OUTLINE • • • • • • •
EGCE 501
REINFORCED CONCRETE STRUCTURES STRUT AND TIE METHOD STRUT-AND-TIE
© 2008 Praveen Chompreda, Mahidol University
• Hardened Concrete History of Strut Strut-and-Tie and Tie Method – Compressive Strength B- and D- Regions – Tensile Strength A li ti Applications – Modulus M d l off Elasticity El i i Construction of Strut-and-Tie Model– Long-Term Deformation Strut Tie Node
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STRUT–AND–TIE METHOD
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STRUT–AND–TIE METHOD
• Strut-an-Tie method came from the truss analogy method introduced in the early 1900s for shear design. • This method uses truss model d l to idealize id li the h flflow of forces in a cracked concrete beam.
• The truss analogy gy method has been validated and improved p considerablyy in 1990’s for full member or sectional design procedures. • In the STM, the complex flow of internal forces is idealized as a truss (called strut-and-tie model) carrying the imposed loading to the supports. • Like a real truss, a strut-and-tie model consists of compression members (called strut) and tension members (called ties) interconnected at joints (called nodes or nodal zones). • Strut Strut-and-Tie and Tie method is suitable for portions of the structures where beam theory is not applicable (called D-Regions).
Truss Analogies S Source: W Wang et. al.l (2007)
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B- AND D- REGIONS
B- AND D- REGIONS
• D-Region or Discontinuous Regions is where flow of forces is complex and the flexural theory is not valid (i.e. plane section does not remain plane) . Generally D-region extends one member depth from the point of Generally, discontinuity • B-Region g or “Beam” Regions g is where flexural theoryy applies pp – ggenerallyy it is the areas outside the D-Regions
B- and D-Regions for various parts of structure 5
B- AND D- REGIONS
Source: www.cee.uiuc.edu/kuchma/
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STRUT–AND–TIE METHOD • Common design g applications pp of strut-and-tie method are: – Deep Beams – Anchorage Zone of prestressed posttensioned members – Corbels and brackets – Dapped end of precast beams – Etc…
B- and D-Regions for various parts of structure
Examples of Strut-and-Tie Models
Source: www.cee.uiuc.edu/kuchma/ 7
Source: www.cee.uiuc.edu/kuchma/
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EXAMPLES
EXAMPLES
Strut-and-Tie Models for Anchorage Zone of Post-Tensioned Beams
Strut-and-Tie Models for Deep Beam
Source: Nawy (2000)
Source: AASHTO (2005) 9
CONSTRUCTION
OF STRUT–AND–TIE MODEL
CONSTRUCTION
• Components of Strut-and-Tie Model
• Forces in struts or ties must be in equilibrium with external forces and reactions • Failure is assumed to occur by Crushing C hi off strut t t in i compression i Yielding of tension tie Anchorage failure Bearing failure of nodal zone
OF STRUT–AND–TIE MODEL
• Based on a chosen model,, we need to keep p the stress in strut,, tie,, and nodal zone within the strength limits. • The STM is based on the lower-bound theorem of plasticity theory. Meaning that the failure load calculated for a given model is always less than or equal to the actual failure load – i.e. we are always be on a conservative side. side • Any statically possible system is a valid strut-and-tie model. But we must design g reinforcements accordingg to the positions p of struts and ties in the chosen model. • The simplest model that uses the least amount of tie is the best model
– Strut (Compression Member) – Tie Ti (T (Tension i M Member) b ) – Nodal Zone (Joint between strut and tie))
– – – –
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Source: ACI (2005)
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CONSTRUCTION
OF STRUT–AND–TIE MODEL
STRUT • Strut is the compression p member,, representing p g compression p stress field in the actual member. • Three basic types of strut.
• Some general rules for construction of the model – The angle between a strut and a tie must be > 25° – Strut must no cross or overlap each other (except at nodal zones) – Tie may cross a strut but the strength of the strut will be reduced compared to th strength the t th off the th strut t t without ith t a tie ti crossing i it
Prismatric Strut (Parallel stress field) 13
STRUT
Bottle-Shaped Strut (Stress field expanded in the middle)
Fan-Shaped Strut (Stress field fan out) Source: Wang et. al. (2007)
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STRUT
• Bottle Shaped p strut is used where there is enough space for strut to expand
Modeling of fan-shaped strut as series of prismatic struts Source: ACI (2005)
Source: Wang et. al. (2007) 15
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STRUT
STRUT
• Strength g of strut depends p on the type (shape) of the strut and whether the strut has any reinforcement i f crossing i iit
Cases where ßs=0.6
Fns = fce Acs = 0.85 βs f 'c Acs
Cross-Sectional Area of Strut
Strut Type
Compressive St Strength th off Concrete
ßs
Uniform Width Strut
1.0
Bottle-Shaped Strut with Reinforcement
0.75
Bottle-Shaped Bottle Shaped Strut without Reinforcement
0.60
Strut in Tension Members or Tension Flanges
0.40
All Other Cases
0 60 0.60 Source: ACI (2005)
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STRUT
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TIES • Tie is the tension member in the STM model • Tie consists of reinforcement and some portion of concrete surrounding it • The centroid axis of the reinforcement coincides with the axis of the tie
• Minimum reinforcement crossingg a strut
Asi
∑b s
sin α i ≥ 0.003
s i
– We can see that we can satisfy the reinforcement requirement by provide either reinforcement in horizontal direction, vertical di direction, i or both b h – If provide in one direction, the g between the reinforcement angle and the strut must be > 40°
• The strength of the tie is assumed to be the yield strength of the reinforcement (nonprestressed reinforcement)
Fnt = Ast fy
Source: ACI (2005)
Area of Reinforcement 19
S off Yield Strength Reinforcement 20
NODES
NODES
• Node is the intersection between struts and ties • There must be at least 3 noncoincide forces coming to a node to satisfy equilibrium • There are four types of nodes, nodes depending on the type of forces comingg to the node: – – – –
• In the strut-and-tie model,, node actually have a finite dimension, depending on the h sizes i off strut or tie, i and sizes of support or bearingg plate. p • Preliminary dimensions may be assumed and may need to be revised if the strength of the node is exceeded at one of the faces
CCC CCT CTT TTT
Source: ACI (2005)
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STRENGTH OF NODES • The strength g of the node depends p on the elements framing into the node • The node can resist compression better than tension
Source: ACI (2005)
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NODES • For nodal zones anchoringg a tie, we need to check whether the tie has enough d l development llength h within i hi the nodal zone • Use the length within the extended nodal zone for checking the development length • If the development length is not enough enough, bars must be bent, hooked, or welded to an anchored plate p
Fnn = fce Anz = 0.85 βn f 'c Anz
Node Type
ßn
CCC
1.0
CCT
0.8
CTT
0.6
TTT
-
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NODES
EGCE 501
REINFORCED CONCRETE STRUCTURES DEEP BEAMS
Source: ACI (2005)
© 2008 Praveen Chompreda, Mahidol University
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DEEP BEAMS
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DEEP BEAMS
• Deep p beam is defined as member having: g – Clear span to depth ratio < 4 – Regions with concentrated load less than 2d away from the face of support
• In deep beams, strain compatibility theory (used in slender beams) is no longer valid. Strain distribution is no longer linear. • Before B f 2002 deep 2002, d beams b are d designed i d using i empirical i i l formula f l • After 2002, ACI uses Strut-And-Tie Method (in Appendix A)
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Source: Macgregor and Wight (2006)
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DEEP BEAMS
DEEP BEAMS
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STRUT–AND–TIE MODELS
Source: ACI (2005)
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MAXIMUM SHEAR • The maximum shear capacity of deep beams
Vn,max = 0.83 0 83 f 'c bw d Web Width
ff Depth Effective
• If exceeded, need to enlarge the section
Strut-and-Tie Models for Deep Beam Source: AASHTO (2005) 31
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MINIMUM REINFORCEMENT
REFERENCES • • • • •
• • •
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AASHTO (2005). LRFD Bridge Design Specifications: Third Edition (2005 Interim Revisions), American Association of State Highway and Transportation Officials, Washington D.C. ACI Committee 318 (2005). Building Code Requirements for Structural Concrete (ACI 318-05), g Hills, MI. American Concrete Institute, Farmington MacGregor, J. G. and Wight, J. K. (2006). Reinforced Concrete: Mechanics and Design, 4th Edition in SI Units, Prentice-Hall, Singapore, 1111 pages. Nawy, E. E G. G (2000) (2000). Prestressed Concrete: A Fundamental Approach, 3rd Edition, Prentice Hall, NJ. NJ Reineck, K. H. (2002). "Modeling Structural Concrete with Strut-and-Tie Models Summarizing Discussion of the Examples as per Appendix A of ACI 318 - 2002." Examples for the Design of Structural Concrete with Strut-and-Tie Models (ACI SP-208), SP-208) American Concrete Institute, 225-242. Schlaich, J., Schäfer, K., and Jennewein, M. (1987). "Toward a Consistent Design of Structural Concrete " PCI Journal, Concrete. Journal 74-150. 74 150 Wang, C. K., Salmon, C. G., and Pincheira, J. A. (2007), Reinforced Concrete Design, 7th Edition, John Wiley and Sons, NJ. Wi h J.J K. Wight, K and d Parra-Montesinos, P M i G. G JJ. (2003). (2003) “S “Strut-and-Tie d Ti M Model d l for f Deep D Beam B Design: D i A Practical Exercise Using Appendix A of the 2002 ACI Building Code.” Concrete International, May 2003, pp. 63-70.
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