Design of Deep Beams and Joints

Concrete Deep Beams, Brackets and Joints O-SCAAD-6 July 12, 2002, AIT, Bangkok

Naveed Anwar Buddhi S. Sharma ACECOMS, AIT

Definition of Deep Members

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

Design of Deep Beams, Brackets and Joints

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The Axial-Flexural Stress Resultants The General Case: Linear or Non-linear Strain Distribution

1 1 N z  1     x, y  dx dy ...  2   1 x y

 Ai i ( x, y ) ...  i 1  n

1  1 n M x   2     x, y  dx dy . y ...   Ai i ( x, y ) yi ...  2 i 1   1 x y  1  1 n M y  3     x, y  dx dy . x ...   Ai i ( x, y ) xi ...  2 i 1   1 x y  Design of Deep Beams, Brackets and Joints

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The Axial-Flexural Stress Resultants Linear Strain Distribution

fs

NA

CL

fc y c

h

f1 f2 fn

ain r t S


for s se and s e Str crete n R/F co

for s se l s e Str Stee Horizontal

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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 “DRegion” • 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 Design of Deep Beams, Brackets and Joints

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Basic Behavior of Deep Members

The Axial Stresses – True Deep Beams

Tension Compression


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The Axial Stresses – Semi Deep Beams

Tension Compression


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The Axial Stresses – Mixed Beam

Tension Compression

D

B


D ACECOMS, AIT

Shear Stresses


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Beam Model for Deep Members

Modeling Using 1D Elements Simple Beam/Column elements Beam elements with rigid ends

Beam elements in “Truss Model”


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Membrane Model

for Deep Members

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 inplane bending or curvature • General Shell Element or Membrane Elements can be used to model Deep Beams Design of Deep Beams, Brackets and Joints

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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), U3, R3

Thickness( t )

U3, R3 U2, R2

Node 3

U2, R2 Node 4

U1, R1 3

2

U1, R1 U3, R3

1

U3, R3

U2, R2

Node 1

U2, R2 Node 2

U1, R1

U1, R1

Shell Design of Deep Beams, Brackets and Joints

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Getting Results From Shell Model Fi  Ai f i

A

n

P   Fi

f1 C

i 1 n

f2

x1

M   Fi xi i 1 n

f3

V   Ai vi

f4

i 1

T

x1

t

f5

A

f1, f2, …..fn are the nodal stresses at section A-A , obtained from analysis Design of Deep Beams, Brackets and Joints

P

M V

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Connecting Beams to Slab

“Zipper”

In general the mesh in the slab should match with mesh in the wall to establish connection Design of Deep Beams, Brackets and Joints

Some software automatically establishes connectivity by using constraints or “Zipper” elements ACECOMS, AIT

Strut and Tie Model for Deep Members

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 Compressive Struts d

L

Real Truss

Ties L

Conceptual Truss

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


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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 Truss in Deep Members

Tension Compression


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The Axial Stresses – Semi Deep Beams

Tension Compression


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The Axial Stresses – Mixed Beam

Tension Compression


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Truss Models and Forces


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Strut Tie Model

a

Effect of Span:Depth Ratio

a

d

d L/d =1 L/a =0.5 L

For L/D < 4 Load transferred by direct Compression

L/d =2 L/a =1 L

For L/D > 4 Auxiliary Ties are required for shear transfer

L/d = 3 L/a = 1.5

For L/D > 5 Beam tends to behave in ordinary Flexure

L/d = 4 L/a = 2

L/d = 5 L/a = 2.5

L/d = 6 L/a = 3


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Strut Tie Model

Not OK: Too Shallow

Effect of Strut Angle Angle < 30 Deg.

Tension in Bottom Chord

Angle = 18 De g

OK: M ost Ecconomical

Too shallow, tension steel not economical, strut too long, anchorage difficult

Angle 35 - 45 Deg

Angle = 34 De g

OK: USed by ACI Code

Gives the most economical and realistic design

Angle > 50 Deg.

Angle = 45 De g

NOT OK: Too Steep and Expensive

Too steep. Requires too much stirrups. Not good.

Angle = 64 De g


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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 Design of Deep Beams, Brackets and Joints

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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

H

C

t

t x 2t


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Simple Vs Modified Truss Model a=1.6

a=1.6 P=10,000 kN

d=1.4

d=1.4 

h=1.6

T

d=1.4

h=1.6

 

L=2.5

T

L=2.5 1

a) Simple "Strut & Tie" Model

  T T

= tan-1 d/0.5L = 48 deg = 0.5P/tan = 4502 kN


c) Modified Truss Model B   T T

= tan-1 d/0.5(L-d1) = 68.5 deg = 0.5P/tan = 1970 kN

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A Space Truss Model for Pilecap P1

a2

a2

P4

P2

P3 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 Compression Member

C xc

P  T  C  D sin( ) M  Txt  Cxc  D sin( ) xd

xd

D

V  D cos( )

xt Tension Member

T

Ast 

T f y P

M V


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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 Design of Deep Beams, Brackets and Joints

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Deep Beams Special Considerations

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:

l n /d  1.25

– Continuous Beam:

l n /d  2.5

• Beam is Deep for Shear:

l n /d  5.0 • Special Case


Deep Beam

P 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 Design of Deep Beams, Brackets and Joints

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Shear Design: ACI Approach • Different for Simple and Continuous Beams • Stirrups Required when – For Single spans – For Continuous spans.

Vu  Vc Vu  0.5Vc

• Critical Sections – Simple Span

0.15 l n  d for UDL 0.15 a  d for Conc. Load

– Continuous Beam: Face of Support Design of Deep Beams, Brackets and Joints

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Shear Design: ACI Approach Allowable shear in concrete Shallow Beams

Deep Beams

Max. Vn  8 f c' bw d

Max. Vn  8 f c' bw d when l / d  2

Vc  2 f c' bw d

Max. Vn 

 Vu d  ' Vc  1.9 f c  2500  w bw d Mu  

Vc  2 f c' bw d

ld  ' 2 10  f c bw d when l / d is 2 to 5 3  d 

 V d Vc  F  1.9 f c'  2500  w u bw d Mu    V d where F  3.5  2.5 u   2.5 Mu  


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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

Negative Moment Truss

Positive Moment Truss

Complete Model Design of Deep Beams, Brackets and Joints

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Brackets and Corbels Special Considerations

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 ACECOMS, AIT

Basic Stresses in Corbels

Tension

Compression


Shear ACECOMS, AIT

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 Design of Deep Beams, Brackets and Joints

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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 [Vua  Nuc(h - d)] and horizontal tensile force of Nuc Strength reduction Factor

  0.85 Design of Deep Beams, Brackets and Joints

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Design of Corbels ACI Method Provide Steel Area Avf to resist Vu

 Vn  0.2 f c b w d Vn  800bw d

Horizontal Axial Tension Force should satisfy Area of Steel provided shall be the greater of the two Strut and tie are should not be less than Ratio shall be Design of Deep Beams, Brackets and Joints

N uc  An f y N uc  0.2Vu

A

f

 An 

2 Av / 3  An  0.5 As  An  f    As / bd  0.04 c f   y    ACECOMS, AIT

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


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Joints Special Provisions

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 Design of Deep Beams, Brackets and Joints

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Basic Stresses in Joints – Gravity

Tension

Compression


Shear

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Basic Stresses in Joints – Lateral

Tension

Compression


Shear

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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 Design of Deep Beams, Brackets and 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


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Summary


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Design of Deep Beams and Joints

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