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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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