deep beam flexure and shear design

A presentation on

Flexural & Shear Design ofFlexural & Shear Design of Deep Beam

Presented by

P N R (P t )Pragya N Roy (Peter)Structural Engineer

Contents

• What is Deep BeamWhat is Deep Beam

• Behavior of Deep BeamBehavior of Deep Beam

• Flexural Design of Deep BeamFlexural Design of Deep Beam

• Shear Design of Deep Beamg p

• Examplesp

What is Deep BeamWhat is Deep Beam• Large depth/thickness ratio • And shear span depth ratio

less than 2.5 for concentrated load

and less than 5.0 for distributed

load. • Where the shear span is the

clear span of the beam for distributed loaddistributed load

ApplicationApplication

• Pile CapPile Cap.• Bridge Girder.

W ll l b d ti l l d• Wall slabs under vertical loads.• Floors slabs under horizontal loads.• Some shear walls.

Behavior of Deep beamBehavior of Deep beam • Two-Dimensional Action.• Plane Section Do Not Remain Plane, in the deep

beam design. • The strain distribution is no longer linear.• The shear deformation cannot be neglected as in

the ordinary beam.• The stress distribution is not linear even in the

elastic stage. • At the ultimate limit state the shape of concrete

i bl k i b li hcompressive stress block is not parabolic shape again.

Behavior of Deep beamBehavior of Deep beam • The distribution of tensile stress at bottom

fiber is constant over the span.• Tensile stress in the bottom fiber at supportTensile stress in the bottom fiber at support

and at mid span is almost the same.• The tension reinforcement must be extended• The tension reinforcement must be extended

to the end of support.Th i t il t t th b tt• The maximum tensile stress at the bottom fiber is far exceed the magnitude of

i tcompressive stress.

Design Criteria for flexure in deep Beams

Simply Supported beams:• The ACI code does not specify a simplified

design procedureg p• ACI procedure for flexural analysis and

design of deep beams follows rigorous nondesign of deep beams follows rigorous non linear approach

• The simplified provisions presented in this• The simplified provisions presented in this section are based on the recommendations of the Euro International Concrete Committeethe Euro-International Concrete Committee (CEB).


• A schematic stressA schematic stress distribution in a homogeneous deep beamhomogeneous deep beam having a span/depth ratio L /h = 1 0Ln/h 1.0

• It was experimentally observed that the momentobserved that the moment lever arm does not change significantly even after initialsignificantly even after initial cracking


Nominal Resisting Moment,Mn = Asfy (moment arm jd)

The reinforcement area ‘As’ for flexure isAs = Mu/φfyjd ≥ 3√fc bd/fy ≥ 200bd/fyAs Mu/φfyjd ≥ 3√fc bd/fy ≥ 200bd/fy

The lever arm as recommended by CEBj 0 2( 2 ) f 1 l/h 2jd= 0.2(l+2h) for 1 ≤ l/h <2jd=0.6l for l/h <1l= Effective span measured from c/c.

The tension reinforcement has to be placed in thelower segment of beam height such that theg gsegment height is Y=0.25h-0.05l < 0.20h

SHEAR DESIGN OF DEEP BEAM

• The shear design of deep beam is similar as shear design of ordinary beam.g y

• The difference is the concrete shear strength.• Limitation of ultimate shear force.• Horizontal and vertical stirrups distribution.

BASIC DESIGN EQUATIONBASIC DESIGN EQUATIONφVn ≥ Vu

• Vn = nominal shear strength• Vn = nominal shear strength• φVn = design shear strength• φ = strength reduction factor (0.85)φ g ( )• Vu = ultimate shear force, factored shear force

SHEAR DESIGN OF DEEP BEAM

• As shear design of ordinary beam, the shear force is resisted by the concrete component and by theshear reinforcement component, as follows :

Vn = Vc + Vswhere :• Vn = Nominal shear strength• Vc = Concrete shear strength without shear• Vc = Concrete shear strength without shear

reinforcement• Vs = shear reinforcement (stirrup) shear strength( p) g

CONCRETE SHEAR STRENGTHCONCRETE SHEAR STRENGTH

The concrete Shear strength of deep beam is taken asThe concrete Shear strength of deep beam is taken as

where :Vc = concrete shear strength (N)M lti t fl t (N )Mu = ultimate flexure moment (Nmm)Vu = ultimate shear force (N)f’c = concrete cylinder strength (MPa)d = effective depthpbw = width of beam webρw = longitudinal reinforcement ratio

CONCRETE SHEAR

Or ,the concrete shear strength can be determined

STRENGTHOr ,the concrete shear strength can be determined

as :

The maximum limit of concrete shear strength is :s

CONCRETE SHEAR STRENGTH

The section must be enlarged if the ultimate shear force does not follow the condition below :

Or,

STIRRUP SHEAR STRENGTHSTIRRUP SHEAR STRENGTH

The strength of horizontal and vertical shear e st e gt o o o ta a d e t ca s eareinforcements is :

LIMITS OF SHEAR REINFORCEMENT

The minimum shear reinforcement area is :The minimum shear reinforcement area is :

Av−min = 0.0015 (bsv)

Avh−min = 0.0025 (bsh)where :where :

Av-min = minimum vertical stirrupsA h i = minimum horizontal stirrupsAvh-min = minimum horizontal stirrups

b = width of beami f ti l tisv = spacing of vertical stirrups

sh = spacing of horizontal stirrups

MAXIMUM SPACING OF SHEAR REINFORCEMENTREINFORCEMENT

The maximum spacing of shear reinforcement is :p g

CRITICAL SECTION IN DEEP BEAM

The critical section to determines the ultimate shear force in the deep beam is :

STEP – BY – STEP PROCEDURE

The followings are the step – by – step procedure used in the shear design for deep beam, as follows :

Determine the critical section to calculate the ultimate shear force Vu.

STEP – BY – STEP PROCEDURESTEP BY STEP PROCEDURE

Check the ultimate shear force, enlarge the section if the condition is not achieved.


Calculate the concrete shear strength Vc:


If Vu < 0.5φVc then no shear reinforcements needed, but for ti l id i i h i f t

STEP BY STEP PROCEDURE

practical reason provide minimum shear reinforcement:

If Vu > φVc then provide the shear reinforcements.


Calculate the ultimate shear force carried by the stirrups Vs


Calculate the ultimate shear force carried by the stirrups Vs.

Ch th ti l d h i t l ti til th diti Choose the vertical and horizontal stirrups until the condition achieved.


Check the spacing of shear reinforcement sv and sh


Check the spacing of shear reinforcement sv and sh.

If necessary check the chosen shear reinforcements for the basic design equation for shear design.


The design procedure above is repeats until the basic design equation for shear design is achievedequation for shear design is achieved

Example: Design ProblemExample: Design Problem

Flexural Design of aFlexural Design of a Simply Supported Beam


• MATERIAL • DIMENSIONMATERIAL• Concrete strength = K

– 300

DIMENSION• b = 500 mm• h = 2750 mm

• Steel grade = Grade 400

h = 2750 mm• Concrete cover = 50

mm• Concrete cylinder

strength = f'c = 0.83×

mm• d = 2700 mm

30 = 24.9 MPa• β1 = 0.85


• Design ForceDesign ForceMu=1.4((1/8)*qL^2

1 4((1/8)*6000*5^2= 1.4((1/8)*6000*5^2=26250 kgm

Deep Beam CheckingL /d = 4700/2700=1 74Ln/d 4700/2700 1.74Where, 1.0 ≤ 1.74 ≤ 5.0 Deep beam action


• Lever Armjd = 0.2(L + 2h)= 0.2(5000 + 2(2750))

=2100mmPositive ReinforcementMu = 262500000Nmm


• SHEAR DESIGN OF SIMPLY SUPPORTED DEEP BEAM


• MATERIAL • DIMENSIONMATERIAL• Concrete strength = K

– 300

DIMENSION• b = 500 mm• h = 2750 mm

• Steel grade = Grade 240

h = 2750 mm• Concrete cover = 50

mm• Concrete cylinder

strength = f'c = 0.83×

mm• d = 2700 mm

30 = 24.9 MPa• β1 = 0.85


• Design ForceDesign ForceX =1.5Ln=0.15x4700 =705mmV =1 4(10770)= 15078 kgVu =1.4(10770)= 15078 kgLimitation Checking

The Section is not enlarged.


• Concrete Shear StrengthConcrete Shear Strength


• Design of StirrupsDesign of Stirrups

V 150780 0 5*Φ*V 872161Vu= 150780 < 0.5*Φ*Vc =872161Provide minimum web reinforcementFor horizontal and vertical stirrups we

choose 2 legs Φ10.gAv=2((1/4)*π Φ2 =2((1/4)* π*102 =157mm2


• Reinforcement arrangementReinforcement arrangement

ReferencesReferences

• Reinforced Concrete by E G NawyReinforced Concrete by E. G. Nawy• http://www.asdipsoft.com/Deep-bm.htm

THANK YOUTHANK YOU

deep beam flexure and shear design

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