SHEAR IN BEAMS

IntroductionIntroduction

Loads applied to beams produce bending moments shearing forces asLoads applied to beams produce bending moments shearing forces as

shown and in some cases torquesshown and in some cases torques

Beams are usually designed for bending moment first thus cross Beams are usually designed for bending moment first thus cross sectional dimensions are evaluated along with the required amounts of sectional dimensions are evaluated along with the required amounts of longitudinal reinforcement Once this is done sections should be longitudinal reinforcement Once this is done sections should be checked for shear to determine whether shear reinforcement is checked for shear to determine whether shear reinforcement is required or not required or not

Shear in Homogeneous Elastic BeamsShear in Homogeneous Elastic Beams

Loaded beam and orientation of cracks

The normal stresses resulting from bending are given by the The normal stresses resulting from bending are given by the following equation as proved by the classicalfollowing equation as proved by the classical bending theorybending theory

xf

x

xx I

yMf

The shearing stresses are given by the following equation proved The shearing stresses are given by the following equation proved in the most classical mechanics of materials booksin the most classical mechanics of materials books

x

bI

QV

x

xxx

Where is the shearing force at the considered section is the Where is the shearing force at the considered section is the moment of the area of the section located between the point where the moment of the area of the section located between the point where the shearing stresses are calculated and the extreme fiber of the section shearing stresses are calculated and the extreme fiber of the section about the neutral axis is the moment of inertia about the neutral axis about the neutral axis is the moment of inertia about the neutral axis and b is the width of the section at the point where shearing stresses and b is the width of the section at the point where shearing stresses are calculatedare calculated

In an attempt to establish the cracking pattern four elements situated at In an attempt to establish the cracking pattern four elements situated at different distances from the neutral axis are studieddifferent distances from the neutral axis are studied

xV xQ

xI

Types of Shear CracksTypes of Shear Cracks

Two types of inclined cracking occur in beams flexure-shear Two types of inclined cracking occur in beams flexure-shear cracking and web-shear crackingcracking and web-shear cracking

A Flexure-Shear Cracks

The most common type develops from the tip of a flexural crack at The most common type develops from the tip of a flexural crack at the tension side of the beam and propagates towards mid depth the tension side of the beam and propagates towards mid depth until it is checked on the compression side of the beam For these until it is checked on the compression side of the beam For these cracks to form the bending moment must exceed the cracking cracks to form the bending moment must exceed the cracking moment of the cross section and a significant shear must existmoment of the cross section and a significant shear must exist

Types of cracks and associated internal forces (a) orientation of cracks (b) shear force diagram (c) bending moment diagram

B Web Shear Cracks

Web shear cracking begins from an interior point in a member at the Web shear cracking begins from an interior point in a member at the level of the centroid of uncracked section and moves on a diagonal level of the centroid of uncracked section and moves on a diagonal path to the tension face when the diagonal tensile stresses produced path to the tension face when the diagonal tensile stresses produced by shear exceed the tensile strength of concrete This type of by shear exceed the tensile strength of concrete This type of cracking is common on beams with thin webs and in regions of high cracking is common on beams with thin webs and in regions of high shear and small moment This combination exists adjacent to simple shear and small moment This combination exists adjacent to simple supports or at points of inflection in continuous beamssupports or at points of inflection in continuous beams

Nominal Shear StressNominal Shear Stress

The only equation available to relate shear stress to shearing force is The only equation available to relate shear stress to shearing force is derived for a beam of constant cross section constructed of a derived for a beam of constant cross section constructed of a homogeneous elastic material Unfortunately Eq (42) can not be homogeneous elastic material Unfortunately Eq (42) can not be applied to reinforced concrete beams for the following reasonsapplied to reinforced concrete beams for the following reasons

1048707 1048707 Reinforced concrete is nonhomogeneous materialReinforced concrete is nonhomogeneous material

1048707 1048707 Concrete is not elastic Concrete is not elastic

1048707 1048707 Variable extent of cracking along the length of a beam Variable extent of cracking along the length of a beam making it impossible to determine cross-sectional propertiesmaking it impossible to determine cross-sectional properties

Therefore the ACI Code has adopted a simple procedure for Therefore the ACI Code has adopted a simple procedure for establishing the magnitude of shear stress v on a cross sectionestablishing the magnitude of shear stress v on a cross section

db

Vv

w

wherewhere = nominal shear stress= nominal shear stress = shearing force at specified section= shearing force at specified section = width of web of cross section= width of web of cross section = effective depth of the section= effective depth of the section

vVwbd

According to ACI Code 1111 design of cross sections According to ACI Code 1111 design of cross sections subject to shear should be based on the following equationsubject to shear should be based on the following equation

Strength of Concrete in ShearStrength of Concrete in Shear

Shear strength of concrete VShear strength of concrete Vcc is evaluated by loading a plain concrete is evaluated by loading a plain concrete

beam to failure Shear stresses are computed by dividing the shearing beam to failure Shear stresses are computed by dividing the shearing force resisted by concrete Vforce resisted by concrete Vcc by b by bwwd Strength of concrete in shear is d Strength of concrete in shear is

directly proportional to the strength of concrete in tension inversely directly proportional to the strength of concrete in tension inversely proportional to the magnitude of bending moment at the section proportional to the magnitude of bending moment at the section under consideration and directly proportional to the reinforcement under consideration and directly proportional to the reinforcement ratio of flexural reinforcementratio of flexural reinforcementFor the sake of simplicity VFor the sake of simplicity Vcc is assumed to be the same for beams is assumed to be the same for beams

with or without shear reinforcementwith or without shear reinforcementFor members subject to shear and bending only ACI Code 11311 For members subject to shear and bending only ACI Code 11311 gives the following equation for evaluating Vgives the following equation for evaluating Vcc

dbfV wcc 530

where Vwhere Vuu is the factored shearing force M is the factored shearing force Muu is the factored is the factored

bending moment occurring simultaneously with Vbending moment occurring simultaneously with Vuu at section at section

considered ρconsidered ρww is the reinforcement ratio of the web and d is is the reinforcement ratio of the web and d is

the effective depth of the beam should not exceed 10 the effective depth of the beam should not exceed 10 and Vand Vcc should not exceed should not exceed u

u

M

dV

Strength Provided by Shear ReinforcementStrength Provided by Shear ReinforcementWhen the nominal shearing force VWhen the nominal shearing force Vn n exceeds the shearing force exceeds the shearing force

that can be resisted by concrete alone Vthat can be resisted by concrete alone Vcc shear reinforcement in shear reinforcement in

any of the forms shown in the following section can be usedany of the forms shown in the following section can be used

Types of Shear Reinforcement

When shear reinforcement is required the following types of When shear reinforcement is required the following types of shear reinforcement are permitted by ACI Code 1151 as shown shear reinforcement are permitted by ACI Code 1151 as shown a Vertical Stirrupsa Vertical Stirrupsb Inclined stirrups making an angle of 45 degree or more withb Inclined stirrups making an angle of 45 degree or more withlongitudinal tension reinforcementlongitudinal tension reinforcementc Longitudinal reinforcement with bent portion making an angle c Longitudinal reinforcement with bent portion making an angle of 30 degree or more with the tension reinforcementof 30 degree or more with the tension reinforcementd Spirals circular ties or hoopsd Spirals circular ties or hoopse Combination of stirrups and bent longitudinal reinforcemente Combination of stirrups and bent longitudinal reinforcementBefore diagonal cracking occurs the stirrups remain Before diagonal cracking occurs the stirrups remain unstressed After cracking the stress in the stirrups increases unstressed After cracking the stress in the stirrups increases as they pick up a portion of the load formerly carried by the as they pick up a portion of the load formerly carried by the uncracked concreteuncracked concrete

Types of shear reinforcement (a) vertical stirrups (b) inclinedstirrups (c) bent-up bars (two groups) (d) bent-up bars (three groups)

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

IntroductionIntroduction

Loads applied to beams produce bending moments shearing forces asLoads applied to beams produce bending moments shearing forces as

shown and in some cases torquesshown and in some cases torques

Beams are usually designed for bending moment first thus cross Beams are usually designed for bending moment first thus cross sectional dimensions are evaluated along with the required amounts of sectional dimensions are evaluated along with the required amounts of longitudinal reinforcement Once this is done sections should be longitudinal reinforcement Once this is done sections should be checked for shear to determine whether shear reinforcement is checked for shear to determine whether shear reinforcement is required or not required or not

Shear in Homogeneous Elastic BeamsShear in Homogeneous Elastic Beams

Loaded beam and orientation of cracks

The normal stresses resulting from bending are given by the The normal stresses resulting from bending are given by the following equation as proved by the classicalfollowing equation as proved by the classical bending theorybending theory

xf

x

xx I

yMf

The shearing stresses are given by the following equation proved The shearing stresses are given by the following equation proved in the most classical mechanics of materials booksin the most classical mechanics of materials books

x

bI

QV

x

xxx

Where is the shearing force at the considered section is the Where is the shearing force at the considered section is the moment of the area of the section located between the point where the moment of the area of the section located between the point where the shearing stresses are calculated and the extreme fiber of the section shearing stresses are calculated and the extreme fiber of the section about the neutral axis is the moment of inertia about the neutral axis about the neutral axis is the moment of inertia about the neutral axis and b is the width of the section at the point where shearing stresses and b is the width of the section at the point where shearing stresses are calculatedare calculated

In an attempt to establish the cracking pattern four elements situated at In an attempt to establish the cracking pattern four elements situated at different distances from the neutral axis are studieddifferent distances from the neutral axis are studied

xV xQ

xI

Types of Shear CracksTypes of Shear Cracks

Two types of inclined cracking occur in beams flexure-shear Two types of inclined cracking occur in beams flexure-shear cracking and web-shear crackingcracking and web-shear cracking

A Flexure-Shear Cracks

The most common type develops from the tip of a flexural crack at The most common type develops from the tip of a flexural crack at the tension side of the beam and propagates towards mid depth the tension side of the beam and propagates towards mid depth until it is checked on the compression side of the beam For these until it is checked on the compression side of the beam For these cracks to form the bending moment must exceed the cracking cracks to form the bending moment must exceed the cracking moment of the cross section and a significant shear must existmoment of the cross section and a significant shear must exist

Types of cracks and associated internal forces (a) orientation of cracks (b) shear force diagram (c) bending moment diagram

B Web Shear Cracks

Web shear cracking begins from an interior point in a member at the Web shear cracking begins from an interior point in a member at the level of the centroid of uncracked section and moves on a diagonal level of the centroid of uncracked section and moves on a diagonal path to the tension face when the diagonal tensile stresses produced path to the tension face when the diagonal tensile stresses produced by shear exceed the tensile strength of concrete This type of by shear exceed the tensile strength of concrete This type of cracking is common on beams with thin webs and in regions of high cracking is common on beams with thin webs and in regions of high shear and small moment This combination exists adjacent to simple shear and small moment This combination exists adjacent to simple supports or at points of inflection in continuous beamssupports or at points of inflection in continuous beams

Nominal Shear StressNominal Shear Stress

The only equation available to relate shear stress to shearing force is The only equation available to relate shear stress to shearing force is derived for a beam of constant cross section constructed of a derived for a beam of constant cross section constructed of a homogeneous elastic material Unfortunately Eq (42) can not be homogeneous elastic material Unfortunately Eq (42) can not be applied to reinforced concrete beams for the following reasonsapplied to reinforced concrete beams for the following reasons

1048707 1048707 Reinforced concrete is nonhomogeneous materialReinforced concrete is nonhomogeneous material

1048707 1048707 Concrete is not elastic Concrete is not elastic

1048707 1048707 Variable extent of cracking along the length of a beam Variable extent of cracking along the length of a beam making it impossible to determine cross-sectional propertiesmaking it impossible to determine cross-sectional properties

Therefore the ACI Code has adopted a simple procedure for Therefore the ACI Code has adopted a simple procedure for establishing the magnitude of shear stress v on a cross sectionestablishing the magnitude of shear stress v on a cross section

db

Vv

w

wherewhere = nominal shear stress= nominal shear stress = shearing force at specified section= shearing force at specified section = width of web of cross section= width of web of cross section = effective depth of the section= effective depth of the section

vVwbd

According to ACI Code 1111 design of cross sections According to ACI Code 1111 design of cross sections subject to shear should be based on the following equationsubject to shear should be based on the following equation

Strength of Concrete in ShearStrength of Concrete in Shear

Shear strength of concrete VShear strength of concrete Vcc is evaluated by loading a plain concrete is evaluated by loading a plain concrete

beam to failure Shear stresses are computed by dividing the shearing beam to failure Shear stresses are computed by dividing the shearing force resisted by concrete Vforce resisted by concrete Vcc by b by bwwd Strength of concrete in shear is d Strength of concrete in shear is

directly proportional to the strength of concrete in tension inversely directly proportional to the strength of concrete in tension inversely proportional to the magnitude of bending moment at the section proportional to the magnitude of bending moment at the section under consideration and directly proportional to the reinforcement under consideration and directly proportional to the reinforcement ratio of flexural reinforcementratio of flexural reinforcementFor the sake of simplicity VFor the sake of simplicity Vcc is assumed to be the same for beams is assumed to be the same for beams

with or without shear reinforcementwith or without shear reinforcementFor members subject to shear and bending only ACI Code 11311 For members subject to shear and bending only ACI Code 11311 gives the following equation for evaluating Vgives the following equation for evaluating Vcc

dbfV wcc 530

where Vwhere Vuu is the factored shearing force M is the factored shearing force Muu is the factored is the factored

bending moment occurring simultaneously with Vbending moment occurring simultaneously with Vuu at section at section

considered ρconsidered ρww is the reinforcement ratio of the web and d is is the reinforcement ratio of the web and d is

the effective depth of the beam should not exceed 10 the effective depth of the beam should not exceed 10 and Vand Vcc should not exceed should not exceed u

u

M

dV

Strength Provided by Shear ReinforcementStrength Provided by Shear ReinforcementWhen the nominal shearing force VWhen the nominal shearing force Vn n exceeds the shearing force exceeds the shearing force

that can be resisted by concrete alone Vthat can be resisted by concrete alone Vcc shear reinforcement in shear reinforcement in

any of the forms shown in the following section can be usedany of the forms shown in the following section can be used

Types of Shear Reinforcement

When shear reinforcement is required the following types of When shear reinforcement is required the following types of shear reinforcement are permitted by ACI Code 1151 as shown shear reinforcement are permitted by ACI Code 1151 as shown a Vertical Stirrupsa Vertical Stirrupsb Inclined stirrups making an angle of 45 degree or more withb Inclined stirrups making an angle of 45 degree or more withlongitudinal tension reinforcementlongitudinal tension reinforcementc Longitudinal reinforcement with bent portion making an angle c Longitudinal reinforcement with bent portion making an angle of 30 degree or more with the tension reinforcementof 30 degree or more with the tension reinforcementd Spirals circular ties or hoopsd Spirals circular ties or hoopse Combination of stirrups and bent longitudinal reinforcemente Combination of stirrups and bent longitudinal reinforcementBefore diagonal cracking occurs the stirrups remain Before diagonal cracking occurs the stirrups remain unstressed After cracking the stress in the stirrups increases unstressed After cracking the stress in the stirrups increases as they pick up a portion of the load formerly carried by the as they pick up a portion of the load formerly carried by the uncracked concreteuncracked concrete

Types of shear reinforcement (a) vertical stirrups (b) inclinedstirrups (c) bent-up bars (two groups) (d) bent-up bars (three groups)

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

Shear in Homogeneous Elastic BeamsShear in Homogeneous Elastic Beams

Loaded beam and orientation of cracks

The normal stresses resulting from bending are given by the The normal stresses resulting from bending are given by the following equation as proved by the classicalfollowing equation as proved by the classical bending theorybending theory

xf

x

xx I

yMf

The shearing stresses are given by the following equation proved The shearing stresses are given by the following equation proved in the most classical mechanics of materials booksin the most classical mechanics of materials books

x

bI

QV

x

xxx

Where is the shearing force at the considered section is the Where is the shearing force at the considered section is the moment of the area of the section located between the point where the moment of the area of the section located between the point where the shearing stresses are calculated and the extreme fiber of the section shearing stresses are calculated and the extreme fiber of the section about the neutral axis is the moment of inertia about the neutral axis about the neutral axis is the moment of inertia about the neutral axis and b is the width of the section at the point where shearing stresses and b is the width of the section at the point where shearing stresses are calculatedare calculated

In an attempt to establish the cracking pattern four elements situated at In an attempt to establish the cracking pattern four elements situated at different distances from the neutral axis are studieddifferent distances from the neutral axis are studied

xV xQ

xI

Types of Shear CracksTypes of Shear Cracks

Two types of inclined cracking occur in beams flexure-shear Two types of inclined cracking occur in beams flexure-shear cracking and web-shear crackingcracking and web-shear cracking

A Flexure-Shear Cracks

The most common type develops from the tip of a flexural crack at The most common type develops from the tip of a flexural crack at the tension side of the beam and propagates towards mid depth the tension side of the beam and propagates towards mid depth until it is checked on the compression side of the beam For these until it is checked on the compression side of the beam For these cracks to form the bending moment must exceed the cracking cracks to form the bending moment must exceed the cracking moment of the cross section and a significant shear must existmoment of the cross section and a significant shear must exist

Types of cracks and associated internal forces (a) orientation of cracks (b) shear force diagram (c) bending moment diagram

B Web Shear Cracks

Web shear cracking begins from an interior point in a member at the Web shear cracking begins from an interior point in a member at the level of the centroid of uncracked section and moves on a diagonal level of the centroid of uncracked section and moves on a diagonal path to the tension face when the diagonal tensile stresses produced path to the tension face when the diagonal tensile stresses produced by shear exceed the tensile strength of concrete This type of by shear exceed the tensile strength of concrete This type of cracking is common on beams with thin webs and in regions of high cracking is common on beams with thin webs and in regions of high shear and small moment This combination exists adjacent to simple shear and small moment This combination exists adjacent to simple supports or at points of inflection in continuous beamssupports or at points of inflection in continuous beams

Nominal Shear StressNominal Shear Stress

The only equation available to relate shear stress to shearing force is The only equation available to relate shear stress to shearing force is derived for a beam of constant cross section constructed of a derived for a beam of constant cross section constructed of a homogeneous elastic material Unfortunately Eq (42) can not be homogeneous elastic material Unfortunately Eq (42) can not be applied to reinforced concrete beams for the following reasonsapplied to reinforced concrete beams for the following reasons

1048707 1048707 Reinforced concrete is nonhomogeneous materialReinforced concrete is nonhomogeneous material

1048707 1048707 Concrete is not elastic Concrete is not elastic

1048707 1048707 Variable extent of cracking along the length of a beam Variable extent of cracking along the length of a beam making it impossible to determine cross-sectional propertiesmaking it impossible to determine cross-sectional properties

Therefore the ACI Code has adopted a simple procedure for Therefore the ACI Code has adopted a simple procedure for establishing the magnitude of shear stress v on a cross sectionestablishing the magnitude of shear stress v on a cross section

db

Vv

w

wherewhere = nominal shear stress= nominal shear stress = shearing force at specified section= shearing force at specified section = width of web of cross section= width of web of cross section = effective depth of the section= effective depth of the section

vVwbd

According to ACI Code 1111 design of cross sections According to ACI Code 1111 design of cross sections subject to shear should be based on the following equationsubject to shear should be based on the following equation

Strength of Concrete in ShearStrength of Concrete in Shear

Shear strength of concrete VShear strength of concrete Vcc is evaluated by loading a plain concrete is evaluated by loading a plain concrete

beam to failure Shear stresses are computed by dividing the shearing beam to failure Shear stresses are computed by dividing the shearing force resisted by concrete Vforce resisted by concrete Vcc by b by bwwd Strength of concrete in shear is d Strength of concrete in shear is

directly proportional to the strength of concrete in tension inversely directly proportional to the strength of concrete in tension inversely proportional to the magnitude of bending moment at the section proportional to the magnitude of bending moment at the section under consideration and directly proportional to the reinforcement under consideration and directly proportional to the reinforcement ratio of flexural reinforcementratio of flexural reinforcementFor the sake of simplicity VFor the sake of simplicity Vcc is assumed to be the same for beams is assumed to be the same for beams

with or without shear reinforcementwith or without shear reinforcementFor members subject to shear and bending only ACI Code 11311 For members subject to shear and bending only ACI Code 11311 gives the following equation for evaluating Vgives the following equation for evaluating Vcc

dbfV wcc 530

where Vwhere Vuu is the factored shearing force M is the factored shearing force Muu is the factored is the factored

bending moment occurring simultaneously with Vbending moment occurring simultaneously with Vuu at section at section

considered ρconsidered ρww is the reinforcement ratio of the web and d is is the reinforcement ratio of the web and d is

the effective depth of the beam should not exceed 10 the effective depth of the beam should not exceed 10 and Vand Vcc should not exceed should not exceed u

u

M

dV

Strength Provided by Shear ReinforcementStrength Provided by Shear ReinforcementWhen the nominal shearing force VWhen the nominal shearing force Vn n exceeds the shearing force exceeds the shearing force

that can be resisted by concrete alone Vthat can be resisted by concrete alone Vcc shear reinforcement in shear reinforcement in

any of the forms shown in the following section can be usedany of the forms shown in the following section can be used

Types of Shear Reinforcement

When shear reinforcement is required the following types of When shear reinforcement is required the following types of shear reinforcement are permitted by ACI Code 1151 as shown shear reinforcement are permitted by ACI Code 1151 as shown a Vertical Stirrupsa Vertical Stirrupsb Inclined stirrups making an angle of 45 degree or more withb Inclined stirrups making an angle of 45 degree or more withlongitudinal tension reinforcementlongitudinal tension reinforcementc Longitudinal reinforcement with bent portion making an angle c Longitudinal reinforcement with bent portion making an angle of 30 degree or more with the tension reinforcementof 30 degree or more with the tension reinforcementd Spirals circular ties or hoopsd Spirals circular ties or hoopse Combination of stirrups and bent longitudinal reinforcemente Combination of stirrups and bent longitudinal reinforcementBefore diagonal cracking occurs the stirrups remain Before diagonal cracking occurs the stirrups remain unstressed After cracking the stress in the stirrups increases unstressed After cracking the stress in the stirrups increases as they pick up a portion of the load formerly carried by the as they pick up a portion of the load formerly carried by the uncracked concreteuncracked concrete

Types of shear reinforcement (a) vertical stirrups (b) inclinedstirrups (c) bent-up bars (two groups) (d) bent-up bars (three groups)

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

The normal stresses resulting from bending are given by the The normal stresses resulting from bending are given by the following equation as proved by the classicalfollowing equation as proved by the classical bending theorybending theory

xf

x

xx I

yMf

The shearing stresses are given by the following equation proved The shearing stresses are given by the following equation proved in the most classical mechanics of materials booksin the most classical mechanics of materials books

x

bI

QV

x

xxx

Where is the shearing force at the considered section is the Where is the shearing force at the considered section is the moment of the area of the section located between the point where the moment of the area of the section located between the point where the shearing stresses are calculated and the extreme fiber of the section shearing stresses are calculated and the extreme fiber of the section about the neutral axis is the moment of inertia about the neutral axis about the neutral axis is the moment of inertia about the neutral axis and b is the width of the section at the point where shearing stresses and b is the width of the section at the point where shearing stresses are calculatedare calculated

In an attempt to establish the cracking pattern four elements situated at In an attempt to establish the cracking pattern four elements situated at different distances from the neutral axis are studieddifferent distances from the neutral axis are studied

xV xQ

xI

Types of Shear CracksTypes of Shear Cracks

Two types of inclined cracking occur in beams flexure-shear Two types of inclined cracking occur in beams flexure-shear cracking and web-shear crackingcracking and web-shear cracking

A Flexure-Shear Cracks

The most common type develops from the tip of a flexural crack at The most common type develops from the tip of a flexural crack at the tension side of the beam and propagates towards mid depth the tension side of the beam and propagates towards mid depth until it is checked on the compression side of the beam For these until it is checked on the compression side of the beam For these cracks to form the bending moment must exceed the cracking cracks to form the bending moment must exceed the cracking moment of the cross section and a significant shear must existmoment of the cross section and a significant shear must exist

Types of cracks and associated internal forces (a) orientation of cracks (b) shear force diagram (c) bending moment diagram

B Web Shear Cracks

Web shear cracking begins from an interior point in a member at the Web shear cracking begins from an interior point in a member at the level of the centroid of uncracked section and moves on a diagonal level of the centroid of uncracked section and moves on a diagonal path to the tension face when the diagonal tensile stresses produced path to the tension face when the diagonal tensile stresses produced by shear exceed the tensile strength of concrete This type of by shear exceed the tensile strength of concrete This type of cracking is common on beams with thin webs and in regions of high cracking is common on beams with thin webs and in regions of high shear and small moment This combination exists adjacent to simple shear and small moment This combination exists adjacent to simple supports or at points of inflection in continuous beamssupports or at points of inflection in continuous beams

Nominal Shear StressNominal Shear Stress

The only equation available to relate shear stress to shearing force is The only equation available to relate shear stress to shearing force is derived for a beam of constant cross section constructed of a derived for a beam of constant cross section constructed of a homogeneous elastic material Unfortunately Eq (42) can not be homogeneous elastic material Unfortunately Eq (42) can not be applied to reinforced concrete beams for the following reasonsapplied to reinforced concrete beams for the following reasons

1048707 1048707 Reinforced concrete is nonhomogeneous materialReinforced concrete is nonhomogeneous material

1048707 1048707 Concrete is not elastic Concrete is not elastic

1048707 1048707 Variable extent of cracking along the length of a beam Variable extent of cracking along the length of a beam making it impossible to determine cross-sectional propertiesmaking it impossible to determine cross-sectional properties

Therefore the ACI Code has adopted a simple procedure for Therefore the ACI Code has adopted a simple procedure for establishing the magnitude of shear stress v on a cross sectionestablishing the magnitude of shear stress v on a cross section

db

Vv

w

wherewhere = nominal shear stress= nominal shear stress = shearing force at specified section= shearing force at specified section = width of web of cross section= width of web of cross section = effective depth of the section= effective depth of the section

vVwbd

According to ACI Code 1111 design of cross sections According to ACI Code 1111 design of cross sections subject to shear should be based on the following equationsubject to shear should be based on the following equation

Strength of Concrete in ShearStrength of Concrete in Shear

Shear strength of concrete VShear strength of concrete Vcc is evaluated by loading a plain concrete is evaluated by loading a plain concrete

beam to failure Shear stresses are computed by dividing the shearing beam to failure Shear stresses are computed by dividing the shearing force resisted by concrete Vforce resisted by concrete Vcc by b by bwwd Strength of concrete in shear is d Strength of concrete in shear is

directly proportional to the strength of concrete in tension inversely directly proportional to the strength of concrete in tension inversely proportional to the magnitude of bending moment at the section proportional to the magnitude of bending moment at the section under consideration and directly proportional to the reinforcement under consideration and directly proportional to the reinforcement ratio of flexural reinforcementratio of flexural reinforcementFor the sake of simplicity VFor the sake of simplicity Vcc is assumed to be the same for beams is assumed to be the same for beams

with or without shear reinforcementwith or without shear reinforcementFor members subject to shear and bending only ACI Code 11311 For members subject to shear and bending only ACI Code 11311 gives the following equation for evaluating Vgives the following equation for evaluating Vcc

dbfV wcc 530

where Vwhere Vuu is the factored shearing force M is the factored shearing force Muu is the factored is the factored

bending moment occurring simultaneously with Vbending moment occurring simultaneously with Vuu at section at section

considered ρconsidered ρww is the reinforcement ratio of the web and d is is the reinforcement ratio of the web and d is

the effective depth of the beam should not exceed 10 the effective depth of the beam should not exceed 10 and Vand Vcc should not exceed should not exceed u

u

M

dV

Strength Provided by Shear ReinforcementStrength Provided by Shear ReinforcementWhen the nominal shearing force VWhen the nominal shearing force Vn n exceeds the shearing force exceeds the shearing force

that can be resisted by concrete alone Vthat can be resisted by concrete alone Vcc shear reinforcement in shear reinforcement in

any of the forms shown in the following section can be usedany of the forms shown in the following section can be used

Types of Shear Reinforcement

When shear reinforcement is required the following types of When shear reinforcement is required the following types of shear reinforcement are permitted by ACI Code 1151 as shown shear reinforcement are permitted by ACI Code 1151 as shown a Vertical Stirrupsa Vertical Stirrupsb Inclined stirrups making an angle of 45 degree or more withb Inclined stirrups making an angle of 45 degree or more withlongitudinal tension reinforcementlongitudinal tension reinforcementc Longitudinal reinforcement with bent portion making an angle c Longitudinal reinforcement with bent portion making an angle of 30 degree or more with the tension reinforcementof 30 degree or more with the tension reinforcementd Spirals circular ties or hoopsd Spirals circular ties or hoopse Combination of stirrups and bent longitudinal reinforcemente Combination of stirrups and bent longitudinal reinforcementBefore diagonal cracking occurs the stirrups remain Before diagonal cracking occurs the stirrups remain unstressed After cracking the stress in the stirrups increases unstressed After cracking the stress in the stirrups increases as they pick up a portion of the load formerly carried by the as they pick up a portion of the load formerly carried by the uncracked concreteuncracked concrete

Types of shear reinforcement (a) vertical stirrups (b) inclinedstirrups (c) bent-up bars (two groups) (d) bent-up bars (three groups)

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

Where is the shearing force at the considered section is the Where is the shearing force at the considered section is the moment of the area of the section located between the point where the moment of the area of the section located between the point where the shearing stresses are calculated and the extreme fiber of the section shearing stresses are calculated and the extreme fiber of the section about the neutral axis is the moment of inertia about the neutral axis about the neutral axis is the moment of inertia about the neutral axis and b is the width of the section at the point where shearing stresses and b is the width of the section at the point where shearing stresses are calculatedare calculated

In an attempt to establish the cracking pattern four elements situated at In an attempt to establish the cracking pattern four elements situated at different distances from the neutral axis are studieddifferent distances from the neutral axis are studied

xV xQ

xI

Types of Shear CracksTypes of Shear Cracks

Two types of inclined cracking occur in beams flexure-shear Two types of inclined cracking occur in beams flexure-shear cracking and web-shear crackingcracking and web-shear cracking

A Flexure-Shear Cracks

The most common type develops from the tip of a flexural crack at The most common type develops from the tip of a flexural crack at the tension side of the beam and propagates towards mid depth the tension side of the beam and propagates towards mid depth until it is checked on the compression side of the beam For these until it is checked on the compression side of the beam For these cracks to form the bending moment must exceed the cracking cracks to form the bending moment must exceed the cracking moment of the cross section and a significant shear must existmoment of the cross section and a significant shear must exist

Types of cracks and associated internal forces (a) orientation of cracks (b) shear force diagram (c) bending moment diagram

B Web Shear Cracks

Web shear cracking begins from an interior point in a member at the Web shear cracking begins from an interior point in a member at the level of the centroid of uncracked section and moves on a diagonal level of the centroid of uncracked section and moves on a diagonal path to the tension face when the diagonal tensile stresses produced path to the tension face when the diagonal tensile stresses produced by shear exceed the tensile strength of concrete This type of by shear exceed the tensile strength of concrete This type of cracking is common on beams with thin webs and in regions of high cracking is common on beams with thin webs and in regions of high shear and small moment This combination exists adjacent to simple shear and small moment This combination exists adjacent to simple supports or at points of inflection in continuous beamssupports or at points of inflection in continuous beams

Nominal Shear StressNominal Shear Stress

The only equation available to relate shear stress to shearing force is The only equation available to relate shear stress to shearing force is derived for a beam of constant cross section constructed of a derived for a beam of constant cross section constructed of a homogeneous elastic material Unfortunately Eq (42) can not be homogeneous elastic material Unfortunately Eq (42) can not be applied to reinforced concrete beams for the following reasonsapplied to reinforced concrete beams for the following reasons

1048707 1048707 Reinforced concrete is nonhomogeneous materialReinforced concrete is nonhomogeneous material

1048707 1048707 Concrete is not elastic Concrete is not elastic

1048707 1048707 Variable extent of cracking along the length of a beam Variable extent of cracking along the length of a beam making it impossible to determine cross-sectional propertiesmaking it impossible to determine cross-sectional properties

Therefore the ACI Code has adopted a simple procedure for Therefore the ACI Code has adopted a simple procedure for establishing the magnitude of shear stress v on a cross sectionestablishing the magnitude of shear stress v on a cross section

db

Vv

w

wherewhere = nominal shear stress= nominal shear stress = shearing force at specified section= shearing force at specified section = width of web of cross section= width of web of cross section = effective depth of the section= effective depth of the section

vVwbd

According to ACI Code 1111 design of cross sections According to ACI Code 1111 design of cross sections subject to shear should be based on the following equationsubject to shear should be based on the following equation

Strength of Concrete in ShearStrength of Concrete in Shear

Shear strength of concrete VShear strength of concrete Vcc is evaluated by loading a plain concrete is evaluated by loading a plain concrete

beam to failure Shear stresses are computed by dividing the shearing beam to failure Shear stresses are computed by dividing the shearing force resisted by concrete Vforce resisted by concrete Vcc by b by bwwd Strength of concrete in shear is d Strength of concrete in shear is

directly proportional to the strength of concrete in tension inversely directly proportional to the strength of concrete in tension inversely proportional to the magnitude of bending moment at the section proportional to the magnitude of bending moment at the section under consideration and directly proportional to the reinforcement under consideration and directly proportional to the reinforcement ratio of flexural reinforcementratio of flexural reinforcementFor the sake of simplicity VFor the sake of simplicity Vcc is assumed to be the same for beams is assumed to be the same for beams

with or without shear reinforcementwith or without shear reinforcementFor members subject to shear and bending only ACI Code 11311 For members subject to shear and bending only ACI Code 11311 gives the following equation for evaluating Vgives the following equation for evaluating Vcc

dbfV wcc 530

where Vwhere Vuu is the factored shearing force M is the factored shearing force Muu is the factored is the factored

bending moment occurring simultaneously with Vbending moment occurring simultaneously with Vuu at section at section

considered ρconsidered ρww is the reinforcement ratio of the web and d is is the reinforcement ratio of the web and d is

the effective depth of the beam should not exceed 10 the effective depth of the beam should not exceed 10 and Vand Vcc should not exceed should not exceed u

u

M

dV

Strength Provided by Shear ReinforcementStrength Provided by Shear ReinforcementWhen the nominal shearing force VWhen the nominal shearing force Vn n exceeds the shearing force exceeds the shearing force

that can be resisted by concrete alone Vthat can be resisted by concrete alone Vcc shear reinforcement in shear reinforcement in

any of the forms shown in the following section can be usedany of the forms shown in the following section can be used

Types of Shear Reinforcement

When shear reinforcement is required the following types of When shear reinforcement is required the following types of shear reinforcement are permitted by ACI Code 1151 as shown shear reinforcement are permitted by ACI Code 1151 as shown a Vertical Stirrupsa Vertical Stirrupsb Inclined stirrups making an angle of 45 degree or more withb Inclined stirrups making an angle of 45 degree or more withlongitudinal tension reinforcementlongitudinal tension reinforcementc Longitudinal reinforcement with bent portion making an angle c Longitudinal reinforcement with bent portion making an angle of 30 degree or more with the tension reinforcementof 30 degree or more with the tension reinforcementd Spirals circular ties or hoopsd Spirals circular ties or hoopse Combination of stirrups and bent longitudinal reinforcemente Combination of stirrups and bent longitudinal reinforcementBefore diagonal cracking occurs the stirrups remain Before diagonal cracking occurs the stirrups remain unstressed After cracking the stress in the stirrups increases unstressed After cracking the stress in the stirrups increases as they pick up a portion of the load formerly carried by the as they pick up a portion of the load formerly carried by the uncracked concreteuncracked concrete

Types of shear reinforcement (a) vertical stirrups (b) inclinedstirrups (c) bent-up bars (two groups) (d) bent-up bars (three groups)

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

Types of Shear CracksTypes of Shear Cracks

Two types of inclined cracking occur in beams flexure-shear Two types of inclined cracking occur in beams flexure-shear cracking and web-shear crackingcracking and web-shear cracking

A Flexure-Shear Cracks

The most common type develops from the tip of a flexural crack at The most common type develops from the tip of a flexural crack at the tension side of the beam and propagates towards mid depth the tension side of the beam and propagates towards mid depth until it is checked on the compression side of the beam For these until it is checked on the compression side of the beam For these cracks to form the bending moment must exceed the cracking cracks to form the bending moment must exceed the cracking moment of the cross section and a significant shear must existmoment of the cross section and a significant shear must exist

Types of cracks and associated internal forces (a) orientation of cracks (b) shear force diagram (c) bending moment diagram

B Web Shear Cracks

Web shear cracking begins from an interior point in a member at the Web shear cracking begins from an interior point in a member at the level of the centroid of uncracked section and moves on a diagonal level of the centroid of uncracked section and moves on a diagonal path to the tension face when the diagonal tensile stresses produced path to the tension face when the diagonal tensile stresses produced by shear exceed the tensile strength of concrete This type of by shear exceed the tensile strength of concrete This type of cracking is common on beams with thin webs and in regions of high cracking is common on beams with thin webs and in regions of high shear and small moment This combination exists adjacent to simple shear and small moment This combination exists adjacent to simple supports or at points of inflection in continuous beamssupports or at points of inflection in continuous beams

Nominal Shear StressNominal Shear Stress

The only equation available to relate shear stress to shearing force is The only equation available to relate shear stress to shearing force is derived for a beam of constant cross section constructed of a derived for a beam of constant cross section constructed of a homogeneous elastic material Unfortunately Eq (42) can not be homogeneous elastic material Unfortunately Eq (42) can not be applied to reinforced concrete beams for the following reasonsapplied to reinforced concrete beams for the following reasons

1048707 1048707 Reinforced concrete is nonhomogeneous materialReinforced concrete is nonhomogeneous material

1048707 1048707 Concrete is not elastic Concrete is not elastic

1048707 1048707 Variable extent of cracking along the length of a beam Variable extent of cracking along the length of a beam making it impossible to determine cross-sectional propertiesmaking it impossible to determine cross-sectional properties

Therefore the ACI Code has adopted a simple procedure for Therefore the ACI Code has adopted a simple procedure for establishing the magnitude of shear stress v on a cross sectionestablishing the magnitude of shear stress v on a cross section

db

Vv

w

wherewhere = nominal shear stress= nominal shear stress = shearing force at specified section= shearing force at specified section = width of web of cross section= width of web of cross section = effective depth of the section= effective depth of the section

vVwbd

According to ACI Code 1111 design of cross sections According to ACI Code 1111 design of cross sections subject to shear should be based on the following equationsubject to shear should be based on the following equation

Strength of Concrete in ShearStrength of Concrete in Shear

Shear strength of concrete VShear strength of concrete Vcc is evaluated by loading a plain concrete is evaluated by loading a plain concrete

beam to failure Shear stresses are computed by dividing the shearing beam to failure Shear stresses are computed by dividing the shearing force resisted by concrete Vforce resisted by concrete Vcc by b by bwwd Strength of concrete in shear is d Strength of concrete in shear is

directly proportional to the strength of concrete in tension inversely directly proportional to the strength of concrete in tension inversely proportional to the magnitude of bending moment at the section proportional to the magnitude of bending moment at the section under consideration and directly proportional to the reinforcement under consideration and directly proportional to the reinforcement ratio of flexural reinforcementratio of flexural reinforcementFor the sake of simplicity VFor the sake of simplicity Vcc is assumed to be the same for beams is assumed to be the same for beams

with or without shear reinforcementwith or without shear reinforcementFor members subject to shear and bending only ACI Code 11311 For members subject to shear and bending only ACI Code 11311 gives the following equation for evaluating Vgives the following equation for evaluating Vcc

dbfV wcc 530

where Vwhere Vuu is the factored shearing force M is the factored shearing force Muu is the factored is the factored

bending moment occurring simultaneously with Vbending moment occurring simultaneously with Vuu at section at section

considered ρconsidered ρww is the reinforcement ratio of the web and d is is the reinforcement ratio of the web and d is

the effective depth of the beam should not exceed 10 the effective depth of the beam should not exceed 10 and Vand Vcc should not exceed should not exceed u

u

M

dV

Strength Provided by Shear ReinforcementStrength Provided by Shear ReinforcementWhen the nominal shearing force VWhen the nominal shearing force Vn n exceeds the shearing force exceeds the shearing force

that can be resisted by concrete alone Vthat can be resisted by concrete alone Vcc shear reinforcement in shear reinforcement in

any of the forms shown in the following section can be usedany of the forms shown in the following section can be used

Types of Shear Reinforcement

When shear reinforcement is required the following types of When shear reinforcement is required the following types of shear reinforcement are permitted by ACI Code 1151 as shown shear reinforcement are permitted by ACI Code 1151 as shown a Vertical Stirrupsa Vertical Stirrupsb Inclined stirrups making an angle of 45 degree or more withb Inclined stirrups making an angle of 45 degree or more withlongitudinal tension reinforcementlongitudinal tension reinforcementc Longitudinal reinforcement with bent portion making an angle c Longitudinal reinforcement with bent portion making an angle of 30 degree or more with the tension reinforcementof 30 degree or more with the tension reinforcementd Spirals circular ties or hoopsd Spirals circular ties or hoopse Combination of stirrups and bent longitudinal reinforcemente Combination of stirrups and bent longitudinal reinforcementBefore diagonal cracking occurs the stirrups remain Before diagonal cracking occurs the stirrups remain unstressed After cracking the stress in the stirrups increases unstressed After cracking the stress in the stirrups increases as they pick up a portion of the load formerly carried by the as they pick up a portion of the load formerly carried by the uncracked concreteuncracked concrete

Types of shear reinforcement (a) vertical stirrups (b) inclinedstirrups (c) bent-up bars (two groups) (d) bent-up bars (three groups)

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

Types of cracks and associated internal forces (a) orientation of cracks (b) shear force diagram (c) bending moment diagram

B Web Shear Cracks

Web shear cracking begins from an interior point in a member at the Web shear cracking begins from an interior point in a member at the level of the centroid of uncracked section and moves on a diagonal level of the centroid of uncracked section and moves on a diagonal path to the tension face when the diagonal tensile stresses produced path to the tension face when the diagonal tensile stresses produced by shear exceed the tensile strength of concrete This type of by shear exceed the tensile strength of concrete This type of cracking is common on beams with thin webs and in regions of high cracking is common on beams with thin webs and in regions of high shear and small moment This combination exists adjacent to simple shear and small moment This combination exists adjacent to simple supports or at points of inflection in continuous beamssupports or at points of inflection in continuous beams

Nominal Shear StressNominal Shear Stress

The only equation available to relate shear stress to shearing force is The only equation available to relate shear stress to shearing force is derived for a beam of constant cross section constructed of a derived for a beam of constant cross section constructed of a homogeneous elastic material Unfortunately Eq (42) can not be homogeneous elastic material Unfortunately Eq (42) can not be applied to reinforced concrete beams for the following reasonsapplied to reinforced concrete beams for the following reasons

1048707 1048707 Reinforced concrete is nonhomogeneous materialReinforced concrete is nonhomogeneous material

1048707 1048707 Concrete is not elastic Concrete is not elastic

1048707 1048707 Variable extent of cracking along the length of a beam Variable extent of cracking along the length of a beam making it impossible to determine cross-sectional propertiesmaking it impossible to determine cross-sectional properties

Therefore the ACI Code has adopted a simple procedure for Therefore the ACI Code has adopted a simple procedure for establishing the magnitude of shear stress v on a cross sectionestablishing the magnitude of shear stress v on a cross section

db

Vv

w

wherewhere = nominal shear stress= nominal shear stress = shearing force at specified section= shearing force at specified section = width of web of cross section= width of web of cross section = effective depth of the section= effective depth of the section

vVwbd

According to ACI Code 1111 design of cross sections According to ACI Code 1111 design of cross sections subject to shear should be based on the following equationsubject to shear should be based on the following equation

Strength of Concrete in ShearStrength of Concrete in Shear

Shear strength of concrete VShear strength of concrete Vcc is evaluated by loading a plain concrete is evaluated by loading a plain concrete

beam to failure Shear stresses are computed by dividing the shearing beam to failure Shear stresses are computed by dividing the shearing force resisted by concrete Vforce resisted by concrete Vcc by b by bwwd Strength of concrete in shear is d Strength of concrete in shear is

directly proportional to the strength of concrete in tension inversely directly proportional to the strength of concrete in tension inversely proportional to the magnitude of bending moment at the section proportional to the magnitude of bending moment at the section under consideration and directly proportional to the reinforcement under consideration and directly proportional to the reinforcement ratio of flexural reinforcementratio of flexural reinforcementFor the sake of simplicity VFor the sake of simplicity Vcc is assumed to be the same for beams is assumed to be the same for beams

with or without shear reinforcementwith or without shear reinforcementFor members subject to shear and bending only ACI Code 11311 For members subject to shear and bending only ACI Code 11311 gives the following equation for evaluating Vgives the following equation for evaluating Vcc

dbfV wcc 530

where Vwhere Vuu is the factored shearing force M is the factored shearing force Muu is the factored is the factored

bending moment occurring simultaneously with Vbending moment occurring simultaneously with Vuu at section at section

considered ρconsidered ρww is the reinforcement ratio of the web and d is is the reinforcement ratio of the web and d is

the effective depth of the beam should not exceed 10 the effective depth of the beam should not exceed 10 and Vand Vcc should not exceed should not exceed u

u

M

dV

Strength Provided by Shear ReinforcementStrength Provided by Shear ReinforcementWhen the nominal shearing force VWhen the nominal shearing force Vn n exceeds the shearing force exceeds the shearing force

that can be resisted by concrete alone Vthat can be resisted by concrete alone Vcc shear reinforcement in shear reinforcement in

any of the forms shown in the following section can be usedany of the forms shown in the following section can be used

Types of Shear Reinforcement

When shear reinforcement is required the following types of When shear reinforcement is required the following types of shear reinforcement are permitted by ACI Code 1151 as shown shear reinforcement are permitted by ACI Code 1151 as shown a Vertical Stirrupsa Vertical Stirrupsb Inclined stirrups making an angle of 45 degree or more withb Inclined stirrups making an angle of 45 degree or more withlongitudinal tension reinforcementlongitudinal tension reinforcementc Longitudinal reinforcement with bent portion making an angle c Longitudinal reinforcement with bent portion making an angle of 30 degree or more with the tension reinforcementof 30 degree or more with the tension reinforcementd Spirals circular ties or hoopsd Spirals circular ties or hoopse Combination of stirrups and bent longitudinal reinforcemente Combination of stirrups and bent longitudinal reinforcementBefore diagonal cracking occurs the stirrups remain Before diagonal cracking occurs the stirrups remain unstressed After cracking the stress in the stirrups increases unstressed After cracking the stress in the stirrups increases as they pick up a portion of the load formerly carried by the as they pick up a portion of the load formerly carried by the uncracked concreteuncracked concrete

Types of shear reinforcement (a) vertical stirrups (b) inclinedstirrups (c) bent-up bars (two groups) (d) bent-up bars (three groups)

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

B Web Shear Cracks

Web shear cracking begins from an interior point in a member at the Web shear cracking begins from an interior point in a member at the level of the centroid of uncracked section and moves on a diagonal level of the centroid of uncracked section and moves on a diagonal path to the tension face when the diagonal tensile stresses produced path to the tension face when the diagonal tensile stresses produced by shear exceed the tensile strength of concrete This type of by shear exceed the tensile strength of concrete This type of cracking is common on beams with thin webs and in regions of high cracking is common on beams with thin webs and in regions of high shear and small moment This combination exists adjacent to simple shear and small moment This combination exists adjacent to simple supports or at points of inflection in continuous beamssupports or at points of inflection in continuous beams

Nominal Shear StressNominal Shear Stress

The only equation available to relate shear stress to shearing force is The only equation available to relate shear stress to shearing force is derived for a beam of constant cross section constructed of a derived for a beam of constant cross section constructed of a homogeneous elastic material Unfortunately Eq (42) can not be homogeneous elastic material Unfortunately Eq (42) can not be applied to reinforced concrete beams for the following reasonsapplied to reinforced concrete beams for the following reasons

1048707 1048707 Reinforced concrete is nonhomogeneous materialReinforced concrete is nonhomogeneous material

1048707 1048707 Concrete is not elastic Concrete is not elastic

1048707 1048707 Variable extent of cracking along the length of a beam Variable extent of cracking along the length of a beam making it impossible to determine cross-sectional propertiesmaking it impossible to determine cross-sectional properties

Therefore the ACI Code has adopted a simple procedure for Therefore the ACI Code has adopted a simple procedure for establishing the magnitude of shear stress v on a cross sectionestablishing the magnitude of shear stress v on a cross section

db

Vv

w

wherewhere = nominal shear stress= nominal shear stress = shearing force at specified section= shearing force at specified section = width of web of cross section= width of web of cross section = effective depth of the section= effective depth of the section

vVwbd

According to ACI Code 1111 design of cross sections According to ACI Code 1111 design of cross sections subject to shear should be based on the following equationsubject to shear should be based on the following equation

Strength of Concrete in ShearStrength of Concrete in Shear

Shear strength of concrete VShear strength of concrete Vcc is evaluated by loading a plain concrete is evaluated by loading a plain concrete

beam to failure Shear stresses are computed by dividing the shearing beam to failure Shear stresses are computed by dividing the shearing force resisted by concrete Vforce resisted by concrete Vcc by b by bwwd Strength of concrete in shear is d Strength of concrete in shear is

directly proportional to the strength of concrete in tension inversely directly proportional to the strength of concrete in tension inversely proportional to the magnitude of bending moment at the section proportional to the magnitude of bending moment at the section under consideration and directly proportional to the reinforcement under consideration and directly proportional to the reinforcement ratio of flexural reinforcementratio of flexural reinforcementFor the sake of simplicity VFor the sake of simplicity Vcc is assumed to be the same for beams is assumed to be the same for beams

with or without shear reinforcementwith or without shear reinforcementFor members subject to shear and bending only ACI Code 11311 For members subject to shear and bending only ACI Code 11311 gives the following equation for evaluating Vgives the following equation for evaluating Vcc

dbfV wcc 530

where Vwhere Vuu is the factored shearing force M is the factored shearing force Muu is the factored is the factored

bending moment occurring simultaneously with Vbending moment occurring simultaneously with Vuu at section at section

considered ρconsidered ρww is the reinforcement ratio of the web and d is is the reinforcement ratio of the web and d is

the effective depth of the beam should not exceed 10 the effective depth of the beam should not exceed 10 and Vand Vcc should not exceed should not exceed u

u

M

dV

Strength Provided by Shear ReinforcementStrength Provided by Shear ReinforcementWhen the nominal shearing force VWhen the nominal shearing force Vn n exceeds the shearing force exceeds the shearing force

that can be resisted by concrete alone Vthat can be resisted by concrete alone Vcc shear reinforcement in shear reinforcement in

any of the forms shown in the following section can be usedany of the forms shown in the following section can be used

Types of Shear Reinforcement

When shear reinforcement is required the following types of When shear reinforcement is required the following types of shear reinforcement are permitted by ACI Code 1151 as shown shear reinforcement are permitted by ACI Code 1151 as shown a Vertical Stirrupsa Vertical Stirrupsb Inclined stirrups making an angle of 45 degree or more withb Inclined stirrups making an angle of 45 degree or more withlongitudinal tension reinforcementlongitudinal tension reinforcementc Longitudinal reinforcement with bent portion making an angle c Longitudinal reinforcement with bent portion making an angle of 30 degree or more with the tension reinforcementof 30 degree or more with the tension reinforcementd Spirals circular ties or hoopsd Spirals circular ties or hoopse Combination of stirrups and bent longitudinal reinforcemente Combination of stirrups and bent longitudinal reinforcementBefore diagonal cracking occurs the stirrups remain Before diagonal cracking occurs the stirrups remain unstressed After cracking the stress in the stirrups increases unstressed After cracking the stress in the stirrups increases as they pick up a portion of the load formerly carried by the as they pick up a portion of the load formerly carried by the uncracked concreteuncracked concrete

Types of shear reinforcement (a) vertical stirrups (b) inclinedstirrups (c) bent-up bars (two groups) (d) bent-up bars (three groups)

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

1048707 1048707 Reinforced concrete is nonhomogeneous materialReinforced concrete is nonhomogeneous material

1048707 1048707 Concrete is not elastic Concrete is not elastic

1048707 1048707 Variable extent of cracking along the length of a beam Variable extent of cracking along the length of a beam making it impossible to determine cross-sectional propertiesmaking it impossible to determine cross-sectional properties

Therefore the ACI Code has adopted a simple procedure for Therefore the ACI Code has adopted a simple procedure for establishing the magnitude of shear stress v on a cross sectionestablishing the magnitude of shear stress v on a cross section

db

Vv

w

wherewhere = nominal shear stress= nominal shear stress = shearing force at specified section= shearing force at specified section = width of web of cross section= width of web of cross section = effective depth of the section= effective depth of the section

vVwbd

According to ACI Code 1111 design of cross sections According to ACI Code 1111 design of cross sections subject to shear should be based on the following equationsubject to shear should be based on the following equation

Strength of Concrete in ShearStrength of Concrete in Shear

Shear strength of concrete VShear strength of concrete Vcc is evaluated by loading a plain concrete is evaluated by loading a plain concrete

beam to failure Shear stresses are computed by dividing the shearing beam to failure Shear stresses are computed by dividing the shearing force resisted by concrete Vforce resisted by concrete Vcc by b by bwwd Strength of concrete in shear is d Strength of concrete in shear is

directly proportional to the strength of concrete in tension inversely directly proportional to the strength of concrete in tension inversely proportional to the magnitude of bending moment at the section proportional to the magnitude of bending moment at the section under consideration and directly proportional to the reinforcement under consideration and directly proportional to the reinforcement ratio of flexural reinforcementratio of flexural reinforcementFor the sake of simplicity VFor the sake of simplicity Vcc is assumed to be the same for beams is assumed to be the same for beams

with or without shear reinforcementwith or without shear reinforcementFor members subject to shear and bending only ACI Code 11311 For members subject to shear and bending only ACI Code 11311 gives the following equation for evaluating Vgives the following equation for evaluating Vcc

dbfV wcc 530

where Vwhere Vuu is the factored shearing force M is the factored shearing force Muu is the factored is the factored

bending moment occurring simultaneously with Vbending moment occurring simultaneously with Vuu at section at section

considered ρconsidered ρww is the reinforcement ratio of the web and d is is the reinforcement ratio of the web and d is

the effective depth of the beam should not exceed 10 the effective depth of the beam should not exceed 10 and Vand Vcc should not exceed should not exceed u

u

M

dV

Strength Provided by Shear ReinforcementStrength Provided by Shear ReinforcementWhen the nominal shearing force VWhen the nominal shearing force Vn n exceeds the shearing force exceeds the shearing force

that can be resisted by concrete alone Vthat can be resisted by concrete alone Vcc shear reinforcement in shear reinforcement in

any of the forms shown in the following section can be usedany of the forms shown in the following section can be used

Types of Shear Reinforcement

When shear reinforcement is required the following types of When shear reinforcement is required the following types of shear reinforcement are permitted by ACI Code 1151 as shown shear reinforcement are permitted by ACI Code 1151 as shown a Vertical Stirrupsa Vertical Stirrupsb Inclined stirrups making an angle of 45 degree or more withb Inclined stirrups making an angle of 45 degree or more withlongitudinal tension reinforcementlongitudinal tension reinforcementc Longitudinal reinforcement with bent portion making an angle c Longitudinal reinforcement with bent portion making an angle of 30 degree or more with the tension reinforcementof 30 degree or more with the tension reinforcementd Spirals circular ties or hoopsd Spirals circular ties or hoopse Combination of stirrups and bent longitudinal reinforcemente Combination of stirrups and bent longitudinal reinforcementBefore diagonal cracking occurs the stirrups remain Before diagonal cracking occurs the stirrups remain unstressed After cracking the stress in the stirrups increases unstressed After cracking the stress in the stirrups increases as they pick up a portion of the load formerly carried by the as they pick up a portion of the load formerly carried by the uncracked concreteuncracked concrete

Types of shear reinforcement (a) vertical stirrups (b) inclinedstirrups (c) bent-up bars (two groups) (d) bent-up bars (three groups)

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

According to ACI Code 1111 design of cross sections According to ACI Code 1111 design of cross sections subject to shear should be based on the following equationsubject to shear should be based on the following equation

Strength of Concrete in ShearStrength of Concrete in Shear

Shear strength of concrete VShear strength of concrete Vcc is evaluated by loading a plain concrete is evaluated by loading a plain concrete

beam to failure Shear stresses are computed by dividing the shearing beam to failure Shear stresses are computed by dividing the shearing force resisted by concrete Vforce resisted by concrete Vcc by b by bwwd Strength of concrete in shear is d Strength of concrete in shear is

directly proportional to the strength of concrete in tension inversely directly proportional to the strength of concrete in tension inversely proportional to the magnitude of bending moment at the section proportional to the magnitude of bending moment at the section under consideration and directly proportional to the reinforcement under consideration and directly proportional to the reinforcement ratio of flexural reinforcementratio of flexural reinforcementFor the sake of simplicity VFor the sake of simplicity Vcc is assumed to be the same for beams is assumed to be the same for beams

with or without shear reinforcementwith or without shear reinforcementFor members subject to shear and bending only ACI Code 11311 For members subject to shear and bending only ACI Code 11311 gives the following equation for evaluating Vgives the following equation for evaluating Vcc

dbfV wcc 530

where Vwhere Vuu is the factored shearing force M is the factored shearing force Muu is the factored is the factored

bending moment occurring simultaneously with Vbending moment occurring simultaneously with Vuu at section at section

considered ρconsidered ρww is the reinforcement ratio of the web and d is is the reinforcement ratio of the web and d is

the effective depth of the beam should not exceed 10 the effective depth of the beam should not exceed 10 and Vand Vcc should not exceed should not exceed u

u

M

dV

Strength Provided by Shear ReinforcementStrength Provided by Shear ReinforcementWhen the nominal shearing force VWhen the nominal shearing force Vn n exceeds the shearing force exceeds the shearing force

that can be resisted by concrete alone Vthat can be resisted by concrete alone Vcc shear reinforcement in shear reinforcement in

any of the forms shown in the following section can be usedany of the forms shown in the following section can be used

Types of Shear Reinforcement

When shear reinforcement is required the following types of When shear reinforcement is required the following types of shear reinforcement are permitted by ACI Code 1151 as shown shear reinforcement are permitted by ACI Code 1151 as shown a Vertical Stirrupsa Vertical Stirrupsb Inclined stirrups making an angle of 45 degree or more withb Inclined stirrups making an angle of 45 degree or more withlongitudinal tension reinforcementlongitudinal tension reinforcementc Longitudinal reinforcement with bent portion making an angle c Longitudinal reinforcement with bent portion making an angle of 30 degree or more with the tension reinforcementof 30 degree or more with the tension reinforcementd Spirals circular ties or hoopsd Spirals circular ties or hoopse Combination of stirrups and bent longitudinal reinforcemente Combination of stirrups and bent longitudinal reinforcementBefore diagonal cracking occurs the stirrups remain Before diagonal cracking occurs the stirrups remain unstressed After cracking the stress in the stirrups increases unstressed After cracking the stress in the stirrups increases as they pick up a portion of the load formerly carried by the as they pick up a portion of the load formerly carried by the uncracked concreteuncracked concrete

Types of shear reinforcement (a) vertical stirrups (b) inclinedstirrups (c) bent-up bars (two groups) (d) bent-up bars (three groups)

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

Strength of Concrete in ShearStrength of Concrete in Shear

Shear strength of concrete VShear strength of concrete Vcc is evaluated by loading a plain concrete is evaluated by loading a plain concrete

beam to failure Shear stresses are computed by dividing the shearing beam to failure Shear stresses are computed by dividing the shearing force resisted by concrete Vforce resisted by concrete Vcc by b by bwwd Strength of concrete in shear is d Strength of concrete in shear is

directly proportional to the strength of concrete in tension inversely directly proportional to the strength of concrete in tension inversely proportional to the magnitude of bending moment at the section proportional to the magnitude of bending moment at the section under consideration and directly proportional to the reinforcement under consideration and directly proportional to the reinforcement ratio of flexural reinforcementratio of flexural reinforcementFor the sake of simplicity VFor the sake of simplicity Vcc is assumed to be the same for beams is assumed to be the same for beams

with or without shear reinforcementwith or without shear reinforcementFor members subject to shear and bending only ACI Code 11311 For members subject to shear and bending only ACI Code 11311 gives the following equation for evaluating Vgives the following equation for evaluating Vcc

dbfV wcc 530

where Vwhere Vuu is the factored shearing force M is the factored shearing force Muu is the factored is the factored

bending moment occurring simultaneously with Vbending moment occurring simultaneously with Vuu at section at section

considered ρconsidered ρww is the reinforcement ratio of the web and d is is the reinforcement ratio of the web and d is

the effective depth of the beam should not exceed 10 the effective depth of the beam should not exceed 10 and Vand Vcc should not exceed should not exceed u

u

M

dV

Strength Provided by Shear ReinforcementStrength Provided by Shear ReinforcementWhen the nominal shearing force VWhen the nominal shearing force Vn n exceeds the shearing force exceeds the shearing force

that can be resisted by concrete alone Vthat can be resisted by concrete alone Vcc shear reinforcement in shear reinforcement in

any of the forms shown in the following section can be usedany of the forms shown in the following section can be used

Types of Shear Reinforcement

When shear reinforcement is required the following types of When shear reinforcement is required the following types of shear reinforcement are permitted by ACI Code 1151 as shown shear reinforcement are permitted by ACI Code 1151 as shown a Vertical Stirrupsa Vertical Stirrupsb Inclined stirrups making an angle of 45 degree or more withb Inclined stirrups making an angle of 45 degree or more withlongitudinal tension reinforcementlongitudinal tension reinforcementc Longitudinal reinforcement with bent portion making an angle c Longitudinal reinforcement with bent portion making an angle of 30 degree or more with the tension reinforcementof 30 degree or more with the tension reinforcementd Spirals circular ties or hoopsd Spirals circular ties or hoopse Combination of stirrups and bent longitudinal reinforcemente Combination of stirrups and bent longitudinal reinforcementBefore diagonal cracking occurs the stirrups remain Before diagonal cracking occurs the stirrups remain unstressed After cracking the stress in the stirrups increases unstressed After cracking the stress in the stirrups increases as they pick up a portion of the load formerly carried by the as they pick up a portion of the load formerly carried by the uncracked concreteuncracked concrete

Types of shear reinforcement (a) vertical stirrups (b) inclinedstirrups (c) bent-up bars (two groups) (d) bent-up bars (three groups)

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

where Vwhere Vuu is the factored shearing force M is the factored shearing force Muu is the factored is the factored

bending moment occurring simultaneously with Vbending moment occurring simultaneously with Vuu at section at section

considered ρconsidered ρww is the reinforcement ratio of the web and d is is the reinforcement ratio of the web and d is

the effective depth of the beam should not exceed 10 the effective depth of the beam should not exceed 10 and Vand Vcc should not exceed should not exceed u

u

M

dV

Strength Provided by Shear ReinforcementStrength Provided by Shear ReinforcementWhen the nominal shearing force VWhen the nominal shearing force Vn n exceeds the shearing force exceeds the shearing force

that can be resisted by concrete alone Vthat can be resisted by concrete alone Vcc shear reinforcement in shear reinforcement in

any of the forms shown in the following section can be usedany of the forms shown in the following section can be used

Types of Shear Reinforcement

When shear reinforcement is required the following types of When shear reinforcement is required the following types of shear reinforcement are permitted by ACI Code 1151 as shown shear reinforcement are permitted by ACI Code 1151 as shown a Vertical Stirrupsa Vertical Stirrupsb Inclined stirrups making an angle of 45 degree or more withb Inclined stirrups making an angle of 45 degree or more withlongitudinal tension reinforcementlongitudinal tension reinforcementc Longitudinal reinforcement with bent portion making an angle c Longitudinal reinforcement with bent portion making an angle of 30 degree or more with the tension reinforcementof 30 degree or more with the tension reinforcementd Spirals circular ties or hoopsd Spirals circular ties or hoopse Combination of stirrups and bent longitudinal reinforcemente Combination of stirrups and bent longitudinal reinforcementBefore diagonal cracking occurs the stirrups remain Before diagonal cracking occurs the stirrups remain unstressed After cracking the stress in the stirrups increases unstressed After cracking the stress in the stirrups increases as they pick up a portion of the load formerly carried by the as they pick up a portion of the load formerly carried by the uncracked concreteuncracked concrete

Types of shear reinforcement (a) vertical stirrups (b) inclinedstirrups (c) bent-up bars (two groups) (d) bent-up bars (three groups)

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

Strength Provided by Shear ReinforcementStrength Provided by Shear ReinforcementWhen the nominal shearing force VWhen the nominal shearing force Vn n exceeds the shearing force exceeds the shearing force

that can be resisted by concrete alone Vthat can be resisted by concrete alone Vcc shear reinforcement in shear reinforcement in

any of the forms shown in the following section can be usedany of the forms shown in the following section can be used

Types of Shear Reinforcement

When shear reinforcement is required the following types of When shear reinforcement is required the following types of shear reinforcement are permitted by ACI Code 1151 as shown shear reinforcement are permitted by ACI Code 1151 as shown a Vertical Stirrupsa Vertical Stirrupsb Inclined stirrups making an angle of 45 degree or more withb Inclined stirrups making an angle of 45 degree or more withlongitudinal tension reinforcementlongitudinal tension reinforcementc Longitudinal reinforcement with bent portion making an angle c Longitudinal reinforcement with bent portion making an angle of 30 degree or more with the tension reinforcementof 30 degree or more with the tension reinforcementd Spirals circular ties or hoopsd Spirals circular ties or hoopse Combination of stirrups and bent longitudinal reinforcemente Combination of stirrups and bent longitudinal reinforcementBefore diagonal cracking occurs the stirrups remain Before diagonal cracking occurs the stirrups remain unstressed After cracking the stress in the stirrups increases unstressed After cracking the stress in the stirrups increases as they pick up a portion of the load formerly carried by the as they pick up a portion of the load formerly carried by the uncracked concreteuncracked concrete

Types of shear reinforcement (a) vertical stirrups (b) inclinedstirrups (c) bent-up bars (two groups) (d) bent-up bars (three groups)

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

Types of shear reinforcement (a) vertical stirrups (b) inclinedstirrups (c) bent-up bars (two groups) (d) bent-up bars (three groups)

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

Minimum Amount of Shear Reinforcement

The instant diagonal crack forms the tension carried by the The instant diagonal crack forms the tension carried by the concrete must be transferred to the stirrups if the beam is not concrete must be transferred to the stirrups if the beam is not to split into two sections To ensure that the stirrups will have to split into two sections To ensure that the stirrups will have sufficient strength to absorb the diagonal tension in the sufficient strength to absorb the diagonal tension in the concrete ACI Code 1155 states that a minimum area of shear concrete ACI Code 1155 states that a minimum area of shear reinforcement is to be provided in concrete members where reinforcement is to be provided in concrete members where the factored shearing force u V exceeds half the shear strength the factored shearing force u V exceeds half the shear strength provided by concrete 050 Φ Vprovided by concrete 050 Φ Vcc except for the following except for the following

1048707 1048707 Slabs ribbed or solidSlabs ribbed or solid1048707 1048707 FootingsFootings1048707 1048707 Beams with total height not greater than 25 cm 25 times Beams with total height not greater than 25 cm 25 times thickness of flange or 050 the width of web whichever is the thickness of flange or 050 the width of web whichever is the greatestgreatest

The exceptions were made because there is a possibility of The exceptions were made because there is a possibility of load sharing between weak and strong areasload sharing between weak and strong areasThis minimum area is given by ACI Code 11553This minimum area is given by ACI Code 11553

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

where Awhere Avv is the area of shear reinforcement within a distance S b is the area of shear reinforcement within a distance S bw w

is the web width S is the spacing of shear reinforcement and fis the web width S is the spacing of shear reinforcement and fysys is is

the yield stress of the shear reinforcementthe yield stress of the shear reinforcementMaximum Stirrup Spacing

The assumption made in Eq (415) is that one or more The assumption made in Eq (415) is that one or more stirrups cross each potential diagonal crack in order to stirrups cross each potential diagonal crack in order to prevent the beam from splitting into two sections between prevent the beam from splitting into two sections between stirrups To ensure that this requirement is satisfied ACI stirrups To ensure that this requirement is satisfied ACI Code 11541 through 11543 specifies the following limits Code 11541 through 11543 specifies the following limits for maximum spacing of shear reinforcementfor maximum spacing of shear reinforcement

A Vertical Stirrups

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

B Inclines stirrups and Bent-up Bars

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

Maximum stirrup spacing

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

Some stirrup shapes (a) open (b) closed

(c) closed (d) two sets (e) two sets

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

Location of critical section for shear

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

Punching shear (a) punching shear failure of an isolated

footing (b) actual failure surface (c) assumed failure surface

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

Punching shear surface for isolated footings

(a) rectangular column (b) circular column

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

Summary of ACI ShearSummary of ACI Shear Design Procedure for BeamsDesign Procedure for Beams

Once the beam is designed for moment thus establishing the Once the beam is designed for moment thus establishing the concrete dimensions and the required longitudinal concrete dimensions and the required longitudinal reinforcement the beam is designed for shear as explained in reinforcement the beam is designed for shear as explained in the next stepsthe next steps1 Draw the shearing force diagram and establish the critical 1 Draw the shearing force diagram and establish the critical section for shearsection for shear

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

2 Calculate the nominal capacity of concrete in shear using any of 2 Calculate the nominal capacity of concrete in shear using any of Equations (47) through (410) as appliesEquations (47) through (410) as applies3 Check whether the chosen concrete dimensions are adequate 3 Check whether the chosen concrete dimensions are adequate for ensuring a ductile mode of failure by satisfying the followingfor ensuring a ductile mode of failure by satisfying the followingequationequation

where u V is the factored critical shearing force acting on the beamwhere u V is the factored critical shearing force acting on the beamIf Eq (422) is not satisfied the concrete dimensions should be If Eq (422) is not satisfied the concrete dimensions should be increasedincreased4 Classify the factored shearing forces acting on the beam 4 Classify the factored shearing forces acting on the beam according to the followingaccording to the following

• Slide 1
• Slide 8

• Slide 1
• Slide 8