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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
QUIZ
A Three-Span Concrete Beam Is Built Monolithically, with Continuous Reinforcement Placed Only in the Bottom of the Beam How Will this Beam Perform Under Service Loads? and at Ultimate?
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UNDER SERVICE LOADS - Uncracked Condition -
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 by Reinforcement (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 concrete members, especially for regions where the plane sections assumption of beam theory does not apply
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A.1 - DEFINITIONS
D-Region - The portion of a member within a distance equal to the member height h or depth d from a force discontinuity or a geometric discontinuity. St. Venants Principle
In the past D-Regions were Designed Based On: Experience or Empirical Rules of Thumb
A.1 - DEFINITIONS
Discontinuity - An abrupt change in Geometry or Loading Daps, Openings Concentrated Loads/Reactions
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A.1 - DEFINITIONS
B-Region - A portion of a member in which the plane sections remain plane assumption of flexure theory from 10.2.2 can be applied.
Bending Region Bernoulli Region
STRUT-AND-TIE MODELS
A Tool for Design/Detailing of D-Regions where Flow of Stresses is Non-uniform Help Visualize Flow of Forces Based on Variable 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 model where the axes of the struts, ties, and concentrated forces acting on the joint intersect.
Nodal Zone - The volume of concrete around a node that is assumed to transfer strut-and-tie forces through the node.
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>
P 2 P 2
P Strut Fill Fill C T P 2 Nodal Zones Tie T P 2
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C Fill C
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P C
A
c
fc
C > u
T P 2
As fy > T
T P 2
A.1 - DEFINITIONS
Strut - A compression member in a strut-and-tie model. A strut represents the resultant of a parallel or a fanshaped compression field.
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BOTTLE-SHAPED STRUT
Crack Tie 2 1 1 Width Used to Compute Ac
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Strut
SPLIT CYLINDER TEST
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A.2.1-2 - STM DESIGN PROCEDURE
Model Member or Regions as an Idealized Truss (Struts, Ties, Nodes) STM Applies to the Entire Member but only Commonly Used at D-Regions STM Transfers Factored Loads to Supports or 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 Strut and Tie at Node = 25
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A.2.6 - STM DESIGN PROCEDURE
Fn Fu
(A-1)
where Fu = Force in Strut/Tie/Node Due to Factored Loads Fn = Nominal Strength of Strut/Tie/Node
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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 Concrete Strength in Strut or Nodal Zone
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A.3.2 STRENGTH OF STRUTS
fcu = 0.85 s fc'
Prismatic Strut Bottle-Shaped Strut - With Reinf. Per A.3.3 - W/o Reinf. Per A.3.3 Strut in Tension Zone of a Member All Others (A-3) s = 1.0
s = 0.75 s = 0.60 s = 0.40 s = 0.60
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A.3.3 REINF. CROSSING STRUTS
Strut Boundary Axis of Strut Strut
1 As1 2
As2 s1
s2
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A.3.3 REINF. CROSSING STRUTS
If fc' 6000 psi Asi bs sin i 0.003 i
(A-4)
- Asi in Orthogonal Directions - Asi in One Direction if > 40
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A.3.5 STRENGTH OF STRUTS
Strut With Longitudinal Reinf. Parallel to Strut Axis, and Enclosed in Ties or Spirals per 7.10
Fns = fcu Ac + A's fs'
(A-5)
For Grades 40 to 60 Use fs = fy
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A.4.1 STRENGTH OF TIES
Fnt = Ast fy + Aps ( fse + fp )
(A-6)
where - ( fse + fp ) fpy - Bonded P/S - Unbonded P/S
fp = 60 ksi fp = 10 ksi
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A.4.2-3 STRENGTH OF TIES
Axis of Reinforcement to Coincide with Axis of Tie Proper Anchorage of Tie Reinforcement at Nodes
Mechanical Device P/T Anchorage Device Standard Hooks Straight Bar Development
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A.4 DEVELOPMENT OF TIES
ws= wtcos + bsin wtcos C Nodal Zone wt
b a bsin
Extended Nodal Zone
T Critical Section
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A.5 STRENGTH OF NODAL ZONES
Fnn = fcu An
(A-7)
where fcu = Effective Concrete Compressive Strength in Nodal Zone per A.5.2 An = Area of:
- Nodal Zone Face Perpendicular to Fu
- Section through Nodal Zone Perpendicular to Resultant Force on Section
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A.5.2 STRENGTH OF NODAL ZONES
fcu = 0.85 n fc'
(A-8)
C-C-C Node C-C-T Node C-T-T Node
n = 1.00 n = 0.80 n = 0.60
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A.1 - DEFINITIONS
Deep Beams See 10.7.1 and 11.8.1
Clear Span (ln ) / Depth 4 Shear 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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