displayed in the drop-down list on the status bar (or within a specific form). If the current length units are m, this option means concentrated areas of reinforcing are in m 2 and distributed areas of reinforcing are in m 2 /m. Note that when using the "current" option, areas of distributed reinforcing are specified in Length 2 /Length units, where Length is the currently active length unit. For example, if you are working in kn and m units, the area of distributed reinforcing is specified in m 2 /m. If you are in Newton and mm, the area of distributed reinforcing is specified in mm 2 /mm. Choice of Units 1-13

20 Chapter 2 Pier Design This chapter describes how the program designs each leg of concrete wall piers for shear using TS Note that in this program shear reinforcing cannot be specified and then checked by the program. The program only designs the pier for shear and reports how much shear reinforcing is required. The shear design is performed at stations at the top and bottom of the pier. This chapter also describes how the program designs and checks concrete wall piers for flexural and axial loads using TS The menu option TS also covers the Specification for Structures to be Built in SEISMIC Areas, Part III - Earthquake Disaster Prevention (TCS 2007). First we describe how the program designs piers that are specified by a Simplified Section. Next we describe how the program checks piers that are specified by a Section Designer Section. Then we describe how the program designs piers that are specified by a Section Designer Section. 2.1 Wall Pier Shear Design The wall pier shear reinforcing is designed for each of the design load combinations. The following steps are involved in designing the shear reinforcing for a particular wall pier section for a particular design loading combination. Determine the design factored forces N d, M d and V d that are acting on the wall pier section. 2-1

21 Shear Wall Design TS Determine the shear force, V c, that can be carried by the concrete. Determine the required shear reinforcing to carry the balance of the shear force. Step 1 needs no further explanation. The following two sections describe in detail the algorithms associated with the Steps 2 and Determine the Concrete Shear Capacity Given the design force set N d, M d and V d acting on a wall pier section, the shear force carried by the concrete, V c, is calculated as. V = 0.65 f A (TS 8.1.3, EDP ) c ctd ch The shear force is limited to a maximum of Vmax = 0.22 f cd A ch (TS 8.1.5b, TCS ) Determine the Required Shear Reinforcing Given V d and V c, the following equation provides the required shear reinforcing in area per unit length (e.g., mm 2 /mm) for both seismic and nonseismic wall piers (as indicated by the "Design is Seismic" item in the pier design overwrites). Note that additional requirements for seismic piers are provided later in this section. where, A sh Vd Vc = (TS 8.1.4, TCS ) f (0.8 L ) yd w Vr = 0.65 fctd Ach + Ash fyd must not exceed 0.22 fcd Ach in accordance with TCS Section , Eqn. 3.17b. H w For walls 2.0, A sh shall be of the gross section area of the wall lw web remaining in between the wall end zones. In cases where H w 2.0, the lw wall web section is the full section of the wall. 2-2 Wall Pier Shear Design

22 Chapter 2 Pier Design 2.2 Wall Pier End Zones This section describes how the program considers the wall end zones requirements for each leg of concrete wall piers using TCS 2007 code. The program uses an approach based on the requirements of Section of TCS Note that the wall end zones requirements are considered separately for each design load case that includes seismic load Details of a Check for Wall End Zone Requirements The following information is available for the wall end zone check: The height of the entire wall, H w, length of the wall pier, l w, and the gross area of the pier, A g. (Refer to Figure 2-5 later in this chapter for an illustration of the dimensions l w and b w, the thickness of the wall pier.) The area of steel in the pier, A st. This area of steel is calculated by the program, or it is provided by the user. The symmetry of the wall pier (i.e., is the left side of the pier the same as the right side of the pier). Only the geometry of the pier is considered, not the reinforcing, when determining if the pier is symmetrical. Figure 2-1 shows some examples of symmetrical and unsymmetrical wall piers. Note that a pier defined using Section Designer is assumed to be unsymmetrical, unless it is made up of a single rectangular shape. Figure 2-1 Example Plan Views of Symmetrical and Unsymmetrical Wall Piers The critical wall height is measured from the foundation or Base level in ETABS using the following criteria: Wall Pier End Zones 2-3

23 Shear Wall Design TS H cr l H w w /6 (TCS 3.6.2, Eqn 3.15a and 3.15b) H cr 2l (TCS 3.6.2) w In structural walls with rectangular cross-sections, the wall end zone length l u is computed as follows: l l u u 2bw 0.2l bw 0.1l w w along the critical wall height, and outside the critical wall height (TCS ) If wall end zones are required, the program calculates the minimum required length of the wall end zone at each end of the wall, l u, in accordance with the requirements of TCS Section Figure 2-2 illustrates the boundary zone length l u. Figure 2-2 Illustration of Wall End Zone Length, lu Reinforcement for Wall End Zones The vertical reinforcement at each wall end zone is required by TCS Sections The program computes and reports the total cross-sectional area of vertical reinforcement as follows: 0.02 bl wu, along critical wall height As (TCS ) 0.01 bl, outside critical wall height wu A s A (TCS ) sv g 2-4 Wall Pier End Zones

24 Chapter 2 Pier Design 2.3 Wall Pier Flexural Design For both designing and checking piers, it is important to understand the local axis definition for the pier. Access the local axes assignments using the Assign menu Designing a Simplified Pier Section This section describes how the program designs a pier that is assigned a simplified section. The geometry associated with the simplified section is illustrated in Figure 2-3. The pier geometry is defined by a length, thickness and size of the edge members at each end of the pier (if any). Figure 2-3 Typical Wall Pier Dimensions Used for Simplified Design A simplified C and T pier section is always planar (not three-dimensional). The dimensions shown in the figure include the following: Wall Pier Flexural Design 2-5

25 Shear Wall Design TS The length of the wall pier is designated l w. This is the horizontal length of the wall pier in plan. The thickness of the wall pier is designated b w. The thickness specified for left and right edge members (DB2 left and DB2 right) may be different from this wall thickness. DB1 represents the horizontal length of the pier end zone. DB1 can be different at the left and right sides of the pier. DB2 represents the horizontal width (or thickness) of the pier end zone. DB2 can be different at the left and right sides of the pier. The dimensions illustrated are specified in the shear wall overwrites (Appendix B) and can be specified differently at the top and bottom of the wall pier. If no specific end zone dimensions have been specified by the user, the program assumes that the end zone is the same width as the wall, and the program determines the required length of the end zone. In all cases, whether the end zone size is user-specified or program-determined, the program reports the required area of reinforcing steel at the center of the end zone. This section describes how the program-determined length of the end zone is determined and how the program calculates the required reinforcing at the center of the end zone. Three design conditions are possible for a simplified wall pier. These conditions, illustrated in Figure 2-4, are as follows: The wall pier has program-determined (variable length and fixed width) end zones on each end. The wall pier has user-defined (fixed length and width) end zones on each end. The wall pier has a program-determined (variable length and fixed width) end zone on one end and a user-defined (fixed length and width) end zone on the other end Design Condition 1 Design condition 1 applies to a wall pier with uniform design thickness and program-determined end zone length. For this design condition, the design algorithm focuses on determining the required size (length) of the end zones, while limiting the compression and tension reinforcing located at the center of 2-6 Wall Pier Flexural Design

26 Chapter 2 Pier Design the end zones to user-specified maximum ratios. The maximum ratios are specified in the shear wall design preferences and the pier design overwrites as Edge Design PC-Max and Edge Design PT-Max. Design Condition 1 Wall pier with uniform thickness and ETABS-determined (variable length) edge members Design Condition 2 Wall pier with user-defined edge members Design Condition 3 Wall pier with a user-defined edge member on one end and an ETABSdetermined (variable length) edge member on the other end Note: In all three conditions, the only reinforcing designed by ETABS is that required at the center of the edge members Figure 2-4 Design Conditions for Simplified Wall Piers Consider the wall pier shown in Figure 2-5. For a given design section, say the top of the wall pier, the wall pier for a given design load combination is designed for a factored axial force N d-top and a factored moment M d-top. The program initiates the design procedure by assuming an end zone at the left end of the wall of thickness b w and width B 1-left, and an end zone at the right end of the wall of thickness b w and width B 1-right. Initially B 1-left = B 1-right = b w. The moment and axial force are converted to an equivalent force set N left-top and N right-top using the relationships shown below. (Similar equations apply at the bottom of the pier.) N N left-top right-top Nd-top Md-top = ( lw B1-left B1-right ) Nd-top Md-top = ( lw B1-left B1-right ) For any given loading combination, the net values for N left-top and N right-top could be tension or compression. Wall Pier Flexural Design 2-7

27 Shear Wall Design TS L p t p 0.5t p 0.5t p t p t p B 1-left B 1-right B2-left B 3-left B 2-right B 3-right L p C L Wall Pier Plan P left-top P u-top Mu-top P right-top Top of pier Left edge member Right edge member M u-bot Bottom of pier P left-bot P u-bot Wall Pier Elevation P right-bot Figure 2-5 Wall Pier for Design Condition Wall Pier Flexural Design

28 Chapter 2 Pier Design Note that for dynamic loads, N left-top and N right-top are obtained at the modal level and the modal combinations are made before combining with other loads. Also for design loading combinations involving SRSS, the N left-top and N right-top forces are obtained first for each load case before the combinations are made. If any value of N left-top or N right-top is tension, the area of steel required for tension, A st, is calculated as: A st N =. f yd If any value of N left-top or N right-top is compression, for section adequacy, the area of steel required for compression, A sc, must satisfy the following relationship. ( max ) Abs ( N) = N Factor [0.85 f ( A A ) + f A ] cd g sc yd sc where N is either N left-top or N right-top, A g = b wb 1, and the N max Factor is defined in the shear wall design preferences (the default is 0.80). In general, we recommend use of the default value. A sc = Abs ( N) ( N Factor) max f yd 0.85 f 0.85 f cd cd A g. If A sc calculates as negative, no compression reinforcing is needed. The maximum tensile reinforcing to be packed within the b p times B 1 concrete end zone is limited by: A = NT b B st-max max w 1. Similarly, the compression reinforcing is limited by: A NC b B sc-max = max w 1. If A st is less than or equal to A st-max and A sc is less than or equal to A sc-max, the program will proceed to check the next loading combination; otherwise the program will increment the appropriate B 1 dimension (left, right or both, depending on which end zone is inadequate) by one-half of the wall thickness to B 2 (i.e., 1.5b w) and calculate new values for N left-top and N right-top resulting in new values of A st and A sc. This iterative procedure continues until A st and A sc are within the allowed steel ratios for all design load combinations. Wall Pier Flexural Design 2-9

29 Shear Wall Design TS If the value of the width of the end zone B increments to where it reaches a value larger than or equal to l w /2, the iteration is terminated and a failure condition is reported. This design algorithm is an approximate but convenient algorithm. Wall piers that are declared overstressed using this algorithm could be found to be adequate if the reinforcing steel is user-specified and the wall pier is accurately evaluated using interaction diagrams Design Condition 2 Design condition 2 applies to a wall pier with user-specified end zones at each end of the pier. The size of the end zones is assumed to be fixed; that is, the program does not modify them. For this design condition, the design algorithm determines the area of steel required in the center end zones and checks if that area gives reinforcing ratios less than the user-specified maximum ratios. The design algorithm used is the same as described for condition 1; however, no iteration is required Design Condition 3 Design condition 3 applies to a wall pier with a user-specified (fixed dimension) end zone at one end of the pier and a variable length (programdetermined) end zone at the other end. The width of the variable length end zone is equal to the width of the wall. The design is similar to that which has previously been described for design conditions 1 and 2. The size of the user-specified end zone is not changed. Iteration occurs on the size of the variable length end zone only Checking a General or Uniform Reinforcing Pier Section When a General Reinforcing or Uniform Reinforcing pier section is specified to be checked, the program creates an interaction surface for that pier and uses that interaction surface to determine the critical flexural demand/capacity ratio for the pier. This section describes how the program generates the interaction surface for the pier and how it determines the demand/capacity ratio for a given design load combination. Note: In this program, the interaction surface is defined by a series of PMM interaction curves that are equally spaced around a 360-degree circle Wall Pier Flexural Design

30 Chapter 2 Pier Design Interaction Surface In this program, a three-dimensional interaction surface is defined with reference to the N d, M 2d and M 3d axes. The surface is developed using a series of interaction curves that are created by rotating the direction of the pier neutral axis in equally spaced increments around a 360-degree circle. For example, if 24 PMM curves are specified (the default), there is one curve every 15 degrees (360 /24 curves = 15 ). Figure 2-6 illustrates the assumed orientation of the pier neutral axis and the associated sides of the neutral axis where the section is in tension (designated T in the figure) or compression (designated C in the figure) for various angles. T C 3 Interaction curve is for a neutral axis parallel to this axis Pier section T C 3 Interaction curve is for a neutral axis parallel to this axis Pier section 2 2 a) Angle is 0 degrees b) Angle is 45 degrees 45 3 Interaction curve is for a neutral axis parallel to this axis Pier section 3 Interaction curve is for a neutral axis parallel to this axis Pier section 2 2 C T a) Angle is 180 degrees b) Angle is 225 degrees 225 C T Figure 2-6 Orientation of the Pier Neutral Axis for Various Angles Note that the orientation of the neutral axis is the same for an angle of θ and θ+180. Only the side of the neutral axis where the section is in tension or compression changes. We recommend use of 24 interaction curves (or more) to define a three-dimensional interaction surface. Each PMM interaction curve that makes up the interaction surface is numerically described by a series of discrete points connected by straight lines. The coordinates of these points are determined by rotating a plane of linear strain Wall Pier Flexural Design 2-11

31 Shear Wall Design TS about the neutral axis on the section of the pier. Details of this process are described later in the section entitled "Details of the Strain Compatibility Analysis." By default, 11 points are used to define a PMM interaction curve. This number can be changed in the preferences; any odd number of points greater than or equal to 11 can be specified, to be used in creating the interaction curve. If an even number is specified for this item in the preferences, the program will increment up to the next higher odd number. Note that when creating an interaction surface for a two-dimensional wall pier, the program considers only two interaction curves the 0 curve and the 180 curve regardless of the number of curves specified in the preferences. Furthermore, only moments about the M3 axis are considered for two-dimensional walls Formulation of the Interaction Surface The formulation of the interaction surface in this program is based consistently on the basic principles of ultimate strength design given in Sections 7.1 of TS The program uses the requirements of force equilibrium and strain compatibility to determine the strength axial load and moment strength (N r, M 2r, M 3r) of the wall pier. For the pier to be deemed adequate, the required strength (N d, M 2d, M 3d) must be less than or equal to the design strength, i.e., (N d, M 2d, M 3d) (N r, M 2r, M 3r) The design strength for concrete and steel is obtained by dividing the characteristic strength of the material by a partial factor of safety, γ mc and γ ms. The values used in the program are as follows: Partial safety factor for steel, γ ms = 1.15, and (TS 6.2.5) Partial safety factor for concrete, γ mc = 1.5. (TS 6.2.5) These factors are already incorporated in the design equations and tables in the code. Although not recommended, the program allows them to be overwritten. If they are overwritten, the program uses them consistently by modifying the code-mandated equations in every relevant place. The theoretical maximum compressive force that the wall pier can carry is designated N oc and is given by: 2-12 Wall Pier Flexural Design N oc = [0.85f cd (A g A st) + f yda st].

32 Chapter 2 Pier Design The theoretical maximum tension force that the wall pier can carry is designated N ot and is given by: N ot = f yda st. If the wall pier geometry and reinforcing is symmetrical in plan, the moments associated with both N oc and N ot are zero. Otherwise, a moment associated will be with both N oc and N ot. The maximum compressive axial load is limited to N r(max), where N r(max) = 0.6 f ck A g for gravity combinations (TS 7.4.1) N r(max) = 0.5 f ck A g for seismic combinations Note: The number of points to be used in creating interaction diagrams can be specified in the shear wall preferences and overwrites. As previously mentioned, by default, 11 points are used to define a single interaction curve. When creating a single interaction curve, the program includes the points at N b, N oc and N ot on the interaction curve. Half of the remaining number of specified points on the interaction curve occur between N b and N oc at approximately equal spacing along the N axis. The other half of the remaining number of specified points on the interaction curve occur between N b and N ot at approximately equal spacing along the N axis. Figure 2-7 shows a plan view of an example two-dimensional wall pier. Notice that the concrete is symmetrical but the reinforcing is not symmetrical in this example. Figure 2-8 shows several interaction surfaces for the wall pier illustrated in Figure " 12'-6" 12 spaces at 1'-0" = 12'-0" 3" # 5@12 o.c., each face, except as noted 2-#9 2-#9 1' 2-#6 f c = 4 ksi fy = 60 ksi Figure 2-7 Example Two-Dimensional Wall Pier With Unsymmetrical Reinforcing Wall Pier Flexural Design 2-13

33 Shear Wall Design TS Figure 2-8 Interaction Curves for Example Wall Pier Shown in Figure 2-7 Note the following about Figure 2-8: Because the pier is two-dimensional, the interaction surface consists of two interaction curves. One curve is at 0 and the other is at 180. Only M 3d moments are considered because this is a two-dimensional example. In this program, compression is negative and tension is positive. The 0 and 180 interaction curves are not symmetric because the wall pier reinforcing is not symmetric. The smaller interaction surface (drawn with a heavier line) has the strength reduction factor, as specified by TS The dashed line shows the effect of setting the N maxfactor to 1.0. The larger interaction surface has both the strength reduction factor and the N maxfactor set to Wall Pier Flexural Design

34 Chapter 2 Pier Design The interaction surfaces shown are created using the default value of 11 points for each interaction curve. Figure 2-9 shows the 0 interaction curves for the wall pier illustrated in Figure 2-7. Additional interaction curves are also added to Figure 2-9. Figure 2-9 Interaction Curves for Example Wall Pier Shown in Figure 2-7 The smaller, heavier curve in Figure 2-9 has the strength reduction factor as specified in TS Details of the Strain Compatibility Analysis As previously mentioned, the program uses the requirements of force equilibrium and strain compatibility to determine the nominal axial load and moment strength (N d, M 2d, M 3d) of the wall pier. The coordinates of these points are determined by rotating a plane of linear strain on the section of the wall pier. Figure 2-10 illustrates varying planes of linear strain such as those that the program considers on a wall pier section for a neutral axis orientation angle of 0 degrees. Wall Pier Flexural Design 2-15

35 Shear Wall Design TS Varying neutral axis locations + ε Varying Linear Strain Diagram ε Plan View of Wall Pier Figure 2-10 Varying Planes of Linear Strain In these planes, the maximum concrete strain is always taken as and the maximum steel strain is varied from to plus infinity. (Recall that in this program compression is negative and tension is positive.) When the steel strain is 0.003, the maximum compressive force in the wall pier, N oc, is obtained from the strain compatibility analysis. When the steel strain is plus infinity, the maximum tensile force in the wall pier, N ot, is obtained. When the maximum steel strain is equal to the yield strain for the reinforcing (e.g., for f yk = 460 MPa), N b is obtained. Figure 2-11 illustrates the concrete wall pier stress-strain relationship that is obtained from a strain compatibility analysis of a typical plane of linear strain shown in Figure In Figure 2-11 the compressive stress in the concrete, C c, is calculated using section 7.2 of TS C c = 0.85f cdk 1cb w (TS 7.1) In Figure 2-10, the value for maximum strain in the reinforcing steel is assumed. Then the strain in all other reinforcing steel is determined based on the assumed plane of linear strain. Next the stress in the reinforcing steel is calculated using the following equation, where ε s is the strain, E s is the modulus of elasticity, σ s is the stress, and f ys is the yield stress of the reinforcing steel Wall Pier Flexural Design

36 Chapter 2 Pier Design T s T s T s T s T s 8 T s 7 T s 6 T s 5 T s 4 C s 3 C s 2 C s 1 C s 0.85f' c C c a = β 1 c 13 ε s 12 ε s Stress Diagram 11 ε s 10 ε s 9 ε s 8 ε s 7 ε s 6 ε s 5 ε s Linear Strain Diagram c 4 ε s 3 ε s 2 ε s 1 ε s t p ε = Plan View of Wall Pier Figure 2-11 Wall Pier Stress-Strain Relationship σ s = ε se s f ys The force in the reinforcing steel (T s for tension or C s for compression) is calculated by: T s or C s = σ sa s For the given distribution of strain, the value of N r is calculated as: N r = (ΣT s C c ΣC s) N max Wall Pier Flexural Design 2-17

37 Shear Wall Design TS In the preceding equation, the tensile force T s and the compressive forces C c and C s are all positive. If N r is positive, it is tension, and if it is negative, it is compression. The value of M 2r is calculated by summing the moments due to all of the forces about the pier local 2 axis. Similarly, the value of M 3r is calculated by summing the moments due to all of the forces about the pier local 3 axis. The forces whose moments are summed to determine M 2r and M 3r are N r, C c, all of the T s forces and all of the C s forces. The N r, M 2r and M 3r values calculated as described in the preceding paragraph make up one point on the wall pier interaction diagram. Additional points on the diagram are obtained by making different assumptions for the maximum steel stress; that is, considering a different plane of linear strain, and repeating the process. When one interaction curve is complete, the next orientation of the neutral axis is assumed and the points for the associated new interaction curve are calculated. This process continues until the points for all of the specified curves have been calculated Wall Pier Demand/Capacity Ratio Refer to Figure 2-12, which shows a typical two-dimensional wall pier interaction diagram. The forces obtained from a given design load combination are N d and M 3d. The point L, defined by (N d, M 3d), is placed on the interaction diagram, as shown in the figure. If the point lies within the interaction curve, the wall pier capacity is adequate. If the point lies outside of the interaction curve, the wall pier is overstressed. As a measure of the stress condition in the wall pier, the program calculates a stress ratio. The 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 (extended outward if needed) intersects the interaction curve. The demand/capacity ratio, D/C, is given by D/C = OL / OC where OL is the "distance" from point O (the origin) to point L, and OC is the "distance" from point O to point C. Note the following about the demand/capacity ratio: If OL = OC (or D/C = 1), the point (N d, M 3d) lies on the interaction curve and the wall pier is stressed to capacity. If OL < OC (or D/C < 1), the point (N d, M 3d) lies within the interaction curve and the wall pier capacity is adequate Wall Pier Flexural Design

38 Chapter 2 Pier Design φp n C P u L Axial Compression Axial Tension O M3 u φm3 n Figure 2-12 Two-Dimensional Wall Pier Demand/Capacity Ratio If OL > OC (or D/C > 1), the point (N d, M 3d) lies outside of the interaction curve and the wall pier is overstressed. The wall pier demand/capacity ratio is a factor that gives an indication of the stress condition of the wall with respect to the capacity of the wall. The demand/capacity ratio for a three-dimensional wall pier is determined in a similar manner to that described here for two-dimensional piers Designing a General Reinforcing Pier Section When a General Reinforcing pier section is specified to be designed, the program creates a series of interaction surfaces for the pier based on the following items: The size of the pier as specified in Section Designer. The location of the reinforcing specified in Section Designer. The size of each reinforcing bar specified in Section Designer relative to the size of the other bars. The interaction surfaces are developed for eight different ratios of reinforcingsteel-area-to-pier-area. The pier area is held constant and the rebar area is mod- Wall Pier Flexural Design 2-19

39 Shear Wall Design TS ified to obtain these different ratios; however, the relative size (area) of each rebar compared to the other bars is always kept constant. The smallest of the eight reinforcing ratios used is that specified in the shear wall design preferences as Section Design IP-Min. Similarly, the largest of the eight reinforcing ratios used is that specified in the shear wall design preferences as Section Design IP-Max. The eight reinforcing ratios used are the maximum and the minimum ratios plus six more ratios. The spacing between the reinforcing ratios is calculated as an increasing arithmetic series in which the space between the first two ratios is equal to one-third of the space between the last two ratios. Table 1 illustrates the spacing, both in general terms and for a specific example, when the minimum reinforcing ratio, IPmin, is and the maximum, IPmax, is After the eight reinforcing ratios have been determined, the program develops interaction surfaces for all eight of the ratios using the process described earlier in the section entitled "Checking a General or Uniform Reinforcing Pier Section." Next, for a given design load combination, the program generates a demand/capacity ratio associated with each of the eight interaction surfaces. The program then uses linear interpolation between the eight interaction surfaces to determine the reinforcing ratio that gives a demand/capacity ratio of 1 (actually the program uses 0.99 instead of 1). This process is repeated for all design load combinations and the largest required reinforcing ratio is reported. Design of a Uniform Reinforcing pier section is similar to that described herein for the General Reinforcing section. Table 2-1 The Eight Reinforcing Ratios Used by the Program Curve Ratio Example 1 IPmin IPmax IPmin IPmin IPmax IPmin IPmin IPmax IPmin IPmin IPmax IPmin IPmin IPmax IPmin IPmin Wall Pier Flexural Design

40 Chapter 2 Pier Design Table 2-1 The Eight Reinforcing Ratios Used by the Program Curve Ratio Example 7 IPmax IPmin IPmin IPmax Wall Pier Flexural Design 2-21

41 Chapter 3 Spandrel Design This chapter describes how the program designs concrete shear wall spandrels for flexure and shear when TS is the selected design code. The program allows consideration of rectangular sections and T-beam sections for shear wall spandrels. Note that the program designs spandrels at stations located at the ends of the spandrel. No design is performed at the center (midlength) of the spandrel. The program does not allow shear reinforcing to be specified and then checked. The program only designs the spandrel for shear and reports how much shear reinforcing is required. 3.1 Spandrel Flexural Design In this program, wall spandrels are designed for major direction flexure and shear only. Effects caused by any axial forces, minor direction bending, torsion or minor direction shear that may exist in the spandrels must be investigated by the user independent of the program. Spandrel flexural reinforcing is designed for each of the design load combinations. The required area of reinforcing for flexure is calculated and reported only at the ends of the spandrel beam. The following steps are involved in designing the flexural reinforcing for a particular wall spandrel section for a particular design loading combination at a particular station. Determine the maximum factored moment M d. 3-1

42 Shear Wall Design TS Determine the required flexural reinforcing. These steps are described in the following sections Determine the Maximum Factored Moments In the design of flexural reinforcing for spandrels, the factored moments for each design load combination at a particular beam station are first obtained. The beam section is then designed for the maximum positive and the maximum negative factored moments obtained from all of the design load combinations Determine the Required Flexural Reinforcing In this program, negative beam moments produce top steel. In such cases, the beam is always designed as a rectangular section. In this program, positive beam moments produce bottom steel. In such cases, the beam may be designed as a rectangular section, or as a T-beam section. Indicate that a spandrel is to be designed as a T-beam by specifying the appropriate slab width and depth dimensions are provided in the spandrel design overwrites (Appendix B). The flexural design procedure is based on a simplified rectangular stress block, as shown in Figure 3-1. The maximum depth of the compression zone, c b, is calculated based on the compressive strength of the concrete and the tensile steel tension using the following equation (TS 7.1): c ε E cu s b = d (TS 7.1) ε cues + fyd The maximum allowable depth of the rectangular compression block, a max, is given by a = 0.85kcb (TS 7.11, 7.3, Eqn. 7.4) max 1 where k 1 is calculated as follows: k 1 = ( 25) f, 0.70 k (TS 7.1, Table 7.1) ck 3-2 Spandrel Flexural Design

43 Chapter 3 Spandrel Design Figure 3-1 Rectangular Spandrel Beam Design, Positive Moment It is assumed that the compression depth carried by the concrete is less than or equal to a max. When the applied moment exceeds the moment capacity at a max, the program calculates an area of compression reinforcement assuming that the additional moment is carried by compression reinforcing and additional tension reinforcing. The procedure used by the program for both rectangular and T-beam sections is given in the subsections that follow Rectangular Beam Flexural Reinforcing Refer to Figure 3-1. For a rectangular beam, the factored moment, M d, is resisted by a couple between the concrete in compression and the tension in reinforcing steel. This is expressed as: a Md = Cc dspandrel 2 (TS 7.1) where Cc = 0.85 fcd at s and d spandrel is equal to h s d r-bot for positive bending and h s d r-top for negative bending. The depth of the compression block, a, is given by: Spandrel Flexural Design 3-3

44 Shear Wall Design TS M a= dspandrel dspandrel 0.85 f d cd t s The program uses the preceding equation to determine the depth of the compression block, a. The depth of the compression block, a, is compared with a max Tension Reinforcing Only Required If a a max, no compression reinforcing is required and the program calculates the area of tension reinforcing, A s = f yd M d d spandrel a 2 The steel is placed at the bottom for positive moment and at the top for negative moment. Note: The program reports the ratio of top and bottom steel required in the web area. When compression steel is required, those ratios may be large because there is no limit on them. However, the program reports an overstress when the ratio exceeds 4% Tension and Compression Reinforcing Required If a > a max, compression reinforcing is required and the program calculates required compression and tension reinforcing as follows. The depth of the concrete compression block, a, is set equal to a = a max. The compressive force developed in the concrete alone is given by C = 0.85 f a t (TS 7.1) c cd max s The moment resisted by the couple between the concrete in compression and the tension steel, M dc, is given by a Mdc = Cc d spandrel 2 max Therefore, the additional moment to be resisted by the couple between the compression steel and the additional tension steel, M ds, is given by 3-4 Spandrel Flexural Design

45 Chapter 3 Spandrel Design Mds = Md M dc The force carried by the compression steel, C s, is given by, C s = d M spandrel ds d r Referring to Figure 3-1, the strain in the compression steel, ε' s, is given by. ( c d ) r ε s = c The stress in the compression steel, f ' s, is given by. σ E ε c d f max s s cu yd c. max The term d r in the preceding equations is equal to d r-top for positive bending and equal to d r-bot for negative bending. The term c is equal to amax k 1. The total required area of compression steel, A' s, is calculated using the following equation. Cs As = ( σ 0.85 f ) s cd The required area of tension steel for balancing the compression in the concrete web, A sw, is: A sw = f yd d M dc spandrel a 2 max The required area of tension steel for balancing the compression steel, A sc, is: A sc = yd M ds ( spandrel r ) f d d In the preceding equations, d spandrel is equal to h s d r-bot for positive bending and h s d r-top for negative bending. In the preceding equations, d r is equal to d r-top for positive bending and d r-bot for negative bending. Spandrel Flexural Design 3-5

46 Shear Wall Design TS The total tension reinforcement A s is given by: As = Asw + Asc Thus, the total tension reinforcement, A s, and the total compression reinforcement, A s'. A s is to be placed at the bottom of the beam, and A s' at the top for positive bending and vice versa for negative bending T-Beam Flexural Reinforcing T-beam action is considered effective for positive moment only. When designing T-beams for negative moment (i.e., designing top steel), the calculation of required steel is as described in the previous section for rectangular sections. No T-beam data is used in this design. The width of the beam is taken equal to the width of the web. For positive moment, the depth of the compression block, a, is given by 2 2M d a= d d 0.85 f b cd f. If a d s, the subsequent calculations for the reinforcing steel are exactly the same as previously defined for rectangular section design. However, in that case, the width of the compression block is taken to be equal to the width of the compression flange, b s. Compression reinforcement is provided when the dimension "a" exceeds a max. If a > d s, the subsequent calculations for the required area of reinforcing steel are performed in two parts. First, the tension steel required to balance the compressive force in the flange is determined, and second, the tension steel required to balance the compressive force in the web is determined. If necessary, compression steel is added to help resist the design moment. The remainder of this section describes in detail the design process used by the program for T-beam spandrels when a > d s. Refer to Figure 3-2. The compression force in the protruding portion of the flange, C f, is given. The protruding portion of the flange is shown crosshatched. ( ) C = 0.85 f b t d f cd s s s 3-6 Spandrel Flexural Design

47 Chapter 3 Spandrel Design Figure 3-2 Design of a Wall Spandrel with a T-Beam Section, Positive Moment The required area of tension steel for balancing the compression force in the concrete flange, A sf, is: A sf C = f f yd The portion of the total moment, M d, that is resisted by the flange, M df, is given by d Mdf = Cf d 2 s spandrel. Therefore, the balance of the moment to be carried by the web, M dw, is given by Mdw = Md Mdf. The web is a rectangular section of width t s and depth h s for which the design depth of the compression block, a 1, is recalculated as: 2 2M a1 = dspandrel dspandrel 0.85 f dw cd ts Spandrel Flexural Design 3-7