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```					   STRUT-AND-TIE
MODEL
based on
AASHTO LRFD Specifications

Week 13

1
Outline

   Background
   AASHTO LRFD Provisions
   Design Example

2
Background

   STM is a Truss Analogy
   Truss Analogy Used in Standard and
LRFD Specifications
Vn = Vc + Vs    Vs = [Asfy/s]d(cot)
- AASHTO Standard
Vs  45º Truss
- AASHTO LRFD
Vs  Variable Angle Truss

3
LRFD 5.2 - Definitions

Strut-and-Tie Model - A model used
principally in regions of concentrated
forces and geometric discontinuities to
determine concrete proportions and
reinforcement quantities and patterns
based on assumed compression struts in
the concrete, tensile ties in the
reinforcement, and the geometry of nodes
at their points of intersection
4
P

P       P
2       2
5
P
Strut
C   C
Fill                      Fill
Fill
C                                    C
T                          T
Nodal
P                           Tie              P
Zones
2                                            2
6
P

C   C
C
>
fc
u

 A   c

C                                  C
T             As fy > T   T

P                                          P
2                                          2
7
Strut-and-Tie Model (STM)

   Valuable tool for the analysis and
design of concrete members,
especially for regions where the
plane sections assumption of
beam theory does not apply

8
STM for D-Regions

Flanged Section

9
STM for D-Regions

Dapped Beam with Opening

10
Past Practice

   D-Regions Designed Based On:
» Experience
» Empirical Rules
» Rules of Thumb

11
Strut-and-Tie Model

12
Basic Concepts

   Visualize a truss-like system to transfer
• Compressive forces are resisted by
concrete “struts”
• Tensile forces are resisted by steel
“ties”
• Struts and ties meet at “nodes”
   For best serviceability, the model should
follow the elastic flow of forces
13
Basic Concepts

   Reinforcement Becomes Active After
Concrete Cracks
   Redistribution of Internal Stresses
Occurs After Concrete Cracks
   After Cracking, Concrete Structures
Behave the Way they Are Reinforced
   For Best Serviceability, the
of Elastic Tensile Stresses
14
Examples of Strut-and-Tie Models

15
Assumptions

   Ties yield before struts crush (for
ductility)
   Forces in struts and ties are uniaxial
   Tension in concrete is neglected
   External forces applied at nodes
* Equilibrium must be maintained *
16
STM Design Procedure

1. Draw Idealized Truss Model and
Solve for Member Forces
2. Check Size of Bearings
3. Choose Tension Tie Reinforcement
4. Check capacity of struts
5. Check anchorage of tension tie
6. Provide crack control reinforcement

17
Examples of Good and Poor STM

18
Factors Affecting Size of Strut

Width of the strut is affected by:
• Location and distribution of reinforcement (tie)
and its anchorage
• Size and location of bearing
19
Strength Limit State for STM

Pr =  Pn           (5.6.3.2-1)
where:
Pr = Factored resistance
Pn = Nominal resistance of strut or tie
 = Resistance factor for tension or
compression (5.5.4.2)

20
Strength of Struts
LRFD 5.6.3.3
Unreinforced strut:
Pn = fcu Acs             (5.6.3.3.1-1)
Reinforced strut:
Pn = fcu Acs + fy Ass     (5.6.3.3.4-1)
where:
 = 0.70 for compression in strut-and-tie models
(LRFD 5.5.4.2.1)
Acs= effective cross-sectional area of strut
(LRFD 5.6.3.3.2)
Ass= area of reinforcement in the strut

21
Limiting Compressive Stress in Strut
LRFD 5.6.3.3.3
fc
fcu                  0.85fc
0.8  170 ε1

where:

ε1  ε s  ε s  0.002  cot 2 α s
fcu  the limiting compressive stress
α s  the smallest angle between the
tension ties (DEG)
ε s  the tensile strain in the concrete
in the direction of the tension tie (IN/IN)
22
Development of Ties (ACI 318)

23
Strength of Tie
LRFD 5.6.3.4.1
Pn = Ast fy + Aps ( fpe + fy )
where
Ast = Total area of longitudinal mild steel
reinforcement on the tie
Aps = Area of prestressing steel
fy = Yield strength of mild steel
longitudinal reinforcement
fpe = Stress in prestressing steel due to
prestress after losses

24
Limiting Stresses for STM Elements
LRFD 5.6.3.3 - 5.6.3.5

Element             Limiting Stress
1 - CCC Node              0.85  fc
2 - CCT Node              0.75  fc
3 - CTT or TTT Node       0.65  fc
4 - Strut                    fcu
5 - Tie               fy or ( fpe  fy )   25
Crack Control Reinforcement

LRFD 5.6.3.6
   Provide orthogonal grid of reinforcement
near each face of D-Region
   Maximum Bar Spacing = 12 in.
   Ratio As / Ag  0.003 in each of the
orthogonal directions
   Crack control reinforcement, located
within tie, considered as part of tie

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DESIGN EXAMPLES

See PCI BDM

27

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