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CE 72.52 - Lecture 7 - Strut and Tie Models

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- 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
- 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
- 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