• Hardened Concrete• History of Strut-and-Tie Method – Compressive Strength
– Tensile StrengthM d l f El i i
History of Strut and Tie Method• B- and D- Regions
A li ti – Modulus of Elasticity – Long-Term Deformation
• Applications• Construction of Strut-and-Tie Model• Strut• Tie• Node
2
S T R U T – A N D – T I E M E T H O DS T R U T – A N D – T I E M E T H O D
• Strut-an-Tie method came from the truss analogy method introduced in the early 1900s for shear early 1900s for shear design.
• This method uses truss d l id li h fl model to idealize the flow
of forces in a cracked concrete beam.
S W l (2007)
Truss Analogies
3
Source: Wang et. al. (2007)
S T R U T – A N D – T I E M E T H O DS T R U T – A N D – T I E M E T H O D
• The truss analogy method has been validated and improved considerably in gy p y1990’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 strut) and tension members (called ties) interconnected at joints (called nodes or nodal zones).
• Strut-and-Tie method is suitable for portions of the structures where beam Strut and Tie method is suitable for portions of the structures where beam theory is not applicable (called D-Regions).
4
B - A N D D - R E G I O N SB - A N D D - R E G I O N S
• 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, D-region extends one member depth from the point of discontinuity
• B-Region or “Beam” Regions is where flexural theory applies – generally it is g g y pp g ythe areas outside the D-Regions
5
B - A N D D - R E G I O N SB - A N D D - R E G I O N S
6Source: www.cee.uiuc.edu/kuchma/
B- and D-Regions for various parts of structure
B - A N D D - R E G I O N SB - A N D D - R E G I O N S
7
Source: www.cee.uiuc.edu/kuchma/
B- and D-Regions for various parts of structure
S T R U T – A N D – T I E M E T H O DS T R U T – A N D – T I E M E T H O D
• Common design applications g ppof strut-and-tie method are:– Deep Beams– Anchorage Zone of
prestressed posttensioned members
– Corbels and brackets– Dapped end of precast
bbeams– Etc…
8Source: www.cee.uiuc.edu/kuchma/
Examples of Strut-and-Tie Models
E X A M P L E SE X A M P L E S
9Source: AASHTO (2005)
Strut-and-Tie Models for Deep Beam
E X A M P L E SE X A M P L E S
10
Source: Nawy (2000)
Strut-and-Tie Models for Anchorage Zone of Post-Tensioned Beams
C O N S T R U C T I O N O F S T R U T – A N D – T I E M O D E LC O N S T R U C T I O N O F S T R U T – A N D – T I E M O D E L
• Components of Strut-and-Tie Model– Strut (Compression Member)
Ti (T i M b )– Tie (Tension Member)– Nodal Zone (Joint between strut
and tie))
• Forces in struts or ties must be in equilibrium with external forces and reactions
• Failure is assumed to occur byC hi f t t i i Source: ACI (2005)– Crushing of strut in compression
– Yielding of tension tie– Anchorage failure
Source: ACI (2005)
Anchorage failure– Bearing failure of nodal zone
11
C O N S T R U C T I O N O F S T R U T – A N D – T I E M O D E LC O N S T R U C T I O N O F S T R U T – A N D – T I E M O D E L
• Based on a chosen model, we need to keep the stress in strut, tie, and , p , ,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 reinforcements according to the positions of struts and ties in the g g pchosen model.
• The simplest model that uses the least amount of tie is the best model
12
C O N S T R U C T I O N O F S T R U T – A N D – T I E M O D E LC O N S T R U C T I O N O F S T R U T – A N D – T I E M O D E L
• 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 t th f th t t ith t ti i it
13
the strength of the strut without a tie crossing it
S T R U TS T R U T
• Strut is the compression member, representing compression stress field in p , p g pthe actual member.
• Three basic types of strut.
14
Prismatric Strut(Parallel stress field)
Bottle-Shaped Strut(Stress field expanded in the middle)
Fan-Shaped Strut(Stress field fan out)
Source: Wang et. al. (2007)
S T R U TS T R U T
• Bottle Shaped strut is used where there pis enough space for strut to expand
15Source: ACI (2005)
S T R U TS T R U T
16
Modeling of fan-shaped strut as series of prismatic struts
Source: Wang et. al. (2007)
S T R U TS T R U T
• Strength of strut depends on the g ptype (shape) of the strut and whether the strut has any
i f i i
= = 0.85 'ns ce cs s c csF f A β f A
reinforcement crossing it
Cross-Sectional Compressive St th f Area of Strut Strength of Concrete
Strut Type ßs
Uniform Width Strut 1.0
Bottle-Shaped Strut with Reinforcement 0.75
Bottle-Shaped Strut without Reinforcement 0.60Bottle Shaped Strut without Reinforcement 0.60
Strut in Tension Members or Tension Flanges 0.40
All Other Cases 0 60
17
All Other Cases 0.60
S T R U TS T R U T
Cases where ßs=0.6
18Source: ACI (2005)
S T R U TS T R U T
• Minimum reinforcement crossing a gstrut
≥∑ sin 0.003sii
s i
A αb s
– We can see that we can satisfy the reinforcement requirement by the reinforcement requirement by provide either reinforcement in horizontal direction, vertical di i b hdirection, or both
– If provide in one direction, the angle between the reinforcement gand the strut must be > 40°
Source: ACI (2005)
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T I E ST I E S
• Tie is the tension member in the STM model
• Tie consists of reinforcement and some reinforcement and some portion of concrete surrounding it
• The centroid axis of the reinforcement coincides with the axis of the tie
• The strength of the tie is assumed to be the yield strength of the reinforcement
=F A f
(nonprestressed reinforcement)
=nt st yF A f
S f
20
Area of Reinforcement
Yield Strength of Reinforcement
N O D E SN O D E S
• Node is the intersection between struts and ties
• There must be at least 3 non-coincide forces coming to a node to satisfy equilibrium
• There are four types of nodes • There are four types of nodes, depending on the type of forces coming to the node:g– CCC– CCT– CTT– TTT
21Source: ACI (2005)
N O D E SN O D E S
• In the strut-and-tie model, ,node actually have a finite dimension, depending on h i f i the sizes of strut or tie,
and sizes of support or bearing plate.g p
• Preliminary dimensions may be assumed and may need to be revised if the strength of the node is exceeded at one of the exceeded at one of the faces
22Source: ACI (2005)
S T R E N G T H O F N O D E SS T R E N G T H O F N O D E S
• The strength of the node depends g pon the elements framing into the node
= = 0.85 'nn ce nz n c nzF f A β f A
• The node can resist compression better than tension Node Type ßn
CCC 1.0
CCT 0.8
CTT 0.6
TTT -
23
N O D E SN O D E S
• For nodal zones anchoring a gtie, we need to check whether the tie has enough d l l h i hi development length within the nodal zone
• Use the length within the Use the length within the extended nodal zone for checking the development length
• If the development length is not enough bars must be not enough, bars must be bent, hooked, or welded to an anchored platep
24
N O D E SN O D E S
25Source: ACI (2005)
E G C E 5 0 1
R E I N F O R C E D C O N C R E T E S T R U C T U R E SDEEP BEAMSDEEP BEAMS
• Deep beam is defined as member having:p 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.B f 2002 d b d i d i i i l f l• Before 2002, deep beams are designed using empirical formula
• After 2002, ACI uses Strut-And-Tie Method (in Appendix A)
27
D E E P B E A M SD E E P B E A M S
28Source: Macgregor and Wight (2006)
D E E P B E A M SD E E P B E A M S
29
D E E P B E A M SD E E P B E A M S
30Source: ACI (2005)
S T R U T – A N D – T I E M O D E L SS T R U T – A N D – T I E M O D E L S
31Source: AASHTO (2005)
Strut-and-Tie Models for Deep Beam
M A X I M U M S H E A RM A X I M U M S H E A R
• The maximum shear capacity of deep beams
= 0 83 'V f b d=,max 0.83n c wV f b d
ff Web Width Effective Depth
• If exceeded, need to enlarge the section
32
M I N I M U M R E I N F O R C E M E N TM I N I M U M R E I N F O R C E M E N T
33
R E F E R E N C E SR E F E R E N C E S
• 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),American Concrete Institute, Farmington Hills, MI.g
• 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 G (2000) Prestressed Concrete: A Fundamental Approach, 3rd Edition, Prentice Hall, NJNawy, E. G. (2000). Prestressed Concrete: A Fundamental Approach, 3 Edition, Prentice Hall, 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) American Concrete the Design of Structural Concrete with Strut-and-Tie Models (ACI SP-208), American Concrete Institute, 225-242.
• Schlaich, J., Schäfer, K., and Jennewein, M. (1987). "Toward a Consistent Design of Structural Concrete " PCI Journal 74 150Concrete. PCI Journal, 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 K d P M i G J (2003) “S d Ti M d l f D B D i • Wight, J. K. and Parra-Montesinos, G. J. (2003). “Strut-and-Tie Model for Deep Beam Design: A Practical Exercise Using Appendix A of the 2002 ACI Building Code.” Concrete International, May 2003, pp. 63-70.
