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NEW APPENDIX A STRUT-AND-TIE MODELS

Introduced in ACI 318-02

Why the New Appendix A?

Definitions

Code Requirements - Design Implications

Design Example

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QUIZ A Three-Span Concrete Beam Is Built

Monolithically, with Continuous ReinforcementPlaced Only in the Bottom of the Beam

How Will this Beam Perform Under ServiceLoads? and at Ultimate?

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UNDER SERVICE LOADS

-Uncracked Condition -

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UNDER SERVICE LOADS-Cracked Condition -

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OBSERVATIONS

After Tensile Cracks Develop in Concrete

Reinforcement Becomes Active

Internal Stresses Redistribute

Crack Propagation is Arrested byReinforcement (Rebars Govern Behavior)

For Best Serviceability, the Reinforcement Must

Follow the Flow of Elastic Tensile Stresses

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STRUT-AND-TIE MODELS(STM)

Valuable tool for the design of concretemembers, especially for regions wheretheplane sections assumptionof beam

theory does not apply

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A.1 - DEFINITIONS

D-Region- The portion of a member within a distanceequal to the member height h or depth d from a forcediscontinuity or a geometric discontinuity.

St. Venants Principle

In the past D-Regions were Designed Based On:Experience or Empirical Rules of Thumb

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A.1 - DEFINITIONSDiscontinuity- An abrupt change in Geometryor Loading

Daps, Openings

Concentrated Loads/Reactions

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A.1 - DEFINITIONS

B-Region- A portion of a member in whichthe plane sections remain planeassumption of flexure theory from 10.2.2can be applied.

Bending Region

Bernoulli Region

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STRUT-AND-TIE MODELS

A Tool for Design/Detailing of D-Regionswhere Flow of Stresses is Non-uniform

Help Visualize Flow of Forces Based onVariable Angle Truss Analogy

Several Solutions Exist for Any Problem

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STM BASIC PRINCIPLE

Concrete is Strong in Compression

Compression Struts Steel is Strong in Tension

Tension Ties

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A.1 - DEFINITIONS

Node- The point in a joint in a strut-and-tie modelwhere the axes of the struts, ties, and concentratedforces acting on the joint intersect.

Nodal Zone- The volume of concrete around anode that is assumed to transfer strut-and-tieforces through the node.

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P2

>

P2

P

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Nodal

ZonesP

2

P

P

2

CC

T T

C CStrut

Fill

Fill

Tie

Fill

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T

C

T

C

C C

P

P2

f

P2

>

Af

C

>A f Ts y

>

Af

C

c

cu

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A.1 - DEFINITIONS

Strut- A compression member in astrut-and-tie model. A strut representsthe resultant of a parallel or a fan-shaped compression field.

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BOTTLE-SHAPED STRUT

Crack

Width Used to Compute Ac

1

2

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Strut

Tie

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SPLIT CYLINDER TEST

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A.2.1-2 - STM DESIGN PROCEDURE

Model Member or Regions as an IdealizedTruss (Struts, Ties, Nodes)

STM Applies to the Entire Member but onlyCommonly Used at D-Regions

STM Transfers Factored Loads to Supportsor Adjacent B-Regions

STM Enforces Equilibrium

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A.2.3-5 - STM DESIGN PROCEDURE

Truss Geometry Based on Size of Struts,Ties, and Nodes

Ties Can Cross/Intersect Ties

Struts Cross Only at Nodes

Minimum Angle Between Axes of Strutand Tie at Node = 25

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A.2.6 - STM DESIGN PROCEDURE

(A-1)

where

Fu= Force in Strut/Tie/Node Due toFactored Loads

Fn= Nominal Strength of Strut/Tie/Node

n uF F

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A.3.1 STRENGTH OF STRUTS

Strut Without Longitudinal Reinf.

Fns= fcuAcs (A-2)

where

Acs= Area at One End of Strut

fcu= Smaller Effective ConcreteStrength in Strut or Nodal Zone

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A.3.2 STRENGTH OF STRUTS

(A-3)

Prismatic Strut s= 1.0 Bottle-Shaped Strut

- With Reinf. Per A.3.3 s= 0.75- W/o Reinf. Per A.3.3 s= 0.60

Strut in Tension Zoneof a Member

s= 0.40

All Others s= 0.60

'

cu s c f 0.85 f

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A.3.3 REINF. CROSSING STRUTS

StrutBoundary

Axis ofStrut

Strut

s2

As2s1

1 As12

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A.3.3 REINF. CROSSING STRUTS

(A-4)

- Asi in Orthogonal Directions

- Asi in One Direction if > 40

'

cIf f 6000 psi

sii

i

Asin 0.003

bs

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A.3.5 STRENGTH OF STRUTS

Strut With Longitudinal Reinf.Parallel to Strut Axis, and Enclosedin Ties or Spirals per 7.10

(A-5)

For Grades 40 to 60 Use fs= fy

' 'ns cu c s s F f A A f +

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A.4.1 STRENGTH OF TIES

(A-6)

where

-

- Bonded P/S fp= 60 ksi- Unbonded P/S fp= 10 ksi

se p py ( f f ) f

nt st y ps se p F A f A ( f f ) + +

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A.4.2-3 STRENGTH OF TIES

Axis of Reinforcement to Coincide withAxis of Tie

Proper Anchorage of Tie Reinforcementat Nodes

Mechanical Device

P/T Anchorage Device

Standard Hooks

Straight Bar Development

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A.4 DEVELOPMENT OF TIES

NodalZone

ExtendedNodal Zone

C

a

b

wt

wtcosws= wtcos + bsin

bsin

TCritical Section

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A.5 STRENGTH OF NODAL ZONES

(A-7)where

fcu= Effective Concrete CompressiveStrength in Nodal Zone per A.5.2

An= Area of:-Nodal Zone Face Perpendicular to Fu- Section through Nodal Zone Perpendicularto Resultant Force on Section

nn cu n F f A

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A.5.2 STRENGTH OF NODAL ZONES

(A-8)

C-C-C Node n= 1.00 C-C-T Node n= 0.80 C-T-T Node n= 0.60

'

cu n c f 0.85 f

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A.1 - DEFINITIONS

Deep Beams See 10.7.1 and 11.8.1

Clear Span (ln ) / Depth 4Shear Span (av) / Depth 2

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ELASTIC ANALYSIS

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STRUT-AND-TIE MODELING

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STRUT-AND-TIE MODELING

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