Design of Deep Beams and Joints

Naveed Anwar

Buddhi S. Sharma

ACECOMS, AIT

Concrete Deep Beams,

Brackets and Joints

O-SCAAD-6July 12, 2002, AIT, Bangkok

Definition of

Deep Members

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Design of Deep Beams, Brackets and Joints

Strain Profile – The Starting Point

• Section Capacity is represented by Stress

Resultants

• Stress Resultants are based on stress

Distribution

• Stress Distribution is based on Strain

Distribution

• Strain Distribution for a particular

deformation is not known for reinforced

concrete sections

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The Axial-Flexural Stress Resultants

...),(1

....,1

...),(1

....,1

...),(1

...,1

121

3

121

2

121

1

i

n

i

ii

x y

y

i

n

i

ii

x y

x

x y

n

i

iiz

xyxAxdydxyxM

yyxAydydxyxM

yxAdydxyxN

The General Case: Linear or Non-linear Strain Distribution

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The Axial-Flexural Stress Resultants

y

h

c

fc

Strain

Stresses fo

r

concrete and

R/F

Stresses fo

r

Steel

f1

f2

fn

fs NACL

Horizontal

Linear Strain

Distribution

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The B and D regions

• If Strain is Assumed Linear then “B” Region– Plane sections remain Plane after Deformation

– “Bernoulli” assumptions apply

• If Strain is Non-linear: “D” Region: Disturbed Region– Zone where ordinary “flexural theory” does not apply

– Plane Sections do not remain plane after deformation

D B D

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Deep or Shallow

• Shallow Members:– Where most of the beam length is “B” Region

• Deep Members: – Where most of the beam length is “D” Region

• Thick Members:– Flexural Deformations are Predominant and shear

deformations can be ignored

• Thin Members:– Shear Deformations are Significant and can not be

ignored

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What is a Deep Member ?

• Member in which most of the length is “D-

Region”

• Members that do not follow the ordinary

flexural-shear theories

• Members in which a significant amount of

the load is carried to supports by a

compression thrust joining the load and the

reaction

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Deep Members: Major Concerns

• Non linear Stress Distribution

• Possibility of Lateral Buckling

• Very Stiff Element

• Very Sensitive to Differential Settlement

• Reinforcement Development (Anchorage)

• High Stresses at Supports and Load Points

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Deep Members

• Deep Beams

• Shear Walls

• Pile Caps

• Brackets, Corbels

• Joints

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Design of Deep Members

• Empirical Methods– ACI Code Method

• The “Tie-Strut” Approach– Truss Analogy Method

– Truss Model Analysis

• Finite Element Analysis– Two Dimensional Analysis using Plane Strain

– Three Dimensional Analysis using Plates or Bricks

– Analysis modes

• Linear Analysis

• Non Linear Analysis

Basic Behavior of Deep Members

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The Axial Stresses – True Deep Beams

Tension

Compression

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The Axial Stresses – Semi Deep Beams

Tension

Compression

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Tension

Compression

The Axial Stresses – Mixed Beam

D B D

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Shear Stresses

Beam Model for Deep Members

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Modeling Using 1D Elements

Simple

Beam/Column

elements

Beam elements

with rigid ends

Beam elements

in “Truss

Model”

Membrane Model

for Deep Members

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Modeling Using 2D Elements

• Deep Beams are subjected to in-plane deformations so 2D elements that have transnational DOF need to be used

• A coarse mesh can be used to capture the overall stiffness and deformation of the beam

• A fine mesh should be used to capture in-plane bending or curvature

• General Shell Element or Membrane Elements can be used to model Deep Beams

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Modeling Using Membrane

Nodes: 4

DOFs: 2 (or 3) DOFs /Node Ux and Uy

2-Translation, 0 or 1 rotation

Dimension: 2 dimension element

Shape: Regular / Irregular

Properties: Modulus of Elasticity(E),

Poisson ratio(v),

Thickness( t )

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Modeling Using Shell Elements

Nodes: 4

DOFs: 5 or 6 DOFs /Node Ux and Uy

3 Translation, 2 or 3 rotation

Dimension: 2 dimension element

Shape: Regular / Irregular

Properties: Modulus of Elasticity(E),

Poisson ratio(v),

Thickness( t )

1

23

U1, R1

Node 3

U3, R3

U2, R2

U1, R1

Node 1

U3, R3 U2, R2

U1, R1

Node 4

U3, R3

U2, R2

U1, R1

Node 2

U3, R3

U2, R2

Shell

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Getting Results From Shell Model

f1, f2, …..fn are the nodal stresses at

section A-A , obtained from analysis

n

i

ii

i

n

i

i

n

i

i

iii

vAV

xFM

FP

fAF

1

1

1

f1

f2

f3

f4

f5

C

T

1x

x1

A

A

t

V

PM

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Connecting Beams to Slab

In general the mesh in the slab

should match with mesh in the

wall to establish connection

Some software automatically

establishes connectivity by using

constraints or “Zipper” elements

“Zipper”

Strut and Tie

Model for

Deep Members

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Tie-Strut Approach: Basic Concepts

• Basic Concept

– The Section is fully cracked

– Concrete takes not tension

– All Tension is taken by steel ties

– All Compression is taken by “struts” forming within

the concrete

– Strut and Tie provide a stable mechanism

– It is a “Lower Bound” solution

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Tie-Strut Approach: Basic Concepts

Conceptual TrussReal Truss

a) Simple Truss Model for V, Mx (Tie and Strut Mode)

L

d

LTies

Compressive

Struts

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Tie-Strut Approach in Use

• Truss analogy already in use– For shear design of “Shallow” and “Deep” beams

– For Torsion design of shallow beams

– For design of Pile caps

– For design of joints and “D” regions

– For Brackets and corbels

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The Truss in Deep Members

Tension

Compression

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The Axial Stresses – Semi Deep Beams

Tension

Compression

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Tension

Compression

The Axial Stresses – Mixed Beam

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Truss Models and Forces

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Strut Tie Model Effect of Span:Depth Ratio

For L/D < 4

Load transferred by direct

Compression

For L/D > 4

Auxiliary Ties are required

for shear transfer

For L/D > 5

Beam tends to behave in

ordinary Flexure

L/d =2

L/a =1

L

d

a

L/d =1

L/a =0.5

L/d = 3

L/a = 1.5

L/d = 4

L/a = 2

L/d = 5

L/a = 2.5

L/d = 6

L/a = 3

L

d

a

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Strut Tie Model Effect of Strut Angle

Angle < 30 Deg.Too shallow, tension steelnot economical, strut toolong, anchorage difficult

Angle 35 - 45 DegGives the most

economicaland realistic design

Angle > 50 Deg.Too steep. Requires toomuch stirrups. Not good.

Angle = 18 Deg

Angle = 34 Deg

Angle = 45 Deg

Angle = 64 Deg

Not OK: Too Shallow

NOT OK: Too Steep and Expensive

OK: USed by ACI Code

OK: Most Ecconomical

Tension in Bottom Chord

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The Basic Elements of Strut and Tie

• Basic Elements

– The Compression Struts in Concrete

– The Tension Ties provided by Rebars

– The Nodes connecting Struts and Ties

• Failure Mechanisms

– Tie could Yield

– Strut can Crush

– A Node could Fail

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Compression Struts

• Struts represent the compression stress field

with the prevailing compression in the

direction of the strut

• Idealized as prismatic members, or uniformly

tapered members

• May also be idealized as Bottled Shaped

members

• Transverse reinforcement is required for

prevention of failure after cracking occurs

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Types of Compression Struts

• Failure of Struts• By Longitudinal

Crushing

• Compression failure

of Struts

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Cracking of Compression Struts

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Tension Ties

• Represents one or several

layers of steel in the same

direction as the tensile force

• May fail due to

– Lack of End Anchorage

– Inadequate reinforcement quantity

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Nodal Zones

• The joints in the strut-and-tie model are know as nodal zones

• Forces meeting on a node must be in equilibrium

• Line of action of these forces must pass through a common point (concurrent forces)

• Nodal zones are classified as:– CCC

– CCT

– CTT

– TTT

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Hydrostatic Nodal Zones

Hydrostatic CCC Node Hydrostatic CCT Node

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Correct and Incorrect Truss

Correct Truss Incorrect Truss

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Using Truss Model

• Draw the beam and loads in proper scale

• Draw Primary Struts and Ties

– Struts angle between 35 to 50 degrees

– Each strut must be tied by “ties”

– The strut and ties model must be stable and determinate

• Assume dimensions of struts and ties

– Not critical for determinate trusses. Any reasonable sizes

may be used

• Make truss model in any software and analyze

• Design Truss Members

– Design rebars for tension members

– Check capacity of concrete compression members

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How to Construct Truss Models

• For the purpose of analysis, assume the main truss layout

based on Beam depth and length

• Initial member sizes can be estimated as t x 2t for main axial

members and t x t for diagonal members

• Use frame elements to model the truss. It is not necessary to

use truss elements

• Generally single diagonal is sufficient for modeling but double

diagonal may be used for easier interpretation of results

• The floor beams and slabs can be connected directly to truss

elements

• Elastic analysis may be used to estimate truss layout

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How to Construct Truss Models

C

t

H

t x 2t

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Simple Vs Modified Truss Model

L=2.5

a=1.6

d=1.4 h=1.6

T

P=10,000 kN

a) Simple "Strut & Tie" Model c) Modified Truss Model B

L=2.5

a=1.6

d=1.4

d=1.4 h=1.6

T

1

= tan-1 d/0.5L

= 48 deg

T = 0.5P/tan

T = 4502 kN

= tan-1 d/0.5(L-d1)

= 68.5 deg

T = 0.5P/tan

T = 1970 kN

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A Space Truss Model for Pilecap

P1

P2

P4

P3

a2

a2

d

L2

L1

Main members

Secondary members

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Iterative Method for Truss Layout

• The truss layout can be found by using a

simple 2D truss analysis

• Draw trial truss using all possible strut tie

members

• Determine forces in the truss system

• Remove the members with small or no

forces and repeat

• Continue until the truss becomes unstable

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Getting Results from Truss Model

V

PM

)cos(

)sin(

)sin(

DV

xDCxTxM

DCTP

dct

C

T

D

Tension

Member

Compression

Member

xc

xt

xd

yst

f

TA

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Assuming Reinforcement

• Assume larger bars on the corners

• Assume more bars on predominant tension

direction/ location

• Assume uniform reinforcement on beam

sides

• Total Rebars ratio should preferably be more

than 0.8% and less than 3% for economical

design

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Interpretation of the Results

• Reinforcement should be provided along all directions where

truss members are in significant tension.

• This reinforcement should be provided along the direction of the

truss member

• The distribution of the reinforcement should be such that its

centroid is approximately in line with the assumed truss element.

• The compression forces in the struts should be checked for the

compressive stresses in the concrete, assuming the same area to

be effective, as that used in the construction of the model.

• The Bearing Stress should be checked at top of piles and at base

of columns

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Drawbacks of the Strut and Tie Approach

• Only guarantees stability and strength

• Gives no indication of performance at

service levels

• In appropriate assumed trusses layout may

cause excessive cracking

• Requires experience in judgment in truss

layout, member size assumption, result

interpretation and rebar distribution

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Designing as A Simple Flexural Member

• Approach– Design the Deep Member as “Big

Beam”

– Follow the normal axial-flexural

concept and provisions

• Input Needed– Mx , V

– Member Dimensions

• Problems– Does not consider the non-linear

strain distribution

– In efficient rebar distribution

– Does not consider Shear transfer

near ends

Deep BeamsSpecial Considerations

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Deep Members

• Behavior of Deep Beams

– What are Deep Beams?

– How do they behave?

• Design of Deep Beams

– The ACI Code Method

– The Tie and Strut Approach

– The Finite Element Analysis

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Deep Beams: ACI definition

• Beam is Deep for Flexure:

– Simple Span:

– Continuous Beam:

• Beam is Deep for Shear:

• Special Case

1.25/dln

5.2/dln

0.5/dln P

Deep

Beam Shallow Beam

ln

d

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Deep Beam or Veirendel Girder

Deep Beam

Deep Beam or

Veirendel Girder

Veirendel Girder

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ACI Approach

• No Detailed Requirements Except “That Non

Linearity of Strain Distribution and Lateral

Buckling Must be Considered”.

• Flexure:

– No Special Requirements for design

– Specifies special limits on minimum steel

• Shear

– Special Provisions for single spans

– Special provisions for continuous beams

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Effect of Load Location

• Behavior of Deep Beams effected by the

application of load to the beam

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Shear Design: ACI Approach

• Ordinary Design Procedure

– When load is applied at the middle or at the bottom edge of the Beam, ordinary shear design provisions for shallow beams are used

• Special Design Procedure– When load is applied at the top, special design

provisions are used because load may form “arching” or “truss” mechanism

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• Different for Simple and Continuous Beams

• Stirrups Required when

– For Single spans

– For Continuous spans.

• Critical Sections– Simple Span

– Continuous Beam: Face of Support

cVuV

cV5.0Vu

Load Conc.for a 0.15

for UDL l 0.15 n

d

d

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Allowable shear in concrete

dbM

dVfV

dbfV

dbfVMax

w

u

uwcc

wcc

wcn

25009.1

2

8.

'

'

'

5.25.25.3

25009.1

2

52/103

2.

2/8.

'

'

'

'

u

u

w

u

uwcc

wcc

wcd

n

wcn

M

dVFwhere

dbM

dVfFV

dbfV

toisdlwhendbfd

lVMax

dlwhendbfVMax

Shallow Beams Deep Beams

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Modeling Openings in Beams

Plate-Shell Model Truss Model

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Nodal Zones within the Interaction of Members

Plastic Truss Model of a

Beam with horizontal

Web reinforcements

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Truss Model for Continuous Beam

Complete Model

Negative Moment Truss Positive Moment Truss

Brackets and Corbels

Special Considerations

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What are Brackets and Corbels

• A short and deep member

connected to a large rigid

member

• Mostly subjected to a

single concentrated load

• Load is within ‘d’ distance

from the face of support

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Brackets or Corbels

• A short member that cantilevers out of a

column or wall to support a load

• Built monolithically with the support

• Span to depth ratio less than or equal to

unity

• Consists of incline compressive strut and a

tension tie

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Basic Stresses in Brackets

Tension Compression Shear

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Basic Stresses in Corbels


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Brackets using Strut and Tie Model

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Corbels using Strut and Tie Model

• Compute distance from column to Vn

• Compute minimum depth

• Compute forces on the corbel

• Lay out the strut and tie model

• Solve for reactions

• Solve for strut and tie forces

• Compute width of struts

• Reanalyze the strut and tie forces

• Select reinforcement

• Establish the anchorage of tie

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Structural Action of a Bracket

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Modes of Failure

• Yield of tension tie

• Failure of end anchorage of the

tension tie, either under the

load point or in the column

• Failure of the compression strut

by crushing or shear

• Local failure under bearing

plate

Failure due to poor detailing

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Design of Corbels ACI Method

• Depth of the outside edge of bearing area

should not be less than 0.5d

• Design for shear Vu, moment

and horizontal tensile

force of Nuc

Strength reduction Factor

850.

d)]-Nuc(h[Vua

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Design of Corbels ACI Method

uuc

ynuc

V.N

fAN

20

Area of Steel provided shall be

the greater of the two

nv

nf

A/A

AA

32

Provide Steel Area Avf to resist Vu

dbV

dbf.V

wn

wcn

800

20

Horizontal Axial Tension Force

should satisfy

Strut and tie are should not be less

than

ns AA. 50

y

cs f

f.bd/A 040

Ratio shall be

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Strut and Tie Method and the ACI Method

• Strut-and-Tie method requires more steel in

the tension tie

• Lesser confining reinforcement

• Strut-and-Tie method considers the effect of

the corbel on the forces of the column

• Strut-and-Tie method could also be used for

span to depth ratio greater than unity

Joints

Special Provisions

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Special Considerations in Joints

• Highly complex state of stress

• Often subjected to reversal of Loading

• Difficult to identify length and depth and

height parameters

• Main cause of failure for high seismic loads,

cyclic loads, fatigue, degradation etc

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Joints

• The design of Joints require a

knowledge of the forces to be

transferred through the joint and

the ‘likely’ ways in which the

transfer can occur

• Efficiency: Ratio of the failure moment of

the joint to the moment capacity of the

members entering the joint

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Basic Stresses in Joints – Gravity


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Basic Stresses in Joints – Lateral


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Strut and Tie Model

Tension Compression Strut and Tie Model

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Corner Joints

Opening Joints:

– Tend to be opened by the applied moment

• Corners of Frames

• L-shaped retaining walls

• Wing Wall and Abutments in bridges

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Corner Joints

• Closing Joints:

– Tend to be closed by the applied moment

• Elastic Stresses are exactly opposite as

those in the opening joints

• Increasing the radius of the bend increases

the efficiency of such joints

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Corner Joints

• T-Joints

• At the exterior column-beam connection

• At the base of retaining walls

• Where roof beams are continuous over

column

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Beam-Column Joints in Frames

• To transfer loads and moments at the end of

the beams to the columns

• Exterior Joint has the same forces as a T

joint

• Interior joints under gravity loads transmits

tension and compression at the end of the

beam and column directly through the joint

• Interior joints under lateral loads requires

diagonal tensile and compressive forces

within the joints

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Design of Joints-ACI

• Type 1 Joints: Joint for structures in non

seismic areas

• Type 2 Joints: Joint where large inelastic

deformations must be tolerated

• Further division into:

– Interior

– Exterior

– Corner

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Design Stages for Type 1

• Providing confinement to the joint region by

means of beam framing into the side of the

joint, or a combination of confinement from

the column bars and ties in the joint region.

• Limiting the shear in the joint

• Limiting the bar size in the beam to a size

that can be developed in the joint

Summary

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Design of Deep Beams and Joints

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