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CE72.52 - Lecture 3a - Section Behavior - Flexure

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CE72.52 - Lecture 3a - Section Behavior - Flexure

  1. 1. 1 CE 72.52 Advanced Concrete Lecture 3a: Section Behavior Flexure Naveed Anwar Executive Director, AIT Consulting Director, ACECOMS Affiliate Faculty, Structural Engineering, AIT August - 2015
  2. 2. Capacity of RC Section subjected to combined Flexural Moment and Axial Force 2
  3. 3. Loads and Stress Resultants 3 Obtained from analysis Depends on Stiffness Dependson SectionsandRebars FOS Loads Actions Deformation Strains Stress Resultants Stresses (Sections & Readers) Advanced Concrete l Dr. Naveed Anwar
  4. 4. The Response and Design 4 Applied Loads Building Analysis Member Actions Cross-Section Actions Material Stress/Strain Material Response Section Response Member Response Building Response Load Capacity FromLoadstoMaterials FromMaterialstoLoadCapacity Advanced Concrete l Dr. Naveed Anwar
  5. 5. 5
  6. 6. Cross-section and DOF 6
  7. 7. Frame/ Linear Member Sections 7
  8. 8. Frame Members and Sections 8
  9. 9. Basic Section Types - Proportions • Slender • Buckling of section parts before reaching material yielding • Cols formed, thin walled metal sections • Compact • Material yielding first, followed by bucking of section parts • Most hot rolled and built-up metal sections • Some thin concrete sections • Plastic • Material failure (yielding, rupture, but no buckling) • Most concrete sections 9
  10. 10. Section Types – Member Usage • Beams • Primarily bending, shear and torsion • Trusses • Primarily tension and compression • Columns • Primarily compression, bending • Shear and torsion also important 10
  11. 11. Cross-section classification based on primary material composition 11
  12. 12. 12 Some of the shapes used for Reinforced and Pre-stressed concrete sections defined in CSI ETABS Section Designer.
  13. 13. Some common cross-sectional types based on materials and geometry 13
  14. 14. 14 (a) C WF S,M H Ell Tee Tube Pipe WF H Ell Tee Tube Pipe (b) (c) I, H Circular Rectangular PipeSquare (d) Tee I Single Tee Double Tee Hollow case Box Some typical standard cross-section shapes used (a) in AISC database, (b) in BS Database, (c) in pre-cast, pre-stressed girders and slabs, (d) in pre-cast concrete piles
  15. 15. Some typical parametrically defined cross-section shapes 15 Square b b a a Db h bf bw tf bf bw h h tf tf Do Di Rectangle Circle Tee I Pipe
  16. 16. 16 (a) (b) (c) Some typical built-up shapes and sections (a) made from standard shapes, (b) made from standard shapes and plates, (c)made from plates
  17. 17. 17 (a) (b) Some typical composite sections. (a) Concrete-Steel composite, (b) Concrete-Concrete composite
  18. 18. Unified Theory for Concrete Design • It is possible to develop a single theory for determining the axial flexural stress resultants of most types of concrete members for all design methods and for most design codes • Unifying Beams and Columns • Unifying Reinforced and Pre-stressed Concrete • Unifying WSD and USD Methods • Unifying different Cross-section Types • Incorporating various stress-strain models 18Advanced Concrete l August-2014
  19. 19. Unifying Beams and Columns 19 Actions Sections Beam Mx or My Rectangular, T, L, Box Column P, Mx and/or My Circular, Polygonal, General Shape Advanced Concrete l August-2014
  20. 20. Unifying Reinforced and Pre-stressed 20 Reinforced Steel Pre-stressing Steel Un-reinforced No No Reinforced Yes No Partially Pre-stressed Yes Yes Fully Pre-stressed No Yes Advanced Concrete l August-2014
  21. 21. Unifying Reinforced and Composite 21 Reinforced Steel Pre-stressing Steel Steel Section Reinforced Yes No No Reinforced-Composite Yes No Yes Partially Pre-stressed - Composite Yes Yes Yes Fully Pre-stressed - Composite No Yes Yes Advanced Concrete l August-2014
  22. 22. Unifying Material Models 22 Strain Stress Linear Whitney PCA BS-8110 Parabolic Unconfined Mander-1 Mander-2 Advanced Concrete l August-2014 Concrete Stress-Strain Relationships
  23. 23. Unifying Material Models 23 Strain Stress Linear - Elastic Elasto-Plastic Strain Hardening - Simple Strain Hardening Park Advanced Concrete l August-2014 Steel Stress-Strain Relationships
  24. 24. Unifying Service and Ultimate State • Service State Calculations • Neutral axis depth controlled by limit on concrete (or steel) stresses directly • Ultimate State Calculations • Neutral axis depth controlled by limit on strain in concrete (or in steel) and indirect control on material stresses • General • Section Capacity based on location of neutral axis, strain compatibility and equilibrium of stress resultants and actions 24Advanced Concrete l August-2014
  25. 25. General Procedure for Computing Capacity • Assume Strain Profile • Assume a specific angle of neutral axis • Assume a specific depth of neutral axis • Assume maximum strain and determine the strain in concrete, re-bars, strands, and steel from the strain diagram • Determine the stress in each component from the corresponding stress-strain Relationship • Calculate stress-resultant of each component • Calculate the total stress resultant of the section by summation of stress resultant of individual components 25Advanced Concrete l August-2014
  26. 26. The General Cross-section 26 y h c fc Strain Stresses for concrete and R/F Stresses for Steel f1 f2 fn fs NA CL Horizontal Comprehensive Case Advanced Concrete l August-2014
  27. 27. The General Stress Resultants 27                                         ...),( 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                Advanced Concrete l August-2014 The Comprehensive Case
  28. 28. Flexural Theory: Stress Resultants 28 The Most Comprehensive Case The Most Simple Case M f A d a n y st        2 0.003 fc() C Strain Stress and Force N.A. OR 0 C 0 0.85 fc ' jd C T b d Section M                                         ...),( 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                y h c fc Strain Stresses for concrete and R/F Stresses for Steel f1 f2 fn fs NA CL Horizontal Advanced Concrete l August-2014
  29. 29. Example: Cross-Section Response • The Section Geometry • Elastic Stresses • Load Point • Neutral Axis • Ultimate Stresses • Cracked Section Stresses • Section Capacity • Moment Curvature Curve 29Advanced Concrete l August-2014
  30. 30. The Governing Equations 30                                         ...),( 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                Nz MxMy Advanced Concrete l August-2014 y h c fc Strain Stresses for concrete and R/F Stresses for Steel f1 f2 fn fs NA CL Horizontal
  31. 31. Axial-Flexural Capacity 31 Nz Mx My The Stress-Resultants for Bi-Axial Bending Advanced Concrete l August-2014
  32. 32. Load Point and Eccentricity 32
  33. 33. Biaxial Elastic Stress Distribution 33Advanced Concrete l August-2014
  34. 34. Neutral Axis and Strain Plane 34Advanced Concrete l August-2014
  35. 35. Ultimate Stress – Rectangular Block 35Advanced Concrete l August-2014
  36. 36. Stresses in Rebars 36Advanced Concrete l August-2014
  37. 37. Cracked Section Stresses 37Advanced Concrete l August-2014
  38. 38. 38Advanced Concrete l August-2014 Axial-Flexural Capacity Nz Mx My +
  39. 39. The Fiber Model and Implementation • In this approach, the section is sub-divided into a mesh, each element called a Fiber. A particular material model is attached to each Fiber and then solved to compute the response. 39 X Y y xx y Origin of Local Axis Origin of Global Axis Rebars Prestressed StrandsOpening Abi Api Shape of different material/properties BendingAxis Plastic Centroid S1 S2 Sn θ Mx xi Ai, fi yi My x y Advanced Concrete l August-2014
  40. 40. Fiber Model - Equations 40 Equilibrium equation based on Integration Equilibrium equation based on Summation Expanded Summation for Complex Models   A iiy A iix A iz dAxfMdAyfMdAfN __ ;; _ 1 _ 11 ;; xAfMyAfMAfN n i iiy n i iix n i iiz                                                                                                 q p l k n j jjj m i yi p y q p l k n j jjj m i xi p x q p l k n j jj m i zi p z xAfMM yAfMM AfNN 1 1 11 3 1 1 11 2 1 1 11 1 1 1 1       Mx xi Ai, fi yi My x y Advanced Concrete l August-2014
  41. 41. Procedure for Computing Stress Resultants • Define the material models in terms of basic stress-strain functions. Convert these functions to discretized curves in their respective local axes; • Model the geometry of the cross-section using polygon shapes and points, called “fibers” • Assign the material models to various fibers • Locate the reference strain plane based on the failure criterion. The failure criterion is a strain in concrete defined in corresponding material model and design code; 41Advanced Concrete l August-2014
  42. 42. Procedure for Computing Stress Resultants • Compute the basic stress profiles for all materials, using the reference strain profile; • Modify the stress profiles for each material based on appropriate material functions, and special factors; • For each material stress profile compute the corresponding stress resultant for the resulting triangles and points in the descretized cross-section. The detailed procedure for determining the resultants is discussed in the next section of this note; 42Advanced Concrete l August-2014
  43. 43. Procedure for Computing Stress Resultants • Modify the stress resultants using the appropriate material specific and strain- dependent capacity reduction factors as defined in design codes; and, • Compute the total stress resultants for all material stress profiles. • Steps 5 to 9 are repeated for other locations of the reference strain plane. The computed sets of Nz, Mx, and My are used to define the capacity surface. 43Advanced Concrete l August-2014
  44. 44. 44 Plain concrete shape Reinforced concrete section Compact Hot-rolled steel shape Compact Built-up steel section Reinforced concrete, composite section Composite section Application of General Equations Advanced Concrete l August-2014
  45. 45. Cross-Section Properties 45
  46. 46. Cross-section Stiffness and Cross- section Properties • As described earlier, the action along each degree of freedom is related to the corresponding deformation by the member stiffness, which in turn, depends on the cross- section stiffness. So there is a particular cross-section property corresponding to member stiffness for each degree of freedom. Therefore, for the seven degrees of freedom defined earlier, the related cross-section properties are: • • uz  Cross-section area, Az • ux  Shear Area along x, SAx • uy  Shear Area along y, SAy • rz  Torsional Constant, J • rx  Moment of Inertia, Ix • ry  Moment of Inertia, Iy • wz Warping Constant, Wz or Cw 46
  47. 47. Basic and Derived Properties • Difference between Geometric and Section Properties • Geometric properties – No regard to material stiffness • Cross-section Properties: Due regard to material stiffness • Cross-sectional properties can be categorized in many ways. From the computational point of view, we can look at the properties in terms of; • Basic or Intrinsic Properties • Derived Properties • Specific Properties for Reinforced Concrete Sections • Specific Properties for Pre-stressed Concrete Sections • Specific Properties for Steel Sections 47
  48. 48. Coordinates and Properties 48
  49. 49. Basic or the Intrinsic Properties • The area of the cross-section, Ax • The first moment of area about a given axis, (A.y or A.x etc.) • The second moment of area about a given axis, (A.y2 or A.x2 etc.) • The moment of inertia about a given axis, I • The shear area along a given axis, SA • The torsional constant about an axis, J • The warping constant about an axis, Wz or Cw • The plastic section modulus about a given axis, ZP • The shear center, SC 49
  50. 50. Derived Properties • The geometric center with reference to the given axis, x0 , y0 • The plastic center with reference to the given axis, xp , yp • The elastic section modulus with reference to the given axis, sx , sy • The radius of gyration with reference to the given axis, rx , ry • Moment of inertia about the principle axis of bending, I11 , I22 • The orientation of the principal axis of bending, J 50
  51. 51. Section Modulus 51 Elastic Plastic y I S xx x  4 2 bh ZPx 
  52. 52. Centroids 52 CG – Center of Gravity SC – Shear Center PC – Plastic Center
  53. 53. The significance of geometric and plastic centroid in columns 53 Pu Pu Pne b h b h h/2 h/2h/2 h/2 Pn GC GC PC Mu = Pu . e (a) (b) (a) Symmetric rebar arrangement, (b) un-symmetric rebar arrangement
  54. 54. Basic Properties about x-y 54
  55. 55. Properties about Axis 2-3 55
  56. 56. Shear Area 56
  57. 57. Torsional Constant, J 57 Circle Square A finite element solution is need for general sections
  58. 58. Warping Constant, Cw 58 A finite element solution is need for general sections
  59. 59. Principal Properties 59
  60. 60. Cracked Section Properties – RC Section 60 Icr = Moment of inertia of cracked section transformed to concrete, mm4 Ie = Effective moment of inertia for computation of deflection, mm4 Ig = Moment of inertia of gross concrete section about centroidal axis, neglecting reinforcement, mm4 Mcr = Cracking Moment, N-mm Ma = Applied Moment, N-mm fc’ = Compressive strength of concrete, Mpa fr = Modulus of rupture of concrete, Mpa λ = Factor for lightweight aggregate concrete yt = Distance from centroidal axis of gross section, neglecting reinforcement, to tension face, mm
  61. 61. Capacity Surface 61 Mx P My
  62. 62. What is Capacity? • The axial-flexural capacity of the cross- section is represented by three stress resultants • Capacity is property of the cross-section and does not depend on the applied actions or loads 62Advanced Concrete l August-2014
  63. 63. What is Capacity? 63 • Capacity is dependent on failure criteria, cross-section geometry and material properties • Maximum strain • Stress-strain curve • Section shape and Rebar arrangement etc Advanced Concrete l August-2014
  64. 64. 64Advanced Concrete l August-2014
  65. 65. 65Advanced Concrete l August-2014
  66. 66. 66Advanced Concrete l August-2014
  67. 67. How to Check Capacity • How do we check capacity when there are three simultaneous actions and three interaction stress resultants • Given: Pu, Mux, Muy • Available: Pn-Mnx-Mny Surface • We can use the concept of Capacity Ratio, but which ratio • Pu/Pn or Mux/Mns or Muy/Mny or … • Three methods for computing Capacity Ratio • Sum of Moment Ratios at Pu • Moment Vector Ratio at Pu • P-M vector Ratio 67 Advanced Concrete l August-2014
  68. 68. Sum of Mx and My • Mx-My curve is plotted at applied axial load, Pu • Sum of the Ratios of Moment is each direction gives the Capacity Ratio 68 Advanced Concrete l August-2014
  69. 69. Vector Moment Capacity • Mx-My curve is plotted at applied axial load • Ratio of Muxy vector to Mnxy vector gives the Capacity Ratio 69 Advanced Concrete l August-2014
  70. 70. True P-M Vector Capacity • P-M Curve is plotted in the direction of the resultant moment • Ratio of PuMuxy vector to PnMuxy vector gives the Capacity Ratio 70 Advanced Concrete l August-2014
  71. 71. Load Point and Eccentricity Vector • The load point location depends on the direction of the eccentricities in the x and y directions 71 Advanced Concrete l August-2014
  72. 72. Interpretation of Capacity Surface 72 +Mx +My  +My -Mx +Mx +My +Mx -My -Mx -My Moment Directions on the M-M Curve -My -Mx Load Point Applied Load Vector +Mx +My  +My -Mx +Mx +My +Mx -My -Mx -My Moment Directions on the M-M Curve -My -Mx Load Point Applied Load Vector Advanced Concrete l August-2014
  73. 73. What is Capacity 73 1- Based on Sum of Moments at Pu 2- Based on Moment Vector at PU 3- Based on True Capacity Vector in 3D Advanced Concrete l August-2014
  74. 74. P-M Interaction Curve 74 • The curve is generated by varying the neutral axis depth                   zi N i si z A cny N i si A cnx dAfdzdafM AfdafN si b si b 1 1 .)( )(   Safe Un-safe Advanced Concrete l August-2014
  75. 75. Mx-My Interaction 75 -Mz Muy (-) Mnz (+) Mnz + My - My + Mz Mx-My Interaction is the basis for many approximate methods Advanced Concrete l August-2014
  76. 76. P-Mx-My Interaction Surface 76 • The surface is generated by changing Angle and Depth of Neutral Axis                                            ...),( 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                Advanced Concrete l August-2014
  77. 77. What is Uni-axial Bending • Uni-axial bending is induced when column bending results in only one moment stress resultants about any of the mutually orthogonal axis. 77Advanced Concrete l August-2014 No Bending Mx = 0, My = 0 Strain StressSection x y e P fc P fs1 fc fs2 P x y P ey P e fs1 fs2 fc Uni-axial Bending Mx <> 0, My = 0
  78. 78. What is Bi-axial Bending • Biaxial bending is induced when column bending results in two moment stress resultants about two mutually orthogonal axis 78 y x ey e ex P x P ey ey P x Advanced Concrete l August-2014
  79. 79. P-M Interaction Diagram 79Advanced Concrete l August-2014
  80. 80. 80Advanced Concrete l August-2014
  81. 81. 81Advanced Concrete l August-2014 Effect of Compressive or Tensile Strength on Interaction curve
  82. 82. 82Advanced Concrete l August-2014
  83. 83. 83Advanced Concrete l August-2014 Symmetrical Column Section Unsymmetrical Column Section Effect of Symmetry of Column Section
  84. 84. Effect of Column Type on Shape of Interaction Diagram 84Advanced Concrete l August-2014
  85. 85. Effect of Reinforcement Ratio on Moment Curvature Curve 85Advanced Concrete l August-2014
  86. 86. 86 Effect of Reinforcement Ratio Effect of Reinforcement Spacing Advanced Concrete l August-2014
  87. 87. Effect of ultimate concrete strain ϵcu 87Advanced Concrete l August-2014
  88. 88. Flexural Design of RC Beam Sections – A Special Case of General Approach 88
  89. 89. First: The Class Project Beam That will not fail ! 89
  90. 90. Simple Beam 90 M V P P
  91. 91. More Interesting Beam 91 M V
  92. 92. Load–Deflection Curve 92
  93. 93. The way to go! • Try to generate the entire “Load-Deformation Curve” • Including “Residual Strength” • Have to rely on “Ductility”, “Plastic hinges” and “Catenary Action” • Make sure beam does not fail in shear 93
  94. 94. Design Process for Class Project • Flexural Design • Shear Design • Ductility and Plastic Hinges • Catenary/Axial Capacity 94
  95. 95. Stress Block – Singly Reinforced Concrete 95 N A x ε’cu=0.0035 εst=0.002 k1fcu 0.87fy BS 8110 β=0.9ε’cu k1= 0.45 fcu βx k1fcu 0.87fy Advanced Concrete l August-2014
  96. 96. Balanced Condition • For balanced condition, the concrete Crushing and yielding of reinforcing bars take place simultaneously 96 ε’cu xu d d E f x xd x E f s y cu cu u u u s y cu                 ' ' '    Xu=Neutral axis for balanced condition Advanced Concrete l August-2014
  97. 97. 97Advanced Concrete l August-2014 ε’cu xu d s y E f Balanced State Concrete and steel reach their failure strain simultaneously x s y E f ε >ε’cu Over reinforced State Concrete reaches failure strain prior to Steel (x>xu) Under reinforced State Steel reaches failure strain prior to concrete (x<xu) x s y E f ε <ε’cu
  98. 98. ACI - Determine Mrc 98 0.003 fc () C Strain Stress and Force N.A. OR 0 C 0 0.85 fc ' jd C T b d Section M • Mrc is a measure of the capacity of concrete in compression to resist moment . • It also ensures someductility by forcing failure in tension • It primarily depends on fc , b, d      c s y s b c s b b c f E c c a c f      0003. , ( , ), ( ) M f a b d a b c b b        (. )' 85 2 M M torc b  , . .05 075 ),,,( ' dbffM ycrc Advanced Concrete l August-2014
  99. 99. Determine Ast for Singly Reinforced Beam 99 0.003 fc() C Strain Stress and Force N.A. OR 0 C 0 0.85 fc ' jd C T b d Section M          bf M dd y f b c f A c u st   22         2 a df M A y u st  where a c fc, . ( )'   M f ab d a n c        (. )' 85 2 a A f f b st y c  .85 This procedure for Ast is iterative b = 0.85 to 0.65 , f =0.9 Advanced Concrete l August-2014
  100. 100. Ast and Asc for Doubly Reinforced Beam 100 0.003 fc() C Strain Stress and Force N.A. OR 0 C 0 0.85 fc ' jd C T b d Section M A A Ast st mrc sc ( )   A A M f d a st mrc stb b y b ( ) .      5 A M M f d dsc u rc y    ( ) ( )'  75.05.0 to Advanced Concrete l August-2014
  101. 101. Reinforcement Limits for Flexure 101 • Minimum Steel • For Rectangular Beams and Tee beams with flange in compression • For Tee beams with flange in tension • (All values in psi and inches) • Maximum Steel y w w y c s f db thanlessnot db f f A 200 3 ' min,  db f f A w y c s ' min, 6  bd A bd A stsc b     ' ' max 75.0 Advanced Concrete l August-2014
  102. 102. Check for Flexural Cracking • The cracking depends on distribution of rebars in the tension zone and on steel stress • Crack width w is given by • The control of cracking is given by • z should be less than 175 kip/in2 for interior • z should be less than 145 kip/in2 for exterior • fs = 0.6 fy • A =Area surrounding the bars • dc = centroid of the bars 102 Adfz cs 3 Adfw cs 3076.0  Advanced Concrete l August-2014
  103. 103. Design for Bending Moment 103Advanced Concrete l August-2014 OK RevisedSectionMaterial OK Mu, fc, fy Section Computer Mrc Doubly Reinforced Beam Compute Ast, Asc Singly Reinforced Beam Compute Ast fMrc > Mu Check Ast (Max) Check Ast (Min) Moment Design Completed Use Ast (min) Determine the Layout of Rebars Y OK
  104. 104. 104

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