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8.1
Strut-and-Tie Model
• Background• AASHTO LRFD Provisions• Design Example
8.2Background
STM is a Truss AnalogyTruss Analogy Used in Standard and LRFD SpecificationsVn = Vc + Vs Vs = [Asfy/s]d(cotθ)- AASHTO Standard
Vs 45º Truss- AASHTO LRFD
Vs Variable Angle Truss
8.3
8.4STM in Codes
CSA 23.3-84OHBDC Third Edition, 1991AASHTO LRFD - First Edition, 1994CHBDC - 2000ACI 318-02 Appendix A
8.5Quiz
A Three-Span Concrete Beam Is Built Monolithically, with Continuous Reinforcement Placed Only in the Bottom of the BeamHow Will this Beam Perform Under Service Loads? and at Ultimate?
8.6As Built
8.7Under Service Loads- Uncracked Condition -
8.8Under Service Loads- Cracked Condition -
8.9Observations
Reinforcement Becomes Active After Concrete CracksRedistribution of Internal Stresses Occurs After Concrete CracksAfter Cracking, Concrete Structures Behave the Way they Are ReinforcedFor Best Serviceability, the Reinforcement Must Follow the Flow of Elastic Tensile Stresses
8.10
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
8.11Deep Beam Stress Trajectories
8.12STM for D-Regions
Tee Beam
Dapped Beam
8.13Past Practice
D-Regions Designed Based On:» Experience» Empirical Rules» Rules of Thumb
8.14Basic Description of theStrut-and-Tie Model
A design tool for “disturbed” regions where the flow of stresses is non-uniform and the usual rules of analysis do not applyA rational approach to visualize the flow of forces at the strength limit state based on the variable-angle truss analogyA unified approach that considers all load effects simultaneouslyA highly flexible and conceptual method that recognizes that several possible solutions may exist for any problem
8.15STM Basic Principle
Concrete is Strong in CompressionCompression Struts
Steel is Strong in TensionTension Ties
8.16
P2
φ >
P2
P
8.17
NodalZonesP
2
P
P2
CC
T T
C CStrut
Fill
Fill
Tie
Fill
8.18
T
C
T
C
C C
P
P2
φ > A f Ts y
P2
φ >
Af
Cc
cu
φ > A f Ts y
φ >
Af
Cc
cu
8.19Basic Concepts
Visualize a truss-like system to transfer load to the supports where:• 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
8.20Strut-and-Tie Model for Simple Span Beam
8.21Examples of Strut-and-Tie Models
8.22Methods for Formulating Strut-and-Tie Models
Stress trajectories from elastic analysis
Load path approach
Experimentally
Standard models
8.23Deep Beam Stress Trajectories
8.24Examples of Strut-and-Tie Models
8.25Examples of Strut-and-Tie Models
8.26
Procedures for Load Path Approach
Find reactionsSubdivide loads and internal forces- Replace stresses with resultants- Replace asymmetrical stresses with
couple and resultantProvide struts and ties to provide load pathLocate ties using practical dimensions
8.27STM from Tests - Dapped Beam
8.28Dapped Beam
8.29Types of Nodes(Schlaich et al. 1987)
C - CompressionT - Tension
TTT
CTT
CCT
CCC
8.30AssumptionsTies yield before struts crush (for ductility)Reinforcement adequately anchoredForces in struts and ties are uniaxialTension in concrete is neglectedExternal forces applied at nodesPrestressing is a load
Equilibrium must be maintained
8.31Strut-and-Tie Model Design Procedure
8.32Examples of Good and Poor Strut-and-Tie Models
8.33Factors 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
8.34Strut-and-Tie vs. Traditional Analysis/Design
Traditional section analysis/designLinear strain over member depthUniform shear stress distributionNot valid for D-regions
Strut-and-tieRegions with nonlinear strain distribution» Deep beams, pile caps» Brackets, beam ledges, P/T anchors» Shear span/member height < 2
8.35
a/d
V/bdfc’
Source: Prestressed Concrete Structures by Collins & Mitchell
8.36LRFD 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
8.375.6.3.1 D-RegionsStrut-and-tie models may be used to
determine internal force effects near supportsand the points of application of concentrated loads at strength and extreme event limit states.
The strut-and-tie model should be considered for the design of deep footings and pile caps or other situations in which the distance between the centers of applied load and the supporting reactions is less than about twice the member thickness.
8.385.8.1.1 D-RegionsComponents in which the distance from
the point of zero shear to the face of the support is less than 2d, or components for which a load causing more than ½ of the shear at a support is closer than 2d from the face of the support, may be considered to be deep components for which the provisions of Article 5.6.3 and the detailing requirements of Article 5.13.2.3 apply.
8.39Strength Limit State for STM
Pr = ϕ Pn (5.6.3.2-1)
where:
Pr = Factored resistance
Pn = Nominal resistance of strut or tie
ϕ = Resistance factor for tension or compression (5.5.4.2)
8.40
LRFD 5.6.3.3Unreinforced strut:
Pn = fcu Acs (5.6.3.3.1-1)
Reinforced strut:
Pn = fcu Acs + fy Ass (5.6.3.3.4-1)
where:ϕ = 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
Strength of Struts
8.41STM for Deep BeamLRFD Fig. C5.6.3.2-1
8.42
LRFD 5.6.3.3.2
Determined by considering available concrete area and anchorage conditions.
When anchored by reinforcement, strut may extend from the anchored bar.
C-T-T Nodea) Strut Anchored by Reinforcement
Effective Cross-Sectional Area of Strut, Acs
8.43Effective Cross-Sectional Area of Strut, AcsLRFD 5.6.3.3.2
C-C-T Nodeb) Strut Anchored by Bearing and Reinforcement
8.44Effective Cross-Sectional Area of Strut, AcsLRFD 5.6.3.3.2
C-C-C Nodec) Strut Anchored by Bearing and Strut
8.45Limiting Compressive Stress in Strut
LRFD 5.6.3.3.3
where:
( )
(IN/IN) tie tension the of direction the in
concrete the in strain tensile the
(DEG) ties tension
adjoining and strut ecompressiv
the between angle smallest the
stress ecompressiv limiting thef
cot0.002
f85.170 0.8
ff
s
s
cu
s2
ss1
c1
ccu
=
==
++=
′≤+′
=
ε
α
αεεε
ε0
8.46Strength of TieLRFD 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 steelfy = Yield strength of mild steel longitudinal
reinforcementfpe = Stress in prestressing steel due to prestress after
losses
8.47Development of Ties
If x < ld fs = fy (x/ld)
Critical Section
= x
8.48Development of Ties (ACI 318)
8.49
Element Limiting Stress ϕ
1 - CCC Node 0.85 fc’ 0.70
2 - CCT Node 0.75fc’ 0.70
3 - CTT or TTT Node 0.65fc’ 0.70
4 - Strut fcu 0.70
5 - Tie fy or (fpe + fy) 0.90 or 1.00
Limiting Stresses for STM ElementsLRFD 5.6.3.3 - 5.6.3.5
8.50Crack Control Reinforcement
LRFD 5.6.3.6
Provide orthogonal grid of reinforcement near each face of D-RegionMaximum Bar Spacing = 12 in.Ratio As / Ag ≥ 0.003 in each of the orthogonal directionsCrack control reinforcement, located within tie, considered as part of tie
8.51Summary
1. Visualize flow of stresses2. Sketch an idealized strut-and-tie model3. Select area of ties4. Check nodal zone stresses5. Check strength of struts6. Provide adequate anchorage for ties
8.52
8.53Strut-and-Tie Model
8.54Strut-and-Tie Model
8.55Design Examples
1. Two Column Bent Cap2. Spread Footing3. Pile Cap4. Dapped-End Beam5. Hammerhead Pier
