E-TN-SWD-BS8110-97

Notation of 8

©COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIFORNIA AUGUST 2002

SHEAR WALL DESIGN BS 8110-97

Technical NoteGeneral and Notation

The Shear Wall Design BS 8110-97 series of Technical Notes describes thedetails of the structural steel design and stress check algorithms used by theprogram when the user selects the 1997 design code, as amended by Issue 2in September 1998 (BSI 1998). The various notations used in this series aredescribed herein.

The design is based on loading combinations specified by the user. To facili-tate the design process, the program provides a set of default load combina-tions that should satisfy requirements for the design of most building typestructures. See Design Load Combinations for Shear Wall Design BS 8110-97.

The program also performs the following design, check, or analysis proce-dures in accordance with BS 8110-97 requirements:

Design and check of concrete wall piers for flexural and axial loads; seeWall Pier Flexural Design for Shear Wall Design BS 8110-97.

Design of concrete wall piers for shear; see Wall Pier Shear Design forShear Wall Design BS 8110-97.

Design of concrete shear wall spandrels for flexure; see Spandrel FlexuralDesign for Shear Wall Design BS 8110-97.

Design of concrete wall spandrels for shear; see Spandrel Shear Designfor Shear Wall Design BS 8110-97.

The program provides detailed output data for Simplified pier section design,Uniformly reinforced pier section design, Uniformly reinforced pier sectioncheck, Section Designer pier section design, Section Designer pier sectioncheck, and Spandrel design.

SI and MKS metric units as well as English units can be used for input. Theprogram code is based on Newton-Millimeter-Second units. For simplicity, all

Shear Wall Design BS 8110-97 General and Notation

Notation of 8

equations and descriptions presented in this series of Technical Notes corre-sponds to Newton-Millimeter-Second units unless otherwise noted.

NotationFollowing is the notation used in the Shear Wall Design BS 8110-97 series ofTechnical Notes.

Acv Concrete shear wall area. Area of section for shear resistance,mm2.

Ag Gross area of a wall pier, mm2.

As Area of tension reinforcing steel, mm2.

Asc Area of reinforcing steel required for compression in a pieredge member, or the required area of tension steel requiredto balance the compression steel force in a wall spandrel,mm2.

Asc-max Maximum area of compression reinforcing steel in a wall pieredge member, mm2.

Ast Area of reinforcing steel required for tension in a pier edgemember, mm2.

Ast-max Maximum area of tension reinforcing steel in a wall pier edgemember, mm2.

Asv Area of reinforcing steel required for shear, mm2 / mm.

A's Area of compression reinforcing steel in a spandrel, mm2.

B1, B2... Length of a concrete edge member in a wall with uniformthickness, mm.

Cc Concrete compression force in a wall pier or spandrel, N.

Cf Concrete compression force in the extruding portion of a T-beam flange, N.


Notation of 8

Cs Compression force in wall pier or spandrel reinforcing steel, N.

Cw Concrete compression force in the web of a T-beam, N.

D/C Demand/Capacity ratio as measured on an interaction curvefor a wall pier, N.

DB1 Length of a user-defined wall pier edge member, mm. Thiscan be different on the left and right sides of the pier, and italso can be different at the top and the bottom of the pier.See Figure 1 of Technical Note Wall Pier Design Sections.

DB2 Width of a user-defined wall pier edge member, mm. This canbe different on the left and right sides of the pier, and it alsocan be different at the top and the bottom of the pier. SeeFigure 1 of Technical Note Wall Pier Design Sections.

DL Dead load.

E The earthquake load on a structure resulting from the combi-nation of the horizontal component and the vertical compo-nent.

Ec Modulus of elasticity of reinforcing concrete, MPa.

Es Modulus of elasticity of reinforcing steel, MPa (assumed as200,000 MPa) (BS 2.4.2.3).

IP-max The maximum ratio of reinforcing considered in the design ofa pier with a Section Designer section, unitless.

IP-min The minimum ratio of reinforcing considered in the design of apier with a Section Designer section, unitless.

K Nondimensional moment, 2dbf

M

cu

, unitless

K' Nondimensional limiting moment in a beam reinforced only fortension, unitless, 0.156.


Notation of 8

Lp Horizontal length of wall pier, inches. This can be different atthe top and the bottom of the pier.

Ls Horizontal length of wall spandrel, mm.

LL Live load

M Factored bending moment at a design section, N-mm.

M3, M2 Applied moments about the major and minor axes of a shearwall, N-mm

Mr3, Mr2 Moment resistance about the major and minor axes of a shearwall, N-mm

Mf In a wall spandrel with a T-beam section and compression re-inforcing, the factored bending moment at a design sectionresisted by the couple between the concrete in compression inthe extruding portion of the flange and the tension steel, N-mm.

Mr Bending resistance, pound-inches.

Msingle Limiting moment capacity of a beam reinforced only for ten-sion, N-mm.

Mtop Factored bending moment at the top of a pier section, N-mm.

Mw In a wall spandrel with a T-beam section and compression re-inforcing, the factored bending moment at a design sectionresisted by the couple between the concrete in compression inthe web and the tension steel, N-mm.

N Ultimate axial load, N.

OC On a wall pier interaction curve the "distance" from the originto the capacity associated with the point considered.

OL On a wall pier interaction curve the "distance" from the originto the point considered.


Notation of 8

Pb The axial force in a wall pier at a balanced strain condition, N.

Pleft Equivalent axial force in the left edge member of a wall pierused for design, N. This may be different at the top and thebottom of the wall pier.

Pmax Factor Factor used to reduce the allowable maximum compressivedesign strength, unitless. This factor is taken as 0.8 by de-fault. This factor can be revised in the preferences.

Pr Design axial strength (resistance), N.

Pr,max The maximum compression force a wall pier can carry, N.

Pt,max The maximum tension force a wall pier can carry , N.

Pright Equivalent axial force in the right edge member of a wall pierused for design, N. This may be different at the top and thebottom of the wall pier.

P Factored axial force at a design section, N.

PCmax Maximum ratio of compression steel in an edge member of awall pier, unitless.

PTmax Maximum ratio of tension steel in an edge member of a wallpier, unitless.

Ptop Factored axial force at the top of a pier section, N.

RLW Shear strength reduction factor as specified in the concretematerial properties, unitless. This reduction factor applies tolight-weight concrete. It is equal to 1 for normal weight con-crete.

RLL Reducible live load.

Ts Tension force in wall pier reinforcing steel, N.

Vc The portion of the shear force carried by the concrete, N.


Notation of 8

Vs The portion of the shear force in a spandrel carried by theshear reinforcing steel, N.

V Factored shear force at a design section, N.

WL Wind load.

a Depth of the wall pier or spandrel compression block, mm.

a1 Depth of the compression block in the web of a T-beam, mm.

bf Width of the compression flange in a T-beam, mm. This canbe different on the left and right end of the T-beam.

d Depth of spandrel beam minus cover to centroid of reinforcing,inches.

dr-bot Distance from bottom of spandrel beam to centroid of thebottom reinforcing steel, inches. This can be different on theleft and right ends of the beam.

dr-top Distance from top of spandrel beam to centroid of the top re-inforcing steel, inches. This can be different on the left andright ends of the beam.

dv The distance from extreme compression fiber to the centroidof tension steel.

fy Characteristically, yield strength of steel reinforcing, MPa. Thisvalue is used for flexural and axial design calculations.

fys Yield strength of steel reinforcing, MPa. This value is used forshear design calculations.

fcu Characteristic concrete cube compressive strength at 28 days,MPa. This value is used for flexural and axial design calcula-tions.

f's Stress in compression steel of a wall spandrel, MPa.


Notation of 8

hf Depth of the compression flange in a T-beam, inches. This canbe different on the left and right ends of the T-beam.

hs Height of a wall spandrel, mm. This can be different on the leftand right ends of the spandrel.

k1 Shear strength enhancement factor.

k2 Concrete shear strength factor, [fcu/25]1/3

pmax Maximum ratio of reinforcing steel in a wall pier with a SectionDesigner section that is designed (not checked), unitless.

pmin Minimum ratio of reinforcing steel in a wall pier with a SectionDesigner section that is designed (not checked), unitless.

sv Spacing of shear rebar, mm.

tp Thickness of a wall pier, mm. This can be different at the topand bottom of the pier.

ts Thickness of a wall spandrel, mm. This can be different on theleft and right ends of the spandrel.

v Shear stress, MPa.

vc Design ultimate shear stress resistance of a concrete beam,MPa.

vc' Design concrete shear stress corrected for axial forces, MPa.

x Neutral axis depth, mm.

xbal Depth of neutral axis in balanced condition, mm

z Lever arm, mm

ΣDL The sum of all dead load cases.

ΣLL The sum of all live load cases.

ΣRLL The sum of all reduced live load cases.


Reference of 8

ε Reinforcing steel strain, unitless.

εc Maximum allowed compression strain in concrete, 0.0035.

εs Reinforcing steel strain in a wall pier, unitless.

ε's Compression steel strain in a wall spandrel, unitless.

γc Partial safety factor for concrete, unitless. The value is 1.5.

γs Partial safety factor for reinforcing steel, unitless. The value is1.05

γm Partial safety factor for shear, unitless. The value is 1.25.

σs Reinforcing steel stress in a wall pier, MPa.

ReferenceBritish Standards Institution (BSI). 1998. BS 8110: Part 1: 1997. Structural

Use of Concrete, Part 1: Code of Practice for Design and Construction.Incorporated Amendment in Issue 2 (September 1998). 389 ChiswickHigh Road. London, U.K., W4 4AL.

Default Design Load Combinations Page 1 of 5



Technical NoteDesign Load Combinations

This Technical Note defines the default concrete shear wall design load com-binations. Use the default shear wall design load combinations, define yourown combinations, or use both default and user-defined combinations. Modifythe default design load combinations and delete them as necessary.

Note:

The program automatically creates code-specific design load combinations for shear walldesign.

Default Design Load CombinationsThe design load combinations automatically created by the program for con-crete shear wall design are given by Equations 1 through 14 (BS 3.1.6,2.4.3.1.1, Table 2.1).

1.4ΣDL Eqn. 1

1.4ΣDL + 1.6(ΣLL + ΣRLL) Eqn. 2

1.2ΣDL + 1.2(ΣLL + ΣRLL) + 1.2WL Eqn. 3

1.2ΣDL + 1.2(ΣLL + ΣRLL) - 1.2WL Eqn. 4

1.4ΣDL + 1.4WL Eqn. 5

1.4ΣDL - 1.4WL Eqn. 6

1.0ΣDL + 1.4WL Eqn. 7

1.0ΣDL - 1.4WL Eqn. 8

1.2ΣDL + 1.2(ΣLL + ΣRLL) + 1.2E Eqn. 9

1.2 [1.2ΣDL + 1.2(ΣLL + ΣRLL) - 1.2E] Eqn. 10

1.4ΣDL + 1.4E Eqn. 11

Shear Wall Design BS 8110-97 Design Load Combinations


1.4ΣDL - 1.4E Eqn. 12

1.0ΣDL + 1.0E Eqn. 13

1.0ΣDL - 1.4E Eqn. 14

In Equations 1 through 14,

ΣDL = The sum of all dead load (DL) load cases defined for the model.

ΣLL = The sum of all live load (LL) load cases defined for the model.Note that this includes roof live loads as well as floor live loads.

ΣRLL = The sum of all reducible live load (RLL) load cases defined for themodel.

WL = Any single wind load (WL) load case defined for the model.

E = Any single earthquake load (E) load case defined for the model.

Dead Load ComponentThe dead load component of the default design load combinations consists ofthe sum of all dead loads multiplied by the specified factor. Individual deadload cases are not considered separately in the default design load combina-tions.

See the description of the earthquake load component later in this TechnicalNote for additional information.

Live Load ComponentThe live load component of the default design load combinations consists ofthe sum of all live loads, both reducible and unreducible, multiplied by thespecified factor. Individual live load cases are not considered separately in thedefault design load combinations.

Wind Load ComponentThe wind load component of the default design load combinations consists ofthe contribution from a single wind load case. Thus, if multiple wind loadcases are defined in the program model, each of Equations 3 through 8 willcontribute multiple design load combinations, one for each wind load casethat is defined.



Earthquake Load ComponentThe earthquake load component of the default design load combinations con-sists of the contribution from a single earthquake load case. Thus, if multipleearthquake load cases are defined in the program model, each of Equations 9through 14 will contribute multiple design load combinations, one for eachearthquake load case that is defined.

The earthquake load cases considered when creating the default design loadcombinations include all static load cases that are defined as earthquake loadsand all response spectrum cases. Default design load combinations are notcreated for time history cases or for static nonlinear cases.

Design Load Combinations That Include a Response SpectrumIn this program, all response spectrum cases are assumed to be earthquakeload cases. Default design load combinations are created that include the re-sponse spectrum cases.

The output from a response spectrum is all positive. Any program shear walldesign load combination that includes a response spectrum load case ischecked for all possible combinations of signs on the response spectrum val-ues. Thus, when checking shear in a wall pier or a wall spandrel, the responsespectrum contribution of shear to the design load combination is consideredonce as a positive shear and then a second time as a negative shear. Simi-larly, when checking moment in a wall spandrel, the response spectrum con-tribution of moment to the design load combination is considered once as apositive moment and then a second time as a negative moment. Whenchecking the flexural behavior of a two-dimensional wall pier or spandrel, fourpossible combinations are considered for the contribution of response spec-trum load to the design load combination. They are:

• +P and +M

• +P and -M

• -P and +M

• -P and -M



where P is the axial load in the pier and M is the moment in the pier. Simi-larly, eight possible combinations of P, M2 and M3 are considered for three-dimensional wall piers.

Note that based on the above, Equations 9 and 14 are redundant for a loadcombination with a response spectrum, and similarly, Equations 9 and 10 areredundant for a load combination with a response spectrum. For this reason,the program only creates default design load combinations based on Equa-tions 9, 11 and 13 for response spectra. Default design load combinationsusing Equations 10, 12 and 14 are not created for response spectra.

Design Load Combinations that Include Time History ResultsThe default shear wall design load combinations do not include any time his-tory results. To include time history forces in a design load combination, de-fine the load combination yourself.

When your design load combination includes time history results, you can ei-ther design for the envelope of those results or you can do a design for eachstep of the time history. You specify the type of time history design in theshear wall design preferences.

When you design for the envelopes, the design is for the maximum of eachresponse quantity (axial load, moment, etc.) as if they occurred simultane-ously. Typically, this is not the realistic case, and in some instances, it may beunconservative. Designing for each step of a time history gives you the cor-rect correspondence between different response quantities, but designing foreach step can be very time consuming.

When the program gets the envelope results for a time history, it gets amaximum and a minimum value for each response quantity. Thus, for wallpiers it gets maximum and minimum values of axial load, shear and moment;and for wall spandrels, it gets maximum and minimum values of shear andmoment. For a design load combination in the shear wall design module, anyload combination that includes a time history load case in it is checked for allpossible combinations of maximum and minimum time history design values.Thus, when checking shear in a wall pier or a wall spandrel, the time historycontribution of shear to the design load combination is considered once as amaximum shear and then a second time as a minimum shear. Similarly, whenchecking moment in a wall spandrel, the time history contribution of momentto the design load combination is considered once as a maximum moment



and then a second time as a minimum moment. When checking the flexuralbehavior of a wall pier, four possible combinations are considered for thecontribution of time history load to the design load combination. They are:

• Pmax and Mmax

• Pmax and Mmin

• Pmin and Mmax

• Pmin and Mmin

where P 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 histo-ries, regardless of what is specified for the Time History Design item in thepreferences.

Design Load Combinations That Include Static Nonlinear ResultsThe default shear wall design load combinations do not include any staticnonlinear results. To include static nonlinear results in a design load combina-tion, define the load combination yourself.

If a design load combination includes a single static nonlinear case and noth-ing else, the design is performed for each step of the static nonlinear analysis.Otherwise, the design is only performed for the last step of the static nonlin-ear analysis.

Overview Page 1 of 16



Technical NoteWall Pier Flexural Design

OverviewThis Technical Note describes how the program designs and checks concretewall piers for flexural and axial loads using BS 8110-97. The Technical Note ispresented into three main sections. First we describe how the program de-signs piers that are specified by a Simplified C & T Section. Next we describehow the program checks piers that are specified by a Section Designer Sec-tion. Then we describe how the program designs piers that are specified by aSection Designer Section. Next, we describe how the program checks and de-signs piers that are specified as a pier with Uniform Reinforcing.

For both designing and checking piers, it is important that you understand thelocal axis definition for the pier.

Designing a Simplified C & T Pier SectionThis section discusses how the program designs a pier that is assigned a Sim-plified C & T Section. The geometry associated with the simplified section isillustrated in "Simplified Pier Design Dimensions and Properties" of TechnicalNote Wall Pier Design Sections Shear Wall Design. The pier geometry is de-fined by a length, thickness and size of the edge members at each end of thepier (if any).

If no specific edge member dimensions have been specified by the user, theprogram assumes that the edge member has the same width as the wall, andthe program determines the required length of the edge member. In allcases, whether the edge member size is user-specified or program-determined, the program reports the required area of reinforcing steel at thecenter of the edge member. This section describes how the program-determined length of the edge member is determined and how the programcalculates the required reinforcing at the center of the edge member.

Shear Wall Design BS 8110-97 Wall Pier Flexural Design

Designing a Simplified C & T Pier Section Page 2 of 16

There are three possible design conditions for a simplified wall pier. Theseconditions, illustrated in Figure 1, are:

1. The wall pier has program-determined (variable length and fixed width)edge members on each end.

2. The wall pier has user-defined (fixed length and width) edge members oneach end.

3. The wall pier has a program-determined (variable length and fixed width)edge member on one end and a user-defined (fixed length and width)edge member on the other end.

Figure 1: Design Conditions for Simplified Wall Piers

Design Condition 1Design condition 1 applies to a wall pier with uniform design thickness andprogram-determined edge member length. For this design condition, the de-sign algorithm focuses on determining the required size (length) of the edgemembers while limiting the compression and tension reinforcing located at thecenter of the edge members to user-specified maximum ratios. The maximumratios are specified in the shear wall design preferences and the pier designoverwrites as Edge Design PC-Max and Edge Design PT-Max.

Design Condition 2

Wall pier with user-defined edgemembers

Design Condition 1

Wall pier with uniform thickness andETABS-determined (variable length)edge members

Design Condition 3

Wall pier with a user-defined edgemember on one end and an ETABS-determined (variable length) edgemember on the other end

Note:

In all three conditions, the onlyreinforcing designed by ETABS is thatrequired at the center of the edgemembers



Consider the wall pier shown in Figure 2. For a given design section, say thetop of the wall pier, the wall pier for a given design load combination is de-signed for a factored axial force Ptop and a factored moment Mtop.

The program initiates the design procedure by assuming an edge member atthe left end of the wall of thickness tp and width B1-left, and an edge memberat the right end of the wall of thickness tp and width B1-right. Initially B1-left =B1-right = tp.

The moment and axial force are converted to an equivalent force set Pleft-top

and Pright-top using the relationships shown in the equations that follow. (Simi-lar equations apply at the bottom of the pier.)

( )right1left1p

toptoptopleft B5.0B5.0L

M

2

PP

−−− −−

+=

( )right1left1p

toptoptopright B5.0B5.0L

M

2

PP

−−− −−

−=

For any given loading combination, the net values for Pleft-top and Pright-top couldbe tension or compression.

Note that for dynamic loads, Pleft-top and Pright-top are obtained at the modallevel and the modal combinations are made, before combining with otherloads. Also for design loading combinations involving SRSS, the Pleft-top andPright-top forces are obtained first for each load case before the combinationsare made.

If any value of Pleft-top or Pright-top is tension, the area of steel required for ten-sion, Ast, is calculated as follows:

syst f

PA

γ= (BS 3.4.4.1)

If any value of Pleft-top or Pright-top is compression, for section adequacy, thearea of steel required for compression, Asc, must satisfy the following relation-ship.

( ) ( ) ]A /f)AA()/f 0.67[(P(P)Abs scscuscgccuFactor max γγ +−= (BS 3.4.4.1)



B1-right

tp

t p

B2-right

B3-right

0.5Lp

0.5tptp

0.5tp

B1-left

B2-left

B3-left

CLWall Pier Plan

Ptop

Mtop

Pbot

Mbot

Pright-botPleft-bot

Pright-topPleft-top

Left

edge

mem

ber

Rig

ht e

dge

mem

ber

Wall Pier Elevation

Lp

Top of pier

Bottom of pier

Figure 2: Wall Pier for Design Condition 1



where P is either Pleft-top or Pright-top, Ag = tpB1 and the Pmax Factor is defined inthe shear wall design preferences (the default is 0.80). In general, we rec-ommend that you use the default value. From the previous equation, it fol-lows that:

( )

( ) ( )ccusy

gccucFactormax

sc f67.0f

Af0.67)(P

(P)Abs

Aγγ

γ

−

−

=φ

If Asc calculates as negative, no compression reinforcing is needed.

The maximum tensile reinforcing to be packed within the tp times B1 concreteedge member is limited by:

1pmaxmax-st BtPTA =

Similarly, the compression reinforcing is limited by:

1pmaxmax-sc BtPCA =

If Ast is less than or equal to Ast-max and Asc is less than or equal to Asc-max, theprogram will proceed to check the next loading combination; otherwise theprogram will increment the appropriate B1 dimension (left, right or both de-pending on which edge member is inadequate) by one-half of the wall thick-ness to B2 (i.e., 1.5tp) and calculate new values for Pleft-top and Pright-top result-ing in new values of Ast and Asc. This iterative procedure continues until Ast

and Asc are within the allowed steel ratios for all design load combinations.

If the value of the width of the edge member B increments to where itreaches a value larger than or equal to Lp/2, the iteration is terminated and afailure condition is reported.

This design algorithm is an approximate but convenient algorithm. Wall piersthat are declared overstressed using this algorithm could be found to be ade-quate if the reinforcing steel is user-specified and the wall pier is accuratelyevaluated using interaction diagrams.

Design Condition 2Design condition 2 applies to a wall pier with user-specified edge members ateach end of the pier. The size of the edge members is assumed to be fixed;


Checking a Section Designer Pier Section Page 6 of 16

that is, the program does not modify them. For this design condition, the de-sign algorithm determines the area of steel required in the center edge mem-bers and checks if that area gives reinforcing ratios less than the user-specified maximum ratios. The design algorithm used is the same as de-scribed for condition 1; however, no iteration is required.

Design Condition 3Design condition 3 applies to a wall pier with a user-specified (fixed dimen-sion) edge member at one end of the pier and a variable length (program-determined) edge member at the other end. The width of the variable lengthedge member is equal to the width of the wall.

The design is similar to that which has previously been described for designconditions 1 and 2. The size of the user-specified edge member is notchanged. Iteration only occurs on the size of the variable length edge mem-ber.

Checking a Section Designer Pier SectionWhen you specify that a Section Designer pier section is to be checked, theprogram creates an interaction surface for that pier and uses that interactionsurface to determine the critical flexural demand/capacity ratio for the pier.This section describes how the program generates the interaction surface forthe pier and how it determines the demand/capacity ratio for a given designload combination.

Note:

In this program, the interaction surface is defined by a series of PMM interaction curvesthat are equally spaced around a 360 degree circle.

Interaction SurfaceGeneralIn this program, a three-dimensional interaction surface is defined with refer-ence to the P, M2 and M3 axes. The surface is developed using a series of in-teraction curves that are created by rotating the direction of the pier neutralaxis in equally spaced increments around a 360 degree circle. For example, if24 PMM curves are specified (the default), there is one curve every 360°/24curves = 15°. Figure 3 illustrates the assumed orientation of the pier neutralaxis and the associated sides of the neutral axis where the section is in ten-



sion (designated T in the figure) or compression (designated C in the figure)for various angles.

Note that the orientation of the neutral axis is the same for an angle of θ andθ + 180°. Only the side of the neutral axis where the section is in tension orcompression changes. We recommend that you use 24 interaction curves (ormore) to define a three-dimensional interaction surface.

Each PMM interaction curve that makes up the interaction surface is numeri-cally described by a series of discrete points connected by straight lines. Thecoordinates of these points are determined by rotating a plane of linear strainabout the neutral axis on the section of the pier. Details of this process aredescribed later in this Technical Note in the section entitled "Details of theStrain Compatibility Analysis."

By default, 11 points are used to define a PMM interaction curve. You canchange this number in the preferences, specifying any odd number of pointsgreater than or equal to 11, to be used in creating the interaction curve. If

a) Angle is 0 degrees 45°

Interaction curve isfor a neutral axisparallel to this axis

3

2

Pier section

b) Angle is 45 degrees


3

2

Pier section

a) Angle is 180 degrees

225°


3

2

Pier section

b) Angle is 225 degrees


3

2

Pier section

T C

T C

C T

C T

Figure 3: Orientation of the Pier Neutral Axis for Various Angles



you input an even number for this item in the preferences, the program willincrement up to the next higher odd number.

Formulation of the Interaction SurfaceThe formulation of the interaction surface in this program is based consis-tently on the basic principles of ultimate strength design given in Sections3.8.4.1 and 3.4.4.1 of the BS 8110-97 code.

The program uses the requirements of force equilibrium and strain compati-bility to determine the design axial load and moment strength (Pr, M2r, M3r) ofthe wall pier. For the pier to be deemed adequate, the required strength (P,M2, M3) must be less than or equal to the design strength, as indicated in thefollowing equation.

(P, M2, M3) ≤ (Pr, M2r, M3r)

The effects of the partial safety factors for concrete (γc = 1.50) and for steel(γs = 1.05) are included in the generation of the interaction curve (BS 1.4, BS2.4.4.1, Table 2.2).

γc = Partial safety factor for concrete. The value is taken as 1.5 (BS2.4.4.1, Table 2.2).

γs = Partial safety factor for reinforcing steel. The value is taken as1.05 (BS 2.4.4.1, Table 2.2).

The theoretical maximum compressive force that the wall pier can carry isdesignated Pr,max and is given by:

Pr,max = 0.67(fcu /γc) (Ag - As) + (fy /γs) As (BS 3.4.4.1)

The theoretical maximum tension force that the wall pier can carry is desig-nated Pt,max and is given by:

Pt,max = (fy/γs)As (BS 3.4.4.1)

If the wall pier geometry and reinforcing is symmetrical in plan, the momentsassociated with both Pr,max and Pt,max are zero. Otherwise, there will be a mo-ment associated with both Pr,max and Pt,max.

In addition to Pr,max and Pt,max, the axial load at the balanced strain condition,i.e., Pb, is also determined. In this condition, the tension reinforcing reaches



the strain corresponding to its specified yield strength modified by the corre-sponding partial factor of safety, fy/γs, as the concrete reaches it assumed ul-timate strain of 0.0035 (BS 3.4.4.1).

Note:

You can specify the number of points to be used for creating interaction diagrams in theshear wall preferences.

As previously mentioned, by default 11 points are used to define a single in-teraction curve. When creating a single interaction curve, the program in-cludes the points at Pb, Pr,max and Pt,max on the interaction curve. Half of theremaining number of specified points on the interaction curve occur betweenPb and Pr,max at approximately equal spacing along the Pr axis. The other halfof the remaining number of specified points on the interaction curve occurbetween Pb and Pr,max at approximately equal spacing along the Pr axis.

Details of the Strain Compatibility AnalysisAs previously mentioned, the program uses the requirements of force equilib-rium and strain compatibility to determine the design axial capacity and mo-ment strength (Pr, M2r, M3r) of the wall pier. The coordinates of these pointsare determined by rotating a plane of linear strain on the section of the wallpier.

Figure 4 illustrates varying planes of linear strain such as those that the pro-gram considers on a wall pier section for a neutral axis orientation angle of 0degrees. In these planes, the maximum concrete strain is always taken as-0.0035 and the maximum steel strain is varied from -0.0035 to plus infinity(BS 3.4.4.1, BS Figure 3.3). (Recall that in this program compression isnegative and tension is positive.) When the steel strain is -0.0035, the maxi-mum compressive force in the wall pier, Pr,max, is obtained from the straincompatibility analysis. When the steel strain is plus infinity, the maximumtensile force in the wall pier, Pt,max, is obtained. When the maximum steelstrain is equal to the yield strain for the reinforcing, Pb is obtained.

Figure 5 illustrates the concrete wall pier stress-strain relationship that is ob-tained from a strain compatibility analysis of a typical plane of linear strainshown in Figure 4.



In Figure 5 the compressive stress in the concrete, Cc, is calculated using thefollowing equation:

Cc = 0.67(fcu/γc)(0.9 x tp) (BS 3.4.4.1, Figure 3.3)

In Figure 4, the value for maximum strain in the reinforcing steel is assumed.Then the strain in all other reinforcing steel is determined based on the as-sumed plane of linear strain. Next the stress in the reinforcing steel is calcu-

lated using the following equation, where εs is the strain, Es is the modulus ofelasticity, σs is the stress, and fy is the characteristic yield strength of the re-inforcing steel.

σs = εsEs ≤ fy/γs (BS 3.4.4.1, Figure 2.2)

The force in the reinforcing steel (Ts for tension or Cs for compression) is cal-culated using the following equation:

Ts or Cs = σs As (BS 3.4.4.1)

Figure 4: Varying Planes of Linear Strain

Varyingneutral axislocations

Varying Linear Strain Diagram

Plan View of Wall Pier

-0.0035

0.000

+ ε

- ε



For the given distribution of strain, the value of Pr is calculated using the fol-lowing equation:

Pr = (ΣTs- Cc - ΣCs) ≤ Pmax (BS 3.4.4.1)

In the previous equation, the tensile force Ts and the compressive forces Cc

and Cs are all positive. If Pr is positive, it is tension, and if it is negative, it iscompression. The term Pmax is taken as Pr,max if Pr is compressive, and as Pt,max

if Pr is tensile.

Figure 5: Wall Pier Stress-Strain Relationship

Linear Strain Diagram

Plan View of Wall Pier

x

ε = 0

.003

5

εs1εs

2εs3εs

4

εs5εs

6εs7εs

8εs9εs

10εs11εs

12εs13

a = 0.9xCc

0.67fcu/γc

c

Stress Diagram

Cs1

Ts5

Cs2

Cs3

Cs4

Ts6

Ts7

Ts8

Ts9

Ts10

Ts11

Ts12

Ts13

t p



The value of M2r is calculated by summing the moments resulting from all ofthe forces about the pier local 2-axis. Similarly, the value of M3r is calculatedby summing the moments resulting from all of the forces about the pier local3-axis. The forces whose moments are summed to determine M2r, M3r and Pr,are Cc, all of the Ts forces and all of the Cs forces.

The Pr, M2r and M3r values calculated as described above make up one pointon the wall pier interaction diagram. Additional points on the diagram areobtained by making different assumptions for the maximum steel stress; thatis, considering a different plane of linear strain, and repeating the process.

When one interaction curve is complete, the next orientation of the neutralaxis is assumed and the points for the associated new interaction curve arecalculated. This process continues until the points for all of the specifiedcurves have been calculated.

Wall Pier Demand/Capacity RatioRefer to Figure 6, which shows a typical two-dimensional wall pier interactiondiagram. The forces obtained from a given design load combination are P andM3. The point L, defined by (P, M3), is placed on the interaction diagram asshown in the figure. If the point lies within the interaction curve, the wall piercapacity is adequate. If the point lies outside of the interaction curve, the wallpier is overstressed.

As a measure of the stress condition in the wall pier, the program calculates astress ratio. The ratio is achieved by plotting the point L and determining thelocation of point C. The point C is defined as the point where the line OL (ex-tended outward if needed) intersects the interaction curve. The de-mand/capacity ratio, D/C, is given by D/C = OL / OC where OL is the "dis-tance" from point O (the origin) to point L and OC is the "distance" from pointO to point C. Note the following about the demand/capacity ratio:

If OL = OC (or D/C = 1), the point (P, M3) lies on the interaction curve andthe wall pier is stressed to capacity.

If OL < OC (or D/C < 1), the point (P, M3) lies within the interaction curveand the wall pier capacity is adequate.

If OL > OC (or D/C > 1), the point (P, M3) lies outside of the interactioncurve and the wall pier is overstressed.


Designing a Section Designer Pier Section Page 13 of 16

The wall pier demand/capacity ratio is a factor that gives an indication of thestress condition of the wall with respect to the capacity of the wall.

The demand/capacity ratio for a three-dimensional wall pier is determined ina similar manner to that described here for two-dimensional piers.

Designing a Section Designer Pier SectionWhen you specify that a Section Designer pier section is to be designed, theprogram creates a series of interaction surfaces for the pier based on the fol-lowing items:

1. The size of the pier as specified in Section Designer.

2. The location of the reinforcing specified in Section Designer.

3. The size of each reinforcing bar specified in Section Designer relative tothe 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

Figure 6: Two-Dimensional Wall Pier Demand/Capacity Ratio

Pr

M3rO

L

C

M3

P

AxialCompression

AxialTension


Designing a Section Designer Pier Section Page 14 of 16

modified to obtain these different ratios; however, the relative size (area) ofeach rebar compared to the other bars is always kept constant.

The smallest of the eight reinforcing ratios used is that specified in the shearwall design preferences as Section Design IP-Min. Similarly, the largest of theeight reinforcing ratios used is that specified in the shear wall design prefer-ences as Section Design IP-Max.

The eight reinforcing ratios used are the maximum and the minimum ratiosplus six more ratios. The spacing between the reinforcing ratios is calculatedas an increasing arithmetic series in which the space between the first tworatios 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 exam-ple, when the minimum reinforcing ratio, IP-min, is 0.0025 and the maxi-mum, IP-max, is 0.02 (BS 3.12.5.3, Table 3.25). Those two ratios can beoverwritten in the Preferences.

Table 1 The Eight Default Reinforcing Ratios Used by the Program

Curve Ratio Example

1 IPmin 0.0025

214

IPminIPmaxIPmin

−+ 0.0038

3

−+

14

IPminIPmax

3

7IPmin 0.0054

4

−+

14

IPminIPmax4IPmin 0.0075

5

−+

14


6

−+

14

IPminIPmax

3

25IPmin 0.0129

7

−+

14


8 IPmax 0.0200


Checking a Pier Section with Uniform Reinforcing Page 15 of 16

After the eight reinforcing ratios have been determined, the program developsinteraction surfaces for all eight of the ratios using the process described ear-lier in this Technical Note in the section entitled "Checking a Section DesignerPier Section."

Next, for a given design load combination, the program generates a de-mand/capacity ratio associated with each of the eight interaction surfaces.The program then uses linear interpolation between the interaction surfacesto determine the reinforcing ratio that gives an demand/capacity ratio of theUtilization Factor Limit, which is equal to 0.95 by default. The Utilization Fac-tor can be redefined in the Preferences. This process is repeated for all designload combinations and the largest required reinforcing ratio is reported.

Checking a Pier Section with Uniform ReinforcingWhen you specify that a Pier Section with Uniform Reinforcing is to bechecked, the program creates an interaction surface for that pier and usesthat interaction surface to determine the critical flexural demand/capacity ra-tio for the pier. The procedure for generating the interaction surface and theprocedure for determining the critical demand/capacity ratio are the same asthose described earlier in a section “Checking a Section Designer Pier Section”of this technical note. The difference is that the pier sections with uniform re-inforcing are planner walls. For these two-dimensional walls, M2 is ignored.

Designing a Pier Section with Uniform ReinforcingWhen you specify that a Pier Section with Uniform Reinforcing is to be de-signed, the program creates a series of interaction surfaces for that pier. Theinteraction surfaces are developed for eight different ratios of reinforcing steelareas. The eight reinforcing ratios used are the maximum and minimum ratiosplus six more intermediate ratios. The procedure of selecting the eight rein-forcing ratios is described in the section “Designing a Section Designer PierSection.” After the eight reinforcing ratios have been determined, the pro-gram develops interaction surfaces for all eight of the ratios using the proce-dure described earlier in this Technical Note in the section entitled “Checkinga Section Designer Pier Section.” The difference is that the pier sections withuniform reinforcing are planner walls. For these two-dimensional walls, M2 isignored.


Designing a Pier Section with Uniform Reinforcing Page 16 of 16

Next, for a given design load combination, the program generates a de-mand/capacity ratio associated with each of the eight interaction surfaces.The program then uses linear interpolation between the interaction surfacesto determine the reinforcing ratio that gives a demand/capacity ratio of theUtilization Factor Limit, which is equal to 0.95 by default. The Utilization Fac-tor can be redefined in the preferences. This process is repeated for all designload combinations and the largest required reinforcing ratio is reported.

General Page 1 of 4



Technical NoteWall Pier Shear Design

This Technical Note describes how the program designs concrete wall piers forshear using BS 8110-97. Note that in this program you cannot specify shearreinforcing and then have the program check it. The program only designs thepier for shear and reports how much shear reinforcing is required. The sheardesign is performed at stations at the top and bottom of the pier.

GeneralThe wall pier shear reinforcing for each leg of the pier is designed for each ofthe design load combinations. The following steps are involved in designingthe shear reinforcing for a particular leg of a wall pier section for a particulardesign loading combination.

1. Determine the factored forces P, M and V that are acting on the wall piersection.

2. Determine the shear force, vc, that can be carried by the concrete. Notethat P and M are required for the calculation of vc.

3. Determine the required shear reinforcing to carry the balance of the shearforce.

Determine the Factored ForcesThe factored forces, P, M, and V that are acting on the individual leg of thepier section are determined from the basic forces for each load case and theload combination factors. Then the leg of the wall pier is designed for thefactored forces.

Shear Wall Design BS 8110-97 Wall Pier Shear Design

Determine the Concrete Shear Capacity Page 2 of 4

Note:

The term RLW is used as a multiplier in the expression for vc as well as on all 'cf terms

in this Technical Note. It is a shear strength reduction factor that applies to lightweightconcrete. It is equal to 1 for normal weight concrete. This factor is specified in the con-crete material properties.

Determine the Concrete Shear CapacityGiven the design force set P, M and V acting on a wall pier section, the shearstress carried by the concrete, vc', is calculated as follows (BS 3.4.5.12,3.4.5.4, Table 3.8):

ccc

cc

'c vA

N1v

MVd

AN

6.0vv +≤+= , (BS 3.4.5.12)

where

41

31

s

m

21LWc d

400bd

A100kk79.0Rv

=

γ, (BS 3.4.5.4, Table 3.8)

RLW is a shear strength reduction factor that applies to lightweight con-crete. It is equal to 1 for normal weight concrete. This factor is speci-fied in the concrete material properties.

k1 is the enhancement factor for support compression and taken con-servatively as 1, (BS 3.4.5.8)

k2 = 3

1

cu

25f

, (BS 3.4.5.4, Table 3.8)

γm = 1.25, (BS 2.4.4.1)

0.15 ≤ bd

A100 s ≤ 3, (BS 3.4.5.4, Table 3.8)

1d400

v≥ , (BS 3.4.5.4, Table 3.8)


Determine the Required Shear Reinforcing Page 3 of 4

1M

Vdv ≤ , (BS 3.4.5.4, Table 3.8)

fcu ≤ 40 N/mm2, (BS 3.4.5.4, Table 3.8)

As is the area of tensile steel and it is takes as half the total reinforcingsteel area, and

dv is the distance from extreme compression fiber to the centroid ofthe tension steel. It is taken as 0.8 LP.

If the tension is large enough that vc' results in a negative number, vc is setto zero.

Determine the Required Shear ReinforcingGiven V and vc, the following procedure provides the required shear reinforc-ing in area per unit length (e.g., mm2/mm or optionally cm2/m) for wall piers(BS 3.4.5.3, Table 3.7).

Calculate the design shear stress from

v = cvAV

, (BS 3.4.5.2)

vmax = min {0.8RLW cuf , 5 MPa}, (BS 3.4.5.2, BS 3.4.5.12)

v ≤ 0.8RLW cuf (BS 3.4.5.2, BS 3.4.5.12)

v ≤ 5 N/mm2, and (BS 3.4.5.2, BS 3.4.5.12)

Acv = tp dv.

If v exceeds 0.8RLW cuf or 5 N/mm2, the concrete section area should be

increased (BS 3.4.5.2, 3.4.5.12). In that case, the program reports anoverstress.

If v ≤ vc' + 0.4, provide minimum links defined by:

y

p

v

sv

f95.0

t4.0

sA

≥ (BS 3.4.5.3, Table 3.7)



else if vc' + 0.4 < v < vmax, provide links given by

( )y

p'c

v

sv

f95.0

t vv

sA −

≥ , and (BS 3.4.5.3, Table 3.7)

else if v ≥ vmax,

a failure condition is declared.

Asv/sv is the horizontal shear reinforcing per unit vertical length (height) ofthe wall pier.

In shear design, fy cannot be taken as greater than 460 MPa (BS 3.4.5.1). Iffy for shear rebar is defined as greater than 460 MPa, the program designsshear rebar based on fy equal to 460 MPa.

Note:

1. The program reports an over stress message when shear stress exceeds 8RLW cuf

or 5 N/mm2 (BS 3.4.5.2, BS 3.4.5.12).

2. You can set the output units for the distributed shear reinforcing in the shear wall de-sign preferences.

Technical Note E-CB-AISC-ASD89-001 Page 1 of 8



Technical NoteSpandrel Flexural Design

This Technical Notes describes how the program designs concrete shear wallspandrels for flexure using the BS 8110-97 requirements. This program al-lows consideration of rectangular sections and T-beam sections for shear wallspandrels. Note that the program designs spandrels at stations located at theends of the spandrel. No design is performed at the center (midlength) of thespandrel.

GeneralThe spandrel flexural reinforcing is designed for each of the design load com-binations. The required area of reinforcing for flexure is calculated and re-ported only at the ends of the spandrel beam.

In this program, wall spandrels are designed for major direction flexure andshear only. Effects caused by any axial forces, minor direction bending, tor-sion or minor direction shear that may exist in the spandrels must be investi-gated by the user independent of the program.

The following steps are involved in designing the flexural reinforcing for aparticular wall spandrel section for a particular design loading combination ata particular station.

• Determine the factored moment M.

• Determine the required flexural reinforcing.

• Determine the required maximum reinforcing.

These steps are described in the following sections.

Determine the Factored MomentsIn the design of flexural reinforcing for spandrels, the factored moments foreach design load combination at a particular beam station are first obtained.

Shear Wall Design BS 8110-97 Spandrel Flexural Design

Determine the Required Flexural Reinforcing Page 2 of 8

The beam section is then designed for the maximum positive and the maxi-mum negative factored moments obtained from all of the design load combi-nations.

Determine the Required Flexural ReinforcingIn 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 suchcases, the beam may be designed as a rectangular section, or as a T-beamsection. You indicate that a spandrel is to be designed as a T-beam by pro-viding appropriate slab width and depth dimensions in the spandrel designoverwrites.

The flexural design procedure is based on a simplified rectangular stressblock, as shown in Figure 1 (BS 3.4.4.1). It is assumed that the compressioncarried by the concrete is less than that which can be carried at the limitingcondition when the depth of neutral axis is half of the effective depth. Whenthe applied moment exceeds the moment capacity associated with the limit-ing depth of the neutral axis, the program calculates an area of compressionreinforcement assuming that the additional moment is carried by compressionreinforcing and additional tension reinforcing.

Figure 1: Rectangular spandrel beam design, positive moment

εc = 0.0035

εsx

h s

ts

Strain DiagramBeam Section

d

a = 0.9 x

0.67fcu/γc

Cc

Ts

Stress Diagram

Csε's

d r-bot

d r-to

p



The procedure used by the program for both rectangular and T-beam sectionsis given below.

Rectangular Beam Flexural ReinforcingFor rectangular beams, the moment capacity as a singly reinforced beam,Msingle, is obtained first for a section. The reinforcing steel area is determinedbased on whether M is greater than, less than, or equal to Msingle. See Figure1. The ultimate moment resistance of the section as a singly reinforced beamis calculated as follows:

Msingle = K'fcutsd2 (BS 3.4.4.4)

K' = 0.156.

If M ≤ Msingle, no compression reinforcement is required. The program cal-culates the area of tension reinforcement, As, as follows:

As = zf95.0

M

y, where (BS 3.4.4.4)

−+=9.0

K25.05.0dz ≤ 0.95 d, and (BS 3.4.4.4)

K = 2scu dtf

M. (BS 3.4.4.4)

This steel is placed at the top if the section is under negative moment, andat the bottom if the section is under positive moment.

If M > Msingle, compression reinforcement is required and the program cal-culates the area of compression reinforcement, As', as follows:

As' = ( ) ( )'dd /f67.0f

MM

ccu's

glesin

−−

−

γ(BS 3.4.4.4)

where d' is the depth of the compression steel from the concrete compres-sion face. It is equal to dr-top for positive moment and dr-bottom for negativemoment. fs' is given by:



fs' = 700

−

dd2

1'

≤ 0.95fy (BS 3.4.4.4)

From equilibrium, the area of tension reinforcement is calculated as:

As = ( )'ddf95.0

MM

zf95.0

M

y

glesin

y

glesin

−−

+ , where (BS 3.4.4.4)

z =

−+9.0'K

25.05.0d = 0.776887 d. (BS 3.4.4.4)

As is to be placed at the bottom of the beam and As' at the top for positive

bending and vice versa for negative moment.

Note:

If the factored moment exceeds 0.156 fcutsd2, the program provides compression steel to

help resist the applied moment.

T-Beam Flexural ReinforcingT-beam action is only considered effective for positive moment. When de-signing T-beams for negative moment (i.e., designing top steel), the calcula-tion of required steel is as described in the previous section for rectangularsections. No T-beam data is used in this design. The width of the beam istaken equal to the width of the web, ts.

For positive moment, the flange is in compression. The program analyzes thesection by considering alternative locations of the neutral axis. Initially, theneutral axis is assumed to be located in the flange. On the basis of this as-sumption, the program calculates the depth of the neutral axis. If the stressblock does not extend beyond the flange thickness, the section is designed asa rectangular beam of width bf. If the stress block extends beyond the flangewidth, the contribution of the web to the flexural strength of the beam istaken into account. See Figure 2.

The T-beam requires only tension reinforcement when the moment is posi-tive, the flange is in compression, the moment is less than βf fcu bfd

2 and theflange depth is less than 0.45d. In those conditions, the tension reinforcingsteel area of the T-beam is calculated as follows (BS 3.4.4.5):



As = ( )

( )fy

fscu

h5.0df95.0hd45.0dtf1.0M

−−+

(BS 3.4.4.5)

where,

βf = 0.45 b

b15.0

d2h

1 bt

1dh wf

f

sf +

−

− . (BS 3.4.4.5)

If the above conditions are not met, the T-beam is designed using the generalprinciple of the BS 8110 code (BS 3.4.4.4, 3.4.4.5).

Assuming the neutral axis to lie in the flange, the normalized moment is com-puted as:

K = 2

fcu dbf

M(BS 3.4.4.4)

Then the moment arm is computed as

z =

−+9.0

K25.05.0 d ≤ 0.95d, (BS 3.4.4.4)

the depth of the neutral axis is computed as

Figure 2: Design of a Wall Spandrel with a T-Beam Section, PositiveMoment

εc = 0.0035

εs

x

h sts

Strain DiagramBeam Section

d

Ts = Tw + Tf

Csε's

d r-b

ot

h fd r

-top

bf

As

A's a 1 =

0.9

x

0.67fcu/γc

Cw

Tw

h f

Tf

Cf

Steel StressDiagram

Concrete WebStress Diagram

Concrete FlangeStress Diagram

0.67fcu/γc



x = 45.0

1(d − z) and (BS 3.4.4.4)

the depth of the compression block is given by

a = 0.9 x. (BS 3.4.4.4)

If a ≤ hf, the subsequent calculations for the reinforcing steel are exactlythe same as previously defined for the rectangular section design. However,in that case, the width of the compression block is taken to be equal to thewidth of the compression flange, bf. Compression reinforcement is requiredif K > K'.

If a > hf, the subsequent calculations for the required area for reinforcingsteel are performed in two parts. First, the tension steel required to balancethe compressive force in the flange is determined, and second, the tensionsteel required to balance the compressive force in the web is determined. Ifnecessary, compression steel is added to help resist the design moment.

The remainder of this section describes in detail the design process used bythe program for T-beam spandrels when a > hf.

In that case, the ultimate resistance moment of the flange is given by:

Mf = 0.67(fcu/γc)(bf − ts)hf (d − 0.5hf), (BS 3.4.4.1)

the balance of moment taken by the web is computed as

Mw = M − Mf, and

the normalized moment resisted by the web is given by

Kw = 2scu

w

dtf

M. (BS 3.4.4.1)

If Kw ≤ K', the beam is designed as a singly reinforced concrete beam. Thearea of steel is calculated as the sum of two parts, one to balance compres-sion in the flange and one to balance compression in the web.

As = ( ) ( ) ( ) z fM

h5.0d fM

sy

w

fsy

f

γγ+

−, where (BS 3.4.4.1)



z =

−+9.0

K25.05.0d w ≤ 0.95d. (BS 3.4.4.1)

If Kw > K', compression reinforcement is required. The compression rein-forcing steel area is calculated using the following procedure:

The ultimate moment of resistance of the web only is given by

Muw = K' fcutsd2. (BS 3.4.4.4)

The compression reinforcement is required to resist a moment of magnitudeMw − Muw. The compression reinforcement is computed as

As = ( ) ( )'dd f67.0f

MM

c'c

's

wuw

−−

−

γ(BS 3.4.4.1)

where d' is the depth of the compression steel from the concrete compressionface, and

fs' = 700

−

d'd2

1 ≤ 0.95 fy. (BS 3.4.4.1)

The area of tension reinforcement is obtained as follows:

As =

−−

++− 'dd

MMd777.0

Mh5.0d

Mf95.0

1 uwwuw

f

f

y(BS 3.4.4.1)

Determine Required Minimum ReinforcingThe minimum flexural tensile reinforcing steel required for a beam section isgiven by the following table, which is taken from BS Table 3.25 (BS 3.12.5.3)with interpolation for reinforcement of intermediate strength:



Table 1 Minimum Percentage of Tensile Reinforcing

Minimum percentageSection Situation

Definition ofpercentage fy= 250 MPa fy = 460 MPa

Rectangular 100bh

As 0.24 0.13

f

w

b

b< 0.4 100

hb

A

w

s0.32 0.18

T-Beam with web intension

f

w

b

b≥ 0.4 100

hb

A

w

s0.24 0.13

T-Beam with web incompression

100hb

A

w

s0.48 0.26

The minimum flexural compression steel, if it is required, provided in a rec-tangular or T-beam section is given by the following table, which is takenfrom BS Table 3.25 (BS 3.12.5.3) with interpolation for reinforcement of in-termediate strength:

Table 2 Minimum Percentage of Compression Reinforcing (if required)

Section SituationDefinition ofpercentage

Minimumpercentage

Rectangular 100bh

As'

0.20

Web in tension 100ff

s

hb

A '

0.40

T-Beam

Web in compression 100hb

A

w

s'

0.20

In addition, an upper limit on both tension reinforcement and compressionreinforcement has been imposed to be 0.04 times the gross cross-sectionalarea (BS 3.12.6.1). The program reports an overstress when the ratio exceed4 percent.

General Page 1 of 3



Technical NoteSpandrel Shear Design

This Technical Note describes how the program designs concrete wall span-drels for shear using BS 8110-97. Note that in this program you cannot spec-ify shear reinforcing and then have the program check it. The program onlydesigns the spandrel for shear and reports how much shear reinforcing is re-quired.

The program allows consideration of rectangular sections and T-beam sec-tions for wall spandrels. The shear design for both of these types of spandrelsections is identical.

GeneralThe wall spandrel shear reinforcing is designed for each of the design loadcombinations. The required area of reinforcing for vertical shear is only cal-culated at the ends of the spandrel beam.

In this program, wall spandrels are designed for major direction flexure andshear forces only. Effects caused by any axial forces, minor direction bending,torsion or minor direction shear that may exist in the spandrels must be in-vestigated by the user independent of the program.

The following steps are involved in designing the shear reinforcing for a par-ticular wall spandrel section for a particular design loading combination at aparticular station.

1. Determine the factored shear force V.

2. Determine the shear stress, vc, that can be carried by the concrete.

3. Determine the required shear reinforcing to carry the balance of the shearforce.

Step 1 needs no further explanation. The following two sections describe indetail the algorithms associated with Steps 2 and 3.

Shear Wall Design BS 8110-97 Spandrel Shear Design

Determine the Concrete Shear Capacity Page 2 of 3

Determine the Concrete Shear CapacityThe shear stress carried by the concrete, vc, is calculated as follows:

vc = 4

13

1

s

m

21LW d

400bd

A100kk 79.0R

γ

(BS 3.4.5.4, Table 3.8)

where,

RLW is a shear strength reduction factor that applies to light-weight con-crete. It is equal to 1 for normal weight concrete. This factor is specified inthe concrete material properties.

k1 is the enhancement factor for support compression, and is conserva-tively taken as 1, (BS 3.4.5.8)

k2 = 3

1cu

25f

≥ 1, (BS 3.4.5.4, Table 3.8)

γm = 1.25, and (BS 2.4.4.1)

As is the area of tensile steel.

However, the following limitations also apply:

0.15 ≤ bd

As100≤ 3, (BS 3.4.5.4, Table 3.8)

d

400≥ 1, and (BS 3.4.5.4, Table 3.8)

fcu ≤ 40 N/mm2 (for calculation purpose only). (BS 3.4.5.4, Table 3.8)

Determine the Required Shear ReinforcingGiven V and vc, the following procedure provides the required shear reinforc-ing in area per unit length (e.g., mm2/mm or optionally cm2/m) for spandrel(BS 3.4.5.4, Table 3.7).

Shear Wall Design BS 8110-97 Spandrel Shear Design


Note:

You can set the output units for the distributed shear reinforcing in the shear wall designpreferences.

Calculate the design shear stress and the maximum allowed shear stress as

v = cvAV

, where (BS 3.4.5.2)

v ≤ 0.8RLW cuf , (BS 3.4.5.2, BS 3.4.5.12)

v ≤ 5 N/mm2, (BS 3.4.5.2, BS 3.4.5.12)

Vmax = min {0.8RLW cuf , 5 MPa}, (BS 3.4.5.2, BS 3.4.5.12)

Acv = tsd, and

d is the distance from the extreme compression fiber to the centroid ofthe tension steel. This is determined from the associated direction ofmoment.

If v exceeds 0.8RLW cuf or 5 MPa, the program reports an overstress. In

that case, the concrete shear area should be increased.

If v ≤ vc + 0.4, provide minimum links defined by

y

s

v

sv

f95.0t 4.0

sA

≥ , (BS 3.4.5.3)

else if vc + 0.4 < v < vmax, provide links given by

( )y

sc

v

sv

f95.0t vv

sA −

≥ , and (BS 3.4.5.3)

else if v ≥ vmax

a failure condition is declared.

In shear design, fy cannot be greater than 460 MPa (BS 3.4.5.1). If fy isdefined as greater than 460 MPa, the program designs shear reinforcingassuming that fy equals 460 MPa.

E-TN-SWD-BS8110-97

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