The document discusses the strut-and-tie approach for analyzing concrete structures. It begins with background concepts such as Bernoulli's hypothesis, St. Venant's principle, and the lower bound theorem of plasticity. It then discusses how axial stresses, shear stresses, and the interaction of stresses affect concrete sections. The document outlines the ACI approach to shear-torsion design and provides equations from ACI 318 for calculating the concrete shear capacity. It introduces the concept of modeling concrete as a truss system and compares this to flexural behavior in beams. The strut-and-tie method is presented as a unified approach for considering all load effects. Guidelines are provided for developing an appropriate strut-and-tie model and

1. 1
CE 72.52 Advanced Concrete
Lecture 5:
Strut and Tie
Approach
Naveed Anwar
Executive Director, AIT Consulting
Director, ACECOMS
Affiliate Faculty, Structural Engineering, AIT
August - 2014

2. Background Concepts
2

3. 3

4. An RC Beam and “Hidden” Truss
A Real Truss
A Truss and a Beam

5. Bernoulli’s Hypothesis
• Bernoulli hypothesis states that: " Plane section
remain plane after bending…"
• Bernoulli's hypothesis facilitates the flexural
design of reinforced concrete structures by
allowing a linear strain distribution for all
loading stages, including ultimate flexural
capacity
5

6. St. Venant’s Principle
• St. Venant's Principle states that:
" The localized effects caused by any load acting
on the body will dissipate or smooth out within
regions that are sufficiently away from the
location of the load"
6

7. “The stress due to axial load and bending approach a linear
distribution at a distance approximately equal to the maximum
cross-sectional dimension of a member, h, in both directions,
away from a discontinuity”
St. Venant’s principale
7(Brown et al. 2006).

8. Lower Bound Theorem of Plasticity
• A stress field that satisfies equilibrium and does
not violate yield criteria at any point provides a
lower-bound estimate of capacity of elastic-
perfectly plastic materials
• For this to be true, crushing of concrete (struts
and nodes) does not occur prior to yielding of
reinforcement (ties or stirrups)
8

9. Material Failure Criterion
For Elasto–Plastic
Materials
• Material Flow based on
Tresca or Von Mises
crireia of shear flow
For Concrete
• Failure of paste or
aggregate in tension
• Loss of Aggregate
Interlock
• Crushing of
Aggregates
9

10. Axial Stress and Concrete Sections
• Concrete is a basically compression based
material
• Concrete in general does not resist much
tension and fails in direct tension
• The failure in concrete is almost always in
tension. The Compression failure is actually a
failure in transverse tension or bursting due to
poison effect

11. Shear Stress and Concrete Sections
• Shear stress contributes to principle stresses
• In the absence of direct compressive stress:
• One of the principle stress will be tension
• Concrete does not take much tensions
• So concrete can not resist much shear stress
• Hence concrete alone can not resist much shear
force or torsion
• Shear and torsion capacity of concrete section
is therefore primarily governed by the tension
capacity of concrete

12. The Axial-Flexural Stress Resultants
• Linear Strain Distribution
y
h
c
fc
Strain
Stresses for
concrete and
R/F
Stresses for
Steel
f1
f2
fn
fs NA
CL
Horizontal

13. The Axial-Flexural Stress Resultants
The General Case: Linear or Non-linear Strain Distribution
 
 
 





























 



...),(
1
....,
1
...),(
1
....,
1
...),(
1
...,
1
121
3
121
2
121
1
i
n
i
ii
x y
y
i
n
i
ii
x y
x
x y
n
i
iiz
xyxAxdydxyxM
yyxAydydxyxM
yxAdydxyxN
















14. ACI Approach to Shear-Torsion Design
• Compute Vc and Tc based on Diagonal Tension
• Optional: Consider Interaction of V + T + M + P for
computing Vc
• Compute reinforcement for Vs = V-Vc and for Ts =
T-Tc
• Limit the maximum Shear/ Torsion stresses on
concrete Compression
Conceptual Design for Pure Shear
Limit Vc on t
( )V fc c 2
c
Steel
Steel
t
c
Limit V on c
and steel ratio
Ast
based on V-Vc
V

15. Interaction of Stresses
• Consider Interaction of V + T + M + P for
computing Vc
Interaction of Shear with other Stresses
V
t1
V
t2
Nt Nc
V
t3
V
t4
T
V
t5
T
V
V & Nt
V & Nc
V, M and P V & T
( )    t t t t t1 2 3 4 5   
M + V (+N) T

16. ACI 318 – 11 Equations for Vc
• The Basic Equation for Concrete Shear Capacity
• Maximum Shear Capacity, With Reinforcement
dbfV wcc

 2
dbfV wcc

 8
Section 11.2.1.1
ACI 318 – 11
Section 11.4.7.9
ACI 318 – 11

17. ACI 318 – 11 Equations for Vc
The Extended Equations for Concrete Shear Capacity
dbf
A
N
V wc
g
u
c










500
12
db
M
dV
fV w
u
u
cc 





 7006.0
For V and -N
For V and Prestress
dbf
A
N
V wc
g
u
c










2000
12
dbfdb
M
dV
fV wcw
u
u
cc w







 5.325009.1 
g
u
wcw
u
u
cc
A
N
dbfdb
M
dV
fV w
500
15.325009.1 









 
For V and N
For V and M
For V, N and M
Equation 11-4, ACI 318 - 11
Equation 11-5,
ACI 318 - 11
Equation 11-7, ACI 318 - 11
Equation 11-8, ACI 318 - 11
Equation 11-9, ACI 318 - 11

18. Axial Stresses – The End Point
• Only axial stress really exists
• Axial stress could be
• Tension
• Compression
• Axial Stresses may be due to
• Direct axial load
• Bending moments
• Principle stresses due to axial and shear stresses

19. The Question
• When designing concrete beams for bending
and axial load, (axial stresses) we ignore the
tension capacity of concrete
• When we design the same beam for shear or
torsion (also producing axial stress), we try to
use that tension capacity
• If we ignore tension capacity of concrete for all
tensile stresses, the design of concrete
members and sections can be simplified
considerably

20. The Truss Analogy for
Concrete Members
20

21. An RC Beam and “Hidden” Truss
A Real Truss
A Truss and a Beam

22. Strut-and-Tie method as a unified approach
• The Strut-and-Tie is a unified approach that
considers all load effects (M, N, V, T)
simultaneously
• The Strut-and-Tie model approach evolves as one
of the most useful design methods for shear critical
structures and for other disturbed regions in
concrete structures
• The model provides a rational approach by
representing a complex structural member with an
appropriate simplified truss models
• There is no single, unique STM for most design
situations encountered. There are, however, some
techniques and rules, which help the designer,
develop an appropriate model
22

23. Evolution of Strut-and-Tie Concept
• The subject was presented by Schlaich et al
(1987) and also contained in the texts by
Collins and Mitchell (1991) and MacGregor
(1992)
• One form of the STM has been introduced in
the new AASHTO LRFD Specifications (1994),
which is its first appearance in a design
specification in the US
• It was first included in ACI 318 Appendix A in
2002.

24. Strut and Tie Approach
Basic Concept
• The section is fully cracked
• Concrete takes no tension
• All Compression is taken by
“Struts”
• All Tension is taken by “Ties”
• “Ties and Struts “ provide a
stable mechanism
• It is a “Lower Bound”
Solution
Application
• Post –cracking Shear
Behavior and Design
• Design of Deep Beam
and Shear Walls
• Design of Corbels,
Brackets, Joints

25. Micro Truss Model
• Micro truss is based on the framework method proposed by
Hrennikoff (1941), in which the structure is replaced by an
equivalent pattern of truss elements.
• One of the patterns proposed by Hrennikoff for modelling plane
stress problems is shown in figure.
• Ah, Ad and Av are the cross sectional areas of horizontal, diagonal,
and vertical truss members, respectively, and they are evaluated
as:
25
Pattern of truss elements for plane stress problems“t” is the thickness of the plate.

26. Micro Truss Model
26
a. The horizontal and vertical members resist the normal stress in the
respective directions and the diagonal members resist the shear stress.
b. As a result, this model can capture both flexure and shear type of
failures in reinforced concrete structures.
c. Steel bars can be easily simulated in this model.

27. Limitation of The Truss Analogy
• The theoretical basis of the truss analogy is the
lower bound theorem of plasticity.
• However, concrete has a limited capacity to
sustain plastic deformation and is not an
elastic-perfectly plastic material.
• AASHTO LRFD Specifications adopted the
compression theory to limit the compressive
stress for struts with the consideration of the
condition of the compressed concrete at
ultimate.
27

28. The Strut and Tie Approach
• The members resist loads through internal stress
resultants
• The internal stress resultants should be in
equilibrium with external actions
• The stress-resultants follow the stress distribution
• The stress resultants form an internal mechanism
• The member should be designed to provide
internal stress resultants at proper location and in
proper direction
• The elastic stress distribution indicates a suitable
stress resultant mechanism

29. The Strut and Tie Approach
29
Strut-and-tie model is a truss model of a structural
member, or of a D-region in such a member, made up of
struts and ties connected at nodes, capable of transferring
the factored loads to the supports or to adjacent B-
regions. (AC1 318-11 definition)

30. Checking Elastic Stress State
• In-Plane analysis of various members can be
carried out using the Shell or Membrane
Elements to determine elastic stress state

31. Design of B & D Regions
• The design of B (Bernoulli or Beam) region is
well understood and the entire flexural
behavior can be predicted by simple calculation
• Even for the most recurrent cases of D
(Disturbed or Discontinuity) regions (such as
deep beams or corbels), engineers' ability to
predict capacity is either poor (empirical) or
requires substantial computation effort (finite
element analysis) to reach an accurate
estimation of capacity
31

32. Deep Members
The need for Strut and Tie Approach

33. The B and D regions
• If Strain is Assumed Linear then “B” Region
• Plane sections remain Plane after Deformation
• “Bernoulli” assumptions apply
• If Strain is Non-linear: “D” Region: Disturbed
Region
• Zone where ordinary “flexural theory” does not
apply
• Plane Sections do not remain plane after
deformation
D B D

34. 34
The figure shows the strain distribution in the deep and slender portion of a beam.
Note:
a = shear span;
B region = region of a structure where the Bernoulli hypothesis is valid;
d = distance from extreme compression fiber to centroid of longitudinal tension reinforcement;
D region = region of discontinuity caused by abrupt changes in geometry or loading;
P = axial load.

35. 35
Principal stresses in an un-cracked concrete beam found by linear elastic analysis
(a)first principal
compression stress
(b) second principal
tension stress
Two-dimensional elastic
finite element analysis

36. 36
(a) Cracking pattern,
(b) Direct arch action
(c) Shear forces in the un-cracked concrete teeth
(d) Interface shear transfer
(e) Residual tensile stresses through the cracks
(f) Dowel effect
(g) Truss action: vertical stirrups and inclined
Struts
(h) Tensile stresses due to (c), (d), (e) and (f)
(i) Final cracking pattern
Shear transfer mechanisms in reinforced concrete

37. Deep or Shallow
Shallow
Members
Where most of the
beam length is “B”
Region
Deep
Members
Where most of the
beam length is “D”
Region
Thin
Members
Flexural
Deformations are
Predominant and
shear deformations
can be ignored
Thick
Members
Shear Deformations
are Significant and
can not be ignored

38. What is a Deep Member ?
• Member in which most of the length is “D-
Region”
• Members that do not follow the ordinary
flexural-shear theories
• Members in which a significant amount of the
load is carried to supports by a compression
thrust joining the load and the reaction

39. Deep Members: Major Concerns
Non linear Stress
Distribution
Possibility of
Lateral Buckling
Very Stiff
Element
Very Sensitive to
Differential
Settlement
Reinforcement
Development
(Anchorage)
High Stresses at
Supports and
Load Points

40. The Axial Stresses – True Deep Beams
Tension
Compression

41. The Axial Stresses – Semi Deep Beams
Tension
Compression

42. Tension
Compression
The Axial Stresses – Mixed Beam
D B D

43. Shear Stresses in Beams

44. The Hidden Truss in Members
Tension
Compression

45. The Hidden Truss in Members
Tension
Compression

46. The Hidden Truss in Members
Tension
Compression

47. Tension
Compression
The Hidden Truss in Members

48. Elastic principal stresses in 4 beams with different span-
to-depth ratios (but constant shear-span-to-depth ratios)

49. Truss Models and Forces

50. Strut and Tie Modeling
50

51. Strut and Tie Modeling – Basic Terminology
• Strut-and-tie modeling (STM) is an approach used
to design discontinuity regions (D-regions) in
reinforced and pre-stressed concrete structures.
• STM reduces complex states of stress within a D-
region of a reinforced or pre-stressed concrete
member into a truss comprised of simple, uni-axial
stress paths. Each uni-axial stress path is
considered a member of the STM.
• Members of the STM subjected to tensile stresses
are called ties and represent the location where
reinforcement should be placed. STM members
subjected to compression are called struts.
51

52. Strut and Tie Modeling – Basic Terminology
• The intersection points of struts and ties are
called nodes. Knowing the forces acting on the
boundaries of the STM, the forces in each of
the truss members can be determined using
basic truss theory.
• With the forces in each strut and tie
determined from basic statics, the resulting
stresses within the elements themselves must
be compared with specification permissible
values.
52

53. Tie-Strut Approach: Assumptions
• Basic Concept and Assumptions
• The Section is fully cracked
• Concrete takes not tension
• All Tension is taken by steel ties
• All Compression is taken by “struts” forming within the
concrete
• Strut and Tie provide a stable mechanism
• Ties yield before struts crush (for ductility)
• Reinforcement adequately anchored
• External forces applied at nodes
• Pre-stressing is a load
• It is a “Lower Bound” solution
Equilibrium must be maintained

54. Tie-Strut Approach: Basic Concepts
Conceptual TrussReal Truss
a) Simple Truss Model for V, Mx (Tie and Strut Mode)
L
d
L
Ties
Compressive
Struts

55. Tie-Strut Approach in Use
For shear design of
“Shallow” and “Deep”
beams
For Torsion design
of shallow beams
For design of Pile
caps
For design of joints
and “D” regions
For Brackets and
corbels

56. Strut Tie Model
• For L/D < 4
• Load transferred by direct
• Compression
• For L/D > 4
• Auxiliary Ties are required
• for shear transfer
• For L/D > 5
• Beam tends to behave in
• ordinary Flexure
L/d =2
L/a =1
L
d
a
L/d =1
L/a =0.5
L/d = 3
L/a = 1.5
L/d = 4
L/a = 2
L/d = 5
L/a = 2.5
L/d = 6
L/a = 3
L
d
a

57. Strut Tie Model
• Angle < 30 Deg.
• Too shallow, tension steel
• not economical, strut too
• long, anchorage difficult
• Angle 35 - 45 Deg
• Gives the most economical
• and realistic design
• Angle > 50 Deg.
• Too steep. Requires too
• much stirrups. Not good.
• Effect of Strut Angle
Angle = 18 Deg
Angle = 34 Deg
Angle = 45 Deg
Angle = 64 Deg
Not OK: Too Shallow
NOT OK: Too Steep and Expensive
OK: USed by ACI Code
OK: Most Ecconomical
Tension in Bottom Chord

58. The Basic Elements of Strut and Tie
Basic Elements
• The Compression Struts
in Concrete
• The Tension Ties
provided by Rebars
• The Nodes connecting
Struts and Ties
Failure Mechanisms
• Tie could Yield
• Strut can Crush
• A Node could Fail

59. Types of Struts

60. Description of Strut and Tie Model

61. Compression Struts
• Struts represent the compression stress field
with the prevailing compression in the direction
of the strut
• Idealized as prismatic members, or uniformly
tapered members
• May also be idealized as Bottled Shaped
members
• Transverse reinforcement is required for
prevention of failure after cracking occurs

62. Types of Compression Struts

63. Failure of Struts
• Struts behave just like columns
• Failure can occur due to
• Compression failure of concrete
• Bursting of struts or transverse tension
• Cracking or struts
• Buckling of Struts
• Prevention of Failure
• Limit kl/r
• Limit compressive stress
• Confine the concrete
• Add compressive reinforcement

64. Tension Ties
Represents one or
several layers of steel
in the same direction
as the tensile force
May fail due to
• Lack of End Anchorage
• Inadequate
reinforcement quantity

65. Anchorage of Ties and Reonforcement
Reinforcement crossing a strut

66. Nodal Zones
• The joints in the strut-and-tie model are know
as nodal zones
• Forces meeting on a node must be in
equilibrium
• Line of action of these forces must pass
through a common point (concurrent forces)
• Nodal zones are classified as:
• CCC
• CCT
• CTT
• TTT

67. Different types of struts
Different types of nodes

68. Hydrostatic Nodes

69. Hydrostatic Nodal Zones
• Hydrostatic CCC
Node
• Hydrostatic CCT Node
The Stresses within the node need to be evaluated

70. 70
This figure illustrates the influence
that a hydrostatic and non-
hydrostatic node has on strut
proportions.
Note: a/d = shear span–to–depth
ratio.
Hydrostatic Node vs. Non-
Hydrostatic Node

71. Extended Nodal Zone
Subdivision of Nodal Zone

72. 72
The diagram details the node geometry of a CCC node and
CCT node.
Note:
CCC = node framed by three or more intersecting struts;
CCT = node framed by two or more intersecting struts and a
tie;
ha = height of the back face of a CCT node;
hs = height of the back face of a CCC node;
ll = length of the bearing plate of a CCC node;
ls = length of the bearing plate of a CCT node;
P = axial load;
α = portion of the applied load that is resisted by the near
support;
θ = angle of the strut measured from the horizontal axis.
d = distance from extreme compression fiber to centroid of
longitudinal tension reinforcement.

73. How to Use Strut-Tie Models

74. Using Truss Model
• Draw the beam and loads in proper scale
• Draw Primary Struts and Ties
• Struts angle between 35 to 50 degrees
• Each strut must be tied by “ties”
• The strut and ties model must be stable and determinate
• Assume dimensions of struts and ties
• Not critical for determinate trusses. Any reasonable
sizes may be used
• Make truss model in any software and analyze
• Design Truss Members
• Design rebars for tension members
• Check capacity of concrete compression members

75. How to Construct Truss Models
• For the purpose of analysis, assume the main truss
layout based on Beam depth and length
• Initial member sizes can be estimated as t x 2t for
main axial members and t x t for diagonal
members
• Use frame elements to model the truss. It is not
necessary to use truss elements
• Generally single diagonal is sufficient for modeling
but double diagonal may be used for easier
interpretation of results
• The floor beams and slabs can be connected
directly to truss elements
• Elastic analysis may be used to estimate truss
layout

76. How to Select the “Correct” Strut-and-Tie Model
• Some researchers suggest using a finite
element model to determine stress trajectories,
then selecting a STM to “model” the stress flow.
• Generally, a STM that minimizes the required
amount or reinforcement is close to an ideal
model.
76

77. Correct and Incorrect Truss
• Correct Truss • Incorrect Truss

78. 78
Complete Model
Negative Moment Truss Positive Moment Truss

80. Introduction
• Pile foundations are extensively used to support
the substructures of bridges, buildings and other
structures
• Foundation cost represents a major portion
• Limited design procedure of Pile cap Design
• Need for a more realistic methods where
• Pile cap size comparable with Columns size
• Length of pile cap is much longer than its width
• Pile cap is subjected to Torsion and biaxial Bending
• Pile cap width, thickness and length are nearly the same

81. Beams, Footings and Pilecaps
• Beam
• L >> (b, h)
• Use “Beam Flexural and
Shear-Torsion Theory”
• Footing
• (b, L) >> h
• Use “Beam/Slab Flexural
and Shear Theory”
• Pile-cap
• b <=> h <=> L
• Use Which Theory ??
h
b
L
h
b
L
h
L

82. Current Design Procedures
• Pile cap as a Simple Flexural Member
• standard specifications (AASHTO, ACI codes) are used.
• Beam/Slab theories or truss analogies are used, and torsion
is not covered for special cases
• The Tie and Strut Model
• More realistic, post cracking model
• No explicit way to incorporate column moments and torsion
• No consideration for high compressive stress at the point
where all the compression struts are assumed to meet.
• Assumption of struts to originate at the center line is
questionable
• The Deep Beam, Deep Bracket Design Approach
• Mostly favored by CRSI, takes into account Torsion, Shear
enhancement
• Complex, insufficient information on its applicability.

83. CE 72.52 – Advanced Concrete Structures - August 2012, Dr. Naveed Anwar
L=2.5
a=1.6
d=1.4 h=1.6
T
P=10,000 kN

a) Simple "Strut & Tie" Model c) Modified Truss Model B
L=2.5
a=1.6
d=1.4
d=1.4 h=1.6
T
1


 = tan-1 d/0.5L
 = 48 deg
T = 0.5P/tan 
T = 4502 kN
 = tan-1 d/0.5(L-d1)
 = 68.5 deg
T = 0.5P/tan 
T = 1970 kN
Simple Vs Modified Truss Model

84. CE 72.52 – Advanced Concrete Structures - August 2012, Dr. Naveed Anwar
Stress in Pile-Cap (SVMax)
Column
Pile Pile

85. CE 72.52 – Advanced Concrete Structures - August 2012, Dr. Naveed Anwar
P1
P2
P4
P3
a2
a2
d
L2
L1
Main members
Secondary members
A Space Truss Model for Pilecap

86. Modeling of Piles
• In truss models, the piles can be modeled as
• Pin supports
• Roller supports
• Spring supports
• As actual members
• Most suitable model is suing actual piles with
soil springs on sides of piles
• Alternately, equivalent springs may also be
used

87. CE 72.52 – Advanced Concrete Structures - August 2012, Dr. Naveed Anwar
Various Options For Modeling Piles
Basic Model Springs
RollerPin

88. CE 72.52 – Advanced Concrete Structures - August 2012, Dr. Naveed Anwar
b
a
2- Pile, Small L L < (3D + b)
L
D
4- Pile
b
L1
D
L1
L2 a
Application of Truss Model

89. CE 72.52 – Advanced Concrete Structures - August 2012, Dr. Naveed Anwar
Applications of Truss Model
5- Pile
b
L1
D
L1
aL2
a
L
D
2- Pile, Large L L > (3D + b)
b

90. CE 72.52 – Advanced Concrete Structures - August 2012, Dr. Naveed Anwar
Applications of Truss Model
L
D
d) Three Pile Case
d) Six Pile Case
P
1
P
2
P
4
P
3
a2
a2
d
L2
L1
a) Two Pile Case
c) Four Pile Case
D
L1
L1 < (3D + b)
L2
< (3D + b)
Main members
Secondary members
e) Sixteen Pile Case (Also for 12 pile, 14 pile, 20 pile)
P
P
P
P

91. C
t
B
t x 2t
t x t

92. Using Trusses to Model Shear Walls
• The behavior of shear walls can be closely
approximated by truss models:
• The vertical elements provide the axial-flexural
resistance
• The diagonal elements provide the shear resistance
• Truss models are derived from the “strut-tie”
concepts
• This model represents the “cracked” state of
the wall where all tension is taken by ties and
compression by concrete

93. CE 72.52 – Advanced Concrete Structures - August 2012, Dr. Naveed Anwar
2
5
10
Comparing Deformation and Deflections of Shell Model with Truss Model

94. CE 72.52 – Advanced Concrete Structures - August 2012, Dr. Naveed Anwar
Comparing Deformation and Deflections of Shell Model with Truss Model
2
5
10

95. CE 72.52 – Advanced Concrete Structures - August 2012, Dr. Naveed Anwar
2
5
10
Comparing Axial Stress and Axial Force Patterns

96. 2
5
10
Truss Models for Shear Walls
Uniaxial Biaxial

98. What are Brackets and Corbels
• A short and deep member connected to a large
rigid member
• Mostly subjected to a single concentrated load
• Load is within ‘d’ distance from the face of
support

99. Notations
99
Reference: Section 11.8, ACI 318-11, Provisions for
brackets and corbels

100. 100

101. Brackets or Corbels
• A short member that cantilevers out of a
column or wall to support a load
• Built monolithically with the support
• Span to depth ratio less than or equal to unity
• Consists of incline compressive strut and a
tension tie

102. Basic Stresses in Brackets
Tension Compression Shear

103. Basic Stresses in Corbels
Tension Compression Shear

104. CE 72.52 – Advanced Concrete Structures - August 2012, Dr. Naveed Anwar
Brackets using Strut and Tie Model

105. Corbels using Strut and Tie Model
• Compute distance from column to Vn
• Compute minimum depth
• Compute forces on the corbel
• Lay out the strut and tie model
• Solve for reactions
• Solve for strut and tie forces
• Compute width of struts
• Reanalyze the strut and tie forces
• Select reinforcement
• Establish the anchorage of tie

106. CE 72.52 – Advanced Concrete Structures - August 2012, Dr. Naveed Anwar
Structural Action of a Bracket

107. Load transfer through the D-region
107
Principal stress trajectories for two different types
of corbels
Quickly sketched load paths throughout D-regions

108. Modes of Failure
• Yield of tension tie
• Failure of end anchorage of the tension
tie, either under the load point or in the
column
• Failure of the compression strut by
crushing or shear
• Local failure under bearing plate
• Failure due to poor detailing

109. Design of Corbels ACI Method
• Depth of the outside edge of bearing area
should not be less than 0.5d
• Design for shear Vu, moment
• and horizontal tensile force of Nuc
• Strength reduction Factor
850.
d)]-Nuc(h[Vua 
Reference: Section 11.8.3, ACI 318 - 11

110. Design of Corbels ACI Method
• Provide Steel Area Avf to resist Vu
• Horizontal Axial Tension Force
should satisfy
• Area of Steel provided shall be the
greater of the two
• Strut and tie are should not be less
than
• Ratio shall be
uuc
ynuc
V.N
fAN
20
 
 
 nv
nf
A/A
AA


32
dbV
dbf.V
wn
wcn
800
20



 ns AA. 50







 

y
c
s f
f
.bd/A 040
Reference: Section 11.8.3, ACI 318 - 11

111. Strut and Tie Method and the ACI Method
• Strut-and-Tie method requires more steel in
the tension tie
• Lesser confining reinforcement
• Strut-and-Tie method considers the effect of
the corbel on the forces of the column
• Strut-and-Tie method could also be used for
span to depth ratio greater than unity

113. Special Considerations in Joints
• Highly complex state of stress
• Often subjected to reversal of Loading
• Difficult to identify length and depth and
height parameters
• Main cause of failure for high seismic loads,
cyclic loads, fatigue, degradation etc

114. Joints
• The design of Joints require a knowledge of the
forces to be transferred through the joint and
the ‘likely’ ways in which the transfer can occur
• Efficiency: Ratio of the failure moment of the
joint to the moment capacity of the members
entering the joint

115. Basic Stresses in Joints – Gravity
Tension Compression Shear

116. Basic Stresses in Joints – Lateral
Tension Compression Shear

117. Strut and Tie Model
Strut and Tie ModelTension Compression

118. Joint geometry (ACI Committee 352)
118
a) Interior
A.1
c) Corner
A.3
b) Exterior
A.2
d) Roof
Interior B.1
e) Roof
Exterior B.2
f) Roof
Corner B.3

119. Corner Joints
• Opening Joints:
• Tend to be opened by the applied moment
• Corners of Frames
• L-shaped retaining walls
• Wing Wall and Abutments in bridges

120. Corner Joints
• Closing Joints:
• Tend to be closed by the applied moment
• Elastic Stresses are exactly opposite as those in
the opening joints
• Increasing the radius of the bend increases the
efficiency of such joints

121. Corner Joints
• T-Joints
• At the exterior column-beam connection
• At the base of retaining walls
• Where roof beams are continuous over column

122. Beam-Column Joints in Frames
• To transfer loads and moments at the end of
the beams to the columns
• Exterior Joint has the same forces as a T joint
• Interior joints under gravity loads transmits
tension and compression at the end of the
beam and column directly through the joint
• Interior joints under lateral loads requires
diagonal tensile and compressive forces within
the joints

123. 123
Joint demands
(a) moments, shears, axial
loads acting on joint
(c) joint shear
Vcol
Ts1 C2
Vu =Vj = Ts1 + C1 -
(b) internal stress resultants
acting on joint
Ts2 =
1.25Asfy
C2 = T
Ts1 =
1.25Asfy
C1 = Ts2
Vcol
Vcol
Vb1 Vb2

124. 124
Classification
/type
interior exterior corner
cont. column 20 15 12
Roof 15 12 8
Values of  (ACI 352)
Joint shear strength
hbfVV jcnu
'
 
 = 0.85

125. Joint Details - Interior
125
hcol  20db

126. Joint Details - Corner
126
 ldh

127. Design of Joints-ACI
• Type 1 Joints: Joint for structures in non seismic
areas
• Type 2 Joints: Joint where large inelastic
deformations must be tolerated
• Further division into:
• Interior
• Exterior
• Corner

128. Design Stages for Type 1
• Providing confinement to the joint region by
means of beam framing into the side of the
joint, or a combination of confinement from
the column bars and ties in the joint region.
• Limiting the shear in the joint
• Limiting the bar size in the beam to a size that
can be developed in the joint

129. 129

130. Extraction of Strut-and-Tie model from Finite Element Analysis
130
• The Struts and Ties can be extracted from concrete members
based on the direction cosines and magnitude of principle
stress trajectories of the concrete members.
• The approach includes two main important steps:
• To extract and display an appropriate STM from the
output of FEA; and
• To refine, analyze and design the extracted appropriate
STM.
• As a sample application, concrete pile cap configuration
demonstrating the capability of this approach is shown in next
slides.
• Results can be verified by comparing the appropriate strut-and-
tie model with the previous theoretical and experimental
studies.

131. 131
Principal Compressive Stress
Contours of Two Piles-Pile Cap
Stress Trajectories Strut and Tie Layout from Program Output

132. 132
Principal Tensile & Compressive Stress Contours
of Four Piles-Pile Cap
Refined Strut and Tie Layout for Four
Piles Pile Cap (Span/Depth=2)

133. 133
Principal Compressive Stress
Contours of Pier Head under
Point Loading
Extracted Strut and Tie Layout
Principal Tensile Stress & Compressive Stress Contours of
Curved Pier Head
Extracted Strut and Tie Layout

134. How to Construct Truss
Models

135. How to Construct Truss Models
C
t
H
t x 2t

136. Iterative Method for Truss Layout
• The truss layout can be
found by using a simple
2D truss analysis
• Draw trial truss using all
possible strut tie members
• Determine forces in the
truss system
• Remove the members with
small or no forces and
repeat
• Continue until the truss
becomes unstable

137. Getting Results from Truss Model
V
PM
)cos(
)sin(
)sin(



DV
xDCxTxM
DCTP
dct



C
T
D
Tension
Member
Compression
Member
xc
xt
xd
y
st
f
T
A



138. Assuming Reinforcement
• Assume larger bars on the corners
• Assume more bars on predominant tension
direction/ location
• Assume uniform reinforcement on beam sides
• Total Rebars ratio should preferably be more
than 0.8% and less than 3% for economical
design

139. Interpretation of the Results
• Reinforcement should be provided along all directions
where truss members are in significant tension.
• This reinforcement should be provided along the
direction of the truss member
• The distribution of the reinforcement should be such
that its centroid is approximately in line with the
assumed truss element.
• The compression forces in the struts should be checked
for the compressive stresses in the concrete, assuming
the same area to be effective, as that used in the
construction of the model.
• The Bearing Stress should be checked at top of piles
and at base of columns

140. 140
PROCEDURE FOR STRUT-AND-
TIE MODELING
Summary:
Delineate D region
from B region
Determine the Boundary
conditionsof the D-region
Sketch the flow of forces
through the D-region
Develop a STM that is
compatible w ith the flow
of forc es
Calculate Strut and Tie
Froces
Select steel for Ties and
Determine its Location
Steel fits in
Assumed STM
Geometery
Chec k stress levelsin
Struts and Nodes
Stresslevels
Okay
Detail Steel Anchorage and
Required Crac k Control Steel
Modify STM by Changing Tie
Locations or inc reasing Bearing
Areas or Increasing Member
Geometery
Change location of Ties
and Modify STM
No
Ye s
No
Yes

141. Drawbacks of the Strut and Tie Approach
• Only guarantees stability and strength
• Gives no indication of performance at service
levels
• In appropriate assumed trusses layout may
cause excessive cracking
• Requires experience in judgment in truss
layout, member size assumption, result
interpretation and rebar distribution

142. ACI Approach to Strut-Tie
Models
Reference: ACI 318 – 11, Appendix A - STRUT-AND-
TIE MODELS

143. Design Procedure
• It shall be permitted to design Concrete members
or
D-regions of such members by modeling the
member as an idealized truss.
• The Model shall be in equilibrium with the applied
loads and reactions
• The geometry of the strut, tie and nodal zone shall
be taken into account
• Ties can cross struts
• Struts shall cross or overlap only at nodes
• Angle at a strut and tie at any node should not be
less than 25 degrees

144. Description of Deep and Slender Members

145. 145

146. Design Procedure
• Design of Strut, tie and nodal zone shall be
based on
• Where
• Fu = Force in a strut, tie or on a face of a nodal
zone
• Fn = Nominal strength
•  = Strength reduction factor (0.75)
un FF  Equation A1, ACI 318 - 11

147. Strength of Struts
• Where
• Ac = Cross-Sectional Area
• fcu shall be taken as the smaller of the effective
compressive strength of the strut and effective
compressive strength of the nodal zone
ccuns AfF 

 cscu ff 85.0

 cncu ff 85.0
Equation A2, ACI 318 - 11
Equation A3, ACI 318 - 11

148. Strength of Struts
• Reduction factors for strut type
• s = 1.0 for uniform cross-sectional Struts
• s = 0.75 for bottle-shaped Struts with
reinforcements
• s = 0.60 for bottle-shaped Struts without
reinforcements
• s = 0.40 for struts in tension members
• s = 0.60 for all other cases
• Compressive reinforcement shall be permitted.
Strength of such Struts shall be

 ssccuns fAAfF
Equation A5, ACI 318 - 11

149. Strength of Ties
• Where
• Ast = Area of Steel
• fy = Yield Strength of Steel
• Aps = Area of Prestressed Steel
• And;
 psepsystnt ffAfAF 
  pypse fff 
Equation A6, ACI 318 - 11
Section A.4.1, ACI 318 - 11

150. Strength of Nodal Zone
• Where;
• An = Area of the face of Nodal Zone at which Fu
acts, taken perpendicular to the axis of Fu
• Or
• An = Area of the section of the nodal zone, taken
perpendicular to the line of action of the resultants
forces on the section
ncunn AfF 

 cncu ff 85.0
Equation A7, ACI 318 - 11
Equation A8, ACI 318 - 11

151. Strength of Nodal Zone
• n = 1.0 for nodal zones bounded by struts of
bearing areas
• s = 0.80 for nodal zones anchoring one tie
• s = 0.60 for nodal zones anchoring two or
more ties

152. The design of a D-region
152
• The design of a D-region includes the following four
steps:
• Define and isolate each D-region;
• Compute resultant forces on each D-region boundary;
• Select a truss model to transfer the resultant forces across
the D-region. The axes of the struts and ties, respectively,
are chosen to approximately coincide with the axes of the
compression and tension fields. The forces in the struts and
ties are computed.
• The effective widths of the struts and nodal zones are
determined considering the forces from Step 3 and the
effective concrete strengths, and reinforcement is provided
for the ties considering the steel strengths. The
reinforcement should be anchored in the nodal zones.

153. 153